eBooks
FREEFALL
STUDENT EDITION
MATHEMATICS
Michael Farlam
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FREEFALL
MATHEMATICS
7
FREEFALL
MATHEMATICS
SUPPORT
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MATHEMATICS
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7
FREEFALL
MATHEMATICS
NUMBER
Operations Tables - Whole Numbers
This sheet is about completing tables. The tables are addition, multiplication and subtraction.
With subtraction ensure the number in the column (up and down) is taken away from the
number in the row, otherwise negative answers will occur.
The question mark starts with the number 13 and then the operation in blue is completed, until
the last square is filled. All answers are whole numbers so fractions or decimal answers mean
that a mistake has been made.
The small tables at the bottom of the page are division. Divide the larger number by the
smaller number and then write the answer in the box below.
Operations Tables - Whole Numbers
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Complete the tables
below.
+ 5 2 1 6 10 7 8 3 4 9
5
10
10
5
5
6
6
8
8
1
1
3
3
2
2
4
4
7
7
9
9
- 15 12 11 16 10 17 18 13 14 19
10
5
6
2
1
Start with 13,
(at the bottom)
follow the
operations and
see which
number you
end with.
-5
8
3
×2
4
÷2
9
+10
×4
-8
÷7
×2
8
6 10 7
÷3
÷2
-5
×9
1
3
÷4
+7
2
÷2
+5
÷5
4
×2
7
+6
9
-2
+3
13
8
2
4
8
12
4
3
6
30
6
5
15
40
16
4
8
2
20
4
10
5
60
6
3
12
100 5 25 10
5
8
10
Addition - Whole Numbers
To perform additions move from right to left and “carry the tens” when the total exceeds 9.
The example below shows how to carry the tens.
First answer the table, then complete the exercises, follow this method:
•
total down the right hand column and if the sum is less than 10 write it in then move to
the next column.
•
If the total is ten or more write the last digit and write the tens digit above the next
column, sometimes said as carry the 1, 2 or so on.
•
Then add the next column including the figure above in your calculation, repeat the
process if the total is 10 or more, and so on, until you are finished.
Carried tens
2 22
12
Question Number
7 997 +
3 567
3 628
8 572
23 764
Answer space
Addition - Whole Numbers
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Complete the table below,
then try the exercises
1
22 +
32 +
2
34
+
5
2
10
3
7
9
6
1
8
67
4
3
3
10
76 +
92 +
4
53
9
75
4
1
5
187 +
6
387 +
39
8
64
5
7
7
2
874 +
8
807 +
232
696
6
9
14
19
383 +
10
140 +
11
705 +
12
783 +
13
378 +
86
855
493
821
674
27
21
287
906
197
1 382 +
15
4 352 +
16
5 030 +
17
8 434 +
18
1 923 +
765
894
1 349
2 159
2 002
43
130
211
3 550
3 641
3 658 +
20
5 337 +
21
2 241 +
22
8 907 +
23
7 066 +
1 291
3 724
5 423
5 024
3 879
1 006
1 467
8 711
7 255
9 054
844
2 440
2 009
3 029
8 777
Further Addition - Whole Numbers
To perform additions move from right to left and “carry then tens” when the total exceeds 9.
The example below shows how to carry the tens.
Follow this method:
•
total down the right hand column and if the sum is less than 10 write it in then move to
the next column.
•
If the sum is ten or more write the last digit and write the tens digit above the next
column, sometimes said as carry the 1, 2 or so on.
•
Then add the next column including the figure above in your calculation, repeat the
process if the total is 10 or more, and so on, until you are finished.
Carried tens
2 22
99
Question Number
7 997 +
3 567
3 628
8 572
23 764
Answer space
Further Addition - Whole Numbers
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
1
Put a space or
comma between
the hundreds and
thousands column
5
10
15
7 429 +
7 397 +
2
8 935
6
8 226 +
1 875 +
3
9 660
7
9 676 +
4 699 +
4
7 065
8
8 220 +
7 331 +
8 059
9
7 499 +
6 578
7 168
3 923
5 707
2 728
8 995
8 490
8 511
9 888
7 493
5 663 +
11
2 760 +
12
9 448 +
13
2 986 +
14
6 905 +
2 880
9 833
8 615
4 443
7 884
7 395
5 809
4 833
9 277
6 593
286
661
49
764
139
7 339 +
16
9 334 +
17
6 775 +
18
1 006 +
19
8 655 +
1 674
5 665
9 034
9 914
9 272
3 770
7 320
6 311
4 788
1 843
5 993
9 758
4 843
4 729
8 702
20 26 951 +
21 85 388 +
22 70 563 +
23 19 335 +
24 73 002 +
18 373
29 744
26 906
86 777
9 677
11 694
36 401
7 824
41 177
39 405
52 205
2 945
67 224
7 355
53 547
4 951
44 989
63 990
93 228
89 430
25 74 359 +
26 11 963 +
27 82 471 +
28 20 773 +
29 98 765 +
36 933
43 811
93 402
58 836
87 654
72 450
19 330
11 639
43 729
76 543
24 302
96 075
72 865
71 090
65 432
52 888
62 797
90 958
33 677
54 321
Subtraction - Whole Numbers
To perform subtractions move from right to left and “borrow tens” when the top number is
smaller than the bottom number. The example below outlines how to show your working.
Firstly answer the table by subtracting the numbers in the column (up and down) from the
numbers in the row. Then start the exercises, follow this method:
•
subtract down the right hand column and if the top number is greater than the bottom
number, or the total of the numbers below it, subtract.
•
If the top number is less than the total of the bottom numbers 'borrow' 1 or more from the
next column, these are 10’s. Write the amount borrowed underneath the column you
borrowed from, below the numbers.
•
Then subtract the next column including the figure you borrowed in your calculation,
repeat the process until you are finished.
99
Question Number
9 636 2 347
1 528
2 333
Borrowed tens
1 12
3 428
Answer space
Subtraction - Whole Numbers
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Complete the table below by
subtracting the column from
the row, then try the exercises.
-
1
67 -
2
24
89 45
10 15 11 20 18 14 19 13 17 12
3
3
10
76 -
4
47
56 29
9
4
5
1
157 -
6
157
68
8
253 -
5
7
7
439 -
8
578
263
2
736 -
6
9
14
19
357 -
10
578 -
11
627 -
12
923 -
13
873 -
134
207
274
467
394
84
76
170
266
387
1 274 -
15
4 880 -
16
6 550 -
17
4 255 -
18
7 638 -
490
967
3 542
1 633
2 952
377
479
876
2 150
2 008
6 557 -
20
5 670 -
21
8 990 -
22
7 553 -
23
9 741 -
1 264
3 622
2 378
1 865
2 503
2 765
1 109
3 482
1 777
3 780
304
576
1 676
2 039
1 986
Further Subtraction - Whole Numbers
To perform subtractions move from right to left and “borrow tens” when the top number is
smaller than the bottom number. The example below outlines how to show your working.
Follow this method:
•
subtract down the right hand column and if the top number is greater than the bottom
number, or the total of the numbers below it, subtract.
•
If the top number is less than the total of the bottom numbers 'borrow' 1 or more from the
next column, these are 10’s. Write the amount borrowed underneath the column you
borrowed from, below the numbers.
•
Then subtract the next column including the figure you borrowed in your calculation,
repeat the process until you are finished.
99
Question Number
9 636 2 347
1 528
2 333
Borrowed tens
1 12
3 428
Answer space
Further Subtraction - Whole Numbers
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Remember to
separate the
thousands
5
167 -
1
588 -
6
20
25
877 -
275 -
11
674 -
7
900 -
3
334 -
12
702 -
8
734 -
4
633 -
13
871 -
9
927 -
927 309
14
219
18
95 57
458
555
17
81 39
169
169
16
62 28
128
394
15
2
27
76
10
73 -
931 133
19
853 -
156
256
183
336
94
273
75
228
481
637
2 345 -
21
6 557 -
22
4 350 -
23
8 005 -
24
7 256 -
755
1 820
1 564
1 621
3 711
430
429
2 319
599
1 888
5 288 -
26
7 450 -
27
4 329 -
28
8 731 -
29
7 601 -
1 969
2 322
1 247
2 933
1 663
2 145
2 069
1 603
1 830
2 937
1 010
1 442
956
3 555
2 450
30 56 730 -
31 31 890 -
32 79 342 -
33 48 067 -
34 82 311 -
4 300
8 635
12 309
8 094
8 538
11 275
4 777
14 117
6 207
17 653
38 795
11 000
34 818
21 955
37 202
Multiplication - Whole Numbers
To perform multiplication move from right to left and carry the tens when the answer is greater
than 9. The example below will show you how to do so.
Firstly answer the table, then when you are ready to start the exercises, follow this method:
•
multiply the first numbers together and write the answer.
•
If the answer is greater than 9 write the “units” digit then the tens digit above the numbers
in the column to the left.
•
Then repeat the multiplication but then add the figure above in your calculation, repeat
the process until you are finished.
•
From question 7 on, two rows are used. The first row is for the units in the bottom
number, the second row is used when you multiply the tens digit. Write a 0 (zero) in the
second row before you start to multiply the tens digit.
•
When the two rows are completed you will have to add them. So carry the tens above the
two rows as shown in the example below.
Carry for ×
23
13
859
436
11 1
5 154
25 770
343 600
374 524
×
Carry for +
Multiplication - Whole Numbers
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
1
Complete the table below,
then try the exercises
5
2
4
12
2
4
8
10
6
3
9
1
2
71
3
7
3
46
4
5
10
92
8
6
5
3
312
6
7
9
716
4
5
7
7
34
8
13
62
46
1
4
8
9
22
10
14
14
311
8 230
27
11
53
15
27
19
71
177
6 758
53
12
34
16
43
20
86
466
2962
47
13
96
17
27
21
73
822
57
18
92
22
7 738
69
67
760
63
23
8 965
82
Further Multiplication - Whole Numbers
To perform multiplication move from right to left and carry the tens when the result is greater
than 9. The example below will show you how to do so.
Follow this method:
•
multiply the first numbers together and write the answer.
•
If the answer is greater than 9 write the “units” digit then the tens digit above the numbers
in the column to the left.
•
Then repeat the multiplication but then add the figure above in your calculation, repeat
the process until you are finished.
•
From questions 5 to 9 two rows are used. The first row is for the units in the bottom
number, the second row is used when you multiply the tens digit. Write a 0 (zero) in the
second row before you start to multiply the tens digit. From Q.10 add two zeros when
using a third row.
•
When the rows are completed you will have to add them. So carry the tens above the
rows as shown in the example below.
Carry for ×
23
13
859
436
11 1
5 154
25 770
343 600
374 524
×
Carry for +
Further Multiplication - Whole Numbers
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
1
Add a zero on
the 2nd line,
two on the
3rd line.
5
3 699
222
6
832
11
743
605
7
308
16
371
12
687
272
5 433
711
17
539
8
866
357
4
6 880
13
459
9
620
14
830
462
787
527
19
373
23
7 889
76
276
18
7 022
8
82
411
22
4 568
3
507
141
21
3
64
234
146
20
4 668
8 758
5
37
198
15
2
4
13
10
1 379
874
453
24
999
999
Division - Whole Numbers
Division is calculating how many times a number ‘goes into’ another. The answer after the
division is called the quotient.
The first 8 questions involve completing tables. A number is given, the large number at the left
side of the box, and this is to be divided by the numbers in the top row. Write the answer
below each number.
Questions 9 through 23 involve divisions that don’t have a remainder. So the quotient will be
a whole number only. While there isn’t a remainder at the end there will be remainders during
the calculation which are carried through. This won’t be necessary in the first 6 questions as
there are no remainders in the calculation stage.
Divide the outside number into the first number on the inside. If the number can divide into the
other number write how many times it can do so, then write the remainder as a small number
above the next number to its right. The remainder becomes the ‘tens’ digit and the number
below it the units (ones) digit, then repeat the process.
Questions 24 through to 41 will have a remainder. Express this as in the example below, with
the answer, a space, a ‘r’, a space, then the remainder.
Questions 42 through 45 are ‘long division’ questions. These all have remainders.
Answer with a remainder
1 153 r 1
21
4
4 613
Carried units become tens
Division - Whole Numbers
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Try the tables first then the exercises,
the first 23 won’t have remainders.
Position your answer on top.
1
2
3
4
16 2 8 4 16 5
45 15 5 3 9
3
369
10 4
844
11 2
4 862
12 5
5 055
13 4
8 480
14 6
6 060
15 6
2 412
16 7
3 577
17 9
6 318
18 4
3 604
19 7
5 642
20 5
4 535
21 3
2 463
22 8
5 616
23 2
1 206
50 25 10 2 5
20 4 5 2 10 6
30 5 15 2 6
9
42 6 14 21 7
7
64 8 4 2 16
8
80 10 20 40 8
1 153 r 1
These have remainders.
Look at the example.
4
21
4 613
24 3
7 984
25 5
8 266
26 8
9 742
27 4
6 830
28 6
2 467
29 7
3 823
30 5
8 788
31 8
9 115
32 7
4 668
33 3
7 133
34 4
5 297
35 9
6 670
36 5
7 239
37 6
2 775
38 4
7 882
39 2
1 875
40 8
8 140
41 7
4 222
45 12
5 563
42 13
2 895
43 18
7 939
44 16
3 924
Further Division - Whole Numbers
Division is calculating how many times a number ‘goes into’ another. The answer after the
division is called the quotient.
Column 1 starts first with dividing by 10, 100 and 1 000. With these, the division process
involves moving the decimal point. When dividing by 10 move the decimal point 1 place to
the left, dividing by 100 moves it 2 places to the left and with 1 000, move the decimal point 3
places to the left. Question 17 is an introduction to writing divisions with remainders as
fractions. Answer the question then select the fraction by circling it.
Column 2 are division exercises which will have a remainder, write the remainder on the top
line of the fraction (the numerator) and write the number you were dividing by as the bottom
number, (the denominator). The example above question 20 shows this.
Division by 10 and Fraction Remainders
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
When you divide by 10 move the
decimal point 1 place to the left.
100 = 2 places and 1000 = 3 places.
÷ 10 1 place left
1 10 5 450
10 34 698
4 100 14 050
5 1 000
6 1 000 73 992
180
10
9
573
100
11
8
5 687
1 000
270
10
10
=
100
13
53 456
=
1 370
692
100
=
12
=
14
9 600
5
2
18 3
5 386
19 2
1 657
20 5
8 791
21 4
3 563
22 8
2 441
23 2
7 265
24 5
4 678
25 6
9 005
26 9
9 121
27 7
76 595
28 9
31 217
29 3
70 303
30 7
56 729
31 4
34 251
32 6
92 335
33 8
43 729
34 7
96 839
35 6
70 207
36 4
35 447
37 8
82 635
38 5
78 118
=
7 808
=
1 000
16 309 ÷ 10 =
17 If 77 jelly beans are shared equally between 2
children, how many is each child given and how
many are left? Solve, then complete the statement.
Each child gets
and there is
jelly beans
extra.
If the remainder was cut up equally, circle the
fraction of a jelly bean received by each child.
1
4
3
=
Read the problem below and
solve in the space provided.
77
5
=
1 000
15 450 ÷ 10 =
2
2 777
2 10 7 330
3 100 27 880
7
6
462 6
3 469.8
3 places left
÷ 1 000
31
Example
2 places left
÷ 100
462 r 5
Instead of writing ‘r’ then
the remainder write it as a
fraction as shown
1
2
1
3
39
11
40
4 655
16
41
5 795
13
4 642
Mental Strategies with Larger Numbers
When you add or multiply or subtract numbers that are larger than you are comfortable with,
you can use a strategy to tackle the problem. Break the operation up so that you can evaluate
the operation in two steps rather than one.
The process breaks the number up into tens and units and uses the first step to deals with the
‘tens’ part and the second step to deal with the units. There are examples at the top of each
Column that show you how to approach the questions.
Mental Strategies with Larger Numbers
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Break the numbers up to perform
these multiplications.
Separate the units from one of
the numbers to make the process
of addition more manageable.
Example
Break up 39 into 30 and 9
27 + 39 = 27 + 30 + 9
= 57 + 9
Add the tens
first
= 66
Then add
the units
Example
14 × 8 = 10 × 8 + 4 × 8
1
7
28 + 47 = 28 + 40 + 7
16 × 7 =
=
Break up 14 into 10 and 4
+
= 80 + 32
Calculate
the 2 parts
= 112
Then add
the 2 parts
× 7 +
× 7
+
Break up 2nd number
54 - 39 = 54 - 30 - 9
Subtract the
‘tens’ first
= 24 -
9
= 15
Then subtract the
units for the answer
=
-
=
=
2
Example
13 46 - 28 = 46 - 20 - 8
=
=
Now try some subtracting.
Use two steps to evaluate.
14 72 - 59 =
-
=
-
-
8
57 + 19 =
+
=
+
+
12 × 9 =
×
=
=
+
×
=
+
=
9
3
88 + 36 =
+
=
+
17 × 8 =
+
×
=
+
×
+
=
=
4
10
117 + 67 =
+
=
+
+
27 × 5 = 20 × 5 +
=
=
×
+
=
5
11
127 + 28 =
+
19 × 9 =
=
+
+
×
=
+
×
+
=
=
6
+
=
+
=
+
26 × 7 =
×
=
=
-
=
-
-
=
16 56 - 18 =
-
=
-
-
=
17 92 - 67 =
-
=
-
-
=
18 74 - 57 =
-
=
-
-
=
12
77 + 77 =
15 81 - 24 =
+
+
×
19 88 - 79 =
-
=
-
=
-
Magic Squares
Magic squares are small puzzles that use your addition and subtraction ability. The total after
the sum of the numbers in a line of a magic square must all be the same, there are 8 lines in all,
3 horizontal (flat), 3 vertical (up and down) and 2 diagonal.
To solve magic squares:
•
One line of numbers will be complete, add the numbers along this line and write the total
in the magic number box, called the ‘Magic Number’ below the magic square.
•
There is one other number not in the line. Use this number together with one of the other
numbers to form another line. Add the two numbers, then subtract the total from the
Magic Number. Write the answer in the square that completes the line, then move to the
next line.
•
Continue this process until all the squares are filled.
Magic Squares
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Find the magic number
and put in in the box, then
fill the missing squares.
5
9
1
8
8
9
5
10
8
1
6
7
The Magic Number is
The Magic Number is
10
6
2
12
The Magic Number is
10
13
7
8
21
10
6
4
6
12
24
7
9
The Magic Number is
30
The Magic Number is
The Magic Number is
7
11
3
4
9
7
10
10
11
16
16
36
15
32
The Magic Number is
20
The Magic Number is
The Magic Number is
8
12
4
13
6
9
12
The Magic Number is
8
9
60
11
40
13
The Magic Number is
45
20
The Magic Number is
Mixed Operations - Whole Numbers
This sheet involves using division, multiplication, addition and subtraction. As there are
several operations in each question there are two methods you can use to answer the questions.
The first is to calculate the entire question mentally. If you can do this great, but if you have
difficulty you should use this method.
•
Split the question up into chunks, in the example at the top of the first column you have
50 - 13 + 4 - 11 = ?
•
Find 50 - 13 first, that’s 37, you then write that answer above the 13 then strike out those
first 2 numbers.
•
Then using 37 find 37 + 4, that’s 41, write that number above the 4 and strike out the 4
•
Then using the 41 find 41 - 11, that is 30, so write the answer in the square.
Column 3 involves reading a sentence and translating it to numbers and operation signs. There
are 3 answers supplied, only one of them is correct. Read the sentence and select the matching
mathematical statement. To select your choice fill the circle. Then solve A, B and C not just
the correct one, answer all three. Use the same method to solve as in the earlier columns.
Some terms you should be familiar with:
•
Sum, total, increase, add - all mean Addition
•
Reduce, decrease, find the difference, less, minus and subtract - all mean Subtraction
•
Divide and ‘find the quotient of’ - mean Division
•
Product, times and ‘lots of’ - mean Multiplication
Mixed Operations - Whole Numbers
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Strike out the numbers
and write the new total as
you go across
20 5 × 4 × 2 =
Read the sentence, select
the matching operation
but solve all three.
Example
21 8 × 2 ÷ 4 =
41 The difference of 12 and 7 plus
37
41
50 - 13 + 4 - 11 =
22 6 × 4 ÷ 8 =
1 3+7-5+6=
23 10 ÷ 5 × 15 =
2 12 - 4 + 20 - 3 =
24 6 × 5 ÷ 10 =
3 20 + 15 - 8 + 3 =
6 6 + 3 - 7 + 50 =
= 30
10 15 + 22 + 18 -
14 21 - 9 - 7 15 42 + 22 - 12 16 53 - 9 + 11 +
17 17 + 8 + 20 18 20 - 8 + 30 19 15 + 20 - 17 -
12 - 7 + 25 =
C
12 × 7 + 25 =
7 + 8 + 15 =
26 6 ÷ 2 × 5 × 3 =
B
8 - 7 + 15 =
27 10 ÷ 2 × 10 ÷ 2 =
C
7 × 8 + 15 =
43 Reduce the quotient of 24 and 6
by 4.
29 42 ÷ 7 × 3 ÷ 9 =
8 14 + 40 - 7 + 8 =
13 14 + 17 - 8 -
B
A
28 20 × 2 ÷ 8 × 3 =
7 29 - 11 + 6 - 9 =
12 16 + 5 + 13 +
12 + 7 + 25 =
by 15.
5 3 + 30 - 6 + 11 =
11 13 + 8 - 6 +
A
42 Increase the product of 7 and 8
25 20 × 3 × 2 ÷ 40 =
4 16 - 10 - 3 + 18 =
9 14 + 7 - 2 +
25.
30
= 40
= 27
= 52
=8
=0
= 35
= 80
= 33
= 36
=8
30 8 × 9 ÷ 12 × 4 =
31 3 × 12 ÷ 4 ×
= 63
A
24 - 6 - 4 =
B
24 ÷ 6 - 4 =
C
24 + 6 - 4 =
44 From the product of 2, 3 and 4
subtract 9.
32 8 × 3 ÷ 4 ×
= 54
33 9 × 5 ÷ 3 ×
= 30
34 10 ÷ 2 × 15 ×
=0
A
2×3×4+9=
B
2+3+4-9=
C
2×3×4-9=
45 Find the total of 22, 33 and 17
35 2 × 2 × 2 ×
= 64
36 5 × 4 × 4 ÷
=4
and reduce the result by 9.
A
22 + 33 + 17 - 9 =
B
22 + 33 - 17 + 9 =
C
22 + 33 + 17 + 9 =
37 6 × 8 ÷ 12 ×
= 32
38 9 × 4 ÷ 18 ×
= 100
39 50 ÷ 5 ÷ 2 ÷
=1
A
21 - 14 - 7 + 22 =
40 6 × 2 × 5 ÷
= 10
B
21 - 14 + 7 + 22 =
C
21 + 14 - 7 + 22 =
46 Reduce the sum of 14 and 21
by 7 then add 22 to the result.
Egyptian Numerals
Egyptian people used symbols to represent numbers. These symbols were of common objects
that the people were familiar with. The table below has the names of these objects, the
symbols themselves and the modern day numbers that these represent.
The system works by having up to 9 of each symbol, so 15 would be 1 heel bone and 5 vertical
staffs, 20 would be 2 heel bones and 6 750 would be 6 Lotus flowers, 7 coiled ropes and 5 heel
bones. The symbols are grouped together in any way you like and in any order you like. But
for our purpose, let the order be from the largest down to the lowest value symbol.
Column 1 has groups of symbols, rewrite these in today’s modern numerals, often referred to
as the Hindu-Arabic number system. The table below will have to used so have it on screen or
copy it in your book. The column ends with a pair of road signs, change the distances from
modern numerals to Egyptian numerals, and write the symbols on the signs.
Column 2 is similar to the road sign questions giving numbers which are to be converted to
Egyptian numerals.
Column 3 has questions that require an operation to be performed with modern numbers, find
the answer, then rewrite that answer in Egyptian numerals.
1
Vertical staff
10
Heel bone
100
Coiled rope
1 000
Lotus flower
10 000
Bent reed
100 000
Fish
1 000 000
Amazed man
Egyptian Numerals
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Change these Egyptian
numerals to our modern
(Hindu-Arabic) numbers
Convert these modern
numerals to Egyptian
numerals
Answer these questions
in Egyptian numerals
20 Write this year in Egyptian
11 32
1
2
12 3040
3
13 15 200
numerals.
21 Ali is offered a sculpture for
he haggles and
is offered a price of:
How much did he
save? Write the
answer in Egyptian
numerals.
-
4
14 420 031
5
22 Slimy Snake World sells asps in
15 4 202
6
boxes of 12. If they have 28
boxes how many asps are there?
Write your answer in Egyptian
numerals.
16 1 032 000
7
8
17 3 040
23 Ali Baba and his forty thieves
Complete the numerical
distances in Egyptian
numerals
18 93 002
9
Nile Ferry
1 100
10
Cairo
Cairo
402
19 4 270 035
each need new socks. If they
buy 8 pair each, how many
socks are bought in all. Answer
in Egyptian numerals.
Roman Numerals
Much like the Egyptian system, Roman numerals build up numbers by combining several
symbols together. The differences are:
•
Instead of pictures, letters are used to represent the numbers, (see the table below)
•
The letters are written in a single row starting from largest to smallest
•
A maximum of 3 of the same letter can be used, instead of 4 letters you subtract the letter
from the next largest letter. Subtraction is the only time that a smaller value letter comes
before a letter of larger value.
Column 1 starts with converting Roman numbers to our modern day (Hindu-Arabic) system.
Use the table below for reference if you need it. Complete the tables first then continue down
the column. There are no ‘subtraction’ style questions in this column.
Column 2 asks you to answer questions in the same way, except this time there are subtraction
style questions present. Remember there is no single Roman numeral for the following:
4, 9, 40, 90, 400 and 900. You have to ‘subtract’ numerals to get these numbers.
Eg. 4 = IV, 9 = IX, 40 = XL, 90 = XC, 400 = CD and 900 = CM.
You can take 1 from 10, 10 from 50 and 10 from 100, 100 from 500 and 100 from 1 000. But
You can’t take 1 from 500 to make 499, or 10 from 500 to make 490. So, 490 = CDXC and
499 = CDXCIX.
Column 3 asks you to write the given numbers in Roman numerals.
1
I
5
V
10
X
50
L
100
C
500
D
1 000
M
Roman Numerals
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Complete the tables below
then answer the questions,
use capital letters.
I
5
X
L
100
500
M
III
VI
XV
LII
CX
DV
MV
Change these Roman
numerals to our
modern numbers.
1 XXV
2 LXVI
3 CCXV
4 CVI
5 LXXXVI
6 CCLI
7 DCL
8 MDI
9 MMDL
10 MDCLX
11 MMMIII
12 DCLXI
13 DXXI
14 MMXX
15 MDXV
Write the Roman numerals
for the modern numbers
shown
Change these to our
modern numbers, these
will have ‘subtraction’
questions among them.
II
III
IV
VII
VIII
IX
XI
L
LX
XL
C
CX
XC
CD
6
12
20
53
110
150
140
16 XXVIII
33 56
17 XXIX
34 456
18 XCI
35 39
19 XCIV
36 74
20 XCIX
37 594
21 CMX
38 381
22 CMXL
39 490
23 MMXC
40 3100
24 MCMXL
41 366
25 XLIV
42 199
26 CXIX
43 673
27 MMCMI
44 1 241
28 DCXCIV
45 895
29 CXXIV
46 394
30 MCXLII
47 1 999
31 CMXLIII
48 2 440
32 CDXCIV
49 1 599
190
502
160
540
901
19
62
Inequalities, Ascending and Descending Order
Inequalities in mathematics refer to the use of < (less than) and > (greater than) signs. This
sheet asks you to look at two numbers, determine which number is the smallest one and put a
< or > in the box that separates them. If you have trouble remembering the ‘less than’ sign (<)
and the ‘greater than’ sign (>) then imagine that they are arrows. Then if you point the arrow
to the smallest number you will always be correct. Just to shake things up a little some require
= signs not just < or >.
Column 1 asks to place < or > between two numbers, then later, instead of two numbers, each
side has a small operation to perform. Find the answer and write it below the operation and
then ‘point the arrows’. The example at the top of the column shows the use of canceling
strokes and finding the total as you move across.
Column 2 is similar to column 1 except that this time there are 3 numbers (or operations)
rather than 2. The other difference is that the signs are given, you write the numbers (and
operations) in the boxes rather than the signs. Fizzle is giving you a warning at question 22
because the signs have changed direction, so watch them! Again look at the example for the
method to be used.
Column 3 involves placing numbers in ascending and descending order, this is actually what
you have already been doing in column 2. How do you remember that ascending goes up
(from lowest to highest) and descending goes down (from highest to lowest). Descending
starts with a D, just like Down. When you go down you go from high to low. I always say:
“What about ascending?...WHO CARES!!” Because you only have to remember one of them,
the other one is just the opposite, so just remember descending.
So for the first part of the column write the smallest number in the row below it, then cancel it
out on the top line with a stroke (so you know you have used it), then look for the next highest
number and so on. The second part of the column is to be written in descending order which
starts with the largest number and goes down, cancel them as before.
Inequalities, Ascending and Descending Order
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Put <, > or = in the
square to make true
Example
39
< 22 + 17
39
30 + 9 - 8
31
1
23
37
2
110
101
3
373
337
4
10 010
5 5 + 23
Put the numbers given
into the correct position,
you may have to do
some calculating first.
26 123, 281, 142, 412, 336
Example
30
30, 18 + 9, 7 × 4
30 27
28
> 7 × 4 > 18 + 9
<
28 9 857, 8 667, 10 023, 9 156
<
18 153, 135, 351
28
<
9+6
7 6 + 22 - 11
27 1 232, 898, 1 199, 959
17 46, 23, 17
10 005
6 23 - 6
Rewrite the numbers
in ascending order
29 43, 103, 79, 110, 130, 101
<
!
19 247, 89, 196
>
22 - 5
30 1 076, 10 760, 1 067, 7 601
>
20 111, 101, 110
8 9 × 4 + 14
70 - 15
9 3×5×2
5×6
10 32 ÷ 8 × 20
9 × 10
<
31 564, 692, 288, 1 206, 1 089
<
Rewrite the numbers
in descending order
21 12 + 4, 6 +3, 14
>
>
32 65, 75, 12, 33, 90, 29, 50
22 14 + 12, 5 × 7, 50 - 19
11 37 + 15 - 9
33 130, 105, 103, 150, 203
7×4
>
12 40 - 12 - 11
13 33 - 15
10 + 5
3×5+8
>
23 37 + 25, 7 × 8, 9 × 6
>
>
24 35 ÷ 7, 45 ÷ 5, 56 ÷ 8
14 4 × 5 × 3
34 297, 369, 403, 1 003, 676
35 8 766, 3 452, 10 299, 5 677
90 - 20
>
15 18 + 11 + 15
5×9
16 90 ÷ 10 × 4
3×5×2
>
36 221, 390, 667, 767, 903, 93
25 200 + 150, 500 - 120, 70 × 3
>
>
37 11, 23, 8, 15, 31, 0, 12, 5, 7
Order of Operations
With a mathematical operation you normally move left to right, with order of operations this
can change. Order of operations requires you to evaluate the question in this order:
•
Evaluate × or ÷ first, then the + and - section of the question.
•
Note × and ÷ have no priority over each other, it’s simply the first one of the two as you
move (to the right) through the question
•
Likewise with + and - they have no priority, it’s the first one you encounter.
Column 1 asks for the answer to each question considering order of operations. Look at each
question as if it is in ‘sections’, the sections will be made by the × and ÷ signs, find totals for
these then carry out the addition or subtraction between the sections. Look at the examples
below.
Column 2 asks to evaluate the exercise each side of a box, then write either <, > or = to make
them true. These are easier as there are 2 small operations rather than large one, so don’t let
the size of the question daunt you. Use the same method as with the first column except:
•
Once you finish evaluating one side write the total in the box below each part of the
question.
•
Repeat for the other side, then using the 2 lower numbers write <, > or = in the box.
Remember think of the < and > as arrows and point the arrow at the smallest number.
The bottom of column 2 has some questions which have a number missing. One side is
complete, find the total for this side then write the answer in both lower boxes. Then find the
value for the box that will give the total.
Column 3 are worded questions, colour the circle in front of the expression that matches the
sentence, but once you have done that, solve all three.
This question doesn’t
require order of operations
consideration because the ×
is first in the exercise.
This does as the × is after
the + so calculate the ×
section first.
This requires the ÷ to be
done first, so evaluate it first
then work through the
question from left to right.
This requires the × and ÷ to
be done first, but because
the × is first you do it first,
get your total then do the ÷.
12
4 × 3 + 2 = 14
6
4 + 3 × 2 = 10
3
40 - 9 ÷ 3 + 2
37
= 39
= 40 - 3 + 2
36
80 - 4 × 9 ÷ 2
18
= 80 - 36 ÷ 2
= 62
Order of Operations
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Try these, remember to
do the and
before
you do the + and –.
Example
12
12 + 3 × 4 = 24
Fill the box with either
<, >, or =. Remember
point ‘the arrow’ to the
smallest number.
Example
16
8
9 + 2 × 4 > 30 - 8 × 2
17 Total each side 14
1 4+6×3=
2 4×6+3=
17 86 - 6 × 11
5 28 - 12 ÷ 2 =
18 10 ÷ 2 × 5
6 4 × 20 ÷ 10 =
7 4×5+3×9
=
15 + 5 × 2
19 6 + 3 × 11
9 + 24 ÷ 3
20 99 - 81 ÷ 9
48 ÷ 6 × 2
=
21 12 + 64 ÷ 8
80 ÷ 5 + 5
=
22 15 + 55 ÷ 5
7 + 39 ÷ 3
10 11 × 6 - 24 ÷ 6
=
11 60 - 20 + 7 × 3
=
=
Find the missing number
that makes these true
23 10 + 5 ×
= 60 - 5
12 9 × 6 - 4 × 6
=
=
24 30 - 12 × 2 = 10 -
13 72 ÷ 6 + 36 ÷ 3
=
=
14 12 + 16 ÷ 4 - 16
=
23 × 2 - 3 × 2
=
=
23 × 2 - 3 =
A
15 + 14 × 3 =
B
14 + 3 × 15 =
C
15 × 3 - 14 =
of his 14 service ribbons and half of
his 32 medal collection to his granddaughter. How many awards does the
child receive?
A
32 + 14 ÷ 2 =
B
14 + 32 ÷ 2 =
C
32 ÷ 2 - 14 =
30 A school music production has
several acts by different groups.
There will be 3 quartets, 4 duos and
one trio. How many students are
performing?
A
3×4×4×2+1
=
25 20 + 16 ÷ 4 = 9 + 5 ×
B
= 12 ÷ 4 + 2
C
=
3×4+4×2+3
=
26 35 - 3 ×
=
28 Jane has a sheep farm. If she has
14 sheep in the barn and three fields
with 15 sheep in each field, how many
sheep does she have?
=
15 30 - 4 × 5 + 17
=
B
29 An ex-soldier wishes to give all
9 30 ÷ 6 + 9 × 5
=
23 - 3 × 2 =
5+3×3
8 10 × 6 - 5 × 5
=
A
C
4 36 - 27 ÷ 9 =
=
27 James is carrying 23 pair of shoes
at once, if he drops three shoes, how
many shoes is he carrying?
40 + 10
16 10 + 15 × 2
3 10 + 15 ÷ 5 =
=
Read the sentence, select
the matching operation but
solve all three.
=
3×4+4×2+1
=
=
Using Brackets
Before attempting this sheet order of operations should be attempted first. This sheet involves
using Brackets and their priority over the 4 operations. Square roots and powers are
occasionally referred to as Orders, these come after brackets. Division and/or Multiplication
comes before Addition and/or Subtraction, so….BODMAS is what you have.
Students forget what O refers to, it also implies that Division comes before Multiplication,
which isn’t correct, they have the same priority. As does Addition and Subtraction, so you can
use BODMAS to help you remember, but it has its problems.
Column 1 starts with questions which have brackets. As brackets come first, evaluate the
operation in the brackets then complete the rest of the question. The first 8 questions have no
order of operation rules to apply, other than brackets. Like the example below.
25
100 ÷ (10 + 15) ÷ 2
4
= 100 ÷ 25 ÷ 2
= 2
Note that the example at the top of the column on the worksheet is an order of operations
question, Q9 onwards are this style of question, testing your knowledge of order of operations.
Column 2 require brackets to be placed if they are required for the operation to equal the
answer given. Questions 18 to 31 require one set of brackets or no brackets, if you put brackets
in a question where they aren’t required, usually due to order of operations, you may get the
question wrong in a test. These questions are challenging. The last seven questions can
either have 1, 2, or no pairs of brackets, so these could also be a challenge. Realise that you can
place the brackets, test and see if it works, and if it doesn’t, try placing them in a different
position. This is a logical way of solving the question.
Column 3 starts with finding the number in the brackets for a given answer. The method is to
find the total that the brackets must equal first, write the answer above the bracketed numbers,
then write the number in the square inside the brackets that will get you the total.
Square roots and division operations are then briefly looked at. With square roots you find the
total in the square root sign, then square root it. When there are operations above and below the
division bar, total the top and total the bottom and then complete the division. If you aren’t
familiar with square roots: a square root is the number which when multiplied by itself equals
the number in the square root sign.
Using Brackets
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Brackets are done before
other operations. Evaluate
these (find the answer).
Put in one set of brackets to
make these true, but only if
they are required.
Example
Example
Find the missing value
to make these correct.
34 (3 +
) × 6 = 36
13 4 × 3 + 2 = 14
35 (2 +
)÷3=6
1 10 × (12 - 7) =
14 6 + 5 × 7 = 41
36 3 × ( 8 -
2 16 ÷ (5 + 3) =
15 10 + 2 ÷ 2 = 6
8
10 + (5 + 3) × 4
5
(3 + 2) × 4 = 20
32
= 10 + 8 × 4
= 42
17 6 + 4 × 5 = 26
4 (13 + 12) × 3 =
18 8 - 3 × 2 = 10
5 80 ÷ (25 + 6 - 11)
=
=
7 (65 - 13 + 4) ÷ 7
=
=
38 40 ÷ ( 3 +
)=8
39 (12 -
=
21 9 + 2 × 5 - 2 × 4 = 47
Example (1)
22 80 ÷ 2 - 3 × 7 - 6 = 25
Example (2)
Remember brackets first,
but then look for order of
operation rules to apply.
29 + 13
=
9= 3
42
7
= 6
Remember this means ÷
40
65 - 40 =
41
4
26 24 ÷ 6 + 3 - 7 × 2 = 24
These may require up to 2
sets of brackets
12 - 3 =
22 - 15
23 48 + 3 × 12 ÷ 4 = 57
25 30 - 4 × 8 - 3 = 1
) × 5 = 25
The division bar and the
square root sign group like
brackets. Try these.
19 12 - 8 ÷ 4 = 10
24 12 + 7 × 2 + 3 = 47
8 4 × (4 + 5) ÷ 12
=
)=5
20 25 - 18 ÷ 3 × 2 = 22
6 12 × (17 + 10 - 23)
=
37 35 ÷ ( 10 -
16 19 - 4 ÷ 3 = 5
3 (20 + 40) ÷ 30 =
=
)=9
42
=
9 =
15 + 45
3×4
=
=
=
9 6 + (9 - 4) × 9
=
=
=
=
12 120 ÷ (13 + 17) + 10
=
29 30 ÷ 6 - 3 × 5 - 1 = 40
44
35 + 2 × 7 =
100 - 8 × 8
=
=
=
=
=
7 + 21× 2 =
=
32 ÷ 4 + 1
30 60 ÷ 20 + 30 - 6 ÷ 2 = 30
11 (34 - 16) ÷ 6 + 3
=
43
28 9 - 6 × 2 × 4 - 3 = 6
10 50 - (12 + 3) × 3
=
27 18 + 24 ÷ 2 + 4 = 7
=
31 3 + 3 × 3 - 3 ÷ 3 = 3
45
6×6÷4
32 10 + 2 × 10 ÷ 1 + 4 = 38
33 50 - 24 ÷ 3 + 1 × 6 = 48
3 + 54 ÷ 9
46
Place Value
Place value is the value of a number in a set position. For example the value of the digit 3 in
the following would be:
•
30 in 1 234
•
300 in 2 346.04
•
30 000 in 1 236 790
•
0.03 in 12.439.
If the number is in front of the decimal point count how many digits are behind it until the
decimal point. Then write the number and count off the zeros behind it.
1 2 34
Example. For the place value of 5 in 457 609.7, write ‘5’ then count the digits behind the 5
back to the decimal point, there are 4 digits so write 4 zeros, 50 000. If the number is behind
the decimal point write ‘0.’ then count the digits between the decimal point and the number.
12
Example. For the place value of 9 in 344.379, write ‘0.’ then count the digits between the
decimal point and the 9. There are 2 digits so write 2 zeros then the 9, 0.009. (which is zero
point zero zero nine or just point zero zero nine).
Column 1 has 10 numbers, 5 written in words, the other 5 in numerals. Start by placing all of
the numbers in the table at the top of the column then find the place value of both the 5 and the
7 in each number.
Column 2 starts with 10 questions on the place value of a number which is given in brackets.
This is the same as the first column except that you are asked to give your answer in words.
There are two acceptable ways as the example shows at the top of the column. The first way is
as used in the first column, but in words. The other way is to write the value of the position. In
the example it is ten thousands.
The last part of the column has three numbers in each question, you are asked to write the
number which gives the largest place value for the number in brackets. Once you have done
this you are asked to write your answer in words. Note that the commas in these questions
separate the numbers, if you write 1 000 as 1,000 don’t get confused with the commas, the
actual numbers don’t have commas in them.
Place Value
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Put the numbers in the questions below, in
the table. Then give the place value of
both the 5 and of the 7 in each number.
The first one has been done for you.
Give the place value of the bracketed
number in words, either of the two
formats shown is acceptable
Example
Units (or ones)
0
1
1
9
Thousandths
Tens
5
Hundredths
Hundreds
4
Tenths
Thousands
2
Decimal Point
Ten Thousands
11 3 583.452 [ 4 ]
Hundred Thousands
364 257.1 [ 6 ] Six ten thousands or Sixty thousand
Millions
Table 1 : Place Values
7
1
2
3
4
5
6
7
8
9
10
12 2 877 643 [ 8 ]
13 9 227.023 [ 3 ]
14 345 002
[4 ]
15 311.129
[9 ]
16 768 909
[6 ]
17 9 076 223 [ 9 ]
[7 ]
18 7.008
19 329 777.3 [ 2 ]
Example (see table also)
Two million, four hundred and fifty thousand, one hundred
555 712.6 [ 1 ]
and nineteen and seven tenths.
5: 50 000 7: 0.7 20
Using the number in brackets, write the
number (and in words) which has the
largest corresponding place value.
1 Five hundred and twenty-seven thousand and three,
point zero four six.
5:
7:
2 Eight million, seventy-six thousand, three hundred
and ten, point two five.
5:
7:
3 Two hundred and thirty-six thousand, seven hundred
and twenty-three, point eight nine five.
5:
7:
4 Five million, four hundred and thirty-seven thousand,
one hundred and two.
5:
7:
Example
400
14 887, 2 300 403, 99 994
4
[ 4 ] 14 887
4 000 is largest place value of 4
Fourteen thousand eight hundred and eighty-seven
21 8 455 237, 322.095, 3 668
[3 ]
22 2 347.8, 554 237, 20.009
[2 ]
23 790, 567 352.96, 378 002
[7 ]
24 9 065, 336 950.8, 93.007
[9 ]
5 One thousand, five hundred and thirty-nine, point
zero two seven
5:
7:
6 117 002.5
5:
7:
7 2 750 032
5:
7:
8 67 025.43
5:
7:
9 29 536.07
5:
7:
10 315 019.7
5:
7:
Expanded Notation
This sheet has some questions written sideways. To read these questions click on the
“View” menu, select the rotate option then “counter clockwise”.
Numbers written in expanded notation are broken up into their place values. Multiplying the
number by the place value of its position.
In column 1 a number has been expressed in expanded notation, this is to be rewritten in basic
numerals. The best way to attempt these problems is to look at them digit by digit, for
example : 7 × 100 000 + 4 × 1 000 + 8 × 100 + 6, follow these steps:
•
The largest unit is 100 000 write the number 7
•
After 100 000’s comes 10 000’s, how many are there? None, so write a zero
•
How many thousands? Write the number 4
•
How many hundreds? Write the number 8
•
How many tens? None, so write a zero
•
How many units (or ones)? Write the number 6
•
You now have 704 806.
After the first 9 questions the sheet is written side ways, so follow the instruction in red above.
Column 2 works in the reverse, remember if there is a zero in the number write nothing for it.
Expanded Notation
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Change from expanded notation
to basic numerals
Now the reverse, write these
numbers in expanded notation
Example
Example
7 × 100 000 + 4 × 1 000 + 3
= 704 003
102 700
= 1 × 100 000 + 2 × 1 000 + 7 × 100
1 5 × 10 000 + 4 × 1 000 + 2 × 10 =
21 593
=
2 3 × 1 000 + 6 × 100 + 3 × 10
=
22 1 270
=
3 9 × 1 000 + 7 × 10 + 8
=
23 50 035
=
4 1 × 1 000 000 + 7 × 10 000 + 7 =
24 9 010 002 =
5 4 × 10 000 + 8 × 1 000 + 5 × 10 =
25 150
=
6 2 × 100 000 + 5 × 10 000 + 9
=
26 70 330
=
7 6 × 1 000 + 7 × 100 + 2 × 10
=
27 8 070
=
8 8 × 1 000 000 + 4 × 10 000 + 1 =
28 3 410
=
9 1 × 100 000 + 6 × 100 + 5
29 7 000 900 =
=
32 85 317
31 105 309
30 17 431
=
=
=
33 5 608 020 =
=
=
10 5 × 100 000 + 2 × 10 000 + 3 × 100 + 7
=
34 4 841
11 9 × 1 000 000 + 4 × 1 000 + 5 × 100 + 3 × 10 + 4
=
35 6 040 902 =
12 8 × 1 000 000 + 3 × 10 000 + 7 × 1 000 + 9 × 10
=
=
13 2 × 100 000 + 3 × 10 000 + 6 × 100 + 4 × 10
=
36 110 882
14 7 × 1 000 000 + 1 × 100 000 + 6 × 100
=
37 7 720 000 =
15 8 × 100 000 + 4 × 1 000 + 4 × 10 + 7
=
=
16 2 × 1 000 000 + 6 × 10 000 + 5 × 1 000 + 9 × 100 + 6
=
38 15 936
17 9 × 10 000 + 3 × 1 000 + 8 × 100 + 8
=
=
18 4 × 100 000 + 5 × 1 000 + 9 × 100 + 3 × 10 + 3
=
39 276 410
19 5 × 100 000 + 9 × 10 000 + 6 × 1 000
=
40 5 059 107 =
20 8 × 1 000 000 + 3 × 100 000 + 5 × 100 + 8
Rounding Numbers
Numbers are often rounded when it is considered the smaller digits are not needed. An
example would be a company’s profits being rounded to the nearest $1 000, or even to the
nearest $1 000 000 for large corporations, as the smaller amounts have little significance in
overall profit.
Column 1 requires numbers to be rounded to the nearest 5. This means the answer will either
end with a 5 or a 0. Note that 2.5 is halfway between 0 and 5 and 7.5 is halfway between 5
and 10. So if the last digit is less than 2.5 (1 or 2) it is changed to a 0. If it is between 2.5 and
7.5 (3, 4, 5, 6, 7) it is changed to a 5. If it is an 8 or 9 it is changed to a zero and 1 is added to
the tens column.
To round to the nearest 10 the halfway point is 5. If the last digit is less than 5, ie (1, 2, 3 or 4)
it is changed to a zero. If the last digit is 5 or more (5, 6, 7, 8 , 9) it is also changed to a zero
but 1 is added to the tens column also.
Column 2 involves rounding to the nearest 100 and then 1 000. The method is the much the
same as with 10’s. For 100 the halfway point is 50 and with 1 000 the halfway point is 500.
This means that all answers will end in ‘00’ for hundreds and ‘000’ when rounded to the
nearest thousand.
An important point is that when a number is rounded it is possible for a number with value to
become 0. For example, 430 when rounded to the nearest 1 000 becomes 0, as it is less than
500.
Column 3 is comparing answers if rounding is done at different stages of an addition. Given
several numbers, add the numbers then round the answer to the nearest 100. Then perform the
addition again except this time round the numbers before the addition.
Rounding Numbers
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Round these numbers
to the nearest 5
In between 50 and 55 and
is closer to 50 (< 52.5)
Examples
i) 52 50
ii) 98 100
In between 95 and 100
and is closer to 100
(≥ 97.5)
1 7
2 33
3 9
4 71
5 99
6 203
7 2
8 91
9 497
10 777
11 106
12 382
Example
1 371 1 400
In between 1 300 and
1 400 and is closer to
1 400 (≥ 1 350)
Example : 776, 345, 881 and 252
33 149
34 670
35 29
36 250
37 1 809
38 5 037
11
776 +
800 +
345
300
881
900
252
300
2 200
2 184
39 3 099
2 300
56 839, 242, 1 487, 3 510
40 14 636
+
41 7 591
13 1 048
Add the numbers and
round the answer to the
nearest 100, then round the
numbers to the nearest 100
and add them afterwards.
Round these to the
nearest 100
+
42 28 370
14 9 497
43 39 889
15 28 498
44 71 656
16 73 444
Round these numbers
to the nearest 10
Example
In between 160 and
170 and is closer to
170 (≥ 165)
167 170
Round these to the
nearest 1 000
Example
1 442
57 112, 5 861, 8 170, 923, 756
1 000
+
+
45 9 027
46 3 641
17 12
18 56
19 104
20 255
21 4
22 86
23 193
24 464
25 298
26 99
50 34 495
27 11
28 67
51 99 199
47 22 555
48 396
49 11 099
29 1 011
52 274 902
30 9 889
53 600 513
31 10 573
54 599 714
32 19 996
55 99 899
57 2 859, 933, 5 211, 43, 1 169
+
+
Number - Find A Word
Look for words in the list at the bottom of the grid. Once you find a word cross it off the list.
A letter could be used more than once so don’t colour it in too dark (using a texta for example)
so that you can still read it.
Number
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Find the words in the puzzle
from the wordlist.
N
J
B
N
R
E
D
U
C
E
S
S
Y
F
G
G
B
Y
C
P
W
Y
A
D
D
I
T
I
O
N
R
T
T
D
P
R
L
N
U
M
E
R
A
L
K
H
C
I
G
O
Z
A
M
I
N
U
S
Z
U
X
B
J
L
M
Q
E
T
N
E
U
O
O
A
N
N
G
H
A
M
N
S
O
O
C
S
D
D
S
N
C
S
N
U
U
O
A
T
I
N
T
D
P
U
L
S
S
J
Q
L
I
E
E
T
E
E
A
R
B
E
U
E
M
E
T
S
R
C
A
R
K
D
O
T
S
L
L
U
N
I
I
C
A
R
E
C
B
D
R
A
P
G
S
I
P
V
N
L
E
F
A
L
U
A
E
R
O
D
Y
L
I
I
P
P
F
R
H
C
C
R
E
C
Y
S
Y
D
B
W
O
I
B
E
T
T
C
D
X
T
I
M
E
S
D
E
D
N
A
P
X
E
R
V
Z
Q
U
O
T
I
E
N
T
B
G
D
D
O
WORDLIST
ADD
QUOTIENT
MINUS
DIVISION
TIMES
DECREASE
SUBTRACT
OPERATION
INEQUALITY
ROMAN
EXPANDED
TOTAL
PLUS
MULTIPLY
BRACKETS
LESS
ADDITION
REDUCE
ORDER
INCREASE
NUMERAL
DIFFERENCE
SUM
PRODUCT
PLACE
7
FREEFALL
MATHEMATICS
SHAPES &
SOLIDS
Naming Plane Shapes
With naming shapes it is important to first count the sides and after that examine the properties
like:
•
parallel sides
•
equal side lengths
•
internal angles
Some descriptions used:
Triangles - right (has a right angle), scalene (unequal sides), equilateral (all sides and angles
equal) and isosceles (two sides and two angles are equal).
Quadrilaterals - square, rectangle, rhombus, kite, parallelogram and trapezium. If a 4-sided
shape doesn’t match these names then label it a ‘quadrilateral’.
Circle - semi-circle (half circle), oval or ellipse (stretched)
Shapes with more than 4 sides - pentagon (5), hexagon (6), heptagon (or septagon) (7),
octagon (8), nonagon (9), and decagon (10).
If a shape with 5 sides or more has unequal sides or internal angles then the word ‘irregular’ is
used to describe it. For example a five sided shape that has sides of different length is called an
irregular pentagon.
Note that some shapes appear more than once.
Naming Plane Shapes
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Name these shapes.
Hey! I’m irregular!
1
2
6
7
3
8
4
5
9
10
15
11
12
13
14
(2 word description)
(2 or 3 word description)
19
16
17
18
20
(2 word description)
(2 word description)
25
21
22
23
24
(2 word description)
27
26
(2 word description)
(2 word description)
28
29
30
(2 word description)
Drawing Plane Shapes
This sheet involves the drawing of 2D shapes, referred to as Plane shapes. In mathematics a
drawn shape gives information about the shape. So that a person can look at the drawing and
not confuse it with another similar shape. For example a square and a rectangle.
There are 3 features to include in all your drawings:
•
Equal or unequal side lengths
Make sure that if a side is the same length as another side you mark it as so in your diagram,
this is done by placing a dash on the sides that are the same length. If there are more than one
pair of equal length sides then use two dashes on the next pair and so on. If a shape has
unequal side lengths then mark the sides as unequal, see below.
Red sides are the same length, blue
sides are the same length. Because the
blue sides use 2 dashes and the red sides
have only 1 dash, then it is known that
they are different lengths
The sides of this
triangle have a different
numbers of dashes, so
all the side lengths are
unequal.
Parallel sides
If pairs of sides are parallel then this should be shown on the diagram, this is done by placing
arrow heads on sides that are parallel. Generally sides aren’t labeled as parallel on shapes with
more than 4 sides, a hexagon for example.
•
Red sides are parallel.
Blue sides are parallel.
Right Angles
If a shape has one or more right angles then show this on the diagram.
•
It may be that you are expected to show matching angle pairs on diagrams also, check with
your teacher.
Drawing Plane Shapes
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
4 Draw a parallelogram and
label each pair of sides as
being equal and parallel.
8 Draw a trapezium and label
two sides as parallel.
1 Draw a rectangle showing
same side lengths and right
angles.
5 Draw an oval which has a
width greater than its height.
9 Draw a pentagon and label
the sides as equal.
2 Draw a scalene triangle,
label the sides as unequal.
6 Draw an octagon and label
all the sides as equal, (think
of a stop sign).
10 Draw a isosceles triangle
and label two sides as equal.
3 Draw an equilateral triangle
and label the sides as equal.
7 Draw a rhombus, label the
sides as equal and each pair
of sides as parallel.
11 Draw a hexagon and label
the sides as equal.
Draw these shapes, show
parallel lines
,same side
lengths
and right
angles.
Example: Draw a square
Right angle symbols used
Side lengths marked to
show they are the same
Naming Solids
When naming solids first decide whether it is a prism or a pyramid. Then look at the base to
name the solid.
Use these words:
Triangular (3 sides), Rectangular (4 sides), Pentagonal (5 sides), Hexagonal (6) and Octagonal
(8 sides). End these words with either the word ‘pyramid’ or ‘prism’.
The most confusion occurs with triangles, is it a pyramid or a prism? If there are only two
triangles in the solid then it is a triangular prism. If there is more than two triangles then the
solid is a pyramid, look at its base to decide what type of pyramid it is.
Other shapes to consider are cube, cone, sphere (and hemisphere), and cylinder.
This sheet repeats a few solids so don’t think that you have made a mistake if you have used
the same name more than once.
Naming Solids
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Name these 3-D
solids
1
2
3
4
5
9
6
7
8
11
13
10
14
15
12
16
17
18
20
19
22
21
23
25
24
27
26
28
29
30
Drawing Solids
Drawing solids can be challenging as showing a 3D solid as a drawing in 2D is not always an
easy task.
The way to approach drawing solids is to complete one face first, for example with a
rectangular prism, draw the rectangle first. Then take each corner back, then join the ends.
With solids that have pentagonal, hexagonal or octagonal faces try to ‘squash’ the faces when
you draw them, for example a octagonal pyramid. Draw the octagon, but squash it. Then pick
a point above the centre of the octagon and connect to its vertices.
If you want to produce a more complicated diagram try to show hidden lines as broken lines.
As shown in the example above and also at the top of the first column of the worksheet.
Drawing Solids
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
4 Draw a triangular prism
8 Draw a hexagonal prism
1 Draw a cube
5 Draw a square pyramid
9 Draw a hexagonal pyramid
2 Draw a rectangular prism
6 Draw a pentagonal prism
10 Draw a triangular pyramid
3 Draw a cylinder
7 Draw a pentagonal pyramid
11 Draw an octagonal prism
When you draw solids
you can show hidden
edges (as dotted lines)
or not show them.
Example: Drawing a cube
Hidden edges not shown
Hidden edges shown
Shapes in the Environment
This sheet deals with the different shapes that you have learnt and where you may find them in
your world.
Column 1 starts with diagrams of objects and asks you to name the shapes/solids that make up
the object. Some objects have more than one shape included in them, there will be two or more
answer lines supplied when this is the case. The shapes can be plane shapes (2D) such as
squares, rectangles, trapeziums, parallelograms etc, as well as solids (3D) such as cones,
spheres and prisms.
Column 2 starts with questions on naming objects that may include the given shape/solid. Try
and name objects that differ from those given in the previous column. List as many as you can
in the space provided.
The bridge at the bottom of column 2 has steel beams which outline many different shapes, see
if you can list 5 different shapes.
Column 3 starts with looking at bricks. Name the different shapes present in each brick then
circle the letter of the brick with the most shapes. Question 16 shows an artists model and
again you are asked to name all the shapes that you can see. Once you have done this, colourin the shapes.
Shapes in the Environment
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Look at the objects
and suggest a
shape or solid.
This time think of
objects that look like
the given shape/solid.
1 Fish tank
8 Cylinder
Name the 2-D shapes in
each brick, then name the
brick with the most shapes
15 Brick A
9 Sphere
2 Balloon
10 Rectangle
Brick B
3 Box of tissues - 2
a) Opening:
11 Cube
Zap Tissues
12 Cone
b) Box:
4 Drink with ice - 2
a) Ice:
Circle A B
13 Rectangular prism
b) Glass:
16
5 Sail boat - 4
047
Name the shapes you
can see in the steel
bridge drawn below.
6 Flag on a pole - 3
14
a)
7 Floor tiles - 2
Name the shapes in the
artist model below,
then colour it in.
b)
c)
d)
e)
Tessellations
Tessellations are patterns made by the repeated use of a shape, without leaving gaps.
Question 1 in the first column features tessellated triangles. Colour the adjacent triangles (that
share a common side) to build the shapes shown. Then colour the square in front of the
shape’s name the same colour. Like a key or legend in a map.
Column 1 then has three tessellated diagrams, complete the patterns. If you want to use
different colours then print the second worksheet, which is unpainted.
Column 2 consists of irregular tessellations. This is when two or more different shapes are
used. Colour them to form patterns.
The final tessellation requires you to complete it by using the dots as guides. Once you have
drawn it colour in the stars that have been formed.
Tessellations
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Create the shapes below on the grid by
colouring the triangles. Then match that
colour in one of the key squares before
the shape name.
Colour-in these irregular tessellations.
5 Triangles and Squares
1
Hexagon
Rhombus
Parallelogram
Trapezium
Complete these regular tessellations
2 Triangle
5 Octagons and Squares
3 Rhombus
Complete this irregular tessellation,
then colour-in the stars
7
4 Hexagon
Tessellations
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Create the shapes below on the grid by
colouring the triangles. Then match that
colour in one of the key squares before
the shape name.
Colour-in these irregular tessellations.
5 Triangles and Squares
1
Hexagon
Rhombus
Parallelogram
Trapezium
Complete these regular tessellations
2 Triangle
5 Octagons and Squares
3 Rhombus
Complete this irregular tessellation,
then colour-in the stars
7
4 Hexagon
Axis of Symmetry 1
An axis of symmetry is a line which divides a shape into two identical shapes. The shapes
have to be a mirror image of each other. To complete this sheet on paper cut the shapes out
and fold the shape on top of itself, the crease made by the fold is the axis of symmetry, count
the creases and write the number in the box inside the shape.
If working on screen, imagine a line cutting the shape and check whether the shape either side
of the imaginary line is an identical image.
Axis of Symmetry 1
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Cut the shapes out and fold them. If the shape doubles back on itself so that the fold mark creates
two shapes that are the same, then that is an axis of symmetry. Repeat until all are found. Then
write the number found in the square of the shape and stick the shape and its name in your book.
1 Isosceles triangle
2 Square
4 Rectangle
3 Pentagon
5 Parallelogram
7 Equilateral triangle
8 Trapezium
10 Octagon
11 Kite
6 Hexagon
9 Scalene triangle
12 Irregular Decagon
Axis of Symmetry 2
The axis of symmetry is just like a fold line where the image on one side of the axis is
reproduced on the other side. Remember that the position of the shape is the same distance
away from the line on the reflected image, so be careful with questions where the shape isn’t
touching the axis.
To complete the first column reflect the shape across to the right hand side of the axis of
symmetry. Using the grid, colour in the squares (or triangles) to make a reflection. Use a pencil
so that you can remove any mistakes easily.
Column 2 has two axes, so reflect to the right then reflect down, then it is your choice how you
get the bottom right image, either by reflecting across or reflecting down.
The shapes made are in black but your reflection can use all the different colours you want,
this isn't an exact reflection but it is more fun. You can even recolour the original shape
different colours then reflect an exact copy.
Column 3 has drawings that you complete with one or several reflected images. Reflect them
and then colour them in!
Axis of Symmetry 2
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
1
Reflect the first pair once
and the second pair twice
then colour them in! Are
you seeing double?
Reflect the shape
both vertically and
horizontally.
Reflect the shape to
the right side of the
axis of symmetry (of
the final shape).
5
9
10
2
6
3
7
11
4
8
12
Rotational and Point Symmetry
Rotational symmetry is shown by rotating a shape and having it appear as the same shape
during the rotation, before completing a revolution. A shape isn't rotationally symmetrical if it
matches only after full rotation.
Column 1 could be answered by tracing the shape and spinning the trace over the original and
counting the number of times it matches the original shape, including when it returns to its
original position. But this can also be attempted mentally by imagining the process.
Column 2 involves identifying point symmetry. Point symmetry is when a shape is identical
after a 180° rotation (half a rotation). These questions can be answered by observation or you
can choose to turn the page upside-down. If the upside-down image is identical to the original
then the object is point symmetric.
To flip the sheet upside-down, select the “View” menu, then the rotate option then
“counter clockwise” or “clockwise” but repeat a second time.
Column 3 involves identifying point symmetry, line symmetry and rotational symmetry. If a
line can be drawn anywhere through the object creating a mirror image on each side then the
object has line symmetry. Point symmetry and rotational symmetry can then be found using
the previous methods. If the type of symmetry exists answer ‘yes’.
Rotational and Point Symmetry
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These objects all have
rotational symmetry state
their order.
1 Order =
Do these objects have
point symmetry?
Answer yes or no.
8
Yes
Do these objects have:
- Line symmetry,
- Point symmetry and
- Rotational symmetry.
15
No
2 Order =
9
Yes
No
3 Order =
Y
N
Point Symmetry
Y
N
Rotational Symmetry
Y
N
Line Symmetry
Y
N
Point Symmetry
Y
N
Rotational Symmetry
Y
N
Line Symmetry
Y
N
Point Symmetry
Y
N
Rotational Symmetry
Y
N
Line Symmetry
Y
N
Point Symmetry
Y
N
Rotational Symmetry
Y
N
16
10
Yes
Line Symmetry
No
4 Order =
11
Yes
No
5 Order =
17
12
Yes
5 Order =
No
13
Yes
No
18
6 Order =
14
Yes
7 Name 2 more objects that
have rotational symmetry.
a)
b)
No
Transformations : Translation, Reflection
and Rotation
When something is transformed it is changed in some way. In mathematics there are 3
different ways to transform a shape.
•
Translation - this simply means to move, on the worksheet the shapes are on a grid and
you are asked to move the object a certain number of squares on the grid.
•
Refection - this is like a mirror image, the shape is reflected across a plane (line). The new
shape will be the same distance away from the plane as the original, only upside-down or
back-to-front.
•
Rotation means to spin around a point, the point could be on the shape or away from the
shape, this type of transformation requires the most thought.
With positioning the letters on the shape, the letters stay in the same position with translation,
reversed in reflection and will be rotated in rotation, look at the examples below.
Translation means to move, the letters stay in the
same position on the image. If you move the shape
6 □'s to the left then the distance between the same
points is 6 □'s. The gap between the object and its
image isn’t 6 □'s, the most frequent mistake made.
Reflection means to copy back-to-front across a
plane, the letters stay the same distance from the
plane which means they appear to be reversed.
Same distance
6 □'s
A
D
A
A
D
object
image
B
B
C
C
B
C
D
D
image
Not 6 □'s
object
B
A
6 □'s
C
Same distance
Rotation means to spin around a point, the point can be on the object or away from the object. The direction of rotation is
given as either clockwise or anticlockwise. The distance of a point on the object from the rotation point remains the same
once rotated.
A
D
A
A
object B
B
90º
C
D
object
image
D
B
C
180º
O
B
C
image
D
A
Transformations : Translation, Reflection and Rotation
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
6 Translate 4 ‰'s down and
reflect in line AB.
Translation means to
move, redraw the shape in
its final position.
11 Rotate anticlockwise 90º at O
then reflect in line AB
A
A
1 Translate 6 ‰'s right
A
B
A
B
C
O
2 Translate 8 ‰'s left
B
B
3 Translate 6 ‰'s left and then
4 ‰'s up
7 Reflect in line AB then
translate 4 ‰'s up then 2 ‰'s
right
12 Rotate clockwise 180º at O
then translate 6 ‰'s right
A
A
B
D
C
Reflection is 'flipping the
shape over' across a line
of reflection.
A
Which transformation
would have occurred to
create the image shown.
B
B
4 Reflect the shape in line AB
show letters on new shape
A
Rotation means to spin
around a point, redraw the
shape in its final position
showing two letters.
13
A
C
O
B
E
B
C
B
14
A
C
A
8 Rotate 180º clockwise at O
B
A
O
A
9 Rotate 180º clockwise at O
D
B
5
A
A
C
15
B
E
B
F
O
A
B
D
A
10 Rotate 90º anticlockwise at O
B
B
O
A
A
A
B
Templates for Building
These templates are designed to be printed, cut out, coloured-in and either:
Sheets 1 & 2 - stick into books to fold out of page
Sheet 3 - 6 - cut out and assemble to make 3D shapes
Constructing Cubes and Square Pyramids
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Stick this
square face
down into your
book
Stick this
square face
down into
your book
Constructing Rectangular Prisms and Triangular Prisms
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Stick this rectangular face
down into your book
Stick this
rectangular
face down
into your
book
Stick this
rectangular
face down
into your
book
Constructing Cubes and Square Pyramids (with tabs)
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Constructing Rectangular Prisms and Triangular Prisms (with tabs)
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Constructing Cylinders and Triangular Prisms (with tabs)
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Constructing Cones and Pentagonal Prisms (with tabs)
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Nets of Solids 1
A net in mathematics refers to the way a solid would look if it were made from paper and
unfolded. So to answer this sheet you have to look at the net and fold it up in your mind and
decide what shape it makes when it is folded up.
For most students the difficulty comes when prisms and pyramids are involved. Think about
this:
•
If a net only has 2 triangles in the net, it will always be a triangular prism.
•
If a net has more that 2 triangles in the net it will always be pyramid.
•
If a net is of a pyramid then the name of the net will come from the shape in the net that
isn’t a triangle.
•
If the net is of a prism then the name of the prism will come from the shape that isn’t a
rectangle or square. (except rectangular prisms)
Remember that a shape can't be a pyramid if it has no triangles in its net. A shape can't be a
prism if it has more than two triangles in its net. To answer the sheet write the solid’s name in
the table at the bottom of the page next to the net’s number.
Nets of Solids 1
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Name the 3D figure than
will be made by the nets,
answer in the table below.
2
4
3
1
5
7
6
8
11
12
10
9
16
14
13
15
18
19
20
17
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Nets of Solids 2
Nets of solids are drawings of solids as though they are made from paper and unfolded. So to
draw a net think of the solid unfolded. The previous sheet “Nets of Solids 1” can be referred to
if you need help.
A prism will always have rectangles in its net and two other shapes as its base shape. So a
pentagonal prism would have two pentagons and five rectangles, an octagonal prism would
have two octagons and eight rectangles in its net.
A pyramid will always have triangles in its net and a different shape as its base shape (except
triangular pyramids. So a hexagonal pyramid would have a single hexagon with six triangles,
while a triangular pyramid would have a triangle with three triangles off it.
Use a pencil so that you can correct mistakes easily.
Nets of Solids 2
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2 Cone
3 Rectangular Prism
4 Triangular Prism
5 Pentagonal Pyramid
6 Hexagonal Prism
7 Octagonal Prism
8 Square Pyramid
9 Pentagonal Prism
10 Triangular Pyramid
11 Rectangular Pyramid
12 Hexagonal Pyramid
Draw the nets of these
solids
1 Cube
Views of Solids
When you look at a solid from the side, front or top you will see a plane (2D) shape. Imagine
looking at a box from the top, side and front. You would have 3 views that would be either
rectangles or squares.
Column 1 starts by giving you 3 views of a solid and asks you to name it. This exercise tests
your visual skills and there aren’t any helpful tips except that pyramids will have more than
one view that includes a triangle, with that triangle being either isosceles or equilateral, not
right or scalene.
Column 2 has diagrams of 5 solids and asks what shape you would see if you were looking
from the direction of the arrow. The answers are all in the box at the bottom of the column,
write in the letter (from A → I) that matches the object in the space provided. Note that the
same letter can be used more than once.
Column 3 asks you to build you own shapes to show the view from the direction that is
arrowed.
Views of Solids
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Match the views shown
with the shapes below,
write the letter.
Name the 3-D solids that
would have these top
and side views.
1
9
Top view
Front view
Top view
Side view
Top view
14
Top
Side view
Sketch the view of the
3-D solids indicated by
the arrow.
Side
2
15
Top view
Front view
Side view
10
Top view
Top
3
Side view
Top view
Front view
Side view
11
Top view
Top
Side view
Side view
Front view
16
Top view
4
Front view
Side
12
Side
Side view
17
Top view
Top view
Top
5
Top view
Front view
Side view
Side view
13
Side
18
Top view
Side view
6
Top
Top view
Side
Front view
19
Top view
20
Top view
Side view
Side view
7
A
Top view
Front view
B
C
Side view
D
8
E
F
Top view
Front view
Side view
G
H
I
Isometric Drawing
Isometric drawing is a method of showing a shape in a way that looks 3 dimensional. The
bottom corner of the drawn shape has a BLUE COLOURED DOT, this is the point that you
should start at and it is shown at the bottom of the page.
There is no ‘best way’ as it depends on how you visualise the solid, but one way is to try to
complete one face first and then move on
1
So to draw the solid to the left, draw the
front face first (right). Then take each
corner back the required depth (bottom
left). Join the edges and don’t forget to
take the two edges back as shown
(bottom right)
2
3
Isometric Drawing - Sheet 1
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Copy the shapes in each question. Make sure that you start your drawing at the
dot, otherwise you may run out of space. Colour them in when you are finished.
2
1
4
3
6
5
8
7
10
9
Isometric Drawing - Sheet 2
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Copy the shapes in each question. Make sure that you start your drawing at the
dot, otherwise you may run out of space. Colour them in when you are finished.
1
2
Drawing Solids from Views
Isometric drawing is a method of drawing a solid in a way that it looks 3 dimensional. The
bottom part of the sheet has a blue dot, this is the point that you should start from. The ‘front
view’ arrow should be pointing at the front view, the object must be drawn in this way or it
may not fit in the page.
Question 1 should be attempted first, Question 2 should be considered challenging.
“15 Isometric drawing” should have been completed before this Worksheet is attempted.
Drawing Solids from Views
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Copy the shapes in each question. Make sure that you start your drawing at the
dot, otherwise you may run out of space. Colour them in when you are finished.
1
TOP VIEW
FRONT VIEW
SIDE VIEW
FRONT
VIEW
2
TOP VIEW
FRONT VIEW
SIDE VIEW
FRONT
VIEW
Euler's Formula
This formula relates the properties of a solid. The properties are the:
F - number of faces on the shape (the sides of the solid)
V - number of vertices on the shape (corners where edges meet)
E - number of edges on the shape (the lines that separate faces)
The Euler formula is: F + V - E = 2
To answer the questions look at each solid, the solids aren't numbered so start where you wish.
Then:
•
Write the name in the first column
•
Count the faces and write the number in the next column
•
Count the vertices and again, write it in the next column
•
Count the edges and write in the next column
•
Solve for F + V - E and it should equal 2.
This formula is a good check to see that you can accurately identify the properties of a shape
as you know that if the answer isn't 2 you have to go back and find your mistake.
Euler's Formula
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Apply the Euler formula to these
3-D solids : F + V - E = 2, by
completing the table below.
1 Cube
Octagonal Pyramid
6 Triangular Prism
2
3
4
5
6
7
8
9
10
11
12
7 Triangular Pyramid
8
10 Pentagonal Prism
NAME OF SOLID
1
4 Pentagonal Pyramid
2 Rectangular Prism
5 Octagonal Prism
9
3 Hexagonal Prism
FACES (F)
Hexagonal Pyramid
11 Trapezoidal Prism
12 Square Pyramid
VERTICES (V) EDGES (E)
F+V-E=
Euler's Formula
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Apply the Euler formula to these
3-D solids : F + V - E = 2, by
completing the table below.
1
5
9
2
3
4
5
6
7
8
9
10
11
12
4
7
8
2
6
10
NAME OF SOLID
1
3
11
FACES (F)
VERTICES (V) EDGES (E)
12
F+V-E=
Cross-sections of Shapes
A cross-sectional view of a solid is like you are slicing through the solid and then looking at
the surface that you have just cut. Imagine slicing a loaf of bread (rectangular prism) and
looking at the slice (a square?). The important rule is that if you slice a prism you get a
uniform cross-section, that is, the cross section is identical wherever you choose to cut through
the solid. Just like slices of bread. But what if the cross sections aren’t the same. That means
the shape isn’t a prism. It doesn’t automatically mean that it is a pyramid though, as you will
find out.
Columns 1 and 2 start with giving three cross sections, these sections are all cut in the same
direction, like sliced bread, not any direction through the solid. You are asked to write the
name of the solid that could have these cross sections. Question 1 has two possible answers.
Important - don’t confuse cross sections with front, end and top views of a solid, all the
sections are views from the one direction.
Column 3 shows a cross section and asks you to give some examples of solids that would have
that section, the last part of the column asks you to name objects that would have cross
sections that are rectangular, circular or triangular.
Cross-sections of Shapes
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Name the 3-D shape(s)
that would have these
cross-sections.
1
1st section
2nd section
8
1st section
2nd section
3rd section
Name some solids that
could have the cross
sections shown below.
16
3rd section
9
1st section
2nd section
3rd section
10 1st section
2nd section
3rd section
11 1st section
2nd section
3rd section
2 solids possible
for these sections
2
3
1st section
1st section
2nd section
2nd section
3rd section
17
3rd section
18
4
1st section
2nd section
3rd section
12 1st section
5
1st section
2nd section
3rd section
13 1st section
2nd section
2nd section
3rd section
3rd section
Name objects in your environment that when sliced
have cross-sections of….
19 Circles
6
1st section
2nd section
3rd section
14 1st section
2nd section
3rd section
20 Rectangles
7
1st section
2nd section
3rd section
15 1st section
2nd section
3rd section
21 Triangles
Parallel, Perpendicular and Skew
Parallel means the same distance apart and not touching. This sheet deals with parallel faces
and parallel edges. The rectangular prism shown below has three pairs of parallel faces.
F
The top face PFLY is parallel to the bottom face NTWD.
The front face PYDN is parallel to the back face FLWT.
The LHS face PFTN is parallel to the RHS face YLWD.
L
Y
P
T
N
W
F
FT
Y
P
D
There are parallel edges as well. On the diagram to
the right, the edges that are the same colour are all
parallel. You can show this using :
PN
L
ND
LW YD
PY
FL
T
W
N
TW
D
PF
YL
DW NT
Perpendicular means ‘at right angles to’ and touching. Again you can have perpendicular faces
and also perpendicular edges. Faces first:
F
L
Y
P
T
N
W
D
The top face PFLY and the bottom face NTWD are
perpendicular to FLWT, YLWD, PYDN and PFTN.
The front face PYDN and the back face FLWT are
perpendicular to PFLY, YLWD, WDNT and PFTN.
The side faces PFTN and YLWD are perpendicular to
PFLY, FLWT, NTWD and PYDN.
Edges can also be parallel. With rectangular prisms and cubes there will be two perpendicular
edges and the end of each edge.
So the edge PY (in red) has perpendicular edges
(in green) YL, YD, PF and PN.
Not the perpendicular symbol is
F
L
Y
P
T
.
W
N
D
Skew edges are edges that aren’t parallel and aren’t perpendicular. So if an edge is skew it
won’t be parallel and it isn’t going to touch the given edge.
Name the skew edges to PN (shown in red).
The skew edges are shown in green and are
FL, TW, YL and DW.
Note that none touch PN, and none are parallel to PN.
F
L
Y
P
T
N
W
D
Parallel, Perpendicular and Skew
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Name the face(s) that ….
E
D
A
F
H
F
G
B
G
E
Y
H
D
12 3 edges parallel to AB are:
AB
BFGC
A
B
A
2 Is parallel to BFGC.
T
C
B
C
1 Is shaded.
What’s skew with you?
Name the skew edges in
these questions.
Name the edges that are parallel
to, & perpendicular to, the edges
on the rectangular prisms below.
YB:
V
GC
,
,
,
.
K
E
3 Is parallel to the shaded face. 13 3 edges parallel to GC are:
21
Name 4 edges
that are skew to
X
22 Name 4 edges that are skew
to VT:
4 Is parallel to ABCD.
14 3 edges parallel to EH are:
23 Name 4 edges that are skew
to KX:
5 Is parallel to AEFB.
15 4 edges perpendicular to AB
24 Name two edges that are
skew to EX and parallel to EB.
,
,
,
6 Is perpendicular to ABCD &
16 4 edges perpendicular to GC
has an edge EH.
7 Are perpendicular to EHGF
& have an edge BF, (2 faces).
Now use the cube shown below.
F
,
9 Are perpendicular to the
shaded face and have edge GF.
E
D
8 Are perpendicular to BFGC
& perpendicular to AEFB.
Q
P
X
S
T
17 Name the face parallel to
QFPS.
18 Name the edges parallel to
FP.
10 Is perpendicular to ABCD &
19 Name the horizontal faces
has vertices C & F.
perpendicular to DFPX.
11 Is perpendicular to DAEH &
20 Name the vertical faces
has edge BC.
perpendicular to PFQS.
25 Name two edges that are
skew to AK and parallel to AT.
26 Name an edge that is skew
to VT and parallel to BY.
27 Name an edge that is skew
to KV and parallel to YX.
28 Name an edge that is skew
to EX and is on the face TBEV.
29 Name two edges that are
skew to AK and perpendicular
to VT.
30 Name two edges that are
skew to KV and perpendicular
to EX.
31 Name two edges that are
skew to XY and are on the face
ATVK.
Shapes - Find A Word
Look for words in the list at the bottom of the grid. Once you find a word cross it off the list.
A letter could be used more than once so don’t colour it in too dark (using a texta for example)
so that you can still read it.
Shapes Find-A-Word
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Find the words in the puzzle
from the wordlist.
X
Q
S
Z
T
R
A
N
S
L
A
T
I
O
N
V
V
E
E
P
E
N
T
A
G
O
N
Z
M
R
E
T
C
Y
L
I
N
D
E
R
V
E
X
A
E
R
A
L
U
G
E
R
R
I
R
R
T
U
R
F
T
T
B
I
H
E
C
O
N
E
Q
S
E
G
L
E
E
R
D
T
T
O
S
V
M
K
H
P
O
E
X
M
L
I
H
A
C
Q
O
E
P
H
R
L
C
T
P
K
M
A
Z
T
D
W
S
Q
H
I
E
T
E
L
M
A
H
N
A
O
O
C
I
E
S
L
I
L
A
H
R
C
Q
G
P
R
A
E
X
M
L
O
C
T
O
Y
U
V
O
L
S
L
R
A
A
A
N
R
E
R
P
B
B
N
L
E
E
A
G
M
R
P
I
S
O
M
E
T
R
I
C
N
U
O
T
A
N
C
R
H
O
M
B
U
S
V
E
Q
N
X
P
E
V
N
O
I
T
A
M
R
O
F
S
N
A
R
T
WORDLIST
AXIS
NET
PYRAMID
SQUARE
REFLECTION
CUBE
CIRCLE
VERTEX
TRIANGLE
TRANSLATION
ISOMETRIC
CYLINDER
CONE
SPHERE
PRISM
OCTAGON
KITE
HEXAGON
TEMPLATE
IRREGULAR
SCALENE
ROTATE
PENTAGON
ISOSCELES
RHOMBUS
PARALLELOGRAM
TRANSFORMATION
SKEW
7
FREEFALL
MATHEMATICS
NUMBER
THEORY
Odd and Even Numbers
Odd and even numbers separate each other and can be quickly identified, as:
•
Odd numbers end in either a 1, 3, 5, 7 or 9
•
Even numbers end in either a 0, 2, 4, 6, 8
•
0 is usually considered neither odd or even
Column 1 starts with 18 questions on classifying numbers as odd or even. The last digit of the
number determines if it is odd or even, so ignore all the digits except the last one. The next
four questions ask you to write the odd number before and after the number given. If the
number is an even number then subtract 1 from it for the odd number before, add one to it for
the odd number after. If the number is odd itself, subtract two from the number for the odd
number before it, add two for the odd number after it. The last 4 questions involve even
numbers before and after. Use the same method, if the number is odd subtract/add one to get
the even numbers before and after and if it is even, subtract/add two.
Column 2 starts with listing all the odd numbers between two given numbers. One of the most
asked questions from students is “does between include the numbers in the question?”, no it
doesn’t. So look at the first number if it is even, the first odd number is one more than it, if it is
odd, the first odd number is found by adding two to it. Then add two to get the next odd
number and so on until the last odd number written is less than the larger number in the
question. Repeat the same process for even numbers. Separate the numbers with a comma.
The last part of column 2 is circling the odd numbers and placing a square around the even
numbers.
Column 3 starts with three questions on completing the pattern, if the numbers are even write
in the next three even numbers, if they are odd put in the next three odd numbers. The rest of
the column is devoted to showing properties of odd and even numbers. These are:
•
Multiply (×) an even number with an even number you get an even number as the answer
•
Multiply (×) an odd number with odd number you get an odd number as the answer
•
Multiply (×) and even number with an odd number (or the reverse) you get an even
number as the answer
•
The sum (+) of an even number with an even number gives an even numbered answer
•
The sum (+) of an odd number with an odd number gives an even numbered answer
•
The sum (+) of an odd number with an even number (or the reverse) will give an odd
number as the answer
When students are asked “What is an odd number times an even number" in a test and don't
answer it they usually say "I forgot the rules", you don't have to remember them, just use 2 as
your even number and 3 as your odd number and answer the question in your head. You get
the answer 6 so its “even”.
Odd and Even Numbers
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Write all the odd numbers
between these 2 numbers
State whether the numbers
below are odd or even.
1
3
2
11
3
8
4
17
5
30
6
57
7
43
8 102
9
95
10 230
11 333
12 507
13 896
27 768 and 776
Complete the
patterns below
37 76, 78, 80,
28 809 and 816
38 185, 187, 189,
,
,
39 293, 295, 297,
,
,
29 495 and 506
Example
30 444 and 455
40
5 +
9
odd
odd
31 291 and 301
41
19
Give the even numbers
between these 2 numbers
32 611 and 622
44
After
45
33 838 and 852
Before
Before
After
Before
After
Now give the even
number immediately
before and after these
711
Before
After
24 1 677
46 an even + an even =
34 992 and 1 003
47 an even × an even =
Circle the odd numbers
and put a square around
the even numbers
198
After
11
181
26 9 999
Before
After
152
776
107
302
308
39
70
663
930
51
odd
17 + 8 =
odd
800
459
221
50
even
8 × 5 =
even
555
Before
7 × 6 =
odd
49
After
25 3 894
48
35 9 895 and 9 903
36
Before
even
From the above you can say:
After
499
even
66 + 14 =
even
22 5 000
23
6 × 12 =
even
20 1 050
21
42 an odd + an odd =
43 an odd × an odd =
234
Before
odd
From the above you can say:
18 400
Give the odd number
before and after the
number given below
odd
33 + 17 =
odd
17 813
= 14 even
11 × 5 =
odd
16 603
,
Carry out the operations
then state if each answer
is odd or even
14 439
15 771
,
even
24 + 13 =
even
odd
From the above you can say:
52 an odd × an even =
an even × an odd =
53 an odd + an even =
an even + an odd =
Triangular Numbers, Square Numbers
and Square Roots
Part of understanding number theory is understanding number patterns. Triangular numbers
are the result of a pattern that can be easily represented graphically by dots which form
triangles (look at the top of the first column). The first line is 1 dot, increase the next line by a
dot so on the next line has 2 dots, the third line increases it by another dot and you get 3 dots,
and so on, adding another dot to each new line. Square numbers could be seen as adding a dot
to its width and adding a dot to its height. See half way down the column, before Q. 4.
In column 1 the first 3 questions deal with triangular numbers. You can work out the number
mentally then 'do the dots' or the reverse, its your choice. Just add a row that is 1 greater than
the previous row. With square numbers you can use a similar method, just add a dot to the
width and a dot to the height to make the next largest square.
Column 2 involves exercises to make sure you understand how you get a square number.
Some students forget and incorrectly think that when you square a number you multiply it by
2, like 4² = 4 × 2 = 8. It is important that you remember that squared means “times itself”
ie. 4² = 4 × 4 = 16. You should know your 10 times table and so these should be just a mental
exercise. Others know higher squares but many don't, so use the working spaces to calculate
the next 10 square numbers.
Column 3 from Q. 28 involves square roots. A square root is the opposite to squaring a
number. It is asking “what number multiplied by itself equals the number in the square root
sign”. You know that 2 × 2 = 4, that would mean that the √4 is 2. Or what number multiplied
by itself equals 4? Well 2 × 2 = 4, so its 2. All the square roots asked for in these questions
are within the first 20 square numbers. That means you have all the answers in the previous
questions, just look at the previous answers and work backwards. Note that you can take
square roots of non-square numbers, the answers will be in decimals however, a job for a
calculator.
Triangular Numbers, Square Numbers and Square Roots
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Find the next 3 triangular
numbers and draw in the
dots to represent them
Add a row, each row increases by 1 dot
List the first 10 square
numbers then calculate
the next 10
24 17² =
25 18² =
17
18
17
18
26 19² =
27 20² =
19
20
19
20
8 1² = 1 × 1 =
9 2² = 2 × 2 =
1
3
6
10 3² =
×
=
11 4² =
1
4th number
12 5² =
13 6² =
2
14 7² =
5th number
15 8² =
16 9² =
3
17 10² =
6th number
Find the next 4 square
numbers and draw them
with dots.
18 11² =
19 12² =
11
12
11
12
4
4th number
4
28
81 =
0
1
Using your previous
answers find the square
root of these numbers
30
9
20 13² =
21 14² =
13
14
13
14
100 =
29
49 =
31
16 =
32
33
121 =
225 =
34
35
400 =
256 =
37
5
5th number
6
6th number
7
7th number
22 15² =
23 16² =
15
16
36
15
16
324 =
81 =
38
39
361 =
289 =
Palindromic, Fibonacci Numbers
and Number Patterns
Palindromic numbers can be read the same way forwards or backwards. For example some
word palindromes are: rotor, radar, mum and dad. The same applies with numbers, examples
are: 121, 54 145, 11, 444 and 7 227.
Column 1 asks you to circle the palindromic numbers. To check them look at the first number
and the last number, if they are different the number isn't palindromic. If they are the same
look at the second number and the second last number and compare. Do this until you reach
the middle number(s). If you have matching pairs of numbers all the way through, (if there is a
middle number alone it is as if it has a matching pair) then the number is palindromic or a
palindrome.
Questions 2 through 17 have numbers missing that you need to fill in to make the numbers
palindromes. Compare the position of the box with the same position from the other end, if it’s
the first number compare it with the last number, if it’s the second number compare it with the
second last position or the reverse. The last question asks you to find the next 11 palindromic
years after 2002, the first one being given to you (2112).
Column 2 introduces Fibonacci numbers. This is an unusual number pattern because it has the
number 1 twice! The numbers are 0, 1, 1, 2, 3, 5, 8, 13, … each number is found by adding
the previous two numbers, this is shown at the top of the second column. So 0 and the first 1
don't fit this rule (they don't have 2 numbers in front of them) but the second 1 is part of the
method and from then on the rule applies. (0 + 1 = 1), (1 + 1 = 2) etc. This column asks you
to find the next 10 numbers. From question 32 working spaces are supplied to help with the
harder addition.
Column 3 is about number patterns. The first section deals with addition or subtraction. Look
at the first two numbers. If they are increasing in size its an addition. To solve, subtract the
first number from the second and that's the amount that you need to add to get the next
number. If the numbers are reducing in size you know it’s a subtraction, subtract the second
number from the first number to find the amount to subtract.
The next group is multiplication and division, if they increase its multiplication, if they
decrease its division. Find the number you multiply the first number by to get the second.
Then, apply this to find the missing number. Division will have the numbers decreasing in size
and so find the number that the first number was divided by to get the second number. Note
that if the numbers are too large or difficult use the 2nd and 3rd numbers.
The last section is more difficult. This time you apply two operations (× or ÷ with + or -).
Instead of just using the first 2 numbers make sure that the solution applies to all the given
numbers. So investigate further into each question when you have a mixture of different types
of patterns.
Palindromic, Fibonacci and Number Patterns
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Find the next 10 numbers
in the Fibonacci pattern
Circle (or oval?) all the
palindromic numbers
644
45 545
59
333
29 13 +
3
5
4
5
114
6
1
8
225
7
=
33
32
11 902
09
16
8
7
2112
[
+
34
93
1
17
34
33
24
19
25
20
26
21
27
,
,
,
,
,
43 36, 29, 22,
,
,
44 93, 82, 71,
,
,
45 0, 80, 160,
,
,
Example
×3
+
=
×3
×3
×3
×3
46 2, 4, 8,
,
,
47 800, 400, 200,
48 15, 30, 60,
35
+
=
22
18
,
1, 3, 9, 27 , 81 , 243
15
23
,
Multiply or divide to find
the next 3 numbers in
these number patterns
33
86
Give the dates of the next
10 palindromic years that
follow 2112.
2002
+
=
Now there's two missing
14 2 3
,
106
43 345 13 6
12
39 2, 4, 6,
42 50, 75, 100
=
1
,
41 15, 12, 9,
31
9 6
,
5+8
=
=
22
38 1, 3, 5,
40 1, 5, 9, 13,
30
7 21
10 1 0
2+3
28 8 + 13 =
Find the missing number
that would make these
palindromes
2
[
1+1
2 002
100
1 212 121
0, 1, 1, 2, 3, 5, 8 , 13
6 565
[
16 016
[
11
3+5
0+1
112 233
88
1+2
[
1 991
[
1
Use addition or subtraction
to find the next 3 numbers
in these number patterns
36
+
=
37
+
=
,
,
,
,
49 1 024, 256, 64,
,
50 ¼, 1, 4,
,
,
,
These use a combination
and are more difficult.
51 2, 5, 11, 23,
,
,
52 4, 7, 13, 25,
,
,
53 10, 15, 25, 45,
54 5, 20, 50, 110,
,
,
,
Factors - Finding the HCF
Factors are numbers that divide into another number without a remainder. All numbers have
factors, the number itself and 1. If a number has only these two factors then it is classified as
being prime. If additional factors are found then the number is called composite.
Column 1 asks you to list the factors of the numbers. Follow this method:
•
Write 1 straight away then see if 2 is a factor and test each number 3, 4, 5, etc
•
Once you get to half of the number you can stop and write the number itself, there are no
factors greater than half the number except itself
•
Perform a check, the first number × the last number = the number, the second number ×
the second last number = the number and work your way into the middle. The middle
two numbers when multiplied = the number. Note that if the number is a square number
then the middle number is the square root.
A common mistake is students finding the smaller factors and missing the larger factors. If
you perform the check you won't miss them. When you list the numbers separate them with a
comma.
Column 2 asks you to find the HCF (Highest Common Factor). When given 2 numbers (or
more) the largest factor that is a factor of both numbers is the HCF. The process is the same as
column 1 only you perform it twice, for the 2 numbers. The bottom of the column asks you to
state whether the numbers are prime or composite. Note that 0 and 1 are usually considered
neither prime or composite and 2 is the only even prime number.
Column 3 is an introduction to factor trees. Factor trees are used to break a number down into
its prime factors. The first nine questions give a number at the top of the 'tree', then another
number is given below it, divide the top number by the lower number and write the answer in
the box. Or ask yourself what number times the lower number equals the top number? The
next four questions use the same process only it is extended a further branch. Find the missing
number as with the earlier questions, then break these numbers down to the two (prime)
factors that multiply together to equal it. If the number is prime it doesn't have any factors so a
single branch and circle are below it, just rewrite the same number in the circle.
To list the prime factors write the numbers that are in the bottom row of the tree, you only
write each number once. That means that if there are, for example, three 2's and a five you
only write 2 and 5, there isn't space for you to write all the numbers.
Factors - Finding the HCF
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
List the factors for
the given numbers
Example
8 1, 2, 4, 8
1
6
2 10
List the factors for each
number then circle the
HCF of the numbers
17 12
The HCF is
18 24
The HCF is
5 40
19 45
6 24
18
9 48
29 18
30 15
6
2
3
31 30
32 56
33 60
10
8
4
34 36
35 12
36 80
6
4
5
56
4 20
8 44
28 24
42
3 12
7 14
Find the missing factor
and write it in the empty
circle.
The HCF is
Use the factor trees to
find the prime factors of
the following numbers
20 28
36
37
The HCF is
Prime factors:
21 16
10 56
30
5
42
The HCF is
11 100
38
22 24
Prime factors:
12
12 60
18
9
9
The HCF is
13 80
14 120
15 150
16 27
Are the numbers below
prime or composite?
Circle the correct answer
23 29
Prime Composite
24 35
Prime Composite
25 33
Prime Composite
26 47
Prime Composite
27 87
Prime Composite
39
100
Prime factors:
10
40
24
Prime factors:
6
Multiples - Finding the LCM
When you think of multiples think of being given a number and multiplying it by 1, 2, 3,.etc.
So the first multiple of a number is itself, as 1 × a number doesn't change it. But while it is
multiplication it is often easier to add. Look at the example below finding the multiples of 32.
3 × 32 =
4 × 32 =
2 × 32 =
5 × 32 =
By Multiplying
128 + 32 =
By Adding
1 × 32 =
32 32, 64, 96, 128, 160
32 =
32 + 32 =
96 + 32 =
64 + 32 =
Column 1 deals with finding the first five multiples of a number. Write the number then add
the number to itself and find the total, then add the number again and find the new total and so
on. Or multiply by 1, 2, 3, 4 and 5, the addition is usually easier.
Column 2 asks for multiples of a number between 25 and 45. Remember between doesn't
include the numbers, if a multiple is 25 or 45 they aren't between. You should be able to
complete the additions mentally. Eg. the number 7...would start with 7 then 14 then 21 then 28
….that’s above 25 so that’s my first number, write in 28 then add 7's….28 + 7 = 35, write
35….35 + 7 = 42, write 42….42 + 7 = 49 that’s above 45 so we don't include it, so you're
done. The next 5 questions ask for the next 3 multiples above 100. The method would be to
count mentally in your head the numbers given until they pass 100, then write the next 3
multiples. Rather than counting from the start you should be able to multiply the numbers by a
number of your choice to get it close to 100. For example if the number was 12...I know that
7 × 12 = 84….. so start from there 84 + 12 = 96…. so 96 + 12 = 108 that is my first number.
Then away you go!
What is the LCM? LCM (Lowest Common Multiple) is the smallest multiple that is shared by
two or more numbers. Column 2, Q. 28 through to the end of the sheet all use the same
method. You are given 2 (or 3) numbers, list the multiples until you think you have the LCM
covered. When you list the multiples for the second number stop once you spot the match.
Then circle the matching LCM’s and write the answer. An example is below.
5
5, 10, 15, 20, 25, 30
List the multiples of 5
4
4, 8, 12, 16, 20
List the multiples of 4, stop when you get
a match (20). Circle the 2 numbers, then
write your answer on the line.
The LCM is 20
Multiples - Finding the LCM
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
List the first 5 multiples
of the numbers below
Example
4 4, 8, 12, 16, 20
1 10
List the multiples of the
numbers below that are
between 25 and 45
18 3
3
20 4
3
5
21 6
4
7
22 12
5 12
6
33 35
21
The LCM is
34 12
List the next 3 multiples
of the given numbers
that are greater than 100
9
7 15
23 5
8 20
24 10
9 11
25 8
10 25
26 9
11 8
27 22
12 100
List the multiples for
each number then circle
the LCM of each.
13 150
14 135
15 160
The LCM is
35 14
8
The LCM is
36 12
15
The LCM is
37 24
10
18
The LCM is
The LCM is
29 8
38 30
6
15
The LCM is
45
The LCM is
33
The LCM is
17 250
20
28 4
30 11
16 202
15
The LCM is
19 5
2
32 9
39 9
31 20
6
15
12
The LCM is
The LCM is
Indexed Numbers
This sheet has some questions written sideways. To read these questions click on the
“View” menu, select the rotate option then “counter clockwise”.
Indexed numbers have a base and an index. The index tells you how many times the base is
multiplied by itself. So 52 is = 5 × 5 and 54 = 5 × 5 × 5 × 5. When a number is shown with an
index it is in ‘index form’, when it is shown with the '×' between the numbers it is said to be in
'expanded' form (that means pulled apart).
Base
5
Index
7
Index Form
57 = 5 × 5 × 5 × 5 × 5 × 5 × 5
Expanded Form
Column 1 involves changing numbers from expanded form to index form. Look at the base
and write this normal size, then count how many of the numbers there are and write that
number next to the base number, as a smaller raised number.
Column 2 is an extension of this, with finding the expansion and then solving for the answer.
An example is at the top of the column. Strokes can be used to total along the line (as in the
example) or answer it mentally. The next section of column 2 deals with indices with a base of
10. With these you don't need to expand, the answer is just 1 with the index number of zeros
after it. For example 105 would be a 1 with 5 zeros or 100 000. That means 101 = 10
(a 1 with 1 zero after it). Numbers are then introduced, for example 6 × 105 would be a 6 with
5 zeros or 600 000, and 9 × 101 = 90.
Column 3 from Q. 43 is the reverse process, look at the number and write it as the first digit in
the answer then a '× 10', then the power. Count the number of zeros after the first digit and
that’s your power!
Indexed Numbers
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Calculate the numbers that
these indexed numbers
represent, you may wish to
expand them first.
Write these numbers in
index form don't multiply
to get the answer
Example
1
2
3
4
Example
5
4
8 × 8 × 8 × 8 × 8 = 85
1 6×6×6
3 5×5
=
=
4 7×7×7×7
=
5 10 × 10 × 10 × 10 × 10 × 10 × 10
=
6 1 × 1 × 1 × 1 × 1 × 1 × 1 × 1× 1
=
7 9×9×9×9×9
=
32
37 9 × 106 =
17 33 =
=
18 103 =
=
19 82 =
=
20 14 =
=
41 2 × 101 =
21 43 =
=
42 7 × 102 =
22 53 =
=
23 24 =
=
Write these as a single
digit number × a base 10
indexed number
38 5 × 101 =
39 2 × 107 =
40 4 × 105 =
Example
8 3×3×3×3×3×3×3×3
6
24 2 =
60 000 = 6 × 104
=
=
Now the reverse,
expand these !
35 5 × 102 =
36 8 × 105 =
16
25 = 2 × 2 × 2 × 2 × 2 =
=
2 4×4×4×4×4
8
34 7 × 103 =
9 512 =
10 109 =
11 1237 =
12 213 =
13 88 =
14 196 =
15 111 =
16 107 =
Write these indexed base
10 numbers as numbers
without working out
43 400
=
44 9 000
=
45 20 000 =
25 103 =
46 4 000 000 =
26 102 =
47 700 000 =
27 106 =
48 3 000 =
28 107 =
49 90 000 000 =
29 1 × 104 =
30 2 × 104 =
50 70 =
51 6 000
=
52 500 000 =
4
31 6 × 10 =
53 200
=
32 2 × 102 =
54 8 000 000 =
33 9 × 102 =
55 30 =
Expanded Notation with Indices
This sheet has some questions written sideways. To read these questions click on the
“View” menu, select the rotate option then “counter clockwise”.
When numbers are written in expanded notation the number is broken up into its place values.
Instead of using numbers to represent the millions, hundred thousands etc. indices are used, a
list of which is below. You should have attempted 'Indexed Numbers' prior to this worksheet.
The Top 8 Indices Chart
105 10 or 101
107
103
So this number would be:
98 765 432
106
9 × 107 + 8 × 106 + 7 × 105 + 6 × 104 + 5 × 103 + 4 × 102 + 3 × 10 + 2
102
104 units
In column 1 a number has been expressed in expanded notation (in index form). You are asked
to write the number in basic numerals. The best way to attempt these problems is to look at
them digit by digit. In the example at the top of the column you have :
7 × 105 + 4 × 103 + 8 × 102 + 6 , follow these steps:
•
Write the 7 straight away, note that its power is 105
•
Is the next number ×104, no it isn’t so write a zero
•
Is the next number ×103, yes it is, so write a 4
•
Is the next number ×102, yes it is, so write an 8
•
Is the next number ×10 (or ×101), no it isn’t so write a zero
•
Are there any single units (or ×1), yes there is, so write a 6
•
You now have 704 806.
If you have trouble remembering the powers, remember that with base 10 indices the power is
the number of zeros behind the one, i.e. 103 = 1 000, a 1 with three zeros.
Column 2 works in the reverse, just start at the first digit and work through, remember if it’s a
zero, write nothing. Count the number of digits behind the first number to find the first base 10
index.
Expanded Notation with Indices
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Change from expanded notation
to basic numerals
Example
105
10
4
Now the reverse, write these
numbers in expanded notation
103
102 10
units
7 × 105 + 4 × 103 + 8 × 102 + 6 = 704 806
1 2 × 103 + 9 × 102 + 5 × 10 + 6
Example
59 003 007 = 5 × 107 + 9 × 106 + 3 × 103 + 6
21 6 610
=
2 3 × 104 + 1 × 103 + 6 × 102 + 3 =
22 271 004
=
3 5 × 105 + 2 × 104 + 7 × 10 + 1
=
23 45 096
=
4 6 × 105 + 2 × 104 + 6 × 102
=
24 5 010 072 =
5 7 × 104 + 3 × 102 + 2 × 10 + 8
=
25 86 090
=
=
18 3 × 107 + 7 × 106 + 2 × 104 + 1 × 103 + 6 × 102 + 9
17 8 × 106 + 5 × 105 + 4 × 104 + 6 × 103 + 5 × 102
16 2 × 107 + 1 × 106 + 5 × 103 + 1 × 102 + 2 × 10 + 8
15 1 × 106 + 3 × 105 + 1 × 104 + 2 × 102 + 2 × 10 + 7
14 9 × 107 + 3 × 106 + 9 × 105 + 3 × 104 + 6 × 102 + 4
13 2 × 106 + 7 × 104 + 7 × 103 + 6 × 102 + 9 × 10 + 6
12 8 × 105 + 6 × 104 + 9 × 103 + 7 × 10 + 5
11 1 × 107 + 4 × 105 + 8 × 104 + 6 × 103 + 3 × 102 + 3
10 3 × 106 + 9 × 105 + 6 × 103 + 8 × 102 + 8 × 10 + 6
=
=
=
=
=
=
=
=
=
=
=
30 2 934 049 =
19 4 × 106 + 5 × 104 + 1 × 102 + 5 × 10 + 3
=
31 5 773 088 =
20 9 × 107 + 6 × 104 + 6 × 102 + 5 × 10 + 4
=
29 7 430 007 =
32 653 310
=
33 52 400 032 =
9 7 × 105 + 7 × 104 + 7 × 10 + 7
34 3 756 407 =
=
35 4 986 020 =
28 107 902
=
8 6 × 104 + 2 × 103 + 6 × 102 + 2 =
36 911 690
=
37 1 804 125 =
27 396
=
38 558 027
7 9 × 107 + 4 × 103
39 8 023 962 =
26 4 700 000 =
40 45 770 000 =
6 8 × 105 + 5 × 104 + 3 × 102 + 5 =
Factor Trees
Factor trees are used to reduce a number down to a multiplication of its prime factors. These
factors can be used to find the LCM and HCF of two numbers. This sheet asks you to find the
answer as a multiplication of the primes and to also write the answer in index form. Factor
trees have the number to be reduced at the top, this is split into two branches and from then on
is split again into 2 branches or extended with a single branch if it is a prime number, until all
the numbers are prime (don't have factors other than 1 and itself).
Column 1 has factor trees already constructed with a number given on the first tier (first split
branch) to help you get started. Here is the method:
•
Look at the number to be reduced and divide it by the number on the first tier, that will
give you the answer for the other first tier box.
•
Then look at the numbers you now have and break them down, if a single branch is
provided that means you can't reduce the number so just rewrite it in the box below, if
you have another pair of branches then select 2 numbers that multiply together to give the
number. Note that you never use 1 in a factor tree.
•
Column 1 only has one solution method, where there is a choice, a number is supplied so
that there is only one answer.
•
Once you have the boxes completed look at the numbers across the bottom row and
multiply them to get the answer, it should equal the number at the top of the tree. Then
write the numbers in the space provided, then rewrite them in index form, which you
should know from the previous sheet.
Column 2 and 3 require you to build the tree without hints. Pick two numbers that multiply to
get the top number, you should pick two numbers that are close together to reduce the number
in the minimum number of steps. So if the number was say 100 you would use 10 and 10 not
25 and 4.
20
5 × 4 is 20
5
Already prime
so rewrite
4
2
5
20 = 5 × 2 × 2
20 = 5 × 22
Rewrite in index
form
2
2 × 2 is 4
Rewrite bottom row
of tree with × signs
between the numbers
Factor Trees
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Complete the factor tree to
reduce the number to its
prime factors, then express
the primes in index form
Ok, do the same thing
but this time build
your own tree
9
500
8
5
12
1
4
500 =
12 =
×
12 =
2
8 =
×
500 =
8 =
×
6
10
180
81
36
2
6
81 =
36 =
×
×
36 =
2
2
×
×
180 =
81 =
7
180 =
48
50
3
11
450
5
50 =
48 =
50 =
450 =
48 =
64
4
8
450 =
120
12
8
4
4
64 =
120 =
1 000 =
64 =
120 =
1 000 =
1 000
Powers, Roots and the Power Key - Calculator
This sheet is about the use of powers and roots on a calculator. There are examples on the next
sheet with a calculator image showing you the buttons. This calculator may be different from
your model but the symbols used on the keys should be the same, just look for the symbols.
Column 1 is all square roots, some calculators require you to type the number then the square
root, the steps for these calculators are different than those shown, try your calculator on the
examples on the next page and see the method you use to get the correct answer. The first part
of the column doesn’t require brackets, the second part does. Your calculator needs brackets to
force order of operations on it, otherwise it will take the square root of the first number only.
Column 2 deals with squares (power of 2) and cubes (power of 3). Some calculators may not
have a cube key, if this is your case then use the power key with a power of 3. Some
calculators experience difficulty with fractions and powers, with the power being considered
on the last number entered only (the denominator), so you may have to put fractions in
brackets before entering the power. The second part of the column deals with using brackets.
Your teacher may ask you to avoid using brackets, instead using the equals key at selected
times, this is just as valid. When dividing, bracket the top operation then divide it by the
bracketed bottom operation then press =.
Column 3 starts with using the power key, look at the examples. Single number cube roots are
then covered briefly, leading to the more complicated cube roots with brackets. Teachers will
ask you to ignore the cube root sign and evaluate the inside first, then cube root it. This
simplifies the operation and should be considered as an easier alternative, done in 2 steps. You
don’t have to solve these calculations in one step, and the use of 3 sets of brackets may lead to
mistakes being made.
Examples of steps in calculating square roots
85 = √ 8 5 = = 9.22
5 2/7 = √ 5 a b c 2 a b c 7 = a b c = 2.30
13.4 + 22 × 3.6
=√
( 1 3
These answers to 2 d.p.
. 4 + 2 2 × 3 . 6 ) = = 9.62
34
45 + 19.7
=√
( 3 4 ÷ ( 4 5 + 1 9 . 7 )
) =
= 0.72
Use the same method with the cube root key
only press Shift first
Examples of steps in calculating squares and cubes
5.62 = 5 . 6 x2 =
= 31.36
(5 2/7)2 = ( 5 a b c 2 a b c 7 ) x2 = a b c = 27.94
Use the same method with the cube key
x3
These answers to 2 d.p.
Examples of steps using the power key
5.68 = 5 . 6 ^ 8 =
= 967 173.12
These answers to 2 d.p.
(5 2/7)9 = ( 5 a b c 2 a b c 7 ) ^ 9 = a b c = 3 220 573.07
Powers and Roots - Calculator Applications
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Use the square root key on
your calculate to answer these.
Find these squares and
cubes, answer to 2 d.p.
where required.
Calc:
4 4 1 =
1 441 =
2 361 =
24
352
=
3 256 =
4 484 =
25
9.72
=
5 289 =
6 729 =
26
20.62
=
8
27
(3½)3
=
28
10.53
=
29
652
=
30
123
=
31 (43/5)3
=
3
=
7
0.04 =
9
0.3136 =
1 369
=
Now round your answers to 1 decimal place
10 32 =
11 77 =
12 205 =
13 888 =
Use the brackets key to
answer these, round to
1 d.p when necessary.
Calc:
(
14
(23 × 16)
15
(27.2 + 48.9) =
32
2 3 × 1 6
)
=
35
17
(106 + 45 × 19) =
=
The brackets aren’t in these questions but
you still need to use them on your calculator.
19
133.6
8.35
20
21
22
=
=
509 - 21 × 18.6 =
17.7 × 30 × 1.8 =
49.3 + 112 + 95 =
=
36 252 - 123
=
3
2
2 293.2
5.2
=
=
48
55
=
49
17
=
50
106
=
51
5.23
=
52
212
=
2 6
53 (1 /7)
=
54 89 - 88
=
56
3
216
=
57
3
343
=
58
3
1 000 =
37 34 + 6.2 × 13 =
38
77 + 252 =
Brackets now required
2
39 ( 66 ) =
40 (11 - 26.2)2
=
41 11 - 26.22
=
2
2
42 11 - 26.2
=
43 183 - 182 - 18
=
44 183 - 182 × 18 =
45 1003 ÷ 103
23
24
Use the cube root key 3
to answer these. Express
your answer to 2 d.p. when
required
=
143
(15 ÷ 3.9)
19 × 42 + 102
52.52
47
55 215 ÷ 213 =
33 19.42 + 10.93 =
34 18.1 × 113
=
16
18
8.09
Use the power key ^
for these. Answer to
2 d.p. if required.
3
3
59
5 375
3
46 100 ÷ 10 ÷ 10 =
=
60 3 452 × 17.3 =
61 3 883 + 883 =
62 3 5 833 - 183 =
Careful with this one
=
3
43
53 + 27
63
3
11²
=
Further Factor Trees - Finding the HCF
This sheet and the one that follows it are challenging.
Finding the Highest Common Factors of larger numbers can be found using factor trees. The
method is as follows:
•
Build a factor tree, you should have completed the previous sheet on factor trees and so
the process of finding prime factors won't be readdressed.
•
Write the number and its prime factors in the table then match off pairs of numbers
against each other, one for one, by circling them.
•
Write one set of the matching prime factors on the first line separated by × signs
•
Evaluate the multiplication and write your answer in the space, this is the HCF.
An example is at the top of the worksheet.
Further Factor Trees - Finding the HCF (or GCF)
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Complete the trees to find the prime factors, put the
prime factors in the table, circle factor pairs in both
lists, multiply these factors to get the HCF (or GCF)
Example
54
9
3
6
3
3
45
2
9
3
1
72
3
10
5
5
2
5
5
2
120
List of Prime Factors
Number
450
54
3 3 3
2
450
3 3 5
5 2
HCF = 3 × 3 × 2
HCF =
Number
18
List of Prime Factors
HCF =
HCF =
2
48
330
Number
List of Prime Factors
HCF =
HCF =
3
180
750
Number
List of Prime Factors
HCF =
HCF =
4
48
270
Number
HCF =
HCF =
List of Prime Factors
Further Factor Trees - Finding the LCM
This sheet is a challenging sheet.
Finding the Lowest Common Multiple of larger numbers can be done by using factor trees.
The method is as follows:
•
Build two factor trees, you should have completed the Factor Tree sheet and so the
process of finding prime factors won't be readdressed.
•
Write the number and its prime factors in the table for both trees then if a number is in the
top list and also in the bottom list strike out the number in the bottom list only.
•
Note that this doesn’t mean if you have a 2 in the top list you strike out all the 2's in the
bottom list, you just strike out one number for one number. If there are 2 threes in the top
list and 4 threes in the bottom then you only strike out 2 threes in the bottom list.
•
Write all the non-struck out numbers from both lists, separated by × signs
•
Evaluate the multiplication and write your answer in the space, this is the LCM.
The multiplication is performed by trying to multiply numbers together to come up with two
large numbers. These larger numbers can then be multiplied. In the example the 3's and the 2's
are multiplied separately, leaving 9 × 72. If the numbers are too large then you may have to
complete a few multiplication steps.
Further Factor Trees - Finding the LCM
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Find the prime factors, put the factors in the table,
then strike out all the numbers in the bottom list that
are in the top list, multiply all non-stroked numbers
Example
18
9
3
1
2
3
2
16
4
8
2
2
2
2
List of Prime Factors
Number
32
4
2
60
2
2
2
18
3 3 2
32
2 2 2
2 2
9
4
8
16 32
LCM = 3 × 3 × 2 × 2 × 2 × 2 × 2
LCM =
Number
288
List of Prime Factors
LCM =
LCM =
2
12
45
Number
List of Prime Factors
LCM =
LCM =
3
24
50
Number
List of Prime Factors
LCM =
LCM =
4
35
120
Number
LCM =
LCM =
List of Prime Factors
Divisibility Tests - 2, 4, 5 and 10
Divisibility tests are rules that check if a number can be divided by another number, without
attempting the division to see if there is a remainder. The rules for this sheet are below.
•
A number is divisible by 2 if its last digit is a 0, 2, 4, 6, or 8
•
A number is divisible by 4 if its last two digits are divisible by 4, this doesn't mean each
digit separately, it is the number formed by the 2 digits. For example 32 is divisible by 4,
so 532, 1 032 and 77 732 would be also.
•
A number is divisible by 5 if its last digit is a 0 or a 5
•
A number is divisible by 10 if its last digit is a 0
So the tests all work on the last digit only, with the exception of the 4 test which requires the
last 2 digits.
These questions have different ways of being answered. You may have to write the answer. If
there is a yes or no answer required, either colour in the circle or strike out the wrong answer.
Divisibility Rules
2 : last digit is 0, 2, 4, 6, 8
4 : last 2 digits form a number
divisible by 4
5 : last digit is either 0 or 5
10 : last digit must be 0
Divisibility Tests - 2, 4, 5 and 10
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
1 28
2 66
3 1 112
4 1 003
5 59 691
6 80 000
7 27
8 185
9 77 772
10 13 574
Yes
Yes
Yes
Yes
No
No
No
23
24 654
6 678
550
454
4 930
92
40 30
41 60
42 110
The numbers below are
divisible by 5. Give the 2
possible numbers it could
be due to the unreadable
last digit.
43 205
24 3▒
45 490
No
Yes
No
Yes
No
Yes
No
25 9▒
Yes
No
26 15▒
Are the following
divisible by 4?
Circle your answer.
772
9 968
No
No
122
490
88 886
Yes
Yes
Test for divisibility of 2, 4,
5 and 10. Fill the oval if
divisibility exists.
Circle the numbers
that are divisible
by both 2 and 4
Test these numbers for
divisibility by 2, fill in
yes or no
44 776
46 2 080
47 32 070
27 21▒
28 1 00▒
48 924
29 86▒
49 74 650
30 9 07▒
11 42
Yes / No
12 50
Yes / No
13 68
Yes / No
14 86
Yes / No
15 532
Yes / No
16 102
Yes / No
33 1 00▒
18 44 448
Yes / No
35 643
19 13 252
Yes / No
36 10 101
20 30 006
Yes / No
37 3 804
21 91 428
Yes / No
38 13 330
22 10 100
Yes / No
39 8 080
10
2
4
5
10
2
4
5
10
2
4
5
10
2
4
5
10
2
4
5
10
2
4
5
10
2
4
5
10
2
4
5
10
2
4
5
10
50 45
Test these numbers for
divisibility by 10, fill
in yes or no
Yes / No
5
Find the next 2 numbers that are
greater than these numbers that
are divisible by 2, 4, 5 and 10:
32 2 11▒
17 5 794
4
Answer the following
questions
31 44 05▒
34 90
2
51 1 00
Yes
No
Yes
No
Yes
No
Yes
No
Yes
No
54 Claudia believes that only
when a number is a multiple of
20, is it divisible by 2, 4, 5 and
10. What do you think?
Yes
No
I stopped listening
to her years ago
52 1 963
53 594
I knew her before
she was President
Divisibility Tests - 3, 6 and 9
Divisibility tests for numbers can be used rather than division. The tests for 3, 6 and 9 are:
•
3: Find the sum of the digits (add them up) that form the number and if the sum is a
multiple of 3 then the number in the question is divisible by 3.
•
6: The number must be divisible by both 2 and 3. The test for 2 is the last digit is even
(0, 2, 4, 6 or 8). So use the same method as with 3 but the number given in the question
must also be even (not the sum).
•
9: Find the sum of the digits and if the number is a multiple of 9 then the number in the
question is divisible by 9.
A card with the rules of divisibility is also below
Divisibility Rules
3 : sum of digits is a multiple of 3
6 : number is divisible by both 3
and 2 (must be even & digit sum
is multiple of 3)
9 : sum of digits is a multiple of 9
Divisibility Tests - 3, 6 and 9
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Sum digits
Test these numbers for
divisibility by 3, fill in
yes or no
8+0+2+2
Example
Sum digits
8 022
16 10 800
12
Yes
Sum digits
1 78
Sum digits
2 93
Sum digits
3 112
Sum digits
4 108
Sum digits
17 60 682
18 99 996
No
Sum digits
Sum digits
20 44 004
Yes
Sum digits
21 88 884
Yes
No
Yes
No
22
Sum digits
Yes
No
35 730
8 10 020
Sum digits
9 84 355
Yes
Yes
No
Yes / No
Circle the numbers that
are divisible by 3, put a
box around them if they
are divisible by 3 and 6.
Sum digits
Sum digits
Yes / No 34 54 738
No
No
7 8 301
Yes / No 33 87 532
No
Yes
6 77 322
Yes / No 32 6 959
19 17 361
Yes
Sum digits
Yes / No 31 47 333
Sum digits
Sum digits
5 3 419
Yes / No 30 88 884
No
Yes
Sum digits
14 053
578
1 809
38
96
8 931
23 54
Sum digits
36 7 893
Test for divisibility of 3,
6 and 9. Fill the oval if
divisibility exists.
10 2 103
Yes / No
11 2 104
Yes / No
12 5 514
Yes / No
13 80 940
Yes / No
24 72
14 30 003
Yes / No
Sum digits
15 11 424
25 105
3
6
9
Sum digits
3
6
9
Sum digits
3
6
9
Sum digits
3
6
9
Sum digits
3
6
9
Sum digits
3
6
9
Sum digits
3
6
9
Sum digits
3
6
9
Sum digits
3
6
9
42 71 712
26 228
43 4 221
44 8 964
27 315
28 882
Sum digits
Yes / No 29 15 597
Sum digits
41 1 452
Sum digits
Sum digits
9
40 2 073
Sum digits
Sum digits
6
39 8 934
Sum digits
Sum digits
3
38 6 003
Sum digits
Sum digits
Sum digits
37 453
Sum digits
Sum digits
Sum digits
Sum digits
Sum digits
Now test for divisibility
by 6. Circle Yes or No.
Sum digits
35 38 007
690
The numbers below are
said to be divisible by 9.
Answer true or false
No
Sum digits
45 72 159
46 5 232
Further Divisibility Tests - 7, 11 and 13
These tests are more difficult than the previous tests. Follow the steps outlined below.
In column 1, to test for divisibility of 7:
•
Take the last digit off the number and double it (×2).
•
Subtract the number (found above) from the original number in the question without its
last digit.
•
The above subtraction can be reversed to avoid subtracting a larger number from a
smaller number (this occurs in question 1).
•
If the subtraction results in a number that is 0 or a multiple of 7 then the number is
divisible by 7.
•
There is an example at top of the column.
Test for divisibility of 11:
•
Counting from left to right add all the odd digits together, that is, the 1st, 3rd, 5th, 7th ….
(write the value in the box below the words 'odd digit')
•
Then add all the even digits together, that is, 2nd, 4th, 6th, 8th ….(write the value in the
box below the words 'even digit')
•
Subtract one from the other, space for subtraction is provided though you probably won't
need it, if the answer is 0 or a multiple of 11 then it is divisible by 11.
•
There is an example at top of the column.
Test for divisibility of 13:
•
Complete the multiplications to give the first 9 multiples of 13, you can use these to help
you identify the multiples
•
This test is similar to the 7 test in that the last digit is removed, only it is multiplied by 9.
•
This number is then subtracted from the number in the question without its last digit, this
subtraction can be reversed if required.
•
If the subtraction results in a number that is 0 or a multiple of 13 then the number is
divisible by 13.
•
There is an example before question 23.
Divisibility Rules
The card to the right is there if you need it.
7 : Remove the last digit off the
number, from it subtract 2 ×
the last digit. Divisible if the
answer is 0 or a multiple of 7.
11 : Add all the odd numbered digits
and then the even numbered
digits. Subtract them, if answer
is 0 or a multiple of 11, it is
divisible by 11.
13 : Remove the last digit off the
number and from it subtract 9
times the last digit. Divisible if
the answer is 0 or a multiple of
13.
Further Divisibility Tests - 7, 11 and 13
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Test these numbers for
divisibility by 7, fill in
yes or no
Example
476
2 × 6 = 12
Yes
No
Test these numbers
for divisibility by
11, fill in yes or no
47 12
35
Yes
No
Yes
odd digit
even digit
24
13
Yes
odd digit
-
Yes
Yes
-
No
-
=
=
=
Example
975
Yes
97 45
No
52
-
Yes
No
odd digit
Yes
No
-
9 × last digit
-
Yes
No
even digit
-
9 × last digit
No
Yes
-
even digit
odd digit
even digit
No
-
26 1 898
9 × last digit
No
13 438 592
-
No
25 1 765
Yes
No
Yes
even digit
Yes
Yes
=
9 × last digit
-
-
7 882
=
23 624
No
12 557 193
Yes
=
24 663
odd digit
6 743
=
45
even digit
11 184 796
Yes
=
20 7 × 13 21 8 × 13 22 9 × 13
No
5 957
=
17 4 × 13 18 5 × 13 19 6 × 13
even digit
10 13 178
odd digit
Yes
11
14 1 × 13 15 2 × 13 16 3 × 13
9 × last digit
No
4 826
24 13
-
9 275
No
3 576
No
8 639
odd digit
-
2 181
88 957
Yes
-
1 119
Example
Find the first 9 multiples
of 13, use the answers to
help test for divisibility by
13, fill in yes or no
No
-
27 1 846
9 × last digit
No
Yes
No
Yes
No
7
FREEFALL
MATHEMATICS
CALCULATOR
THESE SHEETS ARE IN THIS FOLDER AND THEIR CHAPTER FOLDER
Powers, Roots and the Power Key - Calculator
This sheet is about the use of powers and roots on a calculator. There are examples on the next
sheet with a calculator image showing you the buttons. This calculator may be different from
your model but the symbols used on the keys should be the same, just look for the symbols.
Column 1 is all square roots, some calculators require you to type the number then the square
root, the steps for these calculators are different than those shown, try your calculator on the
examples on the next page and see the method you use to get the correct answer. The first part
of the column doesn’t require brackets, the second part does. Your calculator needs brackets to
force order of operations on it, otherwise it will take the square root of the first number only.
Column 2 deals with squares (power of 2) and cubes (power of 3). Some calculators may not
have a cube key, if this is your case then use the power key with a power of 3. Some
calculators experience difficulty with fractions and powers, with the power being considered
on the last number entered only (the denominator), so you may have to put fractions in
brackets before entering the power. The second part of the column deals with using brackets.
Your teacher may ask you to avoid using brackets, instead using the equals key at selected
times, this is just as valid. When dividing, bracket the top operation then divide it by the
bracketed bottom operation then press =.
Column 3 starts with using the power key, look at the examples. Single number cube roots are
then covered briefly, leading to the more complicated cube roots with brackets. Teachers will
ask you to ignore the cube root sign and evaluate the inside first, then cube root it. This
simplifies the operation and should be considered as an easier alternative, done in 2 steps. You
don’t have to solve these calculations in one step, and the use of 3 sets of brackets may lead to
mistakes being made.
Examples of steps in calculating square roots
85 = √ 8 5 = = 9.22
5 2/7 = √ 5 a b c 2 a b c 7 = a b c = 2.30
13.4 + 22 × 3.6
=√
( 1 3
These answers to 2 d.p.
. 4 + 2 2 × 3 . 6 ) = = 9.62
34
45 + 19.7
=√
( 3 4 ÷ ( 4 5 + 1 9 . 7 )
) =
= 0.72
Use the same method with the cube root key
only press Shift first
Examples of steps in calculating squares and cubes
5.62 = 5 . 6 x2 =
= 31.36
(5 2/7)2 = ( 5 a b c 2 a b c 7 ) x2 = a b c = 27.94
Use the same method with the cube key
x3
These answers to 2 d.p.
Examples of steps using the power key
5.68 = 5 . 6 ^ 8 =
= 967 173.12
These answers to 2 d.p.
(5 2/7)9 = ( 5 a b c 2 a b c 7 ) ^ 9 = a b c = 3 220 573.07
Powers and Roots - Calculator Applications
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Use the square root key on
your calculate to answer these.
Find these squares and
cubes, answer to 2 d.p.
where required.
Calc:
4 4 1 =
1 441 =
2 361 =
24
352
=
3 256 =
4 484 =
25
9.72
=
5 289 =
6 729 =
26
20.62
=
8
27
(3½)3
=
28
10.53
=
29
652
=
30
123
=
31 (43/5)3
=
3
=
7
0.04 =
9
0.3136 =
1 369
=
Now round your answers to 1 decimal place
10 32 =
11 77 =
12 205 =
13 888 =
Use the brackets key to
answer these, round to
1 d.p when necessary.
Calc:
(
14
(23 × 16)
15
(27.2 + 48.9) =
32
2 3 × 1 6
)
=
35
17
(106 + 45 × 19) =
=
The brackets aren’t in these questions but
you still need to use them on your calculator.
19
133.6
8.35
20
21
22
=
=
509 - 21 × 18.6 =
17.7 × 30 × 1.8 =
49.3 + 112 + 95 =
=
36 252 - 123
=
3
2
2 293.2
5.2
=
=
48
55
=
49
17
=
50
106
=
51
5.23
=
52
212
=
2 6
53 (1 /7)
=
54 89 - 88
=
56
3
216
=
57
3
343
=
58
3
1 000 =
37 34 + 6.2 × 13 =
38
77 + 252 =
Brackets now required
2
39 ( 66 ) =
40 (11 - 26.2)2
=
41 11 - 26.22
=
2
2
42 11 - 26.2
=
43 183 - 182 - 18
=
44 183 - 182 × 18 =
45 1003 ÷ 103
23
24
Use the cube root key 3
to answer these. Express
your answer to 2 d.p. when
required
=
143
(15 ÷ 3.9)
19 × 42 + 102
52.52
47
55 215 ÷ 213 =
33 19.42 + 10.93 =
34 18.1 × 113
=
16
18
8.09
Use the power key ^
for these. Answer to
2 d.p. if required.
3
3
59
5 375
3
46 100 ÷ 10 ÷ 10 =
=
60 3 452 × 17.3 =
61 3 883 + 883 =
62 3 5 833 - 183 =
Careful with this one
=
3
43
53 + 27
63
3
11²
=
Time Calculations (Calculator)
Time calculations are performed every day, …..how long until lunch?, ….the bus arrives?
When you become a wage earner it is important to be able to check that your hours worked are
correct, but because minutes and hours are in groups of 60 min this is not always straight
forward. But a calculator makes it easy.
The calculator image on the next page shows the DMS key or the 'bubble button' (the key has
an orange border). This key allows you work with hours and minutes.
IMPORTANT YOU MUST ENTER A 0 (THEN DMS KEY) IF DEALING ONLY
WITH MINUTES (NO HOURS).
The calculator will always show the hours, minutes and seconds (we won't be using seconds)
separated by a degree sign (a small raised o) →°.
This reads 8 h 29 min
This reads 4 min
This reads 18 h
In column 1 you are asked to convert the times given in minutes to hours and minutes. This is
done by pressing 0 then the DMS key, then the minutes in the question then the DMS key and
then press = and the answer will be displayed. Example, change 338 min to hours and minutes.
0
3 3 8
=
This reads 5 h 38 min
In the 2nd column times are to be added together, put the first time into the calculator then a +
then the second time in, press =, done. Finding the difference between two times is done by
subtraction, using the same method as above, with one exception YOU MUST CONVERT
P.M. TIME TO 24 H TIME. If they are both a.m. times or both p.m. times no conversion is
necessary, as soon as a question involves both a p.m. and an a.m. time convert the p.m. time to
24 h format. See the example before Q 18.
The time sheet at the bottom of the column 2 requires you to find the time difference for each
day between start and finish times, put them in the spaces provided then add them. Monday is
already done for you, check you get the same answer and don't forget to include it in your
addition.
The 3rd column is using × and ÷ these are done no differently just remember that you are
multiplying by a number not a time and so only use the DMS on the time. The calculation of
an average time (Q 31) can be done in two ways refer to the text box below the calculator
image for the full key stroke method on the next page.
Example of steps in calculating average time
of 11h 12 min, 8 h 57 min and 10 h 9 min
Using Brackets
( 1 1
1 2
+ 8
5 7
Using = to avoid order of operations
1 1
1 2
+ 8
5 7
+ 1 0
+ 1 0
9
9
)
÷ 3 =
= ÷ 3 =
if you get 10 h 6 min for the question above you are correct
Time Calculations (Calculator)
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Rewrite the calculator
displays in h and min.
1
Now use × and ÷ with
these times
Add these times together
using your calculator
15 8 h 27 min + 11 h 36 min
23 3 × 17 min
24 8 × 23 min
16 8 h 19 min + 4 h 48 min +
2
25 7 × 1 h 46 min
2 h 37 min
17 4 h 39 min + 5 h 25 min +
11 h 56 min
3
Find the time difference
between these times that
are on the same day. Use
24 hour time format
26 1 h 55 min ÷ 5
27 19 h 57 min ÷ 9
28 54 min × 6 + 3 h 7 min ÷ 11
+ 1 h 27 min
1853 h in 24 h
4
Example
11.17 a.m. and 6.53 p.m.
1 8
Convert the following
times in min to h and min,
remember to put a zero in
for the hours first.
5 211 min
6 173 min
5 3
1 7
29 8 h 45 min ÷ 5 + 5 min × 17
+ 1 h 31 min
- 1 1
=
7 h 36 min
18 10.30 a.m. and 3.56 p.m.
30 Sean watches 3 movies at a
6 h movie marathon, if two
of the films were 1h 46 min
and 1h 50 min, find the
duration of the other.
19 3.52 a.m. and 11.27 a.m.
20 4.23 a.m. and 6.17 p.m.
31 Julie's time for return travel
to school for 3 days were:
1h 15 min, 56 min and 1h 22
min. Find the average time
for her return trip.
7 245 min
8 727 min
21 1.09 p.m. and 10.57 p.m.
9 341 min
Complete the time sheet
below, then find the total
hours worked.
10 360 min
11 568 min
22
A. Jolie Time Sheet
Day
Start
Mon 9.30 a.m.
12 540 min
13 1331 min
Finish
Hours
4.45 p.m. 7 h 15 min
Tues 10.45 a.m. 2.30 p.m.
h
min
Wed 9.30 a.m.
5.00 p.m.
h
min
Thur 9.30 a.m.
3.45 p.m.
h
min
1.15 p.m. 6.00 p.m.
h
min
Total hours
for the week
h
min
Fri
14 1080 min
Try these time problems
32 It takes Jemma 43 min to
wash a car, and 11 min to
vacuum a car. How long
does she take to wash 5 cars
and vacuum 3 of them?
Jemma has 40 min for lunch,
if she started at 10.25 a.m.
estimate her finish time.
Time taken
Estimated finish time
Comparing Fractions (Calculator)
Using a calculator to compare fractions is done by changing the fractions to decimals and
comparing the decimals instead. If all rounded to the same decimal place, the decimals can be
easily compared, e.g. 0.308, 0.471, 0.235. It is easy to see the largest and smallest decimal, or
if they are equal, if you look at the numbers behind the decimal point.
Column 1 asks you to convert the fractions to decimals, the calculator image on the next page
shows you how to do this. If you have a decimal 0.100 then write it as 0.100 don't change it to
0.1 as this may lead you to make an error when comparing other numbers that are 3 d.p. (You
may look at it as 1 instead of 100).
The rest of the sheet is attempted the same way, by finding the decimal for the fraction then
comparing decimals. In the example below, two fractions are being compared, the method is:
•
Find the decimals for both
•
Compare decimals and put a < or > or = sign in between the decimals
•
If this sign matches the one above it then write true, if not, write false.
4
5
0.800
>
<
7
8
0.875
False
In the 2nd column, from Q. 31 on, use the same method only this time just write the sign into
the box rather than stating true/false.
The third column asks you to compare fractions and arrange them in descending order, use the
decimal system then write the fractions again in descending order. In a test the most common
mistake is rewriting the decimals in descending order, the question asks for the fractions to be
written, not the decimals.
How do you remember the difference between < and >? Imagine they are arrowheads and
point them at the smallest number. E.g. 5 < 6 and 15 > 10, the arrow points to the smallest
one.
Descending order? Remember going down.
Example of how to convert fractions to decimals
6
7
=
?
Don't forget the 'equals' sign
Method 1: Using Fraction Key
6 a bc 7 = a bc
Method 2: Using ÷ Key
6 ÷ 7 =
if you get 0.857 (rounded) you have answered it correctly
Comparing Fractions (Calculator)
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Convert these to decimals
round to 3 d.p.
1
4
1
10
2
3
2
3
4
3
8
5
3
4
6
5
6
7
6
13
8
8
19
9
3
7
24
10
27
11
17
29
12
7
12
13
73
80
14
33
43
1
5
8
<
7
14
36
4
9
17
45
26
8
11
>
5
7
37
4
19
14
60
27
13
17
<
14
19
38
11
18
26
37
28
6
81
<
4
63
39
3
8
21
56
29
35
39
>
60
70
40
12
13
16
19
30
88
93
<
17
19
96
99
17
42
16
17 175
305
18 212
636
19 572
911
20 101
909
15
25
Use the same method
only fill in <, > or =
2
3
22
41
3
7 ,
2
3
3
5
32
3
6
18
36
42
4
10
27
60
Rewrite in
31
Write the decimal (3 d.p)
under the fraction and
then answer true or false
21
Use decimals to arrange
these in descending order
2
3 ,
10
13 ,
,
,
23
60 ,
13
40 ,
,
,
9
10 ,
29
34 ,
,
,
17
40
Rewrite in
Descending
order
39
95 ,
<
3
4
2
5
33
>
3
7
23
8
9
<
9
10
34
3
7
4
9
43
24
6
7
35
8
12
2
3
Rewrite in
>
8
10
Descending
order
13
15 ,
Descending
order
,
1
4
,
18
21
,
Changing Between Mixed Numerals
and Improper Fractions (Calculator)
This sheet is designed for you to learn how to change between mixed numerals and improper
fractions using your calculator. A calculator separates the numerator and the denominator by a
reversed 'L' (or sometimes an ‘r’), for mixed numerals the whole number is also separated by
the 'L' (or ‘r’). See example below.
whole number
numerator
3
6
17
5
18
denominator
denominator
numerator
The first column starts with the way your calculator displays fractions. If there are 2 numbers
the first number is the numerator (number on top) the second number is the denominator
(number on the bottom). These numbers are separated by the "reversed L". Don’t write
fractions the calculator way, write the fractions separated by a line (as above). From Q. 6 on,
you are asked to change mixed numerals to improper fractions. The method required is
described on the next sheet, in words - type the fraction into the calculator, press =, then press
shift and the fraction button.
Column 2 reverses the process and asks you to change improper fractions to mixed numerals,
to answer these put the fraction in the calculator and press =, the mixed numeral will be
displayed. Question 46 through 53 asks you to repeat the same process, if you use the ÷ sign
then you will get a decimal answer, press the fraction key and it will change it to a fraction. Or
a quicker step is to use the fraction key instead of the ÷ sign this will give the answer as a
fraction straight away.
Note: These are all the same:
8
3
3
8
3÷8
The last column asks you to compare the improper fractions with the mixed numerals. Put the
improper fraction into your calculator and press the ‘equals’ key. If the fraction displayed on
the screen is the same as the answer supplied write 'true', if it isn't, write ‘false’. At the bottom
of the column you are asked to show the pairs that ‘match’ (an improper and a mixed number)
by using pairs of shapes (a smaller shape for the improper fractions, larger one for mixed).
Example of how to convert Mixed numerals to Improper Fractions
3
6
7
=
3
a bc
6
?
Don't forget the 'equals' sign
SHIFT
if you get
a bc
7 =
a bc
27
as your answer you have done it correctly
7
Changing Between Mixed Numerals and Improper Fractions (Calculator)
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Change these improper
fractions to mixed
numerals
Write the fraction
represented by the
calculator screens
Change improper to
mixed to answer true
of false to these
8
1
2
1
26 5
=
2
27 7
=
4
48
2
28 3
29 8
49 4
43
3
=
8
8
50 7
51
2
=
7
7
51 8
75
5
=
9
9
2
3
30 16
7
4
5
Change these mixed
numbers to improper
fractions
6 3
8
1
=
2
1
=
2
4
2
=
10 1
5
1
=
3
7 1
9
1
=
1
2
5
=
11 1
12
=
5
31 24
=
7
34 13
=
2
35 19
=
7
36 13
=
2
37 28
=
3
54 1
38 15
=
4
39 17
=
10
55
40 13
=
6
41 53
=
5
56
Write the answers as
mixed numerals
13 2
4
=
7
42 11 ÷ 7
14 2
5
=
6
15 3
1
=
4
43 109 ÷ 10 =
16 1
3
=
4
17 12
1
=
7
20 3
19 3
1
=
8
21 9
4
=
22 4
5
24 10
2
=
9
5
=
11
25 11
=
3
=
4
11
52
6
24
53
7
5
6
= 3
4
7
15
7
=
8
8
17
7
13
11
57 4
= 1
= 2
3
7
= 1
3
11
15
1
=
3
3
Using circles, squares,
triangles or other shapes
draw the same shape
around matching pairs
58
44 3
125
45 20 ÷ 3
=
46 7
321
=
47 47 ÷ 5
=
5
7
4
=
2
=
3
1
=
23 1
10
=
33 16
=
3
2
=
3
1
=
3
2
32 23
=
5
12 2
18 7
=
= 4
16
3
1
5
3
33
7
22
7
6
3
2
3
1
7
5
22
3
31
7
3
7
4
17
3
2
3
20
3
7
1
3
Mixed Operations - Calculator
This sheet is for calculator use so there are no working spaces provided.
The method is simple for this sheet, use your calculator. You don't have to worry about
reciprocals or simplifying, the calculator does it all for you. Mistakes made using the
calculator can occur anywhere due to keystroke error, there is one more common mistake
made though, that is the entering of mixed numbers. Remember that you have to press the
fraction key twice to enter a mixed number.
Your calculator should obey order of operations rules in the exercises in column 3, just don't
press '=' until you key in the entire question otherwise the calculator may give the wrong
answer.
Mixed Operations - Calculator
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Add or subtract
these fractions
Multiply or divide
these fractions
These are all
mixed up
1
1
1
+
=
2
3
17
3
1
×
=
5
4
33
1
3
9
+
×
=
3
4
5
2
4
3
=
5
7
18
7
2
÷
=
12
3
34
1
2
2
÷
+
=
5 15 3
3
4
3
=
5
10
19
2
8
÷
=
3
9
35 5
1
× 3
6
4
3
3
+
=
11
5
20
3
7
×
=
5
6
36 7
3
1
3
÷ 9
+
=
8
4
2
5
3
1
4
+
=
4
6
5
21
4
1
8
÷
×
=
5
3
9
37
6
4
3
2
+
=
5
8 15
22
1
3 18
÷
×
=
7 10 5
38 5 ÷
7
9
5
1
+
=
12
7
4
23
3 17 1
×
÷
=
4 10 8
39
8
17
3
8
+
=
20 10 15
24
4
1
2
÷
÷
=
7 10 5
40 7
5
2
11
× 1 =
6
3
12
9
2
4
7
+2
=
3
5
10
25
1
3
2
× 4
×
=
6
8
3
41 3
3
5
5
÷
=
4
9
12
10 4
6
2
4
=
7
3
9
26 5
11
3
1
÷
×
=
12
4
2
42 2
5
1
× 3 ÷ 5 =
8
6
11 2
4
1
5
- 1
+
=
11
2
6
27 3
1
4
× 6 ×
=
2
5
43 9
1
2
4
- 3 ÷ 1 =
2
3
5
12 4
3
7
2
+
- 1 =
5
10
3
28 7
3
2
2
×
÷ 2 =
4
3
5
44 3
8
2
1
× 1 × 3 =
9
5
2
13 3
5
1
7
- 1 + 4 =
6
4
12
29 7 ÷ 1
14 3
2
7
5
+ 4 + 2 =
5
10
12
30 4
5
3
2
× 3 ÷ 2 =
6
10
5
46 4
7
1
1
÷
÷ 1 =
10
5
4
15 5
1
2
1
- 2
- 1 =
4
3
2
31 5
3
4
÷ 8 × 4 =
14
5
47 2
2
3
1
+
÷
=
5
4
2
16 9
2
6
5
- 3 + 1 =
3
7
6
32 2
3
2
÷ 1 × 3 =
7
3
48 5
7
1
1
÷ 2
- 1 =
8
3
4
1
3
÷ 2 =
4
5
45
- 4
6
=
7
1
5
5
+3
÷
=
2
4
6
7
8
×
=
10 9
1
7
3
×
+
=
2 11 4
2
8
1
×
+
=
5
3
2
Calculating Percentages - Calculator
The method of calculating a percentage from a decimal or fraction is to multiply the decimal
or fraction by 100. To change a percentage to a decimal or fraction you divide by 100.
A picture of a calculator is shown with below with keys highlighted. The methods used for
each column are also listed on the following pages. Where there is a alternative method it is
shown, so you can select the method that makes the most sense to you.
Column 1 asks you to convert fractions to percentages to 2 decimal places, then to a specified
number of decimal places (in brackets). Multiply the fractions by 100 for the answer. Note
that if writing a whole number, such as 45%, when asked to give to 2 d.p. then the answer can
be 45% or 45.00%.
Column 2 asks you to convert the decimals to percentages. Again multiply the question by
100.
Column 3 deals with changing percentages back to fractions or decimals. In this case divide by
100.
Column 2
Example of calculating a decimal percentage from a decimal
Convert 0.206 to a percentage
Using × 100
0 . 2 0 6 × 1 0 0 =
if you get 20.6 for the question above you are correct
Column 1
Example of calculating a decimal percentage from a fraction
Convert 7/8 to a percentage
Using % key (not recommended)
7 ÷ 8 Shift %
Using × 100
7 ÷ 8 × 1 0 0 =
Using fraction key
7 ab/c 8 × 1 0 0 = ab/c
if you get 87.5 (no fix) for the question above you are correct
Example of calculating a fraction percentage from a fraction
Convert 7/8 to a percentage
Using fraction key
7 ab/c 8 × 1 0 0 =
Using × 100
7 ÷ 8 × 1 0 0 = ab/c
if you get 87 ½ for the question above you are correct
note that this style of question isn’t on the sheet
Example of calculating a decimal percentage from a mixed numeral
Convert 2 7/8 to a percentage
Using fraction key
2 ab/c 7 ab/c 8 × 1 0 0 = ab/c
Using × 100
2 + 7 ÷ 8 = × 1 0 0 =
if you get 287.5 for the question above you are correct
Column 3
Example of converting a decimal percentage to a decimal
Convert 56.32% to a decimal
Using ÷ 100
5 6 . 3 2 ÷ 1 0 0 =
if you get 0.5632 (no fix) for the question above you are correct
Example of converting a fraction percentage to a decimal
Convert 56 ¼% to a decimal
Using the fraction key
5 6 ab/c 1 ab/c 4 ÷ 1 0 0 = ab/c
if you get 0.5625 (no fix) for the question above you are correct
Example of converting a decimal percentage to a fraction
Convert 56.25% to a fraction
Using the fraction key Note that the featured calculator can convert
decimals to fractions, this may not be a feature on your calculator
5 6 . 2 5 ÷ 1 0 0 = ab/c
If your calculator has a fraction key but can’t convert decimals to
fractions then use this method (note you need some mental skills)
5 6 ab/c 2 5 ab/c 1 0 0 ÷ 1 0 0 =
if you get 9/16 for the question above you are correct
Example of converting a fraction percentage to a fraction
Convert 56 ¼% to a fraction
Using the fraction key
5 6 ab/c 1 ab/c 4 ÷ 1 0 0 =
if you get 9/16 for the question above you are correct
Calculating Percentages - Calculator
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Convert these fractions to
percentages (by × 100).
Round answer to 2 d.p.
1
2
2
Convert these decimals to
percentages (× 100). Give
your answer to 1 d.p.
Convert these percentages
to decimals (÷ 100). Give
your answer to 3 d.p.
/3
=
% 23
0.15
=
% 46
5
/7
=
24
0.439
=
3
1
/8
=
25
1.266
4
1
/12
=
26
5
4
/13
=
6
1 56/83
7
124
8
85
9
206
10
95%
=
47
102.7%
=
=
48
0.13%
=
0.78841
=
49
11.63%
=
27
0.0606
=
50
225.46%
=
=
28
3.0295
=
51
0.3%
=
/171
=
29
0.0074
=
52
½%
=
/302
=
30
0.4989
=
53
33⅓%
=
/118
=
31
1.0907
=
54
1.03%
=
4 357/502
=
32
0.9
=
55
67 ¾%
=
Continue converting to
percentages but round
your answer to the d.p.
in the brackets.
Round the percentage
answer to the d.p. asked
33
0.021141 [3] =
=
% 34
1.30055 [2] =
11
51
12
131
/165 [2]
=
35
1.08
13
3 81/97 [2]
=
36
0.29992 [2] =
14
1
/3 [3]
=
37
4.00681 [2] =
15
1 144/758 [1]
=
38
0.0096
16
41
=
39
11.0016 [1] =
17
288
/650 [3]
=
40
0.701061 [3] =
18
65
/91 [1]
=
41
0.349211 [0] =
19
2 32/55 [0]
=
42
0.59097 [2] =
20
193
/210 [2]
=
43
0.087026 [3] =
21
5
/8 [2]
=
44
0.389001 [0] =
22
249
=
45
1.999802 [1] =
/75 [1]
/85 [3]
/250 [0]
[1] =
[1] =
Write the percentages
below as fractions
% 56
15%
=
57
68.4%
=
58
12.5%
=
59
75.4%
=
60
117.5%
=
61
42.36%
=
62
½%
=
63
18.8%
=
7
FREEFALL
MATHEMATICS
MEASUREMENT
& PERIMETER
Using Scales
One method of measuring weight is to use scales. The faces of these scales are shown in 100 g
intervals with larger divisions every 500 g. The kilogram readings have a number assigned.
Starting at zero through to 5 kg, the scale travels past 5 kg but doesn't reach 6 kg, the
maximum reading being 5.5 kg.
Column 1 asks you to read the weight displayed on the scales. The pointers are on an exact
division. So answer in kg, but to one decimal place, ie: readings such as 3.2 kg or 4.1 kg.
Column 2 is the reverse, position the pointers so that they represent the given weights. Note
that you are required to approximate the position of the needle as it may have to be positioned
in between divisions, (Q.7 and Q.10). It also asks questions in grams.
Column 3 is similar to Column 2 except that an addition is required first. The table describes a
certain number of 100 g, 250 g and 1 kg weights that are on a particular scale. Add them
across the row of the table and then write the total next to the question number, then point that
pointer.
Using Scales
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Write the weight shown
on the scales below.
For the weights given
draw the pointer in the
correct position
kg
6
1
4.4 kg
0
5
4
5
2
4
3
7
kg
1
5
2
4
8
5
2
4
9
5
2
4
0
1
3
2
3
13
2
-
2
14
1
1
4
11
kg
0
1
5
2
4
2
kg
2
12
kg
0
1
5
2
4
1
kg
2
3
13
kg
0
1
5
2
4
1
kg
2
3
14
0
5
4
1
3
10 5.05 kg
kg
1
3
5
4
12
0
1
3
5
2
2
3 200 g
0
kg
-
3
4
4
3
0
1
3
5
kg
900 g
0
kg
11
3
3
4
1 kg
0
3
5
250 g
1
2.55 kg
0
4
100 g
3
2
5
Scale
0
1
kg
Add the weights in the
table below, put the total
in the space then point the
pointer to the total weight
kg
3
0
1
5
2
4
1
kg
3
2
Reading Water and Electricity Meters
The water and electricity supplied to your home passes through meters which record the
amount used. Electricity meters are gear-driven, the scales changing between clockwise and
anti-clockwise rotation. So the adjacent pointers spin in opposite directions. There is an easy
way to read the scales, just look where the needle is and the number it represents is the
smallest of the two numbers the needle is between. Note that zero is both 10 and zero. If the
pointer is between the 9 and 0 the smallest number is 9, (zero = 10). If the pointer is between
0 and 1 the smallest number is 0, as zero = 0.
Column 1 starts with 4 questions reading the meters shown. The meters give a 5 digit reading
from 10 thousands down to units (1's), the units are in kWh (kilowatt-hours). Read each from
left to right and build the five digit number. The next 4 questions ask you to put the pointers in
the correct position. Start with the 10 000's first and work your way left to the right.
pointer between 4 and 5
n
ee
8
4 1 000
1 0 9
100
9
d
d
an
an
9
3
d
an
ee
n
8
een
tw
be
tw
be
be
tw
poin
ter
te r
in
po
pointer
pointer between 7 and 8
73 884
9 0 1
10
8
2
1 0 9
1
10 000 2
2
8 7
8
3
0
1
9
9 0 1
6
3
3
7
7
5 4
8
8
2
2
4
4
5 6
5 6 7
7
3
3
6
6
Kilowatt Hours (kWh)
5 4
5 4
Column 2 are water meters, these are more standard in layout. The meters measure water
usage in kL (kilolitres). The end 4 digits are inversed (white on red). These end digits
represent 4 decimal places in kL. If measuring in L, the last digit is tenths of a L (1 d.p). The
first 3 questions compare a ‘before’ and ‘after’ reading and then require subtraction to
calculate the amount of water used. Note that these meters deal with large units, some numbers
are not required in the calculation due to them not changing, a short cut - but be careful!
The last 3 questions give the ‘before reading’ and then require this reading to be added to the
usage to get the ‘current reading’. After completing the sum, write the answer in the spaces on
the meter.
7 206.1638 kL
or
7 206 163.8 L
WATER BOARD
72061638
Reading Water and Electricity Meters
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Read the dials below and write
the number they represent
1
1 000
1 0 9
100
9 0 1
Read the 'last bill' reading then the
'current reading' and calculate the
water usage since the last reading.
10
8
2
1 0 9
1
10 000 2
2
8 7
8
3
0 1
9 0 1
9
6
3
3
7
7
5 4
8
8
2
2
4
4
5 6
5 6 7
7
3
3
6
6
Kilowatt Hours (kWh)
5 4
5 4
2
1 000
10 000
8
7
9 0 1
6
5 4
2
2
3
3
3
1 0 9
8
7
9 0 1
6
5 4
2
2
3
3
4
1 0 9
8
7
9 0 1
6
5 4
2
2
3
3
100
9 0 1
1 0 9
100
9 0 1
10 000
8
7
9 0 1
6
5 4
2
2
3
3
6 29 170
10 000
8
7
9 0 1
6
5 4
2
1 0 9
3
7 80 933
9 0 1
1 0 9
100
9 0 1
Last Bill Reading
Current Reading
WATER BOARD
WATER BOARD
07351104
07669563
10
kL
Last Bill Reading
Current Reading
WATER BOARD
WATER BOARD
83962517
91485319
11
10
100
10
9 0 1
8
2
1 0 9
1
10 000 2
8 7
8
3 2
0 1
9 0 1
9
6
3
3
7
7
5 4
8
8
2
2
4
4
5 6
5 6 7
7
3
3
6
6
Kilowatt Hours (kWh)
5 4
5 4
10 000
8
7
9 0 1
6
5 4
2
3
1 000
2
3
1 0 9
100
9 0 1
-
kL
Last Bill Reading
Current Reading
Given the water used and the last reading
add them and write in the current reading
that should be on the meter.
WATER BOARD
WATER BOARD
10
8
2
1 0 9
1
8 7
8
3 2
0 1
9
6
3
7
7
5 4
8
2
4
4
5 6
5 6 7
3
6
Kilowatt Hours (kWh)
5 4
+
92477016
kL
Last Bill Reading
Current Reading
13 175.6907 kL
WATER BOARD
WATER BOARD
+
08631925
kL
1 0 9
8 44 161
-
12 306.5908 kL
8
2
1 0 9
1
2
8 7
8
3
9 0 1
6
3
7
7
5 4
8
2
4
4
5 6
5 6 7
3
6
Kilowatt Hours (kWh)
5 4
1 000
54287976
5 428.7976 -
10
8
2
1 0 9
1
8 7
8
3 2
0 1
9
6
3
7
7
5 4
8
2
4
4
5 6
5 6 7
3
6
Kilowatt Hours (kWh)
5 4
1 000
2
3
100
51064825
10
8
2
1 0 9
1
2
8 7
8
3
0 1
9
6
3
7
7
5 4
8
2
4
4
5 6
5 6 7
3
6
Kilowatt Hours (kWh)
5 4
1 000
WATER BOARD
10
Now the reverse, put in the pointers!
5 11 255
WATER BOARD
kL
10
8
2
1 0 9
1
8 7
8
3 2
9 0 1
6
3
7
7
5 4
8
2
4
4
5 6
5 6 7
3
6
Kilowatt Hours (kWh)
5 4
1 000
10 000
9 0 1
8
2
1 0 9
1
2
8 7
8
3
0 1
9
6
3
7
7
5 4
8
2
4
4
5 6
5 6 7
3
6
Kilowatt Hours (kWh)
5 4
1 000
10 000
100
9
Last Bill Reading
Current Reading
14 785.9707 kL
WATER BOARD
WATER BOARD
+
62958910
kL
Last Bill Reading
Current Reading
Fuel Gauges
Fuel gauges are an indication of how much fuel is in a car’s petrol tank. Most commonly fuel
gauges have divisions at empty, ¼ of a tank, ½ a tank, ¾ of a tank and full. In the city, gauges
aren't that critical, with 24 hr petrol stations in great numbers. In the country away from the
conveniences of the city, the understanding of a fuel gauge is more important.
In column 1 the fuel gauges are shown, put the fraction that the gauge's pointer indicates into
the spaces provided. Note that the pointers are meant to be pointing exactly at a division or
exactly half way between a division. When the pointer is halfway between divisions the
reading will be in 'eighths'. Look at the gauge below, it has the eighths marked.
Column 2 is the reverse, given the fraction draw the pointers to the correct position.
Column 3 asks you to use a fuel gauge to estimate the fuel that remains in the tank as well as
the distance that the car can continue until it will run out of fuel. This requires some
multiplication of fractions skills, the method is as follows:
•
To calculate the litres that remain divide 32 L (tank capacity) by the denominator and
multiply by the numerator. This means divide 32 by the bottom number in the fraction,
then multiply this number by the top number.
•
The same method is used for the distance that the car can still travel. Divide 400 (max
km) by the denominator (bottom number in fraction) and then multiply the answer by the
numerator (top number in the fraction).
For the gauge below the reading is 3/8. This means that if a car has a 32 L tank and a
maximum travel distance on the tank of 400 km (city), then:
Fuel left is 32 ÷ 8 × 3 = 4 × 3 = 12 L
Distance until empty is 400 ÷ 8 × 3 = 50 × 3 = 150 km
8
/8
F
7
/8
6
¾
/8 5
/8
½
4
/8
3
/8
2
¼
/8
1
/8
0
E /8
Fuel Gauges
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Estimate the fraction of a
fuel tank that remains on
the gauges below
1
F
This time the fraction is
given, place the needle in
the correct position.
7
¾
3
F
4
¾
½
13 L:
½
¼
2
7
F
½
8
¼
¾
½
E
½
¼
14 L:
¼
km:
F
E
3
F
¾
E
8
¾
km:
F
¼
E
F
A full tank holds 32 L and
travels 400 km (city). Use
the gauges to estimate the
unused fuel and km that
can still be travelled.
E
9
¾
1
F
4
¾
½
¾
½
¼
½
E
¼
¼
E
4
F
E
10
¾
3
F
8
¼
¼
11
¾
E
E
1
F
2
16 L:
km:
F
¾
½
¾
½
½
¼
¼
¼
E
F
¾
½
E
6
F
½
¼
F
km:
¾
½
5
15 L:
12
¾
½
5
8
F
17 L:
¾
¾
½
¼
E
km:
F
½
¼
E
E
¼
E
E
Using a Ruler - Line Measurement
This sheet has lines of a set length. When you print this sheet you must select
‘No Scaling’ when you print it, otherwise your answers will be wrong.
When a line is measured using a ruler the answer can be expressed in mm or cm. If the ruler
has both units on it you can use the side needed for the answer. However you should also be
able to convert between the two units. You should know that:
1 cm = 10 mm, so 2 cm = 20 mm, and decimals 3.8 cm = 38 mm etc.
Column 1 has horizontal lines and vertical lines. It is the thick (blue) lines you are measuring,
don’t confuse them with the thin black lines for writing your answer on. The first column asks
for answers in cm ( to 1 d.p.), that means 3.7 cm, 11.2 cm and so on. The screw and nut at the
bottom of the page can be measured either across the object or use the lines with the doubleended arrow between them. The second column is much the same except the lines are angled,
note that measurement for this column are in mm.
Use the same method with column 3 only write the size of each side and then add all the numbers together for the perimeter, the distance around the outside.
Using a Ruler - Line Measurement
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Use a ruler to measure
these, express the answer
in cm (to 1 d. p.)
1
2
3
4
Measure the perimeter of
these shapes to the nearest
mm, write the side length
on each side, then total
answer in mm.
Measure these angled lines
to the nearest mm, express
your answer in mm
16
17
23
18
5
19
6
7
8
9
10
11
12
13
24
20
21
25
14
26
22
15
Using a Compass
A compass is used as a measure of direction. A compass needle always points to Magnetic
North, so on this sheet the red section of the needle is pointing North and the white and red
striped end is pointing South. The view is as if you are holding the compass against your chest
looking down at it.
The compasses all have the blue ring with N (North) straight up. To find the direction you are
facing count how many divisions (triangles and circles) that the red pointer is from North.
Then count back the same number of divisions the opposite way from North and that is the
direction you are facing.
The readings on this sheet will be one of the readings on the dial below. If you are not familiar
with the 16 directions of the compass take the time to look at the dial, as the dials on the sheet
don't have all the directions (shown in blue letters) that you need to know.
NNE
NNW
N
W
E
N
N
ENE
E
W
WNW
WSW
ESE
SW
SE
S
SSW
SSE
Using a Compass
4
4
NW
SW
SE
3
3
NE
S
2
2
N
SW
SW
4
E
W
N
NE
W
N
5
NW
S
W
SE
1
SW
S
N
5
9
NE
W
5
E
If the compass was held
in front of you with the
needles as shown which
direction are you facing?
5
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
1
3
1
N
S
2
0
E
4
NW
3
W
5
NE
3
4
N
W
S
N
SE
SE
E
SW
E
N
E
4
E
2
5
10
SE
S
0
5
1
0
SE
N
E
6
2
3
2
NW
1
W
0
5
4
SW
SW
E
3
NE
E
2
SE
2
SE
SE
3
4
S
3
NE
S
N
W
E
N
N
4
NW
NE
W
S
SW
E
W
E
5
11
NW
W
N
3
5
0
1
0
2
1
N
W
SW
7
1
0
N
5
4
3
NW
5
SE
SE
S
3
3
SE
NW
NW
4
12
SW
W
4
S
W
E
8
SW
S
5
0
0
1
1
2
N
N
N
E
2
1
1
0
0
NE
2
2
NE
1
W
0
SW
4
Car Gauges
This sheet asks you to put information on the gauges, a representation of a car’s instrument
cluster.
This is the method:
•
The fuel gauge is on the left hand side of the instrument cluster. Draw a needle to the
amount of fuel specified. Remember that between ¼’s are 1/8’s of a tank, see the fuel
gauge sheet if you need to.
•
The speedometer is numbered 0 - 200 km/h. The speed increases by 10 km/h divisions
with numbers at every 20 km/h. The odometer is the top set of boxes in the speedometer,
while the tripmeter is the bottom set.
•
The tachometer is the next large faced gauge. It displays engine speed with divisions at
every 500 rpm with numbers at every 1 000 rpm. Note that car manufacturers use
‘× 1 000’ so that single numbers are used. So 1 on the dial is 1 000 rpm. 2.5 would be
2 500 rpm. Note there is an LCD readout to show what gear the car is in.
•
The temperature gauge is very simple, it has a green section to show the engine is cold, a
blue section to indicate normal temperature and a red section to show if the engine is hot.
•
The odometer is the record of the distance traveled during the life of your car it is the
6-digit number in your speedometer. So its largest reading is ‘999 999’ km, then it resets
to ‘000 000’. Write the numbers in the spaces.
•
The tripmeter also records the distance traveled but it can be reset. It is often reset when
the petrol tank is filled to calculate fuel economy or the distance covered on holidays. It
has 4 digits but the last digit is tenths of a km. So its maximum reading is 999.9 km
before it resets. Write the numbers in the spaces, note that the last digit is often in inverse
(white on black) to show it is tenths of a kilometre.
•
Some cars have an LCD readout to show the gear you are in, so that you don’t have to
look down at the gear shift. Cars usually have 5 or 6 gears. Colour the LCD bars.
•
When turning at an intersection or overtaking you should use your indicator, colour the
arrow, they are usually green or orange.
Car Gauges
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Fuel : ¾ tank
Speed : 80 km/h
RPM : 3 000
Temp : normal
Place all the details on the
dashboards, watch out for
speed bumps!
Odo : 56 277 km
Trip : 193.4 km
4th Gear
Right indicator flashing
1
100
60
160
6
7
1
¾
180
20
¼
FUEL
0
Km/h
rpm x 1000
0
8
200
E
Fuel : ¼ tank
Speed : 70 km/h
RPM : 4 500
Temp : cold
2
F
3
120
60
40
160
¼
FUEL
0
Km/h
Odo :135 198 km
Trip : 78.2 km
5th Gear
Left indicator flashing
100
3
120
60
40
160
¼
FUEL
E
0
Km/h
5
6
7
1
¾
180
4
2
140
20
TEMP
200
3
½
C
8
Fuel : 1/8 tank
Speed : 105 km/h
RPM : 3 750
Temp : normal
F
TEMP
rpm x 1000
0
E
80
C
7
1
180
20
5
6
¾
½
4
2
140
TEMP
Odo : 79 003 km
Trip : 290.4 km
3rd Gear
Left indicator flashing
100
80
C
0
H
½
5
2
140
40
4
H
F
3
120
H
80
rpm x 1000
8
200
Curved Line Measurement
This sheet outlines the measurement of curved lines. Follow this method:
•
Cut a length of string 30 cm long
•
Use a felt tip pen to mark off 1 cm intervals along the string by lying the string alongside
a ruler.
•
Using a glue stick spread a thin film of glue over the first of the printed lines, the idea
being not to stick the string to the page but to make it tacky enough to hold the string in
place.
•
Count the number of 1 cm intervals to get the number of cm then add the portion of the
last interval, this will be tenths of a cm at the end, i. e: halfway into the last segment = 0.5
etc.
•
Write the answer in the space provided
If you have no glue or string use a compass or set of dividers opened to 1 cm and 'step off' the
curve. Or measure straight line lengths of 1 cm with a ruler.
Note that you can leave the line to make up for a corner you have previously cut (see example
below). Decide if you are going to reduce the measurement at the corner or increase the
measurement at the corner. You can alternate to reduce the error. The answer sheet for this
sheet cuts all corners to give the minimum distance you should get for each line.
Total 10.9 cm
10 whole
intervals = 10 cm
About 0.9 cm
Using String or Intervals - Curved Line Measurement
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Use a piece of string with 1 cm intervals marked on it with a small amount
of glue on the page so that the string doesn't wander about too much. No
string? Use a compass and mark off 1 cm intervals. No compass? Use a ruler
and mark off 1 cm intervals. Find the length and write it in the space next to
the question. Express your answer in cm to 1 d. p.
1
3
2
4
Units of Measurement
When units of measurement are changed, multiply or divide by 10, 100, 1 000 or more. To
change to a smaller unit multiply, to convert to a larger unit divide. Some students have
difficulty with this, they think that if the unit is larger you must multiply. This isn't the case.
Imagine you have 1 m or 100 cm, both are the same distance, there are more of the smaller
unit, 1→100, so:
•
changing to a smaller unit means - multiply
•
changing to a larger unit means - divide
The number used to multiply or divide is the number of small units in the larger unit. For
example 2 m to cm, there are 100 cm in a metre, changing to a smaller unit so multiply…so
multiply by 100. As only 10, 100 or 1 000 are used to get the answer … move the decimal
point. Multiply by 10 moves the decimal place 1 position to the right, 100 move 2 positions
and 1 000 move the decimal place 3 positions to the right.
Division is the same number of moves, only this time it’s to the left. Divide by 10 and move
the decimal place 1 position to the left, 100 move it 2 positions and 1 000 moves 3 positions to
the left. A reminder is at the top of each column.
Column 1 starts with changing mm to cm. You should be able to talk it through to yourself….mm to cm is changing to a larger unit...that means division….there are 10 mm in a cm
that means I divide by 10….divide by 10 means I move the decimal point 1 place to the left.
That means 18 mm = 1.8 cm 113 mm = 11.3 cm and 0.9 mm = 0.09 cm. The next part of the
column is the reverse, multiply by 10 means move the decimal place 1 position to the right. So
54 cm = 540 mm, 1.3 cm = 13 mm and 123 cm = 1 230 mm.
Column 2 deals with converting between cm and m. This time the change is by dividing or
multiplying by 100. That means a 2 decimal place movement. For cm to m (divide by 100)
12 cm = 0.12 m, 106 cm = 1.06 m and 5.78 cm = 0.0578 m. Then the reverse, m to cm means
multiplying by 100 or a 2 decimal place shift to the right: 3 m = 300 cm, 0.7 m = 70 cm and
10.05 m = 1 005 cm.
Column 3 is m to km, this involves multiplying or dividing by 1 000, or a 3 decimal place
movement. That would mean 300 m = 0.3 km, 2 305 m = 2.305 km and 4.7 m = 0.0047 km.
Then the reverse, changing km to m. That’s 3 decimal places to the right: 1.2 km = 1 200m,
0.85 km = 850 m and 0.02 km = 20 m.
Units of Measurement
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Convert these lengths
from cm to m
Convert these lengths
from mm to cm
10 mm = 1 cm
1 place left
100 cm = 1 m
2 places left
Convert these lengths
from m to km
1 000 m = 1 km
1 20 mm
25 200 cm
49 1 500 m
2 50 mm
26 600 cm
50 1 220 m
3 130 mm
27 150 cm
51 4 930 m
4 55 mm
28 370 cm
52 110 m
5 92 mm
29 12 cm
53 550.6 m
6 200 mm
30 86.5 cm
54 1 003 m
7 18.6 mm
31 7 300 cm
55 85 m
8 2 020 mm
32 56.05 cm
56 4.7 m
9 435 mm
33 3 cm
57 309.7 m
10 0.95 mm
34 6.7 cm
58 12 032 m
11 5.47 mm
35 8 002 cm
59 8 771.3 m
12 906.2 mm
36 27.08 cm
60 763.2 m
Convert these lengths
from cm to mm
1 cm = 10 mm 1 place right
Convert these lengths
from m to cm
1 m = 100 cm
2 places right
3 places left
Convert these lengths
from km to m
1 km = 1 000 m
13 5 cm
37 3 m
61 3 km
14 18 cm
38 7 m
62 8 km
15 6.3 cm
39 2.2 m
63 3.2 km
16 74.8 cm
40 5.9 m
64 14.7 km
17 356.7 cm
41 0.3 m
65 10.6 km
18 8.12 cm
42 0.75 m
66 0.9 km
19 0.4 cm
43 0.363 m
67 0.673 km
20 0.566 cm
44 5.02 m
68 5.12 km
21 155 cm
45 10.7 m
69 1.007 km
22 829.3 cm
46 0.006 m
70 0.4503 km
23 1.002 cm
47 5.071 m
71 4.019 km
24 19.91 cm
48 30.262 m
72 0.06 km
3 places right
Further Units of Measurement
The processes are the same as on the previous sheet 'Units of Measurement' except that this
time the conversions aren't in groups of units and multiplication/division. Its all jumbled so
you need to know whether you divide or multiply and how many decimal places you shift the
decimal point. There is also the inclusion of mm to m and the reverse. This is a 3 decimal
place shift (× or ÷ by 1 000).
Further Units of Measurement
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Use the guide
below to help you
mm → cm
1 place left
mm → m
3 places left
cm → mm
1 place right
cm → m
2 places left
m → mm
3 places right
m → cm
2 places right
m → km
3 places left
km → m
3 places right
Convert these to the units
in brackets. They are all
mixed up
18 5.062 km
[m]
47 0.95 m
[mm]
19 0.838 km
[m]
48 1 727.4 m
[km]
20 3.07 cm
[mm]
49 0.052 cm
[mm]
21 121 mm
[m]
50 39 cm
[m]
22 7 777 m
[km]
51 0.2004 km
[m]
23 918 cm
[m]
52 4.52 m
[cm]
24 4.2 cm
[m]
53 80.05 cm
[mm]
25 0.5 cm
[m]
54 293 m
[km]
26 0.89 m
[cm]
55 0.302 m
[mm]
27 765 mm
[cm]
56 13 mm
[m]
28 19.6 cm
[mm]
57 43.9 mm
[cm]
29 16 010 m
[km]
58 0.67 m
[cm]
1 3.2 m
[cm]
30 78 mm
[m]
59 850 mm
[m]
2 4.3 cm
[mm]
31 2.1 m
[mm]
60 2 211 mm
[m]
3 0.98 m
[cm]
32 1.93 m
[cm]
61 549.3 m
[km]
4 200 mm
[m]
33 82.6 cm
[mm]
62 400.3 cm
[m]
5 860 m
[km]
34 16 m
[cm]
63 0.74 km
[m]
6 118 mm
[cm]
35 1.8 km
[m]
64 4.96 cm
[mm]
7 1 033 m
[km]
36 3.8 mm
[cm]
65 113.2 cm
[mm]
8 27.2 cm
[mm]
37 9.021 km
[m]
66 333.7 cm
[m]
9 185 mm
[m]
38 56.5 cm
[m]
67 17 mm
[cm]
10 206.5 mm
[m]
39 28.8 m
[km]
68 0.005 m
[cm]
11 0.35 cm
[mm]
40 0.83 cm
[mm]
69 17 mm
[cm]
12 686 cm
[m]
41 0.6 m
[mm]
70 0.02 km
[m]
13 0.496 m
[mm]
42 0.03 m
[mm]
71 680 mm
[m]
14 4.03 km
[m]
43 41 m
[cm]
72 82.4 cm
[m]
15 65.7 cm
[m]
44 850 mm
[m]
73 0.65 mm
[cm]
16 825 mm
[cm]
45 182.5 cm
[m]
74 32.6 mm
[m]
17 11.3 m
[cm]
46 1 405 mm
[m]
75 0.006 km
[m]
Perimeter of Shapes
The perimeter of a shape is the distance around the outside of the shape. The units of
measurement are mm, cm, m and km. Shapes often have side lengths that are the same, rather
than writing this distance out again a mark is put on the identical sides to show they are the
same length. If there are a number of different pairs of sides that are the same length, 2 marks
together will be used and so on, such as with a rectangle.
To answer columns 1 and 2 use the same method, this is:
•
Look at the side markings and write side lengths on every side. It is up to you if you write
the units as well.
•
The first line of working is the sum of the sides of the shape, write ‘P =’ then the lengths
separated by + signs. Note that to avoid missing sides don't jump around in the addition.
Start on a side then move around the shape in a clockwise or anti-clockwise direction.
•
If the sum is challenging to complete mentally, total the sum as you move through, look
at the example below.
•
The second line is the answer line, again write ‘P =’ then the answer, then the units, either
mm, cm or m.
Column 3 introduces formulae used for squares and rectangles. These are:
•
Squares: P = 4l (which is 4 times the side length)
•
Rectangles: P = 2l + 2b (which is 2 times the length + 2 times the breadth)
The questions are in tables. With squares, multiply the side length by 4 and write the answer
with units. The rectangles are completed in stages, l is given and you double it to get the value
of 2l, b is given and you double it to get the value of 2b. Then add the values for 2l and 2b
together to get the answer, write in the units (all of which are cm.)
Fill in all the side
lengths to avoid
missing them
7m
If you have trouble solving
the sum mentally use
canceling strokes and total as
you go through
7m
7m
7m
7m
14
21
7m
28
35
P=7+7+7+7+7+7
P = 42 m
Perimeter of Shapes
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Put measurements on all
sides then add them to
find the perimeter.
6
Complete the tables, all
measurements are in cm.
1.3 cm
Perimeter Formula for Squares :
1
P = 4l
5 cm
6 cm
P=6+
P=
+
P = 4l
+
=4×5
2.1 m
7
P = 20 cm
12 to 26
cm
25 m
2
where l = side length
Length Perimeter Length Perimeter
(l)
(l)
(4l)
(4l)
11 m
8
5
15 mm
9 mm
22 mm
8 mm
3
30 mm
13 m
9
5m
20 cm
15
8
11
10
14
9
30
3
50
4
80
12
20
55
42
3m
Perimeter Formula for Rectangles :
where : l = length and
b = breadth
5 cm
P = 2l + 2b
4
7m
10
12 m
14 cm
6m
22 m
2m
14
11
16 mm
8 cm
9 mm
P=
×
= 2×14 + 2×5
P = 38 cm
27 to 33
Length
2l
(l)
5
P = 2l + 2b
28
Perimeter
Breadth
2b (2l + 2b)
(b)
5
8
11
12
5
9
6
15
25
20
16
7
19
13
17
10 38 cm
Perimeter - Finding a Missing Side
Sometimes before the perimeter of a shape can be calculated the missing sides have to be
found. The missing sides will either have side markings that indicate that they are the same
size as another side or they will be found by adding or subtracting sides that you know the
value for. Note that this applies due to all the angles being right angles, the corners are not
marked as right angles but you can assume that they are all 90º.
In column 1 Q. 1 - 5, two different ways are used to measure
the same distance. This means that they must be equal. Use
addition or subtraction to find the value. In the example here
a distance is measured as 13 cm. The same distance is then
broken up into to parts, one of those parts is 8 cm. Ask yourself,
'What number adds to 8 to give 13?'
The answer is 5, so the missing length is 5 cm.
8 cm
13 cm
5 cm
From Q. 6, through the column, shapes are given and you use the same method to find the
missing sides. The method is:
•
If the shape has side markings write in all the sides that you know.
•
Use the given lengths to then calculate the other side lengths
Columns 2 and 3 add one step to column 1, the perimeter is calculated. Use 2 lines of working,
the first line being the addition of all the sides and the second line being the answer with the
units (mm, cm, m etc). Writing the side lengths on all sides of the shape will help avoid
mistakes. Some students like to miss this step because they know the value of the side and
don't need to write it, the problem is that when they add the sides to get the perimeter they
often forget to include them. A method to avoid this problem is to label every side with its
measurement then look at the addition and check off the numbers against the numbers on the
sides of the shape. If there are different numbers or there are too many or too few you have to
find the problem.
Note that if you add around the sides of the
shape in the same way, clockwise or an anticlockwise direction, it makes it less confusing.
Be careful with question 19 it is different
and may trick you.
Marks on sides
mean they are the
same, so 5 cm can
be written first.
5 cm
20 - 5
15 cm
20 cm
17 - 5
12 cm
5 cm
17 cm
37
42
57
69
P = 17 + 20 + 5 + 15 + 12 + 5
P = 74 cm
Make sure the
units are the same
as in the question.
Perimeter - Finding a Missing Side
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Find the missing sides
then find the perimeter
Calculate the missing
lengths, all angles are
right angles
1
11 mm
10
2
4 cm
15
8 cm
40 cm
5 mm
12 mm
6m
12 m
7 cm
16
3 mm
P = 3 + 12 + 11 + 5 +
3
4
P=
13 m
+
mm
11
18 m
13 m
25 cm
4 cm
15 mm 22 mm
7 cm
4m 3m
5
13 cm
16 cm
5 cm
17 cm
6m
12 m
17
10 cm
16 m
12
6
7 cm
10 m
20 cm
5 cm
15 cm
7
9m
18
5 m 25 m
8m
30 m
13
17 m
6 mm
8
3 mm
5m
10 m
5 mm
8 cm
14
9
12 cm
19
3 cm
11 cm
35 cm
20 cm
6 cm
2 cm
15 cm
15 cm
9 cm
Perimeter Problems
These problems are an extension of the exercises covered earlier with an inclusion of cost. A
common use of perimeter is in the calculation of cost of fences and walls. To calculate this
cost calculate the perimeter first, then multiply the answer by the cost of the fence/wall by its
unit cost, (the cost per metre).
In all the problems two lines of working are expected. The first line being the sum of the side
lengths:
P = _ + _ + _ + etc
the second line being the total length and the units
P = _____ m (or cm).
Perimeter Problems
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Answer the perimeter
problems below.
1 A rectangular swimming pool
has sides with lengths 12 m and 4 m.
Find the distance around the pool.
5 Three hobby farmers are each
given small garden areas. The plots
all occupy the same land area (16 m²)
but have different side lengths, find
the perimeter of each.
9 Dare-devil Dan is about to jump
2 buses, lined up end to end (see
below). Dan is superstitious and
walks 1 lap around the buses before
the jump, how far does he walk?
11 m
Plot 1 : 16 m by 1 m
The buses are 11 m long and 3 m wide
2 A path 1 m wide surrounds the
Plot 2 : 8 m by 2 m
same pool (Q1) calculate the length
and breadth of the pool and the path
combined, put the dimensions on the
diagram below and find the perimeter
10 An inventor created a synthetic
material that will stretch to 18 times
its length before breaking. If the
original length is 7 m, find its maximum length. Will it stretch around a
rectangular building with sides 25 m
and 35 m? (Find P then circle answer)
Plot 3 : 4 m by 4 m
Max length
3 If the pool above has a
safety fence around the
outside of the path find
the cost of fencing the
pool when the fence costs
$27 per metre.
6 If you were offered a plot and
had to pay to fence the area, would it
be cheaper to have a square area or a
rectangular area?
Yes / No
Circle : square or rectangle
11 Leon's dog 'Oblong' always runs
7 A parcel has 2 strings around it
around the perimeter of the backyard
when he sees Leon arrive home. Find
the distance he runs around the empty
yard. Then calculate the distances he
runs when Leon's car is in position A
or in position B, (not both together).
Leon's car is 2 m wide and 5 m long.
as shown below. Calculate the length
of the string A and string B, then add
them to find the total length.
6 cm
30 cm
A
10 cm
4 In a science experiment Travis
noted that on average an ant stopped
every 5 cm. If the viewing tray was
30 cm by 20 cm, find its perimeter.
17 m
B
A:
B
8m
A
Position A is in the centre of the yard
B:
No car:
How many times would the ant stop:
Car A :
i) in 1 lap
ii) in 5 laps
iii) over a 9 m distance
8 Scott says that when you double
the side length of a square you get 4
times the perimeter, is he right?
For
Not a
Car B :
7
FREEFALL
MATHEMATICS
AREA
Grid Area of Shapes
This sheet uses areas of 1 cm2. When you print this sheet you must select
‘No Scaling’ when you print it, otherwise the grid areas will be scaled and incorrect.
One method of determining area within a shape is to use a grid system in which the space
inside a shape is broken up into squares. This sheet uses a 1 cm grid system. The shapes either
fit the grid exactly or cross the grid diagonally using ½ a grid space.
The top section of the sheet (Q 1 - 9) requires you to calculate the areas of the nine shapes
shown. This is done by counting the number of 1 cm² grid squares inside the shape and
writing the answer. Some of the shapes have diagonals that cut the squares in half. Remember
that ½ + ½ = 1, an even number of ½'s will give a whole number and an odd number of ½
squares will have the answer being a mixed number with a ½ as the proper fraction part. Or
use decimals 0.5 cm² is half a square. Remember to give your answer in cm².
With Q 10 - 13 there are no grid lines on the shapes. This is overcome by either of these two
methods:
•
Use a grid overlay on a transparent sheet (such as an overhead projector sheet), or
•
Rule lines 1 cm apart and create the grid on the paper
A worksheet is supplied that has the grids drawn for you if you can’t print a transparent sheet.
The 2nd column asks you to use the same method with the grid overlay to calculate the area.
But this time write in the side lengths. Then multiply the two side lengths together and see if
this number relates to the area.
The third column then uses the side multiplication method to calculate the area of the shapes in
the questions.
Grids for Transparency Printing
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Grid Area of Shapes
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Find the area
for the shapes
on 1 cm grid
1
1
2
3
cm²
2
3
4
4
6
5
7
5
6
7
8
8
9
9
Use grid overlay to find the
area of these, or use a rule
to make 1 cm spaced lines
A=
cm²
Answer the questions without drawing the shapes.
Calculate the area of…..
14
17 A rectangle with side
cm
cm
10
Use 1 cm grid to find the
area of these, then measure
the side lengths.
A=
cm²
Side =
cm
Area =
cm ×
Side =
cm
Area =
cm²
Multiply sides
lengths 8 cm and 6 cm.
cm
18 A rectangle with side
lengths 7 cm and 9 cm.
15
cm
cm
11
A=
cm²
Side =
cm
Side =
cm
Multiply sides
A=
12
16
cm
Area =
cm ×
Area =
cm²
cm
19 A square with side lengths
of 9 m.
Area =
m ×
Area =
m²
m
20 A square with side lengths
cm
of 7 cm.
A=
Area =
cm ×
Area =
cm²
cm
21 A rectangle with side
13
A=
A=
cm²
Side =
cm
Side =
cm
lengths 19 m and 3 m.
Multiply sides
Area =
m ×
Area =
m²
m
Grid Area of Shapes
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Find the area
for the shapes
on 1 cm grid
1
1
2
3
cm²
2
3
4
4
6
5
7
5
6
7
8
8
9
9
Use grid overlay to find the
area of these, or use a rule
to make 1 cm spaced lines
A=
cm²
Answer the questions without drawing the shapes.
Calculate the area of…..
14
17 A rectangle with side
cm
cm
10
Use 1 cm grid to find the
area of these, then measure
the side lengths.
A=
cm²
Side =
cm
Area =
cm ×
Side =
cm
Area =
cm²
Multiply sides
lengths 8 cm and 6 cm.
cm
18 A rectangle with side
lengths 7 cm and 9 cm.
15
cm
cm
11
A=
cm²
Side =
cm
Side =
cm
Multiply sides
A=
12
16
cm
Area =
cm ×
Area =
cm²
cm
19 A square with side lengths
of 9 m.
Area =
m ×
Area =
m²
m
20 A square with side lengths
cm
of 7 cm.
A=
Area =
cm ×
Area =
cm²
cm
21 A rectangle with side
13
A=
A=
cm²
Side =
cm
Side =
cm
lengths 19 m and 3 m.
Multiply sides
Area =
m ×
Area =
m²
m
Designing Flooring on a House Plan
This sheet deals with area in a practical use, the floor coverings of a house. The depth to which
you complete this sheet depends on your time and creativity. The grid on the third page is used
to represent the floor of your house, looking down through a roof that has vanished.
The method of the sheet is as follows:
•
Wall off the rooms.
•
Try to create: 3 bedrooms, bathroom, kitchen, laundry, lounge room, dining room and a
hallway.
•
Colour the squares in each room a particular colour to represent a different floor
covering, then the basics, fill the house with furniture. This can either be drawn in or the
pieces on the next page can be coloured, cut out and stuck on.
•
Once you have finished the house complete the last page which is used to find the total
cost for flooring. Multiply the length and breadth of the room to get the area, if rooms are
rectangular or square, otherwise just count the squares. Multiply this by the cost of the
floor covering used, to find the cost of each room. Add the costs to get the total.
There are 3 bedroom spaces on the calculation sheet, you may have a 2 bedroom house
though, so leave one blank, or create another room that you may have. There is also one
additional space at the bottom of the page if you wish to put in another room. You may like to
include a garage and park your car in it.
House Pieces
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
BEDROOMS
Double Bed
Single Bed
Wardrobe
DINING ROOM
Chest of
drawers
Round Dining
Table + chairs
Rectangular Dining
Table + chairs
China Cupboard
BATH ROOM
Toilet
Shower with soap
left on floor
Bath
Toilet
Basin
LOUNGE ROOM
Lounge
Arm Chairs
LAUNDRY
Wash Tub
Chair with TV
remote tray and
footrest
Cupboard
Washing Machine
Coffee Table
KITCHEN
GARAGE/CARPORT
Bench with Sink
Bench with Cook-top and
Oven
Bench with wall
cupboards
Refrigerator
Designing Flooring on a House Plan
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Below is a blank floor Plan for a house that is still in design. Consider it your job to put in the walls to make
rooms, naming and fitting out each room (adding furniture). Then using the next page calculate the cost of
flooring for each room and then total. The rooms you need to include are:
Up to 3 Bedrooms (MBR, BR2 and BR3)
Bathroom (Bath)
Kitchen (KN) [+ Pantry (P) optional]
Dining Room (DR)
Laundry (LDY)
Lounge Room (LR)
Hallway (HALL)
Make sure you use a pencil. The separating walls should be on the grid lines, but thick enough to see. If
you have extra room to fill in include rooms like: extra toilet, photographic darkroom, car garage or an
additional bedroom. The scale used is 1 cm = 1 m.
Floor Coverings for the Plan
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Measure the sides (length and breadth) of each room and multiply to get the area. If the room is not square
or rectangular just count the squares. Select the type of flooring, ensure that it matches the type of room you
are dealing with. Then multiply the area by the cost to get the cost for the room. Then add all the rooms to
get total cost. Remember that 1 square = 1 m². Coverings to choose from:
1. Wool carpet
$95 per m²
2. Hard wear carpet
$32 per m²
3. Tiles
$27 per m²
4. Cork tiles
$48 per m²
5. Polished boards
$38 per m²
6. Vinyl
$12 per m²
Master Bedroom
Length
Number of flooring type
Bedroom 2
Length
Number of flooring type
Bedroom 3
Length
Number of flooring type
Dining Room
Length
Number of flooring type
Lounge Room
Length
Number of flooring type
Kitchen
Length
Number of flooring type
Laundry
Length
Number of flooring type
Hallway
Length
Number of flooring type
Bathroom
Length
Number of flooring type
Length
Number of flooring type
× Breadth
=
m²
Cost per m²
× Breadth
Cost for room
=
m²
Cost per m²
× Breadth
Cost for room
=
m²
Cost per m²
× Breadth
Cost for room
=
m²
Cost per m²
× Breadth
Cost for room
=
m²
Cost per m²
× Breadth
Cost for room
=
m²
Cost per m²
× Breadth
Cost for room
=
m²
Cost per m²
× Breadth
Cost for room
=
m²
Cost per m²
× Breadth
Cost for room
=
m²
Cost per m²
× Breadth
Cost per m²
Cost for room
=
m²
Cost for room
TOTAL COST:
Area of Squares
Area is the measurement of space within a shape. The units of measurement are square units,
which can be mm², cm², m² and so on. This sheet deals solely with squares. The formula used
is A = l². The l referring to the side length of the square.
Some students have a difficulty with squaring numbers (²). They forget that it means the
number times itself and instead think it is the number times 2. You can use the formula
A = l × l if this makes it easier to remember. Note that there are 3 lines of working. The first
being the formula, the second being the substitution of values and the third being the answer
with square units.
Column 1 questions have sides with 2 digit numbers. Column 2 has sides with 1 or 2 digits
plus 1 decimal place. Use the same method as with Column 1, just remember that you will
have an answer with 2 decimal places. Column 3 are written problems, solve these in the same
way just without the diagram.
Make sure you place the ² on the units. Some students forget and place the ² on the number in
the answer instead of the units, eg. instead of writing 14 m² they write 14² m, which is
incorrect.
1
43
43 m
A = l²
Show 3 lines of working :
formula, substitution and answer
43
= 43²
129
1 720
A = 1 849 m²
1 849
Area of Squares
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Find the area of these
using decimals
Find the areas of
the squares below
6
1
3.4 cm
15 m
Now instead of a diagram
read the problems
11 A quilt is to be made of 17 cm
side length square patches. Calculate
the area of one of these patches.
A = l²
0
=
A=
m²
7
2
5.6 mm
21 cm
12 A garden shed has a square roof
with 2.3 m side lengths. Calculate the
area of the roof in square metres.
8
3
8.3 cm
19 m
13 A courtyard has a square grassed
section with 27 m long sides. Find
the area for this section.
4
9
37 mm
17.2 cm
14 If turf costs $4/m² what would be
5
53 cm
the cost of relaying the grass surface
above?
10
Cost = area × unit cost
31.5 m
Area of Rectangles
Area is the measurement of space within the shape. The units of measurement are square units,
which can be mm², cm², m² and so on. This sheet deals solely with rectangles. The equation
used is A = lb. The l is the side length, b is the breadth. It doesn't matter which you choose as l
and which as b, the answer will still be the same, but generally l is the longest length.
Note that there are 3 lines of working. The first being the formula, the second being the
substitution of values and the third being the answer with units.
Column 1 questions have sides with 2 digit numbers. Write the first 2 lines of working then
use the multiplication working space. Then answer using the third line.
Column 2 has sides with 1 or 2 digits plus 1 decimal place. Use the same method as with
Column 1 just remember that you will have an answer with 2 decimal places. Multiply the
numbers without the decimal point, then put it back in at the end.
Column 3 has written problems solve these in the same way, just without the diagram.
The most common mistake made is misusing the ², some students forget and put the ² on the
number in the answer instead of the units, instead of 14 m² they incorrectly write 14² m.
18 m
35
18
35 m
A = lb
Show 3 lines of working:
formula, substitution and answer.
1
= 35 × 18
280
350
A = 630 m²
630
Area of Rectangles
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Find the area now
using decimals
Find the areas of the
rectangles below
1
6
56 mm
5.7 m
39 mm
2.3 m
Now find the area in the
following problems
11 A rectangular coffee table has
side lengths 22 cm and 63 cm.
Calculate the area of the table
A = lb
=
A=
2
0
×
mm²
17 cm
7
2.9 cm
6.2 cm
45 cm
12 Find the area of an air hockey
table which has a rectangular playing
area of side lengths 1.1 m and 2.6 m.
18 m
3
2.7 cm
8
9.4 cm
25 m
13 A foyer of a building is to be
tiled. If the area is rectangular with
sides 25 m and 48 m, find the area.
9
4
14 cm
53 cm
34.3 mm
24.6 mm
14 If 2 tilers can lay 80 m² per day,
5
29 mm
74 mm
10
19.6 m
43.6 m
calculate the time taken to complete
the job
Area of Triangles
Area is the measurement of space within a shape. The units of measurement are square units,
which can be mm², cm², m² and so on. This sheet deals solely with triangles. The formula
used is A = ½bh. The b referring to triangle's base length, the h referring to the triangle’s
perpendicular height.
Triangles confuse some students due to there being 3 sides but only 2 measurements being
used in the formula. If given 3 sides some students multiply ½ by all 3 sides, this is incorrect.
Other students use the horizontal measurement always as the base, this isn't the case with
rotated triangles. The critical thing to remember is that the measurements MUST be at right
angles (perpendicular) to each other ALWAYS. So long as you never use sides that don't have
a right angle between them you will be correct.
With Column 1 the triangles have 3 or more dimensions (measurements) on them. Remember
that you only need 2 of the lengths. Circle the sides to be used in the formula. Then find the
area using those 2 sides. Note that there are 3 lines of working, the first being the formula, the
second being the substitution of values and the third being the answer with units.
Column 1 features triangles with sides having 2 digit numbers. Write the first 2 lines of
working then use the multiplication working space. Then write the answer in the third line.
With triangles save yourself time by dividing one of the sides by 2 and using it in the
multiplication. All the questions have at least one even side which can be divided by 2. See
the example below. When you divide a decimal imagine the point isn't there. E.g. 3.4 as 34
half of 34 is 17 so it is 1.7 and 8.8 as 88, half of 88 is 44 so it is 4.4.
Column 2 questions have decimal sides with 1 digit plus 1 decimal place. Use the same
method as with column 1 just remember that you will have an answer with 2 decimal places.
Column 3 are written problems solve these in the same way, just without the diagram.
40 cm
41 cm
A = ½bh
= ½ × 9 × 40
9 cm
20
9
180
A = 180 cm²
Note that you
still write 40
here not 20
40 is even so
find ½ of it
½ × 40 = 20
Area of Triangles
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Select the 2 measurements
to use, circle them, then
find the triangle's area.
mm
1
10
8 mm
Find the area now
using decimals
8.2 mm
6
Now find the area in the
following problems
11 A triangular sun shade has
perpendicular sides of 3.8 m and
6.2 m. Calculate the area
4.6 mm
6 mm
A = ½bh
= ½×
×
A=
2
mm²
7
13 m
5m
12 m
3.8 m
2.4 m
12 A triangular section of glass has
a base of 4.8 m and a perpendicular
height of 7.6 m. Calculate the area.
3
16 cm
6 cm
14 cm
8
9 .6
22 cm
cm
4.4 cm
13 A triangular sail has vertical
height of 5.2 m and a horizontal base
length of 3 m. Calculate its area.
10 m
10 m
4
9
8m
1.2 mm
3. 6
8m
mm
14 A triangular garden bed has
5 15 cm
8 cm
perpendicular sides of 86 cm and
94 cm. Calculate the bed's area.
10
9.2 m
17 cm
9.2 m
Areas of Squares, Rectangles and Triangles
Area is the measurement of space within a shape. The units of measurement are square units,
which can be mm², cm², m² and so on. Note that if you are ever given a question without units,
then express your answer in units². The last questions in each column require you to do this.
Column 1 deals solely with squares. The formula used is A = l². The l referring to the side
length. Some students have a difficulty with squaring numbers (²). They forget that it means
the number times itself and instead think it is the number times 2. You can use the formula
A = l × l if it easier for you to remember. The answer is obtained by multiplying the side length
by itself. Note that there are 3 lines of working. The first being the formula, the second being
the substitution of values and the third being the answer with units.
Column 2 is all rectangles. The equation is A = lb, the length is normally the longest distance,
but it doesn't actually matter which you choose because the answer will be the same. So
multiply the two different sides together in any order.
3 mm
A = l²
10 m
17 m
A = lb
14 cm
Column 3 are all triangles. The equation is A = ½bh, or 'half times the base times the height'.
When you multiply these there is an easier way. The more difficult way is to multiply the two
numbers then divide by 2, only use this method if both numbers are odd. The easier way, if at
least one side is even, is divide one of the numbers by 2 (usually pick the largest even number)
get the answer and then multiply it by the other side. This sheet has been designed to be done
mentally, so no working spaces are provided.
5 cm
A = ½bh
= 3²
= 17×10
= ½×5×14
A = 9 mm²
A = 170 m²
A = 35 cm²
Don't forget to put a 'squared' (²)
in the answer
Calculate like this:
"a half of 14 is 7, 7 times 5 is 35
Areas of Squares, Rectangles and Triangles
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Find the areas of the
rectangles below
2
11
12
11 m
4m
6 cm
21
22
9 cm
6m
8m
8m
A = l²
A = lb
A = ½bh
=
=
= ½×
A=
m²
3
A=
m²
A=
13
4
14
10 cm
5m
5
6
8 cm
11 m
7
A=
18 m
26
12 cm
17 m
4m
9m
7 cm
20
13
6
27
5m
28
20 mm
22 m
19
9
units²
18
12 cm
10
12
24
7 mm
25
12 mm 10 m
17
7 mm
9
m²
5 cm
8
20 m
7m
16
4 mm
=
6m
15 cm
15
×
23 16 mm
20 m
4 cm
9m
10 m
5 cm
30 mm
1
Find the areas of the
triangles below
14 cm
Find the areas of
the squares below
29
21
30
4
20
17
8
13
Finding Area with Different Units
This sheet deals with shapes that have measurements in different units on their sides, for
example one side in metres and the other in cm. With these the method is to change the
larger units to match the smaller units. To convert, multiply the measurement by:
•
10 if converting from cm → mm, e.g. 3 cm = 30 mm, 5.9 cm = 59 mm
•
100 if converting from m → cm, e.g. 3 m = 300 cm, 1.9 m = 190 cm
The reverse applies changing from a smaller unit to a larger unit, convert by dividing the
measurement by:
•
10 if converting from mm → cm, e.g. 56 mm = 5.6 cm, 134 mm = 13.4 cm
•
100 if converting from cm → m, e.g. 35 cm = 0.35 m, 128 cm = 1.28 m
Column 1 asks you first to convert units, note that some ask for a conversion that jumps a unit
m → mm, you may like to do these in 2 stages convert to cm mentally then to mm. The rest of
the column and column 2 are rectangle problems. Change the larger unit to the smaller unit
and write the conversion next to the measurement, then strike out the old measurement. Show
3 lines of working: formula, substitution and answer with units, follow the method shown in
Q. 13. There is a working space provided for your multiplication.
Column 3 deals with triangles. The method is the same except that there is a ½ in the formula.
All the numbers in the questions have at least 1 even number, divide it by 2 and then multiply.
70 cm
0.7 m
112 cm
8 mm
112
70
A = lb
= 70 × 112
A = 7 840 cm²
3.2 cm
32 mm
1
7 840
A = ½bh
= ½ × 32 × 8
A = 128 mm²
4
16
8
128
Finding Area with Different Units
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15
Now lets try
triangles
90 cm
0.3 m
20
1 0.8 m
=
cm
2 7.5 cm
=
mm
3 1.6 m
=
cm
4 30 mm
=
cm
5 1.9 cm
=
mm 16
6 43 cm
=
m
7 0.11 m
=
cm
8 0.4 m
=
mm
9 0.37 m
=
mm
10 273 mm
=
m
11 0.673 m
=
cm
12 2.03 m
=
cm
9 mm
4.8 cm
A = ½bh
=½ ×
5.6 cm
A=
72 mm
×
mm²
21
25 cm
0.8 m
17
20 mm
22
58 cm
Change the measurements
given on the left to the
units on the right hand side
7.3 cm
0.22 m
Change the larger units to
the smaller units of the
other side. Find the area.
13
8 cm
18
9 cm
23
1.6 cm
0.34 m
30 mm
A = lb
=
×
A=
14
mm²
40 cm
1.2 m
19
110 cm
0.8 m
24
54 cm
0.15 m
43 mm
Area Problems
These problems calculate the area of squares, rectangles and triangles and then use the area to
involve money, time or a quantity (such as litres). As with areas of shapes, use three lines of
working: the formula, substitute the values and the answer including units. Don't forget the ²
on the units.
In Column 2, Q. 9 to 16 introduce hectares (ha). Unlike the other units ha doesn't have a ² on it
as it isn't a distance unit it is just an area unit. The unit is used for large areas (usually land)
such as with farms and parkland. The conversion is:
1 hectare (ha) = 10 000 m².
This means that 20 000 m² = 2 ha, 50 000 m² = 5 ha and so on. These questions ask you to
convert the m² units to ha. When you divide by 10 000 you move the decimal point 4 places to
the left.
The third column deals with area problems using measurements in metres. Answer the
questions in m² then convert the answer to ha using the 10 000 division.
Area Problems
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Find the area of the
shapes described below
1 A square with 4 cm sides is cut
from paper, a square with 8 cm is
then cut out. Find the areas of each of
these squares.
A = l²
6 Karen's room is rectangular with
sides 4.5 m and 4 m. Find the floor
area and the cost of re-carpeting it if
the carpet costs $53 per square metre.
Answer these in m² then
convert to hectares
17 A rural property is rectangular in
shape with sides 350 m and 500 m.
Find its area
A = l²
Cost:
2 The side lengths were multiplied
by 2 (4 cm → 8 cm), what number
would you multiply the 4 cm square
area by to get the 8 cm square area?
Land area:
7 Find the area of a field that is
m²:
ha:
120 m long and 80 m wide. If a lawn
18 The owner wants to plant 8 trees
mower covers 1 000 m² in a minute.
How long will it take to mow the field per ha for desalination reasons, how
many trees should be planted?
Trees =
3 Now use 5 cm and 10 cm side
×
length squares. Does the same area
relationship apply as above?
Time (min):
(h:min):
8 Calculate the area of the water
Circle: Yes / No
surface (in m²) of a swimming pool
50m long and 20 m wide
19 A triangular section of paddock
has perpendicular sides of 80 m and
50 m. Calculate its area.
4 David wants to paint his team
emblem on the football field. The
Raiders emblem is a black triangle of
base 5m and height 6 m inside a
yellow square with sides 8m. Find the
shape’s area then the yellow area.
Area:
A hectare (ha) is 10 000 m²
Convert these to ha.
9 30 000 m²
10 40 000 m²
Yellow painted area = ‰ - ∆
11 35 000 m²
=
12 100 000 m²
-
=
m²
5 A jar of grass paint covers 10 m²
how many jars will be needed and
find the total cost. (1 jar costs $3.75)
Jars needed:
black:
Total:
Cost:
m²:
ha:
20 A 16 ha property has a square
free range chicken barn with sides of
200 m. Calculate the area of the barn
13 120 000 m²
14 125 300 m²
15 4 000 m²
yellow:
Land area:
16 300 m²
Barn area:
m²:
ha:
21 What percentage of the land is
taken up by the barn?
Composite Area - Addition
Composite areas are shapes that are made from two or more shapes. To calculate the area the
shapes are broken up into geometric shapes, in this case squares, rectangles and triangles. The
shapes have their areas calculated separately then they are added together to get the total area.
Questions 1 to 7 can be broken into 2 shapes, questions 8 and 9 into three shapes. Note that all
angles that appear to be right angles are right angles.
A1
45 cm
15 cm
30 cm
15 cm
15 cm
The method for the sheet is the same throughout, follow this method:
•
Find all the missing side lengths for the shapes and write them on the diagrams
•
Draw a line to cut the shape up into basic shapes. Label the separate areas A1, A2 … to
show which area is which.
•
Answer the questions showing full working out, follow the method shown in question 1
and the example below.
A2
60 cm
A1 = l ²
A2 = lb
= 15²
= 60 × 15
A1 = 225 cm²
A2 = 900 cm²
A =A1+A2 = 225 + 900
A = 1 125 cm²
Composite Area - Addition
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
4
Find the areas of
the shapes below
A1 = lb
A2 = lb
=
=
A2 =
A = A1+A2 =
32 m
m²
Triples!!!
+
A=
4 cm
8
5
11 cm
?
9 cm
9 cm
11 cm
8 cm
2
m
15 m
4m
m²
14 m
4
5m
4m
5m
A1 =
m
3m
5m
1
4
7
20 m
18 cm
3 cm
A = A1+A2+A3
A = A1+A2 =
+
6
20 mm
30 mm
6 mm
18 mm
9
10 mm
5 mm
3
7 mm
A=
20 mm
Composite Area - Subtraction
Composite areas are shapes that are composed (made up) of two or more shapes. These shapes
can be broken up into geometric shapes, in this case squares, rectangles and triangles to help
calculation of area. The shapes have their areas calculated separately then the smaller shapes
are subtracted from the larger shape. Questions 1 to 7 have one shape subtracted from the
other, Q 8 and 9 have 2 shapes subtracted.
The method for the sheet is the same throughout, follow this method:
•
Find all the missing side lengths for the shapes and write them on the diagrams
•
If the inside shape touches the perimeter to make an opening draw a broken line across
the opening to close it off. Question 3 will need 2 lines.
•
Label the areas A1, A2 (and A3 for questions 8 & 9) to show which area is which. Let A1
be the outside or largest shape area then use A2 and A3 for the other areas.
•
Answer the questions showing full working out, follow the method shown in question 1
or the example below.
5m
A2
20 m
5m
A1
40 m
A1 = lb
= 40 × 20
A1 = 800 m²
A2 = l ²
= 5²
A2 = 25 m²
A = A1 - A2
= 800 - 25
A = 775 m²
Composite Area - Subtraction
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
7
4 cm
25 cm
7 mm
8 cm
4
Find the shaded areas of
the following shapes
20 cm
15 m
1
2m
4 mm
4m
A1 = l ²
A2 = l ²
=
=
A1 =
mm²
A1 =
A = A1 - A2 =
mm²
Now subtract two areas
-
8
A=
5
4 cm
2
4 cm
10 cm
20 cm
12 mm
6 mm
8 cm
8 cm
3 cm
A = A1-A2-A3
A = A1 - A2 =
6
A=
9
7
m
9 cm
3
12 cm
6
3 cm
7 cm
20 m
m
7 cm
2 cm
3 cm
11 cm
15 m
Changing Units of Area
This is perhaps the most difficult part of area calculations, with students often forgetting this
work. The reason is that because there are 10 mm in 1 cm, students automatically use this to
convert mm² to cm², but this isn't the case. There is 100 mm² in 1 cm². The same with
converting cm² to m², there is 10 000 cm² in a m², not 100.
Column 1 first converts areas from cm² to mm². Note the diagram at the top of the column. A
1 cm square is redrawn with grid lines at each millimetre. The square has 10 mm sides and as
you know the area of a square is its side length squared. As 10² = 100 there are 100 1 mm²
squares in a square cm. So to convert the measurements multiply each number by 100. This
means move the decimal point 2 places to the right. So 5 cm² would be 500 mm², 5.5 cm²
would be 550 mm² and 5.05 cm² would be 505 mm². Questions 11- 20 are the reverse, move
the decimal point 2 places to the left. So 8 mm² is 0.08 cm², 130 mm² is 1.3 cm² and
130.02 mm² would be 1.3002 cm².
Column 2 then deals with converting between m² and cm². Look at the diagram and note that
there are (100 × 100) 10 000 cm² in 1 m². So to change cm² to m² multiply the number by
10 000 or move the decimal point 4 places to the right. So 8 m² would be 80 000 cm² and
8.02 m² would be 80 200 cm². Then the method is reversed for the second part of the column.
When you divide by 10 000 move the decimal place 4 places to the left. So 9 cm² would be
0.0009 m² and 136 cm² would be 0.0136 m².
Column 3 are problems that you solve using the units as given in the question. Then after you
answer the question express the answer in the other unit that is requested. Working spaces are
supplied.
Changing Units of Area
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Convert these areas
from cm² to mm²
10 mm
1 cm² = 100 mm²
=
1m
1 m² = 10 000 cm²
21 2 m²
2 8 cm²
22 3 m²
3 7.5 cm²
23 0.3 m²
4 18 cm²
24 0.03 m²
5 29 cm²
25 3.3 m²
6 45.6 cm²
26 0.725 m²
7 65.32 cm²
27 0.203 m²
8 82.09 cm²
28 0.8963 m²
9 100 cm²
29 0.902 m²
10 100.1 cm²
30 0.0009 m²
Convert these areas
from mm² to cm²
1 mm² = 0.01 cm²
41 A rectangular piece of fabric is
size 1.5 m × 2.7 m. Find its area in
both m² and cm².
Fabric area:
m²:
cm²:
42 A square steel plate with sides
8 cm has a rectangular slot cut into it
with sides 35 mm and 15 mm. Find:
i) the size of the plate without the slot
in mm² (A1)
ii) the size of the slot in mm² (A2)
iii) the cut plate area in mm² and cm²
i)
A1 = l ²
Convert these areas
from cm² to m²
100 cm
10 mm
=
Answer these problems
using the methods from
the first two columns
A = lb
1 3 cm²
1 mm²
100 cm
=
1 cm
100 cm
1m
10 mm
1 cm²
10 mm
=
100 cm
1 cm
Convert these areas
from m² to cm²
1 cm² = 0.0001 m²
11 5 mm²
31 7 cm²
12 10 mm²
32 11 cm²
13 25 mm²
33 54 cm²
14 125 mm²
34 154 cm²
15 170.1 mm²
35 296 cm²
16 205.4 mm²
36 875 cm²
17 30.03 mm²
37 1 500 cm²
18 450 mm²
38 8 700 cm²
19 500 mm²
39 12 000 cm²
20 500.6 mm²
40 15 796 cm²
Plate area =
ii)
iii)
Cut Plate area:
mm²:
cm²:
Finding a Side Given Area
This sheet is difficult as it uses skills that are learnt in the Algebra 2 folder. It is essentially
working in reverse, given an area and a side (except with squares) you are asked to find the
unknown side. Note that there is up to 5 lines of working, when you may be able to complete
the problem mentally, so why all the working? These exercises use whole numbers throughout, but the method can be used for decimals also and the use of decimals would make the
exercises too difficult to solve mentally.
Column 1 deals with calculating a square's side length given its area. The first 5 questions test
that you understand what a square root is. Eg. 10² = 100 so √100 = 10. To find the length of a
side given the area you square root the area. So if an area was 4 cm² then its side length is
2 cm. As the √4 = 2. The 3 lines used are again the formula, the substitution then the solution
with units matching the area units only without the ².
Column 2 deals with the area of a rectangle, this time given an area and a side. The first line is
the formula for a rectangle written backwards lb = A. Then the second line has the values
substituted in, in the example at the top of the column: A = 20 mm² and b = 5 mm. The third
line divides each side by b (5) to get l by itself (from Algebra 2 folder).
Column 3 involves triangles, it is an extension of column 2 in that a ½ is used. But the method
is just the same, look at the example. The formula is written backwards, then the substitution is
made, the ½ × the side is calculated then the method is the same as above.
Finding a Side Given Area
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Find the squares of the
following and then
complete the exercises
1 4² = 16
so,
16
Find the unknown side
of the rectangles given
their area and one side
=4
Example
14
A = 35 m²
b=7m
2 5² =
so,
=5
A = 20 mm²
b = 5 mm
3 6² =
so,
=6
lb = A
4 7² =
so,
=7
l × 5 = 20
5 8² =
so,
=8
Find the side length of the
square with the area given
6
7
Area = 16 cm²
Area = 64 m²
l=
A
21
A = 40 mm²
b = 10 mm
A = 48 cm²
b = 6 cm
½bh = A
½×10×h = 40
5×h = 40
l = 20 ÷ 5
h = 40 ÷ 5
l = 4 mm
h = 8 mm
15
16
A = 60 mm²
b = 4 mm
A = 120 m²
b=6m
22
23
A = 60 m²
b=6m
A = 100 mm²
b = 20 mm
24
25
A = 80 cm²
h = 10 cm
A = 120 m²
h=6m
l=
cm
8
l=
9
Area = 100 cm² Area = 25 mm² 17
10
11
Area = 81 m²
Area = 36 mm²
12
Example
l=
l=
l=
Find the height (or base) of
the triangles below given
their area and base (height)
13
18
A = 63 cm²
l = 9 cm
A = 72 cm²
l = 6 cm
19
20
A = 88 mm²
l = 8 mm
A = 140 cm²
b = 7 cm
26
Area = 121 cm² Area = 144 m²
A = 56 mm²
h = 8 mm
7
FREEFALL
MATHEMATICS
TIME
Writing and Reading Time
Writing the time is usually done in one of two ways. The first is the hour then the minutes,
such as seven thirty-eight (7.38) or six fifteen (6.15). The other method is to treat the two
halves of the clock differently.
•
There is a 'past half' (0 - 29 minutes) such as twelve minutes past six (6.12), twenty-five
minutes past nine (9.25).
•
There is a 'to half' (31 - 59 minutes) such ten to eleven (6.50), twenty-five to three (2.35)
•
There is also half past (30 minutes), quarter past (15 minutes) and a quarter to
(45 minutes). Such as a quarter past three (3.15), half past eleven (11.30), quarter to one
(12.45).
Column 1 gives you the time in words and asks for the time in number form as well as placing
and positioning of the hands on the analogue clocks. Then there is the time of day it is, is it
morning, night, afternoon, day, evening etc. These words all have a.m./ p.m. meanings and
you are asked to write in the a.m. or p.m. as well.
Column 2 asks for the time to be written in words. Also give the time of day as morning, night,
afternoon etc.
The 3rd column is using dates, writing the dates in words and then also in numerical form as
the example at the top of the column shows you. The calendars below may be of assistance.
JULY 2013
S
AUGUST 2013
M
T
W
T
F
S
1
2
3
4
5
6
7
8
9
10
11
12
13
4
5
6
14
15
16
17
18
19
20
11
12
21
22
23
24
25
26
27
18
28
29
30
31
25
S
M
T
W
SEPTEMBER 2013
T
F
S
S
M
T
W
T
F
S
1
2
3
1
2
3
4
5
6
7
7
8
9
10
8
9
10
11
12
13
14
13
14
15
16
17
15
16
17
18
19
20
21
19
20
21
22
23
24
22
23
24
25
26
27
28
26
27
28
29
30
31
29
30
Writing and Reading Time
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Write two ways of telling
the time from the clocks
shown. E.g. 7.35 is seven
thirty-five or twenty-five
to eight. Write morning,
afternoon or night as well.
Put the hands on the clock
and write these times
Example
a quarter to eight in
12
the morning
11
1
10
2
9
3
7
4
8
7
1
7.45 a.m.
6
5
AM
a)
2
9
3
4
8
7
2
6
8
5
four thirty-five in
12
the afternoon
11
1
10
3
7
6
10
2
9
3
5
half past three in the
12
morning
11
1
10
2
9
6
T
F
S
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
Example
29 08 13
b)
9
12
AM
PM
14
b)
15 The date 9 days before
AM
PM
a)
5
twenty past six at
12
dusk
11
1
10
13
a)
10
4
8
7
W
Thursday 29th August, 2013
a)
3
5
PM
4
8
6
T
5
twelve to twelve at
12
night
11
1
7
M
AM
4
8
4
S
2
9
3
AUGUST 2013
PM
twenty-five past one
12
in the day
b)
11
1
10
Write these days and their
dates on the calendar in
words then the date in
numerical form.
16 The date 15 days after
b)
2
9
3
4
8
7
6
6
11
5
a quarter to five at
12
dawn
11
1
10
3
4
8
7
6
5
17 The date 10 days after
PM
a)
2
9
AM
18 The date 20 days before
b)
Converting to 24 Hour Time
This time format is used mainly in the armed forces and the travel industry. If you buy
computerised tickets at railway stations the time will be usually written in 24 hour time format.
It reduces confusion particularly with electronic machinery.
The first 2 digits of 24 hour time are the hours, the second 2 digits are the minutes.
The time starts at 0000 hours (midnight) and ends at 2359 (1 minute before midnight). Note
that there is no full stop separating hours and minutes. So:
•
The time is 0000 hours at 12 a.m. (midnight), 12.30 a.m. is 0030 hours
•
Then each hour adds 100 so 1 a.m. is 0100 hours, 2 a.m. is 0200 hours etc
•
Noon is 1200 hours and 1 p.m. is 1300 hours
So once the time reaches 1 p.m. add 12 hours to the normal time to get your answer in 24 hr
time format, e.g. 4.45 p.m. = 1645 hours (hours = 4 + 12 = 16, minutes stay the same)
Note that the word 'hours' is written after the time. When the time is spoken there is no
reference to minutes. E.g. 0400 is "O four hundred hours, Sir!" and the time 1640 is
"sixteen hundred and forty hours, Sir!" (The Sir! is optional)
Time of day in the 3rd column refers to morning, afternoon, evening, night or any other words
that describe the time of day
Converting to 24 Hour Time
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Change these times to 24
hour time format, (from
1 p.m. add 12 hours)
Example
4.45 p.m.
Change the times to a.m./
p.m. format. If the time
starts with 13 or more,
subtract 12 and write p.m.
Write the times below
in 24 hour format
35
Example
1645 hours
11
12
10
1548 hours
1 3 a.m.
18 0400 hours
2 7 a.m.
19 0945 hours
3.48 p.m.
2
9
3
4
8
7
36
XI
5
6
XII
20 1320 hours
II
IX
III
VIII
IV
VII
4 11.25 a.m.
21 1717 hours
5 3.52 p.m.
22 2002 hours
6 5.17 p.m.
23 2211 hours
7 8.47 p.m.
24 1539 hours
8 noon
25 2347 hours
9 4.56 a.m.
26 0016 hours
10 12.19 a.m.
27 0303 hours
11 10.25 p.m.
28 1956 hours
evening
I
X
3 5 p.m.
afternoon
1
V
VI
37 A quarter to ten at night
38 Eleven forty-seven (day)
39 Half past twelve at night
Write/colour the times
in a.m./ p.m. format
40 1556 hours
AM
12 midnight
29 2200 hours
13 8.20 p.m.
30 0000 hours
PM
41 0147 hours
AM
14 9.55 a.m.
31 0222 hours
15 11.39 p.m.
32 1833 hours
16
AM
PM
12
11
1
10
PM
3
4
8
7
6
12
5
Circle: am pm
9
3
4
7
6
12
2218 hours
1
10
2
9
3
4
7
2
8
11
8
1
10
2
9
17
42
33 2008 hours 34 0505 hours
11
AM
PM
6
5
time of day (words)
43 0855 hours
5
Circle: am pm
write answer in words & the time of the day
e.g. afternoon, evening, morning etc
Units of Time
This sheet deals with the changing from one unit of time to another. Time is measured in:
seconds (s), minutes (min), hours (h), days, weeks, years, and so on. In business and especially
in the construction industry units such as years have to be broken into smaller units like weeks
or even days, so that accurate scheduling can occur.
Column 1 outlines common time units and asks you to convert these to the larger unit above it,
or the smaller unit below it. These should be done mentally, no working spaces have been
provided. Some answers will need to be given in decimal or fraction form.
Column 2 requires the conversion of larger time periods to smaller time periods. These
questions involve larger numbers, so working spaces are supplied for each question. The
method is to convert one of the larger unit to smaller units, then multiply this by the number of
larger units. So if converting 6 years to weeks for example, follow these steps:
•
State the number of weeks in one year …. 52
•
Multiply 52 by 6 and you get your answer …. 312 weeks (a working space is provided
with each of these questions)
•
State your answer… “There are 312 weeks in 6 years”, and you are done.
You may think that the working spaces aren’t adequate, Q 23 for example, remember that
when you multiply by 60 it is easier to multiply by 6 and just add a zero, just don’t forget to
add it.
Column 3 adds a step, instead of moving to the next unit down, jump 2 units down. The
questions are the same style as Column 2 only you need to make two conversions and multiply
them.
Units of Time
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Convert these times to the
unit of time on the right.
1 4 weeks =
days
2 1 year =
weeks
3 3 days =
h
4 ¼h=
min
5 26 weeks =
years
6 6 min =
Jumping from unit to unit
requires thought. Change
these units, in steps.
Convert to different units.
These are harder, break
them into parts.
22 Change 7 days → hours
27 Change 1 day → min
1 day =
1 day =
h
1h=
min
h
State your answer
in words
There are
hours
in seven days
23 Change 17 min → seconds
28 Change 8 weeks → hours
s
1 min =
1 week =
days
7 2h=
min
State your answer
in words
8 weeks =
days
8 1 leap year =
days
9 6h=
days
10 3 decades =
years
11 4 min =
s
12 24 months =
years
13 4 centuries =
decades
14 ¼ year =
months
1 day =
h
State your answer
in words
weeks
17 300 s =
min
years
29 Change 1 hour → seconds
1h=
min
1 min =
s
weeks
26 Change 15 years → months
1 year =
months
30 Change 2½ years → days
1 year =
weeks
½ year =
weeks
So 2½ years =
1 week =
19 September =
days
20 2 millennia =
years
21 March + May =
days
h
25 Change 9 years → weeks
s
16 2 fortnights =
18 156 weeks =
1 day =
24 Change 18 days → hours
1 year =
15 ¾ min =
s
State your answer
in words
days
+
weeks
Using Timetables
Reading timetables not only helps you to catch your train but it also avoids wasting time.
Public transport runs whether you use it or not. By using it your travel is environmentally
sound plus it helps maintain the service. The problem with public transport is that it is not
flexible with time. But by using a timetable you can keep wasted time to a minimum. In these
problems it is outlined that the latest time you can arrive at a station is when the train departs.
This is true but not practical, it is always best to arrive early to have spare time in case the
train is early, there is a queue for a ticket or a difference in the driver's watch and your own.
In Column 1 the Angle Line timetable is used, read the times from the timetable to answer the
questions. Alternate Junction and Mathsville are major stations with branch lines and so the
train stops on the platform and waits for 1 minute before departing.
In Column 2 the Symmetry Point Line joins the Angle Line at Alternate Junction station. The
train terminates (stops) at Alternate Junction and returns back to Symmetry Point. It doesn't
continue along the Angle Line. The questions for this column involve travel on the Symmetry
Point Line then changing trains at Alternate Junction to meet the next Angle Line service. So
both timetables will be needed! You are asked to complete missing entries on the timetable,
enter the times (note you don't need to put a.m. or p.m. as they are at the top of the column).
Column 3 deals just with the Symmetry Point Line. This time a return timetable is used for a
return trip. This is the Saturday timetable for the small branch line and only one train operates
the service. While answering these questions keep in mind how the trip could have been made
more enjoyable by using a timetable and using your skills at measuring time.
There is a schematic map of the two lines below.
Obtuseland
Vertical Intersection
t
oin
yP
etr
mm
s
Sy
lat
F
ee
od
es
Tw
lop
sS
bu
are
om
qu
Rh
gS
pin
op
on
Sh
tag
en
eP
Th
f
uf
Bl
Reflex Corner
te
Ki
Alternate Junction
Transversal Valley
Vertex Peak
The Angle Line
Revolution Bend
Mathsville
Adjacent Hill
The Symmetry Point Line
Using Timetables
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Use the timetable below
for this column.
The Symmetry Point Line
connects to the Angle
Line at Alternate Junction
so both timetables will be
used for this column
The Angle Line
Station
Obtuseland
a.m. a.m. a.m. a.m.
7.45 8.05 8.25 8.45
Vertical Intersection
7.49 8.09 8.29 8.49
Alternate Junction arr. 7.53 8.13 8.33 8.53
dep. 7.54 8.14 8.34 8.54
Station
a.m. a.m. a.m. a.m.
Symmetry Point
7.25 7.45 7.55 8.25
8.07 8.27 8.47 9.07
Twodee Flats
Rhombus Slopes
8.10 8.30 8.50 9.10
7.28
7.31
Shopping Square
7.36 7.52 8.06
The Pentagon
7.40 7.56 8.10
Kite Bluff
Alternate Junction
7.44 ...
7.50x 8.02x 8.20x 8.50x
Reflex Corner
7.56 8.16 8.36 8.56
Transversal Valley
8.02 8.22 8.42 9.02
Vertex Peak
Revolution Bend
Mathsville
Adjacent Hill
The Symmetry Point Line
arr. 8.15 8.35 8.55 9.15
dep. 8.16 8.36 8.56 9.16
8.18 8.38 8.58 9.18
1 Will leaves home at 8.00 a.m. and
arrives at Obtuseland Station at
8.08 a.m., find the time:
a) the first train arrives
b) he could have left home to catch
the same train
c) he arrives at Mathsville
d) he arrives at school, a 5 min walk
from Mathsville Stn.
e) the train travel takes
f) the entire trip takes
2 At 8 a.m. Will tries for the earlier
train. He can run to Obtuseland
Stn. in 6 min or to Vertical
Intersection in 9 min, which is
successful?
3 At what time would he arrive at
school now?
4 Find the duration of the total trip
to school.
5 Will wants to see his friend
Jerome. If Jerome catches the
same train as Will at 8.27 a.m., at
what station was he waiting?
6 Will wants to buy a hot chocolate
at the Alternate Junction platform
vending machine. If it is 9 s away
how much time does the machine
have to fill the cup?
...
...
7.58
… denotes doesn’t stop at this station
x denotes terminates here, change here for Angle Line
7 Alicia’s timetable was unreadable
from rain damage. Complete the
table for the Symmetry Point 7.55
and 8.25 a.m. services.
8 Alicia lives at Symmetry Point
and wants to catch the 8.14 a.m.
Angle Line service which train
should she catch?
9 Alicia attends the same school as
Will. At what time would she
arrive at Mathsville?
10 If it takes Alicia 5 min to walk to
school at what time does she
arrive at school?
11 If Alicia misses the 7.45 a.m. train
will she be late if school starts at
9 a.m.?
Circle: Yes / No
12 Calculate Alicia’s total travel time
for both the 7.45 and 7.55 a.m.
services if her house is 4 min
away from Symmetry Point Stn.
(assume no waiting at Symm Pt.)
7.45 a.m. train
On Saturdays one train
operates the Symmetry Pt
line. A return timetable is
included.
Symmetry Pt. → Alternate Jnct
Station
p.m.
p.m.
Symmetry Point
10.30 11.30 12.30
a.m.
a.m.
1.30
Twodee Flats
Rhombus Slopes
10.33 11.33 12.33
10.36 11.36 12.36
1.33
1.36
Shopping Square
10.41 11.41 12.41
1.41
The Pentagon
10.45 11.45 12.45
1.45
Kite Bluff
10.49 11.49 12.49 1.49
Alternate Junction 10.55x 11.55x 12.55x 1.55x
Alternate Jnct → Symmetry Pt.
Station
p.m.
p.m.
p.m.
Alternate Junction
a.m.
11.00 12.00
1.00
2.00
Kite Bluff
The Pentagon
11.06 12.06
11.10 12.10
1.06
1.10
2.06
2.10
Shopping Square
11.14 12.14
1.14
2.14
Rhombus Slopes
11.19 12.19
1.19
2.19
Twodee Flats
Symmetry Point
11.22 12.22 1.22 2.22
11.25x 12.25x 1.25x 2.25x
x denotes terminates here
14 Ian wants to catch a train from
Twodee Flats to Shopping Square.
Reaching the station at 10.50 a.m.
how long does he wait?
15 Does Ian see his train travelling in
the opposite direction? If so, at
what time?
Circle: Yes / No
16 If he spends 2 h at the shops (with
walking time included), when
does the next train arrive to return
home and how long does he wait?
Time of train
Waiting time
17 How long is the train journey
home?
7.55 a.m. train
18 If it takes Ian 15 min to walk
between home and the station, at
13 Ann lives at Isosceles Hill, an 8
what time did he leave home and
min drive from Kite Bluff. If she
arrive back? How much time has
wants to catch the 8.14 a.m. Angle
elapsed during his trip?
Line train find the latest time she
Left home
Arrived home
can arrive at Kite Bluff and the
time she must leave home.
Latest Arrival
Time left home
Elapsed time
Time Calculations 1
Often times are given in a way that you find difficult to use. One example is a movie on video
with its duration given in minutes. If a movie is 173 minutes long, how do you convert it to
hours and minutes?
In Column 1 the exercises involves changing min to h and min. Multiples of 60 are required,
so 1 × 60 = 60 and so on, from then on add 60 to the previous answer to get the next.
To answer Q 2 - 7 find the largest multiple of 60 that is below the number of min in the
question and then put the remainder, the amount left over, in the 2nd box. Then change the
multiple of 60 to hours and then write the remainder in min. Look at the example above Q.2.
Column 2 requires times to be added. Using the addition space add the two times together,
ignoring the fact that the min may exceed 60 min. Then rewrite your answer again in the
smaller outlined box. If the minutes exceed 60 convert the min to hours by deducting 60 min
off the min total and adding 1 h to the hour total. Of course if the min total is greater than 120
min you have to deduct 120 min off and add 2 hours, and so on for each multiple of 60. See
the example at the top of the column.
The 3rd Column requires the method to be reversed. Before subtracting there must be enough
min to perform the subtraction (the first time must have more min than the second time). If it
doesn't, take an hour off the first time and add 60 min to the minutes. For example 4 h 11 min
would become 3 h 71 min, 1 h 35 min would become 95 min etc. There is an example at the
top of the column.
Time Calculations 1
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
List the first 12 multiples
of 60, this will help you
convert min to h and min.
Example
1
1 × 60 =
1
2 × 60 =
9 × 60 =
4 × 60 =
10 × 60 =
5 × 60 =
11 × 60 =
6 × 60 =
1
3 42 +
8 × 60 =
3 × 60 =
Example
3 h 42 min and 7 h 39 min
7 × 60 =
10 h 81 min = 11 h 21 min
7 39
The total is
10 81
11 h 21 min
8 3 h 27 min and 8 h 46 min
7 h 15 min - 4 h 57 min
6 75 4 57
9 11 h 52 min and 7 h 35 min
Example
+
435 min = 420 + 15 min
= 6 h 75 min
Difference is
1
2 18
2 h 18 min
13 8 h 47 min - 5 h 11 min
The total is
Using the answers in
question 1, convert these
times in min to h and min
7 h 15 min
-
+
12 × 60 =
Subtract these times, you
may have to convert an h
to 60 min first (or more).
Add these times together.
If the min exceed 60
change them to h.
Difference is
14 17 h 25 min - 6 h 52 min
-
= 7 h 15 min
2 157 min =
+
min
=
3 203 min =
10 6 h 49 min and 8 h 44 min
+
+
min
+
min
15 26 h 37 min - 13 h 48 min
-
=
4 347 min =
+
=
5 577 min =
+
min
12 3 h 43 min, 3 h 37 min and
+
min
=
7 h 51 min
+
=
7 686 min =
16 16 h 43 min - 9 h 56 min
-
=
6 756 min =
11 7 h 56 min and 2 h 48 min
+
min
17 7 h - 3 h 41 min - 1 h 57 min
-
Time Calculations 2
Often the time until an event occurs is required to be calculated for planning or scheduling.
In column 1 the time in days between two dates is required, count these off to answer them.
With Q 4 and 5 add the separate amounts, the number of days to: get to the end of the month,
the complete month in-between and then the days in the 3rd month.
Column 2, Q 13 to 16, again use the same counting system. But from Q 17 on a new method is
required. The steps are as follows:
•
convert the times to 24 h format (add 12 to hours to times after 1 p.m.)
•
the later time must have more minutes that the earlier time to be able to
subtract, so if the minutes are less take an hour off and add 60 to the minutes column
•
Carry out the subtraction and then write the answer in words.
For example 3.35 p.m. = 1535 h to move an hour across take an hour off the 15 h (15 - 1 = 14)
and add 60 to the minutes (35 + 60 = 95), thus 1495 h. Look at the example before Q 17.
In the third column these require a further step, use the same method but because the times are
on the following day add 24 h for each day that is advanced. For example 5.16 p.m. on the
next day would be 1716 + 24 = 4116 h, then perform the subtraction. See the example at the
top of the column.
With Q 22 and 23 find the sum (addition), then if the minutes are greater than 60 take 60 min
off and add 1 to the hours. For example,
3 h 43 min + 5 h 23 min = 8 h 66 min = 9 h 6 min.
Time Calculations 2
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Find the time difference
in days between the
following dates.
1 8th June and the 23rd June
2 17th September and the 3rd
Find the time difference
in hours between the
following times.
Add 24 hours for every
day that you advance.
These are harder!
13 3 a.m. Tuesday and 11 p.m. Example 7.26 a.m. Friday and
3.52 p.m. Saturday
Tuesday
39 52 -
14 12 p.m. Thursday and
07 26
6 a.m. Friday
September
3 18th March and the 9th
April
Tuesday
+
=
6 02/02/02 and 05/04/02
Example
15 77 -
=
Find the time difference
in weeks between the
following dates.
4 50
+
7 3rd March and 31st March
21 10.08 p.m. Wednesday and
11.27 a.m. and 4.17 p.m.
11 27
+
4.53 a.m. Tuesday
Find the time difference.
Take an hour off and
change it to minutes.
5 15/12/02 and 02/02/03
+
32 h 26 min.
-
a.m. Monday
August
32 26
20 7.17 p.m. Monday and
16 6.15 a.m. Sunday and 4.15
4 16th July and the 28th
1552 = 3952 h (+ 24h)
The difference is
1
15 1 a.m. Monday and 6 p.m.
3.52 p.m. = 1552 h (24h)
4.17 p.m. = 1617 h (24h)
1617 h = 1577 h (1h →min)
1.37 a.m. Friday
-
The difference is
4 h 50 min.
17 12.45 p.m. and 3.16 p.m.
-
3.16 p.m.=
(24h)
(h →min)
Add these times, if the
minutes exceed 60 change
them to hours
22 8 h 47 min and 5 h 39 min
8 20th November and
Christmas day
9 New Years day and 26th
+
The total is
18 7.37 a.m. and 5.28 p.m.
(24h)
-
(h →min)
February
Find the time in years
represented by the
following words.
10 Millennium
11 Century
12 Decade
=
23 6 h 57 min, 3 h 43 min,
2 h 16 min and 4 h 28 min
+
19 4.19 a.m. and 7.53 p.m.
-
(24h)
=
Time Calculations (Calculator)
Time calculations are performed every day, …..how long until lunch?, ….the bus arrives?
When you become a wage earner it is important to be able to check that your hours worked are
correct, but because minutes and hours are in groups of 60 min this is not always straight
forward. But a calculator makes it easy.
The calculator image on the next page shows the DMS key or the 'bubble button' (the key has
an orange border). This key allows you work with hours and minutes.
IMPORTANT YOU MUST ENTER A 0 (THEN DMS KEY) IF DEALING ONLY
WITH MINUTES (NO HOURS).
The calculator will always show the hours, minutes and seconds (we won't be using seconds)
separated by a degree sign (a small raised o) →°.
This reads 8 h 29 min
This reads 4 min
This reads 18 h
In column 1 you are asked to convert the times given in minutes to hours and minutes. This is
done by pressing 0 then the DMS key, then the minutes in the question then the DMS key and
then press = and the answer will be displayed. Example, change 338 min to hours and minutes.
0
3 3 8
=
This reads 5 h 38 min
In the 2nd column times are to be added together, put the first time into the calculator then a +
then the second time in, press =, done. Finding the difference between two times is done by
subtraction, using the same method as above, with one exception YOU MUST CONVERT
P.M. TIME TO 24 H TIME. If they are both a.m. times or both p.m. times no conversion is
necessary, as soon as a question involves both a p.m. and an a.m. time convert the p.m. time to
24 h format. See the example before Q 18.
The time sheet at the bottom of the column 2 requires you to find the time difference for each
day between start and finish times, put them in the spaces provided then add them. Monday is
already done for you, check you get the same answer and don't forget to include it in your
addition.
The 3rd column is using × and ÷ these are done no differently just remember that you are
multiplying by a number not a time and so only use the DMS on the time. The calculation of
an average time (Q 31) can be done in two ways refer to the text box below the calculator
image for the full key stroke method on the next page.
Example of steps in calculating average time
of 11h 12 min, 8 h 57 min and 10 h 9 min
Using Brackets
( 1 1
1 2
+ 8
5 7
Using = to avoid order of operations
1 1
1 2
+ 8
5 7
+ 1 0
+ 1 0
9
9
)
÷ 3 =
= ÷ 3 =
if you get 10 h 6 min for the question above you are correct
Time Calculations (Calculator)
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Rewrite the calculator
displays in h and min.
1
Now use × and ÷ with
these times
Add these times together
using your calculator
15 8 h 27 min + 11 h 36 min
23 3 × 17 min
24 8 × 23 min
16 8 h 19 min + 4 h 48 min +
2
25 7 × 1 h 46 min
2 h 37 min
17 4 h 39 min + 5 h 25 min +
11 h 56 min
3
Find the time difference
between these times that
are on the same day. Use
24 hour time format
26 1 h 55 min ÷ 5
27 19 h 57 min ÷ 9
28 54 min × 6 + 3 h 7 min ÷ 11
+ 1 h 27 min
1853 h in 24 h
4
Example
11.17 a.m. and 6.53 p.m.
1 8
Convert the following
times in min to h and min,
remember to put a zero in
for the hours first.
5 211 min
6 173 min
5 3
1 7
29 8 h 45 min ÷ 5 + 5 min × 17
+ 1 h 31 min
- 1 1
=
7 h 36 min
18 10.30 a.m. and 3.56 p.m.
30 Sean watches 3 movies at a
6 h movie marathon, if two
of the films were 1h 46 min
and 1h 50 min, find the
duration of the other.
19 3.52 a.m. and 11.27 a.m.
20 4.23 a.m. and 6.17 p.m.
31 Julie's time for return travel
to school for 3 days were:
1h 15 min, 56 min and 1h 22
min. Find the average time
for her return trip.
7 245 min
8 727 min
21 1.09 p.m. and 10.57 p.m.
9 341 min
Complete the time sheet
below, then find the total
hours worked.
10 360 min
11 568 min
22
A. Jolie Time Sheet
Day
Start
Mon 9.30 a.m.
12 540 min
13 1331 min
Finish
Hours
4.45 p.m. 7 h 15 min
Tues 10.45 a.m. 2.30 p.m.
h
min
Wed 9.30 a.m.
5.00 p.m.
h
min
Thur 9.30 a.m.
3.45 p.m.
h
min
1.15 p.m. 6.00 p.m.
h
min
Total hours
for the week
h
min
Fri
14 1080 min
Try these time problems
32 It takes Jemma 43 min to
wash a car, and 11 min to
vacuum a car. How long
does she take to wash 5 cars
and vacuum 3 of them?
Jemma has 40 min for lunch,
if she started at 10.25 a.m.
estimate her finish time.
Time taken
Estimated finish time
Regional Time Difference
Australia has 3 time zones:
Eastern Standard Time (EST):
Queensland, New South Wales, Australian Capital
Territory, Victoria and Tasmania.
Central Standard Time (CST):
Northern Territory and South Australia
Western Standard Time (WST):
Western Australia
New Zealand mainland has a single time zone, but the Chatham Islands are +45 min ahead of
NZ mainland time. The Chatham Island’s population are the first in the world to see the
sunrise each new day.
Time is subtracted when moving west, (so it is earlier) and time is added when moving west (it
is later). The amount added/subtracted varies. The arrows at the top and bottom of the map, tell
you how much to add or subtract when you step between adjacent time zones.
Column 1 gives you a time in Perth and asks for the current time in other centres.
Example : Perth local time is 4.45 a.m., the time in Canberra is?
To get from Perth to Canberra add a ½ hr to get to CST then 1½ hr to get from CST to EST, a
total of 2 hours to be added. So the time in Canberra is 4.45 a.m. + 2 hr = 6.45 a.m.
Column 2 selects Wellington (NZ) as the comparison, this time movement west (deduct time)
is involved. Crossing the Tasman Sea to Australia. Remember that the arrows at the top of the
map help you to answer these questions.
The last question in the first 2 columns asks you to write in your location (city or town) and
calculate the local time.
With the 3rd column the time is required to be written and also placed on the analogue clocks.
Remember that a time requires a.m. or p.m. written after it. (unless in 24 hour time format) and
that 12 noon is 12 p.m. and 12 midnight is 12 a.m.
Regional Time Difference
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
- 1½ hr
- ½ hr
Heading
West
- 2 hr
Darwin
NT
Townsville
QLD
Alice Springs
WA
SA
Brisbane
NSW
Adelaide
Perth
ACT
VIC
Auckland
Sydney
Canberra
Wellington
Melbourne
Christchurch
TAS
Hobart
Heading
East
+ 1½ hr
+ ½ hr
If it is 10 a.m. in Perth
calculate the local time in
the cities below (am/pm)
If it is 3 a.m. in Wellington
calculate the local time in
the cities below (am/pm).
1 Sydney
12 Auckland
2 Hobart
13 Sydney
3 Brisbane
14 Perth
4 Auckland
15 Adelaide
+ 2 hr
Show and write the local
time for the city given that
the current time is….
23
11
6 Melbourne
12 1
2
10
2
9
8
3
4
9
8
3
4
7
6
5
Melbourne
7
6
5
Darwin
2.30 p.m.
11
12 1
11
12 1
16 Brisbane
10
2
10
2
17 Hobart
9
8
3
4
9
8
3
4
7 Wellington
18 Darwin
8 Adelaide
19 Christchurch
9 Alice Springs
20 Canberra
10 Townsville
21 Alice Springs
11
22
Your location
11
10
24
5 Darwin
12 1
7
6
5
Perth
6
5
Sydney
8.40 p.m.
25
11
12 1
11
12 1
10
2
10
2
9
8
3
4
9
8
3
4
7
6
5
Brisbane
Your location
7
10.47 p.m.
7
6
5
Auckland
World Time Difference
Time difference in the world is based on GMT, Greenwich Meridian Time. With locations that
are west of GMT time is subtracted. (This means it is earlier than in the U.K.). With locations
east time is added (this means it is later than in the U.K.). GMT is now also referred to as UTC
(Coordinated Universal Time) but this is not as well known.
Column 1 requires use of the map. Cities have been shown with the time ahead (+) or behind
(-) GMT. So to calculate the time in a city in column 1, deduct the time (if -) or add the time
(if +) in hours from/to the GMT time. Note that with New Delhi you add 5½ hours.
With column 2 instead of U.K. local time it is now New York time. If it is 10.00 am in New
York what is the GMT? Once you know the GMT time you can use the same method as the
first column!
Remember that due to cities being up to 12 hours ahead of the U.K. this means than the date in
Auckland (and other cities to the east) could actually be different (ahead) of the U.K. and so
could be the next day. The opposite could be said for cities to the west, the date could be
behind the U.K. and a day earlier.
With column 3, again convert the time in Japan to GMT and then apply the same method as
the previous columns. Remember a time requires a.m. or p.m. written after it, unless in 24
hour time format and that 12 noon is 12 p.m. and 12 midnight is 12 a.m.
World Time Difference
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Anchorage
-9
Moscow +3
Winnipeg
-6
Berlin
+1
London
GMT
New York
-5
Los Angeles
-8
Cairo
+2
New Delhi
+5½
The cities on the
map are ahead (+)
of GMT or behind it
(-) by the number of
hours shown
Buenos Aires
-3
Subtract time when you move West
If it is 10 a.m. in London
calculate the local time in
the cities below (am/pm)
Greenwich Meridian Time
Honolulu
-10
Cape Town
+2
Add time when you move East
Example
2 Sydney
Anchorage
Sydney
+10
Auckland
+12
If it is 10 a.m. on Friday
30/04/02 in New York
calculate the local time
(am/pm), day and date in
the cities below
1 Cairo
Perth
+8
Tokyo
+9
6.00 a.m.
A company in Tokyo
displays times of cities on
digital clocks, show these
times on the clocks below.
Tokyo local time:
AM
PM
3 New York
Friday
30/04/02
17 Sydney Local Time
4 Honolulu
12 New Delhi
PM
5 Buenos Aires
6 Moscow
AM
13 Honolulu
18 London (GMT) Local Time
AM
7 New Delhi
14 Berlin
19 New Delhi Local Time
8 Perth
9 Los Angeles
15 Sydney
10 Winnipeg
AM
PM
20 Los Angeles Local Time
16 Auckland
11 Cape Town
PM
AM
PM
Time - Find A Word
Look for words in the list at the bottom of the grid. Once you find a word cross it off the list.
A letter could be used more than once so don’t colour it in too dark (using a texta for example)
so that you can still read it.
Time Find A Word
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Find the words in the puzzle
from the wordlist.
E
L
B
A
T
E
M
I
T
G
D
D
H
A
M
N
S
N
S
O
C
S
G
N
I
N
E
V
E
D
E
C
N
A
V
D
A
D
G
S
M
P
N
C
S
U
H
T
N
O
M
N
I
U
O
U
A
O
A
T
G
E
N
N
T
D
T
P
R
D
U
R
S
D
L
O
D
S
S
E
A
J
N
Q
A
L
T
E
I
I
L
U
E
E
L
M
I
N
U
T
E
S
V
F
N
A
L
Y
T
E
N
T
E
A
E
N
R
I
F
O
N
E
E
B
G
E
U
N
E
O
M
T
R
E
O
A
T
A
S
R
C
A
R
I
K
D
O
R
R
N
T
S
R
U
O
H
S
T
L
O
L
A
A
E
R
U
N
I
I
E
A
A
R
I
E
P
C
B
N
E
C
E
N
T
U
R
Y
D
R
M
A
P
D
C
T
P
G
A
I
U
P
V
N
O
L
E
E
A
E
F
F
L
L
D
U
D
E
C
A
D
E
A
Y
R
A
WORDLIST
TIME
DURATION
DIFFERENCE
DAY
EVENING
HOURS
DIGITAL
CENTURY
APPOINTMENT
TIMETABLE
ADVANCE
DATE
AFTERNOON
COMPARE
DECADE
ARRIVES
YEAR
ANALOGUE
MINUTES
SCHEDULE
SECOND
MORNING
DEPARTS
MONTH
LATE
7
FREEFALL
MATHEMATICS
FRACTIONS
Fractions of Shapes
Fractions deal with parts of a number or item. The most important thing to remember is:
'take ____ parts from ______' meaning that for the fraction ¾ you take 3 parts from 4.
In column 1 you are asked to find the fraction of the shape that is both shaded and unshaded.
Count the shaded (darkened) sections of the diagram and write this in the top box, then count
the total number of sections (both shaded and unshaded) and put this number in the bottom
box. The same process is done for the unshaded fraction for the question, count the unshaded
sections and put that number in the top box then count the total number of sections (which you
have already done) and put that in the bottom box. An example is below.
There are 4 shaded
(painted) squares
There are 9
squares altogether
There are 5 unshaded
(unpainted) squares
Shaded
Unshaded
4
5
9
9
Column 2 reverses the process, now you are given the fraction and you are asked to shade the
fraction on the diagram. The denominators all match the number of parts in the shape, so that
means you shade the number of parts that match the top number of the fraction.
In Column 3 is the bottom number of the fraction doesn't match the number of sections in the
shape. What you do now is remember to 'take ____ parts from ______". In the example below
there are 20 squares and you are asked to find ¾ of them, this means 'take 3 parts from 4'
which translates to: count off 4 squares and shade 3 of them. Then count off another 4 squares
and shade 3 more and keep going until the you count off the last 4. How many 4's are there in
20? There are 5 lots of 4, and as you are shading 3 out of every 4 that means there is 5 lots of 3
sections to be shaded, so expect to shade 15 squares.
3 out of
3 out of
4 shaded
4 shaded
3 out of
3 out of
3 out of
4 shaded
4 shaded
4 shaded
3
4
Fractions of Shapes
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
1
Shaded
Unshaded
Shade the following
shapes according to
the fraction given
Shade the following
shapes according to
the fraction given
Write the fraction for both
the shaded and unshaded
parts of the shapes below
10
18
1
3
19
3
4
20
1
6
21
2
3
22
6
6
23
5
6
1
2
2
Shaded
Unshaded
11
1
4
3
Shaded
Unshaded
12
3
4
Shaded
Unshaded
4
13
5
Shaded
2
Unshaded
3
6
14
Shaded
Unshaded
6
15
7
Shaded
5
24
6
Unshaded
5
16
8
4
Shaded
Unshaded
20
5
6
25
9
Shaded
Unshaded
17
5
17
100
25
Graphing and Comparing Fractions
This sheet divides a unit length (between 0 and 1) into fractions and asks you to compare the
fractions using the same length interval as a guide. As the fractions are placed directly below
each other, the fractions can be compared more easily.
Column 1 asks you to fill in the missing spaces, the first interval is in two halves, the next in
three thirds and so on. The denominator (the bottom number) stays the same on each line, the
numerator (the top line) counts off starting at 1. Once you have filled the spaces answer the
exercises. If a fraction is to the left of another it is less than it (<), if it is to the right it is
greater than it (>) and if it is in the same position it is equal (=) to it. For some fractions that
are close it may be difficult to tell with your eye, a ruler may need to be used.
With column 2 you are given <, > and = signs. You have to place in the numbers that make the
statement true. Sometimes more than one answer will be possible, if that is the case use the
first fraction that is less than or greater than. Questions 25 through 32 ask you to answer true
or false, again use a ruler if the fractions are difficult to separate with your eye.
Column 3 asks you to arrange the fractions in descending and later ascending order.
Remember that descending starts with a 'd' just like the word 'down'. When you go down you
go from a high position to a lower position, that means that descending order starts with the
highest (largest) fraction and ends with the lowest (smallest) fraction. How do you remember
ascending? "Who cares!" Because if you know descending goes from highest to lowest then
ascending must be the opposite, lowest to highest. It may make it easier if you strike out the
fractions on the top line as you put them in order in the line below.
Remember :
1
=
2
2
=
3
3
=
4
4
=
5
5
=
6
6
=
7
7
Graphing and Comparing Fractions
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Complete the fractions
which divide the
intervals below.
0
1
13
2
0
1
1
15
3
17
0
1
2
4
0
4
19
1
1
5
0
21
1
1
6
0
23
1
1
Now using the intervals
above (and a ruler?),
answer the questions.
25
Fill the box with <, >, or =
to make these true.
3
5
7
9
11
2
<
3
2
5
<
5
3
3
<
4
1
6
>
2
7
=
6
2
16
18
20
22
5
>
5
14
24
7
State if the following are
true or false.
7
1
2
1
3
3
3
1
4
4
1
2
3
6
4
1
7
2
2
4
3
6
2
3
5
7
2
4
6
8
10
12
Arrange these fractions
in descending order.
Find the closest number
that makes these true.
26
5
6
7
7
2
2
27
28
4
3
3
4
5
6 29
6
5
7
6 30
3
1
6
2 31
1
2
2
4 32
1
4
1
2
3
4
2
7
4
4
2
4
4
5
1
3
<
>
=
>
>
=
<
>
4
2
3
3
3
5
1
3
>
=
>
<
=
5
7
4
33
2
7
Rewrite
2
2
34
1
5
Rewrite
4
1
3
Rewrite
=
4
1
Rewrite
7
in order
5
7
37
6
4
3
6
4
6
4
6
,
7
,
1
6
,
4
5
2
,
6
7
,
4
6
,
3
5
,
1
,
,
,
,
2
4
,
2
3
,
38
4
6
Rewrite
Rewrite
in order
3
5
,
1
4
,
,
6
7
,
,
,
6
7
,
1
2
,
in order
39
,
,
in order
7
2
3
Rewrite
3
3
,
3
Arrange these fractions
in ascending order.
7
1
3
,
3
4
,
,
,
,
in order
36
7
,
in order
35
5
,
in order
1
,
,
3
,
,
1
2
,
3
7
,
1
,
,
1
5
,
3
5
,
,
2
4
Types of Fractions
A fraction is composed of one number ‘over’ or ‘on top of’ another. The number on top is
called the numerator, the number on the bottom is called the denominator. How do you
remember these? Remember that denominator starts with a 'd' as does the word 'down'. So you
should remember that the denominator is down below and the numerator is up on top.
Column 1 asks you to write the fraction that has been spelt out then pick the numerator and the
denominator and put these in the boxes. Note that the boxes change their position and so don't
write the numerator in the denominator box. Questions 9 through 13 are the reverse, this tests
your spelling. You can look at the spelling in the questions above to assist you.
Column 2 introduces proper fractions, improper fractions and mixed numerals (or mixed
numbers). A proper fraction is 'the way you would expect a fraction to be' with the numerator
smaller than the denominator, (the number on top smaller than the number below it). An
improper fraction is different because the numerator is larger than the denominator, and a
mixed numeral has a whole number and a proper fraction together, so it's mixed up!
Note that because improper fractions have a larger top number they will have a value that is
greater than 1. Questions 24, 25 and 26 ask you to build fractions using the numbers 2, 4 and
7. The idea being that with the mixed numbers you use 3 different numbers not one number
twice.
The third column has some worded questions about fractions that test your knowledge of
them, then the last 3 questions ask you to sort through the fractions at the bottom of the
column and rewrite them below the type of fraction that they are.
Numerator
(top number)
2
Denominator
(bottom number)
5
Proper Fraction
(smaller numerator
than denominator)
2
Improper Fraction
(larger numerator
than denominator)
7
Mixed Numeral
(whole number and a
proper fraction)
9
5
3
8
10
Types of Fractions
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Write the fraction given in
words as numbers and put
the numerator and the
denominator in the spaces
Example
six sevenths
6
7
Describe the fractions
below as either proper,
improper or mixed
num
denom
6
7
denom
num
1 a half
5
17
num
denom
3 three eighths
18
denom
num
4 seven tenths
20
denom
27 This fraction has a
denominator larger than its
numerator
28 This fraction can have a
value greater than 1
89
43
29 This has a proper fraction
and a whole number.
34
8
10
num
5 ten twelfths
num
1
2
88
19 6
denom
30 An answer of 1 occurs
when the denominator
equals this.
11
7
5
6 five sixths
21 12 6
denom
num
22
7 ten thirds
num
denom
8 six fourths
Now write the fractions
given below in words
11
11
15 4 6
denom
2 two quarters
10
3
16 7
num
9
14
Name the type or part of a
fraction that is being
described below.
5
23
From the fractions below
select the:
12
13
56
31 Proper fractions
13
Use the numbers 2, 4, or 7
to construct:
24 3 proper fractions
32 Improper fractions
25 3 improper fractions
33 Mixed numerals
7
3
12
5
20
12 A fraction has a numerator
of 5 and a denominator of 8
3
26 3 mixed numerals
5
13 A fraction has a numerator
of 9 and a denominator of 4
2
55
8
9
10
56
3
7
67
63
12
5
11
3
13
19
6
2
5
17
12
Converting between Mixed Numerals
and Improper Fractions
Changing between mixed numerals and improper fractions is a skill that must be developed for
operations with fractions (+, -, ×, ÷). The first column is an introduction to the process of
changing mixed numerals to improper fractions. Questions 1 to 3 are connected and then
Questions 4 to 6 are connected. This is to help show you how the action of converting
fractions is done. It is important that you realise that 1 = 2/2 = 3/3 = 4/4 = 5/5 = 6/6 etc.
Questions 7 through 9 are a graphical representation of mixed numerals, with each separate
shape being equal to 1. This means that if the shape is split into 3 parts then it is in 1/3 's. In the
example below there are two shapes fully shaded that means 2, then there are 5 triangles
shaded in the third shape which has 6 parts, that means 5/6 . So we have 25/6 as the mixed
numeral. To write it as an improper fraction look at the shape, it is in 6 parts, so in all we
have ?/6 . Count all the parts up, the first shape has 6 parts, the next another 6, that's 12, then
the 5 makes 17. So the improper fraction would be 17/6 .
6
= 1
6
6
= 1
6
5
6
Mixed Numeral = 2
5
6
Improper Fraction =
17
6
The entire 2nd column is about converting mixed numerals to improper fractions. The top of
the column shows you how this is done. In words the process is as follows:
•
write the denominator for the mixed numeral the same as the denominator in the
improper fraction
•
multiply the denominator of the mixed numeral by the whole number then add it to the
numerator
•
write this number as the numerator, and you're done
Converting improper to mixed is just like dividing, column 3 shows the method with an
example, but in words:
•
write the same denominator in the improper fraction as given in the mixed number
•
divide the numerator by the denominator (it must 'go into it' as it is an improper fraction)
the answer is the whole number in the mixed numeral, the remainder becomes the
numerator.
Converting between Mixed Numerals and Improper Fractions
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Try these, remember that
fractions are equal parts
of a whole number
1 How many ½ 's are there
in 1?
Fill in the missing spaces
then find the improper
fraction for these.
Example
+
10
2 How many ½ 's are there
in a ½?
×
10
3 So how many ½ 's are there
in 1½?
4 How many ¼ 's are there
in 1?
5 How many ¼ 's are there
in ¾ ?
6 So how many ¼ 's are there
in 1¾?
×
11
4
3
=
5
×
12
5
5
=
7
×
13
6
7
=
9
×
3
×
+
7
Improper
+
=
32 What is the
remainder?
+
=
+
=
+
=
Improper
9
Now try converting
improper fractions to
mixed numerals.
35 5
=
3
36 7
=
2
38 22
=
5
17 1 6 =
18 4 3 =
39 17
=
2
40 21
=
4
19 8 2 =
20 6 2 =
41 19
=
6
42 25
=
8
21 4 9 =
22 8 2 =
43 33
=
10
44 38
=
9
23 3 5 =
24 6 3 =
45 41
=
4
46 37
=
8
25 2 6 =
26 4 3 =
47 47
=
12
48 60
=
11
27 2 5 =
28 3 7 =
49 55
=
9
50 60
=
19
7
13
Improper
34 What is the
remainder?
19
=
4
37 11
=
4
4
6
Mixed
33 How many 4 's
go into 19?
16 3 3 =
10
Mixed
9
=
2
15 2 3 =
3
8
30 What is the
remainder?
8
=
3
31 How many 2 's
go into 9?
5
2
Mixed
29 How many 3 's
go into 8?
=
Now change these to
improper fractions without working out spaces
1
4
2
9
92
9 × 10 + 2
=
9
9
2
=
3
14 10 8 =
Write the shaded area
as a mixed numeral and
an improper fraction
Example
2
2
9
Follow the arrows
and put your answer
in the square
16
4
5
7
8
7
20
3
Changing Between Mixed Numerals
and Improper Fractions (Calculator)
This sheet is designed for you to learn how to change between mixed numerals and improper
fractions using your calculator. A calculator separates the numerator and the denominator by a
reversed 'L' (or sometimes an ‘r’), for mixed numerals the whole number is also separated by
the 'L' (or ‘r’). See example below.
whole number
numerator
3
6
17
5
18
denominator
denominator
numerator
The first column starts with the way your calculator displays fractions. If there are 2 numbers
the first number is the numerator (number on top) the second number is the denominator
(number on the bottom). These numbers are separated by the "reversed L". Don’t write
fractions the calculator way, write the fractions separated by a line (as above). From Q. 6 on,
you are asked to change mixed numerals to improper fractions. The method required is
described on the next sheet, in words - type the fraction into the calculator, press =, then press
shift and the fraction button.
Column 2 reverses the process and asks you to change improper fractions to mixed numerals,
to answer these put the fraction in the calculator and press =, the mixed numeral will be
displayed. Question 46 through 53 asks you to repeat the same process, if you use the ÷ sign
then you will get a decimal answer, press the fraction key and it will change it to a fraction. Or
a quicker step is to use the fraction key instead of the ÷ sign this will give the answer as a
fraction straight away.
Note: These are all the same:
8
3
3
8
3÷8
The last column asks you to compare the improper fractions with the mixed numerals. Put the
improper fraction into your calculator and press the ‘equals’ key. If the fraction displayed on
the screen is the same as the answer supplied write 'true', if it isn't, write ‘false’. At the bottom
of the column you are asked to show the pairs that ‘match’ (an improper and a mixed number)
by using pairs of shapes (a smaller shape for the improper fractions, larger one for mixed).
Example of how to convert Mixed numerals to Improper Fractions
3
6
7
=
3
a bc
6
?
Don't forget the 'equals' sign
SHIFT
if you get
a bc
7 =
a bc
27
as your answer you have done it correctly
7
Changing Between Mixed Numerals and Improper Fractions (Calculator)
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Change these improper
fractions to mixed
numerals
Write the fraction
represented by the
calculator screens
Change improper to
mixed to answer true
of false to these
8
1
2
1
26 5
=
2
27 7
=
4
48
2
28 3
29 8
49 4
43
3
=
8
8
50 7
51
2
=
7
7
51 8
75
5
=
9
9
2
3
30 16
7
4
5
Change these mixed
numbers to improper
fractions
6 3
8
1
=
2
1
=
2
4
2
=
10 1
5
1
=
3
7 1
9
1
=
1
2
5
=
11 1
12
=
5
31 24
=
7
34 13
=
2
35 19
=
7
36 13
=
2
37 28
=
3
54 1
38 15
=
4
39 17
=
10
55
40 13
=
6
41 53
=
5
56
Write the answers as
mixed numerals
13 2
4
=
7
42 11 ÷ 7
14 2
5
=
6
15 3
1
=
4
43 109 ÷ 10 =
16 1
3
=
4
17 12
1
=
7
20 3
19 3
1
=
8
21 9
4
=
22 4
5
24 10
2
=
9
5
=
11
25 11
=
3
=
4
11
52
6
24
53
7
5
6
= 3
4
7
15
7
=
8
8
17
7
13
11
57 4
= 1
= 2
3
7
= 1
3
11
15
1
=
3
3
Using circles, squares,
triangles or other shapes
draw the same shape
around matching pairs
58
44 3
125
45 20 ÷ 3
=
46 7
321
=
47 47 ÷ 5
=
5
7
4
=
2
=
3
1
=
23 1
10
=
33 16
=
3
2
=
3
1
=
3
2
32 23
=
5
12 2
18 7
=
= 4
16
3
1
5
3
33
7
22
7
6
3
2
3
1
7
5
22
3
31
7
3
7
4
17
3
2
3
20
3
7
1
3
Simplifying Fractions
Simplifying a fraction is the same as expressing a fraction in its simplest form, the fraction you
are being given is expressed with smaller numbers. The process of simplifying is finding the
largest number that divides into the numerator and the denominator and dividing them both by
it. The largest number is the HCF the Highest Common Factor.
In column 1 questions 1 through 20 ask you to find the HCF of two numbers. Look at the two
numbers and find the largest number that goes into both of them. Questions 21 through 26 use
the HCF to simplify the fraction. In the example (above question 21) the fraction 3/12 is to be
simplified. The HCF is found to be 3, the numerator and denominator are then divided by 3
(the HCF) and the answer placed in the fraction boxes.
Column 2 is all simplification exercises. Find the HCF of the numerator and denominator and
divide through. When you complete each question look at your answer and see whether it can
be further simplified, if it can be, simplify it again or look at the question again to get the HCF
that you have missed and try again from the start.
Column 3 is the same process except the numbers used are larger, make sure that you check
your answer to see whether it can be further simplified. Question 79 through 86 asks you to
draw a line between the fraction and the simplified fraction. After you calculate the simplified
fraction draw a line between it and the question, joining the dots.
Simplifying Fractions
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Find the HCF, the largest
number that goes into
both of these 2 numbers.
Now simplify these. Look
at your answer, if you can
further simplify it you
missed the HCF.
1 3, 9
2 6, 9
3 2, 4
4 8, 10
5 5, 10
6 10, 25
7 8, 16
8 6, 16
9 9, 30
10 12, 30
11 12, 24
12 20, 12
13 4, 14
14 7, 14
15 22, 8
16 50, 15
17 32, 16
18 30, 18
19 14, 42
3
12
21
22
23
24
25
26
3
9
6
18
8
20
10
25
12
30
24
32
HCF
3
HCF
3÷
12 ÷ 3
6÷
18 ÷
HCF
8÷
10 ÷
25 ÷
HCF
HCF
59
120
=
300
60
180
=
200
61
62
12
=
18
90
=
210
140
=
210
63
64
18
=
30
75
=
150
400
=
500
65
110
=
440
66
204
=
612
9
=
15
32
15
=
20
34
35
21
=
35
36
12
=
15
67
68
38
15
=
40
125
=
500
37
16
=
20
80
=
150
69
35
=
120
70
195
=
200
39
16
=
28
40
30
=
50
71
180
=
240
72
150
=
360
41
9
=
12
42
20
=
65
43
19
=
38
44
18
=
54
45
16
=
40
46
47
35
=
42
48
33
20 ÷
HCF
350
=
500
30
31
1
4
58
6
=
9
29
3÷
9÷
HCF
3
100
=
200
28
20 18, 42
Example
57
5
=
15
27
Now lets apply this skill
to simplifying fractions
Now try larger
numbers
2
49
=
50
4
=
8
5
=
20
Draw lines between
fractions and their
simplified match
73
1
2
A
45
=
60
35
40
74
10
12
4
5
B
8
=
12
75
12
24
5
6
C
76
15
45
3
4
D
77
20
28
7
8
E
78
9
12
5
7
F
12
50
=
26
51
15
=
60
52
18
=
27
30 ÷
53
63
=
81
54
42
=
50
79
8
12
2
3
G
24 ÷
56
55
=
64
12
56
=
60
80
60
75
1
3
H
12 ÷
32 ÷
Equivalent and Comparing Fractions
An equivalent fraction is a fraction that is equal to another fraction but has a different
denominator and numerator. For example if you scored 9/10 in a test it would be equivalent to
scoring 18/20 or 90/100 . Equivalent fractions are calculated by multiplying (or dividing) the
numerator and the denominator by the same number. To change 9/10 to 18/20 we multiplied the
top and bottom numbers in the fraction by 2. To change 9/10 to 90/100 we multiplied the top and
bottom numbers by 10.
Column 1 starts by asking you to find two fractions that represent the shaded area. The first
fraction will be found by counting the shaded parts and making that the numerator, and
counting the total number of parts and making that number the denominator. Then either
multiply both of these numbers by any other number, or simplify to find an equivalent fraction.
A hint, do you notice that the shading goes all the way down the columns of the shape? Then
if you cover the rows of the shape except for top row you will get an equivalent fraction. Note
that if the shading is scattered stick to the multiply or divide method.
The next part of column 1 asks you to find equivalent fractions to the one given. Look at the
example below to the left, we are given 2/3 so the denominator is 3. The first question has a
denominator of 6, what do you multiply 3 by to get 6? The answer is 2, so multiply the
numerator by 2 and you get 2 × 2 = 4 so the first answer is 4. The 2nd question has a
denominator of 18, what do you multiply 3 by to get 18? The answer is 6, so multiply the
numerator by 6 and you get 2 × 6 = 12, so the 2nd answer is 12. The same method again will
give you × 10 and thus 20 for the last fraction.
× 10
÷ 10
×6
×2
4
12
20
2
=
=
=
3
6
18
30
÷5
÷2
10
4
2
20
=
=
=
30
15
6
3
Column 2 uses the same theory but with division. To make smaller equivalent fractions divide
the top and bottom numbers in the fraction by another number. Of course when you
divide you must find a number than can divide into both numbers exactly without a remainder.
Questions 16 to 22 give you a fraction and a different denominator to make an equivalent
fraction. Find the number to divide by across the bottom of the fraction and then divide across
the top by the same number. You are then given the numerator of equivalent fractions and
asked to find the denominator. Exactly the same method is used. YOU NEVER ADD OR
SUBTRACT TO GET EQUIVALENT FRACTIONS.
Column 3 asks you to compare fractions, look at the fraction at the right of the empty box,
look at its denominator and change the fraction on the left to match that denominator, then
compare and fill in the box with < , > or =. Remember point the arrow at the smallest fraction
(number). See the example at the top of column 3. Questions 41 to 48 ask you to check if the
fractions are equivalent, check the denominators first and find the number that has been used
to divide or multiply to get the other fraction, then apply the same number to the top and see if
they equal each other. Yes? ……..True, No? ……….False.
Equivalent and Comparing Fractions
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Give two fractions that
represent the shaded part
of each shape
16
1
18
2
20
21
3
22
23
25
9
20
=
50
10
17
19
6
=
18
6
18
=
42
7
20
=
=
=
=
60 6
30 15 12
12
=
=
=
=
24 12
6
2
4
12
=
=
=
=
36 9
6
18
3
1
=
3
12
6
4
=
10
50
8
9
=
10
90
10
6
2
=
3
4
=
12
2
24
26
15
3
=
5
5
=
6
20
6
2
4
12 24
27
=
=
=
=
18
Find the equivalent
fraction with the
given denominator
7
12
=
20
10
Example
4
12
=
9
27
2
=
5
10
3
=
6
18
3
=
20
40
28
29
8
2
40 12
4
=
=
=
=
6
4
16
2
8 24
=
=
=
=
12
Now they are all mixed up
complete for the fractions
3
30
6
30
=
=
=
=
1
10
20
30
12
=
2
50
2
18
6
31
=
=
=
=
1
13
=
=
=
=
9 90
18
2 12 20 18 48
2
24
8 16
32
=
=
=
=
2
14
=
=
=
=
24
12
3 12 30
9
15
30
15 1
33
3
=
=
=
=
15
=
=
=
=
45
15
9
4 20 12 24 32
4
11
=
5
30
10
27
>
34
1
=
2
4
3
4
35
3
=
5
10
6
10
36
8
=
90
45
4
45
37
8
=
12
30
48
38
3
=
7
12
28
39
3
=
4
70
80
40
63
=
70
8
10
Now find the denominator
given the numerator
4
5
Use the equivalent fraction
to compare the fractions.
Use <, > or = to make true
Now use division to find
the equivalent fraction
Test these to see if
these are equivalent.
Answer true or false
41
1
2
=
6
12
42
3
5
=
6
15
43
9
=
30
3
10
44
2
3
=
14
21
45
9
12
=
3
4
=
5
8
46 15
40
47
2
5
=
10
25
48
5
9
=
45
90
Comparing Fractions (Calculator)
Using a calculator to compare fractions is done by changing the fractions to decimals and
comparing the decimals instead. If all rounded to the same decimal place, the decimals can be
easily compared, e.g. 0.308, 0.471, 0.235. It is easy to see the largest and smallest decimal, or
if they are equal, if you look at the numbers behind the decimal point.
Column 1 asks you to convert the fractions to decimals, the calculator image on the next page
shows you how to do this. If you have a decimal 0.100 then write it as 0.100 don't change it to
0.1 as this may lead you to make an error when comparing other numbers that are 3 d.p. (You
may look at it as 1 instead of 100).
The rest of the sheet is attempted the same way, by finding the decimal for the fraction then
comparing decimals. In the example below, two fractions are being compared, the method is:
•
Find the decimals for both
•
Compare decimals and put a < or > or = sign in between the decimals
•
If this sign matches the one above it then write true, if not, write false.
4
5
0.800
>
<
7
8
0.875
False
In the 2nd column, from Q. 31 on, use the same method only this time just write the sign into
the box rather than stating true/false.
The third column asks you to compare fractions and arrange them in descending order, use the
decimal system then write the fractions again in descending order. In a test the most common
mistake is rewriting the decimals in descending order, the question asks for the fractions to be
written, not the decimals.
How do you remember the difference between < and >? Imagine they are arrowheads and
point them at the smallest number. E.g. 5 < 6 and 15 > 10, the arrow points to the smallest
one.
Descending order? Remember going down.
Example of how to convert fractions to decimals
6
7
=
?
Don't forget the 'equals' sign
Method 1: Using Fraction Key
6 a bc 7 = a bc
Method 2: Using ÷ Key
6 ÷ 7 =
if you get 0.857 (rounded) you have answered it correctly
Comparing Fractions (Calculator)
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Convert these to decimals
round to 3 d.p.
1
4
1
10
2
3
2
3
4
3
8
5
3
4
6
5
6
7
6
13
8
8
19
9
3
7
24
10
27
11
17
29
12
7
12
13
73
80
14
33
43
1
5
8
<
7
14
36
4
9
17
45
26
8
11
>
5
7
37
4
19
14
60
27
13
17
<
14
19
38
11
18
26
37
28
6
81
<
4
63
39
3
8
21
56
29
35
39
>
60
70
40
12
13
16
19
30
88
93
<
17
19
96
99
17
42
16
17 175
305
18 212
636
19 572
911
20 101
909
15
25
Use the same method
only fill in <, > or =
2
3
22
41
3
7 ,
2
3
3
5
32
3
6
18
36
42
4
10
27
60
Rewrite in
31
Write the decimal (3 d.p)
under the fraction and
then answer true or false
21
Use decimals to arrange
these in descending order
2
3 ,
10
13 ,
,
,
23
60 ,
13
40 ,
,
,
9
10 ,
29
34 ,
,
,
17
40
Rewrite in
Descending
order
39
95 ,
<
3
4
2
5
33
>
3
7
23
8
9
<
9
10
34
3
7
4
9
43
24
6
7
35
8
12
2
3
Rewrite in
>
8
10
Descending
order
13
15 ,
Descending
order
,
1
4
,
18
21
,
Adding and Subtracting Fractions
(Same Denominator)
Fractions like whole numbers can be added and subtracted. The fractions on this sheet all have
the same denominator so they can be added without finding equivalent fractions.
With column 1 the fractions are added and subtracted to give answers that can't be simplified.
When you add fractions you add the numerator only, that is, the top numbers. The same with
subtraction, you subtract the top numbers not the denominators (bottom numbers). In the
example below an extra working step has been included. The top numbers are added but the
bottom number is unchanged.
9
7
+
=
21
21
9+7
=
21
16
21
Column 2 involves the same style of exercise except this time the fractions can be simplified.
Complete the addition or subtraction then look at your answer. Find the HCF, that is, the
largest number that divides into both the top and bottom numbers. Then divide through by that
number. Use canceling if you wish, or otherwise just write the simplified answer in the next
fraction boxes. Remember that once you have simplified the answer, look at it again to ensure
you can't reduce the fractions any further by dividing again. If you can further simplify that
means that you missed the Highest Common Factor, look back and see whether you can find
the HCF.
7
4
=
15
15
3
15
=
1
5
Column 3 is one extra step more, like the second column you are given exercises that will
result in answers that can be simplified. But this time the answers will be improper fractions.
The question asks for mixed numeral answers so you then have to convert them to mixed
numbers. You should remember how to do this, if not, look at the example of converting to
mixed numbers below. Note that you might like to convert to mixed numbers then simplify
afterwards, both methods will result in the same answer.
How many 3 's
go into 8?
What is the
remainder?
8
2
= 2
3
3
How many 2 's
go into 9?
What is the
remainder?
9
1
= 4
2
2
Adding/Subtracting Fractions (Same Denominator)
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
From now on you
need to simplify
Add or subtract
these fractions
Example
1
2
3
4
5
6
7
8
9
10
11
1
2
+
=
10
10
5
8
-
1
5
+
4
7
-
2
8
=
2
5
=
3
7
=
9
3
=
10
10
16
17
18
2
7
+
=
11
11
19
1
9
+
20
7
5
-
8
6
-
4
9
=
4
5
=
3
6
=
21
22
17
6
=
12
12
23
2
7
+
=
13
13
24
5
4
+
=
10
10
25
12 14
6
9
6
=
26
12
13 15
+
=
32
32
27
14 13
8
28
15
6
8
=
9
4
+
=
20
20
29
3
5
6
10
=
3
1
=
4
4
=
1
3
=
+
6
6
=
5
1
=
+
10
10
=
10
7
=
9
9
=
7
5
=
12
12
=
3
5
=
+
8
8
=
13
7
=
20
20
=
4
3
=
+
14
14
=
7
1
=
9
9
=
8
3
=
10
10
=
13
7
=
+
24
24
=
5
4
=
+
18
18
=
13
3
=
15
15
=
7
4
=
30
30
=
3
5
Add or subtract, simplify
then convert to a mixed
number if possible
Example
9
9
=
+
10
10
Add
9
5
Change to
mixed numeral
30
18
10
Simplify
9
=
5
4
= 1
5
9
5
=
+
6
6
=
=
31
9
7
=
+
12
12
=
=
32
29
5
=
12
12
=
33
17
18
=
+
10
10
=
=
34
44
11
=
9
9
=
=
35
50
11
=
18
18
=
=
36
50
10
=
+
8
8
=
=
Adding and Subtracting Mixed Numerals
(Same denominator)
Adding and subtracting mixed numerals will be challenging for some students, follow the
examples below, then try the exercises.
The entire sheet is done by the one method, you are given 2 mixed numerals and you have to
add or subtract them. The method is:
•
change the first mixed number into an improper fraction and rewrite it in the boxes below
•
Repeat for the 2nd mixed numeral
•
Add the numerators (the top numbers) and write the answer in the top box then write the
denominator straight into the bottom box, don’t add or subtract denominators
•
Find the HCF, the largest number that divides into both the top and bottom numbers
•
Divide through by this number and write the answer in the boxes on the 3rd line, in other
words simplify the fraction
•
Look at your answer, can it be further simplified? If it can try and simplify it and try to
get the HCF that you missed in the first simplification.
•
Change your answer to a mixed number
You are done! It’s a long process and a mistake can be made at many of the stages so be
careful and take your time. Note that you can drop a step, the simplifying, if you use
canceling.
4
Improper:
12 × 4 + 3 = 51
=
3
7
- 2
12
12
51
12
-
31
12
Improper:
12 × 2 + 7 = 31
=
20
12
HCF = 4
20 ÷ 4 = 5
12 ÷ 4 = 3
=
5
3
= 1
5÷3=1r2
2
3
Adding/Subtracting Mixed Numerals (same denominator)
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Solve these, write your
answer as a mixed numeral
The whole sheet uses the same
method. Convert to an improper
fraction , add or subtract, simplify
then change back to mixed numbers.
3
1
1
4
- 1
Improper
=
3
4
Answer
Improper
-
=
=
=
Simplify
4
2
3
3
3
8
=
2
4
5
6
=
6
5
=
1
4
=
=
=
7
8
=
=
=
5
6
1
6
=
3
=
=
=
3
4
3
- 1
1
9
=
=
=
- 1
2
12
-
5
6
=
=
=
7
9
+ 3
12
12
=
5
13
-
5
=
1
=
=
=
=
+ 2
3
8
3
4
3
4
=
=
=
=
8
12
2
=
=
=
7
8
1
-
=
=
=
7
7
+ 2
10
10
3
+
=
=
=
=
=
=
+ 3
3
4
+
=
=
=
-
=
=
=
-
=
=
=
6
9
+ 3
10
10
=
17
=
5
11
16
16
=
16
+
3
8
- 1
20
20
=
15
-
- 1
3
14
+
5
12
=
11
-
3
4
=
10
+
- 2
6
7
9
+
+ 3
=
8
-
+ 2
4
9
Rewrite as mixed
3
7
- 1
10
10
=
3
6
4
9
+
=
=
=
+ 1
8
9
+
=
=
=
Adding/Subtracting Fractions
(Different Denominator)
The main requirement for adding fractions with different denominators is that you know how
to get an equivalent fraction, this means multiplying the fraction's numerator and denominator
(top and bottom number) by a number.
In column 1 you are asked in questions 1 through 7 to find equivalent fractions to the one
given. Look at the example below, we are given 2/3 so the denominator is 3. The first question
has a denominator of 6, what do you multiply 3 by to get 6? The answer is 2, so multiply the
numerator by 2 and you get 2 × 2 = 4 so the first answer is 4. The 2nd question has a
denominator of 18, what do you multiply 3 by to get 18? The answer is 6, so multiply the
numerator by 6 and you get 2 × 6 = 12, so the 2nd answer is 12. The same method again will
give you 20 for the last question.
× 10
×6
×2
4
12
20
2
=
=
=
3
6
18
30
Questions 8 through 10 are additions with different denominators, you have to find the LCM
(Lowest Common Multiple), in other words the smallest number that both numbers divide
into. These three have the LCM found for you with the denominator typed in. In
question 8 only one fraction needs to be changed, (because 2 and 8 both divide into 8),
questions 9 and 10 require both fractions to be changed. Note that you don't just multiply the
denominators together to get the LCM, an example would be 6 & 10 the LCM is 30 not
6 × 10 = 60.
The 2nd column are additions which will give a proper fraction answer, the last two questions
will need to be simplified. In Column 3 the answer will be an improper fraction and so an
extra step is required to change it to a mixed number. An example is below.
Example
Equivalent Fractions
40
6
8
2
=
3
5
15
15
=
34
15
Total
= 2
4
15
Mixed
Adding/Subtracting Fractions (Different Denominator)
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Find the numerator for
these fractions to make
them equivalent fractions.
1
1
=
=
=
2
4
8
20
2
1
=
=
=
3
6
12
18
3
2
=
=
=
3
9
12
21
Where possible simplify
11
1
3
+
2
10
=
+
12
1
1
2
3
-
13
2
1
+
3
4
=
+
14
7
1
10
4
7 Change these to twelfths
=
-
1
=
2
15
3
=
=
=
5
10
15
25
5
4
=
=
=
6
3
30
24
Equivalent Fractions
Total
Equivalent Fractions
=
4
The answer to these will
be a mixed numeral
=
Simplify
19
=
=
1
=
, 4
3
=
, 4
2
=
, 3
3
=
, 4
Add/subtract these. The
denominator is given for
the equivalent fractions
8
=
20
=
21
=
+
1
1
+
2
3
=
+
10
=
6
6
=
=
3
1
5
3
15
-
22
15
=
-
17
2
7
+
5
20
=
+
18
7
1
10
6
=
-
=
=
12
2
=
7
3
=
=
=
1
2
4
9
=
+
10
4
=
3
5
7
1
+
=
9
2
=
8
9
=
Mixed
4
5
+
=
10
6
=
3
1
+
5
6
+
=
=
23
16
1
3
+
2
8
8
=
+
Total
6 Change these to eighths
1
=
2
3
1
+
=
5
2
+
=
=
24
=
=
=
25
=
=
22
1
=
15
4
=
2
8
+
=
3
11
=
+
=
Adding and Subtracting Mixed Numerals
(Different Denominators)
This is a challenging sheet and some students may experience difficulty with it. The method
for each question is the same and is:
•
Change both fractions to improper fractions
•
Find the LCM (Lowest Common Multiple) of the 2 denominators
•
Create equivalent fractions with the matching denominators
•
Add or subtract, you will get an improper fraction as the answer
•
Convert the improper fraction to a mixed numeral, and you are done.
Look at the steps below in the example, the first question has the denominators filled in for
you to get started.
Example
3
Because 3 × 5 = 15
Equivalent Fractions
11 × 5 = 55
Because 5 × 3 = 15
9 × 3 = 27
=
=
11
3
55
15
2
3
-
- 1
9
5
27
15
Create equivalent
fractions LCM = 15
4
5
Change to improper
fractions
=
28
15
Total
= 1
13
15
Write as mixed
numeral
Adding and Subtracting Mixed Numerals (Different Denominators)
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Add or subtract these
fractions, give your answer
as a mixed number.
6
The whole sheet uses the same
method. Convert to an improper
fraction , create equivalent fractions,
add or subtract, simplify if required
then change back to mixed numbers.
1
=
=
2
3
12
2
3
+
+
+ 2
4
12
3
4
=
Create equivalent
fractions
2
=
2
4
5
+ 3
Total
=
+
+
=
3
3
2
4
2
- 1
5
=
-
=
-
=
=
-
=
-
5
4
5
9
=
-
=
-
=
- 2
=
=
+
=
3
4
+
=
+
8
4
3
4
=
-
=
5
5
6
=
+
=
+
4
2
5
=
-
=
-
11
=
2
=
10
2
3
=
+
9
7
4
- 1
3
10
5
=
12
Write as mixed
numeral
1
2
=
4
12
1
5
=
7
Change to improper
fractions
1
3
3
4
=
-
=
-
+ 1
2
7
=
+ 3
=
-
=
=
+
6
3
7
=
-
=
4
1
4
=
-
=
3
2
7
=
+
=
+
5
- 3
2
5
=
=
3
4
+ 2
10
5
+
17
=
4
=
16
4
5
=
=
15
1
2
=
- 1
=
1
3
-
14
1
4
=
- 2
=
5
=
13
7
10
=
+ 2
=
2
12
=
- 2
12
1
2
=
+
=
+
=
- 4
1
2
=
- 2
=
1
3
=
+ 2
=
5
6
=
+ 1
=
=
2
3
=
=
Multiplying Fractions
When fractions are multiplied, multiply straight across the top (the numerators) and straight
across the bottom (the denominators). Unlike addition and subtraction the denominators are
included in the operation.
Column 1 asks you first to multiply a whole number and a fraction. A whole number has no
denominator so just multiply the number and the numerator (top number) of the fraction. A
whole number written as a fraction is just the number over 1. For example 3 = 3/1 and
136 = 136/1. An example is below.
3
×
19
3
=
3×3=9
9
19
From question 4 on, the column is all done the same way. The top numbers multiplied together
and the denominators multiplied together. It doesn't matter if there are two fractions or six
fractions multiplied, the process will still be the same. There will also be whole numbers
mixed in, make sure you multiply the whole number only with the numerators (top numbers),
not the denominator. Note that none of these questions can be simplified.
Column 2 is the same as column 1 only the fraction answer can be simplified.
Column 3 is a further extension, simplify the fractions and convert them to mixed or whole
numbers.
2×4=8
Total
2
3
×
4
7
=
8
21
5
6
×
3
4
=
15
24
=
5
8
4
5
×
10
=
3
40
15
= 2
3 × 7 = 21
HCF = 3
so ÷ top and
bottom by 3
Simplify
8
=
3
2
3
Mixed Numeral
Multiplying Fractions
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Answer these, they
will simplify.
Multiply these
1
2
3
Total
1
3
×
2
=
17 3
8
×
2
=
=
2
7
×
3
=
18 4
×
15
3
=
=
7
2
×
=
15
4
2
3
5
1
2
6
×
1
5
×
3
4
5
7
7
4
9
19
4
=
20 2
3
=
21 5
6
×
2
3
×
8
3
×
=
20
33
=
×
×
4
7
=
22 3
×
10
5
8
=
=
1
4
=
23 5
9
×
3
4
=
=
4
×
11
1
6
=
24 3
8
×
4
6
=
=
9
3
5
×
1
4
=
25 7
×
12
2
5
=
=
10
5
6
×
5
7
=
26 13
×
14
2
3
=
=
1
2
×
1
3
×
27 7
×
10
5
9
=
=
3
5
×
1
4
×
2
3
12
1
5
=
1
2
=
×
2
=
×
4
5
13
5
9
14
3
×
11
4
15
2
5
×
2
3
×
2
=
5
×
3
×
47
2
=
16
×
=
=
4
5
×
6
2
=
=
=
=
=
35
8
15
=
×
5
6
=
=
36
10
13
=
×
7
5
=
=
37
20
5
×
3
4
=
=
=
4
3
28 2
×
×
=
3
5
4
=
4
5
1
×
×
=
5
6
3
=
30
4
3
1
×
×
=
10
4
2
=
31
4
5
× 2 ×
=
7
8
32
3
5
×
× 3 =
10
6
29
=
Mixed Numeral
34
=
4
10
×
=
5
3
Simplify
=
3
5
11
These will result in mixed
or whole numbers.
38
11
10
=
×
5
11
=
=
=
=
39
9
6
×
4
5
=
=
=
Fraction of a Quantity
Quantities can be litres, kg, even $, they are a measurement of anything, not just a volume.
In column 1 two quantities are given and the fraction is required. Treat these problems as just
simplification exercises. Create a fraction with the first value 'over' the second value, then
simplify. Your answer will be a fraction with no units (see below left). Question 7 through 12
require a conversion to be made first so that both quantities are in the same units. Find the
conversion, then follow the same method, form a fraction and simplify. (See below right)
HCF = 8
8÷8=1
24 ÷ 8 = 3
8 hr of 24 hr
=
900 mL of 2.4 L
2.4 L= 2 400 mL =
8
=
24
900
2 400
=
3
8
HCF = 300
900 ÷ 300 = 3
2 400 ÷ 300 = 8
1
3
In column 2 a fraction of a quantity is required. These problems are the same as multiplying
fractions with whole numbers. Multiply the numerator (top number) by the whole number, the
denominator stays the same. The denominator will divide into the numerator and you will be
left with a whole number answer, after you write the answer put the units on the end.
3 × 52 = 156
156
3
= 39 m
× 52 m =
4
4
156 ÷ 4 = 39
Question 18 through to the end of the sheet uses the same method, only a conversion is to take
place first. Given a quantity, change that quantity to a lower unit (to get a whole number
answer). Then multiply by the numerator and divide by the denominator to get the final
answer. The example below is 1.2 t, change this to kg (in brackets) = 1 200 kg, multiply by the
fraction then simplify.
3
of 1.2 t
4
3
× 1 200 kg
=
4
=
3 600
4
[kg]
1.2 t = 1 200 kg
= 900 kg
3 600 ÷ 4 = 900 kg
Fraction of a Quantity
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Express these parts of a
quantity as fractions
1 3 cm of 12 cm =
2 6 L of 20 L
=
Find the new quantity
make sure you show units
=
Example
=
=
14
3
× 60 cm =
4
=
=
4 18 m of 36 m
=
15
2
× 75 kg =
5
=
=
5 66 kg of 90 kg =
16
3
× 16 g =
8
=
6 16 hr of 24 hr =
17
3
× 90 m =
10
=
=
=
=
=
Change the larger units to
the smaller units to solve
sec =
=
18
8 750 kg of 1 t
1t=
kg =
=
4
of 8 cm
5
4
× 80 mm
=
5
=
cm
=
=
19
10 18 hr of a day
1 day =
hr
2
of 2 min
3
=
=
[s]
25
=
11 50 mm of 1.5 m
1.5 m =
mm
=
=
mL =
20
=
12 700 mL of 4.2 L
4.2 L=
3
of 1.2 t
4
=
=
[kg]
=
=
[c]
×
=
2
of 0.6m
3
26
=
×
=
=
=
[mL]
×
7
of $1.10
10
=
×
=
=
=
=
[cm]
×
1
of 3 L
4
=
9 60 cm of 2 m
2m=
=
=
24
[c]
×
2
of 2 m
5
=
[mm]
=
3
of $1.30
10
23
Change to the smaller
units in brackets to solve
7 3 sec of 2 min
2 min =
22
[hr]
×
=
2
× $120 =
3
13
3 $40 of $65
=
40
8
× 5L =
= 4L
10
10
=
5
of 1 day
8
21
×
=
[cm]
Multiplying Mixed Numerals
When fractions are multiplied, multiply across the top numbers and across the bottom
numbers. As mixed numerals have whole numbers in them you first must change them to
improper fractions. Once this has been done the same multiplication method is used.
In column 1 a mixed numeral is multiplied by a proper fraction. The method used is:
•
change the mixed numeral to an improper fraction and leave the proper fraction as it is.
•
multiply the numerators and write the answer then multiply the denominators and write
the answer
•
simplify by finding the HCF (Highest Common Factor), which is the largest number that
goes into both numbers and divide through
•
then rewrite as a mixed numeral or a whole number.
An example is at the top of column 1.
Columns 2 and 3 are done in exactly the same way except there are two mixed numerals so
you have to convert both to improper fractions not just one of them. There is an example at the
top of column 2.
Multiplying Mixed Numerals
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Try these, they will
result in mixed or
whole numbers.
Multiply these fractions
and mixed numbers.
Example
4
=
2
3
5
6
×
14
Simplified: ÷ by 2
5
×
3
6
=
6
70
18
Convert to Improper
2
3
1
=
= 3
1
× 2
2
×
=
=
35
2
3
4
× 1
×
Total
=
=
4
=
3
4
×
×
3
=
1
2
7
1
5
3
1
× 1
2
=
×
=
=
2
5
×
8
2
1
3
1
× 2
4
=
×
=
=
4
5
=
×
1
1
4
× 2
×
2
9
=
=
14
=
6
=
1
4
× 1
×
1
3
=
=
=
15
=
1
=
3
7
× 1
×
4
5
=
=
=
9
2
2
5
1
× 2
2
=
×
=
=
5
3
× 2
12
5
=
=
=
=
16
=
2
=
=
4
=
=
=
×
1
7
Mixed Numeral
=
3
×
=
=
2
× 1
=
13
=
9
1
2
Simplify
9
8
7
=
3
5
Improper Fractions
=
12
2
3
× 1
×
1
4
=
=
=
10
1
3
2
3
× 1
5
=
×
=
=
=
=
17
=
3
=
1
3
× 1
×
2
5
=
=
=
5
6
1
2
×
2
7
=
11
=
×
=
=
=
=
3
1
× 3
1
10
3
×
18
=
=
=
1
3
4
×
× 1
3
5
=
=
=
=
=
Multiplying Fractions Using Cancellation
When fractions are canceled they are simplified before multiplication, but instead of
simplifying the fraction's numerator and denominator you simplify diagonally. This means that
the numerator with one fraction is cancelled against the denominator of the other and vice
versa.
In Column 1 the fractions are either proper or improper, you have to cancel across diagonals
by finding the HCF, the largest number that divides into both numbers and dividing through.
Once you divide, strike out the old number and write its replacement, then multiply and
change to a mixed or whole number.
Questions 1 - 5 can only be cancelled across one diagonal, questions 6 - 12 can be cancelled
across both diagonals.
Column 2 involves proper fractions and mixed numerals, on the line below change the mixed
numeral to an improper fraction, rewrite the proper fraction below itself then start canceling.
The first 4 questions can be cancelled on one diagonal only, then for the rest of the column
canceling will be possible on both diagonals.
Column 3 is a further step again with two mixed numerals having to be multiplied. Convert
both this time to improper, cancel, multiply and change back to mixed numerals. There is an
example below.
Diagonals 6 and 9 have a
HCF of 3:
6÷3=2
9÷3=3
2
Diagonals 5 and 25 have a
HCF of 5:
5÷5=1
25 ÷ 5 = 5
Diagonals 18 and 8 have a
HCF of 2:
18 ÷ 2 = 9
8÷2=4
Diagonals 11 and 33 have a
HCF of 11:
11 ÷ 11 = 1
33 ÷ 11 = 3
5
10
6 25
×
=
= 3
3
1 5
3 9
1
3
Convert improper
fraction to a mixed
numeral
1
9
=
1
7
1
×4
11
8
18
11
3
×
4
33
8
Convert to mixed
numerals
=
27
4
= 6
3
4
Convert improper
fraction to a mixed
numeral
Multiplying Fractions Using Cancellation
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Use cancelling on one
diagonal, then multiply
and change from
improper to mixed
Example
4
4
16 1
×
=
= 1
3 14
3
1
Change to improper
fractions, then answer as
a mixed numeral.
1
4
×
14
7
4
×1
9
5
=
×
3
=
21 4
×
=
5
7
=
3
7 18
×
=
6
5
=
=
×
4
5 11
×
=
2 15
=
16
8
1
×3
11
4
5
9
7
×
=
14 2
=
=
×
2
1
=
5
8
×
=
2 15
=
10 9
×
=
3 20
=
8
12 25
×
=
5
6
9
3 16
×
=
4
9
10
50 7
×
=
7 30
11
55 12
×
=
9 35
=
17 40
×
=
20 34
=
7
12
=
11
=
=
=
=
×
=
20 5 6 × 7
5
=
=
20
3
×3
27
5
=
×
=
=
=
=
=
=
=
=
=
=
=
=
=
=
5
=
×
1
=
×
5
=
×
2
7
28 2 5 × 2 9
=
3
21
×
5
=
×
=
27 1 7 × 2 8
=
=
=
3
4
12
4
7
×1
5
8
=
26 2 5 × 4 6
18 3 4 × 25
19
×
1
=
=
=
25 1 12 × 1 13
=
×
=
1
4
=
=
=
24 2 5 × 1 7
3
3
×
9
=
×
=
23 1 35 × 4 11
17 4 9 × 10
This time cancel across
both diagonals
6
=
15 2 3 × 14
4
3
22 1 26 × 1 10
=
3 11
×
=
2
6
1
19
13 3 2 × 5
=
Both are mixed numerals,
solve these.
=
×
8
7
29 3 9 × 2 10
=
=
=
×
11
11
30 2 12 × 1 21
=
=
=
×
Dividing Fractions And Reciprocal
The reciprocal of a fraction is the fraction turned upside-down. The reciprocal is used to
convert the division process to a multiplication process. When a fraction has a division sign
before it turn the fraction upside and change the ÷ sign to a × sign.
Column 1 requires the reciprocal to be found. This is achieved by turning the fraction upsidedown. In the case of improper fractions they become proper fractions, and proper fractions
become improper fractions. So for example 10/11 would have a reciprocal of 11/10 which
becomes 11/10. Write it as a mixed number for this column because it fully answers the
question, but in the division process we would leave it as an improper fraction for calculations.
Questions 9 through 15 require you to change mixed and whole numbers to improper fractions
then find the reciprocal. A whole number is written as the number over 1, e.g. 23 would be
23
/1, then turn it upside-down and you get 1/23. Look at the example at the top of the column.
Column 2 uses the reciprocal in the division process. Turn the fraction that is behind the
division sign upside-down, then multiply. So there is no division process with fractions
calculations, you always use multiplication. Questions 23 through 25 are an extension of this,
turn the fractions that have a ÷ sign before them upside-down, make sure you don't turn
fractions with a × sign before them upside-down! An example is below, it is done twice, the
first time by simplifying, the second by canceling, if you can cancel then use that method.
Column 3 adds a further step in the process as the answer will be a mixed numeral. There is an
example at the top of the column
Note that this sheet can be done by canceling or by simplifying after multiplication, the choice
is yours. If you do use canceling you will find that the simplifying stage won't be required and
you will have unused spaces.
6
10
60
6
3
÷
=
×
=
25
10
25
3
75
=
4
5
2
6 2 10
4
6
3
÷
=
×
=
25
10 5 25 1 3
5
=
Dividing Fractions And Reciprocal
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Find the reciprocal, answer
as whole or mixed
numbers where possible
Example 1
Example 2
55
9
7
19
9
55
19
7
1
3
4
6
7
8
4
2
3
4
= 2
9
2
=
5
10
5
3
73
5
Find the reciprocal of the
fraction behind the ÷, then
×. Simplify if possible.
This time the answers will
need to be changed to
mixed or whole numbers
16
1
3
÷
=
4
5
×
=
17
3
4
÷
=
8
7
×
=
7
1
5
18
÷
=
2
9
×
=
2
7
÷
=
5
5
×
=
19
Example (with canceling)
3
3
12
9
3
7
÷
=
×
=
4
12 1 4
7
7
Simplify here if
you have difficulty
with canceling
26
27
20
4
=
13
21
1
=
23
7
3
÷
=
11
4
×
4
10
÷
=
9
15
×
8
8
÷
=
5
11
=
28
2
16
÷
=
3
21
×
=
1
3
5
23
÷
÷
2
4
6
=
29
24
3
=
10
=
15 9
1
=
2
=
15
20
÷
=
4
7
=
×
=
9
1
÷
=
8
2
×
×
30
=
25
=
=
×
=
=
=
3
12 5 =
4
14 7 =
×
=
=
13 5
=
=
9 5 =
11 12 =
7
=
=
22
2
=
3
2
=
Change to improper
fractions then find
the reciprocal.
10 2
×
=
=
3
3
6
÷
=
4
11
8
=
1
= 1
=
=
=
5
5
7
÷
÷
4
7
8
×
×
=
=
2
3
3
×
÷
3
10
4
×
×
=
=
=
5
2
÷
÷ 3
12
7
×
×
31
=
=
=
2
11 22
×
÷
3
4
15
×
×
=
=
=
Dividing Mixed Numerals
Dividing mixed numerals is an extension on dividing fractions, the extra step required is to
convert the mixed numeral to an improper fraction.
In column 1 a mixed numeral is divided by a proper fraction. To answer these questions follow
these steps:
•
Change the mixed numeral to an improper fraction
•
Turn the proper fraction upside-down and write the reciprocal underneath the fraction
•
Multiply across the top (numerators) and the bottom (denominators) to get the answer, it
will be an improper fraction
•
These questions can't be simplified so change the improper fraction answer to a mixed
numeral and you are done.
An example is at the top of the column, note that you only invert (turn upside-down) the
fractions that are behind a ÷ sign. In the lower questions multiplication is included to test that
you are aware of this. Don't find the reciprocal of fractions that are behind a × sign.
Columns 2 and 3 are the same as column 1 except both are mixed numerals not just one of the
numbers. An example is below. The steps are the same as above except that:
•
Both must be converted from mixed numbers to improper fractions
•
The answers will often need to be simplified
•
As with previous sheets if you have difficultly with the concept of canceling there are
additional spaces for simplifying your answer, if you are canceling you mostly won't need
to use these spaces.
•
If you don't use canceling you may find questions 16 and 17 difficult to simplify
Both mixed numerals
written as improper
fractions
Diagonals cancelled
HCF of 9 & 9 = 9
9 ÷ 9 = 1 (both same)
HCF of 2 & 4 = 2
2÷2=1
4÷2=2
4
1
=
1
2
9
1 2
÷ 2
2
×
4
1 9
1
4
9
=
=
2
2
1
÷
=
= 2
Changed to ×
then reciprocal
found
9
4
Dividing Mixed Numerals
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Try these, they will
result in mixed or
whole numbers.
Divide these fractions and
whole numbers.
4
3
13 1 5 ÷ 1 10 =
÷
Example
3
=
1
2
7
21
=
2
4
= 5
2
8 42 ÷ 13 =
Reciprocal
3
×
2
1
2
3
÷
1
4
=
×
=
2
=
2
1
1
4
÷
=
1
2
4
5
1
=
=
÷
×
2
×
=
2
3
=
×
×
÷
=
7
14 2 7 ÷ 1 10 =
=
×
3
1
10 3 4 ÷ 2 4 =
7
9
=
=
=
×
=
÷
=
2
15 4 3 ÷ 1 3 =
=
×
5
=
5
=
1
3
÷
×
3
4
=
=
1
=
=
5
3
11 1 7 ÷ 1 7 =
=
×
÷
÷
=
×
×
÷
=
=
×
7
=
×
=
=
×
=
=
1
×
=
5
1
17 1 2 ÷ 1 6 ÷ 1 11
12 9 2 ÷ 3 3 =
=
1
2
1
÷
3 ×
3
3
4
×
1
1
1
=
=
=
6 32 ÷ 3 × 5
=
2
=
=
1
3
=
=
×
=
16 2 3 ÷ 1 4 ÷ 5
2
3
2
×
1 ÷
5
4
3
×
÷
=
1
4
=
=
=
=
÷
=
1
=
÷
3
9 24 ÷ 18 =
=
=
=
=
×
=
3
3
4
=
3
Convert to Improper
1
÷
÷
=
=
=
÷
÷
=
×
×
=
=
=
=
Mixed Operations - Calculator
This sheet is for calculator use so there are no working spaces provided.
The method is simple for this sheet, use your calculator. You don't have to worry about
reciprocals or simplifying, the calculator does it all for you. Mistakes made using the
calculator can occur anywhere due to keystroke error, there is one more common mistake
made though, that is the entering of mixed numbers. Remember that you have to press the
fraction key twice to enter a mixed number.
Your calculator should obey order of operations rules in the exercises in column 3, just don't
press '=' until you key in the entire question otherwise the calculator may give the wrong
answer.
Mixed Operations - Calculator
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Add or subtract
these fractions
Multiply or divide
these fractions
These are all
mixed up
1
1
1
+
=
2
3
17
3
1
×
=
5
4
33
1
3
9
+
×
=
3
4
5
2
4
3
=
5
7
18
7
2
÷
=
12
3
34
1
2
2
÷
+
=
5 15 3
3
4
3
=
5
10
19
2
8
÷
=
3
9
35 5
1
× 3
6
4
3
3
+
=
11
5
20
3
7
×
=
5
6
36 7
3
1
3
÷ 9
+
=
8
4
2
5
3
1
4
+
=
4
6
5
21
4
1
8
÷
×
=
5
3
9
37
6
4
3
2
+
=
5
8 15
22
1
3 18
÷
×
=
7 10 5
38 5 ÷
7
9
5
1
+
=
12
7
4
23
3 17 1
×
÷
=
4 10 8
39
8
17
3
8
+
=
20 10 15
24
4
1
2
÷
÷
=
7 10 5
40 7
5
2
11
× 1 =
6
3
12
9
2
4
7
+2
=
3
5
10
25
1
3
2
× 4
×
=
6
8
3
41 3
3
5
5
÷
=
4
9
12
10 4
6
2
4
=
7
3
9
26 5
11
3
1
÷
×
=
12
4
2
42 2
5
1
× 3 ÷ 5 =
8
6
11 2
4
1
5
- 1
+
=
11
2
6
27 3
1
4
× 6 ×
=
2
5
43 9
1
2
4
- 3 ÷ 1 =
2
3
5
12 4
3
7
2
+
- 1 =
5
10
3
28 7
3
2
2
×
÷ 2 =
4
3
5
44 3
8
2
1
× 1 × 3 =
9
5
2
13 3
5
1
7
- 1 + 4 =
6
4
12
29 7 ÷ 1
14 3
2
7
5
+ 4 + 2 =
5
10
12
30 4
5
3
2
× 3 ÷ 2 =
6
10
5
46 4
7
1
1
÷
÷ 1 =
10
5
4
15 5
1
2
1
- 2
- 1 =
4
3
2
31 5
3
4
÷ 8 × 4 =
14
5
47 2
2
3
1
+
÷
=
5
4
2
16 9
2
6
5
- 3 + 1 =
3
7
6
32 2
3
2
÷ 1 × 3 =
7
3
48 5
7
1
1
÷ 2
- 1 =
8
3
4
1
3
÷ 2 =
4
5
45
- 4
6
=
7
1
5
5
+3
÷
=
2
4
6
7
8
×
=
10 9
1
7
3
×
+
=
2 11 4
2
8
1
×
+
=
5
3
2
Fractions - Find A Word
Look for words in the list at the bottom of the grid. Once you find a word cross it off the list.
A letter could be used more than once so don’t colour it in too dark (using a texta for example)
so that you can still read it.
Fractions Find-A-Word
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Find the words in the puzzle
from the wordlist.
M
D
V
A
R
E
D
N
I
A
M
E
R
N
B
D
Q
U
X
D
L
O
W
E
S
T
R
N
O
C
E
V
Q
T
Z
D
R
T
T
M
I
O
Q
I
I
N
N
V
S
T
L
I
D
E
X
I
M
U
T
M
O
N
U
I
N
R
W
T
D
S
C
P
A
A
P
M
P
D
M
L
E
E
H
I
P
P
M
N
C
R
I
N
T
P
E
Q
R
V
O
O
A
D
T
I
O
N
D
R
L
C
R
I
A
N
L
N
R
I
L
P
A
S
E
I
N
D
A
C
P
I
E
E
T
P
E
T
E
V
F
A
N
V
T
N
M
N
D
Y
I
R
O
P
N
Y
C
X
T
D
O
L
O
U
D
T
B
R
A
O
G
R
E
C
I
P
R
O
C
A
L
P
V
H
C
O
M
M
O
N
N
U
Q
I
G
U
M
M
S
U
B
T
R
A
C
T
I
O
N
V
M
B
B
T
N
E
L
A
V
I
U
Q
E
G
L
L
M
WORDLIST
SUBTRACTION
RECIPROCAL
ADDITION
CANCEL
DIVISION
MULTIPLICATION
CONVERT
INVERT
IMPROPER
REDUCING
DENOMINATOR
REMAINDER
NUMERATOR
SIMPLIFY
MIXED
EQUIVALENT
COMPARE
QUANTITY
SHAPES
WHOLE
PART
COMMON
LOWEST
7
FREEFALL
MATHEMATICS
DECIMALS
Place Value of Decimals
The position of a digit in a decimal could be to the left of the decimal point: units (ones), tens,
hundreds and upwards. Or to the right of the decimal point: tenths, hundredths, thousandths
and so on. This location is its place value.
Column 1 asks you to place decimals that have been broken up into parts into the table. Read
the question and place the digits in the table according to their place value. Once the digits are
in, fill all the spaces between the digits with zeros, except in the decimal point column, just
write in a '.' .
Column 2 asks you to give the place value of the '4' in all the numbers. Look at its location
with the decimal point, the amount of digits either side of it doesn't matter, it is its location to
the decimal point that determines its place value. Use the table if you have difficulty with the
positions or the spelling. Once you have spelt the place value (just write ‘4’ don't spell it) write
the fraction or number that matches the answer.
The next part of column 2 asks you to write the number in decimal form and then expand it.
The first decimal place will be a fraction ‘over’ 10, the second decimal place will be ‘over’
100, the third ‘over’ 1 000. If there is a zero in the decimal then you don’t include it in the
expansion. So watch for zeros!
Column 3 asks you to state the number of decimal places that each number has. Again don't
let the numbers trick you, look for the decimal point and count the number of digits behind (to
the right of) it. If you had a number such as 4.200, while you could write it as 4.2 (which has 1
decimal place) the answer would still be that the decimal has 3 decimal places. Note that
d.p. = decimal places.
The rest of Column 3 uses comparisons of decimals. Look at the numbers and add zeros to the
number with less decimal places, until they have the same number of d.p. (match the number
of decimal places). Then compare them, just imagine that there is no decimal point and they
are whole numbers. This takes the error factor out of comparing. Remember with < and > you
point the arrow to the smallest number (decimal).
A very common mistake when rewriting decimals in order is that you write the altered
decimals in order, not the decimals in the question. So remember write the decimals in the
question in order, not your working out decimals.
Place Value of Decimals
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Complete the table. Fill
blank spaces between the
numbers with zeros. All
questions must have a
zero or whole number in
the units column.
Example
4
100
5.742 = 4 hundredths
Example
2 8 tens and 7 tenths
3 7 hundreds, 8 units and 4
hundredths
4 8 tens, 4 hundredths and
1 thousandth
6 2 hundreds, 7 units, 3 tenths
and 9 hundredths
25 56.244
26 371.9
27 24.01
11 127.4 =
28 12.408
29 100.05
12 15.94 =
30 903.2
31 363.63
13 1.423 =
Add zeros to give all the
decimals the same number
of decimal places then use
< or > to compare them
Example
Write these as decimals,
then expand. Place only
a single digit in the top
(numerator) boxes.
5
Units
Decimal Point
Tenths
Hundredths
100's 10's
1's
.
1
10
1
1
100 1 000
3
.
2
0
Thousandths
Tens
8 3 hundreds, 9 tens and 5
tenths
Hundreds
15 12.471 =
7
7
3
37
+
= 5.37 = 5 +
100
10
100
16
45
= 0.
100
17
572
=
1 000
=
=
10
=
2
19 4 703 =
6
100
+
+
100
1 000
+
=
+
0.04
0.20
32
0.7
0.584
33
0.72
0.8
34
1.25
2.1
35
3.56
3.068
36
1.963
1.9
37
+
20 68 =
=
+
39
=
1 000
=
+
100
7
8
0.2
Rewrite decimals with the
same decimal places, then
arrange in descending order
+
1 000
4
5
+
10
17
18 7
=
100
1
3
0.04 <
Example
7 7 hundreds, 6 units and 8
thousandths
0
3
24 3.7055
14 7.0542 =
5 9 hundreds, 4 tens, 6 units,
5 tenths and 4 thousandths
25.742
23 0.4529
9 14.36 =
1 3 units and 7 thousandths
Example
22 850.65
4 hundreds, 3 units, 2 tenths and
7 thousandths
10 0.034 =
Ex. 4
Give the decimal
places of the
following
Give the value of the 4
in these. Express both
in words and numbers
21
0.72 , 0.3 , 1.004 , 1.3
38 0.555 , 0.5 , 5.05 , 0.55
Rounding Decimals
Rounding is used to shorten decimal answers to a level of accuracy required. Often when
answering questions, you are asked to give your answer to a specific number of decimal
places. This isn’t done by just chopping the decimal off at the number of places required, this
method is followed:
•
Look at the decimal behind the decimal place asked for. For example if rounding to
2 d.p. (2 decimal places) look at the digit in the 3rd decimal place.
•
If this digit is 5 or more then add 1 to the digit in front of it, this means that when
increasing a 3 for example, it would become a 4. But it also means that if it is a 9 it will
become a 0 and will add 1 to the digit in front of it.
•
If the digit is less than 5 the digit is unchanged and the number is just 'chopped' at the
required decimal place.
•
If there isn't a number in the decimal place asked for, then zeros are added to the number
until the decimal place has a zero in the position.
Column 1 asks firstly to round the given decimals and whole numbers to 1 d.p. To do this (an
example is at the top of the column), look at the number in the 2nd decimal place position. If
that number is 5 or more add 1 to the 1 d.p. number then remove all the digits behind it. If the
2 d.p. digit is less than 5 just rewrite the 1 d.p. number unchanged with the digits removed
after it. The second part of column 1 asks you to round to 2 d.p. Look at the 3rd digit and use
the same method, the answer will have 2 digits after the decimal point.
Column 2 asks you to round to 3 d.p. Look at he 4th decimal place digit and apply the same
method. The next part of the column asks you to round to 0 d.p. This could also be asked in
the form : 'round to the nearest unit' or 'round to the nearest whole number'. Look at the digit
after the decimal point and apply the same method then write the whole number without a
decimal point. Note that if a decimal is less than 0.5 then when it is rounded to the nearest
whole number it will become 0.
Column 3 is a combination of the previous 2 columns. The first part asks you to round to the
number of decimal places specified in the brackets. So instead of the column being all the
same rounding, it jumps from one to the next. The second part of the column asks you to round
the one number to 1, 2 and 3 d.p.
Examples of rounding
0.54862
To 1 d.p. : 0.54862 → 0.5 (as 4 < 5)
To 2 d.p. : 0.54862 → 0.55 (as 8 > 5)
To 3 d.p. : 0.54862 → 0.549 (as 6 > 5)
To nearest whole number:
0.54862 → 1 (as 5 = 5)
Example: Whole numbers
65
To 1 d.p. : 65.0
To 2 d.p. : 65.00
To 3 d.p. : 65.000
Example:
71.5
To 1 d.p. : 71.5
To 2 d.p. : 71.50
To 3 d.p. : 71.500
Rounding Decimals
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Round these numbers to
the nearest tenth (1 d.p.)
Example
1.1627
Look at the digit to the right
of the one to be rounded. If
it is 5 or more round it up, if
< 5 the digit is unchanged
1.2
Round these numbers to the
nearest thousandth (3 d.p.)
Example
Look at the digit to the right
of the one to be rounded. If
it is 5 or more round it up, if
< 5 the digit is unchanged
1.1627
1
0.87
22 2.7674
2
0.43
23 0.8436
3
0.90347
24 8.0302
4
0.05956
5
1.163
Round these numbers to
the decimal place given
in the brackets
43 29.6751
[2 ]
44 1 361.54 [1]
45 0.4296
[0 ]
46 9.1
[3 ]
25 6.9969
47 19.9971
[2 ]
18.374
26 6.9996
48 6.0606
[3 ]
6
22.128
27 0.7
49 86
[2 ]
7
67
28 0.70
50 56.0438
[1 ]
8
56.942
29 0.707
9
129.988
30 4
51 11.4099
[0 ]
52 8.8997
[3 ]
53 5.9949
[2 ]
10 99.953
31 9.9999
Round these numbers to the
nearest hundredth (2 d.p.)
Example
1.1627
Look at the digit to the right
of the one to be rounded. If
it is 5 or more round it up, if
< 5 the digit is unchanged
1.16
Round these numbers to
the nearest unit (0 d.p.)
Example
Round each decimal to 1,
2 and 3 decimal places
Look at the digit to the right
of the one to be rounded. If
it is 5 or more round it up, if
< 5 the digit is unchanged
1.1627
1
54 0.1919
[1 ]
[2 ]
[3 ]
55 3.8225
[1 ]
[2 ]
[3 ]
11 0.278
32 4.7
12 0.805
33 9.3
13 0.4246
34 18.78
14 0.9793
35 249.499
15 2.2183
36 7 101.835
56 26.9527
[1 ]
16 73.2
37 83.5004
[2 ]
[3 ]
17 4.95829
38 999.278
57 0.3235
[1 ]
18 9.99811
39 999.872
19 67.9942
40 0.808
[2 ]
[3 ]
20 83
41 0.080
58 9.9994
[1 ]
21 99.9961
42 9 999.5
[2 ]
[3 ]
Addition - Decimals
To perform addition move from right to left and carry tens when the total exceeds 9. The
example below will show the method:
•
If you want to fill the shorter decimal numbers with zeros to make the all the decimals
level on the right hand side you can, if a number is a whole number then write the
decimal point then the zeros.
•
Total down the right hand column and if the total is less than 10 write the answer on the
bottom line.
•
If the number is ten or more write in the units or last digit and write the first digit (tens)
above the next number to the left (carried tens).
•
Add the next column including the carried figure (if any) in your calculation, repeat the
process until you are finished
Carried tens
3 223 2
3
Question Number
7 330.354 +
944.692
7 956.800
8 558.79 0
1 666.900
26 457.536
Answer space
Zeros added to
level off the
right hand side
columns
Addition - Decimals
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Try these additions,
place zeros at the
ends to fill 'shorter'
decimal numbers if
it makes it easier.
1
2.8 +
0.7
2
1.4 +
2.9
3
6.6 +
1.6
4
3.7 +
4.4
5
5.8 +
1.5
6
47.63 +
25.79
7
11.49 +
7.6
8
17.5 +
39.64
9
26.74 +
58.3
10
74.94 +
8.37
11
47.33 +
17.6
12
50.76 +
14.33
12.58
13
8.9 +
30.25
15
14
72.8 +
2.06
5.4
15
36.1 +
48
13.97
16
17.39 +
25.5
40.08
17
63.8 +
20.71
8
18
98.73 +
143.46
288.12
19 447.83 +
106.4
222.6
20 507.9 +
200.85
141.06
21
75.48 +
312.7
458
22 110.79 +
755.63
92.4
+
23 330
115.62
176
24
900.267 +
12.552
0.708
647.320
437.166 +
692.4
240.1
855.83
28
25
329.8 +
588.139
141.7
827.53
26
56.8 +
702.426
879
950.42
27
717.009 +
422.6
196.5
803.72
29 6 720.363 +
9 345.869
823.441
2 995.752
30 3 209.1 +
9 022.803
4 693
8 974.46
31 1 095.736 +
4 577.91
7 211.1
3 820.97
+
32 4 633
2 921.506
5 228.35
9 407
33 5 338.291 +
2 891.537
3 522.4
7 100.22
34 9 352.071 +
1 133.059
6 402.774
3 867.856
5 002.795
35 9 222.5 +
6 731.059
8 605
7 322.41
8 511.977
36 3 476.093 +
767.811
9 491.2
6 333.79
1 874.9
37 5 070.396 +
4 226.5
9 063.9
7 911.26
3 493.12
38 7 336.288 +
8 017.5
2 435.851
499.87
6 513
Subtraction - Decimals
To perform subtractions move from right to left and 'borrow' when the sum of the numbers
below the top digit exceeds 9. The example below outlines the method:
•
If you want to fill the shorter decimal numbers with zeros to make the numbers level on
the right hand side you can, if a number is a whole number then write the decimal point
then the zeros. If there is a decimal point already just write in the required number of
zeros.
•
Subtract down the right hand column if the top digit is large enough to subtract all the
digits below it. Write the answer then move to the next column.
•
If the top digit isn't large enough, add the digits below it and then borrow the required
number of 'tens' from the column to its left, then put the number of tens borrowed below
the column from which it was borrowed.
•
Then subtract the next column, including the figure that you borrowed in your
calculation, repeat the process until you are finished.
14
Question Number
8 438.027 581.400
124 .000
839.45 0
2 211 1
6 893.177
Borrowed Tens
Answer space
Zeros added to level
off the right hand side
columns, note the
decimal point behind
the whole number
Subtraction - Decimals
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Try these subtractions,
place zeros at the ends
to fill 'shorter' decimal
numbers if it makes it
easier.
1
8.6 5.2
2
5.7 3.9
3
7.4 2.1
4
4.3 0.9
6
35.63 18.87
7
59.06 33.4
8
31.42 26.7
9
62.9 48.05
10 82.48 57.9
12
50.76 14.33
12.58
13
73.69 47.5
21.61
14
50
7.8
25.44
15
92.03 46.5
27
16
18 825.02 146.47
482.94
19 213.55 96.7
104
20 495
289.7
96.76
21 629.4 438
177.85
5
9.7 3.5
11 24.1 9.56
28.27 17.19
9.8
17
22 398.45 16.7
145.9
62
37.6
18.71
23 572.93 105.2
359
24
3 290.351 462.783
816.447
658.215
25
4 883.027 177.4
502
938.77
26
1 412.096 364.839
217.6
92.51
27
4 408
950.2
703.152
444.06
28
7 559.824 683.5
297
781.976
29
7 538.293 1 089.476
2 774.502
1 946.229
30
8 356.248 986.51
3 267.382
1 344.8
31
5 963
1 286.9
542.075
1 766.2
32
9 814.763 4 011.6
2 452
1 835.66
33
7 333
2 463.05
1 779.841
856.22
34
8 777.292 2 856.319
683.147
1 936.495
2 351.866
35
7 428
2 441.8
855.561
1 587.04
2 168.7
36
8 059.214 327.162
1 588.3
3 762
2 070.11
37
4 919.5
361.607
1 053.436
1 500.61
1 785
38
9 423.67 3 502.164
54.34
1 823
2 538.2
Multiply and Divide by 10, 100 and 1 000
When you multiply or divide by 10, 100, 1 000, etc a rule applies which allows the calculation
to be done without the usual amount of working. The rule is that when you divide by:
•
10 : move the decimal place one place to the left
•
100 : move the decimal place two places to the left
•
1 000 : move the decimal place three places to the left
The opposite applies with multiplication, when you multiply by:
•
10 : move the decimal place one place to the right
•
100 : move the decimal place two places to the right
•
1 000 : move the decimal place three places to the right
Column 1 are all multiplications, just move the decimal point to the right the same number of
places as there are zeros in the number : 10 = 1 place, 100 = 2 places and 1 000 = 3 places.
Examples are at the top of the column. Remember if a number doesn't have a decimal point
then it is as if there is a decimal point behind the last digit, for example: think of 120 = 120.
and 2 039 = 2 039. and so on.
Column 2 is all division. This time move the decimal point to the left, the same number of
places as there are zeros in the number, 10 (1), 100 (2) and 1 000 (3).
Column 3 are problem-style which use the same approach as the previous 2 columns. The only
difference being an additional calculation is required in most questions.
Multiply and Divide by 10, 100 and 1 000
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
When you divide by 10,
100 or 1 000 you move the
decimal point to the left
When you multiply by 10,
100 or 1 000 you move the
decimal point to the right
× 10
1 place right 5.784 × 10 = 57.84
× 100 2 places right
× 1 000
5.78 × 100 = 578
3 places right
÷ 10
578.4 ÷ 10 = 57.84
1 place left
Now try these, write the
question mathematically
then solve.
40 The airport has a departure levy
of $18.30 on each person. If 1 000
people depart in an hour on average,
578.4
2
places
left
÷ 100
= 5.784 calculate the amount (A) collected in
100
an hour. Then give an estimate on the
amount taken for a 16 h day.
3
places
left
÷ 1 000
A=
1 2.6 × 10
=
22 10
2 19.7 × 10
=
23 1 549 ÷ 1 000
=
3 100 × 100
=
24 4.63 ÷ 10
=
4 5.373 × 1 000
=
2 573.4
5 933.05 × 100
=
6 7 707.7 × 10
=
26
7 0.002 × 100
=
27
8 2.0202 × 10
=
9 2.0202 × 100
=
25
3 82
=
100 63.47
3.4
100
29
=
10 2.0202 × 1 000 =
9.04
1 000
=
30 10
13.06
=
12 0.046 × 1 000
=
31 403.02 ÷ 100
=
13 90.33 × 1 000
=
32 10.7 ÷ 1 000
=
14 12.549 × 100
=
33 10
=
15 528.64 × 10
=
16 0.0003 × 100
=
17 530.2 × 100
=
18 0.07 × 1 000
=
6 059.8
34 1 000 80.5
100
=
=
B=
B=
43 A grain silo holds 9 625 kg of
grain. If it is filled to one hundredth
of its capacity, find the mass of the
grain inside.
19 0.8223 × 10
=
37
20 116.8 × 1 000
=
38 999 ÷ 100
=
21 0.00318 × 10
=
39 999 ÷ 1 000
=
10
+
42 A tennis ball launcher fires a ball
every 6 sec. Find the number of balls
(B) that are fired in a minute, then
find the ball capacity of the machine
if it runs out of balls after 12.5 min.
=
36 702.6 ÷ 10
999
A=
=
=
14.07
water into a dam daily. The next day
after rain it delivers 10 times this
amount. Find the amount (A) for the
wet day, then calculate the amount of
water delivered for the 2 days.
A=
11 17.38 ×10
35
Amount after 16
41 A small stream feeds 19.45 kL of
=
28 25 ÷ 1 000
A=
hours:
=
100
×
=
Multiplication of Decimals
When decimals are multiplied, it is easier to take the decimal point out, that way it is just like
multiplying two whole numbers. Then replace the decimal point in the answer, but where?
Here's where:
•
count the number of digits behind the decimal point in the first decimal
•
then do the same for the second decimal
•
add the two together and that is how many digits will be behind the decimal point in the
answer, so put the decimal point in at the end.
In the example in the top row the first decimal has 2 decimal places, the second decimal has 1
decimal place. Add 2 and 1 and you get 3. So the answer will have 3 decimal places. Write that
in the box provided below the question number. Then carry out the multiplication, get your
answer, then put a decimal point between the 3rd and 4th digit (from the right).
The questions become longer as you work through. Remember to add a zero on the 2nd line
and 2 zeros on the third line (if there is one).
Multiplication of Decimals
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Multiply without
the decimal point
and put it in at
the end. Write
the number of
d.p.'s in the box.
Example
3
25.79 × 0.7
2
4 56
2 579
1
1 6.92 × 0.03
4
7
2 183.7 × 0.04
3 7 × 19.25
692
3
18.053
4 254.6 × 0.9
5 8 257 × 0.05
6 9.003 × 0.8
7 3.411 × 0.9
8 7 644 × 0.009
9 535.3 × 0.71
10 0.1632 × 59
11 2.1 × 37.16
12 5.113 × 0.35
13 112.4 × 0.061
14 37 × 5.036
15 21.55 × 0.49
16 0.5663 × 4.3
17 9.5 × 2.907
18 6.081 × 0.75
19 7.67 × 8.45
20 118 × 0.522
21 41.9 × 0.138
22 8.65 × 24.5
23 94.2 × 2.56
Division of Decimals
Dividing decimals is performed in the same way as with whole numbers except that when you
divide past the decimal point, place a decimal point in the quotient (answer).
Column 1 involves dividing decimals by a whole number. If the number ‘won’t divide’ into the
other number then write a zero and carry the digit to the next number (as tens) and then
attempt the division again.
Column 2 involves longer questions, these involve adding zeros to enable the division to be
completed. The zeros are added until you can remove the remainder. From question 18 answer
the divisions using 4 decimal places. This means that you need a 5 decimal place answer
which can then be rounded to the 4 d.p answer required.
Column 3 involves dividing decimals by other decimals. This column requires you to divide or
multiply both numbers by either 10 or 100 before the division takes place. This is done to
change the divisor (the outside number) to a single digit number, for example:
•
46.4 ÷ 0.8 ….multiply both numbers by 10 to get ...464 ÷ 8
•
29.6 ÷ 60 ….. Divide each number by 10 to get …. 2.96 ÷ 6
•
0.597 ÷ 0.04 … multiply each number by 100 to get 59.7 ÷ 4
Although you are changing the numbers you are using, because you are changing both of them
in the same way the answer stays the same and doesn’t need to be changed.
Division of Decimals
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Divide these decimals
by whole numbers
0.39
Example
1.56 ÷ 4
1 2.56 ÷ 8
4
8
13
1.56
2.56
These are longer, add
zeros to complete these
divisions.
0.43125
Example
3.45 ÷ 8
14 2.63 ÷ 5
Move the decimal point
across to divide by a
whole number (to 3 d.p)
8
5
32124
3.45 000
2.63
65.625
Example
52.1 ÷ 0.8
8
{
52.1 × 10 = 521
0.8 × 10 = 8
23 17 ÷ 0.4
2 9.45 ÷ 3
=
15 19.3 ÷ 4
(3 d.p.)
24 0.72 ÷ 0.5
3 2.28 ÷ 6
16 15.5 ÷ 8
4 0.572 ÷ 4
17 19.81 ÷ 2
25 0.431 ÷ 0.03
=
5 9.275 ÷ 5
Find the answer to 4 dp.
So you will need a 5 d.p
quotient, then round it!
=
(3 d.p.)
(3 d.p.)
26 1.7 ÷ 30
6 45.87 ÷ 3
18 6.5 ÷ 7
=
=
(4 d.p.)
27 8.53 ÷ 0.7
7 0.092 ÷ 4
=
19 14.8 ÷ 3
8 105.6 ÷ 8
9 4.752 ÷ 9
(3 d.p.)
=
(4 d.p.)
(3 d.p.)
28 1.27 ÷ 0.4
=
(3 d.p.)
20 7.26 ÷ 9
=
(4 d.p.)
10 19.06 ÷ 2
29 4.32 ÷ 90
=
(3 d.p.)
21 22 ÷ 7
11 3.584 ÷ 7
=
(4 d.p.)
30 0.5 ÷ 0.06
=
12 59.36 ÷ 8
13 862.5 ÷ 5
22 0.41 ÷ 8
=
(3 d.p.)
31 9.2 ÷ 40
(4 d.p.)
=
(3 d.p.)
54 1 24
521. 000
Changing Fractions to Decimals
When you look at a fraction you can consider it as the top number divided by the bottom
number. To change a fraction to a decimal you carry out the division and the answer after the
division (the quotient) will be in decimal form.
Column 1 starts with changing fractions to decimals that have a denominator of either 10, 100
or 1 000. Remember that this is just like division, when you divide by these numbers move
the decimal point to the left. So for division by 10, move the decimal point 1 position to the
left, 100 - 2 positions to the left and 1 000 moves 3 positions left. So the answer will be the
numerator (top number) with the decimal point moved. There are 9 examples at the top of the
column. Remember that when you move the decimal point, zeros that had meaning on the end
of whole numbers are not required when you convert them to decimals.
Column 2 gives you fractions that require division to change to decimals. First write out the
division with a ÷ sign, this is done for you already in some questions. Then place the numbers
in the working space, add a decimal point and zeros to the number in the division, there is
room to add up to 7 zeros, you won’t need that many though. Then divide. When you divide a
decimal by a whole number add a decimal point in the answer when you cross over it in the
division. There is an example at the top of column 2. Note that questions 29 and 30 are long
division and question 31 asks you to convert a mixed number to an improper fraction then
solve.
Column 3 introduces recurring (or repeating) decimals. These decimals have digits or a
sequence of digits that constantly repeat themselves in the decimal. Because these go on
forever you need to show that this is occurring in your answer. To do this put a dot on the
number or group of numbers that are recurring, for example
•
0.76666666... would be written as 0.76
•
0.7676767676…. would be written as 0.76, because both numbers are recurring
•
0.765876587658…. would be written as 0.7658 (the first and last of the recurring
Numbers have dots above them.
Changing Fractions to Decimals
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Change these fractions
to decimals, make sure
there are no 'end zeros'
Examples
Example
1
1
1
= 0.1
= 0.01
= 0.001
10
1 000
100
10
10
10
= 0.1
=1
= 0.01
1 000
10
100
41
= 4.1
10
352
100
= 3.52
= 0.1
1 000
100
7
=
10
1
2
8
=
100
5
3
=
1 000
16
=
10
37
5
=
100
87
6
=
100
63
=
10
4
7
3
=3÷4
4
0.75
32
4
3 .00
Example
1
=1÷6
6
= 0.16
23 1 = 1 ÷ 2
2
2
1
32 2 =
3
24 1 = 1 ÷ 4
4
4
1
=
33 1 =
25 2 = 2 ÷ 5
5
9
=
26 3 = 3 ÷ 8
8
430
=
1 000
8
9
906
=
1 000
27 4 =
10 408 =
100
Convert these fractions,
which will be recurring
decimals
Use division to change
these fractions to
decimals
34 5 =
6
=
5
11 110 =
100
35 7 =
9
28 7 =
=
8
12 4 076 =
1 000
29 3 =
13 8 008 =
100
20
30 7 =
40
36 5 =
1
14 1 501 =
=
1 000
15
45
=
10
17 550 =
100
60
=
10
37 6 =
18 120 =
=
16
100
Now the reverse,
change these decimals
to fractions
19 0.47 =
21 2.13 =
1
20 1.1 =
22 35.7 =
38 8 =
1
Convert to improper
fraction
31 4 4 =
5
=
=
39 5 =
1
=
0.1666
6
444
1 .0000
7
FREEFALL
MATHEMATICS
ALGEBRA 1
Completing Number Patterns
To complete number patterns either a "rule" is required which tells you how the number
pattern is created, or, enough numbers (usually 4) are needed to determine the rule yourself.
Once the rule has been found the last number is used to find the next one and so on.
In column 1 the rule is given, use the last number given and put it in the rule to get the first
missing number. Then using the new number, put it in the rule and get the next missing value,
and so on.
In column 2 you are asked to create the next pattern by building it with matchsticks. Then find
the missing numbers to extend the pattern. Then write the rule in words as you were given in
the first column.
Column 3 has number patterns, note how the numbers are changing and carry on the process
for the unknown numbers in the boxes. The last few questions are more difficult and require
two operations, for example multiplying by two an subtracting one.
Completing Number Patterns
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Draw the next match-stick
pattern then complete the
numbers and pattern in
words below it.
Complete the patterns
by finding the missing
numbers
12
1 Start with 2, "add" 3
2, 5, 8,
4th
18 1, 2, 4, 8,
pattern
,
4, 7, 10,
3 Start with 5, "add" 5
,
,
Start with
, 20,
,
"add"
, 40,
4th
,
6 Start with 25, "subtract" 5
,
,
7 Start with 2, "multiply by" 2
then add 1
2, 5, 11,
,
,
Start with
3rd
,
,
,
,
10 Start with 3, "multiply by" 2
then add 1
3, 7,
,
,
,
11 Start with 100, "divide by"
2 then add 2
100, 52,
,
,
,
,
,
25 1, 7, 13, 19,
,
,
26 15, 10, 5, 0,
,
,
27 197, 157, 117, 77,
,
,
28 54, 63, 72, 81,
,
6, 10,
9 Start with 6, "multiply by" 2
then subtract 4
6, 8,
,
24 1 000, 100, 10, 1,
14
,
, 70,
,
(decimal answers)
,
,
8 Start with 5, "multiply by" 3 pattern
then subtract 5
5, 10,
,
19 320, 160, 80, 40,
23 87, 91, 95, 99,
3, 7,
25, 20,
,
22 5, 14, 23, 32,
pattern
,
,
21 88, 99, 110, 121,
,
5 Start with 80, "divide by" 2
80, 40,
,
20 1/5, 1, 5, 25,
4 Start with 5, "multiply by" 2 13
5, 10,
,
17 20, 35, 50, 65,
2 Start with 21, "subtract" 3
5, 10,
,
16 4, 9, 14, 19,
,
21, 18, 15,
Complete the patterns you
will at times get answers
which include decimals.
,
,
15
4th
These are harder, you will
have to multiply by a
number then add/subtract.
29 1, 4, 10, 22,
,
30 2, 4, 10, 28,
,
31 1, 4, 13, 40,
,
pattern
,
,
,
,
,
32 2½, 10, 25, 55,
33 1, 2, 5, 14,
34 1, 6, 16, 36,
,
,
Writing Algebraic Expressions
An algebraic expression involves pronumerals (which are letters), like a, b, etc. Why use
letters in mathematics? A letter can be used to represent different numbers, so it is a quick way
to use the same 'formula' over and over.
To write expressions in words there are several words that should be understood:
addition - increase, raise, sum of, plus, and add.
subtraction - decrease, reduce, subtract, minus and take away.
division - divide, over, find the quotient (answer after division).
multiplication - times, product of, multiply, times, lots of
squared - square of, multiplied by itself
In Column 1 write the algebraic expression that the sentence describes, this is sometimes not
straight-forward. E.g. Subtract g² from the product of 9 and m → 9 × m - g²
While g² is written first in the sentence, it is the last mathematical operation to be completed
so it is written last in the expression.
Column 2 is the reverse of column 1, construct the sentences given the expression. Some will
require order of operations to be considered so be careful.
Column 3 requires an expression for the perimeter of the shapes, write each side separated by
a ‘+’ sign, then add the numbers together to get a total but leave the letters as they are.
Different letters are left with plus signs between them, you are about to learn how to add
pronumerals, but for this exercise this is not required.
b+4
Example.
a
a
b+4
P=b+4+a+b+4+a
= b + b + a + a + 8 (leave your answer like this or simplify as below)
= 2b + 2a + 8 (you will learn this on the next sheet)
Writing Algebraic Expressions
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
This time write a sentence
that matches the algebraic
expressions below
Read the sentences and
write algebraic expressions
to describe them.
Example
Increase 15 by t squared
15 + t²
1 The sum of g and 20
Construct an algebraic
expression for the
perimeter of the
shapes below.
Use words like: squared, times,
increase, decrease, raise, reduce, 18
sum of, subtract, product, plus,
divide, multiply, minus and add.
15
w
P=
11 2 × m
h
7
19
2 Decrease w by 29
x
12 k + 5 - d
d
7
3 Multiply 5 by d
* order of operations
20
22
13 a + 2 × b*
4 From 15 subtract u
g
40
x
5 The sum of a, b and c
m
14 854 ÷ 3 × f
6 Raise 16 by r squared
These two are harder
7 Reduce b by 870
8 Subtract 38 from the sum of
w and a
15 9 + h × c
21
b+c
a
16 a² + 25 - q
22
9 Reduce the quotient of 20
and q by 8
h+a
a
h+a
17 x² - 3 × a
10 From the product of m and 9
subtract 6
a
Formulae and Substitution
Substitution means to replace a pronumeral 'letter' with a number. This is just like using a
formula and putting the values in. Just remember that when a number is before a letter it is as
if there is a multiplication sign between them.
Eg. G = 4f, find G when f = 12 min
G = 4f
= 4 × 12
G = 48 min
Note that G is written on the bottom row and includes the units, whatever they may be, in this
case min (minutes).
In the 2nd column you are required to construct a formula that matches the sentence, this may
be challenging. Use the underlined letters to make the formula.
Eg. Chris is 4 times older than Ian, if Ian is 9 years old calculate Chris' age.
C = 4I
=4×9
C = 36 years
The 3rd column requires substitution to be built up in stages, if x² has been found then to
calculate 2x² just double the answer for x². Note that 2x isn’t squared. Calculate x² then
multiply it by 2. Write your answers going down the column and the 3rd row will be some
small combination of the first two rows.
Formulae and Substitution
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Use the formulae below to
substitute then solve
1 Find the perimeter of a
square using P = 4l where
l = 12 cm.
P = 4l
use = signs
= 4 × 12
Answer the top row then
the 2nd row then add the
two rows to help answer
the 3rd row.
Use the underlined letters
to construct a formula for
the following. Then
substitute and solve.
6 Julian is twice as old as
Ann. If Ann is 17 years old
how old is Julian?
show units
11
Use k =
2
3
4
2
4
6
4
7
10
5
10
20
k²
P=
7k
rewrite letter
2 Find the average age of
two girls using: A = x + y
2
where x = 6 years and
y = 16 years.
7 There are 12 pieces in one
family pizza how many
pieces are there in 7?
12
Use g =
x+y
A=
2
=
g²
8 Britney bought a pair of
shoes at a price of $76.85.
Find her change from $100
3 Change minutes to seconds
using T = 60m where
m = 4 min.
4 Find the area of a triangle
using A = ½bh where
b = 16 mm and h = 5 mm
5 Now use b = 12 m and
h = 10 m to find the ∆ area.
k² + 7k
5g
2g² + 5g
13
Use m =
9 Sean has half the money
that he had when he left
home. How much money
has he now if he started
with $8.70.
m²
2m
m² + 2m - 3
14
10 If a car has 5 wheels and a
motorcycle has 2. How
many wheels do 3 cars and
6 motorcycles have?
Use e =
e²
4e
2e² + 8e
Substitution
This sheet doesn’t feature Negative numbers. Sheet 08 in the Algebra 2 folder does.
Substitution means to replace the pronumeral (letter) with a number, then solve the question
just like any other operations exercise. There is only one difference, when the letter is after a
number put a × between them.
E.g. If a = 5 then 7a = 7 × 5 =35, don’t just change the a for a 5, i.e. not 7a = 75. Another
mistake is when a number is squared.
E.g. If a = 9 a² = 9 × 9 = 81, it is not a² = 9 × 2 = 18.
Column 1 asks you to use a = 4 in all the equations. Substitute a into each equation then solve
the equation (get an answer). Remember to watch for order of operations, an example is at the
top of the column.
Column 2 involves squares (²) and there are two letters used, k and e. Use the same process as
with the previous column but remember that a squared number is that number times itself.
When there is a number before a squared term, such as 5m², then only the m part is squared
(= 5 × m²) not the whole line (5m)².
Column 3 involves using brackets, remember that brackets are done first. Otherwise these are
the same style of question as in the previous column.
Substitution
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Use a = 4 to find the value
of the other pronumeral
Example
q = 10a - 11
question
Example
or,
d = e² + 5
d = e² + 5
49
40
= 10 × 4 - 11
1
solution (answer)
b=a+3
2
3
4
h=a+8
m = 2a - 2
w = 6a - 5
9
11
17
9
d=
x = f (m - 3)
18 g = f (m - f )
12
e² - 4
v = 2 (m + 7)
16 c = 9 ( f + 5)
10 m = k² - k
=
5
t = 70
15
g = e² - 5
n=
= 5 × (10 + 4)
d = 54
t = k² + 3
t = 5 (m + f )
14
= 72 + 5
d = 54
8
Example
49
=7×7+5
substitution
q = 29
With brackets solve the
inside of the brackets first,
(order of operations). Let
m = 10 and f = 4
This time use k = 5 and
e = 7. Remember that
k² = 5² or 5 × 5 not 5 × 2
8k²
=
4
19
n = 7 (20 - m)
q = 5a + 7
20 q = m (m - 2f )
6
u = 11a - 23
13
b = 100 - e²
21
7
k = 11a - 10
14 r = 2k²
t = 2m ÷ ( f + 1)
Substitution - Table of Values
Substitution is replacing a 'letter' with a number. A number before a letter means multiply the
number and the substituted number together. So if the equation is n = 2a, and a = 3. Then,
2 × 3 = 6 not 2a =23. Similarly, when you have x² and x = 5 this means 5² = 5 × 5 = 25 not
5 × 2 = 10.
In column 1 an equation or ‘rule’ is given and you are asked to substitute the number given in
the top box for the letter on the right hand side of the equation. Calculate the answer and put
the value for the letter in the box below the number used. An example is at the top of the
column.
Column 2 requires you to state the rule given a completed table, use this method.
•
In a rule: y = ▲x + ■ the value ■ is found when x = 0, (the first value in the bottom row).
In the example below ■ is 3, so that means y = ▲x + 3
•
x
0
1
2
3
4
x
0
1
2
3
4
y
3
7
11
15
19
y
3
7
11
15
19
Look at the next value, for x = 1 we get y = 7. So 7 = ▲ × 1 + 3. Which gives y = 4x +3.
Question 13 and 14 are different, the numbers are decreasing in size, a rule similar to that used
in question 4 applies in this situation.
In the 3rd column complete the table, then to plot the points go across the number in the top
box and up the number in the bottom box. For the circled numbers above, for example, this
would mean that you go across ‘0’ (which means don’t move) then go up 3. Then plot a point.
Then plot the 2nd point, by moving across 1 and up 7, and so on. The points will be in a
straight line, if they aren’t you have made a mistake. Then draw a line through the points.
Extend the line the full size of the graph and place arrow heads on each end.
Substitution - Table of Values
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Example
Use the rule and the value
in the top row, to find the
bottom row value.
7
Rule : m = 2a + 1
h
Rule : h = c²
c
=7
=9
+1
+1
×4
=2
b
=5
x
×3
9
+1
7
=2
5
=3
3
×2
1
+1
m
=2
8
=1
4
×1
3
+1
2
=2
1
×0
0
=2
a
1
0
1
2
3
3
4
15
Rule : b = 2x² + 3
0
1
2
3
Rule : p = 2s + 1
s
4
0
4
Rule : m = 3t + 1
0
5
10
15
20
m
c
0
1
2
3
4
h
0
2
4
6
8
x
0
1
2
3
4
b
2
3
4
5
6
t
0
1
2
3
4
p
0
5
10
15
20
11
3
Rule : d = 3e - 2
2
4
6
8
10
1
2
3
4
p
9
8
7
6
5
4
3
2
1
p
10
2
e
2
9
c
t
1
This time find the rule
Rule : c = 2k
k
0
Complete the table below
then plot the points on the
graph, the numbers in the
top row go across and the
bottom row go up. Then
draw a line through them.
0
1 2 3 4 5 6 7 8 9
s
16 Graphs can be used to find
values that aren't in the table.
Look at the graph in Q15. and
give the value of p when:
s = 2½
p=
d
4
r
2
4
6
8
10
w
Rule : d = 3g - 3
g
50 150 200 250 300
e
6
x
y
20
30
40
0
5
10
15
20
t
h
5
20
35
50
65
v
a
0
2
4
6
8
k
20
18
16
14
12
i
0
1
2
3
4
r
10
8
6
4
2
14
Rule : y = 3x - 2
10
c
13
5
50
Rule : v = 8 - t
17
12
Rule : w = 10 - r
v
0
1
2
3
4
9
8
7
6
5
4
3
2
1
0
1 2 3 4 5 6 7 8 9
t
Adding and Subtracting Pronumerals
Think of algebra as being items. Standing at a street corner you may see trucks, cars and buses.
If you see 3 trucks (3t) and 2 buses (2b) you can't add these together, so: 3t + 2b remains as
3t + 2b. If you see 3 cars then 2 buses then 4 more cars, you have seen a total of 7 cars and 2
buses, or 3c + 2b + 4c = 7c + 2b.
So 'like' terms (the same letter) can be added and subtracted but not different letters.
The same with numbers and terms. Numbers can be collected together and the letters can be
collected together but letters can’t be added/subtracted with numbers.
For example,
3 + 4c + 4 + 5c - 2
= 9c + 5
A common mistake is to write a '1' in front of a single letter, this isn't giving your answer in a
fully simplified way.
For example,
5b - 7 + 2b - 6b
= b - 7 (don't write this as 1b - 7).
Just like numbers, when the same term is subtracted from itself it will give zero.
Example,
5f + 2 - 3f - 2f
= 2 (as 5f - 5f = 0)
Try to write the 'letters' part first when the answer is written, ie: write 5t + 6 rather than
6 + 5t.
Adding and Subtracting Pronumerals
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Simplify these, remember
that you don't write a 1 in
front of a letter, 1m = m
Simplify these, remember
that you can't add/subtract
unlike terms.
Simplify these, remember
that you can't add/subtract
numbers and terms.
1 t+t
21 a + g + a
41 2a + 3 + 2a
2 k + 2k
22 2e + c + e
42 5e + 4 - 4e
3 6m - 4m
23 4h + 3h + 2d
43 10d - 3d - 6
4 12i + 3i
24 10m - 3n - m
44 3 + 4q + 2 - q
5 2j + 5j
25 2c - 7q + 3c
45 4y + 12 - 3y
6 16e - 9e
26 9k + 3k + n
46 4k + 2 + 5 - k
7 9x + 5x + 3x
27 3e + 2f - f
47 a + 2 +2a - 1
8 12b + 9b + 4b
28 m + p - m
48 2d - 5 + 4d
9 6n + 2n + n
29 4d - 2m + d
49 5 + 3e - 2 + e
10 t + 6t - 2t
30 6m + 3b - 5m
50 t + 4 - t - 2
11 3e + 5e - 4e
31 2d + 3d - h
51 3v - 6 - 2v
12 12r - 7r + 2r
32 t + 5e - t
52 10 + 2b - 5
13 11k - 7k - 4k
33 5e - 2e + q
53 a + 8 - a + 8
14 24h - 9h - 14h
34 3a - 2d - 2a
54 6x - 10 + 4x
15 2c + 5c +7c
35 4c + 2b - b
55 9m + 6 - 8m
16 u + 5u - 2u
36 6w + n - 6w
56 5 + b - 4 + b
17 6d - 3d - 2d
37 6n - 3n +2d
57 2f - 10 - f
18 14q - 11q - 3q
38 2f +3f + f - 2d
58 10 + k +15
19 8v - 2v - 3v
39 a + b + a + b
59 h + 3 + h + 1
20 12s + 3s - 14s
40 d + 3d + 3e
60 y + 2y - 3 + y
Further Adding and Subtracting Pronumerals
To add and subtract letters they must be the same. For example:
5a + 2a = 7a,
7bc - 6bc = bc and
3ab + 2ba = 5ba (or 5ab).
However unlike terms can’t be added together:
4c + 3a = 4c + 3a,
7qm - 3q = 7qm - 3q and
3ab - a - 3b = 3ab - a - 3b.
The order of the letters doesn't matter: 4bc = 4cb, 5xqm = 5qxm = 5mqx so terms can be added
when the letters are the same but in a different order.
The same applies to when terms have a power, such as 3b², 5d²f and 2x²y². Terms with different
powers can't be added or subtracted. But you can add/subtract these:
a² + 3a² = 4a² and
5x²y - 2yx² = 3x²y,
but not thse :
a² + 2a = a² + 2a,
3e² - 3ed = 3e² - 3ed.
Remember that 3k²mw² = 3mk²w² = 3w²mk², the order doesn't matter, so long as the powers are
on the same letters.
The final column requires the missing sides to be found and then the sides added together to
get the perimeter. Follow the example at the top of the column.
Further Adding and Subtracting Pronumerals
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Collect like terms and
simplify.
Collect like terms, this
time using powers (²)
Example
Example
bx + b + 5 + 2xb + 3b + 7
50 + a² + 3a + 17 - 2a + 3a²
= 3xb + 4b + 12
1 3a + 5t + 3 - a + 6 + 4t
= 4a² - a + 67
Use your algebra skills to
give an expression for the
perimeter of the shapes
below. Collect like terms.
Example
ac
b+7
b+7
12 17 + 2m² - m + m² +14
Fill in missing
sides
2 12d + 2a + 16 - 5d - 7 + 3a
13 7h² + 3h - 3h² + 4 - 4h²
ac
P = ac + ac + b + 7 + b + 7
P = 2ac + 2b + 14
3 4ux + 3xa - 3xu + 2ax
14 20k² + 14 - 12k - 8k² - 7
22
ha + 4
4 5 + 5m + 21 - 4m + 3ma
15 9r² + 5tr + 7 - 3r² + 16 - 2rt
5 12kd - 5dk - 5dk - 8 - kd
16 13e + 5e² - 8e - 2e² - 5 - 4e
23
6 5 + a + b + 6 + 2ab + 3b
17 20cw + 37 - c² - 18wc - 19
7 5tu + 17 + 12ut - 12t - 12
These are harder, note
which term has the ²
8 20a - 20 - a + 5ta + 11at
2 ed
de + 6
18 9fg² + 3f ²g + 3g²f
24
9 30ue + 15e - 5e - u
x
19 10yt² + 3y²t + 5yt²
6x
10 at + ta + at + ta + a - 4t
20 5a² + 3a²t - 3a² - 2ta²
11 17 + 20as - 3 - 15sa - 3as
21 y² + 2xy² + 3y² - y²x²
2x
Multiplying, Dividing and Using Brackets with Pronumerals
We show multiplication between numbers and pronumerals as the number first then the letter.
For example, 2 × a = 2a, 5 × d = 5d and 4 × a × d = 4ad (or 4da).
When several numbers and letters are mixed up, calculate the product of the numbers first,
then the letters. For example, 5 × e × 2 × g = 10eg. Note that when the same letter is featured
twice a squared term will result. E.g. 3 × q × 5 × q = 15q². A letter times itself is that letter
squared. (a × a = a², c × c = c² etc )
When a pronumeral is divided by a number, write the letter 'over' the number.
r
3m
x
r÷6=
3×m÷4=
6
7
4
Sometimes these can be simplified. Like fractions, if both the top and bottom numbers can be
divided by another number this will simplify the answer.
So, x ÷ 7 =
3
12e ÷ 8
=
(divide by 4)
2
12e
8
=
3e
2
3
12e ÷ 4
=
(divide by 4)
1
12e
4
= 3e
(you don't write a number over 1)
Remember that the largest number (HCF) that divides into both the top and the bottom
numbers (not just any number) is to be found to fully simplify the answer.
Expanding, (removing brackets) is multiplication in steps. The number outside the brackets is
multiplied by the first term inside the brackets and then the second term. These are added
together if the sign between them is a +, subtracted if there is a negative sign - .
For example,
5(y + 4) = 5 × y + 5 × 4
= 5y + 20
e (m - e)
=e×m-e×e
= em - e²
Multiplying, Dividing and Using Brackets with Pronumerals
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Multiply these remember
numbers first then letters.
d × d = d² not 2d
1 2×a
2 c×3
Divide these, remember
that ÷ is the same as
Simplify numbers when
you can
Example
3
9 × m ÷ 3 =1
21 c ÷ 7
4 2×f×5
8 7×t×k
9 g×2×g
22 2 × g ÷ 3
23
15 × k
14 10 × z × q
17 e × e × t × t
20 v × v × 3 × y
35 5 (c - 8)
37 h (h + 2)
25 7 × b ÷ 7
38 e (k + e)
39 5n (n - 11)
26 4 × f ÷ 8
27 18 × a ÷ 6
40 3x (x - 3 )
41 k (m + d)
42 3w (a + 2)
28
45 × y
9
29 50d ÷ 10
43 2q (3q - 2)
44 8e (7e + 3)
45 2m (e + 2m)
30 9e ÷ 6
18 d × a × d × 3
19 9 × 6 × t × t
34 6 (m + 3)
24 22 × e ÷ 4
15 h × 3 × b × 4
16 6 × n × n × b
3a (a - 5)
36 7 (3d + 2)
12 n × 5 × 3 × n
13 x × 4 × t
5 (w + 10)
3
10 a × b × c
11 q × u × q × 7
= 3m
33 4 (c + 6)
6 8×h×2
7 3×x×y
3
Example 1
= 5w + 50
Example 2
3 4×e×2
5 3×4×u
9m
Multiply the outside letter
or number by the inside
parts separately. This is
called 'expanding'.
31
35 y
46 9w (w - 4)
47 a² (b + c)
10
48 x² (d² + t²)
32 48a ÷ 3
49 3u (2 - t²)
= 3a² - 15a
Further Multiplying, Dividing and Using Brackets
with Pronumerals
Column 1 is multiplication between numbers and pronumerals. Always write the number first
then the letter (or letters). For example, 2 × a = 2a, 5 × 3d = 15d and 4 × a × d × 3 = 12ad
(or 12da).
Note that when the same letter is featured twice a squared term will result. A letter times itself
is that letter squared. (a × a = a², c × c = c² etc ) For example, 3q × 5q = 15q², 2c × 3c = 6c²
and abc × bc = ab²c².
The 2nd column are exercises on division. Like fractions, if both the top and bottom numbers
can be divided by another number this will simplify the number part of the answer. Remember
that the largest number that divides into both the top and the bottom numbers (not just any
number) is required to fully simplify.
3
=
12e ÷ 8
12e
2
8
=
3
3e
=
12e ÷ 4
2
(divide top and bottom by 4)
1
12e
4
= 3e
(divide by 4)
With division of terms the following applies. A term divided by itself equals 1, so it cancels
out.
m÷m
=
m
m
4
= 1
8m ÷ 6m
=
3
8m
6m
=
4
3
4
8mb ÷ 6m =3
8mb
6m
4b
=
3
A squared term divided by itself equals the term. (c² ÷ c = c)
d² ÷ d
=
d²
d
3
= d
6d² ÷ 4d
=
2
6d²
4d
=
3d
2
4
8d²b ÷ 6d
=
3
8d²b
6d
=
4db
3
The 3rd column uses terms instead of numbers in area formulae. This is the same as
multiplication (in the first column). The only difference is that triangles have a ½ in the
formula. To allow for this, divide one of the numbers on one of the sides by 2 (hopefully its
even). For example:
½ × 6m × t (= 3m × t )
½ × 6m × 2t (= 3m × 2t)
= 3mt units²
= 6mt units²
Because no units of measurement are mentioned in the question write units² in the answer.
Further Multiplying, Dividing and Using Brackets with Pronumerals
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Multiply these, remember
numbers first then letters.
These are challenging!
Divide these, remember
that ÷ is the same as
Simplify the numbers if
you can, then the terms.
Find the areas, here are the
formulae. A = l² (square)
A = lb (rectangle) and
A = ½ bh (triangle).
1 2h × 3u
Example
4
2 4r × 3f
8e² ÷ 2e
=
Example 1
3a
3 10k × 3k
21 3u ÷ 2
22
5 7md × 2d
6 3e × 2d × 5
2b
2e
= 4e
23 20k ÷ 6
A = lb
A = lb
15x
3
Example 2
3x
5xa
4 4c × 2c
1
8e²
= 3a × 2b
= 3x × 5xa
A = 6ab units² A = 15x²a units²
33
2a
7 8y × 2x × 3y
24 8bc ÷ 3b
8 4de × 3ed × 2
9 5b × 3c × bc
34
25 9ch ÷ 3h
4y
2u
10 ac × a × c × b
11 2t × utx × u
26 14ed ÷ 4d
35
12 d × 2d × u
27 4g² ÷ 2g
5ef
13 cn × nc × ab
14 4v × 2w × 5u
15 abc × abc
28
5b²
36
29 6ck² ÷ 2k
16 2abc × 2cba
17 9k × 2ek × u
20 abc × bcd
A = ½bh
eb
30 16d² ÷ 6d
18 2y × 3d × dy
19 5fn × 3 × fe
9f
15b²
31
9ab²
6ab
32 a²b² ÷ ab²
4ea
37
4abc
6bcd
7
FREEFALL
MATHEMATICS
ANGLES
Naming Angles
Naming angles allows you to explain which angle or part of a shape you're dealing with.
The first column introduces intervals, lines and rays. With intervals and lines you can name
them in either direction. The interval below could be WQ or QW. You are asked to give both
the names in these questions.
W
Q
With rays, name them in the direction of the arrow only. The ray below is called HG only, not
GH.
H
G
In column 2 you are asked to name the rays and also the vertex, this is the meeting point of the
two rays, the ‘elbow’ if you like. The vertex is named by the letter at the point alone. To show
it is a vertex we put a hat on it, like an upside-down ‘v’.
To name an angle we generally use 3 letters, moving from ray-to vertex-to ray. The vertex is
always in the middle. To show that it is an angle you put an angle sign in front of it or write the
letters placing a hat over the vertex.
< ABC
or,
^
ABC
When you name a shape (in column 3) you start at any point then go around the shape the one
way. If the letters are in alphabetical order you might like to start at the earliest letter. No
symbol is put before the name of a shape, though sometimes this is done with triangles.
(e.g. ∆ABC)
Naming Angles
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Give the 2 possible names
for these intervals.
1
2
A
B
K
Z
Name the 2 rays and the
vertex for these angles
or
Y
K
or
or
L
W
4
or
U
ray
T
or
R
6
or
I
H
i)
E
V
R
i)
ii)
or
X
<
^
ray
ray
vertex
ray
ray
vertex
Name these angles, with
the vertex as the middle
letter. Use either ^ or <
16
J
P
Z
17
L
N
T
K
Place the given symbol in
the following angles
B
D
23
F
C
P
G
E
F
T
B
A
I
H
Note E is the point of contact
18
11
K
M
H
E
<
10
12
22
A
X
Y
J
C
B
ii)
or
F
N
E
D
Z
Name these rays
9
F
A
L
E
V
21
or
K
8
vertex
15
A
C
K
ray
A
S
H
ray
S
Q
B
T
20
P
Give the 2 possible names
for these lines.
7
A
vertex
ray
14
C
V
^
T
5
D
19
13
D
3
Name these shapes using
their letters. Then name
the angle with the symbol.
A
M
O
Y
< BCD
< AED
< FED
< AEI
< IHG
< ABC
Classifying Angles
Classification means to group into categories. There are six classifications in all:
•
acute angles are between 0° and 90°
•
a right angle is exactly 90°
•
obtuse angles are between 90° and 180°
•
a straight angle is exactly 180°
•
reflex angles are between 180° and 360°
•
and a revolution is 360°
How do you remember all these? You should be able to remember the straight angle, right
angle and revolution, it is the others you could mix up. Remember that as the angle increases
you move through the alphabet. A (acute) is before O(Obtuse) which is before R(reflex).
There is a guide at the top of the page to refer to. With the angles, the measured side is the side
marked with the arc (part circle) or the right angle symbol.
When asked to name angles remember that you can use either the hat or the angle sign method,
as below.
< ABC
or,
^
ABC
In column 3 the word internal is used. What does internal mean? It means the inside (interior),
so the internal angles are the angles on the inside of a shape.
Classifying Angles
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Obtuse
between
90° and 180°
Acute < 90°
Reflex
between
180° and 360°
Straight 180°
Right 90°
Classify the following
angles
1
Revolution
360°
Complete the
table below
D
E
24
T
20
Classification
Angle
2
X
115°
J
333°
3
acute
73°
4
25
165°
C
O
V
360°
7
Using the ray AB in all
angles name an acute,
obtuse and straight angle
8
21
reflex
K
200°
6
obtuse
L
180°
5
9
For the two shapes
below name internal
angles that are acute,
obtuse or reflex.
B
A
10
E
C
acute
Y
obtuse
reflex
26 A family size pizza has the first
piece eaten forming an acute
angle. Each piece is the same
size and when removed increases
the size of the angle. Find the
number of:
D
11
12
13
acute
obtuse
straight
G
22
exposed tray forms
an acute angle
Q
B
X
14
i) acute angles that can be
A
made
15
acute
16
23
obtuse
S
D
I
A
17
E
18
straight
B
ii) pieces eaten to make a right
angle
iii) pieces eaten to make the
largest obtuse angle
iv) reflex angles that can be
19
acute
obtuse
straight
made
Estimating and Measuring Angles
An angle is used to measure rotation, a protractor is required for this worksheet.
To answer this sheet look at each angle and estimate its size, estimate meaning a skilled guess,
so it is unlikely to guess the exact answer, but within say 15° would be very good. Don't use a
protractor until you have estimated all of the angles, use the guide at the top right of the sheet
if you need the help.
When using a standard protractor ensure you use the correct scale (inside or outside, make sure
the scale used starts at 0°) and if the angle is a reflex (greater than 180°), measure the outside
of the angle then subtract the angle from 360°. If an angle is measured and it is above 90°, then
it must be larger than a right angle, if it isn’t you have made a mistake. Use your estimation as
a guide as well, and then see how close you were.
Estimating and Measuring Angles
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Estimate the size of all the shaded angles and place
your answers in the table. Then using a protractor
measure all the angles. See how well you estimated.
(Within 15° above or below)
90°
270°
180°
360°
1
2
4
3
5
9
6
7
8
11
12
10
13
14
15
16
17
18
No
Estimate
Measured
No
Estimate
Measured
No
1
7
13
2
8
14
3
9
15
4
10
16
5
11
17
6
12
18
Estimate
Measured
Constructing Angles
The previous sheet involved measuring angles, this sheet is about the construction of angles.
The method is as follows:
•
Draw a horizontal line
•
Measure the given angle from the end of the drawn line, plotting a point at the angle
•
Draw a line from the end of the line to the point
•
Label the angle with the letters supplied in the question, remember the vertex is point on
the elbow of the angle.
•
Draw an arc or sector at the vertex to show the angle is the inside or outside of the angle.
•
If the angle is greater than 180° then calculate (360° - the angle) the obtuse/acute angle
and use that angle. Draw the sector on the other side of the angle.
A
B
Vertex
A completed reflex
angle with sector
A completed angle
with sector
A completed angle
A
C
B
A
C
B
C
Constructing Angles
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Draw these acute angles
Draw these obtuse
angles
Draw these reflex
angles
1 Construct a 30° angle
< ABC
5 Construct a 110° angle
< FZT
9 Construct a 240° angle
< BYC
2 Construct a 60° angle
< EVT
6 Construct a 165° angle
< TRE
10 Construct a 190° angle
< XOJ
3 Construct a 37° angle
< NTY
7 Construct a 138° angle
< JPG
11 Construct a 337° angle
< JVA
4 Construct an 73° angle
< ALR
8 Construct an 97° angle
< NJD
12 Construct an 254° angle
< PUK
Creating an Isometric Cube
Isometric drawings are done using 30° angles. These are either drawn with a protractor or a
30°/60° set square. Make sure you don’t use a 45° set square, as then you will be drawing an
oblique drawing. Method:
•
•
From the point on the worksheet, or from a point on your page draw a 30° line to the left
and another to the right.
Measure off 10 cm and draw solid lines
•
Draw 3 vertical lines: from the start point and from ends of both lines you have just
drawn. Make sure that these lines are vertical by measuring the distance from the side
border of the sheet, make sure it is the same distance at the top and the bottom.
•
Join the 3 ends together
•
Draw a light construction line straight up the centre line, and construct the 30° angled
lines from each end. Join up and you are done!
Time Out Activity - Using Angles to Create an Isometric Cube
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Construct an isometric cube with sides 10 cm using
your protractor skills and a ruler. Hint: draw the
bottom edges first then the 3 vertical lines.
A smaller version of
the finished product
30°
30°
Adjacent Angles
Adjacent means next to, in the case of angles this means that the angle shares a common ray
(or arm) and vertex with the angle beside it.
In Column 1 the table requires you to add the two angles together to get a total, the sum of two
adjacent angles. This continues down the column only with graphical representation, note that
these angles aren't to scale, so don’t use the angles as a guide. In Q.3 there are three angles but
only two angles have numbers, the third is a right angle which is 90°. It is suggested that you
write in the '90°', that way you won't forget it in the addition.
In the 2nd Column Q.6 asks you to name the adjacent angle. This requires you to name the two
angles that touch the given angle. As the layout is circular that means each angle has an
adjacent angle on each side of it, don't name one angle then the same angle with the letters
reversed. Q. 7 is another table which this time gives you the total of two angles, and one of the
angles. Subtract the angle from the total and you have the answer.
The same method is used for the rest of the sheet Subtract the known angles from the total to
obtain the unknown angle. Except Q.10 & 12, with these use division! Q.10 gives you the
hint.
Adjacent Angles
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Angles 1 and 2 are
adjacent find the sum
of the two angles
Name two adjacent
angles to the following
A
6
1
J
Angle 1
Angle 2
Total
35°
42°
77°
19°
27°
7°
96°
115°
153°
21°
117°
< ABC :
&
273°
24°
< HBF :
&
311°
46°
< UBC :
&
2
x =
F
7
34°
x =
x
27°
68°
x=
A
< TNI = 68°
T
I
11
e
85° y
240°
Total
Angle 1
38°
15°
12
t t
191°
115°
187°
11°
304°
221°
119°
13
17°
208°
87°
Angle 2
52°
311°
t
c
265°
147°
14
115°
You are given the total
now subtract or divide to
find the missing angles.
< ANI = 77° 8
64°
115°
339°
x=
N
e
=
e
This time subtract. Given
the total and one angle,
find the other angle.
+
3
2e =
H
x
93°
10
B
C
Find the size of the angle
formed by adding these
adjacent angles.
4
U
Keep going!
d
162°
< TNA =
m
m =
k
m =
5
9
41°
65°
29°
15
-
q
84° t
149°
255°
Complementary and Supplementary Angles
Complementary angles are angles that add to 90° and supplementary angles are angles that add
to 180°. How do you remember which is which? C comes before S in the alphabet, as 90°
comes before 180°. Or look at the construction below.
The C can be changed to a 9
(90°) and the S can be changed to
an 8 to remember 180°
C S
In Column 1 the first three questions ask you to verify that the two angles are complementary.
Add the two angles together, if the sum equals 90° then they are.
For the rest of Column 1 you have to find the angle that when added to the angle given, has a
total of 90°. You answer this by subtracting the given angle from 90°. The most common
mistake is thinking that a right angle is 100°, or at least using it in calculations, remember 90°
not 100°!
The 2nd Column has the same layout as the first but instead of the angles adding to give 90°
they should total 180°, to be supplementary. The remainder of the column requires you to
subtract the given angle from 180° to get the answer. When a right angle is involved write in
90°, this process should always be done as the angle may be overlooked.
The 3rd column is a mixture of the first 2 columns.
Complementary and Supplementary Angles
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Are the following angles
complementary?
Are the following angles
supplementary?
24°
45°
55°
156°
22°
y
Circle: Yes / No
Circle: Yes / No
2
17°
20
10
1
Find the missing angle
11
63°
21
141°
127°
68°
Circle: Yes / No
Circle: Yes / No
3
48°
42°
12
22
87°
83°
Find the unknown adjacent
supplementary angle
Find the unknown adjacent
complementary angle
13
d = 90° d
32°
d =
k = 180° k
46°
14
5
37°
23
p
102°
24
k =
25
123°
t
m
Circle: Yes / No
Circle: Yes / No
4
j
61°
d
y
x
9°
15
56°
26
u
6
17°
n
22°
16
y
7
27
39°
a
w
50°
17
p
w
41°
28
136°
h
18
8
29
169°
s
c
25°
19
9
72°
h
152°
e
30
b
33°
k 43°
Complementary Angles
Complementary angles are angles that add to 90°. Supplementary angles add to 180° (the next
sheet). How do you remember which is which? C comes before S in the alphabet, as 90°
comes before 180°. Or look at the construction below.
The C can be changed to a 9
(90°) and the S can be changed to
an 8 to remember 180°
C S
In Column 1 you are asked to find the complement, this means find the angle that when added
to the angle given, has a total of 90°. You do this by subtracting the given angle from 90°. This
continues down the column, subtract the angle given on the diagram from 90° to find the
answer. The most common mistake is thinking that a right angle is 100°, or at least using it in
calculations remember 90° not 100°!
The 2nd Column through to Q 16 in the 3rd Column is an extension of this. The same method
is used only more than one angle is subtracted from 90°. Question 7 shows the method of
working required.
Questions 17 and 18 require you to add the letters together, these equal 90°. Then divide by the
number in front of the letter to get the value of the pronumeral. Note that the value of the
pronumeral is being found, this isn’t necessarily the value of the angle.
Questions 19 and 20 are more difficult questions and require an extra step. The letters are
added on the left of the equals sign, and the subtraction of the angle from 90° is put on the
right side. See the example below.
5d = 90° - 45°
Example
d
45°
4d
5d = 45°
d = 9°
Divide by
5
Complementary Angles
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Find the complementary
angle to those given in
the table below.
Now there are more than
2 angles. Solve these.
14
w
7
1
Angle
Complement
30°
60°
17°
a = 90° - 33° - 33°
33° a
33°
45°
73°
8
15
21°
9°
24°
22°
57°
y
84°
48°
g
17°
9°
9
13°
Find the complementary
angle for the following
t
2
16
14°
11°
x = 90° x
27°
8°
5°
x=
25° j
34°
10
54° m
18°
3
Use division to find the
value of the letters.
11
d
53°
17
8° k
a
75°
4
42°
h
12
e
3d
3d
78°
19
13
40°
e
21°
27°
6
n
16°
b 36°
a
18
31°
c
37°
5
a
e
20
42°
3k
k
Supplementary Angles
Supplementary angles are angles that add to 180°. How do you remember the difference
between supplementary and complementary? C comes before S in the alphabet, just as 90°
comes before 180°. Or look at the construction below.
The C can be changed to a 9
(90°) and the S can be changed to
an 8 to remember 180°
C S
In Column 1 you are asked to find the supplement, this means find the angle that when added
to the angle given, has a total of 180°. You do this by subtracting the given angle from 180°.
This continues down the column, subtract the angle given on the diagram from 180° to get the
answer.
The 2nd Column has a table to complete, this time both the complementary and supplementary
angles are required. Subtract the angle from 180° to get the supplement and from 90° for the
complement. Q 9 through 16 add more angles into the problem, but the problem is still solved
the same way, by subtracting all the given angles from 180°.
Questions 17 and 19 require you to add the letters together, these equal 180°. Then divide by
the number in front of the letter to get the value of the pronumeral. Note that the value of the
pronumeral is being found which isn’t necessarily the value of the angle (see Question 20).
Question 18 is a more difficult question and requires an extra step. The letters are added on the
left of the equals sign, and the subtraction of the angle from 90° is put on the right side. See
the example below.
Example
145° 4d
d
5d = 180° - 145°
5d = 35°
d = 7°
Divide by
5
Supplementary Angles
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Find the supplementary
angle to those given in
the table below.
1
Angle
Supplement
110°
70°
For the angles below find
both the supplement and
the complement.
Angle Supplement Complement
25°
162°
65°
16°
13°
123°
79°
27°
57°
177°
3°
2
x
50°
14
a
42°
25°
15
41°
h
120°
21°
25°
Now there are more than
2 angles. Solve these.
9
x = 180° -
l
55°
16
x =
k
15°
3
22°
45°
35°
b
15°
8
45°
Find the supplementary
angle for the following
13
24°
t
10
w
4
40°
c
Use division to find the
value of these unknowns.
17
117°
g
g
11
5
d 63° 20°
18
4e
164°
a
20°
6
u
38°
19
12
4x
106° 40°
m
7
p
g
x 2x
2x
20 With the value of x found
above, find the value of:
2x =
4x =
Angles at a Point
Just like a circle, the angle sum at a point is 360°. To find the size of an unknown angle
subtract all the given angles from 360°.
In Column 1 the unknown angle is given a letter, just write the letter then '=' and subtract all
the given angles from 360°. Some angles are right angles, write '90°' in these so that you don't
forget to include the right angle in your calculation.
Column 2 requires you to use your algebra skills, add the letters and then divide 360° by the
number in front of the letter. Questions 13 through 15 have an additional step, use the same
method to find the value of the letter, but then multiply that value by the number in front of the
letter for each angle. For example if you find x = 25° and the angle in the question is 3x, then
3x = 3 × 25° = 75°. If the question asks 'to solve for x', then you don't need to do this, it is
only when you are asked for the angle, as the angle is 3x you must find the size of the angle, as
the size of x isn't sufficient.
Column 3 is a harder column and some students may experience difficulty with these. The
steps are a combination of the earlier problems.
•
add the letters together and write it then an equals, (in question 16. e + e = 2e)
•
then subtract the given angles (remember the right angles) from 360°
•
the next line write the letters sum again only evaluate 360° - the angle
•
then use division to solve like earlier problems
•
the last two questions are harder, they require multiplication to get the angle (like Q13-15)
Angles at a Point
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Use subtraction from 360°
to find the unknown angle
This time use division
1
8
d
k
k
d = 360° -
x
15
k =
2e = 360° -
9
y y
2
3k =
e
2e =
d =
305°
k
Use subtraction from
360° then division to find
the unknown angles
y y
165°
280°
e =
y
16
10
123° 117°
y
200°
17
11
aaa
a
a
aaa
4
109°
f
Use the same method but
this time find the value of
the letter and the angles
2k 3k
k
145° 60°
135° g
189°
7
n n
n
Now the angles
are different
20
5c =
2c
3c
285°
2c =
14
65° b
48°
152°
n
5c =
2d 3d
3d
56°
156°
k =
3k =
13
m
19
6k =
2k =
6
a a
a
120°
12
5
q
q
xxx
xx x
3
e
3c =
21
3k
5k 4k
3d
2d
Vertically Opposite Angles
Vertically opposite angles are equal when formed by intersecting straight lines. The diagram
below illustrates this. Don't let the word 'vertical' confuse you, as you can see if angles are
opposite each other horizontally they are also considered vertically opposite (equal).
180°
In Column 1 identify the vertically opposite angle, by drawing a star in the angle and then also
by naming it. Write the name of the angle on the line and colour the star beside it the same
colour as the one you placed on the diagram. Q 5 is more difficult as many angles are all on
the one diagram so take care with this question.
Column 2 asks you to find the value of the letter, this is just identification, (the answer is in
front of you) for Q. 6 through 9. But from Q 10 on…. some mathematics is required, this will
always be subtraction or division look at the examples below.
With column 3 Questions 15 and 16 use the same method as the example above them,
establishing two angles from identification then the 3rd by supplementary angle methods, look
at the example at the top of the column. The last two questions are more difficult and require
you to choose vertically opposite or supplementary methods to solve them.
Example 1
62°
Example 2
Example 3
Example 4
Find total
first
This angle is the
supplement 180°
- 62° = 118°
62°
42°
a 30°
This angle is
vertically
opposite = 62°
62°
62°
a = 62° - 30°
a = 32°
a a
2a = 62°
a = 31°
a
Divide by 2
20°
30°
a = 62° - 30°
a = 32°
Vertically Opposite Angles
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
m
U
H
20°
b
70° c
n
67°
23°
10
C
D
X
O
16
y
Y
B
11
I
i 11°
d
67°
r 33°
Name the vertically
opposite angle to the
following angles
A
2p
54°
A
T
5
15
a
w
F
P
71°
x
9
X
L
u = 77°
x = 62°
57°
e
A
g = 41°
x = 180° - 41° - 77°
8
Y
S
135°
k
G
4
x
41° u
K
N
77° g
7
T
3
Example
B
<
Colour this star the
same as the one you
put in the angle
m =
82°
E
A
2
6
C
D
1
Use vertically opposite
and supplementary angle
properties to find the
value of these letters
Find the value of the
letters below
Name and label with a star
the vertically opposite
angle to the one marked
with a star
82°
17
12
n
G
F
b
x
122°
85°
55°
B
C
D
13
H
E
< IBG
< HBD
< ABC
< CBD
vertically
opposite
v
v
38°
18
14
26°
39°
56°
c
k 114°
q
48°
Parallel Lines and the Transversal
When parallel lines are crossed by a line the angles formed are related to each other by
properties. This will require you to remember certain words and their meaning.
The first word is 'transversal' this is the name of the line that cuts across parallel lines. In
column 1 you are asked to name the transversal and the parallel lines, write the 2 letters that
represent the line, this can be done either way as the example below shows. The naming of
angles uses the 3 point method, there is 4 possible answers for some questions as the example
below shows, realise that if the answer doesn't match your answer you still may be correct, but
the middle letter (vertex) must always be the same.
F
TRANSVERSAL
F
V
Y
LINES
T
V
A
Y
T
M
D
B
D
B
VAM or
VAB or
MAV or
BAV
Lines are VY or YV and TD or DT
Transversal is either FB or BF
AMD or
FMD or
DMA or
DMF
TMB or
BMT
When lines cross, pairs of angles are made. Co-interior angles (called C angles) are the pair of
angles formed on one side of the transversal inside the parallel lines, a 'c' can be formed
around the angles (one is back to front). There are only two co-interior angle pairs in a
standard 2-line 1 transversal problem. Remember the word interior means on the inside (of the
parallel lines and the transversal).
Alternate angles (called Z angles) are also on the inside of the parallel lines but unlike cointerior angles they are on opposite sides of the transversal. A 'Z' can be made by the lines that
include these angles.
Both alternate and co-interior angles are inside the two parallel lines. Corresponding angles
(called F angles) require one point to be on the inside of the parallel lines and the other to be
on the outside. An 'F ' or back to front ‘F’ is made by the lines that include these angles. With
corresponding angles the angle is in exactly the same position on both points of intersection.
Column 3 asks you to write either 'alternate', 'co-interior' or 'corresponding'. Questions 32
through 37 are more challenging but the same method is used.
Corresponding Angles
Alternate Angles
Co-interior Angles
Parallel Lines and the Transversal
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Show the co-interior
angle by colouring the
circle (the numbers are
for marking purposes)
Using the letters name the
parallel lines (L) and the
transversal (T)
1
L:
A
C
D
L:
F
E
1
10
Q
T
H
J
M
3
6
L:
X
Z
P
12
M
K
L:
B
V
X
C
16
5
C
F
A
D
4
15
2
4
7
6
4
3
28
29
2
1
2
30
31
32
33
34
35
36
37
7
6
17 3
7
2 1
6
7
5 4
1
19
5
1
7
3
7
2
3
1
5
4
2
6
Now colour the
corresponding angle
X
20
W V
T
B
21
1
2
H
U
5
7
1
6
23 2
1
4
6
22
1
M
6
2
3
2
4
E
D
2
6
P
S
27
1
B
A R
7
6
3
4
G
D
F
3
5
5
H
6
3 6
1 4
7
6
4
18
E
26
1
13
3
1
3
Using the assigned
letters name the 2
angles with symbols
5
25
5
2
5
6
L:
T:
2
24
Now colour the
alternate angle
14
I
6
3
5
1
U
4
1
5
T:
F
11 2
7
4
L:
D
7
7
L:
2
7
4
5
4
3
L:
T:
3
9
2
5
7
4
6
T:
B
2
8
State if these angles are
corresponding, alternate
or co-interior.
3
7
5
4 7
4
5
7
3
3
6 5
Parallel Line Angle Properties
When two parallel lines are crossed by a transversal, pairs of angles are made that we can use
mathematically. The previous sheet taught you alternate, co-interior and corresponding angles,
now we can use their angle properties, which are:
•
corresponding angles are equal if lines are parallel
•
alternate angles are equal if lines are parallel
•
co-interior angle sum is 180° if lines are parallel (the two angles add to 180°)
In Column 1 an angle is given and an unknown angle is required. The unknown angle will be
either the alternate, corresponding or co-interior angle to the given angle. The first step is
identifying which. Once identified if the angle is the alternate angle or the corresponding angle
it is the same as the given angle, no working required. If the relationship is co-interior then
subtract the given angle from 180° to obtain the answer.
Column 2 gives both angles, identify if the relationship between the 2 angles is either alternate
or corresponding. If it is, the angles must be the same for the lines to be parallel, if the angles
aren't equal then the lines aren't parallel. If the relationship between the 2 angles is co-interior
then the sum of the two angles must equal 180° for the lines to be parallel. If they don't add to
180º then the lines aren't parallel. This includes if the angles are the same, the same co-interior
angle means the lines aren't parallel, unless the angles are 90°. See the example at the top of
the column.
Column 3 is the same as column 1 in procedure, the only difference being that you have to
divide the angle by the number in front of the letter to get your answer. The last 3 questions
require a subtraction to take place before the division.
Corresponding Angles
Alternate Angles
Co-interior Angles
Corresponding angles are
equal if the lines are parallel
Alternate angles are equal
if the lines are parallel
Co-interior angles add to
180° if the lines are parallel
Parallel Lines Angle Properties
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State if alternate, co-interior
or corresponding. Then
find the value of the letter.
Example
Find if the lines below are
parallel, give a brief reason
for your decision.
Example
x
57°
91°
corresponding
89°
x = 57°
117°
e
3x
3x = 60°
60°
x = 20°
15
130°
2x
so lines aren't parallel.
10
88° 88°
2
Example
Parallel?
Circle: Yes / No
Corresponding angles not equal
1
Find the value of x
153°
d
Parallel?
Circle: Yes / No
16
4x
120°
17
3
y
87°
11
92°
88°
k
4
6x
Parallel?
Circle: Yes / No
18
3x 75°
17°
19
5
12
112°
a
82°
6
3x
Co-interior angles require
an extra step, find the
value of x in these
m
77°
13
7
85°
85°
c
Parallel?
Circle: Yes / No
20 2x = 180° 2x =
2x
70 °
21
8
13 5°
w
143°
9
66°
Parallel?
Circle: Yes / No
98°
g
14
93°
93°
Parallel?
Circle: Yes / No
5x
22
3x
15 3°
Further Parallel Lines
If two parallel lines are crossed by a transversal eight angles are formed. Given one angle, all
of the other angles can be found by using these properties:
•
supplementary, the angles adjacent (next to it) to the angle can be found by subtracting
the given angle from 180°
•
vertically opposite, the angle opposite the given angle is the same size as the angle
•
then parallel line properties are used. Alternate, corresponding and co-interior can be
used (it isn't necessary to use them all)
In column 1 a single angle is given, you are required to find the value of the black dotted angle
in two solution 'moves'. The first step requires you to use supplementary or vertically opposite
methods then for the second step use one of the parallel line properties.
The example below shows the angle given is 108°, vertically opposite was used to find 5 then
co-interior to the black dotted angle. Note we could also use two other methods. Supplement
to 4 then alternate to black dot or supplement to 6 and corresponding to black dot. Either way
is correct. Note that the dots can be coloured, ensure that your colour choice is matched on the
solution line.
Column 3 is an extension, this time the seven other angles are required. Use any method you
like just make sure you give a reason, like the example below.
1
Example
2
Example
1 3
4
108°
2
5
6
112°
3
5
4
7
6
= 68° as supplement
= 112° as vertically opposite
= 68° as supplement
= 108° as vertically opposite
= 68° as co-interior
= 72° as co-interior to ●
= 112° as alternate
= 68° as supplement to ●
= 112° as corresponding
Further Parallel Lines
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Find the vertically opposite
or supplementary angle and
use it to solve for the unknown angle, give a reason
Example
5
2
3
1
5
88°
4
65°
4
1
3
Now using the same skills
find all the angles on the
diagram, with reasons.
10
6
1
2
5
7
6
38°
3
= 115° supplement to 65°
= 115° alternate to •
1
6
2
1
55°
3
1
5
4
148°
6
5
2
3
11
1
6
4
3
103° 6
2
4 5
7
7
2
3
1 3
1
4
5
1
8
126°
5
3
7
4
6
4
2
4
5
4
3 165°
6
1
12
2
3
5
2
1
24°
1
6
4
6
2
3
6
2
102°
2
5
9
4
1 2
3
6
77°
5
132°
3
5
4
7
6
Measuring Angles in Triangles
This exercise is an introductory exercise to show that the angle sum of a triangle is 180°. Use a
protractor to measure each angle in the triangles, note you should expect to experience some
error in any measurement exercise, your total may be out by a degree.
Follow these steps:
•
•
measure the angles with a protractor and then using the addition spaces in the bottom left
corner add them and compare the total to 180°
Repeat for all the triangles
Why are there errors? As the angles are not exactly a whole degree you may round them up or
down, this will affect your total.
Measuring Angles in Triangles
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Measure the 3 angles in
each of the triangles, then
add them to get
180°,allow for small
1
2
3
5
4
1
+
2
+
3
+
4
+
5
+
6
+
6
Angles in Triangles
The angle sum of a triangle is 180°. This property holds true regardless of the type of triangle.
Types of triangles are:
•
Equilateral - all sides equal length, all (internal) angles are 60°
•
Isosceles - two sides equal length, two angles equal, note that the equal sides are opposite
the equal angles
•
Right angled triangle - has a right angle in the triangle with the other two angles being
unequal
•
Right Isosceles triangle - is a right angled triangle with the other two angles being equal
(they must be 45°)
•
Scalene triangle - all sides and all angles unequal, the words acute and obtuse can be used
to further the description
•
Acute triangle - all angles are less than 90°
•
Obtuse triangle - one angle is greater than 90°, note that you can't have more than one
obtuse angle in a triangle
Column 1 requires you to find the missing angle. Two angles are always given, subtract these
from 180° to get the answer, (example at top of column). Once you have the 3 angles describe
the type of triangle it is from the selection at the top of the sheet. Often the case is to use the
word ‘obtuse’ in describing a triangle but not ‘acute’. So that if obtuse isn't used the triangle
must be acute. This classification will only apply to scalene and isosceles triangles, right
angled triangles and equilateral triangles can't contain an obtuse angle.
Column 3 tests your knowledge of isosceles and equilateral triangles. With equilateral
triangles you know the angle is always 60°. So if the angle is 12w then that means 12w = 60°
(÷ by 12) and w = 5°. If it was 15t then 15t = 60° (÷ by 15) and t = 4°. Note that in these
questions you asked to find the value of the letter, this is different than finding the angle. For
example if an angle is 20k in the corner of an equilateral triangle, k = 3° but the actual angle is
still 60°.
With isosceles triangles the process is the same except that instead of 60° the angle will match
another angle on the diagram, question 18 is more challenging.
Angles in Triangles
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Use the side lengths to help
you choose a method to
find the value of the letters
60°
60°
60°
Equilateral
Isosceles Right Angled
Scalene
Find the missing angle,
then classify (name the
type of ) the triangle.
Right Isosceles Acute
Obtuse
6
x = 90°
x
28°
1
124°
23°
Type:
13
7
64°
a
h
32°
66°
y
14
24°
d
Use the same method to
solve, identify the type of
triangle formed. Use 'x ='
15
16°
Type:
u
12
28°
Type: Right angled triangle
x
q
Example x = 180° - 62° - 28°
62°
11
45°
Type:
2
a
36°
72°
8 A triangle has angles 23°
and 46°, find the other angle
Type:
2x =
2x
16
3
90°
Type:
b
45°
84°
9 A triangle has angles 56°
and 68°, find the other angle
Type:
4
These are harder
17
k
4w
4c
5t
60°
60°
Type:
5
39°
3n
Type:
10 Two identical angles in a
triangle total 120°, find the
other angle
18
53°
Type:
4v
m
Type:
v
Isosceles Triangles
Your knowledge of isosceles triangles and their properties will be tested throughout your
mathematics studies, the 3 methods of finding missing angles are dealt with on this sheet. In
column 1 your are given one angle and asked to find another. With a scalene triangle this is
impossible because you require 2 angles to find the remaining angle, but because these are
isosceles triangles you know that the angles opposite the sides with markings are equal.
Column 1 is done all the same way, place the number in the empty corner then you have your
2 angles. Then subtract the 2 angles from 180°. Look at question 1, the unlabelled angle is also
48°, so the working is s = 180° - 48° - 48° or s = 180° - 2 × 48°. The most common mistake
when answering these questions is you will forget the unlabelled angle and subtract 48° from
180° instead of 48° doubled.
In column 2 the second type of problem is encountered, this time you are given the non-paired
angle. This time 2 × (the letter) = 180° - (the given angle). An example is at the top of the
column, the entire column is done using the same method. It is up to you if you write the letter
in the unlabelled corner, but it is recommended.
Column 3 has written problems, 17 - 19 are the same as the previous columns, you just have to
decide which method is used. The last 2 questions are harder, an example is below.
Example (questions 20 and 21)
How this appears as a diagram:
The working required:
7x + 7x + x =15x
A triangle has a pair of angles which are
seven times the size of the other angle,
find the size of all the angles. Let x = the
smallest angle.
x
15x = 180°
÷ 15
x = 12°
7x
7x
x = 12° and 7x = 84°
Isosceles Triangles
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Find the value of the
letter, these will take
3 lines to solve.
Find the value of the
pronumeral
Solve these, you might
not always need 3 lines.
Use 'x=' in your working.
Example
1
s
2m = 180° - 52°
s = 180° -
17 An isosceles triangle has
one angle of 102°, find the
size of the other angles
52°
2m = 128°
48°
m
m = 64°
2
79°
m
m+m=2m
10
40°
18 An isosceles triangle has a
pair of angles of 17°, find
the size of the other angle
e
3
h
d
11
61°
28°
4
9°
c
k
19 An isosceles triangle has a
pair of angles of 55°, find
the size of the other angle
12
122°
5
45°
y
q
34°
13
33°
These are harder!
6
20 An isosceles triangle has a
pair of angles that are twice
the size of the other angle
find the angles (Hint: use x
and 2x)
g
x
7
14
f
85°
j
i
15
8
n
x=
21 Repeat the above question
only this time the angles are
4 times the size of the other.
136°
71°
16
9
8°
53°
y
2x =
p
Exterior Angles of a Triangle
There is a relationship between the exterior angle of a triangle and the 2 opposite interior
angles, the sum of the 2 opposite interior angles = the exterior angle. In other words when you
add the 2 opposite interior angles you get the exterior angle. The diagram below shows this
graphically. It is important that you realise that the two opposite angles aren’t adjacent to
(don't touch) the exterior angle.
Column 1 Q.1 - 5 require you to add the two interior angles together to get the exterior angle.
The adjacent angle is kept out of these questions. An example is at the top of the column. The
next 3 questions show that if you have the adjacent internal angle you just take the supplement
to get the external angle (subtract from 180°). So if the adjacent angle is one of the 2 angles,
then just take the supplement (subtract it from 180°) of the adjacent angle. This is where the
most common mistake is made, if the adjacent angle is given don't add it to the other angle to
get your answer as you will be wrong.
Adjacent angle
This angle is ignored
180° -
Column 2 works in reverse, you are given the exterior angle and have to calculate the missing
interior angle. Do these by subtracting the given interior angle from the exterior angle, see the
example at the top of the column.
The third column is challenging as it requires you to use your skills with isosceles and
equilateral triangles as well as algebra. There is an example below.
Example : Find the angle
Because the triangle is
isosceles the other base
angle is also e
130°
e
2e = 130° (opposite angle sum = exterior)
e = 65°
e
e
Solution:
130°
Exterior Angles of a Triangle
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Find the exterior angle for
the following triangles.
Example
x = 57° + 59°
57°
59°
Now the exterior angle is
given, use subtraction to
find the unknown angle.
Example
1
70°
35°
9
111°
h
2
x = 35°
u
19
k
31°
47°
10
51°
18
g
42°
56°
x = 70° - 35°
x
x = 116°
x
Use your knowledge
of triangle properties
to find the value of
the angles
k
20
73°
b
144°
n
11
3
s
21 Hint: Find a then b then c
20°
63°
128°
e
49°
c
12
4
14°
35°
b
a
28°
a
110°
121°
t
5
13
132°
22
158°
q
17°
v
Find the exterior angle
by finding the supplement, ignore any extra
angles
6
14
47°
117°
b
15
z
41°
19°
x
16
a
138°
17
36° c
60°
23
126°
86°
8
3a
d
55° b
7
12a
71°
28°
2a
n
3a
Angles in Quadrilaterals
The angle sum of a quadrilateral is 360°, this means that when all the angles inside the shape
are added, they total 360°. If you are given 3 angles then to find the missing angle subtract the
3 angles from 360° and what is left is the your answer.
In column 1 through to question 11 the exercises are all done the same way, you are given 3
angles (remember a right angle is 90°), subtract these from 360°. If you write in ‘90°’ in the
space next to the right angle sign you have less chance of forgetting it in your calculations.
The question at the top of the 1st column shows the setting out for the exercises.
The rest of column 2 involves a small amount of algebra. The exercises are done in the same
way as the first column in that the missing angles equal 360° minus the given angles. Because
in the questions the 2 unknown angles are the same then you can add the letters together,
i.e. a + a = 2a, x + x = 2x. An example is below. Question 20 is a similar style of question.
Column 3 is the same as the first column only one of the angles given is an exterior angle. To
change the exterior angle to an interior angle subtract it from 360°. Then you have 3 interior
angles which subtracted from 360° gives you the missing angle. The most common mistake is
forgetting to change the exterior angle it to an interior angle in these questions.
Example: Find the value of e
Solution:
2e = 360° - 80° - 60°
2e = 220°
80°
e
e
60°
e + e = 2e
e = 110°
Angles in Quadrilaterals
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Find the missing angle in
these quadrilaterals
Convert the exterior angles
to interior angles to find x
then solve for y.
9
1 x = 360° - 90° - 90° - 90°
x=
142°
16 x = 360° -
98°
x=
q
x
52°
10
40°
155°
x
72°
m
2
78°
a
y
y=
y=
17
11
62°
e
3
65°
d
65°
32°
230°
38°
x
40°
125°
These have 2
unknown angles
292°
115°
12 2t = 360° -
4
18
2t =
98° 117°
n
y
118°
y
t
t
t=
63°
x
142°
68°
72°
5
v
43°
13
19
m
y
27°
70°
x
6
35°
65°
m
70°
95°
105° 14
y
63°
7
b
85°
k
8°
32°
b
112°
38°
120°
84°
w
x
2x =
x
102°
3x
20
2x
63°
15
8
Find the 2 angles
46°
3x =
7
FREEFALL
MATHEMATICS
INTEGERS
Directed Numbers
Directed numbers are numbers that have direction assigned, a number will have a positive
value (+) in one direction and a negative value in the opposite direction. These are related to
the number plane, where up and to the right are positive and down and to the left are negative.
So up, North, East, a gaining of something or time after something has occurred are all
positive. These can be denoted by a positive (+) sign, but this isn't necessary, just the number
alone is sufficient. Down, South, West, a loss of something or time before something is to
occur are all negative. These are denoted by a negative (minus) sign (-).
Column 1 uses words that imply either a positive or negative answer. Use the above as a guide
if you need to and then write in the answer, note that you don't write the units, for example:
$450 profit → 450, $50 loss → -50. So no km, cm, $, min or any other unit is required. A
number with a minus sign when necessary is it.
Column 2 asks you to match opposites, the match is between those on the left hand side with
those on the right hand side. Draw a line to connect the dots. The second part of the column
asks you to make short statements using the number and topic supplied. Show units with these.
Use the sign with the number to help you decide on which word to use. (Up, down, before,
after etc.)
Column 3 introduces addition and subtraction. The first section asks you to imagine a change
in temperature, you are given the starting temperature and then are told the size that the
temperature rises or falls. You can use the thermometer diagram beside the questions as a
guide. The second section gives you a 'before and after' comparison, you are asked to calculate
the amount of the drop or rise in temperature that has occurred. If there is a drop use a
negative sign to show that there was a fall in temperature.
Directed Numbers
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Match the words used
in column 1 with its
opposite, with a line
Use a directed number to
represent the given action
Example
26
Lost
Penalise
1 Up 60 m
North
Increase
2 Loss of $95
Fall
Withdrawal
3 Increase of 13%
Decrease
Found
Bank fees
Profit
Up
South
Deposit
Down
Left
Rise
Assist
Interest
Loss
Right
A gain of 300 points
4 Deposit of $20
5 South 400 km
6 Left 15º
7 Dismantling 61 cars
8 Bank fees of $3.80
300
Construct sentences that
describe the directed
numbers, given a subject
for the sentence
9 Down 4 storeys
10 Fall of 23ºC
11 Profit of $712
12 East 650 m
13 Lost 6 nuts
14 12 s before lift-off
Example
ºC
8
7
6
5
4
3
2
1
0
-1
-2
-3
-4
-5
-6
-7
-8
31 From 6ºC
drops 8ºC
32 From 4ºC
drops 11ºC
33 From -3ºC
rises 7ºC
34 From -5ºC
rises 3ºC
35 From -7ºC
rises 7ºC
Give the temperature
difference between the two
thermometers, if there is a
temperature drop, use '-'.
ºC
ºC
8
7
6
5
4
3
2
1
0
-1
-2
-3
-4
-5
-6
-7
-8
-13: Time
The space shuttle will
commence lift off in 13 seconds
27 -4 000: Distance
15 Withdrawal of $193
16 12 knot wind assist
17 West 5 paces
Find the new
temperature
when the given
change occurs
ºC
8
7
6
5
4
3
2
1
0
-1
-2
-3
-4
-5
-6
-7
-8
ºC
8
7
6
5
4
3
2
1
0
-1
-2
-3
-4
-5
-6
-7
-8
8
7
6
5
4
3
2
1
0
-1
-2
-3
-4
-5
-6
-7
-8
28 -220: Money
18 Penalise 13 strokes
A
19 Found 14 bolts
20 Constructing 4 buses
29 40: Percentage
21 Account interest $16
36 From A to C
37 From B to C
38 From C to D
22 4 min after ignition
23 Right 85 cm
B
30 -6: Temperature
39 From D to B
40 From A to B
24 North 33 km
41 From D to A
25 Rise of 56%
42 From B to A
C
D
Number Line and Magnitude
This sheet deals with the size and position of negative numbers. Positive numbers are all
greater than zero and increase with the size of the numbers used. Negative numbers are the
opposite, they are all less than 0 and as the number part (ignoring the negative sign) increases
the number decreases in size. For example, 200 > 30 but -200 < -30.
Column 1 deals with plotting points on the number line and then reading a number line. You
are first asked to plot numbers, this is done by selecting the unfilled circle above the number
required and filling it in. You are asked to plot odd and even numbers, an even positive number
will also be even if negative. The same applies for odd numbers. The case of zero (0) is that it
is neither even or odd. Questions 5 through 8 asks you to write the plotted numbers in
ascending order, remember descending is down (highest to lowest), ascending is the opposite.
With a number line the lowest values are to the left.
Column 2 asks you to use < and > signs to make the mathematical statement true. Remember
that you point the ‘arrow’ at the smallest number. Then arrange the numbers in descending
order in the next section of the column.
Column 3 asks you to circle the largest number and put a square around the smallest number.
The last section of the column deals again with inequalities (the use of < and >). Instead of
drawing the sign, a sign has been included, if the sign is correct (if it 'points' to the smallest
number) then write 'true', if it doesn't, write 'false'.
Number Line and Magnitude
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Plot the numbers on the
number lines provided
(by filling the circles)
1 -3, -1, 0, 1, 3
-5 -4 -3 -2 -1 0 1 2 3 4 5
2 All numbers between
0 and -5
-8 -7 -6 -5 -4 -3 -2 -1 0 1 2
3 All even numbers between
-4 and 4.
Circle the largest number,
a square for the smallest.
Fill the box with < or >, to
make these true
9
-3
4
10 10
5
-5 -4 -3 -2 -1 0 1 2 3 4 5
Write the numbers
marked on the number
lines in ascending order.
5
-20
-18 32
-3
-6
0
3
33
-4
-2
-10
0
0
15 -7
-23 16 15
-40 34
-3
-6
2
-1
17 -60
-3 18 -10
0
35
0
-1
-5
3
19
6
20 3.5
-7
36
-3
-7
-10
-5
8
14 12
-4
21 -12
-9 22
8
-45 37
-34
0
-2
-5
23 -56
-9 24
0
-3
38
-10
-5
30
-20
39
3
0
9
-10
40 -200
-50
5
199
41
3
-6
6
-3
42
-21
22
0
20
Arrange these in
descending order
25 6, -17, 0, 20, -100
,
,
,
,
Write 'true' or 'false' for
the following
26 40, -5, 14, -35, 2
43
5
44
-3 > -5
45
-7 < -6
46
-5 >
Arrange these in
ascending order
47
8
> -10
28 -5.9, -9, 34, -30, 242
48
6
< -16
49
-1 > -6
50
2
< -1
51
0
< -3
52
3
>
0
53
0
>
3
,
,
-5 -4 -3 -2 -1 0 1 2 3 4 5
-5 -4 -3 -2 -1 0 1 2 3 4 5
15
13 -3
,
,
,
27 -7, 0, 127, 13, -6.5
7
-2
-22 12 -25
-5 -4 -3 -2 -1 0 1 2 3 4 5
6
10
11 16
-5 -4 -3 -2 -1 0 1 2 3 4 5
4 All odd numbers between
-4 and 4.
31
,
,
,
,
,
,
,
29 23, 107, -14, 56, -2
8
,
,
,
30 -20, -85, -37, 87, 4
,
,
0
,
-5 -4 -3 -2 -1 0 1 2 3 4 5
,
< 10
,
Addition of Integers
Unlike positive numbers (1, 2, 3 …) which when added always give a larger number, the
addition of negative numbers (-1, -2, -3….) results in smaller numbers. Negative numbers can
challenge students so don't feel frustrated if you don't take to them straight away, it may mean
counting to yourself out loud or using your fingers to assist you.
In Column 1 both positive and negative numbers are involved in additions with positive
numbers. Questions 1 to 9 and questions 10 to 18 are in 2 groups. These questions give
answers that form a number pattern, the pattern can be used to check your answer as you move
through the question groups. Note also there is a number line at the top of the page to assist
you. Remember that when you subtract you move to the left and addition moves to the right.
Questions 21 to 29 mostly won't fit on the number line so you have to understand the concept
of negative numbers for these questions. All the questions start with a negative number and
have a positive number added to them. Answer these questions by solving them and then circling the answer.
Column 2 involves the addition of both positive and negative numbers with negative numbers.
Don’t get confused by the two signs (+ and -) next to each other. When a plus and minus sign
are together the result is a minus sign. So + and - = - (or - and + = -). It is even more easy if
you remember unlike signs together = minus. As this rule will be built upon in the next sheets.
So really, while we are adding the numbers, the result is a subtraction, I hear you ask 'then why
show the + sign?', the answer is that when you substitute negative numbers into positive
equations the two signs will occur. This column has the same layout, two groups of questions
which have a pattern. Use the pattern or the number line to assist you.
Column 3 involves 3 numbers, that means that you answer the questions in 2 stages. Look at
the first 2 numbers, complete the sum, strike them out and write the total above them. There is
an example at the top of the column.
Addition of Integers
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
-8
-7
-6
-5
-4
Look at the pattern formed
by your answers and the
number line to assist you
-3
-2
-1
0
1
2
3
4
5
6
7
8
Now find the
answer for these
When you add a negative
number it is like subtracting a positive number
Example
1 3+2
2 2+2
3 1+2
30 5 + 1
31 5 + 0
32 5 + -1
=
=
=
=
=
=
4 0+2
5 -1 + 2 6 -2 + 2
=
=
=
=
7 -3 + 2 8 -4 + 2 9 -5 + 2
=
=
=
13 -5 + 3 14 -5 + 4 15 -5 + 5
=
=
16 -5 + 6 17 -5 + 7 18 -5 + 8
=
=
=
Circle the correct answer
59 8 + 10 + -6 =
60 -3 + 9 + -2 =
=
=
=
=
61 -8 + -3 + 6 =
62 11 + -5 + -8 =
63 8 + -12 + 4 =
=
46 -3 + -7 64 -6 + 5 + -9 =
45 -2 + -7
=
=
58 10 + -5 + 3 =
42 1 + -7 43 0 + -7 44 -1 + -7
=
=
=
39 4 + -7 40 3 + -7 41 2 + -7
=
=
=
4 + - 10 + 6 =
36 5 + -5 37 5 + -6 38 5 + -7
=
=
10 -5 + 0 11 -5 + 1 12 -5 + 2
=
33 5 + -2 34 5 + -3 35 5 + -4
-6
=
65 12 + 8 + -15 =
Circle the correct answer
66 -15 + 3 + 10 =
19 -5 + 4 =
-9
9
-1
47 -6 + -6 =
0
12 -12
20 -4 + 3 =
-1
7
-7
48 3 + -5 =
-8
-2
21 -10 + 8 =
2
-2 -18
49 -5 + -7 =
2
-12 -2
22 -15 + 3 =
-12 12 -18
50 8 + -5 =
3
-3
13
69 -10 + -10 + 50 =
23 -30 + 25 =
-55 -5
5
51 10 + -12 =
-2 -22
2
70 -30 + 5 + 17 =
24 -14 + 14 =
0
28 -28
52 -4 + -10 =
6
67 -9 + -9 + 20 =
8
68 8 + -15 + -6 =
-6 -14
71 9 + -15 + 8 =
25 -5 + 10 =
-15 15
5
53 -15 + -8 =
-23 -7
7
26 -7 + 3 =
-4
10
54 14 + -12 =
-2
26
27 -20 + 60 =
-40 -80 40
55 20 + -50 =
30 -70 -30
28 -15 + 27 =
-42 -12 12
56 -5 + -25=
20 -20 -30
29 -11 + 4 =
-7 -15
57 12 + -12 =
0
4
7
2
72 7 + 12 + -20 =
73 13 + -20 + 10 =
74 18 + -20 + -2 =
-24 24
75 -25 + 35 + -10 =
0
Subtraction of Integers
Unlike positive numbers which when subtracted always give a smaller number, the subtraction
of two negative numbers results in larger number. That is because when two ‘-’ signs are
together the result is a ‘+’.
In column 1 both positive and negative numbers are involved in subtractions with positive
numbers. Questions 1 to 9 and questions 10 to 18 are in 2 groups. Note that sometimes
numbers have a '+' in front of them and sometimes they don't, but both mean the same thing.
E.g. -10 - 2 = -10 - +2 = -12. As you learnt on the previous sheet unlike signs equals a minus,
so the minus and the plus next to each other equals a minus. These questions give answers that
form a number pattern, this pattern can be used to check your answer as you move through the
question groups. Note also there is a number line at the top of the page to assist you.
Questions 21 to 29 mostly won't fit on the number line so you have to understand the concept
of negative numbers for these questions. All the questions subtract a positive number. This
means they will decrease, circle the correct answer.
Column 2 involves the subtraction of both positive and negative numbers with negative
numbers. Don’t be confused by the two minus signs (- -) next to each other. When a minus and
another minus sign are together the result is a plus ‘+’ sign. So - and - = +. And as you know
a + and + = +. It is even more easy if you remember like signs = plus. This column has the
same layout, two groups of questions which have a pattern. Use the pattern or the number line
to assist you. Then for the lower part of the column circle the correct answer.
Column 3 involves 3 numbers, that means that you answer the questions in 2 stages. Look at
the first 2 numbers, complete the sum strike them as you move through. There is an example at
the top of the column.
Subtraction of Integers
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
-8
-7
-6
-5
-4
Look at the pattern formed
by your answers and the
number line to assist you
1 5 - +2 2 4 - 2
=
3 3 - +2
=
-3
-2
-1
4 2 - +2 5 1 - +2 6 0 - +2
=
=
7 -1 - 2
8 -2 - 2
9 -3 - 2
=
=
=
10 2 - 0
11 2 - 1
12 2 - +2
=
=
=
3
=
=
=
=
=
5 - - 10 - 6 =
58 20 - +8 - 4 =
60 -40 - 14 - - 6 =
=
39 -4 - -2 40 -4 - -3 41 -4 - -4
=
=
42 -4- -5 43 -4- -6 44 -4- -7
=
61 -12 - +7 - -10 =
62 -8 - -13 - -15 =
=
=
16 2 - 6
17 2 - 7
18 2 - 8
45 -4- -8
46 -4- -9
=
=
=
=
=
=
63 8 - +12 - - 4 =
64 -20 - - 5 - - 8 =
65 20 - 35 - - 3 =
Circle the correct answer
20 5 - +9 =
7
8
15
59 6 - -10 - 18 =
=
-18 -2
6
Example
=
19 8 - +10 =
5
36 -4 - -3 37 -5 - -3 38 -6 - -3
=
Circle the correct answer
4
Now find the
answer for these
33 -1- -3 34 -2 - -3 35 -3- -3
=
13 2 - +3 14 2 - +4 15 2 - +5
2
30 2 - -3 31 1 - -3 32 0 - -3
=
=
1
When you subtract a
negative number it is like
adding a positive number
=
=
0
66 4 - -4 - -8 =
2
47 5 - -2 =
-3
3
4
-13 -4
48 -3 - -7 =
4
-4 -10
21 -2 - +6 =
-4
-8
4
49 -10 - -10 =
-20 10
0
22 -10 - 20 =
-30 10 30
50 -14 - - 6 =
-20 -8
8
69 -2 - -17 - +9 =
23 15 - 30 =
-45 -15 15
51 3 - - 13 =
-10 16 -16
70 10 - 20 - 30 =
24 -9 - +12 =
-21
3
21
52 -24 - -12 =
-12 -36 12
25 12 - 12 =
-24 24
0
53 5 - - 17 =
-22 -12 22
26 13 - +15 =
2
-2 -28
54 -20 - - 9 =
-29 -11 11
27 -20 - 7 =
-27 -13 13
55 15 - - 7 =
-8
22
8
28 -14 - 14 =
-28
0
28
56 8 - - 8 =
16
-8
0
29 -11 - 9 =
-2
2
-20
57 0 - - 6 =
-6
0
6
7
67 4 - -4 - +8 =
68 -10 - 40 - - 90 =
71 -9 - -10 -4 =
72 8 - 19 - - 11 =
73 16 - 20 - - 7 =
74 -7 - -20 - - 13 =
75 -6 - 11 - - 17 =
9
Subtraction/Addition and Using Brackets
with Integers
This sheet mixes subtraction and addition together and introduces the use of brackets with
integers.
Remember negative numbers are on the left hand side of the number line and as you should
know as you move to the left the 'value' of the numbers decrease. So instead of 0 being the
smallest number it now lies in the middle of positive and negative numbers. So:
•
0 > (greater than) all negative numbers e.g. 0 > -1, 0 > -8, 0 > -1 008
•
The larger the value of the number part of the negative number the smaller it is,
e.g. -8 < (less than) -3, -204 < -176 and -82 > -106
•
All positive numbers are greater than negative numbers
Column 1 asks you to place a <, > or = sign in the boxes. The first two involve 2 numbers on
each side. Find the answer for each side and write it in the box below it, then 'point the arrow'
at the smallest number. From then on there are 3 numbers on each side. Look at each side
separately, add/subtract the first two numbers, strike them out and total them. Repeat for the
other side and write in the <, > or = signs. An example is to the left, below.
Column 2 introduces brackets with negative numbers. When a number is in brackets on its
own then treat the brackets as if they aren't there. For example -(-10) = - -10 = 10 and
-(15) = -15. When there are two or more numbers within brackets evaluate the inside of the
brackets first to get a single number, then the above applies again, remove the brackets and
solve. For example
-(10 - 13) = -(-3) = 3, -(5 - -11) = -(16) = -16 and -(-6 -7) = -(-13) = 13.
Column 3 is an extension on this with 4 numbers involved so total as you move across.
Brackets are also introduced. Again it isn't obvious why we need brackets for single numbers,
it will become clear when powers and substitution are introduced. The last 5 questions involve
two numbers in brackets, remember with brackets you solve the inside of the brackets first,
then the tricky part, how does the sign in front of the brackets affect the number inside? An
example is to the right, below.
Remember : Unlike signs equal minus
19 - 40 = -21
Example
-8
-21
2 + -3 = 2 - 3
2 + (-3) = 2 - 3
-15 - -7 + -4 < 19 - 40 - -6
2 - +3 = 2 - 3
2 - (+3) = 2 - 3
-12 < -15
Remember : Like signs equal plus
2 + +3 = 2 + 3
2 - (-3) = 2 + 3
2 - -3 = 2 + 3
-15 - -7
= -15 + 7
= -8
-8 + -4
= -8 - 4
= -12
-21 - -6
= -21 + 6
= -15
Example
-10
9
-(-7 - 3) - (4 - -5)
= 10 - 9
=1
(4 - - 5)
- -10 (-7 - 3) = (4 + 5)
= 10 = -10
=9
Subtraction/Addition and Using Brackets with Integers
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Fill the box with <, >, or
= to make these true
Example
-67
-75 - -8 + -4 > -180 + 100
-71 > -80
1
20 - -4
-5 + 22
Express the following
without the brackets
12 -(-3)
13 -(-8)
14 -(7)
=
=
=
15 -(-20) 16 -(-30) 17 -(19)
=
=
18 (-5 - 10)
2
9 - 14
-8 + 14
=
4
5
6
= -(
13 + -6 - -4
4 + -7 - -12
)
=
22 -(9 + 3)
20 - -6 + 8
-17 - -9 + 8
-8 - 7 - -11
-8 - -40 + 7
11 - 17 + 6
-17 - -8 - -3
14 - 9 - -20
18 - -12 - 5
19 (-9 - 14)
21 -(-5 - 6)
=
=
23 -(8 + 3)
=
=
=
=
24 -(2 - 7)
Example
12
18
5 - (- 7) + 6 - 10
= 8
34 -8 - (-3) - 7 + 12
=
35 -19 + 8 - (3) + 7
=
36 0 - (-7) - 10 + 3
=
37 -(-7) + 3 - 20 + 4
=
38 13 - 20 + -6 - -10
=
39 -11 - 7 + (-2) + 5
=
40 -1 - -1 - (+1) - 1
=
41 37 -(-20) + 9 - 4
=
42 -15 - 8 - -9 - -7
=
25 -(10 + 7)
=
=
=
=
26 -(4 - 9)
7
=
=
20 -(-2 - 13)
3
These are longer and
may include brackets ( ).
27 -(9 + 4)
=
=
=
=
These are harder!
43 6 + (-3 -7) - -8
=
=
44 -14 - (-6 -2) + 6
8
9
-2 + 5 - 9
-6 - -10 - -4
-5 - 8 - -7
28 (8 - -9)
=
=
=
=
-3 - +7 + 11
30 -(8 - -30)
10
4 + -7 - +3
-20 - -10 - 4
-15 - -8 + 6
31 -(6 - -10)
=
=
=
=
32 -(-7 - -4)
11 13 - 50 - -8
29 (10 - -15)
33 -(-10 - 9)
=
=
=
=
=
=
45 9 - (-2 -15) + 17
=
=
46 (-2 - -3) + (3 - 5)
=
=
47 -(5 - 12) - (2 - 5)
=
=
Multiplication of Integers
As with addition and subtraction, multiplication has rules concerning negative numbers also.
These are:
•
a positive × a positive = a positive, this you already know ( + × + = +)
•
a negative number × a negative = a positive ( - × - = +)
So this means that when like signs are multiplied you get a positive answer, and:
•
a negative × a positive = a negative ( - × + = -)
•
a positive × a negative = a negative ( + × - = -)
So this means that when unlike signs are multiplied you get a negative answer.
In column 1 you are asked to multiply 2 numbers together, one is a negative number, the other
a positive, this means the answer will always be negative.
In column 2 down to question 44 you are asked to multiply 2 negative numbers. This means
the answer will always be positive. The rest of the 2nd column is a mixture, some are like
signs others are unlike. The method is to look at the question and determine the sign, talk it
out in your head '….a minus times a minus is a plus' or '…..a minus times a plus is a minus'.
Then write the sign and then go back and multiply the numbers as if the signs aren’t there.
In column 3 the first 5 questions are to test you know the sign. As above you talk it through
mentally, for -5 × -2 × 4 × -3 you would say this: 'a minus times a minus is a plus...a plus
times a plus is a plus ... a plus times a minus is a minus. So the sign is negative, circle it. You
can use the same method for the next set of questions to get the sign first, then multiply the
numbers ignoring the signs. Using the same example above, 5 × 2 is 10, × 4 is 40 and × 3 is
120. So you have -120! There is an example at the top of the column.
Multiplication of Integers
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Try these, use the rules:
- × + = - and + × - = -
Try these, use the
rule: - × - = -
Circle the sign that the
answer would have,
don't try to solve them
1 -2 × 3
=
2 4 × -2
=
31 -2 × -2
=
32 -4 × -5
=
3 -3 × 4
=
4 5 × -3
=
33 -10 × (-3)
=
34 -7 × -3
=
5 5 × -7
=
6 -5 × (+10) 35 -6 × -21
=
=
36 -8 × -4
=
7 8 × -4
=
8 -3 × 7
=
37 (-5) × -9
=
38 -11 × -8
=
9 -2 × (+8)
=
10 -5 × 5
=
39 -4 × (-13)
=
40 -22 × -5
=
11 3 × -25
=
12 9 × -5
=
41 - 1.7 × -2
=
13 -6 × (+4)
=
14 5 × -11
=
43 -14 × -5
=
× - = -, then - × + = - and
42 -31/2 × -2 Example +then
- × - = +. So positive
=
10
30
5 × -2 × 3 × -2
= 60
44 -3 × -15
=
64 5 × -4 × -4
=
15 13 × -10
=
16 -12 × 2
=
17 15 × (-4)
=
18 6 × -9
=
45 -30 × 6
=
46 -2 × -16
=
19 -8 × 3
=
20 -7 × +20
=
47 -9 × -4
=
48 11 × -7
=
21 -3 × 12
=
22 7 × -6
=
49 -10 × -23
=
50 -60 × 4
=
23 9 × (-11)
=
24 -16 × 2
=
51 -15 × -5
=
52 5 × -12
=
25 -35 × +3
=
26 12 × -7
=
53 8 × -4
=
54 -3 × -11
=
27 -8 × 21
=
28 -4 × 13
=
55 -8 × -7
=
56 -5 × 50
=
29 -7 × +8
=
30 34 × (-3)
=
57 -2 × -45
=
58 7 × -21
=
These are
mixed up
Example
-×+=-
-×-=+
-5 × 2 × -3
- or +
59 -2 × -2 × 3
- or +
60 4 × -2 × 5
- or +
61 -2 × -4 × -4
- or +
62 -5 × -7 × -2 × -1
- or +
63 -(-7) × 4 × -2
- or +
With these, find the sign
of the answer first, then
multiply for the answer.
65 -2 × -6 × -5
=
66 8 × 3 × -2
=
67 -2 × 5 × -2 × -3
=
68 5 × 3 × -1 × 1
=
69 20 × -2 × -2 × 5
=
70 -3 × -4 × -2 × -2
=
71 10 × -2 × -1 × 2
=
72 -4 × -2 × -4 × 2
=
73 2 × -2 × 2 × -2
=
74 -1 × -1 × -1 × -1
=
75 4 × -2 × -5 × 0
=
Division of Integers
As with addition, subtraction and multiplication, division has rules concerning negative
numbers also. The good news is that it is just the same as the other rules, that is:
•
a positive ÷ a positive = a positive, this you already know ( + ÷ + = +)
•
a negative number ÷ a negative = a positive ( - ÷ - = +)
So this means that when like signs are divided you get a positive answer, and:
•
a negative ÷ a positive = a negative ( - ÷ + = -)
•
a positive ÷ a negative = a negative ( + ÷ - = -)
So this means that when unlike signs are divided you get a negative answer.
In column 1 you are asked to divide 2 numbers, one is a negative number, the other a positive,
this means the answer will always be negative. So don't let the signs confuse you, write the
negative sign then divide the two numbers, ignoring the signs, to complete the answer.
In Column 2 down to question 40 you are asked to divide two negative numbers. This means
the answer will always be positive, so divide the two numbers together ignoring the signs. The
rest of the 2nd column is a mixture, some are like signs others are unlike. The method is to
look at the question and determine the sign, talk it out in your head '….a minus divided by a
minus is a plus' or '…..a minus divided by a plus is a minus'. Then write the sign. Then divide
the numbers as if the signs aren’t there.
In Column 3 the first 5 questions are to test you know the sign. As above you talk it through
mentally, for -40 ÷ -2 ÷ 5 ÷ -2 you would say this: 'a minus divided by a minus is a plus...a
plus divided by a plus is a plus ... a plus divided by a minus is a minus. So the sign is
negative, so circle it. Use the same method with the rest of the column, find the sign first by
talking it through, write it in, then staying with the example above, ignore the signs e.g.
40 ÷ 2 is 20, ÷ 5 is 4 and ÷ 2 is 2. So you have -2! There is an example at the top of the
column.
Division of Integers
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Find the quotient
for these divisions.
Now apply the
rule : '- ÷ - = +'
1 20 ÷ -4
2 15 ÷ -5
27 -25 ÷ -5
28 -35 ÷ -7
=
=
=
=
3 30 ÷ -5
4 45 ÷ -9
29 -39 ÷ -13
=
=
=
6 34 ÷ -2
5
-3
519
7 64 =
-8
-6 - 942
8 54 ÷ -6
=
9 72 ÷ -6
-8
11 28 ÷ -4
176
12 99 =
-11
=
Now find these.
Use: '- ÷ + = -'
13 -12 ÷ 3
14 -60 ÷ 4
=
=
15 -72 ÷ 9
16 -90 ÷ 6
=
=
17
4 - 556
19 -63 ÷ 7
33 -75 ÷ -15
35 -81 =
-9
36 -28 ÷ -7
37 -48 ÷ -6
38 -80 ÷ -20
=
=
-8 - 504
41 -56 ÷ 7
42 18 ÷ -3
=
=
18 -39 ÷ 3
43 -40 ÷ -5
44 55 ÷ -11
=
=
=
20
22 -68 ÷ 2
47 38 =
-2
=
24 -24 =
3
26 -85÷ 5
=
-÷+=-
-÷-=+
-12 ÷ 6 ÷ -3
- or +
53 -30 ÷ -5 ÷ -2
- or +
54 -24 ÷ 2 ÷ -4
- or +
55 -18 ÷ -6 ÷ -3
- or +
56 -60 ÷ 5 ÷ -3 ÷ -2
- or +
57 -9 ÷ -3 ÷ -3 ÷ -1
- or +
- 344
49 -20 ÷ -2
Find the sign for the
answer first, then divide
for the number.
Example
20
4
40 ÷ -2 ÷ 5 ÷ 4 =
58 -45 ÷ -3 ÷ -5 =
59 -60 ÷ -15 ÷ -2 =
60 -100 ÷ -5 ÷ -2 ÷ -5 =
61 -200 ÷ 5 ÷ -4 ÷ 2 =
62 70 ÷ 2 ÷ -7 ÷ -1 =
63 90 ÷ -2 ÷ -5 ÷ -3 =
46 -66 ÷ -3 64 -150 ÷ -3 ÷ -5 ÷ -2 =
=
65 300 ÷ 5 ÷ -12 ÷ 5 =
48 80 ÷ -5
45
21 -75 =
5
7 - 406
=
Try these they
are mixed up.
4
25
=
40 -27 ÷ -3
39
9 - 423
=
34
Example
-3 - 258
=
23 -84 ÷ 7
=
=
10
=
32 -44 ÷ -4
31
=
30 -42 =
3
Circle the sign that the
answer would have,
don't try to solve them
=
50
=
-6 - 258
51 120 =
-4
52 -100 ÷ -4
=
66 -24 ÷ -2 ÷ -3 ÷ -2 =
67 500 ÷ -5 ÷ 10 ÷ -5 =
68 -80 ÷ -4 ÷ -5 ÷ -2 =
-1
Mixed Operations and Brackets with Integers
This sheet involves integers in exercises with the four operations. This means that order of
operations rules will have to be considered when solving these problems. Questions using
brackets are also involved, remember brackets are done first before any other operation.
Column 1 involves integers and +, -, × and ÷. Remember order of operations require that the ×
and ÷ part of the exercise be done before the + and - part. An example is at the top of the
column.
Column 2 has division questions with the numerator and denominator containing operations.
Calculate the top row and the bottom row separately first. Then perform the division. The
questions involve order of operations from Q.20 so an extra line is supplied for these to
perform the order of operation part first. An example is below.
11 × -6 + 3
90 ÷ 10 ÷ -3
=
=
-66 + 3
9 ÷ -3
-63
-3
= 21
Column 3 involves using brackets. The
method for these is:
•
1st working line: evaluate the brackets and rewrite
100 + 3 × (9 - 15)
the first line with the brackets removed
= 100 + 3 × -6
•
2nd working line : perform the order of operations
part of the question
= 100 - 18
•
Solve
= 82
Mixed Operations and Brackets with Integers
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
These may have order of
operations rules to take
into account.
Example
-12
15 - 3 × -4
Now divisions, calculate
the numerator and
denominator then divide
Example
Example
-18
-21 - -3
= 15 - -12
Bring out the brackets!!!
It’s brackets first, then
order of operations
=
-2 × -3
-18
6
=
-3
13 -50 - -2
1
2
-9 - 17 × 2
24 ÷ -3 × -4
20 - 8
14
44 ÷ -2
-5 - -6
3
4
-55 + 15 ÷ 5
5 × -10 ÷ 2
15
8 × -7
-9 - 5
16
-18 - -3
10 - 13
5
6
16 ÷ -4 × -2
7 - -3 × -4
=
8
=
=
=
=
-40 - 32
8 + -9 × 2
-60 ÷ -15 × 4
-6 - 5
=
=
=
=
=
=
20 17 - 3 × -5
20 ÷ -2 - 6
9
10
100 ÷ -20 + 5
7 + 13 × -5
21 -7 + 6 × -3
26 (-11 - 9) - 3 × -2
=
19 + 25 × -3
11 × -5 - 10
22 80 ÷ -4 + 6
30 + 8 × -2
=
27 (12 - 30) ÷ (27 ÷ -3)
=
=
12
25 (1 - 7) × 4 - 9
=
=
-50 + 9 × 5
11
= 42
24 15 - 7 × (-3 + 8)
15 - -10
19 -84 ÷ -4
=
=
-4 × -2
7
-(13 - 7) × (-2 - 5)
23 -3 + (-5 -7) × 4
17 -100 + 25
18
-7
= -6 × -7
6
= 27
6
=
28 (-65 + 40) × (-3 -8)
=
=
=
Powers and Integers
When a negative number is raised to a power the answer is either positive or negative. When
the power is even the answer will be positive and when the power is an odd number the answer
will be negative. So as a negative number is multiplied by itself, over and over, the answer
will alternate (change) between positive and negative.
Column 1 asks you to express the number raised to a power in expanded form. This means no
powers and separated by × signs. Write the number in the brackets then a × sign, then the
number again and so on, until there are the same amount of numbers as the power. Then solve,
find the sign first, then multiply the numbers without signs. Questions 6 to 11 attempt to show
you how the answer changes with the power. The last two questions require a single word
response.
Column 2 starts with you identifying the rule regarding odd and even powers and negative
numbers. If the power is an odd number then the answer will be negative. If the power is
even the answer will be positive. Column 2 then returns to expanding and finding the solution.
The difference is that the numbers are larger and they require using the multiplication working
spaces provided. ‘Build up’ two numbers from the expansion by replacing 2 or 3 numbers with
one larger number, then use the working space. An example of this is above Q 22. When you
multiply the two numbers don’t include the signs as you should know the sign of the answer
from the power in the question being odd or even. These questions continue to the end of the
sheet.
Powers and Integers
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Would the answer to these
be a positive or a negative
number?
Show these in expanded
form, then evaluate
Example
Example
3
(-2) = -2 × -2 × -2 = -8
Odd power so negative
(-58)9
Positive
Negative
Negative
1 (-3)3 =
=
14 (-853)3
Positive
2 (-5)3 =
=
15 (-17)8
Positive
Negative
3 (-2)4 =
16 (-11)2
Positive
Negative
=
17 (-35)10
Positive
Negative
4 (-10)3
18 (-81)95
Positive
Negative
=
19 (-60)607
Positive
Negative
=
20 (-2 319)12
Positive
Negative
5 (-2)6
21 (-48 646)5
Positive
Negative
=
24 (-7)3
25 (-5)5
26 (-4)6
Expand these and then
make two numbers to
multiply for the answer
=
6 (-1)2 =
Example
(-3)6
-27
3
-27
7 (-1) =
= -3 × -3 × -3 × -3 × -3 × -3
8 (-1)4 =
= -27 × -27
27 (-9)4
27
1
=
When you multiply ignore
the - signs and multiply 2
positive numbers. The
power of 6 tells you the
answer will be positive
10 (-1)6
=
22 (-6)3
=
11 (-1)7
=
=
12 A negative number raised to an
answer
13 A negative number raised to an
odd power gives a
27
= 729
9 (-1)5 =
even power gives a
1
answer
23 (-8)3
189
540
729
28 (-8)4
Absolute Value
An absolute value of a number is the positive value of that number. Absolute value is shown
by ‘bars’ | | , these are straight lines not brackets. These bars mean that you want a positive
answer for the number (or operation) inside the bars. For example:
•
|5| = 5 → absolute value has no effect on numbers that are already positive
•
|-10| = 10 → remove the negative sign and 10 is the absolute value of -10.
Column 1 starts with 10 questions on numbers inside ‘bars’. If the number is a positive
number then you just rewrite the same number without the bars. If the number is a negative
number, write the number without the minus sign. The rest of the column requires you to
perform a calculation and then find the absolute value. These questions involve two lines of
working. In the first line write the bars and inside them the answer to the operation. Then the
third line write the answer. The answer will be the same as the previous line, but no bars, and
if there is a minus sign, it is removed.
Column 2 introduces a negative sign outside of the bars. As the sign is outside of the bars it is
not affected by them so: -|10| = -10 and -|-10| = -10. So if there is a negative sign before the
bars the answer must be negative. The first 10 questions test you understand this, then the next
10 ask you to apply this to some operations. There are two examples, before question 29,
showing you how to answer the questions.
Column 3 involves order of operations in the questions. Complete the order of operations step
first, then use the next 3 lines to solve the question, use the example at the top of the column as
a guide.
Absolute Value
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
The - sign outside the
brackets is not affected
Find the absolute value
of these numbers
Examples
i) |-12| = 12
Examples
ii) |25| = 25 i) -|17| = -17 ii) -|-76| = -76
1
|30| =
2
3
|-73| =
5
Example
15
-|8| =
20 -|-61| =
|12 + 5 × 3| - |-5 + 7|
4 |-1.6| =
21 |-5.9| =
22 -|81| =
= |27| - |2|
|-19| =
6 |141| =
23 -|-11| =
24 |-35| =
= 27 - 2
7
|-83| =
8
|39| =
25 -|-20| =
26 -|47| =
9
|0| =
10
|-1| =
27 -|-5| =
28 |-94| =
|65| =
Now find the absolute
value of these operations
Examples
Examples
i) |43 - - 16|
ii) |18 - 30|
= |59|
= |-12|
= 59
= 12
11 |15 - 25|
12 |67 - 14|
13 |18 - -6|
19
These may require order
of operations rules to be
considered. Solve these.
= 25
39 |15 + 9 ÷ 3| - |26 - 50|
Now try these
operations
i) |-11| - |-4|
ii) -|13| - |-9|
= 11 - 4
= -13 - 9
=7
= -22
29 |29| - |-39|
30 |12| + |-26|
31 -|-2| × |-7|
32 -|-8| ÷ |-4|
33 -|-3| + |-3|
34 |-20| - |17|
40 -|30 ÷ 3 × 2| × |-9 + 5|
41 -|-12 ÷ 6| × |5 - 4 × 7|
14 |17 - 200|
42 |16 ÷ 8 - 4| - |-3 + 7 × 6|
15 |5 × -8|
16 |-4 × -12|
35 |58| - |-12|
17 |-27 + 9|
36 |-41| + |-9|
43 -|6 - 4 × 9| ÷ |12 - 2 × 3|
18 |-5 - - 3|
37 -|-11| × |9|
38 -|-60| - |20|
7
FREEFALL
MATHEMATICS
ALGEBRA 2
Solving One Step Equations
Addition and Subtraction
One Step equations are equations you solve in one line of working. They will involve a letter
(pronumeral) with two numbers, you have to solve the equation for the value of the letter.
In column 1 you are given one step equations that you can solve with no working, talk them
through if it helps you. Here is an example:
d + 20 = 30 …. So something plus 20 equals 30
Try d = 10 …. 10 plus 20 equals 30, so d = 10
Another way to do these is to think of the equation as scales, as each side is equal you have to
maintain balance by finding what number would keep the scales balanced.
Column 2 is the same as column 1 except you are asked to now show working. The goal is to
get the letter by itself. One of the examples at the top of the 2nd column is b + 15 = 85. To get
the letter by itself you have to remove the + 15, how do you remove + 15? You subtract 15.
So you minus 15 from the left hand side, and because it's like a set of scales, you must do the
same thing on the other side of the equals sign. So you subtract 15 from b + 15 and you get b
on its own. You subtract 15 from 85 and you get 70. The other example is a - 9 = 27. You have
to get a by itself, to get rid of the - 9 you have to add 9. So you add 9 to each side and you get
a = 36.
So the questions are the same as the first column but you are showing you are following a
procedure by showing your working out. You may be wondering why you have to do this
when often you can see the answer without working. You have to practice the process so that
when you encounter more complicated problems in the future, that you can't just see the
answer for, you know the correct method to use.
Column 3 is the same as column 2 except the letter is on the right hand side of the equals sign.
Solve it in the same way and you get the number equal to the letter. Then rewrite your answer
so that the letter is first. Again there are examples at the top of the column.
Solving One Step Equations - Addition/Subtraction
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Look at these equations
and give the value for
the pronumeral
1 k + 8 = 10
2 b + 3 = 15
3 d+4=9
4 x + 20 = 60
Now solve these
equations showing
working
k=
The letter is on the right and
so these need an extra line.
Find the value of the letter.
Examples
Examples
b + 15 = 85
a - 9 = 27
-15
+9
-15
b = 70
+9
a = 36
24 c + 15 = 22
25 y + 11 = 28
26 x - 9 = 34
27 t - 7 = 18
16 = 5 + c
35 = e - 7
-5
+7
-5
11 = c
c = 11
+7
42 = e
Then reverse
the equation
e = 42
44 20 = f - 11
45 19 = m + 4
46 83 = r + 16
47 12 = n - 36
28 q - 20 = 17
29 b + 10 = 11
30 n + 26 = 26
31 d - 20 = 77 48 64 = 19 + k
49 11 = m - 17
32 t - 14 = 33
33 a - 11 = 59
5 t - 10 = 5
6 m - 8 = 20
7 h + 9 = 30
8 l - 20 = 70
9 w - 15 = 104
10 s + 12 = 13
11 u - 9 = 9
12 q + 7 = 23
13 n - 100 = 40
14 y + 22 = 30
34 15 + g = 33
35 w - 9 = 8
36 x - 12 = 38
37 18 + c = 27
38 e + 7 = 103
39 i - 56 = 11
50 87 = u - 51
51 100 = a - 9
52 33 = 18 + s
53 56 = d - 24
54 75 = p + 55
55 86 = 37 + x
56 11 = j - 11
57 62 = 16 + t
15 a + 16 = 50
16 e - 7 = 6
17 z - 11 = 5
18 c - 10 = 94
19 j + 3 = 101
20 v + 35 = 100
40 37 + h = 80
41 k - 8 = 109
21 t - 4 = 24
22 m - 17 = 4
23 x + 11 = 100
42 d - 90 = 1
43 u - 15 = 63
Further One Step Equations
Adding and Subtracting Negative Numbers
This sheet involves one step equations that involve integers (positive and negative numbers).
This sheet may challenge some students. Before you attempt this sheet you should have
completed the previous one step equation sheet and the folder "Integers".
Column 1 begins with 10 questions, you are asked to write in the number that satisfies the
equation (makes both sides equal). This is a quick refresher on using negative numbers. The
next 11 questions are much the same, except the box has been replaced by a pronumeral
(letter). If you have trouble with these questions pick a number and replace the letter with it
and see if it works. If it doesn't, use another number that you think is a better choice and so on.
Make sure you write the letter, an ‘=’ sign and the answer. Not the answer alone.
The second column follows the same method used with the previous one step equation sheet.
See the example at the top of the column. The second part of the column has the letter on the
right hand side of the equation, solve in the same way then use the additional line to reverse
the answer so that the letter is on the left hand side. This is the same method used in the third
column on the previous sheet.
Column 3 is different in that the sign in front of the letter is a minus sign. When you solve this
equation you will have a negative sign in front of the letter. Remove it by multiplying each
side by -1, this will change the signs on both sides. There are a pair of examples at the top of
the column.
Just a reminder:
+- =-+ =(a plus sign before or after a minus sign is the same as a minus sign alone)
--=+
(a minus sign together with another minus sign is the same as a plus sign alone)
Further One Step Equations - Add/Sub Negative Numbers
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Now solve these
equations showing
working
Find the solution to
these and put the
number in the box
1
2
3-
6 + z = -5
=5
3
-6 +
= 11
4
-5 +
= -1
5
- 8 = -6
6
+ 3 = -10
7
+ -7 = 4
8
- 8 = -8
9
+5=7
10
+ -3 = -1
-6
12 n - 5 = -3
13 j - 2 = -10
14 v + 4 = -7
-6
+9
+9
t=5
22 m + 7 = 3
23 b - 4 = -6
24 n + 11 = -5
25 a - 3 = -2
7 - j = -6
-7
-h - 4 = 10
-7
+4
-j = -13
j = 13
+4
-h = 14
Multiply
throughout
by -1
h = -14
38 8 - m = -11
39 -v - 6 = 5
40 -t - 6 = 0
41 -7 - a = -3
42 5 - e = -2
43 -x - 11 = 2
44 14 - p = 40
45 -d - -9 = 12
46 -k - 8 = -15
47 -t - -6 = 4
34 -20 = x + 9
35 -3 = 10 + u 48 14 - u = 20
49 -5 - c =2
36 5 = -3 + b
37 -1 = k + 6
51 -80 - f = 35
26 q + 6 = -1
27 9 + r = -5
28 t - 4 = -3
29 c - -5 = -5
30 w - -2 = 0
31 n - 7 = -3
g=
Get the letter on its own
then rewrite the equation so
that the letter is on the left.
32 15 = -7 + d
15 m + 6 = -2
33 -3 = y - 3
Then
reverse the
equation
16 y + 8 = -8
17 d + 10 = -7
t - 9 = -4
z = -11
Now try these without
working, test numbers
to help you if necessary.
11 g + 6 = 4
Examples
Examples
=-1
8+
Solve these then change the
sign on both sides of the
equation so that it's positive
18 q + 2 = 0
19 a - 7 = -2
20 e + -5 = 10
21 s - 7 = -7
50 8 - a = -15
Solving One Step Equations - Multiplication
This sheet deals with one step equations that involve multiplication. The sheet starts with a
column of questions that you should be able to answer by inspection without showing your
working out. From column 2 onwards you are expected to show working, even if you can still
solve the problems mentally.
Column 1 involves simple one step equations in that the answers are all cardinal numbers. One
way to solve these is to talk them through, for example:
8m = 24 …. say 8 times something equals 24
Try m = 3 …. say 8 times 3 is 24 so write m = 3
Column 2 asks you to show working and also includes fractions. These fractions won't
simplify until Q 34. Follow the examples at the top of the column by dividing the number on
the right hand side by the number on the left hand side (in front of the letter). Show your
working as the example shows.
From Q 34 it's fractions only, this time they simplify. So use the same method to get the first
line, then find the HCF (Highest Common Factor) or the largest number that goes into the top
and bottom numbers and divide through.
The third column changes with the last 6 questions. These involve answers that are mixed
numerals (number + fraction). Use the same method to get to the first line by dividing through.
You will have an improper fraction (the numerator or top number is larger than the
denominator). Then convert to mixed numerals….do you remember how? Ask yourself:
•
how many times does the bottom number go into the top number, write this in the whole
number box at the front
•
then write the remainder (amount left over) in the top box.
•
the bottom box has the same number as the bottom box in the line above.
7k = 17
÷7
k=
16e = 14
÷16
÷16
7
HCF = 2
e=
8
e=
14
k=
÷7
17
7
2
3
7
16
7
8
7 goes into 17
twice (14)
remainder 3
(17 - 14 = 3)
just rewrite the 7
from the line above
Solving One Step Equations - Multiplication
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Look at these equations
and give the value for
the pronumeral
1 4d = 8
2 6h = 12
Solve these equations
only this time show your
working out.
d=
39 12v = 10
40 24f = 6
41 20s = 16
42 30h = 12
43 35x = 28
Examples
6p = 54
5d = 3
÷6
÷5
÷6
p=9
d=
3 5x = 45
4 9g = 36
38 40a = 25
÷5
3
5
24 3e = 36
25 9u = 4
26 8a = 40
27 2w = 54
5 10n = 120
6 7e = 42
7 8y = 56
8 5t = 55
28 11q = 9
29 17t = 7
30 12c = 60
31 15x = 13
32 20m = 9
33 7r = 42
9 11a = 44
10 9q = 18
These can be changed
from improper fractions
to mixed numerals
11 3z = 39
12 2w = 30
13 5r = 15
44 4t = 9
45 3c = 11
46 8n = 35
47 14d = 33
48 11z = 50
49 4k = 33
14 15p = 45
15 4b = 28
16 6c = 36
17 2k = 48
These will have fraction
answers that can be
simplified
34 6j = 3
35 8d = 6
36 15k = 9
37 20e = 12
18 9y = 90
19 30u = 180
20 15x = 60
21 2v = 34
22 7j = 84
23 5t = 0
Further One Step Equations - Multiplication
This sheet involves the use of negative numbers in one step equations. To answer this sheet
you should have first completed the 'Integers' folder or already have learnt about negative
numbers. The methods used in this sheet are identical to the previous sheet except that the
rules for operations with negative numbers are included.
Column 1 begins with revision of operations with negative numbers, you are asked to write in
the answer (with the sign) in the box provided. Remember that:
•
negative × (or ÷) negative = positive
•
negative × (or ÷) positive = negative
•
positive × (or ÷) negative = negative
The second part of column 1 is the same as above except the box is replaced by a letter and
you have to specify the value of the letter. Talk it through to yourself if it makes it easier. So if
you had:
8b = -40: say→ eight times something gives me -40. I know 8 times 5 gives
me 40, so eight times -5 must give me -40.
so b = -5.
Note that you can test numbers to see if they work and modify your number until you get your
answer also.
Column 2 uses division by the number (including the sign) that is in front of the letter. Look at
the examples at the top of the column. The questions have the letter on the right hand side in
the second part of the column. Use the same method then reverse your answer, as shown in the
example at the base of this sheet.
Column 3 is the same as the third column on the previous sheet. Simplify the fraction on the
first line by dividing through by the HCF. From Q 44 the answers are to be expressed as mixed
numerals. Have a look at the examples below.
-35 = -7x
-32u = 56
-5w = -19
÷ -7
÷ -32
÷ -5
5=x
÷ -7
÷ -32
7
56
u=
x=5
-4
u=
-32
-7
4
w=
÷ -5
-19
-5
w= 3
4
5
Further One Step Equations - Multiplication
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Find the solution to
these and place the
number in the box
1
3×
= -6
2
5×
= -20
3
-7 ×
= -35
4
-9 ×
= 45
5
-6 ×
= -66
6
3×
= -15
7
2×
= -2
8
-5 ×
= -50
9
4×
= -36
10 1 ×
= -1
Examples
Now try these without
working, test numbers
to help you if necessary.
11 6r = -24
12 -9e = 72
Solve these equations,
only this time show your
working out.
6x = -54
-3h = -33
÷6
÷-3
÷6
38 -20e = -8
39 16v = -12
40 -45t = 20
41 30h = -24
42 -6g = -3
43 60p = -48
44 -3a = -8
45 -4w = 11
46 5n = -12
47 -7b = 24
÷-3
x = -9
h = 11
22 12q = -24
23 -7t = -70
24 15d = -45
25 -4a = 36
26 -10h = 50
27 -b = 80
28 3c = -39
29 -8m = -48
30 15i = -90
31 -2y = 54
r=
32 -5e = -75
The fraction answers will
need to be simplified.
From Q 44 mixed numeral
answers are required.
33 4s = -100
13 -5q = -25
14 2y = -4
15 -8v = 32
16 -3j = -33
Solve these then reverse
the answer so that the
letter is on the left.
34 38 = -2x
35 -28 = -4r
36 -56 = 7w
37 75 = -3f
17 7k = -28
18 -5c = 0
19 -4n = -60
20 13g = -52
21 4p = -4
48 -12c = -17 49 -3x = 5
Solving One Step Equations - Division
These one step equations have the letter being divided by a number (or decimal). As with
addition, subtraction and multiplication you will use the opposite operation to solve these
exercises. The opposite sign to ÷ is ×, so you will use multiplication to solve these questions.
Column 1 starts with questions that you should be able to complete mentally. Talk it through to
yourself, for example if you have t/6 = 4:
t
/6 = 4 ….. say something divided by 6 equals 4
t = 24 ….. say 24 divided by 6 is equal to 4, so t = 24
Column 2 then uses multiplication to solve the problems. Look at the number that the letter is
being divided by and multiply both sides by it. Examples are at the top of the column.
Column 3 uses the same solving method except decimals are involved. Follow the examples at
the top of the column then see if you can answer the first 4 questions mentally. From then on
when you multiply, use the multiplication working space to get your answer. Write your
answer and remember to write the letter then ‘=’ before your answer.
Solving One Step Equations - Division
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Find the value for the
letter that will make
these equations true.
1
d
=5
2
2
t
=3
6
3
b
=3
4
4
c
=3
5
5
f
=4
6
6
n
=4
10
7
p
=3
7
8
q
=8
6
9
k
= 10
5
10
j
=9
2
11
v
=8
3
12
m
=3
12
13
y
=4
8
h
14
=6
7
15
r
= 17
2
16
u
=8
20
d=
Now solve these equations
showing working
This time the questions
involve decimals. Try to
answer without a calculator,
spaces are provided.
Examples
d
=8
×7 7
×7
v
=5
×11 11 ×11
d = 56
v = 55
x
=6
9
18
q
= 10
12
20
c
=9
7
k
= 15
5
22 w = 4
20
n
= 21
6
24
25 m = 2
53
26
17
19
21
23
27
y
=3
31
28
h
=3
15
b
= 17
10
Examples
k
= 3.2
×4 4
×4
n
=5
2.1
×2.1
×2.1
k = 12.8
v = 10.5
33 a = 3.4
2
34
e
=3
4.3
35
h
=6
7.1
36
t
= 5.2
3
37
n
=7
2.9
38
q
=5
4.7
39
t
= 9.6
3
40
b
=4
0.8
d
= 25
4
a
= 40
5
29
p
= 12
4
30
e
= 13
10
31
f
=8
9
32
h
= 30
6
41 k = 7.4
5
2.9
7
Further One Step Division Equations and
Solutions to Equations
This sheet involves using negative numbers, you should have covered 'Integers' before you
attempt this sheet.
Column 1 carries on from the previous sheet only this time negative numbers are involved.
Examples are at the top of the column, the method is the same as the previous sheet.
Column 2 uses substitution, (when the letter is replaced by a number). These questions aren't
to find the value for the letter that makes the equation true. A value is already given and you
are asked to test to see whether it is a solution. This is done by replacing the letter with the
value given in each question, simplifying the left hand side of the equation (L.H.S.) and
checking if it equals the right hand side (R.H.S.) of the equation.
If the same number is on both sides of the = sign then the value 'satisfies' the equation (it is a
solution). If the numbers aren't the same on either side of the = sign then you write ≠ (not
equal to) and say that it isn't a solution. As in the example at the top of column 2.
Column 3 follows on from column 2 except that brackets are introduced. You have to evaluate
the brackets before you multiply by the outside number.
Test for b = 6
2(b - 9) = -6
-3
2 × (6 - 9) = -6
-6 = -6
LHS = RHS
so 6 satisfies the
equation.
Further One Step Division Equations and Solutions to Equations
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Find the value of the
letter that make these
true.
Examples
n
= -5
-8
× -8
× -8
j = -72
n = 40
3
23 Test for a = -3
-6a = 18
Example
j
= -12
6
×6
×6
1
Substitute the value for the
letter to see if it is a
solution for the equation
a
= -6
5
2
h
= -15
-3
4
t
=9
-7
y
= -12
5
g +17 = 40
Try g = 33
33 + 17 = 40
LHS ≠ RHS so 33
50 ≠ 40
is not a solution
17 Test for n = 31
24 Test for k = -8
k2 = 64
n - 12 = 19
25 Test for g = 5
18 Test for t = 16
7(g + 3) = 56
t + 37 = 63
5
t
= -1
7
6
m
= -3
-6
26 Test for d = 12
19 Test for m = 7
7
u
= -5
9
8
b
= -2
16
5(9 - d) = -2
3m = 23
27 Test for x = -27
9
d
= -11
-3
10
g
= 60
-5
20 Test for u = -4
x
= -9
-3
8u = 32
11
f
= -12
20
12
q
= -8
-7
21 Test for c = -6
13 m
= -3
17
14
e
= -5
15
15
16
k
= -1
-2
x
= -2
19
19 - c = 25
28 Test for e = -9
2(3 - e) = 24
22 Test for q = 9
29 Test for v = 3
-5 - q = 14
5(v - 8) = -30
Powers and Expansion
This sheet has some questions written sideways. To read these questions click on the
“View” menu, select the rotate option then “counter clockwise”.
When you use powers with pronumerals (letters) it is important that you understand what the
power means. Unlike numbers raised to a power, you can't use a calculator to get an answer
with letters.
Column 1 gives you expanded expressions (expressions don't have an =, equations have an =)
that you are asked to simplify. Expanded form is when the expression is broken up so that
there are no powers, with each piece being separated by an × sign. The way you answer these
is:
•
Multiply all the numbers together and write the number part of the answer
•
Count the number of same letters then write the letter and the number straight after it,
small and raised, e.g. a2. If there is only 1 letter then you don't need to write a number.
•
Are there other letters in the question? If there are, count the number of those letters and
repeat the process.
Column 2 asks you to expand the expressions. Use the opposite method. Write the number
then an ‘=’ sign. Then the letters separated by × signs, the number of letters being the power,
no power means one letter.
Column 3 is a brief introduction to multiplying with powers. The column starts with asking
you to expand out the multiplication, then simplify. It’s the same as the previous work. You
may notice a rule that applies to multiplying indices. This rule is that you add the powers when
you multiply. This is dealt with by the second part of the column, there are examples in the
column. Just remember when a letter is by itself, it is as if there is a 1 as its power. So m = m1.
Powers and Expansion
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Simplify these expressions
by removing the × signs
and rewriting using powers
Example
24
Now the opposite, expand
these removing powers
and inserting × signs
j3
Expand these and
then rewrite them as
simplified indices
Example
6am2 = 6 × a × m × m
h2 × h2 = h × h × h × h
4×j×v×6×j×j =
24vj3
1 y×y×y×y
=
18 7b2 =
39 a2 × a3
2 m×m×m×m×m =
19 10k4 =
=
3 a×a
=
3
20 2t
24
23
57ud3 =
4r3b4 =
=
=
8y4s3
10 8 × i × i × h × i × i
25
=
3r2d2 =
9 n×n×c×c×c
=
=
=
=
8 a×x×x
26
30 12x2y2 =
7 2×b×b×3×b×b =
42 3c2 × c2
ab2c3 =
=
Examples
5x5 × 4x7
13 c × c × 5 × c × b × c × c × 4
31 10d3aw =
dv4h2 =
32 3m2qk3 =
33
2s2rx3 =
=
34
17 y × y × 3 × a × 2 × x × a × 4
35 38q2vb2 =
=
36 9k3g2e =
16 s × t × s × t × v × 4 × s × s
37 2cgpq2 =
=
38 x2atf 3 =
15 8 × c × b × a × c × b × c × 5
=
7 + 9 = 16
h7 × h9 = h16
=
=
43 3y3 × 4y
Sum the powers to find
the answer to these
12 e × e × 3 × q × e × e × 7 × q
14 6 × g × b × b × g × g × b × 9
=
=
11 d × a × d × d × a × a =
=
=
27
6 u×u×4×u×u
=
6nr3t
22 d2c4 =
28
=
41 n3 × n
y2ib3
5 6×t×t×t
=
29
21 5qx =
=
=
3
=
= h4
40 b4 × b2
=
4 h×h×h
h2 = h × h
h2 = h × h
Example
k = k1
k × k6
= 20x12
43 u3 × u8 =
44 t9 × t11 =
45 c5 × c17 =
46 3x14 × x17 =
47 5d3 × 7d
=
48 2s5 × 4s5
=
49 7q6 × 9q3 =
50 5w4 × 2w × 3w2 =
51 4e4 × 3e5 × e2
=
1+6=7
= k7
Substitution
Substitute means to replace with. In mathematics you are often asked to replace a letter with a
number and find the answer. There are two important things to remember, the first is when
there is no sign between letters then it is as if there is a × between them.
For example wr is w × r, so if w = 4 and r = 2 then wr = 4 × 2 = 8 not 42.
The second point is powers, so if w = 4 then : w2 = w × w = 4 × 4 = 16 (or 42 = 16),
not w2 = 4 × 2 = 8. A frequent mistake.
Column 1 starts with questions that ask you to find the values of two pronumerals. Substitute
in the numbers on the second line (after an equals sign), then on the third line write the letter =
and your answer. Question 1 has the format shown for you to start you off. From question 4
you may encounter questions that order of operations rules apply. Remember × and ÷ before
+ and -, be careful.
Column 2 is the same as column 1 except you may find the numbers a little more challenging.
Find the values of a and b
given y = 12 and x = 7
a)
b)
a = y + 3x
b = 2y - x
24
21
= 12 + 3 × 7
a = 33
= 2 × 12 - 7
b = 17
Column 3 tests your knowledge of negative numbers, remember:
•
a negative number × or ÷ a negative number gives a positive number
•
a positive number × or ÷ a negative number gives a negative number
•
a negative number × or ÷ a positive number gives a negative number
•
+ then a - = - (+ - = -)
•
- then a + = - (- + = -)
•
- then a - = + (- - = +)
Substitution
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Substitute the values for the
letters and find the answer
1 Find the value of k and r
when x = 7 and m = 6
a) k = xm
b) r = x + m
=
=
×
k=
6 Solve for q, v and b given
e = 6 and x = 3
a)
b)
q = 2xe
v = q - 2e
These questions use
negative numbers
10 Evaluate for h and n when
t = -5 and c = 3
a) h = c + t
b) n = 3t
+
r=
c) b = q + v - x
2 Evaluate for h and e when
a = 5 and t = 3
11 Find the values of s and k
given x = -8 and d = -4
a) h = at
a)
b)
s=x-d
k = d2
b) e = 2a + t
3 Solve for b and v given
w = 7 and c = 2
a)
b)
b = w2
7 Find the values of r, c and
t given u = 10 and a = 4
a)
b)
r = 3u - a
c = a2 - u
12 Find the values of n and b
given r = -10 and e = 6
a)
b)
v = 3cw
c) t = 2r - 3c
4 Find the values of g and d
given n = 12 and q = 5
a)
b)
g = n + 8q
d = n - 2q
5 Solve for c, v and y given
t = 6 and e = 3
a)
b)
2
c = te
c) y = c + 2v
8 Solve for x, y and z given
a = 6 and b = 2
a)
b)
x = ab2
y = 2x - 18
n = r - 2e
b = 3r + e
13 Solve for a and b given
u = -4 and y = -7
a)
b)
a = 2u2
b=a+y
c) z = ( y - x )2
v = t + 5e
9 Find the values of d and p
given c = 24 and v = 3
a)
b)
d = c + 3v
p = 2d - c
14 Find the values of a and k
given p = -3 and q = 9
a)
b)
2q
pq
a=
k= 2
p
p
7
FREEFALL
MATHEMATICS
GEOMETRY
Bisecting an Interval
Bisecting something is cutting it into two equal pieces. This sheet deals with bisecting
intervals. An interval is a straight line but instead of going on forever it has specific length.
The intervals on this sheet have markers at their ends for you to position instruments more
effectively.
Column 1 starts with asking you to bisect these five intervals with a ruler. Measure the length
of the interval, write your answer in the space, give the answer in mm. Divide this number by
2 and write that answer in the ‘half length’ space. Mark off half-way with a small stroke.
Questions 6 through 10 involve the same 60 mm interval, the questions ask you to break the
interval up into a specific number of parts. Divide 60 by the number of parts required in the
question and that is the measured gap required between 'marks'.
Columns 2 and 3 are the same, it is asking you to bisect the given intervals using a compass.
Below is the process outlined step-by-step.
1
2
Draw an arc from one end of
the interval (red circle). Open
the compass more than half the
interval’s length.
Repeat for the other
end of the interval.
Keep the compass set
at the same distance.
3
Draw a line through the two
intersection points (red circles).
Use a ruler to measure the lengths
as a check.
6
7
8
9
10
20
5
10
4
0
3
MATHEMA
TICS
50
60
70
2
40
1
30
0
FREEFALL
80
90
100
Bisecting an Interval
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Using your ruler, measure
then bisect these intervals
1
Use a compass to bisect
these intervals. Give the
measurement from one
end to the midpoint (cm)
Interval length :
Half length :
2
14 Half length :
11 Half length :
Interval length :
Half length :
3
Interval length :
Half length :
15 Half length :
12 Half length :
4
Interval length :
Half length :
5
Interval length :
Half length :
Divide the 60 mm
intervals into 3, 4,
5, 6 and 10 parts
6 3 parts, interval =
mm
7 4 parts, interval =
mm
8 5 parts, interval =
mm
9 6 parts, interval =
mm
10 10 parts, interval =
mm
13 Half length :
16 Half length :
Perpendicular and Parallel Lines
Perpendicular lines are lines that are at right angles to each other, so the angles formed when
the lines intersect are 90º. Parallel lines are lines that are the same distance apart throughout
their length and so they never intersect (touch each other). This sheet requires the use of a
compass and a set square to create parallel and perpendicular lines.
Column 1 - Perpendicular from a point on a line - Set Square
90
80
90
80
3
10
F RE
20
30
Place right angle
markings to show it
is a perpendicular.
0
0
10
F RE
20
30
40
E FA
50
LL
MA
60
TH E
70
MA
TICS
2
Draw the line
along the edge of
the set square.
40
E FA
LL
50
MA
60
TH E
70
MA
TICS
1
Place the set square
on the line with its
corner on the point.
Column 1 - Perpendicular from a point away from a line - Set Square
1
2
10
0
F RE
20
30
40
E FA
LL
50
MA
60
TH E
70
MA
TICS
80
80
40
E FA
LL
50
MA
60
TH E
70
MA
TICS
30
F RE
20
10
0
3
Place right angle
markings to show it
is a perpendicular.
90
Draw the line
along the edge of
the set square.
90
Place the set square
on the line with the
point on its edge.
Column 2 - Perpendicular from a point away from on a line - Compass
1
Position the compass in the
point and draw an arc that
cuts the interval in two places
3
Draw a line through the
intersection points of the
green arcs and the point
2
Then place the compass at the
points where the blue arc cuts
the interval, create 2 arcs.
4
!
Place right angle
markings to show it
is a perpendicular
Green arcs and
blue arcs don’t
have to be the
same size
Column 2 - Perpendicular from a point away from on a line - Compass
1
Position the compass at the
point and draw an arc cutting
the interval each side of the
point (blue).
2
Position the compass where
the blue arcs cut the interval
and draw 2 arcs as shown,
(green circles).
3
Draw a line through the two intersection points and the original
point. Add a right angle symbol
to complete.
Column 3 - Drawing a Parallel line - Set Square
1
2
Position the set square marked with a circle on the
line. Place another set square (square) or ruler on
its edge, slide it to the required length (2.2 cm).
Slide the set square (circle) up to the zero then draw
your line. To show the lines are parallel draw an
arrow-head on each line.
0
10
0
20
10
30
0
20
20
40
10
0
50
40
30
30
50
60
70
20
30
40
50
60
40
60
10
80
70
50
90
60
80
70
90
80
90
Column 3 - Drawing a Parallel line - Compass
1
Open your compass the required distance and draw
two arcs from any position on the interval.
2
Draw a line across the edges of the circles. Place
arrow heads on the lines to show they are parallel.
70
80
Perpendicular and Parallel Lines
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Use a set square to draw
a perpendicular to the
interval at the X, in the
direction of the arrow.
Use a compass and a ruler
to draw a perpendicular
from the X to the interval
1
7
2
8
Use a set square and ruler
to draw a parallel line to
those below, separated by
the given distance
13 2 cm
14 3 cm
3
9
15 1.7 cm
10
Use a set square to draw a
perpendicular from the X
to the interval or intervals
4
Keep going only this time
use a compass and a ruler
16 2.2 cm
Now the X is on the line
5
11
17 4.3 cm
6
12
Geometry - Spider's Web
This sheet is a line drawing exercise which will make a spider's web. Join the numbers that add
up to 8, in branches that face each other, 1 and 7, 2 and 6 and so on. If you prefer to have no
numbers on your worksheet, the second worksheet has the numbers removed.
Geometry - Spider's Web
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Join the numbers that add up to 8 in
the branches that face each other, 1
and 7, 2 and 6, 3 and 5 and so on
7
6
7
7
5
6
6
4
5
5
3
4
4
2
3
2
3
1
4
1
1
4
5
2
2
1
3
2
1
1
5
5
5
3
4
4
3
3
2
6
6
4
5
6
7
7
7
6
1
1
7
2
3
2
6
7
Geometry - Spider's Web
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Join the numbers that add up to 8 in
the branches that face each other, 1
and 7, 2 and 6, 3 and 5 and so on
Equilateral and Scalene Triangles
Equilateral triangles are triangles with all sides equal and all interior angles equal (all 60º).
This sheet is about constructing equilateral triangles with a compass (and a ruler). This is best
described in steps, follow the steps outlined below.
Scalene triangles have no equal sides. Follow the steps on the next page for constructing these
triangles. Note that when a scalene triangle has an interior angle that is 90°, it isn't a scalene
triangle it is a right (angled) triangle.
Column 1 - Constructing Equilateral Triangles - Compass
1
Position the compass at the
end of the interval and open it
to the interval length and
draw an arc.
2
Repeat for the other
end of the interval
to create a point of
intersection.
3
Draw a line from the end of the interval to
the intersection point and then repeat for
the other end of the interval. Mark the
sides to show sides are all equal.
Column 2 - For the second part of the
column rule a line first then repeat the
above steps.
Column 2 - Constructing Scalene Triangles - Compass
1
Position the compass at the
end of the interval and open it
to the given length and draw
an arc.
2
Repeat for the other
end of the interval
to create a point of
intersection.
3
Draw a line from the end of
the interval to the intersection
point and then repeat for the
other end of the interval.
Column 2 - For the second part of the
column rule a line first then repeat the
above steps. Rule the longest line first,
then use your compass to create the
other two lines.
Equilateral and Scalene Triangles
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Make an equilateral triangle given a side.
Use a compass and a ruler. Construct the
triangle in the direction of the arrow.
Now make scalene triangles
with one side drawn and the
other two sides given.
1
2
9 4 cm and 3 cm
10 2.5 cm and 3.1 cm
3
4
11 2.2 cm and 4.3 cm
12 1.9 cm and 2.7 cm
5
6
13 1.3 cm and 4.7 cm
14 5.4 cm and 4.9 cm
Now construct equilateral triangles
with the side lengths given below
7 5.5 cm
Construct scalene triangles with
the side lengths given below
15 3 cm, 1.7 cm and
2.4 cm
8 6.3 cm
16 3.8 cm, 2.9 cm
and 4.1 cm
Creating Isosceles and Side-Angle-Side Triangles
Column 1 - Constructing Isosceles Triangles - Compass
1
Repeat for the other end of the line
to create a point of intersection.
Note that the second line may not
be the same length as the first.
3
4
Open your compass to the
given length then position the
compass at the end of the line
and draw an arc.
Using a ruler, draw lines
from the intersection
point to the ends of the
line.
2
Label the two
sides of equal
length with
dashes.
Questions 7 & 8 - Rule a line first then
repeat the above steps. Rule the longest
line first, then use your compass to
construct the other two lines.
Creating Isosceles and Side-Angle-Side Triangles
17
10
0
20
30
40
50
60
70
80
90
EE
FR
2
L
FA
L
7
40
6
50
5
60
4
70
3
80
70
65
50
55
45
11
12
11
60
85
13
10
9
1
90
80
0
8
0
95
5
7
6
5
4
3
2
1
0
10
0
0
10
10
18
10
9
8
7
6
5
4
3
2
1
15
0
20
40
60
80
0
20
17
30
50
M
70
90
5
Complete the final
side by using a
ruler between the
two points
LL
TH
25
17
10
20
60
80
10
0
EE
FR
FA
A
30
5
0
0
10
30
50
M
70
LL
TH
90
EE
FR
FA
A
35
14
15
15
16
10
4
CS
TI
A
EM
14
16
Rule the other line
given in the question.
It doesn’t have to pass
through the point.
40
CS
TI
A
EM
40
13
20
15
16
10
12
25
16
90
10
75
3
80
30
FREEFALL MATHEMATICS
100
Remove the protractor and
place a ruler with the zero at
the end of the line and the
point on the ruler’s edge.
85
10
10
9
65
60
8
70
7
6
45
5
50
4
55
3
95
11
13
2
40
35
15
18
1
11
12
12
14
14
15
17
0
10
2
Place a protractor at the
end of the line and plot a
point at the given angle.
In our example it is 45º.
13
1
Draw the line of
longest line first,
the 4 cm line in
our example.
75
Example:
4 cm, 3 cm
& 45° angle
Column 2 - Constructing Triangles - Protractor & Ruler
Making Isosceles and Side Angle Side Triangles
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Make an isosceles triangle, one side is
already drawn for you. Use the two
other measurements to complete it.
1 3 cm & 3 cm
2 2.3 cm & 1.6 cm
3 2.6 cm & 2 cm
4 4.2 cm & 4.2 cm
Here are two sides with the angle
between them. Use a protractor
and a ruler to draw them, then
join the ends for the third side
9 4 cm, 3 cm, 60º
10 2.5 cm, 3.2 cm, 40º
5 3.3 cm & 3.3 cm
6
Now construct isosceles triangles
with the side lengths given below
1.2 cm & 3 cm
11 5.3 cm, 4.7 cm, 25º
7 2.6 cm, 4.1 cm and 4.1 cm
12 7.1 cm, 6.3 cm, 38º
8 2.7 cm, 3.8 cm and 3.8 cm
Angle-Angle-Side Triangles and Properties
A triangle can be constructed when you are given two angles and a side. This sheet requires a
protractor and a ruler. The steps are below:
Example:
45º and 60º
Base Angles
Column 1 - Constructing AAS Triangles - Protractor & Ruler
2
70
85
80
75
95
10
65
55
50
11
12
11
12
45
3
15
10
13
13
17
17
5
17
5
18
0
18
0
2
1
EE
FR
0
3
4
100
90
2
M
LL
30
25
80
20
15
13
5
18
0
18
0
50
5
TIC
MA
60
17
4
17
5
HE
AT
10
10
17
3
16
70
12
45
35
14
15
15
16
FA
40
14
1
13
10
55
50
11
12
11
60
75
70
65
85
80
95
10
EE
FR
80
85
20
15
65
25
45
17
30
70
50
16
55
16
60
13
15
75
13
40
35
Erase the extended
side length. All done!
0
95
11
10
12
10
11
12
14
14
15
90
A
5
Use a ruler to draw the
final side. Stop when
the line intersects the
other side.
Repeat for the
other end of the
line.
FA
M
S
10
0
LL
EM
TH
C
TI
90
10
60
20
16
10
A
80
17
4
25
70
20
15
30
60
25
16
35
14
15
15
16
50
14
5
40
45
35
30
16
6
40
30
50
40
8
7
20
85
55
13
65
60
13
70
75
12
14
15
15
10
9
10
95
80
10
10
11
11
12
14
90
Place a ruler at the end of the
line and draw a line through
the dot, the extra length can be
removed later with an eraser.
0
1
Using your protractor place
it at one end of the line and
measure the first angle. Plot
a dot at the first angle.
40
S
6
30
7
20
8
10
9
0
10
Column 2 - Q.8 - Q.11.
Use the same method
as above after first
ruling the line.
The second part of the sheet aims to outline the relationship be-
tween the largest angle of a
triangle and the triangle’s longest side. Name the longest side and largest angle and in Q.15
explain how the angle and side are always located on a triangle.
Angle-Angle-Side Triangles and Properties
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Use a protractor and a ruler to
create triangles with base internal
angles as given below
This time draw the given side then
use the same method. Be careful
where you choose to place the line.
1 40º & 50º
2 23º & 55º
8 2.5 cm, 30º, 40º
9 3.7 cm, 22º, 47º
3 70º & 80º
4 52º & 47º
10 4.4 cm, 53º, 35º
11 1.7 cm, 75º, 75º
Name the largest angle and the
largest side in these triangles
5 15º & 38º
A
c
12 Side :
b
Angle : <
B
a
C
6 117º & 29º
D
13 Side :
Angle : <
j
f
J
14 Side :
7 60º & 60º
Angle : <
N
F
d
k
V
v
n
K
15 What is the relationship between the largest
angle and the longest side?
Bisecting Angles and Making 45° Angles
This sheet is about bisecting an angle, this means cutting the angle in half. The second part of
the sheet is about creating 45° angles using a compass and a ruler. When you make a 45° angle
you are also making a 135° angle, it’s the angle’s supplement, it will be the adjacent obtuse
angle beside the 45° angle.
Column 1 - Bisecting Angles - Protractor & Ruler
1
2
Place the compass
point at the vertex
and draw an arc of
any size.
Using the intersection of the
arc and the ray (red circle),
create another arc. This arc
also can be of any size.
3
4
Repeat for the other intersection point (red). Use
the same compass opening
as the previous arc.
Using the vertex and
the arc intersection
point (red circles),
draw the bisecting ray.
40
50
60
70
80
90
100
65
11
10
95
10
70
85
75
80
80
85
30
75
90
AT
20
10
95
10
60
65
70
50
45
0
25
5
20
55
30
18
35
17
40
15
15
10
11
14
15
16
16
17
11
13
12
12
13
14
5
Use a protractor to measure
the size of the bisection and
original angle to check that
your method is correct.
ALL M
11
60
12
55
12
13
13
14
14
50
45
40
35
30
15
25
15
16
20
16
15
17
17
18
10
5
0
Note Q.4 is a reflex
angle so a working
space is provided.
10
F
FR E E
TICS
H E MA
0
1
0
5
4
3
2
10
9
8
7
6
Bisecting Angles and Making 45° Angles
A 45º angle can be constructed by first making a 90º angle and bisecting it. So a right angle is
created, then the same process from the previous column is repeated.
Column 2 - Constructing 45º Angles - Compass & Ruler
2
4
Using the intersection
points, create two
more circles (or arcs).
Use the intersection points
created by the two circles
(red), draw a perpendicular
line. This is a 90º angle.
The first circle (green) intersects
the perpendicular and the interval.
Using these point create two arcs
or circles (red circles).
6
70
65
45
12
13
50
13
55
45
11
12
50
55
60
85
80
75
85
65
70
60
13
16
40
14
35
20
14
15
15
16
15
30
25
16
75
13
40
35
11
95
80
11
10
12
10
11
12
14
14
15
90
95
Check your work
with a protractor.
10
5
Draw a line through the
vertex and intersection
point (red circle) and its
done!
10
1
3
Using a compass
draw a circle near
the centre of the
interval.
30
25
20
15
15
16
10
17
17
5
17
5
18
0
18
0
17
10
Bisecting Angles and Making 45° Angles
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
To make a 45° angle create a 90°
angle (perpendicular) then bisect it.
A 45° angle could also be 135°!
Bisect the angles below using a compass
and a ruler. Verify that the bisection is
correct using a protractor.
1 Original angle
°
Bisection angle
° 5
2 Original angle
Bisection angle
6
3 Original angle
Bisection angle
7
4 Original angle
Bisection angle
8
360 -
Constructing 60° and 30° Angles
This sheet is about constructing a 60° angle and a 30° angle (a bisected 60° angle), to complete
this sheet you require a compass and a ruler. One important thing to remember is that if you
make a 60° angle you are also making a 120° angle, it’s the supplementary angle. It will be the
adjacent obtuse angle beside the 60° angle. Likewise when you make a 30° angle the adjacent
obtuse angle will be 150°.
Column 1 - Constructing 60º & 120º Angles - Compass & Ruler
1
Using the intersection of the arc
and the interval (red circle),
keep the compass at the same
opening and draw another arc.
2
3
4
Place the compass at
the end of the interval,
open to any size and
draw an arc.
60
50
5
IC
AT
18
40
S
6
7
30
20
8
9
10
10
0
20
15
80
70
65
55
50
45
11
12
40
13
13
11
60
85
80
75
95
10
10
90
12
65
4
17
25
45
EM
TH
17
30
70
50
3
16
55
35
15
16
60
13
40
15
75
13
14
MA
70
10
80
85
12
12
14
2
LL
FA
95
11
10
11
1
EE
10
FR
90
Check your work
with a protractor.
0
0
Use a ruler to connect
the two intersection
points (red circles).
14
35
30
14
15
15
16
25
20
16
15
10
10
17
5
17
5
0
18
0
Constructing 60° and 30° Angles
Column 2 - Constructing 30º & 150º Angles - Compass & Ruler
1
2
Create a 60º angle as outlined
on the previous page. A 30º
angle is made by bisecting a
60º angle.
Draw an arc from the
vertex of the angle (red
circle) of any length.
3
Then repeat for the other
intersection point (red
circle). Keep the compass
at the same opening.
4
Draw an arc from
one intersection
point (red circle).
6
5
Use a ruler to connect
the intersection point
and the vertex (red
circles). All done!
7
8
40
9
30
10
70
65
12
45
11
12
50
55
60
85
80
75
95
13
13
10
85
17
17
5
17
5
18
0
18
0
17
20
10
0
S
45
10
16
6
30
16
30
25
16
AT
IC
35
20
3
5
40
14
15
15
50
TH 4
EM
65
50
40
35
14
15
15
16
2
60
MA
55
70
FA
LL
60
13
14
15
70
13
14
1
75
12
EE
11
95
80
11
10
10
11
12
0
90
80
FR
90
10
Using a protractor,
check your work.
25
20
15
10
10
0
Constructing 60° and 30° Angles
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
To make a 60° angle use the same
method as creating an equilateral ∆.
A 60° could also be a 120° angle.
To make a 30° angle create a 60°
angle then bisect it. A 30° angle
could also be 150°!
1
5
2
6
3
7
4
8
Flower Design
This is a simple design that creates a flower-like effect. Draw lines starting at the smallest
circle then moving to the next circle and one position clockwise. Complete all the way to the
outside circle then repeat starting from the next position on the inside circle, and so on.
After you have a full revolution you repeat the process only you move anti-clockwise instead.
There are two worksheets, the first has one completed clockwise step and a completed anticlockwise step. The second sheet is completely blank if you don’t need the help.
A image of the completed flower is below.
Flower Design
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Complete the line pattern, move one circle out and one line clockwise. Go all the way around, then repeat only move one circle out
and one line anti-clockwise. Two lines are already drawn for you.
Flower Design
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Complete the line pattern, move one circle out and one line clockwise. Go all the way around, then repeat only move one circle out
and one line anti-clockwise.
Making a Hexagon
This sheet involves constructing a hexagon using a compass and ruler, if printing this sheet use
the dot in the middle of the circle to open your compass to the required radius or use a ruler
and open your compass to 7 cm.
1
Position your compass at any
position on the circumference
and draw a circle.
3
Repeat until six circles
are completed
2
Place the compass at one of the points
where the blue circle intersects the first
circle and create another circle.
4
Join all intersection
points to complete a
hexagon (yellow).
Making a Hexagon
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Using a compass and
ruler make a hexagon in
the circle below
Drawing Quadrilaterals
This sheet involves the construction of four different plane (2-D) shapes. These are squares,
rectangles, rhombi (plural of rhombus) and parallelograms.
Column 1 - Constructing Squares & Rectangles - Set Square & Ruler
10
9
8
7
90
0
100
5
4
3
10
10
60
70
80
90
20
30
40
50
60
70
80
90
5
10
90
3
2
80
1
80
70
100
90
0
70
60
100
5
Place right angle markings
in each corner and mark
the opposite equal sides
with dashes. All done!
60
4
FREEFALL MATHEMATICS
FREEFALL MATHEMATICS
9
8
7
6
50
50
FREEFALL MATHEMATICS
40
50
40
5
40
30
4
30
20
3
20
10
2
10
30
FREEFALL MATHEMATICS
0
1
0
20
80
90
90
10
70
4
0
0
60
Use a ruler to draw the
remaining two sides. In our
example they are 3 cm.
Slide the set square along the ruler edge
until its corner is at zero, (red circle). Then
draw the same length line as previously
drawn, 4 cm in our example
0
50
80
2
40
70
1
30
60
0
20
FREEFALL MATHEMATICS
50
FREEFALL MATHEMATICS
40
3
10
9
80
8
70
FREEFALL MATHEMATICS
7
60
6
50
30
40
30
20
20
10
10
0
0
2
Place a ruler on the set square’s
edge so that the corner is at the
other side length, 3 cm in our
example, (red circle).
6
1
Example:
4 cm × 3 cm
Rectangle
Using your set square
draw a line the required
length. In our example,
4 cm in length.
Drawing Quadrilaterals
Column 2 - Constructing a Rhombus - Compass & Ruler
1
2
Open your compass to the side length (4
cm in our example) and draw a circle.
Then draw 2 lines from the centre to the
circumference, at any angle you like.
Draw a point
where you
wish to start.
Example:
4 cm Rhombus
3
Using the compass open at
the same length, draw 2 arcs
from the intersection points
shown below in green.
4
Use your ruler to
complete the
rhombus.
5
Show the parallel
sides with arrows.
Label all the equal
sides with dashes.
Column 2 - Constructing a Parallelogram - Set Square & Ruler
1
0
10
Start by using the same method as with the construction
of a square or rectangle. When drawing the second line,
instead of starting from zero start from another number,
in the example below 2 cm was chosen.
8
20
7
h
10
9
0
30
70
6
5
4
3
60
80
2
50
90
1
40
100
0
20
30
40
50
60
70
80
90
80
90
FREEFALL MATHEMATICS
0
FREEFALL MATHEMATICS
10
10
20
b
30
40
50
60
70
FREEFALL MATHE-
2
Complete the sides and place arrows on parallel
sides. You should also put dashes on the sides to
show that they are the same length, that way a
rhombus isn’t confused with a parallelogram.
Drawing Quadrilaterals
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Construct squares using a ruler and a set
square. Place side length markings and
perpendicular symbols on the shape.
Use a compass to draw rhombi
with the side lengths below, mark
the parallel sides.
1 l = 2.5 cm
2 l = 4.1 cm
9 l = 2.5 cm
10 l = 1.9 cm
3 l = 37 mm
4 l = 1.9 cm
11 l = 2.8 cm
12 l = 14 mm
Now construct rectangles with the same
length sides marked and perpendicular
markings in each corner
Construct parallelograms with parallel
side markings. The base length and the
perpendicular height are given
5 l = 4 cm, b = 2.5 cm
6 l = 2.2 cm, b = 3.1 cm 13 h = 1.5 cm, b = 4 cm
14 h = 3.1 cm, b = 3.6 cm
7 l = 34 mm, b = 6 cm
8 l = 1.3 cm, b = 5.2 cm 15 h = 0.8 cm, b = 2.6 cm
16 h = 1.8 cm, b = 3.2 cm
The Circle
This sheet deals with naming parts of a circle, these are areas of circles created by particular
lines and also the lines themselves.
The sheet has 2 columns with diagrams at the top of each. In the spaces provided, name the
parts of the circle that are being pointed to.
The questions below the diagrams all refer to the parts that have just been named, so look at
the diagrams below if you need help. A word you may not be familiar with is radii - this is the
plural of radius (so 1 radius, 2 radii). Also segments may be named minor segments when
they are smaller than a semi-circle and major segments when they are larger than a semi-circle.
Though for this sheet the word ‘segment’ alone is adequate, ask your teacher if minor/major is
also required.
Segment (minor)
Quadrant
Sector
Column 1 - Circle Areas
Semi-circle
Circumference
er
met
Dia
Column 2 - Circle Lengths
Radius
Chord
Arc
Tangent
The Circle
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Name the parts of
the circle below.
The answers for these
will be found on the
diagram above
1 Pizzas are sliced into these
2 Two of these makes a semi-circle
3 This is the only shape that doesn't touch the
centre of the circle
4 When two or more of these are combined
they could make a quadrant
Name the lengths
on the circle
The answers for these
will be one of the
lengths shown above
11 This distance is half the diameter
12 If a wheel rolled one revolution, the distance
traveled would be the
13 A right angle is formed where the radius
meets this line
14 A segment is formed by an arc and one of
these
A
5 A chord is used to make this shape
6 This shape has the circle's diameter as one of
its sides
7 This shape has a right angle
8 When an arc is combined with two radii the
shape that is made is a
9 If 6 identical sectors form a quadrant what
C
3
Name lengths/areas
described by these
letters
15 AC:
D
O
1
B
16 OB:
17 AB:
18 CD:
19 OD:
20 BC:
21 1:
22 3:
shape is formed by 12 sectors ?
10 If 6 identical sectors form a quadrant what
shape is formed by 3 sectors ?
23 1 + 2:
2
Geometry - Find A Word
Look for words in the list at the bottom of the grid. Once you find a word cross it off the list.
A letter could be used more than once so don’t colour it in too dark (using a texta for example)
so that you can still read it.
Geometry Find a Word
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Find the words in the puzzle from the wordlist
P
E
R
P
E
N
D
I
C
U
L
A
R
I
I
S
P
A
T
T
E
R
N
Q
B
D
R
E
N
S
U
T
N
E
G
N
A
T
X
E
Z
C
T
T
O
I
B
F
R
M
Q
A
E
G
T
P
D
E
E
S
D
L
V
E
K
X
U
R
C
N
R
T
M
R
C
A
A
B
C
D
F
E
S
D
O
M
O
A
V
E
R
R
K
T
V
E
X
E
H
A
M
P
I
A
L
H
E
X
A
G
O
N
C
Q
Q
U
P
D
L
E
M
T
P
N
S
E
C
T
O
R
J
Q
A
L
S
E
A
N
G
V
E
S
I
E
B
F
M
S
S
C
R
L
T
L
C
Q
K
O
T
L
I
L
G
H
S
T
I
K
E
U
N
F
N
P
T
C
S
Q
P
T
N
U
X
A
U
Q
E
T
A
T
O
R
E
U
X
E
Q
R
P
A
R
A
L
L
E
L
V
I
C
N
C
E
C
N
E
R
E
F
M
U
C
R
I
C
T
WORDLIST
BISECT
CHORD
CENTRE
HEXAGON
SQUARE
DIAMETER
INTERVAL
TANGENT
RADIUS
ISOSCELES
QUADRANT
PARALLEL
SECTOR
COMPASS
RECTANGLE
CIRCLE
ROTATE
ARC
PATTERN
DEGREE
CIRCUMFERENCE
PERPENDICULAR
INTERSECTION
EQUILATERAL
7
FREEFALL
MATHEMATICS
GRAPHS &
TABLES
Picture Graphs
Picture graphs are similar to bar graphs, the data is displayed by pictures or symbols of an
item. Graphs usually have an axis to read off the value, Picture graphs don't require this scale
as each picture of an item represents a specified number of items. Picture graphs are as
accurate as their scale allows, but they are used mainly for quick reference rather than an exact
display of data.
Column 1 of this sheet has the graphs drawn for you, you are asked to read the graph and
answer the questions. Note the 'key' or 'scale', this tells you the value of each symbol. To
increase accuracy, portions of a symbol can be used.
The computer in the first column is either: fully drawn, ¾ drawn, ½ drawn or ¼ drawn. If a
fully drawn computer represents 20 computers then part drawn computers equal that fraction
of 20.
Graph 2 is a representation of a Café's sales of fresh juices. This time each symbol represents
200 glasses of juice. Note that again the same fractions ¼, ½ and ¾ are used. Questions 6 and
7 are about rounding. While 2 symbols may represent 400 glasses, those same 2 symbols could
also represent 390 glasses. Some students will have difficulty with this concept.
Column 2 asks you to construct graphs from data in the tables. The data can be constructed
exactly by adding whole and part pieces of a symbol, you don't need to worry about rounding.
Remember that fractions of a picture are always on the end (right hand side) and there can
only be one in each line. Don't use several fractions of a symbol, the purpose of a picture graph
is that they are easy and quick to read, not confusing!
Picture Graphs
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Study the graphs then answer
the questions
Now place the details in
the table on the graph
Graph 1 : Computer Sales in 2002
Table 1 : Jeans Shop Sales Assistant Figures
Jan
Feb
= 20
Mar
Apr
Name
Ian
Deb
Mike
Luke
April
Sales
240
80
140
170
150
Graph 3 : Jeans Shop Sales Assistant Figures
May
June
Ian
July
Deb
Aug
Mike
Sept
Luke
Oct
April
Nov
8 Give the values represented by these symbols:
= 40
Dec
=
=
=
=
1 Give the values represented by these symbols:
=
=
=
=
Table 2 : Students Using Bus Transport
2 Name the months with the same sales figures and give
School Name
the number of sales
Student Numbers
Morcombe
250
3 Give the lowest sales month & sales
St Marks
300
4 Give the highest sales month & sales figure, give a
Macquarie
175
Dusty Flats
550
Te Akuna
50
Knox
225
Twin Peaks
475
Karori
325
reason why this month could have the most sales.
Graph 2 : Average Monthly Juice Sales (glass)
Apple
Carrot
Pineapple
= 200
= 200
= 200
= 200
Orange
Graph 4 : Students Using Bus Transport
Morcombe
= 100
St Marks
Macquarie
5 Name and give sales figures for the:
Dusty Flats
a) least popular juice
Te Akuna
b) most popular juice
Knox
6 Actual sales figures for carrot juice could range from
975 - 1 024 glasses. Briefly explain why.
Twin Peaks
Karori
9 Give the values represented by these symbols:
7 Give the largest and smallest possible sales figures for
pineapple juice
=
=
=
=
Column Graphs
Column graphs show data with columns which are read off a vertical scale, with the horizontal
axis giving the meaning of the column. This sheet has two columns, the first being reading
graphs and the second being constructing graphs (from reading a table).
Column 1 starts with a column graph on the fundraising activity of a charity. It outlines the
different methods of fundraising along its horizontal axis and their financial success on the
vertical axis. This graph can be read to the nearest $5 000.
The second graph is about the conversion of a chicken battery farm to a free range farm. This
time there are two columns for each category. The free range egg figure and the battery laid
figure. Note also that the columns are rectangular prisms, these give the graph a different
appearance, but be careful when reading the columns. Read the columns off the top edge that
is ‘closer’ to you, not the edge at the back of the column. In other words, use the front face of
the prism for measurement.
Column 2 starts with a question about dice. When you roll 2 dice you have 36 different
combinations of the two faces of each die. The first table is just like a 6 by 6 addition table,
just add the two numbers. Then count how many of each number there are for the next table.
Note both tables have been started for you. Then graph the result. Note that the first column is
already done for you.
Question 6 involves a survey of traffic, it requires you to read the data straight from the table
and place it on the graph.
Note the use of the word frequency, frequency means the number of times something occurs.
Column Graphs
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Study the column graphs below
then answer the questions
Complete the tables for the totals
possible for 2 dice, then graph them.
Graph 1 : Fundraising by a Charity
5
Table 1 : Total of 2 dice
10
Face
No.
1
2
3
1
2
3
4
6
2
3
4
5
3
4
4
4
8
7
3
6
Total
2
3
4
Freq
1
2
3
6
6
a) is the third most successful
5
Frequency
2 Which method of fundraising:
Graph 2 : Ingall’s Hen Farm Egg Production
4
3
2
1
Key:
4
Free Range
3
Battery
2
2
3
4
5
6
Year
3 Give the total number of eggs for: a) 1999
6 Transfer this data
60
Survey
50
Freq
4 Give the year that free range production was:
Cars
63
a) 1 000 000 eggs less then battery laid eggs
Trucks
11
b) increased and battery laid was unchanged
Buses
17
c) three times more than battery laid eggs
M-cyc
9
Bikes
4
d) increased by 1 500 000 eggs
9 10 11 12
Graph 4 : Vehicle Survey
Table 3 : Vehicle
Type
c) 2001
8
Face Total
to the graph
1
7
40
30
20
10
Vehicle Type
Bikes
d) made $5 000
2002
9 10 11 12
M-Cyc
c) more than triples the street donations
2001
8
Buses
b) matches Government donations
1999 2000
7
Graph 3 : Face Total : 2 dice
c) Shop Sales
5
5
Trucks
b) Regular Donors
Table 2 : Frequency for Total of Two dice
Cars
Annual
Appeal
Government
Shop Sales
Street
donations
Door Knock
Open Day
Regular
Donors
Corporate
Method
1 Give amount raised by: a) Door Knock
Eggs (× 1 000 000)
5
6
1
b) 2000
4
5
2
Frequency
Amount ($ × 10 000)
9
Bar Graphs
Bar graphs display data in the same way as column graphs except the graph is on its side.
These graphs all have:
•
A title which describes the purpose of the graph
•
Axes headings, these describe the scale on each axis
•
Scales, these let you to read or construct the bars with reasonably accuracy
Column 1 asks you to extract information from constructed graphs. Both graphs have columns
that are either exactly on the number or half way between two numbers on the horizontal scale.
The first graph deals with Blue Whale population, Questions 1 and 2 ask you to read figures
off the graph, question 3 asks you to write the years that correspond to the change in
population.
The second graph refers to the way a student may spend their day. This graph uses ½ hours so
write either 0.5 h or ½ h.
The second column asks you to create bar graphs. Write the title, axes headings and complete
the scale which has been started for you. Then construct the bars.
Bar Graphs
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Answer these questions
on the use of bar graphs
Now put this data on
the bar graph below
7
Graph 1 : Blue Whale Population Estimation
Population (× 1 000)
1
Year
2
3
5
4
7
6
8
Table 1 : Least Painful Home Duty
Student Approval
Home Duty
9 10
Mowing Grass
15%
1976
Washing Dishes
10%
1981
Tidy Bedroom
5%
1986
Washing Car
25%
1991
Meal Preparation
45%
1996
Graph 3 :
2001
1 Give the estimated number of blue whales in:
a) 1976
b) 1986
10
20
c) 1996
2 Give the increase in whale numbers between
a) 1976 - 1991
b) 1981 - 2001
3 Between which years is there a population increase of:
a) 500
b) 1 500
c) 3 000
d) 4 000
8 Table 2 : Timing Sunscreen Protection
Graph 2 : Analysis of a School Day
Hours Allocated
1
2
3
4
5
6
7
8
Meals
Activity
Travel
Sleep
School
Entertainment
Home duties
4 Give the number of hours spent: a) Sleeping
c) At school
d) Travelling
5 Give the difference in time spent on these activities:
a) Entertainment and.. Homework
Eating
b) Time at School and.. Travel
Sleeping
c) Home duties and.. School time
Travel
6 How many hours could the above student be at home
each weekday
Effective Time (min)
A
105
B
130
C
125
D
110
E
180
F
135
G
115
Graph 4 :
Homework
b) Eating
Sunscreen
100 110 120 130
Line Graphs
Line graphs show data as a continuous line, this allows for estimations to be made between
data. For example if population data is assembled every 5 years by using a line graph you can
estimate the population in between those years. You should realise that there are limitations to
this, as the population won’t change in a straight line, it could rapidly rise and fall between
actual readings. Another common use of line graphs is to relate two values against each other.
This relationship may be a straight line and so values at any point on the line will be correct.
Column 1 starts with a town’s population being graphed. Note the scale on the vertical axis is
× 10 000, so 1 = 10 000, 1.4 = 14 000 and so on. You would use a ruler to accurately read off
the graph. The second graph involves converting currency, between NZ$ and A$ (or NZD and
AUD).
Column 2 questions 5, 6 and 7 refer to the column 1 graph, only you are asked to find one
value from the graph and then using your number skills, find other conversions without using
the graph. Remember when you multiply by 10 you move the decimal point one place to the
right, 100 = 2 places and so on. For division move the decimal point to the left.
Graph 3 compares US$ with both Australian and New Zealand currencies. To convert US$ to
A$ use a ruler horizontally (flat) then where the ruler intersects the black line, read off the
amount on the bottom scale. To convert to from US$ to NZ$ use the same method but use the
red line. Note that you could use this graph to convert between A$ and NZ$, you would go up
to the red or black line, then across to other line, then down.
Line Graphs
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Study the line graphs below
then answer the questions
Convert the first figure and use it,
not the graph, to find the others.
5 a) A$38 = NZ$
b) A$3.80 = NZ$
c) A$380 = NZ$
d) A$3 800 = NZ$
5
6 a) A$50 = NZ$
b) A$0.50 = NZ$
4
c) A$100 = NZ$
d) A$5 000 = NZ$
3
7 a) NZ$100 = A$
b) NZ$200 = A$
2
c) NZ$1 000 = A$
d) NZ$1 001 = A$
6
Use this graph to convert
between US$, A$ and NZ$
2000
1998
1996
1994
1992
1990
1988
1986
1984
1
1982
Population (× 10 000)
Graph 1 : Population of Mathsville 1982-2000
Graph 3 : Converting US$ to A$ and NZ$
Year
100
b) 1992
c) 1996
U.S. Dollars ($US)
1 Give the population in: a) 1988
d) 2000
2 Estimate the population in a) 1985
b) 1991
c) 1995
d) 1999
3 Give the year that may have had a population of:
a) 8 000
b) 32 000
c) 13 000
90
AUD
80
NZD
70
60
50
40
30
20
10
Graph 2 : NZ$ and A$ Conversion
Australian Dollars (A$)
100
10 20 30 40 50 60 70 80 90 100
90
A$ and NZ$
80
8
70
60
US$ 20 14
50
40
A$
30
NZ$
80
32
50
60
70
9 a) US$10 = NZ$
b) US$100 = NZ$
c) US$1 000 = NZ$
d) US$1 = NZ$
Complete the table to the nearest $
10 a) US$50 = A$
b) US$100 = A$
Table 1 : NZ$ and A$ Conversions
c) US$1 000 = A$
d) US$5 000 = A$
New Zealand Dollars (NZ$)
NZ$ 40 70
30
80
63
56 14
98
22 42
11 a) A$40 = US$
39 c) A$0.40 = US$
18
60
Use the graph for the first conversion
then your number skills for the rest.
10 20 30 40 50 60 70 80 90 100
A$
40
20
10
4
Table 2 : US$, A$ & NZ$ Conversions
b) A$400 = US$
d) A$400.40 = US$
80
Composite Graphs
Composite graphs stack different pieces of information together to form the total column (or
bar). This method of displaying data allows access to the total as well as the basic parts to
form the total.
Column 1 starts with a composite column graph showing a family’s phone call history for the
past 6 months. The calls are categorised as local calls, national calls, international calls and
calls to mobile phones. All columns are rounded to the nearest $10 so that they are easier to
read. Note that each part starts where the other stops, so to obtain a value for a specific part
read off the amount at its base and subtract that from the amount at its top.
Another way of showing composite graphs is with just a single bar (or column) which is
divided up into parts. These are usually to scale so that a ruler can be used to get information.
Time lines are commonly in this format. Our graph is a time line about the construction of a
hamburger in a fast food outlet. Use a ruler to measure each part and the total of the bar, note
that 1 cm = 4 s (seconds). Q 5 shows employees being faster than standard time in certain
aspects of burger building, replace the standard time with the new time and then give the
reduced total time for the complete build. Q 6 refers to an employee who prefers a better
quality of finish and is slower in certain phases of the build.
Column 2 asks you to construct a composite column graph comparing 5 different dentists in
the procedure of putting a filling in a tooth. Position columns on top of the first part (the
needle injection) which is in place for each column and build them up.
The same method is use for the composite bar graph, note that 1 cm = 10 animals (or 1 mm =
1 animal). You can colour the key the same colour as the blocks and you can write the names
on as well.
Composite Graphs
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Construct a composite column
graph for these dentists
Study the composite graphs then
answer the questions
Graph 1 : Forsythe Family Monthly Phone Bill
200
Action
Key:
180
Local Calls
160
National Calls
140
Dentist A Dentist B Dentist C Dentist D Dentist E
Injecting
6
4
4
7
2
Drilling
13
8
12
15
14
120
International Calls
Filling
5
7
5
4
5
100
Mobile Calls
Grinding
9
6
11
10
12
80
All times above are in min
60
Graph 3 : Time Comparison for Filling a Tooth
40
7
Round all answers
to the nearest $10
Month
1 Give the total amount due in:
a) Jan
b) Feb
c) May
2 Which month had the lowest number of calls to:
a) Mobile phones
40
Key:
35
June
May
April
Mar
Feb
Jan
20
Time (min)
Amount due ($)
Table 1 : Time Comparison for Filling a Tooth
30
Injecting
25
Drilling
20
Filling
15
Grinding
10
b) Local Numbers
5
3 Give the amount spent on international calls in:
a) Jan
b) Feb
c) March
d) April
A
Fire Sauce
Cannon
Position
Pickles
Slap on
Patty
a) ‘Apply’ lettuce
D
E
Construct a composite bar graph
for the details in the table below
Top bun and
wrap
1 cm = 4 seconds
4 Give the time taken to:
C
Dentist
Graph 2 : Standard Times for Building a Burger
Throw
lettuce
B
b) Build burger
5 If some actions can be done faster, calculate total time
a) Jackie can sauce burgers in 3 s, new total time:
Table 2 : Treated Animals for Day
Animal
Dogs
Cats
Number
16
25
Birds Rabbits Other
9
7
23
Graph 4 : Treated Animals for Day
8
b) Oscar can place pickles in 2 s, new total time:
c) Jason cannons the sauce in 5 s: new total time:
d) Marie slaps on a patty in 4 s, new total time:
6 Ryan treats burgers as art, find his total time when he
1 cm = 10 animals
Key:
takes 16 s to position the pickles and 12 s to apply sauce,
Dogs
Birds
all other times are unchanged
Cats
Rabbits
Other
Composite Graphs
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Construct a composite column
graph for these dentists
Study the composite graphs then
answer the questions
Graph 1 : Forsythe Family Monthly Phone Bill
200
Action
Key:
180
Local Calls
160
National Calls
140
Dentist A Dentist B Dentist C Dentist D Dentist E
Injecting
6
4
4
7
2
Drilling
13
8
12
15
14
120
International Calls
Filling
5
7
5
4
5
100
Mobile Calls
Grinding
9
6
11
10
12
80
All times above are in min
60
Graph 3 : Time Comparison for Filling a Tooth
40
7
Round all answers
to the nearest $10
Month
1 Give the total amount due in:
a) Jan
b) Feb
c) May
2 Which month had the lowest number of calls to:
a) Mobile phones
40
Key:
35
June
May
April
Mar
Feb
Jan
20
Time (min)
Amount due ($)
Table 1 : Time Comparison for Filling a Tooth
30
Injecting
25
Drilling
20
Filling
15
Grinding
10
b) Local Numbers
5
3 Give the amount spent on international calls in:
a) Jan
b) Feb
c) March
d) April
A
Fire Sauce
Cannon
Position
Pickles
Slap on
Patty
a) ‘Apply’ lettuce
D
E
Construct a composite bar graph
for the details in the table below
Top bun and
wrap
1 cm = 4 seconds
4 Give the time taken to:
C
Dentist
Graph 2 : Standard Times for Building a Burger
Throw
lettuce
B
b) Build burger
5 If some actions can be done faster, calculate total time
a) Jackie can sauce burgers in 3 s, new total time:
Table 2 : Treated Animals for Day
Animal
Dogs
Cats
Number
16
25
Birds Rabbits Other
9
7
23
Graph 4 : Treated Animals for Day
8
b) Oscar can place pickles in 2 s, new total time:
c) Jason cannons the sauce in 5 s: new total time:
d) Marie slaps on a patty in 4 s, new total time:
6 Ryan treats burgers as art, find his total time when he
1 cm = 10 animals
Key:
takes 16 s to position the pickles and 12 s to apply sauce,
Dogs
Birds
all other times are unchanged
Cats
Rabbits
Other
Sector Graphs
You may know of them as Pie Graphs, Circle Graphs or Sector Graphs, they are all the same,
just different names. Sector graphs display data as portions of a circle, so they are most
commonly used with percentages or portions of a whole. At this stage we will deal only with
percentages, all of which will have values that are multiples of 5%.
As a circle covers 360º in a revolution, with percentages that means 100% is 360º. So 10% is
36º and so 5% is 18º. Column 1 asks you to give the sector angle for all the multiples of 5%.
To do this add 18º to the previous angle and that’s your answer. You should be able to check
you addition at 50% because it will equal ½ of 360º, which is 180º.
Graph 1 is a sector graph describing travel to school, you are asked to find the percentages of
each. You need a protractor to do this, the percentages are multiples of 5 and so your angle will
match a percentage given in Q.1. Note that the lines inside the circle may have to be extended
to allow you to read them properly, (depending on your protractor type). The second
worksheet is supplied for students without protractors, markings are on the graph at every 5º.
Graphs 3 and 4 are the reverse, creating sectors this time. Before you start you may like to
change all the percentages to angles, remember use the values in Q.1. Once you have finished
each sector remember to label it.
Graphs 5 and 6 are sector graphs that you can colour. They don’t need labels as the square in
front of the category is coloured the same colour as the sector. It is usually referred to as a
‘key’. Look at the example below, it shows a completed sector graph.
Example : Household Expenditure
Car 15%
Colour the sectors, each sector is 5%, all
percentages given are in multiples of 5%.
Colour the square, which is the key, the same
colour as its respective sector.
Repayments 35%
Utilities 10%
Food 25%
Other 15%
Sector Graphs
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Convert these percentages of a circle to
Angles by adding 18º for each 5%
Place this data on the sector graphs below
Graph 3 : Batman’s Vehicle Usage (%)
1 5% = 18°
40% =
75% =
10% = 36°
45% =
80% =
Place these on the circle:
15% = 54°
50% =
85% =
Car 60%
20% =
55% =
90% =
25% =
60% =
95% =
30% =
65% =
100% =
35% =
70% =
Now try
these
Graph 1 : Type of Transport to School (%)
2 Give the % who travel by:
Train
Bike 20%
Plane 5%
7
Graph 4 : Karl’s Music Preference (%)
Place these on the circle:
Alternative 25%
Rap 15%
b) Car
Dance 20%
c) Bus
Walk/Ride
Copter 15%
Mainstream 35%
a) Train
Bus
Car
6
Classical 5%
d) Foot or Bike
Each sector is worth 5%. Colour the
sectors the same colour as the key squares
3 Give the % of students that:
a) Don’t catch a bus
b) Use a car or bus
8 Graph 5 : How Tina spends her Spare Time
c) Use public transport
Graph 2 : Type of Movie Last Seen by Students
Read 15%
4 Give the % who saw a:
a) Comedy
Comedy
Computer 5%
Thriller
Sport 20%
b) Thriller
c) Sci-fi
Watch TV 20%
Sci-fi
Action
d) Drama
With Friends 40%
Drama
e) an Action film
f ) an Animated film
Animated
9 Graph 6 : What Scares Students the Most
Heights 10%
Spiders 15%
5 Give the % of students that:
a) Saw either a Science fiction or Action film
b) Didn’t see a Thriller
c) Last saw a comedy or drama:
d) Didn’t see an animated or Sci-fi movie
Other 20%
Rats 5%
Sharks 15%
Ventriloquist
Dummies 35%
Sector Graphs
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Convert these percentages of a circle to
Angles by adding 18º for each 5%
Place this data on the sector graphs below
Graph 3 : Batman’s Vehicle Usage (%)
1 5% = 18°
40% =
75% =
10% = 36°
45% =
80% =
Place these on the circle:
15% = 54°
50% =
85% =
Car 60%
20% =
55% =
90% =
25% =
60% =
95% =
30% =
65% =
100% =
35% =
70% =
Now try
these
Graph 1 : Type of Transport to School (%)
2 Give the % who travel by:
Train
Bike 20%
Plane 5%
7
Graph 4 : Karl’s Music Preference (%)
Place these on the circle:
Alternative 25%
Rap 15%
b) Car
Dance 20%
c) Bus
Walk/Ride
Copter 15%
Mainstream 35%
a) Train
Bus
Car
6
Classical 5%
d) Foot or Bike
Each sector is worth 5%. Colour the
sectors the same colour as the key squares
3 Give the % of students that:
a) Don’t catch a bus
b) Use a car or bus
8 Graph 5 : How Tina spends her Spare Time
c) Use public transport
Graph 2 : Type of Movie Last Seen by Students
Read 15%
4 Give the % who saw a:
a) Comedy
Comedy
Computer 5%
Thriller
Sport 20%
b) Thriller
c) Sci-fi
Watch TV 20%
Sci-fi
Action
d) Drama
With Friends 40%
Drama
e) an Action film
f ) an Animated film
Animated
9 Graph 6 : What Scares Students the Most
Heights 10%
Spiders 15%
5 Give the % of students that:
a) Saw either a Science fiction or Action film
b) Didn’t see a Thriller
c) Last saw a comedy or drama:
d) Didn’t see an animated or Sci-fi movie
Other 20%
Rats 5%
Sharks 15%
Ventriloquist
Dummies 35%
7
FREEFALL
MATHEMATICS
NUMBER
PLANE
Grid Reference - The Hidden Friend
A grid reference system is used to pinpoint locations on a map or diagram. This sheet uses grid
referencing to plot diagrams of 2 hidden friends. Grid referencing gives a letter then a number,
the letter being along the horizontal axis (the top and bottom) and the number being on the
vertical axis, which are the two sides.
Some maps may use a zero, these maps start at 1. To plot the points move across to the letter
required then up the column to the number, then colour the square. Then plot the next grid
position, after the points are plotted your friends should be revealed.
The co-ordinates are all in the same colour except where you are told to change colour, but if
you want to make a multi-coloured one go ahead! You should strike out each grid reference as
you plot them incase you need to remember where you are up to.
Grid Reference - The Hidden Friend
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Co-ordinates:
Colour the following squares and
you will find a smiling friend
A B C D E F G H I
J K L M N O P Q R S
D3
I2
C7
N5
B6
F2
D8
C4
L6
G2
E9
E3
J8
K11
L4
F12
M4
Q7
F4
F9
H12
I13
H2
J3
O8
P9
B5
13
12
11
10
9
13
12
11
10
9
8
7
6
8
7
6
5
4
3
5
4
3
M6
H9
D12
E13
P6
Q8
2
1
2
1
J12
K3
S8
G11
O5
K7
I9
R9
A B C D E F G H I
J K L M N O P Q R S
Colour the following squares in blue
(or black and red when indicated) and
you will find an enthusiastic friend
A B C D E F G H I
E6
Different
colour
Co-ordinates: G11 A4 D1 F7 K12 L3
P6 N8 Q8 A9 I12 S8
C2 O2 E11 S9
H8 L12
M6 M11 N7 R10 O5
J K L M N O P Q R S
13 P12 R3 Q2 B5 E2
12 F8 N10 A7 M4 O8
11 Q1 J13 O11 M5 G4
10 C7 D8 C8 G5 E5
9
H12 B10 E3 F10 A8
8
G6 D12 E4 S4 P1
7
6 P8 S6 H3 Q11 S7
5 C1 B3 O3 C11 Q7
13
12
11
10
9
8
7
6
5
4
3
4
3
2
1
2
1
A B C D E F G H I
J K L M N O P Q R S
L9
E8 A6 L8 D6 O4
R5
Change Colour:
K2 J1 J2 I2
Change Colour:
J5
I5
J4
H9 K5
Using a Street Map
A street map uses grid co-ordinates to specify the location of streets and points of interest. The
reference system generally uses letters across its horizontal axis (top and bottom) and numbers
on its vertical axis (sides). To give a co-ordinate you move across first, then up, so the letter
first then the number.
For example D7 would be the green area. Note that some areas could be given more than one
co-ordinate, due to its size it could cross into another reference area. If you are attempting this
sheet on paper use a ruler or just the eye.
Column 1 asks for the co-ordinates for the black symbols on the map. Look at the map key
(sometimes called a ‘legend’) so that you know what you are looking for. Then using the grid
read off the locations.
Column 2 asks you to place buildings and points of interest on the map. So show that artistic
talent!
Column 3 asks you to place post boxes, traffic lights and phone boxes in given locations.
You can print either a map with a grid over it or a map with no grid.
Using a Street Map - No Grid
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
C
D
et
t
sS
R
Be
l-A
ir
ar
k 's
t
nt S
Mou
Key:
Police Station
8
i Tourist Information
E
Give the grid references
for all of the following
1 Petrol Stations
F
G
H
I
St
er
Si
lv
Co
n
Luck Av
3
St
Jo
h
St
J
K
L
M
N
5
4
O
Draw these buildings or
make symbols for them in
the co-ordinates below.
P
2
it
n
Petrol Station
n' s
1
rc
u
Ca
ve
rs
o
D
6
Ci
t
ffe
C
7
Camping Area
co
rd
tit
ut
io
n
ns
Co
v
11
9
St
Rd
eS
am
eA
12
10
Bl
v
School
Rd
13
de
Je
B
S
Hospital
Bl
El
li
St
St
vd
e
Li
nk
Rd
Circus Rd
e
riv
D
ic
M
St
Fl
m
as
St
Ea
ste
rn
Rd
dr
ew
An
St
Ru
se
Th
o
Sc
en
Q
Rd
tS
t
St
Sm
all
Rd
St
re
Pl
A
P
Ca
rlt
on
St
Cl
ar
k
Fo
re
s
e
W
ate
r
e
Be
ll
O
t
d
St
ov
Gr
1
N
S
ee
Tr
M
ai
n
Pd
4
3
M
g
Fi
on
m
ich
Bi
rd
Av
et
ol
Vi
le
att
W
5
L
St Clair Rd
First Avenue
6
K
v
A
R
7
2
J
ria
8
I
nd
xa
Pa
cif
ic
i
St
H
le
A
9
Cl
i ff
G
Park Pde
Bluff Av
10
F
Cl
12
11
E
Olive Blvde
13
B
Du
nc
an
A
Q
R
S
Put traffic lights, post
boxes and phone booths
in these locations
10 Veterinarian
K9 20 Traffic Lights Co-ordinates
2 Schools
11 St Mary's Cathedral
K5
P3
F8
H3
3 Camping Grounds
12 Palace Hotel
N11
N7
K10
F2
4 Tourist Information
13 Imam Ali Mosque
I1
14 Sunshine Circus
C1
5 Hospitals
6 Intersection of First Av
and Bird St
7 Intersection of Carlton
Rd and Alexandria Av
15 Council Car Parks
M1
A5
Q9
21 Post Boxes Co-ordinates
I9
R10
S3
K2
B3
M4
16 Fuzzy's Café
P12 22 Phone Booths Co-ordinates
17 Community Centre
O3
R1
J12
A12
8 Intersection of Scenic
Drive & Wattle Grove
18 Council Parks
H11 D7
S9
L7
I2
9 Bell Place
19 Aquatic Centre
P10
E4
H8
O6
Using a Street Map - Grid
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
C
D
et
t
sS
R
Be
l-A
ir
ar
k 's
t
nt S
Mou
Key:
Police Station
8
i Tourist Information
E
Give the grid references
for all of the following
1 Petrol Stations
F
G
H
I
St
er
Si
lv
Co
n
Luck Av
3
St
Jo
h
St
J
K
L
M
N
5
4
O
Draw these buildings or
make symbols for them in
the co-ordinates below.
P
2
it
n
Petrol Station
n' s
1
rc
u
Ca
ve
rs
o
D
6
Ci
t
ffe
C
7
Camping Area
co
rd
st i
tu
tio
n
Co
n
v
11
9
St
Rd
eS
am
eA
12
10
Bl
v
School
Rd
13
de
Je
B
S
Hospital
Bl
El
li
St
St
vd
e
Li
nk
Rd
Circus Rd
e
D
riv
ic
M
St
Fl
m
as
St
Ea
ste
rn
Rd
dr
ew
An
St
Ru
se
Th
o
Sc
en
Q
Rd
tS
t
St
Sm
all
Rd
St
re
Pl
A
P
Ca
rlt
on
St
Cl
ar
k
Fo
re
s
e
W
ate
r
e
Be
ll
O
t
d
St
ov
Gr
1
N
S
ee
Tr
M
ai
n
Pd
4
3
M
g
Fi
on
m
ich
Bi
rd
Av
et
ol
Vi
le
att
W
5
L
St Clair Rd
First Avenue
6
K
v
A
R
7
2
J
ria
8
I
nd
xa
Pa
cif
ic
i
St
H
le
A
9
Cl
i ff
G
Park Pde
Bluff Av
10
F
Cl
12
11
E
Olive Blvde
13
B
Du
nc
an
A
Q
R
S
Put traffic lights, post
boxes and phone booths
in these locations
10 Veterinarian
K9 20 Traffic Lights Co-ordinates
2 Schools
11 St Mary's Cathedral
K5
P3
F8
H3
3 Camping Grounds
12 Palace Hotel
N11
N7
K10
F2
4 Tourist Information
13 Imam Ali Mosque
I1
14 Sunshine Circus
C1
5 Hospitals
6 Intersection of First Av
and Bird St
7 Intersection of Carlton
Rd and Alexandria Av
15 Council Car Parks
M1
A5
Q9
21 Post Boxes Co-ordinates
I9
R10
S3
K2
B3
M4
16 Fuzzy's Café
P12 22 Phone Booths Co-ordinates
17 Community Centre
O3
R1
J12
A12
8 Intersection of Scenic
Drive & Wattle Grove
18 Council Parks
H11 D7
S9
L7
I2
9 Bell Place
19 Aquatic Centre
P10
E4
H8
O6
Plotting Points - Positive Quadrant
This sheet involves the plotting of Cartesian co-ordinates. When plotting points on a number
plane you are given 2 numbers, referred to as an ordered pair. The first number tells you how
much you move across and the second number tells you how much you move up or down.
This sheet only moves across to the right and up, negative numbers (not on this sheet) move to
the left and down.
Move across first along the x axis.
Then move up.
Eg. To plot the co-ordinate (2, 3)
3
2
Note that the plane consists of an x-axis (horizontal or across) and a y-axis which is vertical
(up and down). The plural of axis is axes (pronounced ak-seas). The co-ordinate (0, 0) is
commonly referred to as the 'origin', the place where measurement starts from. Teachers often
ask for the co-ordinates of the origin.
The first column asks you to locate the points and write in their co-ordinates. This is done by
opening a bracket writing the x co-ordinate, writing a comma, then the y co-ordinate and close
bracket. You should always place the co-ordinates in brackets.
The second column asks you to plot points. Once the group of points are plotted draw a line
between points (in alphabetical order) to form a shape. Then write in the name of the shape
plotted in the space provided.
Plotting Points - Positive Quadrant
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Give the co-ordinates for the
following points, place them in
brackets separated by a comma
y
12
9
B
9
8
H
7
N
6
S
C
G
R
4
I
O
4
6
5
A
5
3
T
3
2
D
2
0
10
Q
E
10
1
F
P
11
7
13
11
12
8
y
J
13
Place the groups of points given below on
the grid, join the points and name the shape
they create, make sure you label the points.
1
L
K
M
x
x
1 2 3 4 5 6 7 8 9 10 11 12 13
1 Point A
11 Point K
2 Point B
12 Point L
3 Point C
13 Point M
0
1 2 3 4 5 6 7 8 9 10 11 12 13
21 Point A
(10, 4)
Point B
(11, 2)
Point C
(12, 4)
Point D
(11, 5)
Shape formed:
22 Point E
(3, 5)
Point F
(0, 8)
Point G
(5, 13)
Point H
(8, 10)
Point K
(9, 6)
Shape formed:
4 Point D
14 Point N
5 Point E
15 Point O
6 Point F
16 Point P
7 Point G
17 Point Q
8 Point H
18 Point R
9 Point I
19 Point S
23 Point J
(5, 4)
Point L
(7, 8)
Shape formed:
10 Point J
20 Point T
Two word description
24 Point M
(2, 3)
Point N
(8, 3)
Point O
(6, 0)
Point P
(0, 0)
Point S
(13, 13)
Shape formed:
25 Point R
(10, 9)
Point T
(13, 9)
Shape formed:
Two or three word description
Drawing Animals - Positive Quadrant
This sheet uses your skills in plotting points from the previous sheet. The series of points
makes a pair of animals when lines are joined between the points. The way you attempt this
exercise is, for the first animal:
•
Plot one point at a time and then draw a line between it and the previous point. Don’t try
to plot all the points and then draw the lines afterwards as the drawing is too complicated.
•
When you reach stop, restart at the new position and start again, DON'T JOIN THE
POINTS BETWEEN WHERE YOU STOPPED AND WHERE YOU RESTARTED.
For the next animal follow the same process except use 2 different colours.
Move across first along the x axis.
Then move up.
Eg. To plot the co-ordinate (2, 3)
2
3
Drawing Animals - Positive Quadrant
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
y
Join the points on the grid with
the following co-ordinates and
you will find a circus friend.
Co-ordinates:
START: (2, 12) , (3, 12) , (4, 11) , (5, 12) ,
(7, 12) , (8, 11) , (9, 12) , (10, 12) ,
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
13
13
12
12
11
10
11
10
9
8
7
6
5
4
3
2
1
9
8
7
6
5
4
3
2
1
(11, 11) , (11, 6) , (10, 3) ,
(9, 3) ,
(8, 7) ,
(8, 5) ,
(7, 7) ,
(7, 3) ,
(6, 2) ,
(4, 2) ,
(4, 3) ,
(5, 3) ,
(6, 4) ,
(5, 7) ,
(4, 5) ,
(4, 7) ,
(3, 3) ,
(2, 3) ,
(1, 6) , (1, 11) ,
(2, 12)
STOP!
START:
(11, 10) , (12, 11) , (16, 12) , (18, 10) ,
(19, 7) , (18, 9) , (18, 3) , (15, 3) ,
(16, 4) , (16, 6) , (14, 5) , (11, 5) ,
(11, 4) , (13, 4) , (13, 1) , (12, 2) ,
(9, 2) ,
(8, 3) , (7, 6) , STOP!
x
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
0
y
Join the points and this time you
will find a friend with which you
will have to be very patient.
Co-ordinates:
START: (1, 11) , (2, 10) ,
(2, 9) ,
(1, 8) ,
(1, 7) ,
(2, 6) ,
(2, 5) ,
(3, 3) ,
(5, 1) ,
(7, 0) , (13, 0) , (16, 3) ,
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
13
12
13
12
(17, 5) , (15, 3) , (12, 2) ,
(8, 2) ,
(6, 3) ,
(4, 5) ,
(4, 7) ,
11
10
11
10
(4, 8) ,
(3, 9) , (3, 10) , (4, 11) ,
9
8
7
6
5
4
3
2
1
9
8
7
6
5
4
3
2
1
0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
(3, 6) ,
STOP!
START:
(11, 6) , (12, 6) , (11, 7) , (10, 6) ,
(11, 5) , (12, 5) , (14, 6) , (14, 7) ,
(12, 9) , (10, 9) ,
(8, 7) ,
(8, 5) ,
(10, 3) , (12, 3) , (14, 4) , (15, 6) ,
(15, 7) , (14, 9) , (13, 10) , (10, 10) ,
(8, 9) ,
(7, 7) ,
(8, 3) , (11, 2) ,
(12, 2) , (15, 3) , (17, 6) , (17, 7) ,
x (15, 11) , (12, 12) , (10, 12) , (7, 11) ,
(5, 9) ,
(4, 6) ,
(4, 5) , STOP!
Plotting Points - 4 Quadrants
This sheet involves the plotting of Cartesian co-ordinates. When plotting points on a number
plane you are given two numbers. The first number tells you how much you move across and
the second number tells you how much you move up or down. As the numbers are in order,
across first, up/down second, these numbers can be referred to as ‘ordered pairs’, as their order
is important. Positive numbers move across to the right if the first number, negative numbers
move to the left if the first number. Positive numbers move up if the second number, negative
numbers move down if the second number.
The first column asks you to locate the points and write in their co-ordinates. This is done by
opening a bracket writing the x co-ordinate, a comma, then the y co-ordinate and close bracket.
Always place co-ordinates in brackets.
The second column asks you to plot points. Draw filled in circles at the given location and
label them with the letter supplied in each question.
Once you plot the point record down which quadrant it is in. Quadrants start with the 1st
Quadrant (top right) and then moving anti-clockwise, are called II (2nd), III (3rd) and IV (4th). If
the point is on the axis then record it by filling in the oval around the word ‘Axis’.
(2, 3)
Move across first along the x axis.
Negative to the left, Positive to the right.
3
Then move up or down in the y direction.
Negative is down, Positive is up.
2
(2, -3)
(-2, 3)
2
(-2, -3)
2
3
3
2
3
Plotting Points - 4 Quadrants
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Give the co-ordinates for the points
on the number plane below.
P
6
y
W 5
E
Z
Plot the following points on the number plane
below, label each point and give the quadrant
that the point lies in.
T
II
F
6
H
N
4
V
3
Y
2
J
1
U -1
O
-2
A
2
3
4
5 6
T
1
x
Q
1
2
S
K
-6 -5 -4 -3 -2 -1 0
-6 -5 -4 -3 -2 -1 0
R
-4
M
x
1
2
3
4
5 6
-1
I
-2
X
G -3
S
I
5
4
3 B
y
-3
D
L
-4
-5 C
-5
-6
1 Point A
14 Point N
2 Point B
15 Point O
3 Point C
16 Point P
4 Point D
17 Point Q
5 Point E
18 Point R
6 Point F
19 Point S
7 Point G
20 Point T
8 Point H
21 Point U
9 Point I
22 Point V
-6
III
27 Point A
(-5, -2)
IV
36 Point J
(1, -4)
Quadrant I II III IV Axis
Quadrant I II III IV Axis
28 Point B
37 Point K
(2, 4)
(-3, -4)
Quadrant I II III IV Axis
Quadrant I II III IV Axis
29 Point C
38 Point L
(4, 0)
(5, -3)
Quadrant I II III IV Axis
Quadrant I II III IV Axis
30 Point D
39 Point M
(3, -3)
(-4, 2)
Quadrant I II III IV Axis
Quadrant I II III IV Axis
31 Point E
40 Point N
(-3, 6)
(3, -5)
Quadrant I II III IV Axis
Quadrant I II III IV Axis
32 Point F
41 Point O
(0, 0)
(0, -3)
Quadrant I II III IV Axis
Quadrant I II III IV Axis
33 Point G
42 Point P
(6, -5)
(-6, -5)
10 Point J
23 Point W
Quadrant I II III IV Axis
Quadrant I II III IV Axis
11 Point K
24 Point X
34 Point H
43 Point R
12 Point L
25 Point Y
13 Point M
26 Point Z
(-2, 4)
(-2, 1)
Quadrant I II III IV Axis
Quadrant I II III IV Axis
35 Point I
44 Point S
(6, 2)
Quadrant I II III IV Axis
(2, -2)
Quadrant I II III IV Axis
Drawing Animals - All Quadrants
This sheet uses your skills in plotting points from the previous sheet. The series of points
makes a pair of animals when lines are joined between the points. The way you attempt this
exercise is, for the first animal:
•
Plot one point at a time and then draw a line between it and the previous point. Don’t try
to plot all the points and then draw the lines afterwards as the drawing is too complicated.
•
When you reach stop, restart at the new position and start again, DON'T JOIN THE
POINTS BETWEEN WHERE YOU STOPPED AND WHERE YOU RESTARTED.
(2, 3)
Move across first along the x axis.
Negative to the left, Positive to the right.
3
Then move up or down in the y direction.
Negative is down, Positive is up.
2
(2, -3)
(-2, 3)
2
(-2, -3)
2
3
3
2
3
Drawing Animals - 4 Quadrants
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Co-ordinates:
Plot the points and join them
with lines to meet one of
your more colourful friends
(7, 4) ,
START: (-1, 3) , (-1, 4) ,
(1, 6) ,
(2, 6) ,
(6, 5) ,
(7, 2) , (6, -1) , (3, -2) , (5,-4) , (5, -5) , (3, -6) , (1, -6) ,
(-2, -4) , (-4, -6) , (-5, -6) , (-6, -5) , (-6, -4) , (-5, -2) , (-6, -1) , (-8, 2) ,
y
(-8, 4) , (-7, 6) , (-6, 6) , (-4, 4) , (-4, 3) , STOP!
6
START: (-2, 1) ,
5
(1, 4) ,
(4, 4) ,
(5, 2) , (3,-1) , (0, -2) , (3, -3) ,
4
(3, -5) , (1, -5) , (-1, -4) , (-2, -3) ,
3
(-4, -5) , (-5, -5) , (-5, -3) , (-3, -2) ,
2
(-5, -1) , (-7, 2) , (-6, 4) , (-5, 4) ,
1
x
-9 -8 -7 -6 -5 -4 -3 -2 -1 0
-1
1
2 3
4
5
6
7
(-3,1) , STOP!
START: (-4, 3) , (-3, 4) , (-2, 4) ,
8 9
(-1, 3) , (-2, 2) , (-2, -4) , (-3, 2) ,
(-4, 3) , STOP!
-2
-3
START: (-3, 4) , (-4, 6) , STOP!
-4
-5
START: (-2, 4) , (-1, 6) , STOP!
-6
Co-ordinates:
Plot the points and join them
with lines to share your lunch
with your vegetarian friend.
START: (2, -1) , (3, -3) , (3, -4) , (2, -5) , (0, -6) ,
(-1, -6) , (-3, -5) , (-4, -4) , (-4, -3) , (-3, -1) , (-6, 2) , (-7, -4) , (-8, 4) ,
(-7, 6) , (-4, 5) , (-2, -1) , (0, -1) ,
y
(3, 4) , (5, 6) ,
(8, 6) ,
START:
(7, 5) ,
(6, 3) , (2, -1) ,
(4, 4) ,
(5, 6) , STOP!
6
STOP!
5
4
START:
(2, 0) ,
3
(5, 3) ,
(2, 0) , STOP!
2
-2
-3
-4
-5
-6
(6, 5) ,
START: (-3, 0) , (-5, 2) , (-6, 4) ,
1
-9 -8 -7 -6 -5 -4 -3 -2 -1 0
-1
(9, 4) ,
x
1
2 3
4
5
6
7
8 9
(-4, 3) , (-3, 0) , STOP!
START: (-2, -4) , (-6, -4) , STOP!
START: (-2, -4) , (-5, -2) , STOP!
START: (-2, -4) , (-5, -3) , STOP!
START: (1, -4) , (5, -4) , STOP!
START: (1, -4) , (5, -3) , STOP!
START: (1, -4) , (5, -5) , STOP!
Warship Rules
This game uses your grid reference skills to play a game. This game is called ‘Warship’ and it
involves marine military vessels.
The game:
•
The bottom grid is for placing one of each of the 5 craft supplied either horizontally or
vertically. For a shorter game don't use the 'zodiac'.
•
Your opponent does the same on their separate grid sheet and then you decide who has
first shot.
•
You pick a co-ordinate and call it out, letter first then number, (like… B3) if it hits you
opponent's craft they say 'Hit', if it misses they say 'Miss'. On the top grid you then place
a cross in the square if it hits or a dot in the square if its a miss. The dot ensures you don’t
call the same co-ordinate again.
•
Your opponent then has a shot, if their co-ordinates hit part of a craft then you call 'Hit'
and place a cross on the bottom grid, if they miss you don’t have to record their shot.
•
Note that your next try will be above/below or to the left/right of the hit if you hit your
opponent, until you sink the craft. Your opponent must call out 'Sunk' if the craft is
destroyed (a cross over each square that the craft is in). If you didn't hit a craft in your
previous try then guess another 'random' location, away from the point called earlier.
•
Shots are alternated until one player loses all his or her fleet. You may like to cross out
your sunken opponents craft as a reminder of the craft still remaining.
•
Note: Sometimes you score a hit and then get 4 more hits but your opponent doesn't call
'sunk'. If they have placed two craft in a line this is possible don't jump to the conclusion
that they have made a mistake.
•
If you lose a zodiac early you are unlucky and may lose the game because of it, so think
carefully about its placement. Also if the zodiac is hit while your opponent is hitting a
craft next to your zodiac then you must say that it is the zodiac that was sunk.
•
Another way to speed up the game is to give a player another turn if a hit is made
including after the craft is sunk.
Don't cheat by changing the location of your craft during play as your opponent will quickly
tire of playing you and it spoils the fun for both of you. You would also be surprised at how
long one particular craft can be unsuccessfully hunted.
7
6
5
4
3
2
1
7
6
5
4
3
2
1
9
8
7
6
5
4
3
2
1
9
8
7
6
5
4
3
2
1
A B C D E F G H I J
10
10
1
2
3
4
5
6
7
8
9
A B C D E F G H I J
A B C D E F G H I J
5 squares
4 squares
3 squares
2 squares
1 square
A B C D E F G H I J
Aircraft Carrier
Cruiser
Submarine
Sub Hunter
Zodiac
A B C D E F G H I J
GAME 2 : YOUR SHIPS
10
1
2
3
4
5
6
7
8
9
10
GAME 1 : YOUR SHIPS
5 squares
4 squares
3 squares
2 squares
1 square
8
8
Aircraft Carrier
Cruiser
Submarine
Sub Hunter
Zodiac
9
9
A B C D E F G H I J
10
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
A B C D E F G H I J
A B C D E F G H I J
10
GAME 2 : OPPONENT'S SHIPS
GAME 1 : OPPONENT'S SHIPS
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
5 squares
4 squares
3 squares
2 squares
1 square
A B C D E F G H I J
A B C D E F G H I J
GAME 3 : YOUR SHIPS
Aircraft Carrier
Cruiser
Submarine
Sub Hunter
Zodiac
A B C D E F G H I J
A B C D E F G H I J
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
GAME 3 : OPPONENT'S SHIPS
Warship - Printable Sheet
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Latitude and Longitude
Similar to the number plane, when you give locations on the globe you do so by using 2
co-ordinates. Latitude lines are the horizontal or flat 'lines' (remember lat is flat). Longitude
lines are the vertical lines (remember they aren't flat!).
Note that there are no negative numbers, instead direction is used, the equator is when the
latitude is 0°. Then moving up (northerly), latitudes increase and are denoted by an N for
north. Moving down (southwards) they are denoted by an S. Longitudes are measured from
the 0° 'prime meridian' which matches GMT, Greenwich Meridian Time. Then when you move
left (West) you denote the movement with a W, moving right (East) you use an E.
When you give these locations unlike a number plane you give the latitude first, you probably
remember the compass points by saying 'North South East West' well this applies to the coordinates also, give the North or South co-ordinate first then the East or West co-ordinate. You
also use a degree sign, separate the co-ordinates with a comma and put them in brackets.
If you look at the point N at the top left of the globe you would give its position as:
(50°N, 120°W).
While the sheet differs from column to column the use of co-ordinates is all that is being dealt
with on this sheet. The first column asks you to give the co-ordinates for the given point, open
a bracket and write in the latitude, then a comma, then the longitude and close the brackets.
Note that 0° latitude or longitude doesn't require a letter, just write it as 0°.
Column 2 is the reverse, given a co-ordinate find the point. Write in the letter.
Column 3 takes this one step further. A word (which will be a city) will be formed by using
the co-ordinates to find each letter in order, write the letter as you locate the points. If the city
is unrecognisable then you need to go back and check your points. Note that the letters are in
order they aren't jumbled.
The most common mistake made apart from giving the co-ordinates reversed is with the
longitude. Most people get the North or South right but some mix up their east with their west.
Latitude and Longitude
70
ºN
80º
N
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Q
N
60
ºN
ºN
50
H
D
40º
O
P
S
K
N
30ºN
I
20º N
E
10ºN
C
G
U
0º
A
10ºS
L
T
B
V
J
20º S
M
40º
20ºE
40ºE
60ºE
80ºE
100
ºE
120
ºE
140
ºE
16
0ºE
20ºW
0º
60ºW
40ºW
0º
18
ºW
160
ºW
140
ºW
120
W
100º
80ºW
30ºS
R
60
F
S
50
ºS
ºS
70
ºS
80º
S
Give the co-ordinates
of these points:
Give the letter at the
co-ordinates below:
1
A
12
(30°S, 120°W)
2
B
13
(40°N, 40°W)
3
C
14
(30°N, 60°E)
4
D
15
(20°S, 100°W)
Give the cities that
these co-ordinates
spell
23
(40°N, 40°W)
(10°S, 160°W)
(60°S, 20°E)
(30°N, 100°E)
(30°N, 60°E)
24
5
E
16
(0°, 120°E)
6
F
17
(60°S, 20°E)
7
G
18
(50N°, 120°W)
8
H
19
(20°S, 140°W)
9
I
20
(70°N, 0°)
10
J
21
(40°N, 140°E)
11
K
22
(20°S, 20°E)
Name City:
Name City:
(40°N, 80°W)
(20N°, 20°E)
(20°S, 140°W)
(50°N, 40W°)
(30°N, 100°E)
25
Name City:
(50°N, 40°W)
(40°N, 140°E)
(0°, 120°E)
(30°N, 60°E)
(20°S, 100°W)
(40°N, 140°E)
(50°N, 120°W)
7
FREEFALL
MATHEMATICS
PERCENTAGES
Fractions to Percentages
Often a result of a test or survey is expressed as a fraction. Fractions can be difficult to
compare and often these are converted to percentages, allowing easier comparison of the
result. A fraction is converted to a percentage by multiplying it by 100. This is more easily
achieved if the fraction already has a denominator of 100, so finding the equivalent fraction
with a denominator of 100 is the first step:
7
×4
25
×4
=
28
100
= 28 %
Why is it 28%?…. Because
28
100
× 100 = 28%
In the example above 7/25 is changed to 28/100 by looking at the denominators. Ask yourself
what you would multiply 25 by to get 100?… 4! If you multiply the bottom by 4 you have to
multiply the top by 4 …. 7 × 4 = 28. Then if you multiply 28/100 by 100, you are left with 28%.
Note that this method only works for factors of 100, denominators of 2, 4, 5, 10, 20, 25, 50. It
can also work for multiples of 100, this time by dividing, as long as the numerator also allows
the division.
Column 1 starts with writing the shaded fraction of a shape as a fraction with a denominator of
100 (don't simplify), and then as a percentage. Count the painted squares, this is the numerator
of the fraction. The same number is also the percentage.
Column 2 is an extension of this. A fraction is given and you are asked to make an equivalent
fraction with a denominator of 100. Look at the denominator (bottom number) and find the
number you need to multiply it by to get 100. You can write this number above/below the
equals sign if you like (see the example, top of column). Then multiply the numerator (top
number) by the same number. This will give you the percentage. Note questions 12 and 13
require division.
You may think that 100% is all that you can give. With things like 'effort', 'determination' or
'support', this is the case. With mathematics it is possible you can find 300% of a number, it's 3
times the number.
Column 3 involves converting mixed numbers to percentages. If 1 = 100/100, 2 = 200/100,
3 = 300/100 etc, this means that 1 = 100%, 2 = 200% and 3 = 300%. So with this column first
split the whole number from the fraction and write each separately as a fraction over 100, then
add them together and change to a percentage. An example is at the top of the column.
Fractions to Percentages
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Write the shaded part of
the (hundredths) grid as a
fraction 'over a hundred',
and then as a percentage.
Rewrite the fraction
changing the denominator
to 100. Then write as a %.
Example
×5
3
1
=
=
=
5
%
6
7
=
10
2
=
=
=
20
100
%
9
=
50
3
=
4
3
=
=
7
11
=
4
100
% 12
13
12
=
200
=
=
100
%
100
100
2
10
=
200
100
+
25
=
100
= 225 %
=
% 17 4 10 = 100 + 100 = 100
7
%
%
=
100
1
2
=
%
100
+
100
=
=
=
100
100
100
=
%
9
19 7
=
20
100
+
100
%
=
=
100
=
%
=
% 20 1 200 = 100 + 100 = 100
34
%
%
15
21 9
=
500
100
+
100
=
%
8
22 6
=
25
100
+
100
=
=
=
=
=
100
%
100
=
100
2
=
100
%
%
16 Ken forgets his lunch 1 day
in 5, write as a percentage
=
100
%
15 Ellen scored 17/20 in a test,
write this as a percentage.
5
100
%
=
=
100
=
=
100
100
%
=
100
225
=
14 David is 3/10 of the way
home, write as a percentage
3
1
4
=
350
500
4
= 15 %
18 5
17
10
Example
=
7
8
100
×5
4
6
15
=
20
100
Change these mixed
numbers to fractions with
100 as the denominator,
then simplify them.
%
23 4
=
50
100
+
100
=
=
100
%
Percentages to Fractions
To use percentages in calculations the percentage will usually be converted to a decimal or a
fraction. This sheet deals with converting percentages to fractions.
Column 1 asks you to write the percentages in simplified form. Start by rewriting the
percentage as a fraction over 100. Then fully simplify, this is done by finding the largest
number that goes into both numbers, their HCF, then dividing through by that number. An
example is at the top of the first column, use the same method throughout the column.
Column 2 is in problem form, start with the same method (except question 16) by writing the
percentage as a fraction with 100 as its denominator. Then find the equivalent fraction
changing the denominator to the number given in the question. In question 16 you have to find
the percentage first, though you may see a quicker way.
Column 3 is much the same as column 1 with the difference being that the percentages are
greater than 100% and so the fraction will have a value greater than 1. Start with the same first
step, writing the number over 100. Then look for the HCF. Divide through and write the
answer as an improper fraction. The answer could be left this way if the question just asked for
a fraction, but the question asks for mixed numerals. So convert the improper fraction to a
mixed numeral. Look at the mixed numeral when you have finished and make sure it can't be
further simplified.
Percentages to Fractions
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Change these percentages
to fractions with 100 as
the denominator, then
simplify them fully.
Example
(÷ 5)
11
55 % =
1
2
30 % =
20 % =
55
20 100
100
100
=
11
20
=
=
12 Michael achieved 68% in a
test. If the test was out of
50, what was his mark?
4
5
6
7
8
60 % =
12 % =
34 % =
44 % =
15 % =
85 % =
100
100
68 % =
100
100
100
100
64 % =
100
=
=
100
=
11 Write a 16% levy on water
rates as a fraction
%=
100
=
120 % =
19
250 % =
%=
=
2
1
= 1
100
2
=
100
=
110 % =
=
=
21
180 % =
100
=
=
125 % =
=
=
100
%
17 Ian falls out of bed in his
sleep 8% of nights. How
many times would he fall
out of bed in a 25 night
period?
%=
100
3
=
=
16 Jane scored 12 out of 20 in
her previous test. If the next
test is out of 50 what must
she score for the same %?
=
150
=
15 How many apartments are
still available?
20
improper
=
22
10 Ryan drank 35% of his
drink, express as a fraction
%=
Rewrite as a
mixed numeral
50
20
=
3
=
100
%=
12
9
=
100
14 A new apartment building
has 300 apartments, 40%
have been sold, how many
apartments is this?
=
=
÷ 50
150 % =
13 Carla achieved 80% in a test
which had a maximum mark
18
of 20, find her mark.
=
=
Example
2
80 % =
3
These percentages are
larger than 100%. Express
them as mixed numerals.
Use equivalent fractions
to solve these
100
=
=
23
105 % =
100
=
=
Decimals and Percentages
When percentages are used in calculations such as with bank loans and government taxes, they
are often converted to decimals. This is done, as with fractions, by dividing by 100. With
decimals this step is replaced with moving the decimal point 2 places to the left (dividing by
100). If converting a decimal to a percentage then you multiply by 100, move the decimal
point 2 places to the right.
Column 1 has 100 square grid and you are asked to express the shaded area as a decimal and
percentage, then repeat for the unshaded part. To express the shaded part as a decimal count
the squares and write the number behind 0._ _. For example 53 squares would be 0.53, don't
write in a zero when it isn’t required, for example 30 squares is 0.3 rather than 0.30. The
percentage shaded is the number of shaded squares with the % sign after it. For example 10
squares = 10%. Once you have answered the shaded part then complete the unshaded portion.
You can count the white squares, but it would be easier and faster to subtract the number of
blue squares from 100. If you want to be really fast then count the smallest number of squares
(unshaded or shaded) and answer it first. Your teacher may ask you what the sum (addition) of
the two decimals equals and what the sum of the two percentages equals, you should know
that this is 1 (decimals) and 100% (percentages).
Column 2 involves changing decimals to percentages, answer by multiplying by 100, so move
the decimal point 2 places right. If there isn't a decimal point it's because the number is a
whole number, so you assume it's behind the last digit. That means you add 2 zeros to the end.
There is an example at the top of the column.
Column 3 is the reverse, converting percentages to decimals. This time move the decimal point
2 places to the left, as you are dividing by 100. Again there is an example at the top. Possibly
the most asked question and the one that tricks the most students is to write 0.5% (or ½%) as a
decimal, this question is there (Q.45), see how you go.
Decimals and Percentages
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Write the both the shaded
and unshaded part of the
hundredths grid as a
decimal and a percentage.
Decimal: write the number of squares
behind decimal point ie, 23 sq = 0.23
Percentage: write the number of
squares in front of the % sign. (23%)
1
31.8% = 0.318
2 places left
% 30 45%
=
6 0.28
=
31 11%
=
7 0.135
=
32 1%
=
8 0.217
=
33 10%
=
9 3.54
Unshaded:
10 0.4
=
=
34 120%
=
=
35 430%
=
=
36 5.7%
=
12 1.22
=
37 60.9%
=
13 8.09
=
38 0.4%
=
14 3.064
=
39 12.7%
=
Unshaded: 15 2.077
=
40 463%
=
Shaded:
=
%
% 11 2
Shaded:
=
=
%
=
16 4.002
=
41 800.6%
=
=
% 17 1.01
=
42 100%
=
18 0.006
Shaded:
19 7.309
=
=
43 40.7%
=
=
44 110%
=
% 20 10
=
45 0.5%
=
Unshaded: 21 0.407
=
22 2.6004
=
%
23 0.0801
=
46 20.75%
=
=
47 10.03%
=
=
48 9.1%
=
Shaded: 24 0.1001
=
49 80.6%
=
=
25 0.3509
=
50 0.02%
=
% 26 3.05
Unshaded:
27 0.099
=
28 5.0002
=
%
29 1.7071
=
51 309.1%
=
=
52 22.6%
=
=
53 0%
=
=
54 1.001%
=
=
4
2 places right
=
=
3
0.14 = 14%
Now change these
percentages to decimals,
by dividing by 100
5 0.04
=
2
Convert these decimals
to percentages.
Multiply by 100.
=
Further Fractions to Percentages
Changing fractions to percentages with denominators that are factors of 100 has been covered,
but what of the fractions with denominators that aren't factors of 100, this sheet deals with
these.
Column 1 deals with the first step in changing a fraction to a percentage, you should look at
the fraction and see if you can express it with a denominator of 100. All the fractions in this
column can be, and the process is:
•
Look at the fraction and change it to have a denominator that is a factor of 100 (2, 4, 5,
10, 20, 25, 50)
•
Rewrite the equivalent fraction with the new denominator
•
Then write the equivalent fraction with a denominator of 100
•
The percentage is the numerator (top number).
An example is at the top of the column. Note that these fractions have been carefully selected,
this process only works sometimes, but it is an important method to learn.
Column 2 outlines the method for all other fractions. Using division, change to a decimal, then
to a percentage by moving the decimal point 2 places to the right. The method is identical to
converting fractions to decimals with the decimal point shifting being the only difference. The
answers are to be percentages to 1 d.p. so that means you require a 4 d.p. answer. Why 4?
Because the 4th decimal digit is needed for rounding, if the 4th decimal digit is 5 or above
then you round up. In the example at the top of the column the 4th d.p. digit is a 0 so there is
no rounding.
Column 3 has long division problems, again the percentage is required to be to 1 d.p. so you
need a 4 d.p. answer then round up or down as required. An example is shown, in the example
the answer is rounded up due to the 4th d.p. digit being a 5.
Further Fractions to Percentages
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Use you knowledge of the
factors of 100 to write
these fractions as decimals
Example
Example
÷3
9
15
1
=
6
8
2
3
12
3
32
40
4
12
16
5
36
80
6
24
120
7
39
60
8
18
24
9
33
110
10
24
30
11
66
150
12
30
75
13
27
45
Convert these fractions
to decimals, then to
percentages (to 1 d.p.)
=
=
=
=
=
× 20
3
5
4
4
20
4
=
0.6250
×100
60
100
=
=
=
=
=
5
= 8
8
= 60 %
100
100
100
100
100
=
=
add zeros
multiply by 100
move • 2 places
1
= 6
6
22
%
%
%
3
= 8
8
3
=
These fractions to
percentages will require
long division (1 d.p.)
Example
15
4
= 9
9
4
5
= 31.3%
16
23 2 =
16 5.0000
48
15 2
0.3125
=
=
%
=
% 16
5
= 6
6
5
=
=
=
=
=
=
=
=
=
=
=
=
=
100
100
100
100
100
100
100
=
=
16
40
%
%
17
2
= 7
7
=
100
2
80
80
0
%
18
=
%
1
= 3
3
=
%
=
%
=
% 20
=
%
24
1
=
19
7
= 8
8
7
=
1
= 9
9
=
=
32
=
=
15
20
=
=
1
=
14
=
change to
decimal
5
24
5.0000
= 62.5%
=
5
= 9
9
21
1
25
4
=
17
9
=
11
17 4
11 9
Fractions - Decimals - Percentages
This is a combination of converting between percentages, fractions and decimals. It is assumed
that earlier sheets have been completed before this sheet is attempted.
Column 1 asks you to:
•
Convert a decimal to a %. To answer these multiply by 100 or move the decimal point 2
places to the right.
•
Converting a % to a decimal is the opposite, divide by 100 this time so move the decimal
point 2 places to the left.
•
Converting fractions to percentages requires multiplying by 100. This will involve 2
steps. First rewrite the fraction (form an equivalent fraction) with a denominator of 100.
The % will be the numerator (top number) of the fraction. Rewrite it with a % sign.
•
Converting a % to a fraction is the reverse. Write the % over 100 (dividing by 100) then
simplify the fraction. Expect improper fraction answers at times.
Column 2 is the same as Column 1 except it is in table form and is jumbled. You are given
either a decimal, percentage or fraction and you have to find the other two. An example is at
the top of the column.
Fractions - Decimals - Percentages
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Express these decimals as percentages,
multiply by 100, so move the decimal
point 2 places right.
Complete the table below filling all the
missing spaces, so that the fraction,
decimal and percentage are equivalent.
1 0.34
2 1.24
Example
3 0.05
4 0.4
32
5 7.09
6 0.091
7 0.002
8 2
9 0.708
10 20.06
FRACTION
37
Now the reverse, change these percentages
to decimals. Move • 2 places to the left.
38
11 86%
12 17%
13 9%
14 44.6%
15 110%
16 0.5%
17 30.02%
18 400.5%
19 300%
39
40
20 80.3%
Express these fractions as percentages,
change denominator to 100, numerator = %
21
23
25
27
7
10
3
5
150
200
110
500
=
=
=
=
22
100
24
100
26
100
28
100
1
4
9
20
18
25
11
5
=
=
=
=
41
42
100
43
100
44
100
45
100
Now change these percentages to fractions
46
29 60%
100
=
30 15%
100
=
47
31 350%
33 ½%
35 104%
100
100
100
=
32 32%
=
34 120%
=
36 42%
100
100
100
%
=
=
=
48
49
100
100
100
100
1
1
2
100
9
50
100
18
25
100
100
=
10
=
8
25
%
=
0.5
%
=
0.8
%
75
=
=
=
%
%
100
0.95
=
%
%
100
120 %
44
=
=
=
100
32
DECIMAL PERCENTAGE
=
100
100
0.32
=
=
%
%
100
2.5
%
0.08
%
65
%
Calculating Percentages - Calculator
The method of calculating a percentage from a decimal or fraction is to multiply the decimal
or fraction by 100. To change a percentage to a decimal or fraction you divide by 100.
A picture of a calculator is shown with below with keys highlighted. The methods used for
each column are also listed on the foillowing pages. Where there is a alternative method it is
shown, so you can select the method that makes the most sense to you.
Column 1 asks you to convert fractions to percentages to 2 decimal places, then to a specified
number of decimal places (in brackets). Fix your calculator to the correct d.p. then multiply
the fractions by 100 for the answer. Note that if writing a whole number, such as 45%, when
asked to give to 2 d.p. then the answer is 45.00%. Even though the decimal portion adds
nothing to the answer, it fulfils the question. If the question said ‘show to 2 d.p. where
appropriate (or necessary)’ then you wouldn’t have to show the decimal part.
Column 2 asks you to convert the decimals to percentages. Again multiply the question by
100.
Column 3 deals with changing percentages back to fractions or decimals. In this case divide by
100.
Column 2
Example of calculating a decimal percentage from a decimal
Convert 0.206 to a percentage
Using × 100
0 . 2 0 6 × 1 0 0 =
if you get 20.6 for the question above you are correct
Column 1
Example of calculating a decimal percentage from a fraction
Convert 7/8 to a percentage
Using % key (not recommended)
7 ÷ 8 Shift %
Using × 100
7 ÷ 8 × 1 0 0 =
Using fraction key
7 ab/c 8 × 1 0 0 = ab/c
if you get 87.5 (no fix) for the question above you are correct
Example of calculating a fraction percentage from a fraction
Convert 7/8 to a percentage
Using fraction key
7 ab/c 8 × 1 0 0 =
Using × 100
7 ÷ 8 × 1 0 0 = ab/c
if you get 87 ½ for the question above you are correct
note that this style of question isn’t on the sheet
Example of calculating a decimal percentage from a mixed numeral
Convert 2 7/8 to a percentage
Using fraction key
2 ab/c 7 ab/c 8 × 1 0 0 = ab/c
Using × 100
2 + 7 ÷ 8 = × 1 0 0 =
if you get 287.5 for the question above you are correct
Column 3
Example of converting a decimal percentage to a decimal
Convert 56.32% to a decimal
Using ÷ 100
5 6 . 3 2 ÷ 1 0 0 =
if you get 0.5632 (no fix) for the question above you are correct
Example of converting a fraction percentage to a decimal
Convert 56 ¼% to a decimal
Using the fraction key
5 6 ab/c 1 ab/c 4 ÷ 1 0 0 = ab/c
if you get 0.5625 (no fix) for the question above you are correct
Example of converting a decimal percentage to a fraction
Convert 56.25% to a fraction
Using the fraction key Note that the featured calculator can convert
decimals to fractions, this may not be a feature on your calculator
5 6 . 2 5 ÷ 1 0 0 = ab/c
If your calculator has a fraction key but can’t convert decimals to
fractions then use this method (note you need some mental skills)
5 6 ab/c 2 5 ab/c 1 0 0 ÷ 1 0 0 =
if you get 9/16 for the question above you are correct
Example of converting a fraction percentage to a fraction
Convert 56 ¼% to a fraction
Using the fraction key
5 6 ab/c 1 ab/c 4 ÷ 1 0 0 =
if you get 9/16 for the question above you are correct
Calculating Percentages - Calculator
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Convert these fractions to
percentages (by × 100).
Round answer to 2 d.p.
1
2
2
Convert these decimals to
percentages (× 100). Give
your answer to 1 d.p.
Convert these percentages
to decimals (÷ 100). Give
your answer to 3 d.p.
/3
=
% 23
0.15
=
% 46
5
/7
=
24
0.439
=
3
1
/8
=
25
1.266
4
1
/12
=
26
5
4
/13
=
6
1 56/83
7
124
8
85
9
206
10
95%
=
47
102.7%
=
=
48
0.13%
=
0.78841
=
49
11.63%
=
27
0.0606
=
50
225.46%
=
=
28
3.0295
=
51
0.3%
=
/171
=
29
0.0074
=
52
½%
=
/302
=
30
0.4989
=
53
33⅓%
=
/118
=
31
1.0907
=
54
1.03%
=
4 357/502
=
32
0.9
=
55
67 ¾%
=
Continue converting to
percentages but round
your answer to the d.p.
in the brackets.
Round the percentage
answer to the d.p. asked
33
0.021141 [3] =
=
% 34
1.30055 [2] =
11
51
12
131
/165 [2]
=
35
1.08
13
3 81/97 [2]
=
36
0.29992 [2] =
14
1
/3 [3]
=
37
4.00681 [2] =
15
1 144/758 [1]
=
38
0.0096
16
41
=
39
11.0016 [1] =
17
288
/650 [3]
=
40
0.701061 [3] =
18
65
/91 [1]
=
41
0.349211 [0] =
19
2 32/55 [0]
=
42
0.59097 [2] =
20
193
/210 [2]
=
43
0.087026 [3] =
21
5
/8 [2]
=
44
0.389001 [0] =
22
249
=
45
1.999802 [1] =
/75 [1]
/85 [3]
/250 [0]
[1] =
[1] =
Write the percentages
below as fractions
% 56
15%
=
57
68.4%
=
58
12.5%
=
59
75.4%
=
60
117.5%
=
61
42.36%
=
62
½%
=
63
18.8%
=
7
FREEFALL
MATHEMATICS
VOLUME,
CAPACITY &
MASS
Centicube Volumes
Volume is a measure of the amount of space an object occupies. The units used in measuring
volume are cubic units: mm3, cm3 and m3. The solids on this sheet are all constructed from
centicubes (each 1 cm3). This allows the volume to be found by counting the cubes with the
volume being the total number of cubes (in cm3).
The first 2 columns ask you to count the cubes, you have to accept that the cubes that you can't
see, due to cubes being in front, are still there. A cube must be on top of another cube to be
above the first layer. When you write the volume you must include the units (cm3).
Column 3 introduces the Volume formula : V = Ah. This applies to any shape with a uniform
cross-section, (this means the blue area is the same throughout the height of the shape). Count
the number of squares that make up the blue area (A) and multiply this by the number of red
squares that make up the height (h), this will give you the volume in cm3.
Don't let the word 'height' confuse you. You may feel that height must go up and down but the
word height is used to describe the direction of the uniform cross-section. When you think
about it if you turn the page sideways it will be up and down, so the answer will be the same.
Centicube Volumes
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Calculate the volume of these
solids built from identical
cubes, assume 1 cube = 1cm3.
1
Find the area of the front face
then multiply it by the height
(the distance the shape goes
back into the page).
13
V=
2
14
21 Face Area (A) =
Shape height (h) =
cm3
V=
3
4
cm3
cm
cm3
A×h=
V=
cm2
V=
15
V=
V=
V=
5
6
22 A =
V = Ah =
V=
7
8
cm3
16
V=
V=
cm2 h =
23 A =
h=
V = Ah =
17
V=
V=
V=
9
10
24 A =
18
h=
V = Ah =
V=
19
V=
V=
11
12
25 A =
V = Ah =
V=
20
V=
V=
V=
h=
cm
Volume, Area and Height
When a solid has a uniform cross section (same shape throughout the height of the solid) then
the formula V = Ah can be used to obtain the volume of the solid. The area will be given in
square units (mm2, cm2, m2) and the height of the solid will be given in mm, cm and m.
Multiply the two together to get your answer. The answers will be in cubic units mm3, cm3, m3.
The entire sheet uses the same method, this is : writing the equation (V = Ah), substituting the
values (A and h), with an '=' sign before them and a '×' sign between them, then the answer
with the cubic units after it.
Volume, Area and Height
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
9
9.6 mm2
3.1 mm
A =17 m2
5
Multiply the area by the
height to find the volume
1
h =15 m
Area = 12 m2
h=3m
V = Ah
=
10
×
2
V=
A =2.17 cm
6
3
m
58 cm2
2
72 cm
h =0.8 cm
A=
26 cm2
h=
9 cm
A = 44 mm2
7
11
3m
h =23 mm
8.7 m2
3
A=
35 mm2
14
h=
mm
8
12
2
A = 17.6 cm
66
mm 2
cm
4.5
=
h
2
h = 53 cm
=
cm
A
83
4
47 mm
Volume of Rectangular Prisms
Rectangular prisms are the most common form of packaging, from shipping containers to
match boxes they are everywhere. To calculate their volume you multiply the 3 sides together,
in any order. Because we have V = Ah as the standard equation for volume with the 'area' part
being a rectangle we can replace the A with l × b. So we have V = l × b × h, or V = lbh.
This sheet uses the same method for the entire sheet. Using 3 lines of working:
•
Write the equation V = lbh
•
Substitute in the values (order isn't important), separated by × signs
•
Strike out 2 numbers that you can multiply together mentally and write the total above the
two numbers
•
Use the working space to multiply the non-stroked number with the number above
•
Write this answer in the 3rd line with 'V =' at the start and the cubic units at the end
(either mm3, cm3 or m3)
An example is at the top of the 1st column.
Volume of Rectangular Prisms
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Use V = lbh to find the
volume of these prisms,
remember units!
8
4
7 cm
8 cm
Example
22 cm
4 cm
11 cm
16 cm
6 cm
4 cm
8 cm
3
V = lbh
48
48
=8×6×4
4
V = 192 cm3
192
5
10 mm
9
8 mm
1
37 mm
5m
48 m
18 cm
12 m
5 cm
4 cm
6
10
2
23 cm
4 mm
13 mm
7 cm
9 mm
4 cm
33 mm
9 mm
15 mm
7
7 mm
3
11
9m
7m
7m
7m
9 mm
17 mm
12 m
5m
Further Volumes of Prisms
The method of the sheet is to first calculate the area of the face (uniform cross section area)
then multiply it by the height of the shape. Note that because this sheet deals with triangular
prisms you must not confuse the h used in A = ½bh with the h used in V = Ah.
There are six lines of working as the example in column 1 shows. Calculate the area of the
triangular face, then using the working space multiply the area with the shape’s height to get
its volume. Write 3 lines of working for the volume. Don't forget you will have square units
(mm2, cm2, m2) for the area and cubic units (mm3, cm3, m3) for the volume.
The 3rd column involves composite shapes. You have to add the 2 areas to get the total area.
So find the sum of the two areas (add them) and multiply by the height to get the volume.
There is an example at the top of column 1.
Further Volumes of Prisms
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Find the uniform cross
section area, use V=Ah
to find the volume
3
6
18 m
25 m
Example
55 m
+
14 m
12 m
7 cm
4 cm
A2 = lb
A1 = lb
13 cm
16 m
5m
A = ½bh
2
14
=½×4×7
13
A = 14 cm2
V = Ah
= 14 × 13
V = 182 cm3
42
140
4
182
15 mm
A=
m2
31 mm
6m
7
23 m
6 mm
9m
+
V = Ah
40 mm
1
A=
6 mm
17 mm
12 mm
43 cm
5
2
17 mm
16 mm
12 mm
85 cm
60 cm
+
Converting between Volume and Capacity
When you calculate the volume within a solid it's measured in cubic units such as mm3, cm3,
or m3. These units are used usually to define empty space such as storage space, shipping
containers or packaging, usually when they are to be filled with solid material. Capacity is
measured in millilitres (mL), litres (L), kilolitres (kL) and Megalitres (ML). Each unit is 1 000
times larger than the unit before it. While capacity is usually associated with liquids it is also
used with air. For example the space inside a car will have its capacity measured in litres.
Column 1 starts with comparing mL and cm3. They are the same, so a measurement in mL,
say 5.72 mL is the same in cm3, 5.72 cm3. So rewrite the number and change to the other unit.
From question 7 you are asked to convert cm3 to L. As 1 cm3 is 1 mL, and there are 1 000 mL
in a L, then 1 cm3 must be 1/1000 of a L. So divide by 1 000 or move the decimal point 3 places
to the left to get your answer. Remember that litres is written with an upper case L.
A common object where both capacity and volume are used is with car motors. The size of the
engine could be given in c.c. (cubic centimetres) which is an accepted way of writing cm3, or
litres (L). For example a 1 800 c.c. engine could also be described as a 1.8 L engine.
Column 2 starts with the reverse converting from L to cm3. This time you multiply by 1 000 or
move the decimal point 3 places to the right. The last 6 questions involve the conversion
between kL and m3. Just as with mL and cm3, they are identical so 1 kilolitre (1 000 L) =
1 cubic metre (m3). Just rewrite the number and change the unit.
Column 3 asks you first to convert from m3 to L. Remember that 1m3 = 1 kL which is 1 000 L.
So multiply by 1 000 or move the decimal point 3 places to the right. The next section of the
column reverses the process, converting L to m3. Reverse the process by dividing by 1 000 or
moving the decimal place 3 places to the left.
This sheet doesn't deal with Megalitres (ML) as they are rarely used. But there are times when
their use is essential when dealing with large capacities, can you think of where ML would be
used?
DAMS & WATER SUPPLIES
Converting between Volume and Capacity
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Change the units from
mL to cm3 or the reverse
Example
73 mL
=
73 cm3
Now reverse, change
these from L to cm3
3 places right
4.3 L = 4 300 cm3
Change these volumes
from m3 to litres.
3 places right
2.53 m3 = 2 530 L
1 350 mL
=
24 2 L
=
47 7.2 m3
=
2 105 mL
=
25 0.5 L
=
48 0.5 m3
=
3 56.3 cm3
=
26 3.1 L
=
49 3.7 m3
=
4 850 mL
=
27 0.2 L
=
50 0.25 m3
=
5 2.43 cm3
=
28 3.5 L
=
51 16.403 m3
=
6 600 cm3
=
29 1.75 L
=
52 0.02 m3
=
30 0.04 L
=
53 5.62 m3
=
31 4.03 L
=
54 1.01 m3
=
32 0.05 L
=
55 7.9035 m3
=
Convert these volumes
in cm3 (c.c.) to litres
850 cm3 = 0.85 L
3 places left
7 1 000 cm3
=
33 0.01 L
=
56 18.006 m3
=
8 800 cm3
=
34 0.2063 L
=
57 105.12 m3
=
9 450 cm3
=
35 2.02 L
=
10 2 100 cm3
=
36 0.25 L
=
11 1 700 cm
=
37 4.06 L
=
12 750 cm3
=
38 0.96 L
=
13 1 250 cm3
=
39 3.007 L
=
14 5 000 cm3
=
40 4.55 L
=
15 85 cm3
=
16 4 302 cm3
=
17 120.4 cm3
=
2.9 m3
18 333 cm3
=
19 2 006 cm3
3
Now change these from
litres to cubic metres
3 places left
2 650 L = 2.65 m3
58 2 000 L
=
59 1 700 L
=
60 800 L
=
61 3 075 L
=
62 175 L
=
2.9 kL
63 422 L
=
41 7 m3
=
64 9.06 L
=
=
42 3.84 kL
=
65 45.6 L
=
20 500 cm3
=
43 43 kL
=
66 115 L
=
21 25.1 cm3
=
44 0.3 m3
=
67 2 L
=
22 1 007 cm3
=
45 204 m3
=
68 500.5 L
=
=
46 7.06 kL
=
69 3 000.6 L
=
3
23 10 cm
Change the units from
kL to m3 or the reverse
Example
=
Units of Capacity
Capacity is another way of expressing volume. Rather than using length measurements,
capacity is expressed using measurements normally associated with liquids. Usually mL, L
and kL. This sheet deals with the changing of units between these three. As with length, (mm,
m and km), capacity units are × 1 000 apart (cm is the exception with length). So you multiply
by 1 000, or as hopefully you know, you move the decimal point 3 places to the right
(multiplying) and to the left (dividing).
Column 1 deals with changing from a smaller unit to a larger unit. With these you divide by
1 000. So move the decimal point 3 places to the left with these, remember to write the new
unit after your answer, it will be the unit inside the brackets. Watch that you don't have
unnecessary zeros in your answers when decimal points are involved.
Column 2 is the reverse, this time moving to a smaller unit so you have more of them, so you
are multiplying by 1 000. Move the decimal point 3 places to the right to make the number
larger. Again remember to write the new units after the number.
Column 3 asks you to find the volume of the prisms, the answer will be in cubic units, either
cm3 or m3. Convert these to either mL (if cm3) or kL (if m3), rewrite the number and change
the units. Then rewrite the answer in litres (L), using your skills from the previous 2 columns.
Capacity is volume, so you still write 'V =' in your answer.
Note that L is the symbol for litres, it is a capital letter, this may seem unusual but many metric
units use capital letters, an example being ‘N’ (Newtons, a measure of force) or ‘W’ (Watt, a
measure of electrical power).
Units of Capacity
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Change these measurements to the larger units
shown in the brackets
mL → L
L → kL
÷ 1 000 → 3 places left
÷ 1 000 → 3 places left
Convert the measurements
to the smaller units shown
in the brackets
kL → L
L → mL
× 1 000 → 3 places right
Calculate the volume in
either cm3 or m3, write the
answer as mL or kL, then
rewrite converting to litres
× 1 000 → 3 places right
45
5 cm
4 cm
1 2 000 mL [L] =
23 1 L
[mL] =
10 cm
2 5 000 L
[kL] =
24 6 L
[mL] =
3 1 500 L
[kL] =
25 4.5 kL
[L] =
4 3 400 mL [L] =
26 1.25 L
[mL] =
5 900 L
[kL] =
27 0.75 L
[mL] =
6 500 mL
[L] =
28 0.8 kL
[L] =
7 1 375 mL [L] =
29 3.2 L
[mL] =
8 250 mL
[L] =
30 0.01 kL
[L] =
9 505 L
[kL] =
31 1.12 kL
[L] =
10 50 L
[kL] =
32 3.05 kL
[L] =
11 10 mL
[L] =
33 0.079 L
[mL] =
12 35 mL
[L] =
34 5.03 L
[mL] =
13 1 017 L
[kL] =
35 0.871 kL
[L] =
14 110 mL
[L] =
36 5.64 kL
15 4 070 L
[kL] =
37 0.0025 L [mL] =
V=l×b×h
V=
cm3
V=
mL
V=
L
46
15 m
kL
L
47
38 2.004 kL
[L] =
17 60 mL
[L] =
39 0.07 L
[mL] =
18 15 L
[kL] =
40 0.003 kL
[L] =
19 410 mL
[L] =
41 0.0005 kL [L] =
20 5 030 L
[kL] =
42 1.9 L
21 303 L
[kL] =
43 0.0087 kL [L] =
22 1 L
[kL] =
44 0.053 L
8 cm
20 cm
[L] =
16 1 400 mL [L] =
2m
5m
5 cm
48
[mL] =
[mL] =
10 m
3m
2m
Mass
Mass is a measure of weight, so why is it called mass? The difference is that mass is constant,
if you weigh 53 kg at sea level you won't actually weigh 53 kg on top of Mt Everest. This is
due to gravity being weaker when you are further from the centre of the planet, so as you
move up, you technically weigh less! But your mass is still 53 kg, it stays unchanged from
gravitational or outside effects. This is exaggerated when space or moon travel are involved as
each planet/moon has a different gravitational pull. So the technical word is mass but unless
you are going to Mars or Mt Everest the word weight is usually more commonly used.
Column 1 deals with changing from one unit to a larger unit. In all circumstances the
difference is a factor of 1 000, and when you move to a larger unit you divide. But when you
divide by 1 000 it is easier to move the decimal point 3 places (count the zeros) to the left.
Examples are at the top of the column, don't forget to write the units (mg, g, kg, t). If in your
answer you have a zero on the end of the number and it is behind the decimal point, it
shouldn't be there, decimal answers should always be in their shortest form.
43.70 kg = 43.7 kg
Don't write the mass with an unnecessary
zero, always write decimals in their most
simplest form.
Column 2 is the reverse, you are moving to a smaller unit and so you multiply by 1 000. This
means you move the decimal point 3 places to the right. Examples are there as well as the
'arrowed' guide, again don't forget the units.
Column 3 is in problem form and asks you to then express (state or give) the answer in two
different units. The last two questions ask for the unit weight. The unit weight is the weight of
1 item, so you will have to divide the total mass by the number of items to get the unit weight.
Mass
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
Express these masses in
the units given in the
brackets. All are ÷ 1 000.
50 mg = 0.05 g
950 g = 0.95 kg
4 250 kg = 4.25 t
3 places left
3 places left
3 places left
Now the reverse, multiply
by 1 000 to change to the
units in the brackets
3.7 t = 3 700 kg
0.52 kg = 520 g
1.06 g = 1 060 mg
3 places right
3 places right
3 places right
Calculate the combined
mass of these, then
convert to kilograms.
45 7 cans of salmon, each 450g
g:
kg:
1 2 000 kg
[t] =
23 1.2 kg
[g] =
2 3 600 g
[kg] =
24 4.3 g
[mg] =
3 900 g
[kg] =
25 3.85 t
[kg] =
4 150 mg
[g] =
26 0.7 kg
[g] =
5 305.7 kg
[t] =
27 5.02 t
[kg] =
6 100 mg
[g] =
28 11 kg
[g] =
7 1 750 mg [g] =
29 0.05 g
[mg] =
47 A dozen oranges each having
an average mass of 270g
8 80.2 g
[kg] =
30 1.01 g
[mg] =
g:
9 220 kg
[t] =
31 10.01 kg
[g] =
kg:
10 3 510 kg
[t] =
32 0.04 kg
[g] =
11 5 300 mg [g] =
33 13.14 t
[kg] =
12 95 g
[kg] =
34 0.35 t
[kg] =
13 750 g
[kg] =
35 9.04 g
[mg] =
14 11 mg
[g] =
36 46 kg
[g] =
15 8 070 kg
[t] =
37 0.01 g
[mg] =
16 610 kg
[t] =
38 6.702 t
[kg] =
17 8 g
[kg] =
39 0.65 g
[mg] =
18 400 mg
[g] =
40 7.15 kg
[g] =
19 16 300 kg [t] =
41 6.06 t
[kg] =
20 7 070 mg [g] =
42 0.05 kg
[g] =
21 30 090g
[kg] =
43 0.2 g
[mg] =
22 101.3 g
[kg] =
44 3.71 g
[mg] =
46 3 cartons of milk, each one
having a mass of 830 g.
g:
kg:
Now find the average
unit weight for the
following
48 Eight tablets with a total
mass of 36.48 g
g:
mg:
49 Four cars with a total mass
of 4.3 t.
t:
kg:
Net and Gross Mass
When you buy a product from a shop you want to know how much product you are getting, so
that you can compare. For example frozen corn kernels will be in a plastic bag while unfrozen
corn kernels will be in a can, the can may weigh more than the bag, but is that because of the
can or is there more corn inside? Instead of products having their total weight on them they
have their Net weight, this is the weight of the actual product inside the can or bag.
So Net weight is the weight of the contents. The total weight, product + container is called the
Gross weight. Gross weight is rarely on the item but you have to calculate it if you are
sending an item by mail or paying for its transport. Gross weight is also important in industry
which sets a maximum weight that a person can lift, the packaging must be factored into
calculations. Remember the environment when you buy an item, try to buy one that minimises
the use of packaging.
These three equations are involved with this worksheet:
•
Net weight
= Gross weight - Container weight
•
Gross weight
= Net weight + Container weight
•
Container weight = Gross weight - Net weight
Column 1 asks you to calculate the gross weight, add the packaging weight to the net weight to
get the answer. You should be able to complete the table mentally, watch the units asked for in
the answer. Problem style questions then follow, use the working spaces provided then give
your answer in the units asked for.
Column 2 asks you to calculate the net weight. This is found by subtracting the packaging
weight from the gross weight. This column is in the same format as the previous column, you
should be able to do these mentally, then the problems have working spaces provided.
Column 3 asks you to calculate the weight of the packaging, subtract the net weight from the
gross weight to answer these. The last question has a table that gives two values in each row,
you have to use your skills with these to calculate the third value. Do you add or do you
subtract…?
DOLPHIN SAFE
TUNA
225g NET
The weight given on a product is the net weight.
This is the weight of the tuna inside the can
without the can’s weight. You only want to
know this weight so… ‘the net is what you get.’
Net and Gross Mass
© FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE
What's in the net is what
you get.. calculate the net
mass of these
Calculate the gross
weight of the items
below in the given units
4 Gross - Container = Net
1 Net + Container = Gross
Net
150 g
225 g
480 g
1 kg
20 kg
13.2 kg
Container
Gross
g
30 g
g
70 g
g
90 g
g
50 g
kg
750 g
kg
1050 g
Net
Gross
Container
305 g
60 g
g
495 g
25 g
g
120 g
35 g
g
75 kg
17 kg
kg
1 750 kg
300 kg
t
8.7 t
250 kg
t
5 Tegan has caught a rare
finger-swallowing tiger slug. If
2 A cable car weighing 1.4 t
has 8 people board it with an the bug catcher weighs 469 g
and with the slug weighs 718 g,
average weight of 77 kg.
find the mass of the slug.
i) Calculate the combined
weight of the passengers
kg:
6 A variety pack of cereals
has 8 different selections. If the
ii) Calculate the current gross
net mass of the pack is 600g
mass of the cable car in kg and t find the mass of each selection.
Now calculate the weight
of the containers.
9 Gross - Net = Container
Gross
Net
Container
45 g
18 g
g
300 g
225 g
g
705 g
620 g
g
8.3 kg
7.6 kg
g
850 kg
65 kg
t
19.2 t
1.6 t
t
10 Harry's glasses weigh 248g,
in their case the weight is 437g.
Find the weight of the case.
-
11 Han weighs 75 kg, when he
is encased in carbonite he tips
the scales at 722 kg. Find the
weight of the carbonite.
-
+
kg:
7 A man on scales weighing
86 kg lifts his daughter up and
the new reading is 133 kg. Find
3 A jar of vitamin pills
contains 35 pills at 7 g each. the weight of his daughter.
If the jar itself weighs 83 g
find the gross weight.
t:
Fill in the blanks… may
the gross be with you!
12
Net
Packaging
200 g
55 g
+
17 g
8 A rain gauge jar weighs
57 g, after rain it weighs 67 g.
The mL of water inside is…
1
10
100
1 000
20.3 kg
Gross
80 g
21.9 kg
0.83 t
150 kg
5.7 t
80 kg
97 kg
1.3 t
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