CAPS
Mathematics
a
oo
Le
6
rn
k
Grade
e r ’s B
L. Bowie • C. Gleeson-Baird • R. Jones
H. Morgan • K. Morrison • M. Smallbones
1 Term 1
Kites are very popular for relaxation and enjoyment
2
Platinum Maths Gr6_Term 1_CAPS.indd 2
09/02/13 1:12 PM
Topics 1– 8
Starting off
Kites are very popular in different cultures. The
Chinese have been making and flying kites for
over 3 000 years. Early kites were made out of
silk and bamboo. Since then, kites have been
used in scientific experiments, as well as for
relaxation and enjoyment.
One of the special quadrilaterals is called a
kite, and has the shape of a kite. Real kites have
many different shapes, including box kites,
power kites and many other decorative shapes.
Look at the diagram of a kite below. This is a 2D
shape that you will study in later grades.
1. What do the two pairs of markers mean?
2. What does the little square in the centre of
the kite mean?
3. How would you explain to a learner who
cannot see the diagram what a kite looks
like? Use simple language and write a few
sentences to describe the shape of a kite.
Content covered in Term 1
Topic 1: Whole numbers, Topic 2: Number sentences, Revision, Topic 3:
Addition and subtraction, Topic 4: Common fractions, Revision, Assignment,
Topic 5: Time, Topic 6: Properties of 2D shapes, Revision, Topic 7: Data
handling, Topic 8: Numeric patterns, Revision
3
Platinum Maths Gr6_Term 1_CAPS.indd 3
09/02/13 1:12 PM
Topic
1
Count, order, compare and represent
whole numbers
Maths ideas
• Count in large
numbers.
• Recognise the
place value of
digits in up to
six-digit numbers.
• Order and
compare up to
six-digit numbers.
Key words
• whole numbers
– zero and all
positive numbers
with no fractions
Read and write large numbers
In Grade 5 you worked with whole numbers with up to six digits.
To read a larger number, read each group of digits as a three-digit
number. We use spaces to separate the groups of three digits when
we write large numbers.
Example
456 789: four hundred and fifty-six thousand, seven hundred and
eighty-nine.
572 391: five hundred and seventy-two thousand, three hundred
and ninety-one.
Numbers have zeros as place holders when there is no digit in that
place. The zeros help you to write the other digits in the correct places.
Example
230 006: two hundred and thirty thousand and 6.
405 016: four hundred and five thousand and 16.
You can expand large numbers into their separate place values as an
addition calculation.
Example
456 102 = 400 000 + 50 000 + 6 000 + 100 + 2
ExERCiSE 1.1
1. Write the following numbers in digits:
a ) four hundred and fifteen thousand, six hundred and eighteen
b ) eight hundred thousand and fifty-two
c ) eight thousand and twelve.
2. Write the following numbers in expanded form and then write
down how you would read them:
a ) 47 561
b ) 386 276
c ) 129 560
d ) 50 600
e ) 450 017
f ) 202 011
g ) 600 192
h ) 900 280
4
Term 1
Platinum Maths Gr6_Term 1_CAPS.indd 4
09/02/13 1:12 PM
Count large numbers
Now that you can read large numbers, you can also count forwards
and backwards with them.
Example
100 005 is 6 more than 99 999.
99 999
100 005
+1 +1 +1 +1 +1 +1
100 000
99 999 is 101 less than 100 100.
99 999
−1
100 000
− 100
100 100
ExERCiSE 1.2
1. Count forwards in hundreds from:
a ) 2 800 to 3 700
b ) 23 493 to 24 293 c ) 68 544 to 69 344
2. Count backwards in thousands from:
a ) 31 050 to 25 050 b ) 49 980 to 42 980 c ) 78 999 to 69 999
3. Fill in the missing numbers in the number chains below:
a ) 91 999
→ –10 → □ → –10 → □ → –10 → □ → –10 → □ → –10 → □
b ) 24 701
→ –200 → □ → –200 → □ → –200 → □ → –200 → □
4. Write the following numbers:
a ) 5 more than 9 999
c ) 40 more than 9 980
e ) 8 more than 99 999
g ) 200 more than 99 998
i ) 300 less than 990 000
k ) 1 000 more than 99 999
Challenge
How much do you
need to add to these
numbers to get
100 000?
1. 99 999
2. 88 888
b ) 6 less than 10 004
d ) 8 less than 10 181
f ) 9 less than 100 005
h ) 150 less than 100 102
j ) 10 000 less than 890 000
l ) 2 000 less than 200 000
3. 77 777
4. 66 666
5. 55 555
6. 44 444
7. 33 333
Topic 1: Count, order, compare and represent whole numbers
Platinum Maths Gr6_Term 1_CAPS.indd 5
5
09/02/13 1:12 PM
Game
• Play in pairs. Cut
out (or fold and
neatly tear) 10
squares of paper.
On each piece of
paper write one of
these instructions:
+ 50; – 50; + 100;
– 100; + 150; – 150;
+ 200; – 200; + 250;
– 250.
• Shuffle these
instructions and
place them face
down on the table.
• On another page,
write down the
number 100 000.
• Now take turns
picking up an
instruction and
applying it to your
number, until all
the instructions
have been used.
Check each other’s
working. The
highest number
wins.
Did you know?
• The population
of a place is the
number of people
living there.
• There are more
than 900 million
people in Africa.
• The population
of South Africa is
about 50 million.
6
Use place value to order numbers
The value of any digit depends on the place it has in a number. A
six-digit number has hundred thousands, ten thousands, thousands,
hundreds, tens and units.
Example
In the number 456 129, the 4 has a value of four hundred thousand;
the 5 has a value of fifty thousand; the 6 has a value of six thousand;
the 1 has a value of one hundred; the 2 has a value of twenty;
the 9 has a value of nine.
ExERCiSE 1.3
1. Write down the place value of each digit in words:
a ) 34 678
b ) 346 781
c ) 346 701
2. Give the values of the digits 2 and 5 in each number:
a ) 53 925
b ) 124 589
c) 597 230
d ) 582 307
3. Give the values of the digits 7 and 3 in each number:
a ) 34 781
b ) 752 136
c) 375 100
d ) 703 945
To order large numbers, compare the place values of the digits,
working from left to right.
Example
Compare the numbers 456 987 and 456 132.
The first three digits are the same in both numbers.
The first number has 9 hundreds and the second number has only
1 hundred, so 456 987 is greater than 456 132.
ExERCiSE 1.4
1. Below are different sets of population figures. Order each set from
the greatest to the smallest:
a ) 964 972 999 402 964 217 974 135
b ) 473 419 437 419 443 789 409 090
2. Order these amounts of money from the smallest to the largest:
a ) R456 875 R455 674 R465 876
R455 898
b ) R34 589
R34 876
R35 987
R35 876
Term 1
Platinum Maths Gr6_Term 1_CAPS.indd 6
09/02/13 1:12 PM
Use symbols to compare numbers
Do you remember the symbols for less than and greater than? You
need to remember the rules for ordering numbers when using these
symbols.
< means less than:
Did you know?
245 457 < 254 358 means
245 457 is less than 254 358
> means greater than: 486 574 > 486 386 means
486 574 is greater than 486 386
In the numbers above, which digit tells you what the greater number is?
In the first example, 245 457 < 254 358, it is the ten thousands digit. In
the second example, 486 574 > 486 386, it is the hundreds digit.
ExERCiSE 1.5
1. Compare each pair of numbers below and fill in the symbol < or >.
Identify the digit that tells you which number is greater or which
number is smaller.
a ) 245 695 □ 245 726
b ) 823 617 □ 822 399
c ) 467 990 □ 709 008
d ) 731 420 □ 731 402
e ) 905 056 □ 900 863
f ) 999 999 □ 999 899
Ancient Egyptian
mathematician
• Mathematics in
Africa started
much earlier than
the first written
numerals of
ancient Egypt,
around 3 100 BC.
• Ancient Africans
used the concept
of zero.
• Zero is a very
important number.
• Using zero as
2. Compare each pair of numbers below. Write the number
a place holder
helps us to see
represented by the expanded form and then fill in the
the difference
symbol < or >.
between 11, 101,
a ) 60 + 3 000 + 500 000 + 400 □ 30 000 + 400 + 600 000 + 60
1 001, 10 001 and
b ) 9 + 800 000 + 40 000 □ 400 000 + 90 000 + 80
100 001.
c ) 50 + 400 000 + 20 000 □ 400 000 + 500 + 2 000
d ) 500 + 6 000 + 30 + 400 000 + 1 + 20 000 □ 400 000 + 500 + 20 + 6 000 + 20 000 + 1
Challenge
Write these six numbers in order from the smallest to the greatest using the
< symbol:
• three million, four hundred and fifty thousand, and ninety-six
• 4 356 700
• 400 000 + 90 + 1 000 + 3 000 000 + 50 000
• 300 000 + 600 + 90 + 4 000 000 + 1 + 6 000 + 50 000
• 3 399 999
• four million, nine hundred and ninety-nine
Topic 1: Count, order, compare and represent whole numbers
Platinum Maths Gr6_Term 1_CAPS.indd 7
7
09/02/13 1:12 PM
Topic
2
Number sentences
Maths ideas
Solve number sentences
• Solve number
sentences
– by trial and
improvement
– by inspection.
Some number sentences have brackets. Always calculate what is in
brackets first. If there are no brackets in a sentence, the rule is: do
division and multiplication first, from left to right, then do addition
and subtraction from left to right.
• Identify equivalent
number sentences.
Key words
• trial and
improvement –
finding a missing
number by making
more accurate
guesses based on
your results
• inspection –
finding a missing
number by looking
at each part of the
number sentence
Challenge
Use one pair of
brackets to make
each number
sentences true:
a) 4 + 18 ÷ 2 = 11
b) 4 + 18 ÷ 2 = 13
c) 3 × 9 – 8 + 10 = 9
d) 3 × 9 – 8 + 10 = 13
One method for solving a number sentence is called trial-andimprovement. Another method is inspection where we break up the
sentence into its parts.
Example
Solve the following number sentence by trial and improvement:
(□ × 3) – 2 = 19
□ represents a number.
Try 5: (5 × 3) – 2 = 13. Too small. Try a bigger number.
Try 8: (8 × 3) – 2 = 22. Too big. Try a smaller number.
Try 7 and substitute: (7 × 3) – 2 = 19. So the missing number is 7.
Example
Solve the following number sentence by inspection:
12 – (2 × □) = 2
□ represents a number.
12 – (10) = 2
So 2 × □ = 10
2 × 5 = 10
Check your answer by substitution: 12 – (2 × 5) = 2
ExERCiSE 2.1
1. Complete the following number sentences:
a ) (17 – 5) × (6 – 2) = □
b ) (63 – 41) ÷ 2 = □
2. Find the missing values to complete the number sentences:
a ) 12 × 7 = 7 × □
b ) 7 + 37 = □ + 7 c ) 36 ÷ 9 × 9 = □
d ) 37 × 5 ÷ 5 = □
e ) 102 × 1 = □
f ) 27 ÷ □ = 1
3. Solve the following. Check each answer by substitution.
a ) (□ × 4) – 4 = 20
b ) (8 – 6) + 12 = □
c ) □ + (12 ÷ 2) = 14
d ) (14 ÷ □) + 4 = 11
e ) □ – (6 ÷ 3) = 11
f ) (3 × □) × 1 = 3
8
Term 1
Platinum Maths Gr6_Term 1_CAPS.indd 8
09/02/13 1:12 PM
Equivalent number statements
Two number statements are equivalent if you get the same answer
when you calculate each statement, or if you can change the one
statement to be the same as the other.
Example
Are these number statements equivalent?
3(15 – 2) × (20 + 7) and 39 × 27
Simplify the first statement: 3(15 – 2) × (20 + 7)
= 3(13) × 27 → 3(13) means 3 × 13
= 39 × 27
Yes, these are equivalent statements.
Key words
• equivalent
number
statements
– number
statements that
give the same
answer
ExERCiSE 2.2
1. 35 × 47 is equivalent to which number statements below? Show how
you chose your answers by either calculating the number statements,
or changing them to be the same as the given number statement.
a ) 47 × (3 × 5)
b ) 35(40 + 7)
c ) 30(40 + 7) + 5(40 + 7)
d ) (40 + 7) × (30 + 5)
2. 38 × 16 is equivalent to which number statements below?
a ) 28 × (10 × 6)
b ) (30 + 8) × (10 + 6)
c ) (20 – 4) × (40 – 2)
d ) 10(40 – 2) + 6(40 – 2)
3. Fill in the missing numbers in the following equivalent statements:
a ) 6 × (3 + 7) = (□ × 3) + (6 × □)
b ) (4 × 3) + (4 × □) = 4 × (□ + 5)
Challenge
By working out each side of the sentence,
show that 3 × (5 + 2) = (3 × 5) + (3 × 2).
Now use the diagram to also show that
the two sides are equivalent statements.
Draw your own diagram to show that:
2 × (4 + 3) = (2 × 4) + (2 × 3)
Topic 2: Number sentences
Platinum Maths Gr6_Term 1_CAPS.indd 9
9
09/02/13 1:12 PM
ExERCiSE 2.3
For each question, complete the sentences then write down what
you noticed:
1. a ) 27 × 8 = □ 8 × 27 = □
b ) 16 × 4 = □ 4 × 16 = □
c ) 56 ÷ 8 = □ 8 × □ = 56
d ) 252 ÷ 6 = □ 6 × □ = 252
2. a ) (17 × 4) × 2 = □
b ) (16 × 25) × 4 = □
c ) (23 × 50) × 2 = □
(4 × 2) × 17 = □
(25 × 4) × 16 = □
(50 × 2) × 23 = □
3. a ) 32 – 6 + 6 = □
b ) 721 + 5 – 5 = □
c ) 74 – 18 + □ = 74
Example
What you learnt in Exercise 2.3 can help you to solve the following
kind of number sentence: 32 + 8 – □ = 31.
If we substitute 8 for □, we get 32 + 8 – 8 = 32.
We want the answer to be 31, so we need to subtract an extra 1.
This means that □ = 9. Substitute 9 into the number sentence to
make sure that it does make the sentence true.
ExERCiSE 2.4
Now use this new idea to solve these number sentences:
a ) 82 + 17 – □ = 84
b ) 73 – 12 + □ = 76
c ) 27 + 17 – 17 = □
d ) 67 + 34 – □ = 70
e ) 187 + 33 – □ = 190
f ) □ + 23 – 23 = 83
Example
Challenge
I think of a number,
add 50 and divide by
2. Write this as a
number sentence. If
my final answer is 35,
which number was I
thinking of?
10
9 × 300 = (9 × 3) × 100 = 27 × 100 = 2 700
5 000 ÷ 100 = 50 × (100 ÷ 100) = 50 × 1 = 50
ExERCiSE 2.5
Complete the following as in the example above:
a ) 7 × 800 = □
b ) 17 × 300 = □
c ) 14 × 4 000 = □
d ) 2 700 ÷ 100 = □
e ) 79 000 ÷ 100 = □ f ) 82 000 ÷ 1 000 = □
Term 1
Platinum Maths Gr6_Term 1_CAPS.indd 10
09/02/13 1:12 PM
Revision
1. Write the following numbers in words:
a ) 467 715
b ) 300 002
(1)
(1)
2. Write the following words in number form:
a ) three hundred and ninety-six thousand, four hundred and eighteen
b ) forty thousand, nine hundred and twelve
(1)
(1)
3. Write down the number that is:
a ) 40 more than 356 440
b ) 9 less than 500 000
(1)
(1)
4. Give the values of the 5 and the 8 in each of these numbers:
a ) 978 597
b ) 562 080
(2)
(2)
5. Order the following numbers from the smallest to the greatest:
270 065; 99 914; 454 867; 974 661; 947 814; 455 080
(2)
6. Complete the following number sentences:
a ) (24 – 9) × (3 + 6) = □
b ) 32 ÷ (3 + 5) × 8 = □
c ) (84 ÷ 12) + (72 ÷ 9) = □
(1)
(1)
(1)
7. Solve the following number sentences. Check each answer by substitution.
a ) (□ × 5) – 7 = 23
b ) □ + (42 ÷ 2) = 27
c ) (32 ÷ □) + 7 = 15
(2)
(2)
(2)
8. Fill in the missing numbers to make this number statement true:
□ × 34 = (5 × 30) + (5 × □)
(2)
9. Which statements are equivalent to 24 × 84?
Write down only the letters of the correct answers.
a ) 84 × (2 × 4)
b ) 24(80 + 4)
c ) 20(80 + 4) + 4(80 + 4)
d ) (80 + 4) × (20 + 4)
(2)
Total marks: 25
Revision
Platinum Maths Gr6_Term 1_CAPS.indd 11
11
09/02/13 1:12 PM
Topic
3
Addition and subtraction
Maths ideas
Estimation
• Round numbers to
the nearest 1, 10,
100 or 1 000.
It is important to estimate an answer by rounding before doing exact
calculations.
• Estimate answers.
• Add and subtract
numbers with up
to six digits.
• Use inverse
operations to
check answers.
• Use a calculator.
• Solve addition
and subtraction
problems.
• When rounding to the nearest 10, look at the units digit.
In the number 473, 3 is less than 5 so you round down to the
nearest 10, to get 470.
• When rounding to the nearest 100, look at the tens digit.
In the number 463, 6 is greater than 5 so you round up to the
nearest 100 to get 500.
• When rounding to the nearest 1 000, look at the hundreds digit.
In the number 1 633, 6 is greater than 5, so you round up to the
nearest 1 000 to get 2 000.
Example
An estimate of 45 202 + 12 842 is 45 000 + 13 000 = 58 000.
When adding two numbers that are very close to each other, you can
use doubling (multiplying by two) to estimate the answer.
Example
12 308 and 12 255 are both close to 12 300. By doubling,
12 300 × 2 = 24 600, so this is a good estimate for 12 308 + 12 255.
Key words
ExERCiSE 3.1
• estimate – to
approximate
the result of a
calculation
1. Estimate the answers by rounding to the nearest 100:
a ) 8 097 + 10 909
b ) 9 034 – 2 114
c ) 11 676 + 14 230
• rounding –
reducing or
increasing a
number so that it
is a multiple of the
number you are
rounding to
12
2. Estimate the answers by rounding to the nearest 1 000. Then use a
calculator to work out the exact answers:
a ) 31 100 – 25 820
b ) 21 807 + 34 654
3. Use doubling to estimate. Find the exact answer on your calculator.
a ) 40 323 + 40 288 b ) 10 209 + 10 210 c ) 89 546 + 89 554
Term 1
Platinum Maths Gr6_Term 1_CAPS.indd 12
09/02/13 1:12 PM
Have fun with calculators
Using a calculator can help make calculations much easier, but it is
important to know how your calculator works. Sometimes you can
press the wrong key on your calculator and get the wrong answer
without realising it. This is why you should always estimate the answer
before you work it out on the calculator.
Calculators are useful for checking answers, and to see how accurate
your estimates are.
Example
Estimate and calculate 14 506 + 6 257.
First round off each number: 14 000 + 6 000.
Then calculate the estimated answer: 14 000 + 6 000 = 20 000.
Now use the calculator to see how close your estimation is.
Since the exact answer is 20 763, your estimated answer is close to
the correct answer.
ExERCiSE 3.2
1. Copy and complete the table below.
Calculation
a)
b)
c)
d)
e)
f)
g)
h)
Rounded
Estimated
answer
Actual
answer
(calculator)
Difference
between the
two answers
5 389 + 4 788
12 477 + 9 099
18 686 + 17 544
45 233 + 39 233
45 233 – 39 233
18 686 – 17 544
12 477 – 9 099
5 389 – 4 788
2. Using the calculator, work out which operation sign (+ or –) needs
to be in each block. For which number sentences could you not
use either + or – ?
a ) 58 610 □ 34 457 = 93 067
b ) 130 567 □ 74 356 = 56 211
c ) 248 344 □ 153 446 = 94 898
d ) 679 488 □ 335 221 = 344 267
Topic 3: Addition and subtraction
Platinum Maths Gr6_Term 1_CAPS.indd 13
13
09/02/13 1:12 PM
Use the column method to add
Key words
• column method
– writing numbers
with the same
place value
underneath each
other to do the
calculation
In Grade 5 you learnt to add larger numbers by using the expanded
vertical column method and the column method.
Example
Expanded vertical column method
Calculate 67 988 + 45 876.
67 988 = 60 000 + 7 000 +
900
45 876 = 40 000 + 5 000 +
800
100 000 + 12 000 + 1 700
= 100 000 + 12 000 + 1 700
= 113 864
+ 80
+ 70
+ 150
+ 150
+ 8
+ 6
+ 14
+ 14
Example
Column method – with no ‘carrying over’
Calculate 54 453 + 2 526.
Write digits with the same place value underneath each other.
54 453
+ 2 526
54 453 + 2 526 = 56 979
56 979
Example
Column method – with ‘carrying over’
Calculate 54 675 + 32 526.
The groups of tens, hundreds
5 14 16 17 5
and thousands are carried over.
+ 3 2 5 26
8 7 2 01
54 675 + 32 526 = 87 201
ExERCiSE 3.3
Estimate the answers. Then use any method above to find the exact
answers.
1.
3.
5.
7.
9.
14
85 269 + 56 912
132 453 + 143 232
564 702 + 33 467
443 560 + 245 439
345 654 + 243 343
2. 188 344 + 144 353
4. 246 322 + 223 433
6. 644 733 + 245 258
8. 23 344 + 185 736
10. 785 467 + 89 453
Term 1
Platinum Maths Gr6_Term 1_CAPS.indd 14
09/02/13 1:12 PM
Use the column method to subtract
You can also use the vertical column method and the expanded
vertical column method to subtract large numbers.
Example
Expanded vertical column method
Calculate 175 687 – 132 443.
175 687 = 100 000 + 70 000 + 5 000 + 600 + 80 + 7
– 132 443 = 100 000 + 30 000 + 2 000 + 400 + 40 + 3
0 + 40 000 + 3 000 + 200 + 40 + 4
= 43 244
Key words
• exchange – break
down larger units
into smaller units
Example
Column method without exchanging numbers
175 687
– 132 443
43 244
Example
Column method with exchanging numbers
Subtract 24 138 from 132 563.
Write digits with equal value
1 23 12 5 56 13
underneath each other
– 24138
10 000 becomes 10 thousands
108425
ExERCiSE 3.4
1. Estimate the answers. Then do the subtractions.
a ) 24 375 – 18 547 b ) 342 534 – 240 768 c ) 756 458 – 346 769
2. The amounts below show how much money eight businesses had
at the start of the year and how much they spent during the year.
Estimate and then calculate how much money is left at the end of
the year.
a ) start: 178 796; spent: 76 563 b ) start: 256 988; spent: 124 755
c ) start: 645 756; spent: 414 542 d ) start: 988 673; spent: 765 453
e ) start: 456 323; spent: 324 121 f ) start: 678 877; spent: 443 216
g ) start: 578 908; spent: 367 901 h ) start: 799 344; spent: 633 144
Topic 3: Addition and subtraction
Platinum Maths Gr6_Term 1_CAPS.indd 15
15
09/02/13 1:12 PM
Key words
• inverse operation
– opposite
operation
Addition and subtraction are inverse
operations
Do you remember that addition and subtraction are opposite
operations? We say that addition is the inverse of subtraction and
subtraction is the inverse of addition. This means that you can
check subtraction calculations with addition, and check addition
calculations with subtraction.
Example
Calculate 403 215 + 232 731.
Use the column method: 4 0 3 2 1 5
+232731
635946
Check: 635 946 – 232 731 = 403 215 or 635 946 – 403 215 = 232 731
Example
Calculate 473 235 – 232 131.
Use columns: 4 7 3 2 3 5
–232131
241104
Check: 241 104 + 232 131 = 473 235
ExERCiSE 3.5
1. Use the column method to find these answers:
a ) 345 765 + 202 387 b ) 766 219 + 34 387 c ) 3 34 938 + 231 493
2. Use subtraction to check your answers in Question 1.
3. Use the column method to find the answers to these subtractions:
a ) 345 765 – 202 387 b ) 766 219 – 34 387 c ) 334 938 – 231 493
4. Do column additions to check your answers in Question 3.
5. Complete these number sentences. Check each answer by
substitution.
a ) 23 405 + □ = 42 000
b ) □ + 350 000 = 500 000
c ) □ – 45 347 = 950 000
16
Term 1
Platinum Maths Gr6_Term 1_CAPS.indd 16
09/02/13 1:12 PM
Addition and subtraction problems
We can use addition and subtraction to solve many different types
of problems in mathematics. Sometimes you need to use both
operations in a multi-step problem. You can also use the inverse
calculations to check your answers.
ExERCiSE 3.6
Use addition or subtraction to solve the following word problems.
Check your answers with an inverse calculation where possible.
1. The population of two towns is 265 846 and 486 720. How much
less than 1 000 000 is this?
Key words
• multi-step – you
need more than
one calculation to
solve a problem
• difference –
the result of
a subtraction
operation
2. A survey found 234 690 people like peanut butter, 190 355 like
jam and 321 100 people like cheese. How many people were
interviewed in the survey?
3. During a rugby contest, 54 342 people watched the first game,
while 49 990 people watched the second game. If 156 330 people
watched all three games, how many watched the third game?
4. What is the difference between nine thousand and the sum of
5 470 and 1 388?
5. A farmer has 2 050 sheep, 1 475 pigs, 4 875 cattle and 478 horses.
If his neighbour has 10 134 animals, how many more animals does
the neighbour have?
6. The difference between two numbers is 350 578. If the first
number is 405 322, what is the second number?
7. Use the facts in the table to answer the questions.
Country
South Africa
Namibia
UK (United Kingdom)
Ukraine
Area in square kilometres (km²)
1 219 912
825 418
244 820
603 700
Length of coastline (km)
2 798
1 572
12 429
2 782
a ) How much greater, in area, is Namibia than the UK?
b ) How much more coastline does the UK have than Ukraine?
c ) How much smaller than 5 000 km is the total coastline of
South Africa and Namibia together?
Topic 3: Addition and subtraction
Platinum Maths Gr6_Term 1_CAPS.indd 17
17
09/02/13 1:12 PM
Topic
4
Common fractions
Maths ideas
• Work with
fractions as parts
of a whole or as
grouping.
• Find equivalent
fractions.
What is a fraction?
A fraction of a whole is one or more parts of that whole. A fraction
like __34 (three-quarters) is the same as saying 3 ÷ 4, so a fraction means
that you are dividing. The denominator 4 means that you are dividing
the whole into 4 equal parts. The numerator 3 means that you have
chosen 3 out of the four equal parts.
• Order and
compare common
fractions.
Example
Imagine that you cut a pie into 4 equal quarters.
If you eat three of the pieces, then you have
eaten __34 of the pie (three out of four equal
pieces).
• Add and subtract
fractions with the
same denominator.
• Find fractions of
whole numbers.
• Solve problems
with fractions.
Make sure you know the following fraction names.
1
__
2
1
__
3
a
half
a
third
1
__
4
1
__
5
1
__
6
1
__
7
1
__
8
1
__
9
a quarter
a
a
a
an
a
(fourth) fifth sixth seventh eighth ninth
1
__
10
a
tenth
Example
Key words
• denominator –
the number below
a fraction line
• numerator – the
number above a
fraction line
3
__
of the rectangle is red; three sevenths; 3 out of 7 blocks
7
4
__
of the rectangle is blue; four sevenths; 4 out of 7 blocks
7
The whole rectangle is __77 blocks or 1 whole.
ExERCiSE 4.1
1. Write down what fraction of each rectangle is red and what
fraction is blue:
a)
e)
b)
c)
d)
f)
2. a) What fraction of this grid is green, what fraction is
blue and what fraction is red?
b) Which colour makes up half of the rectangle?
Why?
18
Term 1
Platinum Maths Gr6_Term 1_CAPS.indd 18
09/02/13 1:12 PM
You can also draw your own fraction diagrams which you can use to
answer fraction questions.
Example
a) What fraction of the orange is each piece?
There are eight equal pieces, so each
piece is __18 (one eighth) of the orange.
b) If you eat 4 pieces, what fraction of the
orange have you eaten?
4 pieces is half of the orange, because
there are 8 pieces and 4 is half of 8.
ExERCiSE 4.2
1. Draw diagrams to show the fraction __23 (two-thirds) in these three
different ways:
a ) This circle has been divided into three equal parts. Copy the
diagram and shade __23 of the circle.
b ) Draw a rectangle made of three equal sized blocks. Shade _23_ of
your rectangle.
c ) Use your ruler to draw a line that is 6 cm long. Divide your line into
three equal lengths. Now use a different colour pen to show __23 of
the length of your line. How many centimetres is __23 of your line?
2. a )
b)
c)
5
6
3
6
2
5
Copy these diagrams into your books and colour the given fraction.
3. Draw a diagram of your choice to show each fraction.
3
(three-tenths)
a ) __14 (one-quarter) b ) __27 (two-sevenths) c ) __
10
4. On the right is a pizza divided into 12 equal slices.
a ) Are eight pieces more or less than half of the pizza? Why?
6
of the pizza.
b ) Copy the diagram and shade __
12
c ) If you eat six pieces, have you eaten half of the pizza? Explain.
3
of the pizza.
d ) Use a different colour to shade __
12
e ) If you eat three pieces, how many quarters of the pizza have
you eaten? Explain.
Challenge
Look at the pizza
above. How many
slices are in __13 of __12 of
the pizza?
Topic 4: Common fractions
Platinum Maths Gr6_Term 1_CAPS.indd 19
19
09/02/13 1:13 PM
Fractions by grouping
A fraction can also represent a group of items.
Example
Zodwa has 20 sweets. She places them on the table in four equal
groups, so that there are 5 sweets in each group. Each group is __14
(one-quarter) of the total, because she has divided the sweets
into four equal groups. Notice that 20 ÷ 4 = 5, so __14 of 20 is 5. If she
eats three of the groups, then she will have eaten 3 × 5 = 15 of her
sweets. This means that __34 of 20 is 15.
ExERCiSE 4.3
1. a ) Draw a diagram to show how you can group 12 coins into four
equal groups.
b ) How many coins are in each group? What fraction of the coins
is in one group?
c ) How many coins are in two of the groups? What fraction of the
money is this?
Challenge
Without a calculator,
find __12 of __13 of __12 of 144.
Can you find the
answer with only one
calculation? If you
can, explain why this
works.
2. Draw diagrams to show these fractions, and then state how many
coins each fraction represents:
a ) __13 of 6, __23 of 6 and __33 of 6
b ) __15 of 10; __25 of 10 and __45 of 10
3. A teacher has 36 pencils to share equally. If there are 12 learners,
what fraction of the pencils will each get? How many pencils will
each get?
Example
How many of 27 oranges would make up __16 of the total? You can
make 6 groups of 4 oranges each, but there will still be 3 oranges
left over. This remainder must also be divided into 6 parts, which
is __36 . The answer is 4 + __36 oranges.
ExERCiSE 4.4
Calculate the following:
1
of 37
1. __
12
20
5
2. __
of 36
12
3. __16 of 19
4. __15 of 32
Term 1
Platinum Maths Gr6_Term 1_CAPS.indd 20
09/02/13 1:13 PM
Sometimes you must find the total before you can work out a fraction
of the whole.
Example
Ismael runs 11 km on Monday, 9 km on Tuesday and 10 km on
Wednesday. To find what fraction of the total distance he ran on
Wednesday, first add the distances: 11 + 9 + 10 = 30. On Wednesday
10
as a fraction.
he ran 10 km out of the total 30 km, which is __
30
ExERCiSE 4.5
1. In a class each learner plays one sport, 4 learners play cricket,
18 are soccer players and 8 play hockey.
a ) How many learners are in the class?
b ) What fraction of the class plays hockey?
c ) What fraction plays cricket?
d ) If half the soccer players are girls, what fraction of the class is this?
Ismael is a keen runner
2. In your group of friends, 3 ride to school on their bicycles, 2 walk
and 5 take the school bus.
a ) How many friends are in your group?
b ) What fraction of your friends walk to school?
c ) How do half of your friends get to school? Explain how you got
your answer.
3. In a game, 12 red balls, 8 purple balls and 4 green balls are used.
a ) How many balls are used in the game?
b ) What fraction of the balls are green?
c ) What fraction of the balls are purple?
d ) How many balls make up half of the total? Can you write the
fraction __12 as a different fraction in this question?
4. A biscuit recipe uses the following ingredients: 3 cups of flour, 1
cup of sugar and 2 cups of chocolate powder.
a ) What fraction of the ingredients is flour?
b ) What fraction of the ingredients is sugar?
c ) What fraction of the ingredients is chocolate powder?
d ) Can you write any of your answers as a different fraction?
One of each of the green,
red and purple balls
5. There are 10 cars in the parking yard at school. Half of the cars are
black, three are white and the rest are red. What fraction of the
cars is red?
Topic 4: Common fractions
Platinum Maths Gr6_Term 1_CAPS.indd 21
21
09/02/13 1:13 PM
Key words
Equivalent fractions
• equivalent
fractions –
fractions with the
same value
By now you might have seen that different fractions may have the
same value. Fractions with the same value are called equivalent
fractions.
Example
This cake is divided into sixths. If you eat 3 slices you have eaten __36
of the cake, but this is the same as eating __12 of the cake. This means
that __36 and __12 are equivalent fractions.
Challenge
• A mystery fraction
is equivalent to __13 .
• If I subtract 3 from
the numerator
of the mystery
fraction and I
subtract 6 from the
denominator, the
value of the new
fraction is half of
the value of the
mystery fraction.
• What is the value
of the mystery
fraction?
ExERCiSE 4.6
This circle is divided into twelfths.
1. How many twelfths make half of the circle?
□
Complete: __12 = 12
2. How many twelfths make a quarter of the circle?
□
Complete: __14 = 12
3. How many twelfths make a third of the circle?
□
Complete: __13 = 12
ExERCiSE 4.7
Use the diagram below to answer these questions:
1. Find all the fractions equivalent to __12 . Do you notice anything
useful about the numbers in these equivalent fractions?
2. Find all the fractions equivalent to __23 . What do
you notice?
3. Complete these equivalent fraction statements:
a ) __13 = □
6
□
b ) __14 = 12
d ) __68 = □
4
□
e ) __12 = 10
2
c ) __
=□
10
5
4. Are these fractions equivalent? If so, add one more equivalent
fraction.
8
3
4
and __23
b ) __
and __15
c ) __
and __13
a ) __
12
10
12
22
Term 1
Platinum Maths Gr6_Term 1_CAPS.indd 22
09/02/13 1:13 PM
By now you might have seen that you can multiply or divide the
numerator and denominator of a fraction by the same whole number
to find an equivalent fraction. We say that fraction is in its simplest
form if no whole number can divide into both the numerator and
denominator.
Example
5
Here are three fractions that are equivalent to __
:
25
5÷5
5×2
10 _____
5×4
20
1 _____
_____
__
__
___
= ;
= ;
=
25 ÷ 5 5 25 × 2 50 25 × 4 100
5 __
10
20
, 1 , __
and ___
are equivalent fractions, and __15 is in
The fractions __
25 5 50
100
its simplest form.
Fraction snap. Play
with a friend:
• Cut out (or fold
and neatly tear) 30
squares of paper
or card.
1. Fill in the missing numbers to form equivalent fractions:
□ □
3
□
__
d ) 4 = 100 = 9
□
• simplest form
– a fraction is in
its simplest form
when no whole
number can divide
into both the
numerator and
denominator
Game
ExERCiSE 4.8
25
= 5 = 1
a ) __
75
Key words
b ) __25 = 10 = 40
□ □
□
18
1
__
e ) 36 = □
= 12
60 □ 15
c ) __
= 6 =
72
□
4
2
__
f ) 52 = = 1
□
□
2. Write down the fractions from question 1 that are in their simplest
form.
3. Divide the numerator and denominator by the same number to
write the following fractions in their simplest form:
56
39
65
36
84
42
b ) __
c ) __
d ) ___
e ) __
f ) __
a ) __
66
84
99
48
100
96
4. Write each of the following fractions as equivalent fractions with a
denominator of 10:
3
12
4
b ) __14
c ) __
d ) __
e ) __15
f ) __25
g ) __
a ) __12
30
30
20
5. Write each of the following fractions as equivalent fractions with a
denominator of 100:
3
1
7
g) __
h) __
a ) __12 b) __14 c ) __24 d ) __34 e ) __15 f ) __
10
10
10
3
5
3
5
25
1
1
1
j ) __
k) __
l ) __
m) __
n) __
o) __
p) __
i ) __
20
20
25
25
50
50
50
50
6. Create 3 fraction chains of your own with 5 different equivalent
fractions in each. Give the simplest form of the fraction in each chain.
• Write one fraction
on each card with
a denominator up
to 12, so that some
of the fractions are
3
,
equivalent, like __
12
2
1
__ and __.
4
8
• Shuffle the cards
and deal 15 cards
each.
• Take turns to place
a card face up on
the desk.
• Whoever sees
two cards with
equivalent
fractions calls
‘Snap’ and takes
the cards that are
on the desk.
• Set a time and see
who has the most
cards at the end.
Topic 4: Common fractions
Platinum Maths Gr6_Term 1_CAPS.indd 23
23
09/02/13 1:13 PM
Compare and order fractions
Key words
• ascending order
– from the smallest
to the greatest
• descending order
– from the greatest
to the smallest
If fractions have the same denominators, it is easy to see which
fraction is greater or smaller.
Example
Which fraction is greater: __25 or __35 ? Because the denominators are the
same and 3 is greater than 2, it is easy to say that __35 > __25 .
If fractions have different denominators, you must first use equivalent
fractions to make the denominators the same.
Example
7
Which fraction is greater: __34 or __
?
12
9
Change the __34 into twelfths, to get the equivalent fraction __34 × __33 = __
.
12
9
7
7
Now it is easy to see that __
> __
, so __34 > __
.
12 12
12
To order fractions from the smallest to the greatest (ascending order)
or from the greatest to the smallest (descending order), first write all
the fractions with the same denominator.
ExERCiSE 4.9
1. Fill in the < or > symbol:
Challenge
You have won a prize
in a raffle, but you
may choose how
much you want to
win. Would you
65
of
prefer to win __
75 __
9
the total prize or 10
of the total prize?
24
89
9
99
7
11
17
b ) __
□ __
c ) ___
□ ___
d ) ___
□ ___
a ) __27 □ __67
10
10
100
100
100
100
2. First make sure the fractions have the same denominators, and
then compare them by filling in <, > or = symbols:
5
b ) __38 □ __
16
9
e ) __
□ __37
21
9
a ) __45 □ __
10
13
□ __23
d ) __
18
11
c ) __56 □ __
12
6
12
f ) __
□ __
11
33
3. Arrange these fractions in descending order, by first writing them
all with a denominator of 48:
5
__
;
6
1
__
;
3
11
__
;
12
19
__
;
24
3
__
;
4
7
__
;
8
11
__
;
24
12
__
24
Term 1
Platinum Maths Gr6_Term 1_CAPS.indd 24
09/02/13 1:13 PM
Add and subtract fractions
If fractions have the same denominators, then you can add and
subtract them easily.
Example
A cake is cut into ninths. You are given 2 pieces (two-ninths) and
then 3 pieces (three-ninths), but you give away 1 piece (one-ninth).
What fraction of the cake do you still have?
You have __29 + __39 – __19 = __49 . You still have four-ninths of the cake.
A mixed number is a number that has two parts: a whole number and
a fraction. If there are mixed numbers, you add or subtract the whole
numbers first.
Key words
Example
• mixed number –
a number that is
made up of two
parts: a whole
number and a
fraction
2__35 + 3__45 – 4__15 = (2 + 3 – 4) + (__35 + __45 – __15 ) = 1 + __65 = 1 + __55 + __15
= 1 + 1 + __15 = 2 __15
Example
6__35 – 2__45 = (6 – 2) + (__35 – __45 ) = 4 + __35 – __45
To subtract __45 from __35 , use 1 = __55 to get more fifths.
4 + __35 – __45 = 3 + __55 + __35 – __45 = 3 + __85 – __45 = 3__45
ExERCiSE 4.10
Use the above methods to do the following additions and subtractions.
Write your answers as mixed numbers with fractions in their simplest form.
1. __35 + __45 – __25
4. 6__34 – 5__14
4 __
7
2. __
– 2 + __
15 15 15
5. 12__12 + 1__12 – 6__12
3. 2__13 + 3__13 + 4__23
6. 1__16 + 3__56 – 2__16
7. 6__35 + 1__45 + 2__15 – 7__25
8. 12__19 – (2__49 + 3__29 ) (Remember to do the brackets first.)
9. 12__19 – 2__49 + 3__29
10. 4__14 – 3__34 + 4__14
Topic 4: Common fractions
Platinum Maths Gr6_Term 1_CAPS.indd 25
25
09/02/13 1:13 PM
Solve problems with fractions
Use your knowledge of fractions to solve these problems.
ExERCiSE 4.11
3
1. There are 600 people in a concert audience. __
are male and __56 are
10
older than 10 years of age.
a ) What fraction of the audience is female?
b ) What fraction of the audience is younger than 10 years of age?
c ) How many people in the audience are female?
d ) How many people are younger than 10 years of age?
2. Jane has walked 2__45 km from her home towards the shops. She still
has to walk a further 1__35 km to get there. What is the total distance
from her house to the shops?
3. We are erecting safety fencing around our swimming pool. The
total length of fencing is 20__38 m. The builders have already erected
two pieces. One is 3__28 m, the next is 4__58 m. How much fencing do
they still need to erect?
4. a ) What fraction of an hour is 40 minutes, in its simplest form?
b ) I sleep 8 hours a night. What fraction of a day is this?
c ) How many hours is __14 of a day? How many hours is __34 of a day?
5. Mila has saved R208. She decides to spend __58 of her money on a
new skirt.
a ) How much did the skirt cost?
b ) How much money has she got left?
6. Thando had 24 sweets. He ate one-quarter of his sweets and gave
half of what was left to a friend.
a ) How many sweets did he eat?
b ) How many sweets did he give to his friend?
7. Joe had R150. He spent __25 of the money on a shirt, and then he
spent __23 of the money that was left to buy a cap.
a ) How much did the shirt cost?
b ) How much did he have left after buying the shirt?
c ) How much did the cap cost?
d ) What fraction of money did he have left after buying the cap
and the shirt?
26
Term 1
Platinum Maths Gr6_Term 1_CAPS.indd 26
09/02/13 1:13 PM
Revision
1. Round to the nearest 100 to estimate the answers to each of the following calculations:
a ) 13 598 + 12 435
b ) 27 987 + 23 743
(1)
(1)
2. Use the column method to do the following additions:
a ) 13 458 + 31 247
b ) 145 832 + 129 305
(1)
(2)
3. Use the column method to do the following subtractions:
a ) 35 789 – 19 997
b ) 456 589 – 234 960
(1)
(2)
4. Use the figures in the table to answer the questions below:
Country
Area in square
kilometres
Length of
coastline (km)
Ghana
Tanzania
Cameroon
239 460
945 087
475 440
539
1 424
402
a ) How much longer is Tanzania’s coastline than Ghana’s?
b ) Find the total area of Ghana, Tanzania and Cameroon.
c ) What is the difference between the area of Tanzania and the area of Ghana?
(2)
(3)
(2)
5. Which of these fractions are equivalent to __46 ?
14
__
;
16
12
__
;
18
22
__
;
33
16
__
22
(2)
6. Write the following as fractions in their simplest form:
14
a ) __
35
(1)
48
b ) __
72
(1)
7. Compare these fractions by filling in >, < or = symbols:
9
a ) __45 □ __
10
5
b ) __38 □ __
12
(1)
(1)
8. Do these calculations:
a ) __25 of 1 575ml
(1)
b ) 17__14 – 13__38
9. __35 of a class of 35 children have not handed their projects in on time. How many
learners handed their projects in on time?
(1)
(2)
Total marks: 25
Revision
Platinum Maths Gr6_Term 1_CAPS.indd 27
27
09/02/13 1:13 PM
Assignment
Different number systems
The need to count gave rise to number systems in different parts of
the world as early as the 4th century. The Incas, Mayans, Babylonians,
Romans, Greeks and Chinese cultures all had number systems.
The Roman number system
1
I
2
II
3
III
11 = XI
4
IV
5
V
6
VI
7
VII
8
VIII
9
IX
10
X
20
XX
23 = XXIII
1. What difficulty would you have using this number system? (2)
2. Write the symbols for the numbers in your home address.
The Roman number syste
m
is based on the Greek
alphabet and originally
consisted of 600 differen
t
symbols. The Romans
discarded many symbols
because they took up to
o
much space on a stone
tablet and took too muc
h
time to carve. Greeks no
w
use only 27 symbols that
they join together to form
numbers.
(3)
Today we use numbers in many different forms, such as
sign language for numbers, binary numbers, identity numbers,
cellphone numbers and serial numbers.
The binary number system
In the binary number system, you multiply by 2 to get to the next
place value column. You then choose the columns that make the
number in a decimal number system.
Decimal Th
H
T
U
3
Binary
8
4
2
1
1
1
2×1+1×1=3
To write 3 as a binary number
you have to use the binary place
values. 3 is 1 × 2 + 1 × 1. Write
a 1 in the 1’s place and a 1 in the
2’s place. 3 in binary is 112.
The small 2 shows the number
is a binary number and not
11 (eleven).
28
Binary
8
4
2
1
system
The binary number
Decimal Th
H
T
U is used in computer
number
programming. Our
2
e of 10
system uses a bas
the
2×1+1×0=2
and therefore uses
inary
digits 0 to 9. The b
What number does 10 in the
es a base
number system us
binary system mean in our
0 and
of 2 and the digits
decimal system?
onic
1 only. Many electr
Write 1 in the second column
ked 1
appliances are mar
and 0 in the f irst column.
‘Off ’.
and 0 for ‘On’ and
So 1 × 2 + 0 × 1 = 2.
1
0
Term 1
Platinum Maths Gr6_Term 1_CAPS.indd 28
09/02/13 1:13 PM
iven
ntry is g
u
o
c
a
izen in
The
Every cit umber at birth.
n
ation
a unique t stores inform
en
at
governm person using th tity
ut that
an iden
(2) abo er. A South Afric
s.
f 13 digit ual
numb
o
s
t
is
s
n
co
ivid
number
its or ind t
ig
d
f
o
p
u
ou
(3)
Each gro information ab
es
digits giv .
on
the pers
3. Write these binary numbers in the decimal number
system:
a ) 101
b ) 110
4. Write these numbers as binary numbers:
a) 1
b) 4
c) 7
identity numbers
Look at the identity number 990117 5134 094.
990117
5
134
0
9
4
The first six digits
are the date of
birth. The year is
first – 1999, then
the month – 01,
and then the day
– 17.
This shows the
person is male or
female.
0 – 4 indicates
female and
5 – 9 indicates
male.
People born on the
same day all have
different digits
here.
This number is
always 0 for
South African
citizens and 1 for
foreigners.
This is always
either an 8 or a 9.
This is a validation
number for
checking authentic
documents.
5. What is the date of this person’s birthday?
(2)
6. Is this person male or female?
(2)
7. How old is this person?
(3)
8. Is this person a South African citizen?
(1)
9. Why does the government need to keep information about its
citizens?
(2)
10. Your identity number is printed on your birth certificate. In a
small group, write each of your identity numbers on a separate
piece of paper and fold them. Mix up the pieces of paper from
all the members of your group. Each person should take one
piece of paper and use the identity number to identify the
person it belongs to.
(3)
Serial numbers are unique
numbers that are placed
on objects for identifying
them. Each serial number
Serial numbers
has details about the
item, such as where it
11. Find the serial number on a bank note and compare it
factured, and a
with another learner’s bank note.
(2) was manu
batch number. We record
12. Why do you think it is important to prevent counterfeit
serial numbers to try to
notes being made?
(2) guard against theft and
13. Record the serial numbers of any three items in your house. (3) counterfeiting. Every
banknote has a serial
Total marks: 30number for this purpose.
Assignment
Platinum Maths Gr6_Term 1_CAPS.indd 29
29
09/02/13 1:13 PM
Topic
Time
5
Maths ideas
Read and write the time
• Read and write
time in 12-hour
and 24-hour
formats.
In previous grades, you learnt to tell time on analogue clocks and
digital clocks. Working accurately with time is one of the most
important skills you will ever learn.
• Convert units of
time.
• Understand the
concept of time
zones.
Example
What time is shown on the clocks below?
11
10
• Read and use
calendars.
• Calculate durations
of time.
Key words
• analogue clocks –
clocks with hands
that point to the
numbers
• digital clocks
– clocks that
show the time by
numbers only
Fourteen minutes
and fifty-nine
seconds past ten
11
10
12
1
2
9
3
8
4
7
6
5
Ten minutes and
fifteen seconds
past one
30
12
1
2
9
3
8
4
7
6
5
Both clocks show 7 hours and
25 minutes. You can read or
write the time as 07:25 or as
25 minutes past 7.
When you write time in 12-hour notation, remember to write a.m. or
p.m. to show the difference between morning and afternoon. When
you write time in 24-hour notation, the day runs from midnight to
midnight, which is 24 hours. This means that 12:00 is 12 noon and
midnight is written as 00:00.
Example
Explain the meanings of these 24-hour times:
• 18:00 reads as 18 hours, or six o’clock in the evening, or 6.00 p.m.
• 08:45 is 8.45 a.m. or quarter to nine in the morning.
Sometimes it is possible to read seconds from clocks. Look at the
clocks on the left.
ExERCiSE 5.1
1. Write the following in 24-hour time:
a) three o’clock in the morning
b) 12 noon
c)
10 p.m.
2. Write the following 24-hour times in 12-hour time:
a)
b)
c)
d)
e)
Term 1
Platinum Maths Gr6_Term 1_CAPS.indd 30
09/02/13 1:13 PM
Convert units of time
You have already learnt about the different units that we use to
measure time. It is important to be able to convert correctly between
the different units of time, especially when you do time calculations.
Remember that a century is 100 years.
•
•
•
•
•
60 seconds = 1 minute
60 minutes = 1 hour
24 hours = 1 day
7 days = 1 week
1 century = 100 years
•
•
•
•
1 month = about 4 weeks
12 months = 1 year
1 year = 365 days
1 leap year = 366 days
Example
b) How many years are there in
994 days?
994 ÷ 365 = 2 years and
264 days left over
1 year = 365 days
There are 2 years in 994 days
There are 1 095 days in 3 years.
with 264 days left over.
a) How many days are there in
3 years?
3 × 365 days = 1 095 days
Key words
• century – 100 years
Did you know?
• A leap year is
a year that has
an extra day in
February.
• Every fourth year is
a leap year.
• Leap years are
multiples of four,
for example, 2012
is a leap year.
FEBRUARY 2012
S M T W T
5 6
7
F S
1
2
8
9 10 11
3 4
12 13 14 15 16 17 18
Example
a) How many years are there in b) How many centuries are there
5 centuries?
in 1 207 years?
5 × 100 = 500 years
1 207 ÷ 100 = 12 with 7
1 century = 100 years
remaining.
There are 12 centuries in
There are 500 years in
1 207 years and 7 years
5 centuries.
are left over.
ExERCiSE 5.2
19 20 21 22 23 24 25
26 27 28 29
Challenge
If you take one
second to count each
number, how long
will it take you to
count to one million
without resting?
Complete the following sentences:
1. There are □ seconds in 5 minutes.
2. There are □ minutes in 400 seconds.
3. There are □ minutes in 28 hours.
4. There are □ hours in 107 minutes.
5. There are □ hours in 2 460 minutes.
Topic 5: Time
Platinum Maths Gr6_Term 1_CAPS.indd 31
31
09/02/13 1:13 PM
Did you know?
Earth completes a full
orbit around the sun
in one year (about
365 days). This
journey around the
sun gives us the four
seasons. In the past,
keeping track of the
seasons was very
important to people
because they needed
to know when to
plant and harvest
crops, and how to
manage their
livestock in response
to the colder climate.
Cultural and religious
practices also
depend on the time
of year, so time was
important.
Challenge
Jen has had only four
birthdays but she is
in Grade 10 at school.
Can you explain how
this is possible?
ExERCiSE 5.3
1. Choose the correct answer from Column B for each statement in
Column A.
Column A
Column B
There are □ in 5 days.
9 weeks
b) There are 63 days in □.
105 days
40 weeks is equal to □.
12 years
a)
c)
d) 15 weeks is equal to □.
e)
There are 144 months in □.
120 hours
about 10 months
2. a ) How many days are there in 1 year?
b ) How many days are there in 2 years?
c ) How many days are there in 3 years?
3. a )
b)
c)
d)
How many years are there in 467 days?
How many years are there in 1 245 days?
How many days are there in 1 leap year?
How many days are there in 3 leap years?
4. Complete these statements:
a ) 1 century = □ years
b ) 8 centuries = □ years
c ) 100 years = □ century
d ) 1 000 years = □ centuries
e ) 25 centuries = □ years
f ) 245 years = □ centuries
5. Roshaan is 3 years old. She is □ days old.
6. Ann spends 14 days visiting her aunt.
a ) Ann spends □ weeks with her aunt.
b ) Ann spends □ hours with her aunt.
c ) Ann spends □ minutes with her aunt.
7. Thandi studies at university for 3 full years.
a ) Thandi is a student for □ days.
b ) Thandi is a student for □ months.
c ) Thandi is a student for about □ weeks.
32
Thandi on her
graduation day
Term 1
Platinum Maths Gr6_Term 1_CAPS.indd 32
09/02/13 1:13 PM
Time zones
A long time ago, communities used their own timekeeping system.
As travel became easier with new railways, the need for standard time
increased to allow for the scheduling of trains.
Sunday
Sunday
The rotation of the earth causes some parts of the world to be in
darkness while others have daylight. This means that time is different
A steam train engine
for different places around the world. The world is divided into 24 time from 1910
zones. The lines that run from the North Pole to the South Pole to divide
time zones are called zone meridians. The Greenwich Meridian, which
passes through London, became the main meridian for timekeeping
where standard time would be read. The zone boundaries are not
straight lines because adjustments were made according to the needs of
the local people living within some time zones.
Time zone map
Example
At the top of each column on the map, you can see that there is a one hour difference
between each time zone – one hour in the day earlier to the west of Greenwich Meridian and
one hour later to the east.
It is 13:00 in Colombia. Find the time in South Africa by counting the zones on the map. Then
add an hour for every zone. South Africa is 7 hours ahead of Colombia, so the time is 20:00.
Topic 5: Time
Platinum Maths Gr6_Term 1_CAPS.indd 33
33
09/02/13 1:13 PM
Did you know?
• In many cultures,
people watched
the stars to find
the time of year.
They also watched
for signs in nature,
such as plants
blossoming or
birds migrating.
• The ancient
Egyptians tracked
a year by watching
the length of the
shadow cast by an
obelisk.
• In England, it
is likely that
Stonehenge was
built for the same
reason. The English
people measured
the time of year by
the angle of the
setting or rising
sun.
ExERCiSE 5.4
1. Look at the map on page 33. Find any two countries that are:
a ) GMT + 1
b ) GMT + 2
c ) GMT
d ) on the Greenwich Meridian
2. Answer these questions based on the map on page 33:
a ) What does it mean for Papua New Guinea and part of Australia
to be in the same zone?
b ) Why does the International Date Line appear on both the left
and the right side of the map?
c ) What does it mean that Australia is divided into three zones?
3. Name a country that is:
a ) 2 hours ahead of South Africa in time
b ) 1 hour behind South Africa in time
c ) 7 hours ahead of South Africa in time
d ) 6 hours behind South Africa in time.
4. Use the time zones to work out the answers.
a ) It is 08:00 in Sweden. Calculate the time in China.
b ) It is 14:00 in Madagascar. Calculate the time in Mauritania.
5. Use the time zones to work out the answers.
a ) It is 13:00 on Wednesday in Papua New Guinea. Find the day
and time in South Africa.
b ) It is 22:00 on Tuesday in Angola. Find the day and time in
Venezuela.
6. a )
Stonehenge in
England
South Africa
11
12
11
1
10
8
7
6
5
12
3
9
4
8
11
1
10
2
9
34
Mongolia
How many hours is Columbia behind South Africa? How many
hours is Mongolia ahead of South Africa?
b)
China
Angola
Sweden
Challenge
The time is 02:00 on a
Sunday in Venezuela.
Calculate the time
and day in Papua
New Guinea.
Colombia
6
5
1
10
2
7
12
3
9
4
8
2
3
4
7
6
5
How many hours is Angola behind China? How many hours is
Sweden ahead of Angola?
Term 1
Platinum Maths Gr6_Term 1_CAPS.indd 34
09/02/13 1:13 PM
interpret calendars
Key words
A calendar date consists of the day of the month, the month and
the year. You should be able to calculate time intervals using your
knowledge of calendars.
• time interval –
length of time that
passes
Example
Calculate how much time passes from 12 June at 09:00 until 23 July
at 18:00.
Number of days in June (until 30 June) = 18
Number of days in July = 23
Total number of days = 41 days
Time in hours: 09:00 until 18:00 = 18 hours – 9 hours
= 9 hours
The time interval is 41 days and 9 hours.
ExERCiSE 5.5
1. Look at a calendar for this year. How many months and days pass
from one date to another?
a ) 4 March until 9 September b ) 23 January until 7 July
c ) 12 October until 9 December d ) 15 March until 15 November
e ) 23 June until 31 December
2. Look at the calendar for January of a certain year. Answer the
questions that follow.
a ) How many weeks are there in January?
b ) On what day of the week does 12 January fall?
c ) Luke has a birthday on 25 January. On which day does it fall?
d ) How much time
passes from 1 January
until 23 January?
S
M
T
W
T
F
S
e ) How much time
1
passes from 06:00 on
2
3
4
5
6
7
8
5 January until 14:00
9
10 11 12 13 14 15
on 7 January?
16 17 18 19 20 21 22
f ) How much time
23 24 25 26 27 28 29
passes from 07:13 on
30 31
12 January until 23:45
on 29 January?
JANUARY
Did you know?
Different cultures use
different calendars.
We call the Ethiopian
calendar the Ge’ez
calendar. The Ge’ez
calendar is 7 years
and 8 months behind
the Gregorian
calendar that we use
in South Africa. The
year 2001 in Ethiopia
began on 11
September 2008,
according to our
calendar.
Topic 5: Time
Platinum Maths Gr6_Term 1_CAPS.indd 35
35
09/02/13 1:13 PM
Topic
Topic
Properties of 2D shapes
6
Maths ideas
• Identify and name
2D shapes.
• Identify sides and
angles.
• Investigate the
properties of 2D
shapes.
• Identify differences
between
rectangles and
parallelograms.
• Draw 2D shapes
on grid paper.
• Compare and sort
2D shapes.
identify and name 2D shapes
Two-dimensional (2D) shapes have two dimensions, length and
width. The sides of 2D shapes can be curved or straight. Some
2D shapes have special names. A circle is a round 2D shape. A
polygon is a closed shape formed by straight lines. We identify a
polygon by the number of sides it has. Some polygons have all sides
equal in length and all angles equal in size.
A triangle has three sides. Here are four different triangles.
All three sides are
equal.
Two sides are equal.
One angle is a right
angle.
All three sides have
different lengths.
A quadrilateral has four sides. Some quadrilaterals have special names.
Parallelogram
Kite
Trapezium
Rectangle
Square
Some polygons have more than four sides.
Key words
• two-dimensional
(2D) shapes –
shapes made up of
length and width
only
• dimensions – the
lengths of the
sides of a shape
• polygon – a 2D
shape enclosed
by three or more
straight sides
• circle – a 2D shape
in which all points
are the same
distance from the
centre of the shape
36
Pentagons are
polygons with five
sides.
Hexagons are
polygons with six
sides.
Septagons or
heptagons are
polygons with seven
sides.
Octagons are
polygons with eight
sides.
ExERCiSE 6.1
Give the correct name of each shape below. State which polygons
have all sides equal and all angles equal. You may need to measure
the lengths of the sides to be sure.
1.
6.
2.
3.
7.
11.
12.
8.
13.
4.
5.
9.
14.
10.
15.
Term 1
Platinum Maths Gr6_Term 1_CAPS.indd 36
09/02/13 1:13 PM
Angles and lines in 2D shapes
When we work with 2D shapes, it helps if we can identify different
types of angles. We classify angles according to their size. A right
angle is like the corner of a page, or the corners of a classroom door.
Angles that are smaller than right angles are called acute angles.
Angles larger than right angles are called obtuse angles.
These angles are obtuse angles.
An acute angle is
smaller than a right
angle.
The corner of a square
fits exactly into a right
angle.
An obtuse angle is
larger than a right
angle.
These angles are acute angles.
Key words
• angle – the
amount of turn
around a fixed
point enclosed by
two lines which
meet each other
• right angle – an
angle like the
corner of a page
A revolution is a full turn.
A straight angle is like half a turn, or
half a revolution. If you face the front
of the class and then turn to face the
back of the class, you have turned
through one straight angle.
• acute angle – an
angle smaller than
a right angle
• obtuse angle – an
angle that is larger
than a right angle
A reflex angle is bigger than
a straight angle.
• straight angle –
half a turn, forming
a straight line
• reflex angle – an
angle bigger than
a straight angle
ExERCiSE 6.2
Identify the types of angles below. You can use the corner of a page to
check for right angles, acute angles and obtuse angles.
• revolution – a full
turn
8
2
7
11
1
5
3
4
12
6
9
10
Topic 6: Properties of 2D shapes
Platinum Maths Gr6_Term 1_CAPS.indd 37
37
09/02/13 1:13 PM
Parallel lines are straight lines that are always the same distance
apart, like the lines of a railway track. Can you see that the opposite
sides of a rectangle are parallel to each other? The sides of a triangle
can never be parallel, because they always touch at a corner. A
parallelogram is a quadrilateral with opposite sides parallel.
Example
We can use the squares on grid paper to draw
2D shapes. The two parallel sides of a 2D shape
are opposite each other and fall on the grid
lines.
b)
a)
Rectangle
• parallel lines –
straight lines that
are always the
same distance
from each other
Parallelogram
ExERCiSE 6.3
• parallelogram – a
quadrilateral with
opposite sides
parallel and equal
in length
Rectangle
Square
Parallelogram
Square
The three shapes above are all parallelograms because their opposite
sides are parallel. A square and a rectangle are parallelograms with
right angles. Other parallelograms do not have right angles, but the
opposite angles are equal to each other.
Key words
Name of
quadrilateral
c)
1. Copy the quadrilaterals from the above example onto square grid
paper and cut them out.
2. Work in small groups to investigate the properties of your cut out
quadrilaterals. Record your findings in a table like the one given
below:
Number of
parallel sides
Number of
equal sides
Number of right
angles
Number of
equal angles
2 pairs of parallel sides
3. Here is a hexagon with equal sides and equal angles. Are the
opposite sides parallel? Is this shape a parallelogram? Explain.
Challenge
Try to guess the name of each quadrilateral correctly by the clues given.
1. A quadrilateral that has two pairs of parallel sides but no right angles.
2. A quadrilateral that has four right angles but is not a square.
3. A quadrilateral that has two pairs of equal sides but is not a rectangle.
38
Term 1
Platinum Maths Gr6_Term 1_CAPS.indd 38
09/02/13 1:13 PM
Compare and sort 2D shapes
A polygon has the same number of sides, angles and vertices. Vertices are
the corners of a polygon, where two sides meet. The singular of vertices is
a vertex. To compare two polygons, you should list the features that are
the same in both polygons and also the features that are different.
Example
Compare a square
and a pentagon with all sides equal
.
Similarities: Both are polygons with all the sides the same length
and all the angles the same size.
Differences: A square has four equal sides and four angles that are
right angles. It has four vertices. The pentagon has five
equal sides and five angles that are obtuse angles. It
has five vertices.
Key words
• vertex – a corner
point where two
sides of a polygon
meet
• vertices – the
plural of vertex
ExERCiSE 6.4
Game
1. List the similarities and differences of:
a ) a square and a rectangle
b ) a rectangle and a parallelogram
c ) a hexagon with equal sides and a square.
2. Fill in the missing words in each of the sentences below:
a ) A pentagon has ________________ more vertex/vertices than
a square.
b ) A hexagon has twice as many sides as a ________________.
c ) A parallelogram has half as many angles as a
________________.
d ) The sum of the number of vertices of a triangle and the
number of vertices of a pentagon is the same as the number
of vertices of an ________________.
3. Name quadrilaterals that have:
a ) two pairs of equal sides
b ) four equal angles
c ) two acute angles and two obtuse angles.
4. Name three different quadrilaterals that have two pairs of
parallel sides.
• In pairs, take
turns to make a
statement about
a quadrilateral,
like ‘has four equal
sides’.
• Each player must
write down all the
quadrilaterals that
have this property,
and then discuss
your answers
together.
• Score plus 1 for
each correct
answer and minus
1 for each incorrect
answer.
• After an agreed
time, see who has
the highest score.
Topic 6: Properties of 2D shapes
Platinum Maths Gr6_Term 1_CAPS.indd 39
39
09/02/13 1:13 PM
Draw and identify 2D shapes
ExERCiSE 6.5
1. On grid paper, draw a polygon with five vertices.
a ) How many sides does your polygon have?
b ) What is the name of your polygon?
c ) Does your polygon have all sides equal?
d ) Did everyone in the class draw the same polygon? How are
they the same or different?
2. Is it possible to have each of the following 2D shapes? If you think a
shape is possible, draw a neat clear sketch of the shape on grid paper.
a ) a parallelogram with only one right angle
b ) a triangle with one right angle
c ) a triangle with two angles larger than a right angle
d ) a pentagon with two right angles
e ) a quadrilateral with all four angles smaller than a right angle
f ) a parallelogram with exactly two right angles
g ) a quadrilateral with two angles larger than right angles
h ) a parallelogram with opposite angles that are not equal
3. Look at all the real life examples of two-dimensional shapes
that you may be familiar with. Identify each of the shapes in the
pictures. Draw them on square dotted grid paper and then explain
where you would see each of these objects or what the purpose of
the object is.
b)
a)
c)
40
d)
Term 1
Platinum Maths Gr6_Term 1_CAPS.indd 40
09/02/13 1:13 PM
Example
To draw two parallel lines, use a ruler and a set square:
Step 1: Use your ruler to draw a straight line on one of the grid lines.
Step 2: Without moving your ruler, place your set square firmly against a short edge of your
ruler. Then slide your ruler up along the edge of your set square.
Step 3: Remove your set square. Use the new position of your ruler on the grid to draw a
second straight line.
Step 1
Step 2
Step 3
If you worked carefully, you will have drawn two parallel lines.
Example
To draw a right angle, follow these steps:
Step 1: Use your ruler to draw a straight line on one of the grid
lines.
Step 2: Place your set square on top of your ruler at the end of
the line.
Step 3: Draw a second straight line up against the vertical edge of
your set square.
If you worked carefully, you will have drawn a right angle. We say
that the two lines are at right angles to each other.
ExERCiSE 6.6
a ) Measure the length of each line in this diagram.
b) Identify all parallel lines and right angles in the
diagram.
c ) What 2D shapes do you see in the diagram?
d) Draw parallel lines and right angles to copy the
diagram accurately.
Topic 6: Properties of 2D shapes
Platinum Maths Gr6_Term 1_CAPS.indd 41
41
09/02/13 1:13 PM
Example
To draw a rectangle, you need to know the length and the width of
the rectangle. Draw a rectangle with a length of 9 cm and a width
of 6__12 cm:
Step 1: Draw a straight horizontal line exactly 9 cm long.
Step 2: At both ends of the line, draw a line exactly 6__12 cm long, at
right angles to the first line.
Step 3: Join the ends of the two 6__12 cm lines with another 9 cm
horizontal line.
Step 1
Step 2
ExERCiSE 6.7
1. Draw two parallel lines, one 6 cm long and the other 10 cm long.
Join the ends of your parallel lines. Name the 2D shape that you
have drawn.
Step 3
2. Draw a straight line exactly 8 cm long. At one end of this line, draw
a line of 6 cm at right angles to the first line. Join the ends of your
lines. Name the 2D shape that you have drawn.
3. Draw a rectangle with a length of 8 cm and a width of 4__12 cm.
4. Draw a square with a side length of 6 cm.
5. a ) Draw a parallelogram with the longer sides 6 cm, and the
shorter sides any length you choose . The length of these sides
should be the same, so use the grid squares to help you.
b ) What do you notice about the angles of a parallelogram?
6. Draw a line of 6__12 cm.
a ) At each end of the line, draw a line of 4 cm at right angles to
the first line. Are these two lines parallel to each other? Why or
why not?
b ) Join the ends of the two lines you drew in a. What 2D shape
have you drawn?
42
Term 1
Platinum Maths Gr6_Term 1_CAPS.indd 42
09/02/13 1:13 PM
Revision
1. Give the following 24-hour times in 12-hour time (a.m or p.m.):
a ) 15:30
b ) 3:45
(1)
(1)
2. Read and write the times shown on each clock face:
(2)
a)
b)
11
10
12
1
9
8
2
3
7
6
5
4
3. Write ‘a quarter past nine in the evening’ in 24-hour time.
(1)
4. Complete the following statements:
a ) □ weeks = 63 days
b ) □ seconds = 18 minutes
c ) 1 leap year = □ days
d ) 50 years = □ decades
(1)
(1)
(1)
(1)
5. Taliep left South Africa to work in Australia on 1 February 2008. He returned on
9 January 2010. How many years and days was he away from home?
(4)
6. List three similarities between rectangles and parallelograms.
(3)
7. Use grid paper to draw the following polygons, if they are possible. If they
are not possible, explain why.
a ) a quadrilateral with five vertices
b ) a pentagon with two right angles
c ) an octagon with two reflex angles
(2)
(2)
(2)
8. Use grid paper to draw a square with sides 7 cm in length.
(1)
9. Give the name of a shape that has:
a ) only straight sides
b ) only straight sides with all sides equal.
(1)
(1)
Total marks: 25
Revision
Platinum Maths Gr6_Term 1_CAPS.indd 43
43
09/02/13 1:13 PM
Topic
7
Data handling
Maths ideas
Collect and organise data
• Use tallies, tables
and questionnaires
to collect and
organise data.
You already know that you can collect data by counting things, by
asking questions and by doing surveys. You also know that you can
use tallies and tables to organise the data that you collect.
• Draw pictographs
and bar graphs to
represent data.
• Read and interpret
data that is
presented in
different ways.
Example
Zodwa and Nikki collected information about the clubs that the
learners in their class belong to. They used this table to collect and
organise the data.
Club
Tally
Total
Music
12
Drama
14
Choir
6
Chess
7
Key words
Environment
1
• table –
information
arranged in rows
and columns
None
5
Total
45
• Summarise data.
• Work out the
mode and median
of data sets.
• questionnaire –
a form for asking
questions and
recording answers
• data – information
that you collect
44
You can also collect data using a questionnaire. A questionnaire
is a form that has a list of questions printed on it. People fill in the
questionnaire by ticking blocks, circling their choices or writing answers.
Example
Anita is worried about the amount of litter left in the school
grounds after break each day. These are the questions she asks
herself:
• Are other learners worried about it?
• Does the school need more bins?
• Would they be prepared to do something about the litter?
• What else can the school do to stop littering?
Term 1
Platinum Maths Gr6_Term 1_CAPS.indd 44
09/02/13 1:13 PM
She draws up this questionnaire to collect the data she needs.
Litter questionnaire
Please tick the correct box.
1. Is litter a problem at our school? Yes □ No □ Don’t know □
2. Does our school need more bins? Yes □ No □ Don’t know □
3. Would you take part in a project to stop littering? Yes □ No □ Don’t know □
4. What can the school do to stop littering? (Tick more than one if necessary).
More bins
□ Close the tuck shop
□
Fine people who litter
□ Punish people who litter
□
Elect litter monitors
□ Start a recycling project
□
5. Do you have any other suggestions for stopping litter? Write them here.
________________________________________________________________________
ExERCiSE 7.1
1. Do you think Anita’s questionnaire is a good one? Why?
Did you know?
2. Work with a partner. You are going to collect data to find out what
Grade 6 learners think about homework.
a ) Write down three questions that you could investigate.
b ) Design a short simple questionnaire to collect data to answer
your questions.
c ) Discuss how you could use your questionnaire to collect the
data and how you could organise and represent the data.
The government uses
questionnaires to
collect data about
every person in the
country during a
census. In South
Africa, a census is
taken every 10 years.
The most recent
census was in 2011.
3. A newspaper reported that learners as young as Grade 6
sometimes experiment with smoking. Denise and Ahmed decide
to find out if this is a real problem in their class of 45 learners.
a ) What questions do you think they should ask?
b ) Draw up a simple questionnaire that they could use to collect
the data they need to answer their questions.
4. a ) Draw up a questionnaire and use it to find out whether
learners in your school usually bath, shower or use other
methods of washing themselves.
b ) Use your questionnaire to collect data from at least 20 learners.
c ) Organise your results in a table.
d ) Draw a suitable graph to display your results.
e ) Write one sentence describing what you found out.
Topic 7: Data handling
Platinum Maths Gr6_Term 1_CAPS.indd 45
45
09/02/13 1:13 PM
Read and draw pictographs
Pictographs use symbols, or pictures, to show data. A key tells you what
the symbols represent. You need to use the key to interpret a pictograph.
Example
This pictograph shows the number of young children in different parts of the world who do not
attend primary school.
The children of primary school age who do not receive education
Industrialised countries
Key:
South Asia
2 000 000
West/Central Africa
1 000 000
Eastern/Southern Africa
Middle/North Africa
500 000
East Asia/Pacif ic
100 000
According to the key, how many children don’t attend primary school in industrialised
countries?
1 000 000 + 500 000 + 200 000 = 1 700 000 or 1,7 million children.
Key words
ExERCiSE 7.2
• pictograph – a
graph that uses
symbols and a key
to represent data
• key – an
explanation of
what the symbols
mean
Learner numbers in
each Grade 6 class
Number of
Class
learners
6A
6B
6C
6D
6E
6F
6G
46
16
24
18
21
20
22
20
Term 1
Platinum Maths Gr6_Term 1_CAPS.indd 46
1. Use the key in the above example to represent the number 7,3 million.
2. Thando drew this pictograph showing how many tickets were sold
for the school concert:
Day
Number of tickets sold
Monday
Tuesday
Wednesday
Thursday
Friday
Saturday
Sunday
Key:
= 120 tickets
a) How many tickets were sold
for the Saturday show?
b) Each ticket costs R65. How
much money did the school
make on ticket sales on
Thursday night?
c ) How many seats did the
school sell in total during the
week?
d) What was the total amount
of money made by the
school?
3. Draw a pictograph to show the data in the table on the left. Give
your graph a heading. Choose a suitable symbol and remember to
draw a key.
09/02/13 1:13 PM
Read and draw bar graphs
Bar graphs are a useful way to compare sets of data.
Rap
0
Type of music
Key words
Double bar graphs
A double bar graph has two bars for each item in the data. Each bar
shows a sub-group of the data item. A double bar graph has a key to
show what the bars represent.
Example
Favourite type of music
10
8
• bar graph – a
graph with
a vertical or
horizontal bars
each representing
one set of data
• double bar graph
– a bar graph with
two bars per set of
data
6
4
Gospel
Rap
Pop
0
Rock
2
Hip Hop
Number of Grade 6 learners
Luke wanted know if there was
a difference between the music
that boys liked and the music
that girls liked. He used his
earlier data to draw this graph
to show the data for boys
and girls.
• Hip hop, rock and rap
music was more popular
with girls than boys.
• Pop and gospel music was more
popular with boys than girls.
• Very few girls chose pop music
compared with boys.
• Very few boys chose rock music
compared with girls.
5
Type of music
Key:
Boys
Girls
Topic 7: Data handling
Platinum Maths Gr6_Term 1_CAPS.indd 47
Gospel
A total of 45 learners took part in the survey.
10
Pop
The horizontal axis shows the type of music. The vertical axis
shows the number of learners that chose each type. If you read
the data, you will see that most learners liked hip hop. Rap
music was least liked because only six learners liked rap. You
add the number of learners to find out how many learners took
part in the survey.
Favourite type of music
Rock
Luke did a survey to find out which type of music was most
popular in Grade 6. He asked each learner to choose only one
type of music. The bar graph shows the results of the survey.
Number of Grade 6 learners
15
Hip hop
Example
47
09/02/13 1:13 PM
ExERCiSE 7.3
1. The table below shows how much a school paid for water each
month for five consecutive months:
Month
Amount paid (R)
a)
b)
c)
d)
e)
Type of book
Poetry
Non-f iction
Biography
Science f iction
Adventure
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
Drama
Number of readers
Reading preferences
March
April
May
June
July
300
390
350
420
420
Draw a bar graph to illustrate this data.
In which month(s) did they pay the most?
In which month(s) did they pay less than in the previous month?
Suggest one reason why they may have paid less in this month.
In which two months did they pay the same amount per month?
2. Look at the bar graph on the left.
a ) What information is shown on the graph?
b ) Draw a tally table to show the information that is on the
graph.
c ) How many readers took part in this survey?
d ) What type of books do most people enjoy reading?
e ) How many people like reading science fiction?
f ) How many more people read drama than poetry?
g ) The librarian needs to order more books. Which type of
books do you suggest she order? Explain your answer.
3. The table below shows the number of girls and boys in Grade 6
involved in five different clubs at school:
Club
Choir
Maths
Chess
Community outreach
Environment
Girls
4
3
3
8
2
Boys
3
0
7
4
2
a ) Draw a double bar graph that separately shows the number of
Grade 6 girls and boys involved in each club at the school.
b ) Which club has the highest number of Grade 6 learners? Why
do you think this might be so?
c ) Does this table tell us that there are 36 learners in this Grade 6
class? Explain your thinking.
4. Carry out a survey to find out what clubs girls and boys in your
class belong to. Draw a double bar graph to show your results.
48
Term 1
Platinum Maths Gr6_Term 1_CAPS.indd 48
09/02/13 1:13 PM
The mode and median in a set of
data
Last year you learnt that the mode is the number or item that appears
most often in a set of data.
Example
Ms Smith recorded these test marks for Mathematics:
Wanita
Musa Themba Fadiela Thandi
15
18
15
12
18
Leroy
Nazley
Lana
Lila
Mandla
15
15
16
19
18
Find the mode of the test marks for Mathematics.
15 is the mark that appears most often. It appears four times.
So 15 is the mode of this data set.
• A data set may have more than one mode. For example, in
the data set 2; 2; 3; 3; 4; 5, the numbers 2 and 3 appear twice.
Therefore 2 and 3 are both modes of this data set.
• If no items in the data set appear more than once, then there is
no mode for the data set. For example 2; 3; 4; 6; 8; 9; 10 has no
mode.
The median is the middle number in an ordered data set.
These are the rules to find the median:
• First arrange the numbers in ascending order.
• Select the middle number.
Key words
• mode – the
number or the
data item that
appears most
often in a data set
• median – the
middle number in
an ordered data
set
Example
Mthunzi is collecting money for the hospice.
He collects the following amounts:
R5; R12; R3; R9; R17; R10; R8
He wants to find the median of the amounts he collected, so he
writes the amounts from smallest to greatest (in ascending order):
R3; R5; R8; R9; R10; R12; R17
The number in the middle of the data is R9.
So the median amount of money Mthunzi collected is R9.
Mthunzi collecting
money
Topic 7: Data handling
Platinum Maths Gr6_Term 1_CAPS.indd 49
49
09/02/13 1:13 PM
ExERCiSE 7.4
1. Find the mode and the median of each set of numbers:
a ) 5; 7; 4; 2; 3; 5; 6; 3; 5; 6; 5
b ) 12; 12; 11; 12; 12; 13; 14
c ) 21; 15; 12; 19; 13; 14; 16; 17; 18
d ) 34; 23; 23; 31; 27; 23; 19; 23; 24
e ) 10; 14; 8; 12; 6; 10; 12; 12; 8
f ) 6; 6; 5; 3; 4; 5; 6; 7; 5; 2; 8
Did you know?
It is estimated that it
takes 1 100 litres of
water to produce one
hamburger, and 145
litres of water to
produce one can of
fizzy cold drink.
2. The Kumar household used the following amounts of water in 7 days:
Monday Tuesday Wednesday Thursday Friday Saturday Sunday
180 ℓ
220 ℓ
150 ℓ
260 ℓ
230 ℓ
180 ℓ
250 ℓ
a ) On which day did the Kumar household use the most water?
b ) Find the median amount of water that they used in 1 week.
c ) Why do you think the water usage of a family differs from day
to day?
3. These are some results of a Mathematics test:
Kerry Tim
16
Pete
Ntombi
Bev
Lara
Sonny
Siwe
Andrew
Thabang
Rajan
16
16
11
12
10
16
11
16
14
13
a ) Find the median result.
b ) Find the mode of the test results.
Challenge
Find the median of
this population data
from mid-2011:
Botswana 2 300 500
Lesotho 2 200 100
Namibia 2 350 000
South Africa 50 500 200
Swaziland 1 300 500
Sudan 44 600 900
Egypt 82 690 000
Mauritius 1 305 000
Zimbabwe 12 150 000
Malawi 15 980 900
50
4. Zama measured the air temperature each day for a week:
Monday Tuesday Wednesday Thursday Friday Saturday Sunday
28 °C
29 °C
27 °C
32 °C
30 °C
29 °C
31 °C
a ) What was the modal temperature?
b ) What was the median of these temperatures?
c ) In which season do you think Zama measured these
temperatures? Why do you say so?
5. Find the modal age of learners in your class, and also which
month is the mode of their birth months. Write a short paragraph
explaining:
a ) how you collected the data
b ) how you organised the data.
Term 1
Platinum Maths Gr6_Term 1_CAPS.indd 50
09/02/13 1:13 PM
interpret and make sense of graphs
To interpret, or understand data, we study graphs and tables to find
patterns in the data. We then look for factors that might influence or
cause these patterns. However, you must remember that you cannot
always be sure exactly what is causing the results.
Example
The pictograph shows the number of pieces of litter found in a school’s grounds before, during
and after the school introduced an anti-litter project. Write a paragraph describing what the
graph shows.
BEFORE
Day 1 - 5
DURING
Day 6 - 7
Key:
= 5 pieces
AFTER
Day 11 - 15
The graph shows that from day 1 to day 5, learners littered a lot. The anti-litter project
started on day 6 and learners responded immediately. The result was that there was
very little littering from day 6 to day 10.
The data also shows that from day 11, littering increased. This suggests that learners
probably forgot about the anti-litter project and started to litter again.
Example
The pie graph shows how South Africa uses its
water supply.
Write a short paragraph summarising what the
graph shows.
This graph shows that most of the water is used for
irrigation. Industry, power stations and mines use about
1
__
of the water supply. Homes and municipalities use less
4
than a quarter. Irrigation is the biggest sector because
South Africa is quite a dry country and farmers have to
irrigate their crops to make them grow well.
How water is used in South Africa
Mining
Electricity
Industry
Homes
and
municipality
Irrigation
Topic 7: Data handling
Platinum Maths Gr6_Term 1_CAPS.indd 51
51
09/02/13 1:13 PM
ExERCiSE 7.5
1. The infant mortality rate of a country tells you
how many children die before they reach the
= 5 deaths
age of five. The figures are given per thousand
= 1 death
children born alive. This pictograph shows the
infant mortality rates for South Africa and her
neighbours.
a ) Which country has the highest infant
mortality rate?
b ) Which country has the lowest infant mortality rate?
c ) What is the median mortality rate?
d ) Suggest one possible factor that might help to cause a higher
infant mortality rate in Lesotho than in South Africa.
Key:
Botswana
= 10 deaths
South Africa
Lesotho
Namibia
Zimbabwe
Mozambique
Swaziland
Main causes of death in children under 5 (South Africa, 2010)
Infection or
meningitis
2. This pie graph shows the main causes of death in children
under the age of five in South Africa.
a ) What is the main cause of death in children under five?
b ) What fraction of under-fives dies from birth complications?
c ) The Medical Research Council reports that 75 000 children
under the age of five die each year. Use the graph to
estimate how many of these children die from diarrhoea.
d ) How many under-fives are killed accidentally by other
children?
Injuries
Accident
involving
other
Diarrhoea child
Birth
complications
leading
to death
Pneumonia
HIV and AIDS
3. This bar graph shows the top ten causes of
death in children in South Africa in 2010 and
how many children died from each cause.
HIV/AIDS
Low birth weight
Cause of death
Diarrhoea
Pneumonia
Malnutrition
Write a paragraph summarising what this graph
shows you.
Newborn infections
Birth complications
Road accidents
Heart diseases
Major causes of death in children under five (world 2008)
Fires
0
5
10
15
25
20
30
35
Deaths (thousands)
4. This pie graph shows the main
causes of death in under fiveyear-olds for the whole world.
Other infectious and
parasitic diseases
HIV/AIDS
Measles
Malaria
Compare this pie graph with
the graph for similar statistics in
South Africa.
a ) How is the data similar?
b ) How is the data different?
c ) Summarise the main differences.
52
Noncommunicable
diseases (postneonatal)
Injuries (postneonatal)
Diarrhoeal
disease
(postneonatal)
Neonatal
deaths
Acute respiratory
infections
(postneonatal)
Source: WHO The Global burdan of disease. 2004 update (2008)
Term 1
Platinum Maths Gr6_Term 1_CAPS.indd 52
09/02/13 1:13 PM
Work through a data cycle
Read this diagram to remind
yourself what steps you need
to follow to complete a
data cycle on your own.
1. Ask a
question
2. Collect data
to answer
question
3. Record and
organise the
data
5. Interpret and
summarise
the data
4. Represent
the data
graphically
ExERCiSE 7.6
Key words
In this exercise you will find out how much water is used by different
families in your class each day. You will use this to work out how much
each family uses in a month. Collect your data from at least five learners.
• data cycle – the
process of asking
a question,
collecting and
organising data
and summarising
the results
The table shows how much water is used for different activities.
Amount of water used for everyday activities
Flushing toilet 8 ℓ
Washing hands 2 ℓ
Brushing teeth 1 ℓ
Washing yourself 4 ℓ
5-minute shower 30 ℓ
Washing dishes 5 ℓ
Cooking 2 ℓ
A bath 150 ℓ
A glass of water 250 ml
1. Plan your survey and decide how you will collect the data. You can
use a questionnaire or a table. Draw this up in detail.
2. Carry out your survey and choose between five and ten learners to
answer your questions. Choose a mixture of boys and girls.
3. Organise and record your data.
Use a table to summarise your data. Your table could look
something like this:
Learner
Water used in a day
Water used in a month
4. Draw a bar graph to show how much water each family uses.
5. Write a paragraph summarising and interpreting your data.
Topic 7: Data handling
Platinum Maths Gr6_Term 1_CAPS.indd 53
53
09/02/13 1:13 PM
Topic
8
Numeric patterns
Maths ideas
• Use flow diagrams
to demonstrate:
• inverse
operations
• associative
property
• multiplication
strategies.
• Use multiple
operations in flow
diagrams.
Key words
• flow diagram – a
diagram to show
how a number
changes after a
calculation
• input number
– the number
that goes into a
calculation
• output number
– the number that
comes out of a
calculation
• rule – a number
calculation that
changes an input
number to an
output number
Flow diagrams
A calculation usually changes a number. You can show a calculation
in a flow diagram. The number that you start with is called the input
number, and the answer to the calculation is called the output
number. The calculation is also called the rule for the flow diagram.
A rule can have more than one operation.
Example
Here is a flow diagram, with the
2
Rule
input numbers on the left, the rule 4
in two instruction boxes, and the
6
–1
×3
output numbers on the right.
8
You apply the rule in the correct
order to get the output numbers. 10
(2 × 3) – 1 = 6 – 1 = 5; (4 × 3) – 1 = 12 – 1 = 11; (6 × 3) – 1 = 17;
(8 × 3) – 1 = 23; (10 × 3) – 1 = 29
5
11
17
23
29
You can describe a rule in different ways.
Example
Here is a rule in words: “add 2 and
multiply by 6”.
You can also write the rule as a number
statement, using brackets to show
which operation is done first: (□ + 2 ) × 6.
You can show the same rule in a flow diagram.
Rule
+2
×6
ExERCiSE 8.1
1. Write each rule in two different ways, and then find the output
numbers from the input numbers 2, 4 and 10.
a ) multiply by 4
b ) + 30
d)
c)
Rule
Rule
×10
+2
×6
2. Waheeda says she thinks that the rule “×3 then +2”, is the same
rule as “+ 2 then × 3”. Use a number statement with 10 as the input
to see if she is correct.
54
Term 1
Platinum Maths Gr6_Term 1_CAPS.indd 54
09/02/13 1:13 PM
interesting properties of
multiplication
Flow diagrams can remind you of useful properties of multiplication
and division.
ExERCiSE 8.2
3
1. Copy the flow diagram and fill in the missing output numbers:
7
15
What do you notice about the input and output numbers? Why
does this happen?
Rule
+0
×1
2. Copy the following two flow diagrams and complete them:
3
5
10
3
Rule
×4
15
×5
5
10
Rule
×5
×4
15
What do you notice about the output numbers of these diagrams?
Why does this happen?
Rule
Rule
3
3
3. Copy these two flow diagrams
×100
÷4
×25
5
5
and complete them:
10
10
Write down how these diagrams give you an easy
way to multiply by 25.
4. Copy and complete flow diagram A.
From this flow diagram, write down an easy way to multiply a
number by 70.
5. Copy and complete flow diagram B.
From this flow diagram, write down an easy way to multiply a
number by 300.
6. Copy and complete flow diagram C.
From this flow diagram, write down an easy way to multiply a
number by 9 000.
A
5
9
10
B 2
15
20
Rule
×7
Rule
×3
C 5
8
12
×10
×100
Rule
×9
×1 000
Topic 8: Numeric patterns
Platinum Maths Gr6_Term 1_CAPS.indd 55
55
09/02/13 1:13 PM
Multiplication and division are
inverse operations
Remember that multiplication and division are inverse operations.
Flow diagrams can show this inverse relationship.
Challenge
ExERCiSE 8.3
What three divisions
will form the
opposite of
multiplying by 75?
1. Copy the following two flow diagrams and then complete them.
Write down what you notice, and explain why this happens.
1
3
5
7
5
Rule
15
25
35
×5
9
Rule
÷5
45
2. Copy and complete the following two flow diagrams. Check the
division by multiplying, and check the multiplication by dividing.
21
63
Rule
÷7
4
2
5
11
Rule
×12
63
72
×9
10
3. Copy the following flow
5
diagram and then fill in the
10
×20
missing outputs. What do you
15
notice about the input and
output numbers? Why does this happen?
5
9
18
18
Rule
Rule
÷100
×5
5 4. What is the missing part of the rule in the flow diagram on the
left?
9
18 5. Draw a flow diagram to show that × 8 and ÷ 8 are inverse
operations. Also write your flow diagram as a number sentence.
6. Use a flow diagram with three rule boxes to show that × 10 is the
opposite of doing two divisions. Also write your flow diagram as a
number sentence.
7. Use a flow diagram to show that × 4 × 25 is the opposite of divide
by 100. Also write your flow diagram as a number sentence.
56
Term 1
Platinum Maths Gr6_Term 1_CAPS.indd 56
09/02/13 1:13 PM
Find missing rules and numbers
You should be able to work out numbers that are missing in the
inputs, the outputs or the rule of a flow diagram.
Example
What is the missing part of the rule? Look at the 3
Rule
11
two inputs and their outputs.
6
20
×3
(3 × 3) □ = 11 and (6 × 3 ) □ = 20
3 × 3 = 9 and 6 × 3 = 18, so it is easy to see that
32
the missing part of the rule is +2.
What is the missing input for the output number 32? Write it as a number sentence:
(□ × 3) + 2 = 32. Remember that you can use inspection or trial and improvement to work
out the missing number.
If (□ × 3) + 2 = 32, then (□ × 3) must be 30. What multiplied by 3 gives 30? The missing input
number must be 10.
ExERCiSE 8.4
First find the missing operation in each rule. Then write the flow
diagram as a number sentence, and solve it to find the missing input
number.
1.
18
Rule
6
÷6
60
13
2.
Rule
8
10
×3
21
3.
5
26
41
5
Rule
10
20
× 20
÷ 100
7
10
50
4. Here is a rule in words: “multiply by 100 and subtract □”.
a ) Show this rule as a flow diagram.
b ) If the input numbers 3 and 7 have output numbers 290 and
690, find the missing number in the rule.
c ) Which input number will have the output number 990?
Topic 8: Numeric patterns
Platinum Maths Gr6_Term 1_CAPS.indd 57
57
09/02/13 1:13 PM
Challenge
I am thinking of a
number. When I add
3 to my number,
multiply the answer
by 5, multiply again
by 20 and divide by
25, I get 32. Show this
on a flow diagram.
What number was I
thinking of at the
beginning? What will
the output number
be if I start with the
number 10? Simplify
the flow chart to get
the same output
numbers with only
two operations in the
rule.
You can also show input and output numbers in a table, and work in
the same way with rules as you did in the flow diagrams.
Example
This table uses the rule: ×3 and then +2
Input number
Output number
RULE
×3 +2
2
3
4
5
6
8
11
14
17
20
Notice that in the bottom row of the table, you can get from each
output number to the next output number by adding 3. The next
three output numbers would be 23, 26 and 29.
ExERCiSE 8.5
For questions 1, 2 and 3:
a ) Use the given rule to complete the row of output numbers.
b ) Look at the row of output numbers. How do you get each output
number from the previous output number?
c ) Use your answer in (a) to write down the next three numbers in
the output row.
d ) Use the rule in the table to check your new output numbers.
1. Rule: (Input number – 3) × 5
Game
Work in pairs. Each
draw a flow diagram
that uses two of
these operations in
the rule: +5; ×3; –2;
÷2; ×10; –7; +4. Write
in five input numbers
between 1 and 100,
and the five correct
output numbers, but
don’t write in the rule
that you used. Swap
your flow diagrams
and see who can first
guess the two
operations used in
the rule.
58
Input
Output (input –3) × 5
4
5
6
7
8
9
10
7
8
9
10
11
12
17
19
21
23
25
5
2. Rule: (Input number × 2) + 4
Input
Output (input × 2) + 4
6
3. Rule: (Input number – 3) ÷ 2
Input
Output (input – 3) ÷ 2
15
4. Work out the missing part of the rule and complete the table:
Input
Output (input – 3) ?
5
4
6
7
8
9
8
10
12
20
5. Work out the missing part of the rule and complete the table:
Input
Output (input ?) + 1
5
11
13
7
8
9
15
17
19
29
Term 1
Platinum Maths Gr6_Term 1_CAPS.indd 58
09/02/13 1:13 PM
Revision
1. Below are some of the test scores for a
Grade 6 Mathematics test:
Zandi Mark Jabu Patti Seth Lyn Rajan Unathi
19
15
11
16
16
13
16
14
a ) Find the median test mark for these
scores.
(2)
b ) Find the mode for these scores.
(1)
3. a ) Use the given rule to complete
the row of output numbers:
2. Use this graph to help you answer the
questions that follow:
Excuses for why yesterday’s
homework was not done
Number of learners
8
7
Key
Grade 6
Grade 7
Input
Output
4
3
2
1
I left it at home
I can’t f ind it in my bag
I did not know I had any
I forgot
I went out until too late
0
I did not understand
(2)
(2)
(3)
5
8
11
14
17
20
4. a ) Complete this table:
(3)
3 6 9 12 15 18 21
Input
Output (input –3) ÷ 3
Reasons
a ) What kind of graph is this?
b ) How many Grade 7 learners used
the reason that they got home too
late the night before?
c ) How many Grade 6 learners used
the reason that the dog ate their
homework?
2
b ) Write down the next three input
numbers in the top row.
(2)
c ) Look at the row of output numbers.
How do you get the next output
number from the previous output
number?
(2)
d ) Write down the next three numbers
in the output row without using
the rule.
(2)
5
(1)
(1)
b ) Explain the pattern in the numbers in
the output row.
(2)
Total marks: 25
(1)
Revision
Platinum Maths Gr6_Term 1_CAPS.indd 59
(1)
Rule: (Input number × 3) – 2
6
Dog ate my homework
d ) How many learners used the
reason that they forgot to do
their homework?
e ) Which reason did the learners use
most often?
f ) How many Grade 6 learners did
not do their homework?
59
09/02/13 1:13 PM
2 Term 2
Pluto
Neptune
Uranus
Saturn
Jupiter
Mars
Earth
Venus
Mercury
Planets revolve around the sun
60
Platinum Maths Gr6_Term 2_CAPS.indd 60
12/02/13 12:50 AM
Topics 9–16
Planet
Distance from the sun (km)
Mercury
57,9 million
Venus
108,2 million
Earth
149,6 million
Mars
227,9 million
Jupiter
778,3 million
Saturn
1 427 million
Uranus
2 871 million
Neptune
4 497,1 million
The distance of the planets from the sun
Planet
Ageing factor
Mercury
4,155
Venus
1,625
Mars
0,532
Jupiter
0,084
Saturn
0,034
Uranus
0,012
Neptune
0,006
Planets and their ageing factors
Starting off
The sun is the centre of our solar system with
planets, asteroids, meteorites, rocks and comets
revolving around it. The eight planets that are
closest to the sun are Mercury, Venus, Earth,
Mars, Jupiter, Saturn, Uranus and Neptune.
Pluto is a dwarf planet.
1. What type of three-dimensional object is
Earth?
2. Saturn is a planet with rings around it. Do
you think the rings are two-dimensional or
three-dimensional?
3. Which planet is closest to the sun?
4. Which planet is furthest from the sun?
5. How close is Earth to the sun?
The length of a year on each planet depends on
the time that the planet takes to revolve around
the sun. Earth takes about 365 days to complete
one trip around the sun. So a year is about
365 days long.
Mercury takes about 88 Earth days to revolve
around the sun.
The further away from the sun a planet is, the
longer the length of its year.
That means that your age would be dif ferent
on dif ferent planets. Multiply your age by the
ageing factor to work out your age on any
planet.
6. If you are 10 years old on Earth, will you be
younger or older on Venus?
Content covered in Term 2
Topic 9: Whole numbers, Topic 10: Multiplication, Revision,
Topic 11: Properties of 3D objects, Topic 12: Geometric patterns, Revision,
Topic 13: Symmetry, Topic 14: Division, Revision, Topic 15: Decimal fractions,
Topic 16: Capacity and volume, Revision
61
Platinum Maths Gr6_Term 2_CAPS.indd 61
12/02/13 12:50 AM
Count, order, compare and represent
whole numbers
Topic
9
Work with nine-digit numbers
Maths ideas
• Recognise place
value in a ninedigit number.
• Order and
compare nine-digit
numbers.
• Round whole
numbers to the
nearest 5, 10, 100
and 1 000.
Key words
• million – one
thousand
thousand
In Term 1 you worked with six-digit numbers. Now you will work with
nine-digit numbers. In a whole number the ninth digit from the right is
the hundred millions digit. Remember that zeros can be placeholders.
Example
You read the number 198 234 112 as one hundred and ninetyeight million, two hundred and thirty-four thousand, one hundred
and twelve.
You write it in expanded form as
100 000 000 + 90 000 000 + 8 000 000 + 200 000 + 30 000 + 4 000 +
100 + 10 + 2.
ExErCiSE 9.1
1. Read the following numbers and write them in expanded form:
a ) 443 092 101
b ) 210 100 324
c ) 599 001 001
2. Give the place value of the digit 4 in the following numbers:
a ) 546 909 852
b ) 439 011 200
c ) 354 987 102
To compare and order large numbers with the same number of digits,
compare the digits starting from the left.
ExErCiSE 9.2
Did you know?
There are
approximately 50
million people in
South Africa.
If you lay one million
rand coins on top of
each other, the pile
will be higher than a
30 storey building!
62
1. Use the < or > symbols to show which number is larger:
a ) 345 234 756
347 234 756
b ) 512 035 781
602 155 723
c ) 734 680 255
352 908 476
2. Order the following groups of numbers in descending order:
a ) 231 345 435
233 345 435
210 231 211
243 345 211
250 211 345
b ) 175 370 899
205 372 980
198 245 123
210 218 900
211 218 901
Term 2
Platinum Maths Gr6_Term 2_CAPS.indd 62
12/02/13 12:50 AM
round off whole numbers
Rounding numbers makes them easier to work with. In topic 3 you
rounded numbers to estimate answers to calculations.
Example
1 234 367 rounded to the nearest 10 is 1 234 370.
1 234 367 rounded to the nearest 100 is 1 234 400.
1 234 367 rounded to the nearest 1 000 is 1 234 000.
← Rounding up
← Rounding up
← Rounding down
When rounding up a 9 in one place value, you need to round to the
next place value.
Example
Round these numbers to the nearest 100 and 1 000.
23 987 rounded to the nearest 100 is 24 000.
129 638 rounded to the nearest 1 000 is 130 000.
← Rounding up 900 to 1 000
← Rounding up 9 000 to 10 000
If you have to round a number to the nearest 5, decide which multiple
of 5 is closest to your number.
Challenge
Example
Round 137 to the nearest 5.
The multiples of 5 that are closest to 137 are 135 and 140.
137 – 135 = 2
140 – 137 = 3
137 is closest to 135.
so, 137 rounded to the nearest 5 is 135.
ExErCiSE 9.3
1. Round the following numbers to the nearest 5:
a ) 13
b ) 27
c ) 58
d ) 105
e ) 5 429 f ) 34 103
A local minibus taxi is
deciding how many
seats to put in its
new f leet of up to
20 vehicles, together
with 20 drivers.
They regularly have
to transport groups
of between 250 and
256 passengers.
How many seats
should they put in
the buses to keep all
the drivers in work?
2. Round the following numbers to the nearest 10, 100 and 1 000:
a ) 56 937
b ) 145 278
c ) 745 762
d ) 9 165 894
e ) 39 482
f ) 126 826
g ) 388 417
h ) 6 199 318
Topic 9: Count, order, compare and represent whole numbers
Platinum Maths Gr6_Term 2_CAPS.indd 63
63
12/02/13 12:50 AM
Topic
Multiplication
10
Maths ideas
• Find factors and
multiples of twoand three-digit
numbers.
• Estimate answers.
• Break up numbers
to multiply.
• Use factors to
multiply.
• Round off and
compensate.
• Use the column
method to
multiply.
• Solve problems
with multiplication.
Key words
• multiple – the
result when you
multiply whole
numbers
• factor – a number
that divides exactly
into another
number
Challenge
Using only the
numbers 2, 3, 4 and
6, make these four
digit numbers: a
multiple of 4, a
number with 6 as a
factor, and an odd
multiple of 3.
Factors and multiples
Factors and multiples are useful for breaking down and forming
numbers. Remember that two numbers multiplied together form
a multiple. A number is a factor of another number if it can divide
exactly into that number.
Example
10 × 40 = 400, so 400 is a multiple of 10 and also a multiple of 40.
10 and 40 are called factors of 400.
10 = 5 × 2, so 5 and 2 are also factors of 400.
All the factors of 400 are: 1, 2, 4, 8, 10, 16, 20, 25, 40, 50, 80,100, 200
and 400.
You can write a number as a multiplication of a factor pair. Here are
all the factor pairs for the number 18: 9 × 2; 18 × 1; 3 × 6.
What happens when you multiply numbers by 10, 100 and 1 000?
Example
325 × 1 000 = 325 000 (add three extra zeros).
500 × 100 000 = 50 000 000 (add five extra zeros).
To calculate 324 × 3 000, use (324 × 3) × 1 000 = 972 × 1 000 = 972 000.
ExErCiSE 10.1
1. Find as many factors as you can for each number:
a ) 17
b) 20
c ) 21
d ) 29
e ) 55
2. Find at least two multiples and four factors for each:
a ) 98
b) 99
c ) 100 d ) 125
e ) 250
3. Which of the whole numbers 1 to 20:
a ) are factors of 27
b ) are multiples of 4
c ) form factor pairs for the number 24?
4. Copy this table and fill in the missing numbers:
× 100
× 1 000
× 10 000
× 100 000
234
567
64
Term 2
Platinum Maths Gr6_Term 2_CAPS.indd 64
12/02/13 12:50 AM
Break up numbers to multiply
You can use factors to make multiplication easier.
Example
Calculate 342 × 42
342 × 42 = 342 × 6 × 7 = 342 × 2 × 3 × 7
Now do each part in turn:
(342 × 2) × 3 × 7 = (684 × 3) × 7
= 2 052 × 7
= (2 000 × 7) + (50 × 7) + (2 × 7) break up into parts
= 14 000 + 350 + 14
= 14 364
ExErCiSE 10.2
Use factors to do the following multiplications in more than one step:
1. 245 × 12
2. 303 × 25
3. 125 × 30
4. 326 × 24
5. 980 × 36
6. 250 × 240
Remember that you can break up large numbers, and then multiply
smaller numbers in more steps. Using brackets helps you to keep track
of what you are doing.
Example
238 × 35 = 238 × (30 + 5)
= (238 × 30) + (238 × 5)
= (238 × 3 × 10) + (238 ÷ 2 × 10)
= (714 × 10) + (119 × 10)
= 7 140 + 1 190
= 8 330
ExErCiSE 10.3
Challenge
Draw up a 3 × 3
square grid of 9
squares. In each
square, write one of
the numbers 1, 2, 3, 4,
6 or 12 so that the
three numbers in
each row or column
always equals 36
when multiplied. You
can use each number
as many times as you
like!
Do these multiplications by breaking down one of the numbers as in
the example above:
1. 343 × 25
2. 4 001 × 43
3. 453 × 72
4. 3 200 × 66
Topic 10: Multiplication
Platinum Maths Gr6_Term 2_CAPS.indd 65
65
12/02/13 12:50 AM
Estimate answers
It is very important to estimate before you do a calculation, so that
you know how big your answer should be. To estimate, you can round
one or more of the numbers.
Example
Here are three different estimates for 3 843 × 102:
4 000 × 100 = 400 000
4 000 × 102 = (4 000 × 100) + (4 000 × 2) = 400 000 + 8 000 = 408 000
3 800 × 100 = 380 000
ExErCiSE 10.4
1. Round to estimate if the following answers will be in the
thousands, ten thousands, hundred thousands or millions:
a ) 3 989 × 44 b ) 909 × 322 c ) 1 201 × 591 d ) 1 009 × 291
2. Round to estimate these answers and then use a calculator to see
how accurate your estimate is:
a ) 2 432 × 987
b ) 7 351 × 255
c ) 4 003 × 105
d ) 1 414 × 580
e ) 1 040 × 337
f ) 3 838 × 838
You can also compensate for rounding in order to find the exact
answer to a multiplication.
Example
4 327 × 99 is almost the same as 4 327 × 100.
4 327 × 99 = 4 327 × (100 – 1)
= (4 327 × 100) – (4 327 × 1)
= 432 700 – 4 327
= 428 373 (by column subtraction)
This is an exact answer. (–1) compensates for rounding off 99 to 100.
ExErCiSE 10.5
Round one number and then compensate to find the exact answer:
1. 322 × 98
66
2. 2 350 × 101 3. 45 × 32
4. 320 × 204
Term 2
Platinum Maths Gr6_Term 2_CAPS.indd 66
12/02/13 12:50 AM
Use the column method to multiply
You will now learn how to set out multiplication in columns.
Remember that in the place value system, 10 of one unit makes
1 unit in the column on the left. For example, 10 units = 1 ten,
10 tens = 1 hundred, and 10 hundreds = 1 thousand. This means that
if you get more than ten when you multiply, you write the tens part in
the column on the left.
Example
Find 2 468 × 231.
An estimate is 2 500 × 200 = 25 × 2 × 100 × 100 = 500 000.
2 4 6 8
×
2 3 1
2 468 × 1
2 4 6 8
2 468 × 30
7 4 0 4 0
2 468 × 200
+ 4 9 3 6 0 0
5 7 0 1 0 8
ExErCiSE 10.6
First estimate the answers to these calculations. Then find the exact
answer by using the column method of multiplication.
1. 234 × 65
2. 1 378 × 14
3. 2 534 × 75
4. 1 903 × 566
5. 3 220 × 480
6. 2 097 × 707
Challenge
To multiply by a four-digit number, use the same method. Now you will have
four answers to add together for the total. For the fourth line, multiply the top
number by the thousand digit of the second number.
1. 1 396 × 1 063
2. 1 632 × 1 397
3. 1 560 × 1 893
4. 1 712 × 1 612
5. 2 756 × 3 715
Topic 10: Multiplication
Platinum Maths Gr6_Term 2_CAPS.indd 67
67
12/02/13 12:50 AM
Solve multiplication problems
You can use any of the multiplication methods that you know to solve
these problems. Remember to estimate your answer first so that you
know more or less what your answer should be.
ExErCiSE 10.7
1. There are 25 cartons of juice in a box. How many cartons will there
be in 2 380 boxes?
2. Each month an average of 172 people stay in a six-bed hut in
Lower Sabie, Kruger Park. How many people will stay in the hut
over a period of a year?
3. A family saves R959 per month.
a ) How much will they save in one year?
b ) How much will they save in 10 years?
4. There are 365 days in one year, and each day has 24 hours. How
many hours are there in one year?
5. Lukhanyo planted 137 cabbages in each row on his farm. If there are
2 015 rows of cabbages, how many cabbages were planted?
Lukhanyo’s cabbages
grow in rows
Challenge
Some calculations
form beautiful
patterns with large
numbers.
11 × 11 = 121
11 × 111 = 1 221
11 × 1 111 = …..
Use a calculator to
find the next few
numbers in this
pattern and make a
prediction about
11 × 111 111.
68
6. A new employee is packing tins onto supermarket shelves. He has
been instructed to pack 1 550 tins per day. How many tins will he
pack out in four weeks if he works six days a week?
7. Trucks are taking wood to a saw mill. Each truck carries
approximately 1 225 planks of wood.
a ) In one week, 125 trucks offload wood. How many planks are
offloaded?
b ) Each plank is 435 cm in length. What is the total length of wood
delivered to the mill per week? Give your final answer in metres.
8. Population density measures how many people on average live in
each square kilometre (a square area of length 1 km). In Gauteng,
there are on average 576 people in each square kilometre, and in
the Western Cape there are on average 37 people in each square
kilometre.
a ) If Gauteng covers an area of 17 027 square kilometres, find the
population of this province.
b ) If the Western Cape covers an area of 129 462 square
kilometres, find the population of this province.
c ) Find the difference in population of these two provinces.
Term 2
Platinum Maths Gr6_Term 2_CAPS.indd 68
12/02/13 12:50 AM
revision
1. Arrange the following numbers in descending order:
a ) 1 546 823; 1 564 823; 1 465 823; 1 645 238; 1 546 328
b ) 5 890 314; 5 890 413; 5 089 314; 5 908 413; 5 890 134
(1)
(1)
2. Use the symbols < and > to compare these numbers:
a ) 2 897 243 □ 2 987 342
b ) 5 765 664 □ 5 576 664
(1)
(1)
3. Round each of the following numbers to the nearest 10:
a ) 276 935
b ) 2 856 486
(1)
(1)
4. Round each of the following numbers to the nearest 100:
a ) 28 675
b ) 8 468 981
(1)
(1)
5. Round each of the following numbers to the nearest 1 000:
a ) 657 104
b ) 1 003 571
(1)
(1)
6. Write down the values of the underlined digits:
a ) 325 895 032
b ) 587 908 300
(1)
(1)
7. List the following sets of numbers:
a ) The multiples of 7 between 107 and 120.
b ) The factors of 18 that are also factors of 27.
(2)
(2)
8. Write down a factor pair for each number.
a ) 140
b ) 42
(1)
(1)
9. Find all the factors of the following numbers:
a ) 32
b ) 45
(2)
(2)
10. Use the column method of multiplication to do the following calculations.
First estimate your answer by rounding.
a ) 4 019 × 534
b ) 8 976 × 398
(2)
(2)
11. Use the shortest possible method to calculate the following multiplications:
a ) 1 298 × 3 000
b ) 5 261 × 4 000
(2)
(2)
Total marks: 30
Revision
Platinum Maths Gr6_Term 2_CAPS.indd 69
69
12/02/13 12:50 AM
Topic
Properties of 3D objects
11
Maths ideas
identify 3D objects
• Identify and name
3D objects.
We live in a three-dimensional (3D) world, and the solid objects
that you see around you are 3D objects.
• Use features
to distinguish,
describe, sort and
compare objects.
Some 3D objects have special names.
• Make models of 3D
objects from nets.
• Interpret drawings
of 3D objects.
Key words
• threedimensional (3D)
objects – objects
that have three
dimensions: length,
width (or breadth)
and height
• prism – a 3D
object that has two
identical, parallel
faces that are
polygons
• pyramid – a 3D
object that has a
polygon as a base
and all its other
faces are triangles
• tetrahedron –a
pyramid made
of four identical
triangles
• face – a flat surface
of an object
Sphere:
a 3D object that is the shape
of a ball
Cone:
a 3D object that has a circular
base and a curved surface
that comes to a point
A prism is a 3D object with two parallel and identical faces that are
polygons. The other faces are rectangles, which are all at right angles
to the parallel faces. Prisms are an important family of 3D objects.
Cube:
a prism that has six identical
square faces
Rectangular prism:
a prism that has a
rectangular base
Triangular prism:
a prism that has a triangular
base
Pentagonal prism:
a prism that has a pentagon
as its base
Hexagonal prism:
a prism that has a hexagon
as its base
Octagonal prism:
a prism that has an octagon
as its base
A pyramid is a 3D object with a base that is a polygon. All the other
faces are triangles that meet at a common point.
Triangular-based pyramid:
a pyramid that has four faces, all in the shape of a
triangle (tetrahedron is another name for this pyramid)
70
Cylinder:
a 3D object with two parallel
circular bases and one
curved surface
Rectangular-based pyramid:
a pyramid with a
rectangular base
Term 2
Platinum Maths Gr6_Term 2_CAPS.indd 70
12/02/13 12:50 AM
Construct 3D objects
Key words
A net is a flat shape that you can fold up to make a model of 3D
object. A net can help you to see the 2D shapes that make up a 3D
object. The flat shapes that make up a solid are called faces. For
example, a cube is made up of six square faces.
• net – a flat pattern
that you can cut
out, fold and glue
together to make
a model of a 3D
object
Example
Which of these are nets can you fold to make a cube?
A
B
C
D
E
F
All of these nets have six identical squares, but not all of them will
fold up to form a cube. Copy each net, cut it out and fold it up.
Can you see that nets A, B and D are nets of a cube, but C, E and F
are not? What do nets A, B and D have in common? Each of these
nets has four squares in a column, with one square on either side of
the column of four squares. It does not matter where the two side
squares are placed.
ExErCiSE 11.1
1. a ) Use one of the cube nets A, B or D to draw a net of a cube with
sides of 3 cm, on 1 cm square grid paper.
b ) Add flaps to your net so that you can glue the net together.
Cut out your net and check that it will fold up to form a cube.
(Do not glue your cube together yet.)
2. a ) Based on your net in question 1, draw the net of a rectangular
prism with a length of 4 cm, a width of 3 cm and a height of
2 cm, on 1 cm square grid paper. Draw a rough sketch first to
plan the dimensions of each rectangle in the net. Remember
to add flaps for gluing the net together.
b ) Cut out your net. Construct your rectangular prism by folding
the flaps in place. If your net does not work, draw and cut out
another one that you think will work.
Topic 11: Properties of 3D objects
Platinum Maths Gr6_Term 2_CAPS.indd 71
71
12/02/13 12:50 AM
c ) Once you are sure that your rectangular net is correct, fold and
glue your rectangular prism and your cube.
3. Look at the diagram of a triangular prism on page 70. What 2D
shapes make up this solid? Design a net of a triangular prism so
that its base is a triangle with each side equal to 5 cm, and when it
stands on its base it is 10 cm high. Draw your net accurately on
1 cm triangular grid paper. Remember to put in flaps for gluing the
net together. Cut out your net and construct your triangular prism.
Assembling a
rectangular prism
from its net
Did you know?
The huge pyramids
in Egypt are squarebased pyramids, and
are over 5 000 years
old.
4. Look at the diagrams of pyramids on page 70. Describe the 2D
shapes that make up these solids. Choose either a triangular
pyramid or a square-based pyramid, and construct a model of the
pyramid using a net. Choose your own dimensions.
Keep your models to use later.
ExErCiSE 11.2
1. Identify and name the solids described below:
a ) It has a round base and a curved surface that comes to a point.
b ) It has a pentagonal base and five of its faces are triangles.
c ) It has seven faces.
The three
Pyramids at Giza,
near Cairo in
Egypt
72
2. Use the picture to answer these questions:
a ) Name two different cylinders in the picture.
b ) List three different objects with curved surfaces.
c ) Name all the prisms in the picture.
d ) Name all the pyramids in the picture.
Term 2
Platinum Maths Gr6_Term 2_CAPS.indd 72
12/02/13 12:50 AM
Describe, sort and compare
3D objects
By now you should be able to identify the shape and number of faces
a solid has. Some solids have only flat surfaces, like a prism, and some
have curved surfaces, like a cone, cylinder or sphere. Only flat surfaces
are called faces. You can also identify solids by the number of edges
and vertices.
vertex: the point at
which three or more
faces of a 3D object
meet (the plural of
vertex is vertices)
face: a flat surface of a
3D object
edge: the line where
two faces meet
Key words
• edge – the line
where two faces of
a solid meet
Did you know?
The rectangular prism has 8 vertices and the square-based pyramid has
5 vertices. The rectangular prism has 12 edges and the square-based
pyramid has 8 edges.
Bees make
honeycombs in the
form of hexagonal
prisms
ExErCiSE 11.3
1. Use your model of a cube to answer the following questions:
a ) How many faces does the cube have?
b ) How many edges does the cube have?
c ) How many vertices does the cube have?
2. Use your model of a triangular prism to write down how many
faces, edges and vertices it has.
3. Here are the names of some solids:
sphere; cone; cylinder; rectangular prism; triangular prism;
pentagonal prism; triangular-based pyramid; square-based
pyramid. From the list, which object(s) have:
a ) no edges
b ) 1 edge
c ) 2 edges
d ) 8 edges
e ) 9 edges
f ) 12 edges?
4. From the list of 3D objects in question 3, which object(s) have:
a ) no vertices
b ) 1 vertex
c ) 4 vertices
d ) 5 vertices
e ) 6 vertices
f ) 8 vertices?
Challenge
Complete these
sentences:
1. The number
of edges of a
pyramid is □
times the number
of sides of the
base.
2. The number of
edges of a prism
is □ times the
number of sides
of the base.
Topic 11: Properties of 3D objects
Platinum Maths Gr6_Term 2_CAPS.indd 73
73
12/02/13 12:50 AM
You can describe 3D objects in different ways, by using the following
features:
• the surfaces (are they only curved, only flat faces, or curved and
flat?)
• the number of faces and the shapes of the faces
• the number and size of the angles on each face
• the number of edges
• the number of vertices.
ExErCiSE 11.4
1. Use the models you have made and the diagrams on page 70 to
complete the table below. The first one has been done for you.
3D object
Number
of edges
Number
of
vertices
Number
of square
faces
Number of
rectangular
faces
Number of
triangular
faces
Number
of
circular
faces
6
4
None
None
4
None
Triangular-based
pyramid
Cube
Rectangular prism
Cylinder
Triangular prism
Square-based
pyramid
Challenge
Can you draw the net
of a cone? The circular
base is easy but what
will the curved
surface look like?
74
2. Which of the 3D objects above have curved and flat surfaces?
Did you know?
Rural people build the cone–cylinder
house in many parts of Africa. The roof
is the shape of a cone while the walls
look like a cylinder.
Term 2
Platinum Maths Gr6_Term 2_CAPS.indd 74
12/02/13 12:50 AM
interpret drawings of 3D objects
When you look at a 2D picture of a 3D object, the shapes of the faces and
the sizes of the angles might not look the same as they do in real life.
Example
Look at the angles that have been coloured pink,
green, yellow and blue. What is the shape of the front
of the house? It is a rectangle, so in real life the pink
angle must be a right angle, but does it look like a
right angle in the picture? Use the corner of a page
to check the size. What is the shape of the part of the
roof that is shown? It is a rectangle, so that means
the yellow and green angles are right angles, even
though they are not the same size as the corner of a
page. What is the shape of the blue angle? The blue
angle is the corner of a rectangular window, so the angle is a right angle. In real life these
four angles are all the same size even though they don’t look the same in the picture.
ExErCiSE 11.5
1. Look at the picture of the fish tank A on the right.
a ) Identify the 3D object.
b ) The base of the object is a regular 2D shape. What does this
tell you about the sides and the angles of the base? What is
the shape of the base?
c ) Classify the size of each angle of the base.
d ) Do the angles on the diagram have the same size? Is this the
same as the real 3D object?
2. Look at the picture of the birthday cake B on the right.
a ) Identify the 3D object.
b ) The sides of the cake are covered in a special type of icing
called marzipan. What shape should the piece of marzipan be
to cover the sides of the cake?
A
B
3. Match
the nets below with
the fish tank and the birthday cake:
a)
b)
a)
b)
b)
a)
Topic 11: Properties of 3D objects
Platinum Maths Gr6_Term 2_CAPS.indd 75
75
12/02/13 12:50 AM
Topic
Geometric patterns
12
Maths ideas
• Extend geometric
patterns.
• Look for rules
and relationships
in geometric
patterns.
Extend patterns and look for rules
Example
Use the pattern to work out what the next diagram will look like.
• Represent patterns
in different ways.
• Extend sequences
involving constant
difference.
• Extend sequences
involving constant
ratio.
• State the rules
for geometric
patterns.
Key words
• sequence – a set
of numbers or
shapes in a pattern
Did you know?
We find patterns not
only in numbers, but
also in nature and in
culture. In culture,
patterns are used in
beadwork, fabric
patterns and
paintings.
76
The number of matchsticks in the first three diagrams is 6; 12; 18,
so each new diagram uses six more matchsticks than the previous
diagram. You can describe this sequence of numbers in words: add
6 to get the next number.
Example
Describe the rule in this sequence: 3; 1; 4; 2; 5; 3; 6…
The numbers are increasing and decreasing, so there must be more
than one operation in the pattern. The rule is: subtract 2, and then
add 3. Check: 3 – 2 = 1; 1 + 3 = 4; 4 – 2 = 2; 2 + 3 = 5; and so on.
ExErCiSE 12.1
1. Write down the next three numbers and describe each pattern:
a ) 3; 6; 9; 12; □
b ) 1; 8; 15; 22; □
c ) 7; 9; 8; 10; 9; □
d ) 4; 1; 16; 1; 64; 1; □ e ) 2; 4; 8; 16; 32; □ f ) 3; 9; 27; 81; □
2. Draw the next diagram in each pattern. Explain the rule.
a)
b)
3. The diagrams below show artwork that the Nguni people paint on
their houses. Describe the pattern in the sequence of artwork.
Term 2
Platinum Maths Gr6_Term 2_CAPS.indd 76
12/02/13 12:51 AM
Tables and flow diagrams
You can use a table to help you understand how a pattern works.
Example
This pattern is made by adding squares to the previous figure.
1
2
3
4
2
4
+2
6
+2
8
+2
The same difference makes it easy to find the next term. But to predict the number
of squares used in the 10th or 20th figure, we need a rule. To work out a rule, find the
relationship between the figure number and the number of squares. In some patterns, such
as the example above, you can find the relationship by inspection.
Figure number × 2 = number of squares
Check that the rule works for all terms. 4th figure: 4 × 2 = 8 squares ← check 6 + 2 = 8.
To find the number of squares in the 10th figure: 10 × 2 = 20 squares.
1
Example
Look at the pattern made with matchsticks.
Figure number
Number of matchsticks
1
2
3
4
7
10
+3
6
3
2
4
7
37
10
+3
Use the trial-and-improvement method to work out the rule when you cannot find it by inspection.
Each figure in this pattern is made by adding 3 more match sticks to the previous pattern. Try
multiplication or division by 3 first and then adjust your answer by adding or subtracting from it.
1 × 3 + or − □ = 4;
3+1=4
So, figure number × 3 + 1 = number of matchsticks.
Try substituting other values from the table to check the rule.
You can use flow diagrams to check whether the rule you found works.
number of match sticks
Figure number
×3
+1
6
×3
+1
19
Topic 12: Geometric patterns
Platinum Maths Gr6_Term 2_CAPS.indd 77
77
12/02/13 12:51 AM
Example
In the example on page 77, find which figure number will have 37
matchsticks.
×3
Figure number
+1
37
If we need to calculate the figure number from the number of
matches used in that figure, we do the inverse of all the operations.
Challenge
• Make a table
showing the total
number of blocks
in each building
in the question
you have just
answered.
• Can you see a
pattern?
• Write a rule for the
pattern.
• Fill in on your
table the total
number of blocks
in buildings with
6, 10, 12, 20 and 37
levels.
Figure number × 3 + 1 = the number of matchsticks.
The number of matchsticks – 1 ÷ 3 = figure number.
12
37
−1
÷3
37 – 1 = 36; 36 ÷ 3 = 12
The figure number using 37 matchsticks is 12.
ExErCiSE 12.2
Blocks are used to model buildings. The rows of blocks are labelled
“level 1”, “level 2”, and so on.
1
Level 1
2
3
Level 2
Level 3
1. Draw the next building in the sequence.
2. Count the blocks in level 1 of each building. In the table below, fill
in the number of blocks in level 1 for buildings 1, 2, 3 and 4:
Building number
Blocks in level 1
1
2
3
4
6
10
12
20
73
3. Write down the rule for the pattern of numbers.
4. Test your rule for buildings 3 and 4 using number sentences.
5. Use the rule to fill in the number of blocks in level 1 for the
buildings with 6 levels, 10 levels, 12 levels and 20 levels.
6. Calculate the number of levels that a building will have, if it has 73
blocks in level 1. Enter this in the table.
78
Term 2
Platinum Maths Gr6_Term 2_CAPS.indd 78
12/02/13 12:51 AM
Look for patterns and rules
In this section you can have fun finding a rule for a pattern, and then
use the rule to work out diagrams further along the pattern.
ExErCiSE 12.3
1
2
3
Study the pattern and answer the following questions:
1. Describe the pattern in your own words.
2. What must you do to make the next diagram in the pattern?
3. Copy the table and fill in the number of matches needed to make
each of the first 4 diagrams:
Diagram number
Number of matchsticks
1
2
3
4
5
7
12
101
4. Find a rule to calculate the number of matchsticks needed for any
diagram number. Write your rule like this:
Diagram number
Rule
Challenge
number of matches
5. Show how you tested your rule for diagram numbers 2 and 3.
6. Use your rule to fill in the number of matches used for 5, 7 and 12
houses joined together.
7. Now use your rule to calculate how many houses are joined together
when you use 101 matches to make them. Show your working.
ExErCiSE 12.4
Study the following number sequences. Write down the next three
numbers in each sequence, and write down the rule for getting from
one number to the next:
1. 3; 6; 9; 12; □
2. 0; 2; 4; 6; □
3. 78; 67; 62; □
4. 3; 9; 27; □
5. 1; 3; 6; 10; 15; □
6. 128; 64; 32; 16; □
7. 60; 63; 66; 69; □
8. 1; 1__14 ; 1__12 ; 1__34 ; □
Six matchsticks have
been used to make
this triangle. Add
another three
matchsticks to the
diagram, so that
there are five
triangles altogether.
Topic 12: Geometric patterns
Platinum Maths Gr6_Term 2_CAPS.indd 79
79
12/02/13 12:51 AM
More fun with patterns and rules
When you look at geometric patterns, it is often quite easy to see
what the next diagram in the pattern will be. It can be more difficult to
work out a rule that tells you about a diagram much further along the
pattern, without you having to draw it.
ExErCiSE 12.5
Here is a pattern of dots on blocks:
1
2
3
1. Describe the pattern in words.
2. How many dots must be added to each block to get to the number
of dots on the next block? Find a pattern in these added numbers.
3. Draw the fourth block.
4. How many dots must be added to the third block to get the fourth
block?
5. Now complete this table for block numbers 3, 4 and 5:
Block number
Number of dots added to the
previous block
2
Rule
3
4
5
2
3
4
5
11
15
3
59
6. Let the “block number” be the input number in a flow diagram.
The “number of dots added to the previous block” is the output
number. Work out a rule to change input numbers to output
3
numbers. Put this rule into the flow diagram and complete it.
7. Use your rule from question 6 to work out which block number
will need to have 59 dots added to its previous block in the
pattern. Write this number in the table.
11
8. Now count the number of dots that are in each of the first 4 blocks.
Block number
Number of dots
1
1
2
3
4
5
7
11
144
9. Write down the rule for finding the number of dots from the block
number. Use the rule to complete the above table for block 5, 7 and 11.
10. Which block number will have 144 dots? Fill this in on the table.
80
Term 2
Platinum Maths Gr6_Term 2_CAPS.indd 80
12/02/13 12:51 AM
Challenge
ExErCiSE 12.6
Write the rule for each of these patterns. Use substitution or flow
diagrams to complete the tables.
1.
1
2
Eight matchsticks
have been used to
make this square.
Add another two
matchsticks to the
diagram, so that
there are two
squares.
3
Diagram number
Number of beads
1
2
3
4
9
45
2.
1
2
Diagram number
Number of matchsticks
3
1
2
3
4
10
6
54
3.
1
2
Diagram number
Number of sweets
3
1
2
3
4
17
3
47
4.
1
Diagram number
Number of counters
2
1
3
2
9
3
4
8
49
Topic 12: Geometric patterns
Platinum Maths Gr6_Term 2_CAPS.indd 81
81
12/02/13 12:52 AM
5. This one is not easy. Try different ways to get to the rule.
1
2
3
Figure number
Number of lines
1
2
3
4
14
9
600
6.
1
2
Diagram number
Number of triangles
3
1
2
3
4
11
7
65
7.
1
1
Block number
Number of dots
2
3
2
3
1
2
3
0
4
13
64
Challenge
Look at this pattern. Ring 1 is red. Ring 2 is the surrounding
yellow ring. Draw a table to show how many hexagons there
will be in each of the next 5 rings. Write the rule for this
pattern. Where would you find this pattern in nature?
82
Term 2
Platinum Maths Gr6_Term 2_CAPS.indd 82
12/02/13 12:52 AM
revision
1. Name the following 3D objects from the descriptions that are given:
a ) I have four faces, four vertices and six edges. What am I?
b ) I have one circular face and one vertex. What am I?
c ) I have two parallel faces that are pentagons and five rectangular faces. What am I?
d ) All my faces have only right angles. What am I?
(1)
(1)
(1)
(1)
2. Match the nets shown below with each of the objects described in question 1:
(4)
A
B
C
D
3. Fill in the missing numbers:
a ) A triangular-based pyramid has □ faces, □ vertices and □ edges.
b ) A triangular prism has □ faces, □ edges and □ vertices.
(6)
4. Study the pattern below and then answer the questions that follow:
1
2
3
a ) What must you do in order to draw the next diagram in the pattern?
b ) Copy the table and fill in the number of dots needed to make the first four diagrams:
Diagram number
Number of dots
1
2
3
4
5
7
(2)
(4)
12
130
c ) Write a rule to calculate the number of dots needed for any diagram number.
d ) Use your rule to fill in the number of dots used to make diagrams 5, 7 and 12.
Show all your working.
e ) Now use your rule to calculate which diagram number in the sequence will
need 130 dots to form the circle.
(2)
(3)
(2)
5. Write down the rule to get from each input to each output number:
Input number
Output number
2
5
4
9
6
13
8
17
10
21
(3)
Total marks: 30
Revision
Platinum Maths Gr6_Term 2_CAPS.indd 83
83
12/02/13 12:52 AM
Topic
Symmetry
13
Maths ideas
• Recognise lines of
symmetry.
• Draw and describe
lines of symmetry.
• Recognise and
describe the
order of rotational
symmetry.
• Use line and
rotational
symmetry to
describe patterns.
Key words
• line symmetry
– a shape has
line symmetry if
it can be divided
into two identical
halves by a straight
line
• line of symmetry
– a line that divides
a shape into two
identical halves
• infinite – without
a limit
Line symmetry
You already know that some shapes have symmetry. A
shape has line symmetry if it can be divided into two
identical halves by a straight line. A line that divides
a shape into two identical halves is called a line of
symmetry.
A shape that has line symmetry may have one or more
lines of symmetry.
A circle is a very special 2D shape in many ways. One
example of this is its lines of symmetry. You can divide
a circle into two identical halves by drawing a line from
any point on the edge of the circle, through the centre
of the circle. This means that there is no limit to the
number of lines of symmetry of a circle. We say that a
circle has an infinite number of lines of symmetry.
A rectangle has only two lines of symmetry,
as shown in the diagram on the right.
ExErCiSE 13.1
1. Look at the shapes on the left.
a ) Write down the mathematical name of each shape.
b ) Complete the table below for the given shapes.
Name of shape Number of sides Number of lines of symmetry
A.
B.
C.
D.
E.
2. Use your answers to question 1 to complete the following
sentence: A polygon with all sides the same length and all angles
equal has the same number of lines of symmetry as the number
of its …
3. Why does a rectangle not fit the rule in question 2?
4. A diagonal line goes from one corner to the opposite corner of a
shape. Can a diagonal be a line of symmetry in:
a ) a rectangle
b ) a square?
84
Term 2
Platinum Maths Gr6_Term 2_CAPS.indd 84
12/02/13 12:52 AM
ExErCiSE 13.2
1. Look at the operators and letters below:
+–MATHS×÷
a ) Which letter has no line symmetry?
b ) Which operators have four lines of symmetry each?
c ) Find the total number of the lines of symmetry in all the
operators and letters.
2. Copy the shapes and draw in all the lines of symmetry:
b)
a)
c)
d)
Challenge
3. Copy and complete these shapes if the dotted lines are lines
of symmetry:
b)
a)
c)
e)
d)
f )
Look back at the
polygons in question
1 of Exercise 13.1.
Imagine drawing lots
of polygons, and
increasing the
number of sides each
time.
Imagine you could
draw a polygon
whose sides are all
the same length and
with infinitely many
sides.
a) What shape
would it finally
look like?
b) How many lines
of symmetry
would it have?
Topic 13: Symmetry
Platinum Maths Gr6_Term 2_CAPS.indd 85
85
12/02/13 12:52 AM
Key words
• rotational
symmetry –
a shape has
rotational
symmetry if you
can rotate (turn)
it so that it fits
onto itself before
it completes a full
turn
rotational symmetry
A shape has rotational symmetry if one can rotate (turn) it so that it
fits onto itself before it completes a full turn.
Example
• order of
rotational
symmetry – the
number of times
that a shape fits
onto itself when it
turns through a full
revolution
Imagine that you stick a pin
in the centre of this shape,
and then you turn the shape
around the pin point.
After a half turn, the shape will look exactly the same as it started.
We say that the shape has rotational symmetry.
The order of rotational symmetry of a shape is the number of
times that the shape fits onto itself when it turns all the way around
through a full revolution. The shape in the example fits onto itself
twice in a full turn, so its order of rotational symmetry is 2.
Challenge
Example
African bead designs
often have both line
symmetry and
rotational symmetry.
Find at least two
objects with line
symmetry and two
objects with
rotational symmetry
and bring them to
class. Describe the
symmetry in your
objects to your
friends.
A square has rotational symmetry of order 4, because it fits onto
itself 4 times in a full revolution.
Rotation 1
Rotation 2
Rotation 3
Rotation 4
Notice that you do not count the first and the last positions twice,
because they are identical.
A rectangle has rotational symmetry of order 2, as shown in the
diagram below:
A circle is again a very special 2D shape in terms of its rotational
symmetry.
You can rotate a circle through any angle and it will always look
exactly the same. We say that a circle has an infinite order of
rotational symmetry.
10°
86
10°
Term 2
Platinum Maths Gr6_Term 2_CAPS.indd 86
12/02/13 12:52 AM
ExErCiSE 13.3
1. Write down the order of rotational symmetry of each shape:
a)
b)
c)
d)
2. Describe what line or rotational symmetry you can see in each
pattern:
b)
a)
Challenge
Draw any shape that
has:
ExErCiSE 13.4
1. Look at the diagrams of polygons whose sides are all the same length
in question 1 of Exercise 13.1 again. Now complete the table below:
Name of
shape
Number of
sides
Number
of lines of
symmetry
Order of
rotational
symmetry
2. What do you notice about the number of sides, the number of
lines of symmetry and the order of rotational symmetry of a
polygon whose sides are all the same length?
3. Write down the name of a quadrilateral that has:
a ) line symmetry and rotational symmetry
b ) rotational symmetry, but not line symmetry.
4. What shape has an infinite number of lines of symmetry and an
infinite order of rotational symmetry?
a) four lines of
symmetry
and rotational
symmetry of
order 4
b) two lines of
symmetry
and rotational
symmetry of
order 2
c) one line of
symmetry
and rotational
symmetry of
order 2
d) one line of
symmetry and
no rotational
symmetry
e) no symmetry.
Think of shapes that
you have not seen in
this chapter. Be as
creative as possible,
and have fun!
Topic 13: Symmetry
Platinum Maths Gr6_Term 2_CAPS.indd 87
87
12/02/13 12:52 AM
Topic
Division
14
Maths ideas
• Divide by multiples
of 10.
• Find prime factors.
• Revise division
with remainders.
• Divide a fourdigit number by a
two-digit number.
Check solutions
using calculators
and inverse
operations.
• Estimate before
dividing.
• Work with ratio
and rate.
• Solve division
problems.
Work with factors
Earlier this year you learnt about factors and multiples. Here are some
useful rules for divisibility, to help you find factors of large numbers.
rules of divisibility
2 – The last digit must be an even number
3 – The sum of the digits must be a multiple of 3
4 – The last two digits must be divisible by 4
5 – The last digit must be 0 or 5
6 – The number must be divisible by 2 and by 3
8 – The last 3 digits must be a multiple of 8
9 – The sum of the digits must be a multiple of 9
10 – The last digit must be zero
Example
Is the number 4 014 divisible by 3, 9 or 4?
4 + 0 + 1 + 4 = 9 which is divisible by 3 and by 9, so 3 and 9 are
both factors.
The last two digits form the number 14 which is not divisible by 4,
so 4 is not a factor.
You should also remember the rules for dividing by multiples of 10:
the digits move 1 place to the right
10
the digits move 2 places to the right
100
1 000
the digits move 3 places to the right
ExErCiSE 14.1
1. Use the rules of divisibility to check which of the following
numbers are multiples of (are divisible by) 2, 3, 4, 5, 6, 8, 9 and 10:
a ) 552
b ) 315
c ) 620
d ) 426
2. Do the divisions and then fill in < , > or = to make each number
sentence true:
a ) 34 000 ÷ 1 000 □ 3 400 ÷ 100
b ) 40 000 ÷ 100 □ 74 000 ÷ 1 000
c ) 9 200 ÷ 100 □ 19 200 ÷ 10
88
Term 2
Platinum Maths Gr6_Term 2_CAPS.indd 88
12/02/13 12:52 AM
Work with prime numbers
A prime number has only two factors, 1 and itself. A number with
more than two factors is called a composite number.
The number 1 has only one factor and is therefore not a prime
number nor a composite number.
Example
The factors of 7 are only 1 and 7, so 7 is a prime number.
The factors of 8 are 1, 2, 4 and 8, so 8 is not a prime number. We say
that 8 is a composite number.
Key words
• prime number – a
number with only
two factors, 1 and
itself
• composite
number – a
number with more
than two factors
To find all the prime factors of a number, start with one factor pair and
then keep splitting into more factor pairs until you cannot divide any
further.
Example
Write the prime factors of 1 140.
×
114
1 140 = 10
= 2×5 ×
3 × 38
= 2 × 5 × 3 × 2 × 19
So 1 140 = 2 × 2 × 3 × 5 × 19
ExErCiSE 14.2
1. Which number is the only even prime number?
2. Write down the numbers 1 to 16 in your workbook. From that list
write down the numbers that are:
a ) neither composite nor prime b ) factors of 27
c ) multiples of 4
d ) prime numbers
e ) composite numbers.
3. Which of these numbers are prime and which are composite?
a) 64
b ) 78
c ) 111
d) 54
e) 38
f ) 37
g ) 121
h) 172
i ) 441
j ) 324
Did you know?
To find some prime
numbers, the Greek
mathematician
Eratosthenes drew a
10 × 10 grid with the
numbers 1 to 100. He
crossed off the
multiples of 2, then
the multiples of 3,
then the multiples of
4, and so on until
only prime numbers
were left. This
method can help you
to identify prime
numbers smaller
than 100.
4. Mirror primes have digits in the reverse order, like 17 and 71.
Which of these prime numbers has a mirror prime?
a) 13
b ) 57
c ) 73
d) 97
e) 23
5. Write the prime factors of each number, as in the example above:
a) 18
b ) 32
c ) 60
d) 100
e) 65
f ) 42
g ) 99
h) 110
i ) 346
j ) 420
Topic 14: Division
Platinum Maths Gr6_Term 2_CAPS.indd 89
89
12/02/13 12:52 AM
Multiplication and division are
inverse operations
Multiplication facts help you with division if you break down the
division into smaller steps. Remember to first estimate the answer.
Example
Estimate and then use multiplication facts to find 497 ÷ 19.
An estimate is 500 ÷ 20 = 25.
Now write down some simple multiplication facts for 19. This is
called a clue board.
19 × 10 = 190
19 × 20 = 380 (190 × 2)
19 × 2 = 38
19 × 5 = 95 (half of 190)
The closest multiplication to 497 is 380, so start with 19 × 20.
multiply
19 × 20 = 380
19 × 5 = 95
19 × 1 = 19
subtract
497 – 380 = 117
117 – 95 = 22
22 – 19 = 3
497 ÷ 19 = 20 + 5 + 1 remainder 3 = 26 remainder 3.
The estimate was quite close! Finally check by multiplying:
19 × 26 plus remainder 3 = (19 × 20) + (19 × 6) + 3
= 380 + 114 + 3 = 497
ExErCiSE 14.3
Challenge
Find 12 500 ÷ 250 in
these different ways:
12 500 ÷ (10 × 5 × 5)
12 500 ÷ (500 ÷ 2).
Explain why you get
the same answer
both times.
90
1. Use clue boards to do these divisions. First estimate your answers.
a ) 868 ÷ 14
b ) 636 ÷ 21
c ) 908 ÷ 52
2. Calculate the missing number:
a ) 320 ÷ □ = 4
b ) 312 ÷ □ = 12
c ) □ ÷ 29 = 31
3. Calculate the answers. Remember to do brackets first.
a ) 12 × 8 ÷ 4
b ) 55 ÷ 11 × 600
c ) 12 ÷ 4 + 63 ÷ 9
d ) 34 – 21 ÷ 7
e ) (42 + 16) × 19
f ) 84 – (16 × 4) ÷ 8
4. I am thinking of a number. If I multiply my number by 30, the
answer is 2 400. What is my number?
5. I am thinking of a number. If I divide my number by 60, the answer
is 50. What is my number?
Term 2
Platinum Maths Gr6_Term 2_CAPS.indd 90
12/02/13 12:52 AM
Use the long division method
When you used a clue board to divide, you worked with multiplication
facts. You can also set out that work in columns, in a method called
long division.
Example
Find 8 643 ÷ 36.
Write down some multiplication facts for 36:
36 × 100 = 3 600; 36 × 200 = 7 200
36 × 10 = 360; 36 × 20 = 720; 36 × 40 = 1 440
240 remainder 3
36 8 6 4 3
7200
← 36 × 200 = 7 200
1443
1440
← 36 × 40 = 1 440
3
So, 8 643 ÷ 36 = 240 remainder 3.
Key words
• long division
– a method for
dividing using
multiplication facts
and subtraction
You can also use long division to divide by larger numbers.
Example
Find 4 813 ÷ 213.
Write down some multiplication facts for 213:
2130 × 10 = 2 130; 213 × 20 = 4 260; 213 × 2 = 426
22 remainder 127
213 4 8 1 3
4260
← 213 × 20 = 4 260
553
426
← 213 × 2 = 426
127
So, 4 813 ÷ 213 = 22 remainder 127.
ExErCiSE 14.4
Challenge
Fela, an egg farmer, is
trying to decide
which size packaging
to use.
He can choose packs
of 12, 14, 15, 16, 18
or 20.
Find out which is the
best size packaging
for 1 265 eggs, so
that the least number
of eggs will be left
over.
Use long division for the following calculations. Estimate your answer
first.
1. 456 ÷ 16
2. 987 ÷ 28
3. 843 ÷ 27
4. 3 726 ÷ 35
5. 5 781 ÷ 41
6. 4 976 ÷ 31
7. 9 430 ÷ 225
8. 3 007 ÷ 521
9. 5 356 ÷ 213
Topic 14: Division
Platinum Maths Gr6_Term 2_CAPS.indd 91
91
12/02/13 12:52 AM
Solve division problems
Use any method to solve the following division problems. Check
your answers by doing multiplication or by using a different division
method. Remember to estimate the answer first.
ExErCiSE 14.5
Challenge
A water tank holds
10 000 litres of water.
How many 500 ml
bottles can be filled
from the tank?
1. Mr Hendricks has 630 eggs. He packs the eggs in trays. Each tray
holds 48 eggs.
a ) How many complete trays can he fill?
b ) How many eggs will be left over?
c ) How many groups of 630 does he need so that the trays are
full with no remainder?
2. Timbavati Primary School made R5 096 from ticket sales for their
talent show. A total of 98 people bought tickets. Calculate the
price of each ticket.
3. A charity organisation needs to raise R4 125 for an HIV/AIDS home.
There are 75 volunteers. How much money does each volunteer
have to raise?
4. Thandi works for an organisation that packs food parcels for the
poor. A company has donated 19 boxes of assorted tins, each
containing 55 cans. Calculate how many cans each family will
receive if there are 95 families to feed.
5. A teacher paid R8 320 for new textbooks. If she ordered 52 books,
how much did each textbook cost?
6. Mpho runs 19 km each day for 28 days during his training. How
many days will it take Kholeka to run the same total distance if she
runs 16 km each day?
7. Richard and Imran run every day. Imran runs 24 km on Tuesday,
and Richard runs __56 of this distance. How far does Richard run?
92
Term 2
Platinum Maths Gr6_Term 2_CAPS.indd 92
12/02/13 12:52 AM
Solve problems by comparing
quantities of the same kind
Example
A recipe says you must use 100 g of sugar and 600 g of flour. If you
double the amount of sugar to 200 g, you must also double the
amount of flour to 1 200 g. If you halve the amount of flour to
300 g, you must also halve the amount of sugar to 50 g.
Example
Maria works for 8 hours in the garden and Lebo works for 6 hours
in the garden. If they are paid R168 in total, how much should they
each get so that the money is fairly divided?
In total the girls worked for 8 + 6 = 14 hours. R168 ÷ 14 = R12.
Maria should get 8 × R12 = R96, and Lebo should get
6 × R12 = R72. Check that R96 + R72 = R168.
ExErCiSE 14.6
1. A recipe to make 15 biscuits asks for the following ingredients:
300 g flour
180 g sugar
200 g butter
How much of each ingredient will you need to make:
a ) 5 biscuits
b ) 10 biscuits
c ) 30 biscuits?
2. A cool drink is made from the following ingredients:
40 ml lime juice
120 ml orange juice 180 ml soda
Calculate how much soda and lime juice you will need for:
a ) 30 ml of orange juice
b ) 360 ml of orange juice
3. Use the recipe from question 1 to answer the following:
a ) If 600 g of flour is used, how much of each ingredient will be
needed?
b ) If 90 g of sugar is used, how much of each ingredient will be
needed?
4. You can make purple paint by mixing blue paint with three times
as much red paint.
a ) With 300 ml red paint, how much blue paint do you need?
b ) With 200 ml blue paint, how much red paint do you need?
Topic 14: Division
Platinum Maths Gr6_Term 2_CAPS.indd 93
93
12/02/13 12:52 AM
Solve problems by comparing two
different quantities
Example
A car maker says that you can drive 100 km on 20 ℓ of petrol.
Calculate how far you can drive on 55 ℓ of petrol.
First find how a far you can drive on one litre of petrol:
100 km
______
= 5 km on one litre of petrol.
20 ℓ
When you know how many kilometres you can drive on 1 ℓ of
petrol, it becomes easy to work out how many kilometers you can
drive on 55 ℓ of petrol:
5 km × 55 ℓ = 275 km
ExErCiSE 14.7
1. Yusuf is paid R115 for working for five hours.
a ) How much is he paid for each hour of work?
b ) How much would he get paid for working for seven hours?
2. A vehicle increases in price by R4 800 over one year.
a ) If the price increased by the same amount each month, what
was the monthly increase?
b ) If this continues, how much more will the vehicle cost in
another three month’s time?
3. A truck travelled 160 km on 10 litres of diesel.
a ) How many kilometres does the truck travel on one litre of diesel?
b ) How far can the truck drive on four litres of diesel?
Would you like a family holiday?
FIVE DAYS
ing
az
at ansaemaside resort
for only
R1 200
(maximum of four people)
94
4. A machine makes 984 spanners every four hours.
a ) How many spanners does the machine make each hour?
b ) How many spanners could it make in 10 hours?
5. A five-day holiday package for a small family costs R1 200.
a ) How much does it cost per day for the family?
b ) Calculate the cost if the family uses this package for a sevenday holiday.
c ) The family consists of three people who each pay the same
amount. What would each person pay per day for the sevenday package?
Term 2
Platinum Maths Gr6_Term 2_CAPS.indd 94
12/02/13 12:52 AM
revision
1. What do you know about the number of lines of symmetry of:
a ) a polygon that has all sides of equal length
b ) a circle
c ) a rectangle?
(2)
(2)
(2)
2. What do you know about the order of rotational symmetry of:
a ) a polygon with equal sides
b ) a circle?
x(2)
(1)
3. Look at the symbols below:
A
B
#
€
C
%
D
E
Write down the letters of the symbols that have:
a ) no line symmetry or no rotational symmetry
b ) line symmetry, but not rotational symmetry
c ) rotational symmetry, but not line symmetry
d ) line symmetry and rotational symmetry.
(1)
(1)
(1)
(1)
4. List the following sets of numbers:
a ) The composite numbers between 20 and 30.
(1)
b ) The prime numbers smaller than 10.
(1)
5. Write each of the following fractions in its simplest form:
4
a ) __
12
(1)
125
b ) ___
250
(1)
6. A recipe uses 500 g of flour and 200 g of sugar.
a ) Write the ratio flour : sugar in its simplest form.
b ) How much flour must you use with 100 g of sugar?
c ) If this recipe makes 20 biscuits, how much flour and how much sugar will
be needed to make 10 biscuits?
(2)
(1)
(2)
7. A printer can print 15 pages in 3 minutes. What is its printing rate in pages per minute?
8. Use the long division method to do the following divisions:
a ) 3 234 ÷ 154
b ) 1 554 ÷ 240
(3)
(3)
Total marks: 30
Revision
Platinum Maths Gr6_Term 2_CAPS.indd 95
(2)
95
12/02/13 12:52 AM
Topic
Decimal fractions
15
Maths ideas
• State the place
value up to two
decimal places.
• Multiply and divide
decimal fractions
by 10 and by 100.
• Recognise
equivalence
between common
and decimal
fractions.
• Count, order and
compare decimals
to two decimal
places.
• Round off decimal
numbers.
• Add and subtract
decimals.
• Do mixed
operations with
decimals.
• Solve problems
with decimals.
Key words
• decimal fractions
– fractions written
as decimal
numbers with a
decimal comma
• decimal places –
the digits after the
decimal comma
read and write decimal fractions
Decimal fractions are used every day when calculating with money
or when measuring length. Both common fractions and decimal
fractions describe parts of a whole. The same number can be
represented as a common fraction (__12 ) or as a decimal fraction (0,5).
You know that 700 ÷ 10 = 70, and 70 ÷ 10 = 7, so what is 7 ÷ 10? You
7
, or you can add more columns
can write this answer as the fraction __
10
to the place value table.
Example
Here is the decimal number 892,73.
We read it as eight hundred and ninety-two comma seven three.
÷10
÷10
÷10
÷10
H
T
U
,
t
h
hundreds
8
800
tens
9
90
units
2
2
,
,
,
tenths
7
hundredths
3
7
__
10
3
___
100
The digits after the comma are called decimal places. They are
smaller than units so we also call them decimal fractions. We can
expand the number 892,73 as 800 + 90 + 2 + 0,7 + 0,03.
7
(seven tenths).
The decimal fraction 0,7 means __
10
3
___
The decimal fraction 0,03 means 100 (three hundredths).
73
(seventy-three hundredths).
The decimal fraction 0,73 means ___
100
1
1
__
1 decimal place: 0,1 = 10 2 decimal places: 0,01 = ___
100
There must be at least one digit in front of the decimal comma.
ExErCiSE 15.1
1. Write the following as decimal fractions:
a ) thirty-one comma one two b ) seventeen comma two three
c ) forty-three comma four five d ) fifty-nine comma six one
2. Write the following numbers in expanded notation:
a ) 24,6
b ) 12,63
c ) 72,3
d ) 47,09
e ) 56,90
f ) 75,03
g ) 89,23
h ) 63,91
96
Term 2
Platinum Maths Gr6_Term 2_CAPS.indd 96
12/02/13 12:52 AM
Count in decimals
Did you know?
To count forwards in decimals, we add the same decimal to each
previous number. You can use a number line to help you.
The word ‘decimal’
means ‘based on ten’.
Our number system
is called the decimal
system because each
place value is ten
times smaller than
the one to its left.
Example
Start at 10 and count forwards in 0,5s.
+ 0,5 + 0,5 + 0,5
10 10,5 11 11,5 12
To count backwards in decimals, you subtract the same decimal
from each previous number.
Example
Start at 6 and count backwards in 0,4s.
− 0,4 − 0,4 − 0,4
4,4 4,8 5,2 5,6
6
Challenge
ExErCiSE 15.2
1.
0 0,1
0,5
1
1,3
• Start at 16 and
count backwards
in 1,5s five times.
What number do
you end at?
2
Copy the number line and mark in the numbers 0,3; 0,7; 1,1; 1,5
and 1,8. Use the number line to write the next four numbers in
the following sequences:
a ) 0,1; 0,2; 0,3; □; □; □; □
b ) 1,7; 1,6; 1,5; □; □; □; □
c ) 0,2; 0,4; 0,6; □; □; □; □
• Start at 16 and
count forwards
in 1,5s five times.
What number do
you end at?
• What is the
difference
between the two
numbers you
ended with?
2. List the next four numbers in the following sequences:
a ) 0,01; 0,02; 0,03; □; □; □; □
b ) 5,51; 5,50; 5,49; □; □; □; □
c ) 25,5; 25,10; 25,15; □; □; □; □
3. Fill in the missing numbers in the following number chains, then
check your final answer with a calculator:
a ) 0 → +0,3 → □ → +0,3 → □ → +0,3 → □ → +0,3 → □ → +0,3 → □
b ) 2 → –0,2 → □ → –0,2 → □ → –0,2 → □ → –0,2 → □ → –0,2 → □
c ) 0 → +0,03 → □ → +0,03 → □ → +0,03 → □ → +0,03 → □
Topic 15: Decimal fractions
Platinum Maths Gr6_Term 2_CAPS.indd 97
97
12/02/13 12:52 AM
Convert decimal fractions
To convert a decimal to a fraction, we write it with denominator 10 or
100, then we change it into its simplest form.
Did you know?
Example
You can use your
calculator to convert
common fractions to
decimal fractions:
75
75 ÷ 5
0,75 = ___
= ______
= __3
100 100 ÷ 5 4
_3 = □
4
3 ÷ 4 = 0,75
If a fraction has a denominator of 10 or 100, it is easy to write it as a decimal.
Example
25
4
Write __
and ___
as decimal fractions.
10
100
4
__
is 4 tenths, so we write it as 0,4 (zero comma four).
10
25
___
is 25 hundredths or 0,25 (zero comma two five).
100
ExErCiSE 15.3
1. Write the following decimals as fractions in their simplest form:
a ) 0,5
b) 0,05 c ) 0,25 d) 0,8
e ) 0,2
f ) 0,4
2. Write the following fractions as decimals:
26
53
4
7
b) __
c) ___
d) ___
a ) __
10
10
100
100
70
f ) ___
100
44
e ) ___
100
20
as a decimal fraction? First write the
How will you write __12 or __45 or __
25
fraction as an equivalent fraction with 10 or 100 as the denominator.
Example
Write the fractions __15 and __34 as decimals.
1 ____
2
__
Remember: what you do with
= 1 × 2 = __
= 0,2
5 5 × 2 10
the denominator, you must
3 _____
75
__
= 3 × 25 = ___
= 0,75
4 4 × 25 100
also do with the numerator.
ExErCiSE 15.4
1. Convert each common fraction to a decimal fraction. Use a
calculator to check your answers.
20
8
2
b) __14
c ) __25
d) __
e ) __
f ) __
a ) __12
25
20
50
2. On the number line, fill in the missing fractions and decimals in
the boxes: 0
1
3
10
0,0
98
4
1
0,25
Term 2
Platinum Maths Gr6_Term 2_CAPS.indd 98
12/02/13 12:52 AM
Multiply and round off decimals
Multiply decimals by 10 and 100
You know how to multiply whole numbers by 10 and by 100. Now you
will see how to do this with decimals.
ExErCiSE 15.5
1. Use your calculator to complete the following sentences:
0,025 × 10 = □
0,25 × 10 = □
2,5 × 10 = □
Challenge
Which is the largest
answer?
Now write down in your own words what happens when you
multiply a decimal fraction by 10.
4,56 × 100
2. Use what you now know to do the following multiplications
yourself. Then check your answer on a calculator.
a ) 0,7 × 10
b ) 1,5 × 10
c ) 0,5 × 10
d ) 22,3 × 10
e ) 12,08 × 10
f ) 1,78 × 10
4 960 × 100
3. Use your calculator to find the answers to the following
multiplications. Write down what you notice.
a ) 340 × 10; 34 × 10; and 3,4 × 10
b ) 3 400 × 100; 340 × 100; and 34 × 100.
0,46 × 1 000
45,5 × 10
476,7 × 10
456 000 × 1 000
round off decimal numbers
You can also round off decimal numbers to estimate answers. When
you round off to units, decimals greater than or equal to 0,5 become 1,
and decimals smaller than 0,5 become zero.
Example
43,42 rounds down to 43, because 0,4 is less than 0,5.
43,5 rounds up to 44, and 43,62 rounds up to 44.
ExErCiSE 15.6
1. Round the following decimals to the nearest unit:
a ) 16,4
b ) 28,7
c ) 71,3
d ) 35,5
e ) 3,28
f ) 74,21
g ) 3,83
h ) 69,72
2. Round the following amounts to the nearest Rand:
a ) R45,32 b ) R5,71
c ) R10,10 d ) R5, 90
e ) R10,50
Topic 15: Decimal fractions
Platinum Maths Gr6_Term 2_CAPS.indd 99
99
12/02/13 12:52 AM
Compare and order decimals
You can work out which decimal numbers are greater by comparing the
place values of the decimal places. This is useful when you are working
with measurements. Just like you have done with whole numbers, you
compare each digit in the numbers, starting from the left.
Example
Which is the longest and which is the shortest:
3,46 km; 3,5 km or 3,45 km?
6 km
4 km + ____
3,46 km = 3 km + ___
10
100
5 km
3,5 km = 3 km + ___
10
5 km
4 km + ____
3,45 km = 3 km + ___
10
100
5 , but 3,45 km and 3,46 km both have only ___
4.
3,5 km has ___
10
10
So 3,5 km is the longest.
6
4 , but 3,46 km has ____
3,45 km and 3,46 km both have ___
10
100
5 . So 3,45 km is the shortest.
and 3,45 km has only ____
100
The distances in order from longest to shortest are: 3,5 km;
3,46 km; 3,45 km.
ExErCiSE 15.7
Who lives closest to school and who lives furthest from school? List
these learners in order from closest to school to furthest from school:
1. Mpho 7,43 km; Tumo 7,23 km; Phil 7,32 km; Lucy 7,24 km
2. Zwane 6,17 km; Ben 6,27 km; Anna 6,21 km; Vusi 6,12 km
3. Sam 8,81 km; Mapelo 8,18 km; Joel 8,88 km; Lufuno 8,17 km
4. Thembi 9,99 km; Sipho 9,09 km; Thuli 9,9 km; Paul 9,19 km
Challenge
Five learners walk to school each day from their
homes. Can you work out where each learner lives?
Nkhandla 5,21 km
Nkwalini 5,02 km
• Phindi lives closer than Chloë but further than Vuyo.
• Lindy lives less than 5 km away.
• Chloë lives more than 5 km away, but not as far
away as Thuli.
Westclif f 4,96 km
Fernkloof 4,89 km
Bergsig 5,12 km
• Vuyo lives closer than Lindy.
100
Term 2
Platinum Maths Gr6_Term 2_CAPS.indd 100
12/02/13 12:52 AM
Add and subtract decimals
When you add and subtract two decimals, you use the same column
method you use for whole numbers. Remember to line up the digits
correctly, according to their place value and including the decimal
comma. Fill in the zeros at the end as placeholders if the number of
decimal places in the two numbers is not the same.
Example
Use the column method of addition and subtraction to find the
total of, and the difference between, 820,45 and 95,3.
Addition
820,45
+ 95,30
915,15
1
Subtraction
8 112 100 , 4 5
−
9 5 ,3 0
7 2 5 ,1 5
7
Game
To add more than two decimals, write the numbers below each other
with the decimal comma in the same place.
Example
F ind the sum of 45,78; 23,6 and 27,45.
4 15,17 8
2 3, 6 0
+ 2 7, 4 5
9 6, 8 3
1
• In pairs, cut out
(or fold and neatly
tear) ten equal
squares of paper.
• On each square,
write down any
number between
0 and 100 with one
or two decimal
places.
• Shuffle the
numbers and place
them face down.
ExErCiSE 15.8
1. First write an estimate for each addition, and then use the column
method to find the exact answer:
a ) 1,2 + 0,9
b ) 2,75 + 0,4
c ) 32,1 + 12,09
d ) 124,89 + 70,33
2. First write an estimate of each subtraction, and then use the
column method to find the exact answer:
a ) 1,2 – 0,9
b ) 2,75 – 1,4
c ) 32,1 – 12,09
d ) 124,89 – 70,33
e ) 21,32 – 8,45
f ) 340,23 – 122,78
3. First write an estimate for each addition, and then use the column
method to find the exact answer:
a ) 21,32 + 8,45 + 0,6
b ) 340,23 + 122,78 + 89,09
• Each player picks
up one paper and
writes down the
number on it.
• Now take turns to
pick up another
paper and add that
to your number to
get a new total.
• Check each other’s
work.
• After you have
picked up all the
cards, the winner is
the player with the
highest final total.
Topic 15: Decimal fractions
Platinum Maths Gr6_Term 2_CAPS.indd 101
101
12/02/13 12:52 AM
More calculations with decimals
Mixed operations
Remember the correct order for mixed operations. Work out brackets
first, then do multiplication and division (from left to right), and then
addition and subtraction (from left to right).
ExErCiSE 15.9
Do the following calculations in the correct order:
1. (3,45 + 2,09) × 10
2. 3,45 + (2,09 × 10)
3. 23,01 – (2,44 + 3,01)
4. 23,01 – 2,44 + 3,01
5. (23,01 – 2,44) + 3,01
6. 24,3 + (2,9 × 10)
Check calculations using inverse operations
In the same way as for whole numbers, you can check addition and
subtraction of decimals with an inverse calculation.
Example
Dumisani adding and
subtracting decimals
Challenge
I am thinking of a
decimal number. I
multiply the number
by 100, then subtract
22,5 to get 90,9. Draw
a flow diagram to
show these
operations, and then
use inverse
operations to work
out the number that I
started with.
102
180,91 + 966,73 = 1 147,64 can be checked with
1 147,64 – 180,91 = 966,73 or
1 147,64 – 966,73 = 180,91
275 – 98,37 = 176,91 can be checked with
176,91 + 98,37 = 275,28
ExErCiSE 15.10
First estimate the answers to the following decimal calculations, and
then find the exact answers. Then check your answers to questions
a) – d) with an inverse calculation.
a ) Find the sum of 5,27 and 12,25.
b ) How much more is 3,46 than 1,5?
c ) How much is 2,3 and 1,25 altogether?
d ) How much less than 5,27 is 2,2?
e ) Find the sum of 3,45; 1,25; 1,5 and 2,3.
f ) Find the sum of 2,2; 5,27 and 12,25.
Term 2
Platinum Maths Gr6_Term 2_CAPS.indd 102
12/02/13 12:52 AM
Solve problems with decimal
fractions
You can use your knowledge of decimals to solve many different types
of problems.
ExErCiSE 15.11
For each problem below, first estimate the answer. Then solve the
problem and finally use a different calculation to check your answer.
1. You have R18,50 and two of your friends have R12,42 and R15,03. How
much more money do you need to hire a taxi for a trip that costs R50?
2. Your mother is mixing fruit juice for a party. The container can hold
20 litres. She pours in 10,4 litres of soda, 3,63 litres of lime juice
and 2,4 litres of pineapple juice. How much orange juice can she
still pour in?
3. You buy two shirts for R35,20 each and a cap. How much did the
cap cost if you get R2,44 change from a R100 note?
4. You want to buy a bicycle from a friend for R150, and so far you
have saved R95. If you earn R16,50 an hour working at a local shop,
will you earn enough extra money in three hours?
5. In your training, you run 8,5 km on each of the first two days, then
7,95 km on the third day and 12,25 km on the fourth day. How
many kilometres have you run in total?
6. Nina brought items costing R45,78, R13,12 and R42,99. Round the
final total to the nearest 5c to calculate how much she paid.
7. Zareenah bought three pencils at R1,84 each from a supermarket
that rounds up totals to the nearest 5c. Calculate how much
change she received from a R10 note.
Challenge
Make a set of digit cards (0 to 9) and a decimal comma card.
• Put the digit cards face down in a pile and then take four cards.
• Use these cards together with the decimal comma card to
make a number with two decimal places that is close to 50.
• Use the same four cards to make a dif ferent number that is
close to 50.
• Which number is closest to 50?
Topic 15: Decimal fractions
Platinum Maths Gr6_Term 2_CAPS.indd 103
103
12/02/13 12:52 AM
Topic
Capacity and volume
16
Maths ideas
The capacity of containers
• Estimate, measure,
compare, record
and order
capacities.
Capacity is the total amount of space inside a container. It tells you
how much the container can hold. Volume is the amount of space
that something takes up.
• Read volumes
off different
containers.
• Convert between
units of capacity.
• Do calculations
and solve
problems involving
capacity.
Key words
Remember that containers are not always filled up. A container
can have space for 1 litre (a capacity of 1 litre) but it may only have
half a litre inside it. In this case, we say the volume of water in the
bottle is __12 litre.
To estimate the capacity of a container you have to think about how
many litres (ℓ) or millilitres (ml) it can hold. You also need to think
about which of these units is the best to use. The capacity of larger
containers is given in litres and the capacity of smaller containers is
given in millilitres. Remember that there are 1 000 ml in 1 litre.
• capacity – the
maximum amount
a container can
hold
• volume – the
amount of space
something takes up
250 ml and one litre
This box has sides of 1 cm and
holds exactly one millilitre.
ExErCiSE 16.1
A 20 ℓ water container
Measuring spoons
104
1. Choose the approximate capacity of each of these objects:
a ) small tea cup: 20 ml, 200 ml, 2 ℓ, 20 ℓ
b ) bottle of medicine: 15 ml, 150 ml, 1,5 ℓ, 15 ℓ
c ) petrol tank of a medium-sized car: 40 ml, 400 ml, 4 ℓ ,40 ℓ
d ) small bottle of eye drops: 10 ml, 100 ml, 1 ℓ, 10 ℓ
e ) large carton of juice: 10 ml, 100 ml, 1 ℓ,10 ℓ
f ) amount of water you should drink each day: 20 ml, 200 ml, 2 ℓ,
20 ℓ
2. Which units (millilitres or litres) would you use to measure the
capacity of the each of these objects?
a ) a glass of juice
b ) a large bottle of cola
c ) a can of cola
d ) a fish tank
e ) a bucket
f ) a rainwater tank
Term 2
Platinum Maths Gr6_Term 2_CAPS.indd 104
12/02/13 12:53 AM
Estimate and measure capacity
You can use a measuring jug to measure liquids in millilitres or litres.
Example
How much liquid is in the container on the right?
First look at the scale. Each litre is divided into 10 equal intervals.
1 000 ml ÷ 10 = 100 ml. So each interval represents 100 ml.
The liquid measures between 1 ℓ 500 ml and 1 ℓ 600 ml, which is
about 1 ℓ 550 ml.
ExErCiSE 16.2
How much liquid is in each jug?
1.
2.
4.
3.
5.
Did you know?
Containers of
different shape can
have the same
capacity. A short fat
container may hold
the same amount as
a tall thin container.
6.
ExErCiSE 16.3
Find six waterproof containers of different shapes, such as a sandwich
box, margarine tub, juice bottle or pudding bowl.
Draw up a table like the one below, for your six containers:
Container
My estimate
Actual capacity
Difference
1. Estimate the capacity of each container.
2. Measure the capacity of each container using a measuring jug.
3. Calculate the difference between your estimate and the actual
measurement. How close were you?
4. Which of your containers holds the most? Which holds the least?
5. Which two containers hold approximately the same amount of
liquid, although they are different shapes?
Challenge
Find four household
containers that are
different shapes but
which each hold
approximately 1 litre.
Topic 16: Capacity and volume
Platinum Maths Gr6_Term 2_CAPS.indd 105
105
12/02/13 12:53 AM
Key words
• kilolitre – one
thousand litres
• convert – change
from one unit into
another unit
Convert units of capacity
The capacity of most of the containers we use every day can be
measured in millilitres or litres. But some very big containers can
hold thousands, or even millions of litres. The capacity of very big
containers is measured in kilolitres (kl). One kilolitre is 1 000 litres.
Greater to smaller: multiply
Smaller to greater: divide
litres
millilitres
× 1 000
millilitres
÷ 1 000
litres
Example
Convert from litres to millilitres:
2,5 ℓ × 1 000 = 2 500 ml = 2 ℓ 500 ml
0,75 ℓ × 1 000 = 750 ml
0,04 ℓ × 1 000 = 40 ml
Convert from millilitres to litres:
2 860 ml ÷ 1 000 = 2,86 ℓ = 2 ℓ 860 ml
500 ml ÷ 1 000 = 0,5 ℓ = _12 ℓ
70 ml ÷ 1 000 = 0,07 ℓ
Example
Convert from litres to kilolitres:
2 700 ℓ ÷ 1 000 = 2,7 ℓ = 2 ℓ 700 ml
450 ℓ ÷ 1 000 = 0,450 ℓ
90 ℓ ÷ 1 000 = 0,09 ℓ
Convert from kilolitres to litres:
2,56 kl × 1 000 = 2 560 ℓ
0,85 kl × 1 000 = 850 ℓ
0,03 kl × 1 000 = 30 ℓ
ExErCiSE 16.4
Petrol storage tanks like
these have a capacity
of 2 000 kl. This is
2 000 000 litres.
106
1. Convert the following capacities to millilitres:
a) 5 ℓ
b) 8 ℓ
c)
d ) 5 ℓ 750 ml
e ) 4,68 ℓ
f)
g ) 4,05 ℓ
h ) 3 ℓ 500 ml
i)
j ) 2,19 ℓ
k ) 8,05 ℓ
l)
3 ℓ 456 ml
7,39 ℓ
3,56 ℓ
0,14 ℓ
2. Convert following capacities to litres:
a ) 4 960 ml
b ) 8 260 ml
d ) 1 280 ml
e ) 1 ℓ 450 ml
g ) 5 ℓ 500 ml
h ) 7 ℓ 70 ml
j ) 900 ml
k ) 450 ml
2 000 ml
5 ℓ 280 ml
300 ml
93 ml
c)
f)
i)
l)
Term 2
Platinum Maths Gr6_Term 2_CAPS.indd 106
12/02/13 12:53 AM
Compare and order capacities
Challenge
To compare measures of capacity, first convert them to the same unit
of measurement.
You can convert a mixed measure, such as 4 ℓ 500 ml, to either litres
expressed as a decimal, or to millilitres.
Example
Order these capacities from the smallest to the greatest:
4 509 ml; 4 ℓ 450 ml; 4,49 ℓ; 4,5 ℓ
Convert all the measurements to millilitres:
4 509 ml
4 ℓ 450 ml
4,49 ℓ
4 509 ml
4 450 ml
4 490 ml
During a flood all the
sluice gates at the
Gariep Dam were
opened and water
flowed out of them at
a rate of 2 300 kl per
second.
a) How many litres is
this per second?
b) How many
kilolitres is this
per minute?
4,5 ℓ
4 500 ml
Compare and write them in order from smallest to greatest:
4 450 ml
4 500 ml
<
4 509 ml
Change them back to the original measurements:
4 ℓ 450 ml
<
4,49 ℓ
<
4,5 ℓ
<
4 509 ml
<
4 490 ml
<
You can also use your knowledge of fractions to compare capacities.
Example
The labels on three containers give the capacities as
1__12 litres, 1,8 litres and 1 litre 750 ml. To compare these
5
, so 1__12 litres is 1,5 litres.
capacities, use the fact that __12 = __
10
Study the capacity fractions.
1
100 ml __
of a litre 0,1 litres
10
750
75
As a fraction of 1 litre, 750 ml is _____
= ___
= 0,75, so
1 000 100
1 litre 750 ml is the same as 1,75 litres.
In ascending order, the capacities are 1,5; 1,75; 1,8, or
1__12 litres; 1 litre 750 ml and 1,8 litres.
250 ml __14 of a litre
0,25 litres
500 ml __12 of a litre
0,5 litres
750 ml __34 of a litre
0,75 litres
ExErCiSE 16.5
Write each set of measurements in order from the smallest to the greatest:
1.
2.
3.
4.
5.
2 960 ml
3,560 ℓ
550 ml
67,8 ℓ
2__14 ℓ
6. 7__34 ℓ
7. 3,04 ℓ
3,49 ℓ
3,490 ℓ
0,5 ℓ
6 780 ml
2,3 ℓ
1__12 ℓ
3_12 ℓ
0,56 ℓ
6__34 ℓ
2,28 ℓ
2 ℓ 560 ml
3 400 ml
99 ml
6,09 ℓ
2 200 ml
7 800 ml
3__12 ℓ
7 ℓ 90 ml
3 390 ml
7,89 ℓ
3,45 ℓ
Topic 16: Capacity and volume
Platinum Maths Gr6_Term 2_CAPS.indd 107
107
12/02/13 12:53 AM
Solve problems involving capacity
Example
a) Find the total of 3,4 ℓ; 2 850 ml and 5 ℓ 378 ml.
Convert to millilitres:
3 400 ml + 2 850 ml + 5 378 ml
1 1
3 4 00
Add:
2 8 50
+ 5 3 78
1 1 6 2 8 ml
b) What is the capacity of f ive
750 ml bottles of water?
Give the answer in litres.
Multiply: 27 5 0
+
5
3 7 5 0 ml
Convert to litres:
3 750 ml ÷ 1 000 = 3,75 ℓ
ExErCiSE 16.6
Challenge
Amount of water used
for everyday activities
Flushing toilet 8 ℓ
Washing yourself 4 ℓ
Washing dishes 5 ℓ
Washing hands 2 ℓ
Five-minute shower 30 ℓ
Cooking 2 ℓ
Brushing teeth 1 ℓ
A bath 150 ℓ
A glass of water 250 ml
Estimate how much water
your family uses in a
week.
Start by making a list of all
the ways in which your
family uses water. Then
use the table to estimate
how much they use.
1. One container holds 1,6 ℓ, a second holds 1 400 ml and a third
holds 2 ℓ 350 ml. What is their total capacity?
2. Jenna drank 1 800 ml of water today and John drank 1,45 ℓ. How
much more water did Jenna drink than John?
3. A large container of washing-up liquid holds 5 ℓ.
a ) How many 250 ml bottles can you fill from this?
b ) If a 5 litre container costs R47,25 what is the price per litre?
4. One glass holds 250 ml of juice. How much juice will I need for 24
learners if each learner gets one glass of juice?
5. Sindi makes 3 ℓ of fruit punch with lemonade, pineapple juice
and apple juice. She uses 450 ml of pineapple juice and 1,8 ℓ of
lemonade. How many litres of apple juice are in 3 ℓ of fruit punch?
6. Which is more: five bottles of 275 ml or 1,5 ℓ?
7. A medicine bottle holds 150 ml. A teaspoon measure is 5 ml. The
dosage of the medicine is 2 teaspoons three times a day. How
many days will the medicine last?
8. Phindi drank five 350 ml glasses of water and her friend drank
three 500 ml bottles of water. Who drank more water?
9. A small farm dam holds 900 kl of water. In the summer, about
200 ℓ is lost each day due to evaporation. How much water will
be left in the dam after five days at this rate of evaporation?
108
Term 2
Platinum Maths Gr6_Term 2_CAPS.indd 108
12/02/13 12:53 AM
revision
1. Write the following fractions as decimals:
5
a ) __
10
(1)
71
b ) ___
100
(1)
2. Write the following decimals as common fractions in their simplest form:
a ) 0,8
b ) 0,03
(1)
(1)
3. Write the value of each digit in this number: 315,24.
(2)
4. Which is the greater in the following pairs of decimals?
a ) 76,79 or 76,81
b ) 24,31 or 24,4
(1)
(1)
5. Order the following decimals from smallest to largest:
a ) 7,24; 7,43; 7,34; 7,54; 7,3
b ) 28,76; 28,6; 28,67; 28,09; 28,69
(2)
(2)
6. Add the following decimals:
a ) 23,5 + 24,2
b ) 127,4 + 185,8
(1)
(1)
7. How much liquid is in each of these containers?
a)
b)
(2)
8. Convert the following capacities to millilitres:
a ) 2,569 ℓ
b ) 0,05 ℓ
(1)
(1)
9. Convert the following capacities to litres:
a ) 3 600 ml
b ) 30 ml
(1)
(1)
10. Write the following capacities in order from smallest to greatest:
a ) 3 967 ml; 3,98 ℓ ;3 ℓ 950 ml; 3,9 ℓ
b ) 235 ml; 0,23 ℓ ;0,224 ℓ 231 ml
(2)
(2)
11. Solve the following word problems:
a ) Three jugs hold 1,38 ℓ; 2 ℓ 730 ml and 1 500 ml. What is their total capacity?
(3)
b ) How many 5 ℓ buckets can you fill from a 12 695 ℓ water tank?
(3)
Total marks: 30
Revision
Platinum Maths Gr6_Term 2_CAPS.indd 109
109
12/02/13 12:53 AM
3 Term 3
This photograph shows that about 90% of an iceberg is under water.
110
Platinum Maths Gr6_Term 3_CAPS.indd 110
02/02/13 1:41 AM
Topics 17–27
Starting off
The North and South poles of the Earth are
covered by ice. Some of this ice breaks off and
floats away. The floating chunks of ice are called
icebergs. The word ‘berg’ means mountain in a
number of languages, including Afrikaans.
The ratio of the part of the iceberg that is
above water to the part that is under water
is about 1 : 9.
1. If the total mass of an iceberg is 10 000
tonnes, what mass of the iceberg is:
a ) above water?
b ) under water?
2. a ) What do you think the phrase ‘the tip of
the iceberg’ means?
b ) Write a sentence in which you use this
phrase in an everyday situation.
The ice at the North and South poles is slowly
melting, because the Earth is gradually getting
warmer and warmer. We call this problem
global warming.
3. Find out how recycling can help to slow
down global warming.
Content covered in Term 3
Topic 17: Whole numbers, Topic 18: Mass, Revision, Topic 19: Addition and
subtraction, Topic 20: Viewing objects, Revision, Topic 21: Properties of 2D
shapes, Topic 22: Transformations, Revision, Topic 23: Percentages,
Topic 24: Temperature, Revision, Topic 25: Data handling, Project,
Topic 26: Numeric patterns, Revision, Topic 27: Length
111
Platinum Maths Gr6_Term 3_CAPS.indd 111
02/02/13 1:41 AM
Count, order, compare and represent
whole numbers
Topic
17
Maths ideas
• Recognise place
value in nine-digit
numbers.
• Write nine-digit
numbers in
expanded form.
• Read, write and
order nine-digit
numbers.
Place value and expanded notation
Do you remember how to read the number 321 206 493? We say three
hundred and twenty-one million, two hundred and six thousand, four
hundred and ninety-three.
You can use place value to write large numbers in expanded form.
Example
Write 127 567 834 as an expanded addition:
127 567 834 = 100 000 000 + 20 000 000 + 7 000 000 + 500 000 +
60 000 + 7 000 + 800 + 30 + 4
If the expanded form is not in place value order, look carefully at the
number of digits in each part. Write the parts in descending order
(from the largest to the smallest). In the examples below, the numbers
in brackets show how many digits are in the number.
Example
Write this expanded addition as a whole number:
7 000 + 5 000 000 + 6 + 40 000 + 200 000 + 30 + 200 + 10 000 000 + 200 000 000
(4)
(7)
(1)
(5)
(6)
(2) (3)
(8)
(9) ← Number of digits
= 200 000 000 + 10 000 000 + 5 000 000 + 200 000 + 40 000 + 7 000 + 200 + 30 + 6
= 215 247 236
ExErCiSE 17.1
Challenge
How much must you
add to each number
to get 100 000 000?
• 99 999 999
• 88 888 888
• 77 777 777
• 66 666 666
• 55 555 555
112
1. Write down each number in its expanded form, and give the value
of each underlined digit:
a ) 12 458 239
b ) 53 568 210
c ) 184 203 176
d ) 215 007 814
e ) 309 141 758
f ) 593 087 915
2. Write each expanded addition as a whole number, and then write
the number in words:
a ) 60 000 000 + 4 000 000 + 400 000 + 10 000 + 3 000 + 500 + 30 + 9
b ) 900 000 000 + 90 000 000 + 4 000 000 + 300 000 + 7 000 + 400
+ 60 + 6
c ) 8 000 + 90 + 3 + 100 000 + 5 000 000 + 40 000 + 60 000 000 +
700 + 900 000 000
d ) 600 + 200 000 000 + 4 + 30 000 000 + 9 000
Term 3
Platinum Maths Gr6_Term 3_CAPS.indd 112
02/02/13 1:41 AM
read, write and order large numbers
When you write large numbers, remember to group the digits in
threes from the units digit.
ExErCiSE 17.2
1. Write down the following numbers:
a ) 8 more than 99 999
b ) 9 less than 100 005
c ) 200 more than 99 998
d ) 175 less than 100 030
e ) 2 more than 999 999
f ) 5 less than 1 000 003
g ) 1 000 more than 9 999 000 h ) 30 less than 99 999 990
i ) 300 less than 10 000 000
j ) 2 000 less than 10 000 000
k ) 100 000 less than 10 000 000 l ) 800 000 less than 10 000 000
2. Complete the following sequences:
a ) 19 999 980; 19 999 985; □; 19 999 995; □
b ) 120 000 200; □; 120 000 600; □; 120 001 000; □
c ) 155 200 143; 155 200 147; □; □; 155 200 159; □
d ) 250 250 250; □; 250 250 350; □; □; 250 250 500
Challenge
Here is an interesting
pattern with large
numbers.
11 × 11 = 121
111 × 111 = 12 321
1 111 × 1 111 = …
Use a calculator to
find the next few
numbers in this
pattern and make a
prediction about
111 111 × 111 111.
Remember that when you order numbers, you compare the place
value of digits starting from the left.
ExErCiSE 17.3
1. Write the following numbers in descending order:
a ) 18 764 321; 18 674 321; 18 674 231; 18 764 123; 18 467 321; 18 467 123
b ) 124 689 135; 124 869 135; 124 968 153; 124 896 135; 124 189 635; 124 168 539
c ) 278 356 281; 278 536 281; 278 235 681; 278 512 638; 278 635 182; 278 135 628
2. Write the following numbers in ascending order:
a ) 18 764 321; 18 674 321; 18 674 231; 18 764 123; 18 467 321; 18 467 123
b ) 224 689 135; 224 869 135; 224 968 153; 224 896 135; 224 189 635; 224 168 539
c ) 478 356 281; 478 536 281; 478 235 681; 478 512 638; 478 635 182; 478 135 628
Topic 17: Count, order, compare and represent whole numbers
Platinum Maths Gr6_Term 3_CAPS.indd 113
113
02/02/13 1:41 AM
Topic
Mass
18
Maths ideas
• Read scales.
• Estimate and
measure mass.
• Compare and
order mass.
Units of measure for mass
You have worked with two units of mass: grams and kilograms. Very
light objects like the eraser on the top of a pencil, a button, a large
paperclip or a piece of paper have a mass of approximately one gram.
Heavier objects like a one-litre bottle of water or six bananas have a
mass of approximately one kilogram.
• Read and
record mass
measurements.
1 kg (kilogram) = 1 000 g (grams)
• Do calculations
with mass.
Example
• Convert different
units of mass.
• Solve problems
about mass.
Key words
• mass – the
amount of matter
contained in an
object
One of these masses is the approximate mass of a man:
95 g; 950 g; 95 kg; 190 kg.
Which mass is most likely to be correct?
95 g is less than a block of butter, so it is too light.
950 g is less than a kilogram bag of sugar, so it is too light.
190 kg is about the mass of a male lion, and while some people may
have this mass, it is too much to represent the mass of most men.
So 95 kg is the approximate mass of a man.
ExErCiSE 18.1
How much can you remember about the approximate masses of
everyday objects? What is the approximate mass of each of these
objects? Choose the closest mass.
Did you know?
In everyday
language, we often
use the word weight
for mass.
In physics and
engineering, weight
strictly means how
heavy the object is.
We measure weight
in units of force.
114
1.
15 g
150 g
1,5 kg
15 kg
2.
270 g
2,7 kg
27 kg
2 700 kg
3.
8g
80 g
800 g
8 kg
4.
10 g
100 g
1 kg
10 kg
5.
50 g
500 g
5 kg
50 000 kg
6.
35 g
350 g
3,5 kg
35 kg
Term 3
Platinum Maths Gr6_Term 3_CAPS.indd 114
02/02/13 1:41 AM
read measuring scales
When you need to weigh something accurately, you use scales like the
ones in the pictures.
The type of scale you use depends on the size or type of the object
whose mass you want to find. An analogue scale has a pointer that
points to the mass of the object on the scale. Digital scales show the
mass in numbers, and are more accurate for measuring mass.
A
B
ExErCiSE 18.2
1. Match the objects below with one of the scales shown on
the right:
a)
b)
c)
C
D
d)
e)
E
2. How many of each item would you
need to balance the scale?
a ) paper clips
b ) 250 g tubs of margarine
c ) 100 g packets of nuts
3. Choose weights from the ones
shown on the right to balance a scale
containing each of the following
objects:
a ) 350 g of nuts
b ) 1,8 kg of rice
c ) 1,25 kg of sugar
Topic 18: Mass
Platinum Maths Gr6_Term 3_CAPS.indd 115
115
02/02/13 1:42 AM
Estimate and measure mass
It is very useful to be able to find the mass of objects accurately. A
gram is very small, so you can only work out an approximate mass
on some scales. Make sure that nothing else is pressing down on the
object when you read the scale, especially your own hands.
Example
What is the approximate mass of the apples in the picture on the left?
Each kilogram is divided into 10 equal sections, or 100 g. The
pointer is about __12 of the way between 900 g and 1 kg. So the mass
of the apples is about 950 g.
A kitchen scale
ExErCiSE 18.3
1. Copy and complete the table below, by first estimating and then
using a measuring scale:
Object
Grams or kilograms?
Estimated mass
Actual mass
10 pencils
5 maths books
Your shoe
A chair
2. Choose two more objects to weigh. Write them into the table.
Challenge
Look around your
classroom and try to
find an object to put
into each of the rows
in this table.
Measure the mass of
each object using a
scale and round it to
the nearest 10 g:
Less than 100 g
3. Decide which is the best unit of measure for each object: grams
or kilograms.
4. Hold each object in your hands to estimate its mass.
5. Weigh the object carefully. Choose the best scale for the object.
ExErCiSE 18.4
Write these objects in order from lightest to heaviest:
1.
100 g–250 g
250 g–500 g
500 g–750 g
750 g–1 kg
1 kg
2.
More than 1,5 kg
116
Term 3
Platinum Maths Gr6_Term 3_CAPS.indd 116
02/02/13 1:42 AM
Convert between units of mass
You can convert measurements of mass from one unit to another.
Greater to smaller: multiply
Smaller to greater: divide
tonnes
kilograms
kilograms
grams
× 1 000
kilograms
grams
÷ 1 000
tonnes
kilograms
Example
Multiply or divide by 1 000 to convert these measurements:
2 860 g ÷ 1 000 = 2 kg 860g = 2,86 kg
4,56 kg × 1 000 = 4 560 g
5 700 g ÷ 1 000 = 5 kg 700g = 5,7 kg
2,5 kg × 1 000 = 2 500 g
350 g ÷ 1 000 = 0,35 kg
0,75 kg × 1 000 = 750 g
70 g ÷ 1 000 = 0,07 kg
0,04 kg × 1 000 = 40 g
ExErCiSE 18.5
Did you know?
1. Convert from kilograms to grams:
a ) 5,65 kg
b ) 7,4 kg
c ) 0,75 kg
d ) 0,09 kg
2. Convert from grams to kilograms:
a ) 4 070 g
b ) 8 200 g
c ) 220 g
d ) 750 g
3. How many grams are in each of the following?
b ) __12 of a 1 kg c ) 0,1 of 10 kg
a ) __34 kg
You can also compare masses that are whole numbers or decimals. It
is helpful to first convert the masses to the same unit of measurement.
The Babylonians
were the first people
to use a standard
measure for mass.
They used a set of
stones of different
masses to compare
to the mass of other
objects. The ‘stone’ is
still used today as a
measure of mass in
some countries.
ExErCiSE 18.6
1. A supermarket prints out labels for some fruit and vegetables.
Order the masses from lightest to heaviest.
a)
b)
c)
Fruit
Mass
Vegetable
Mass
Fruit
Mass
pears
3,5 kg
carrots
2,4 kg
pineapples
475 g
bananas
peaches
plums
350 g
0,36 kg
39 g
onions
potatoes
pumpkins
2 050 g
2,35 kg
2 450 g
f igs
oranges
apples
0,48 kg
450 g
0,5 kg
Topic 18: Mass
Platinum Maths Gr6_Term 3_CAPS.indd 117
117
02/02/13 1:42 AM
Did you know?
• The Egyptians and
Greeks used wheat
seed as a standard
measurement.
• The Arabs started
a weight standard
for gold, silver and
precious stones
to allow them to
trade or barter.
Solve problems involving mass
ExErCiSE 18.7
Solve the following word problems about mass. Remember to convert
the measurements to the same units before you begin:
1. What is the total mass of 2,3 kg, 500 g and 3 kg 450 g? it should be in kg
2. What mass is 500 g less than 4,32 kg?
3. How much lighter is 345 g than 1,1 kg?
4. The heaviest mass allowed on a ride at the fairground is 55 kg.
Diana’s mass is 28,1 kg and Dave’s mass is 27 000 g. Can they go on
the ride together?
5. a ) The average mass of one peach is 175 g. What is the mass of
nine peaches? Give your answer in kilograms.
b ) If 57 peaches cost R19, what is the cost of 1 peach?
6. a ) The mass of eight apples is 1 200 g. What is the average mass
of one apple in kilograms?
b ) A 5 kg bag of apples costs R65. What is the cost of 500 grams
of apples?
c ) Approximately how many apples will have a mass of 1 kilogram?
7. Use your answers in questions 5 and 6 to answer the questions.
a ) A shopkeeper stocks 1 980 peaches and 120 five-kilogram
bags of apples. What is the total mass of this fruit in kilograms?
b ) What income will she get if she sells all of this fruit?
8. Which has a greater mass: twenty-eight 35 g packets of chips or
three 400 g bags?
9. Dried dog food comes in 5 kg bags. A dog eats 400 g each day.
How long will the food last?
10. The masses of the items in Nombulelo’s shopping are 450 g,
0,25 kg, 650 g and 1,3 kg. The doctor said that she must not
carry more than 4 kg. She still needs some potatoes. What is the
heaviest mass of potatoes she can add to her bag?
Nombulelo’s shopping
118
Term 3
Platinum Maths Gr6_Term 3_CAPS.indd 118
02/02/13 1:42 AM
revision
1.
2.
3.
4.
5.
6.
7.
8.
9.
Write down the number that is 5 more than 8 009 998.
Write down the number that is 200 less than 5 162 170.
Write down the number that is 70 more than 45 569 788.
Write down the number that is 1 000 less than 210 000 000.
Write the following numbers in ascending order:
a ) 1 376 475; 1 376 500; 1 367 475; 1 367 574
b ) 95 505 505; 95 550 550; 95 550 505; 95 505 550
Write the following numbers in expanded notation:
a ) 123 787 224
b ) 368 000 909
Write each of the following expanded additions as a whole number:
a ) 400 000 000 + 3 000 000 + 70 000 + 4 000 + 800 + 70 + 5
b ) 600 + 60 + 60 000 + 6 + 600 000 + 6 000 000
What is the most likely mass of each of the following objects?
a ) an onion: 10 g; 100 g; 1 kg; 10 kg
b ) 10 pencils: 6 g; 60 g; 600 g; 6 kg
c ) a new-born baby: 3 g; 30 g; 3 kg; 30 kg
d ) a man: 8 g; 80 g; 8 kg; 80 kg
What masses do these scales show?
a)
b)
(1)
(1)
(1)
(1)
(2)
(2)
(1)
(1)
(1)
(1)
(1)
(1)
(1)
(1)
(2)
10. Write each of the following masses in grams:
a ) 2,595 kg
b ) 7,6 kg
c ) 0,32 kg
11. The mass of five pears is 900 g.
a ) What is the mass of one pear?
b ) What is the mass of 12 pears? Give your answer in kilograms.
12. If 1 kg of cheese costs R55, what will 200 g of cheese cost?
13. Nomfundo buys these vegetables and fruits: a cabbage with mass __14 kg ; a bag of potatoes
(1)
(1)
(1)
(1)
(1)
(2)
with mass __12 kg; avocados in a bag with mass __25 kg; a green pepper with mass __15 kg; a bag of
tomatoes with mass __34 kg.
a ) Order the objects from the heaviest to the lightest, and find the total mass.
(3)
b ) Round off all the masses to the nearest 100 g.
(2)
Total marks: 30
Revision
Platinum Maths Gr6_Term 3_CAPS.indd 119
119
02/02/13 1:42 AM
Topic
Addition and subtraction
19
Maths ideas
Estimate answers
• Estimate answers
by rounding.
It is useful to be able to round large numbers. This will help when you
want to estimate an answer to a calculation with large numbers.
• Add and subtract
whole numbers.
• Work with multiple
operations.
• Use inverse
operations.
• Work with a
calculator.
• Solve multistep problems
involving addition
and subtraction.
Example
Round the numbers 34 909 812, 210 328 426 and 301 286 502 to
the nearest 1 000.
Look at the hundreds digit each time.
800 is larger than 400, so 34 909 812 rounds up to 34 910 000.
400 is smaller than 500, so 210 328 426 rounds down to 210 328 000.
500 rounds up, so 301 286 502 becomes 301 287 000.
ExErCiSE 19.1
Round the following numbers to the nearest 1 000:
1. 4 560 909
2. 39 567 190
3. 256 299 589 4. 382 726 840
5. 30 987 340
6. 3 294 128
7. 95 817 230
8. 491 939 573
ExErCiSE 19.2
1. Estimate the answers to the following calculations. Then use a
calculator to check how accurate your estimate was.
a ) 1 234 322 + 1 144 564
b ) 2 998 322 – 1 775 645
c ) 30 265 661 + 120 103 454
d ) 3 447 921 – 1 999 200
e ) 556 342 843 + 572 543 354 f ) 600 875 545 – 300 875 344
2. Round to estimate the answers to the following calculations.
Remember to calculate the brackets first. Then use a calculator to
check how accurate your estimate was.
a ) (34 909 100 – 12 456 870) + 8 950 120
b ) 34 909 100 – (12 456 870 + 8 950 120)
c ) (420 455 320 + 654 389 761) – 330 567 103
d ) 420 455 320 + (654 389 761 – 330 567 103)
3. a ) List the prime numbers between 20 and 40.
b ) Round off each of these prime numbers to the nearest 10.
c ) Add all these prime numbers together. Is this total number a
prime number?
120
Term 3
Platinum Maths Gr6_Term 3_CAPS.indd 120
02/02/13 1:42 AM
Use the column method to add
In Topic 3, you used the column method of addition. This is a
convenient method for adding large numbers. Line up the numbers
according to their place value, one under the other, and add each
column starting with the units.
Example
Column method – with ‘carrying over’:
If numbers in a column add up to more than ten, carry over the
tens to the column on the left.
Calculate 230 954 675 + 45 670 023 + 201 632 526.
2 3 20 19 5 14 16 17 5
4 5 67 0 0 23
+20 1 63 2 5 26
47 8 25 7 2 24
The answer is 478 257 224.
Example
10 345 732 + 91 299 855.
To estimate the answer, round to the nearest 1 000.
10 346 000 + 91 300 000 =
10 346 000
+ 91 300 000
101 646 000
The estimated answer is 101 646 000.
Now calculate the exact answer to see how reasonable the
estimate was:
10¹3¹4¹5 732
+ 91 2 9 9 855
101 6 5 5 587
The answer is 101 655 587.
ExErCiSE 19.3
1. First estimate each answer and then find the exact answer using
the column method:
a ) 176 249 + 0 + 890 385
b ) (340 357 + 119 238) + 0
c ) 32 046 290 + 43 748 341
d ) 69 290 453 + 94 990 909
e ) 117 454 454 + 11 909 909 + 190 231 044
f ) 230 394 506 + 116 460 079 + 56 043 921
2. What do you notice when you add 0 to a number?
Topic 19: Addition and subtraction
Platinum Maths Gr6_Term 3_CAPS.indd 121
121
02/02/13 1:42 AM
Use the column method to subtract
You can also use the column method to subtract large numbers.
Remember that if you want to subtract a larger digit from a smaller digit,
you can exchange a unit in one column for 10 in the column on its right.
Example
Find the difference between 212 198 431 and 439 239 511.
Write the larger number first.
4 3 9 12 13 9 45 11 1
–
21 2 1 9 8 4 31
=
227 0 4 1 0 80
The answer is 227 032 080.
ExErCiSE 19.4
Estimate each answer, and then do the exact calculations using the
column method:
1. 176 249 – 90 385
2. 5 046 290 – 3 748 341
3. 7 454 454 – 1 990 909
4. 42 943 605 – 91 604 709
5. 110 544 356 – 80 246 642
6. 120 298 003 – 10 842 114
7. 230 357 634 – 189 324 883
8. 365 556 873 – 215 783 147
You can also use a calculator to work with large numbers.
ExErCiSE 19.5
Copy and complete the table. Round the numbers and then estimate
the answer. Use your calculator to find the actual answer and then the
difference between the two answers.
Calculation
1.
2.
3.
4.
5.
Rounding
Estimated
answer
Actual
answer
Difference between
the two answers
245 389 + 124 788
5 120 477 + 993 099
10 518 586 + 30 317 544
90 818 546 – 77 516 524
655 715 077 – 100 237 099
122
Term 3
Platinum Maths Gr6_Term 3_CAPS.indd 122
02/02/13 1:42 AM
Addition and subtraction are
inverse operations
Remember that you can use addition to check the answer to a
subtraction, and subtraction to check an addition answer, because
they are inverse (opposite) operations.
Example
Choose an inverse operation to check this addition:
5 170 914 + 4 962 758 = 10 133 672
This addition calculation can be checked with:
10 101 133 23 166 67 122
− 5 1 7 0 9 1 4
4 9 6 2 7 5 8
9
or
10 101 133 23 166 67 122
− 4 9 6 2 7 5 8
5 1 7 0 9 1 4
9
ExErCiSE 19.6
For each calculation, first write down an estimate, then calculate the
exact answer, and finally use an inverse operation to check your answer.
If you are short of time, you can use a calculator to quickly check your
answer.
1. 375 343 + 894 505
2. 5 583 244 – 2 900 378
3. 4 599 836 + 3 170 356
4. 46 388 567 – 34 074 744
5. 41 395 211 + 28 882 646
6. 13 564 854 – 10 435 399
7. 150 056 645 + 119 004 881
8. 318 437 066 – 95 894 329
ExErCiSE 19.7
Use a calculator to complete each number statement:
1. 12 759 164 + □ = 56 278
2. 5 816 581 – □ = 978 657
3. □ – 3 485 728 = 6 925 245
4. □ + 23 759 232 = 98 752 394
Challenge
South Africa
produces 229 945
barrels of oil a day
and uses 502 180
barrels a day. How
many barrels does
South Africa have to
import into the
country in a 30-day
month?
Topic 19: Addition and subtraction
Platinum Maths Gr6_Term 3_CAPS.indd 123
123
02/02/13 1:42 AM
Add and subtract with brackets
When working with brackets, remember to first do the calculation
inside the brackets.
ExErCiSE 19.8
1. First estimate and then do the following calculations:
a ) (453 942 + 275 931) – 317 908
b ) (1 580 342 + 1 385 211) – 488 011
c ) 30 689 755 – (10 221 007 + 199 676)
d ) 902 879 355 – (302 345 757 + 122 288 533)
2. Work out the answers to the following calculations:
a ) (154 899 + 176 807) – (109 344 + 89 003)
b ) (340 376 933 – 280 175 698) + (301 438 544 – 110 237 843)
c ) (7 543 128 – 3 241 352) – (2 123 473 – 0)
d ) (16 428 100 + 160 189 655) – (20 274 203 – 9 189 999)
ExErCiSE 19.9
1. Use a calculator to do the following calculations:
a ) (94 562 093 + 64 572 365) – 26 340 526
b ) 94 562 093 + (64 572 365 – 26 340 526)
c ) (94 562 093 + 64 572 365) + 26 340 526
d ) 94 562 093 + (64 572 365 + 26 340 526)
e ) (94 562 093 – 64 572 365) + 26 340 526
f ) 94 562 093 – (64 572 365 + 26 340 526)
2. Which of the calculations in question 1 give the same answer?
Why?
3. Calculate the following to work out which number sentence is not
correct:
a ) 13 632 178 + 11 921 345 = 11 921 345 + 13 632 178
b ) 21 934 521 + 9 881 259 = 9 881 259 + 20 934 501
c ) 56 823 465 + 31 034 574 = 31 034 574 + 56 823 465
124
Term 3
Platinum Maths Gr6_Term 3_CAPS.indd 124
02/02/13 1:42 AM
Solve addition and subtraction
problems
You can use the methods that you have learnt to solve addition and
subtraction problems with larger numbers. Remember to set out
your work correctly and show all your calculations. First estimate your
answer before working out the exact calculation. Check your answer
by using an inverse operation.
ExErCiSE 19.10
1. A bakery bakes 2 625 000 bread rolls for distribution. The bakery
sells 250 585 to one supermarket, 755 250 to another and
1 325 200 rolls to a third supermarket.
a ) How many bread rolls have been sold?
b ) How many bread rolls are left?
2. A space shuttle has enough fuel left to travel 153 300 km. If it
travels 46 710 km in its next orbit around the earth, and then
59 900 km in the following orbit, will it have enough fuel for a third
orbit of 50 000 km?
3. A box is packed with four packets with the following masses:
102 450 g; 450 987 g; 202 340 g and 90 345 g. If the box can
carry 1__12 tonnes, how much mass can still be packed in?
4. A pear weighs 164 g. A banana weighs 175 g. A pineapple weighs
523 g. A watermelon weighs 965 g. If a truck carries 250 of each
fruit, what is the total mass of all the fruit?
5. In a country of 320 126 547 people, 103 053 462 are women and
96 148 398 are men. How many children are there?
6. A wholesaler sells a fridge for R2 399 and a stove for R3 179. How
much money will it receive if it sells 10 000 of each of these?
7. Three cycling clubs are training for a marathon. In the first week,
the members of Club A have cycled a total of 350 543 km and
the members of Club B have cycled a total of 404 220 km. If the
total distance cycled by all three clubs is 1 000 000 km, find the
difference cycled between the club with the largest total and the
club with the smallest total.
Topic 19: Addition and subtraction
Platinum Maths Gr6_Term 3_CAPS.indd 125
125
02/02/13 1:42 AM
Topic
Viewing objects
20
Maths ideas
• Link the position
of the viewer to
views of single or
composite objects.
• Work with different
views of geometric
objects.
Different viewpoints
Objects look different depending on where you are standing. Your
house would look different from usual if you looked down at it from
an aeroplane. How an object looks depends on your viewpoint, or
where you are looking from.
Imagine what you would look like to a person standing on your lefthand side, or on your right-hand side, or directly above your head or
directly in front of you. Use a mirror or camera to see if your thinking
was correct.
ExErCiSE 20.1
1. Examine the picture of a library building, and then study the views
that are shown below:
A
B
C
a ) In each case, identify where the person was standing when
they drew that view of the library.
b ) Explain how you made your choice in question a).
2. Look at the pictures below. Match each view to the position of the
person drawing the picture.
a)
Key words
• viewpoint – the
position from
which you view an
object
• position – the
place where
someone or
something can be
found
A
A
B
B
C
C
C
A
A
126
B
A
b)
B
B
CC
Term 3
Platinum Maths Gr6_Term 3_CAPS.indd 126
02/02/13 1:42 AM
Draw views from different viewpoints
From a picture, you can imagine what the object looks like from
different positions.
ExErCiSE 20.2
1. Look at the objects below:
A
B
C
D
a ) Draw a view of each object as if you were standing above each
object.
b ) Draw a view of each object as if you were standing in front of
each object.
c ) Draw a view of each object as if you were standing on the
right-hand side of the object.
d ) What have you noticed about the left-hand and right-hand
sides of each of these objects?
2. Name two 3D geometric objects that would have the same top
view, side view and front view.
3. a ) Pick an object in your class and draw the front view, side view
and top view.
b ) Ask your friend to identify the object you have drawn.
c ) Ask your friend to identify where you were standing when
drawing each different view.
Topic 20: Viewing objects
Platinum Maths Gr6_Term 3_CAPS.indd 127
127
02/02/13 1:42 AM
ExErCiSE 20.3
1. Look at the picture of a jewellery box on the left.
a ) Draw the shape that you would see if you were looking at the
base of this jewellery box.
b ) Draw the view you would see from the side.
c ) Draw the view you would see from the top if the lid was lifted
off the jewellery box.
2. Look at the picture on the left.
a ) Identify the object in the picture.
b ) Which sketch below represents a side view of this object?
A
B
C
c ) Explain how you made your choice.
3. Look at the picture on the left.
a ) Which object is featured in the picture?
b ) Where was the person standing to take the picture?
c ) There is more than one possibility for the side view of this
object. Draw one possible side view.
4. Look at the picture of a
skateboard ramp on the right.
Identify where the person
drawing the picture was
standing to draw each of the
following views:
a)
128
b)
c)
Term 3
Platinum Maths Gr6_Term 3_CAPS.indd 128
02/02/13 1:46 AM
ExErCiSE 20.4
Write down the letters of the correct views of each stack of cubes.
(Some views are not correct.)
A
B
C
front view
a)
b)
c)
top view
a)
b)
c)
side view
a)
b)
c)
Topic 20: Viewing objects
Platinum Maths Gr6_Term 3_CAPS.indd 129
129
02/02/13 1:46 AM
ExErCiSE 20.5
Here are the diagrams of five solid objects:
a)
b)
d)
e)
c)
1. Identify the 3D objects.
2. Draw what each object will look like when you look at it directly
from the front.
3. Draw what each object will look like when you look down at it
from directly above.
4. Which of the objects look like this from directly below:
5. Which of the objects look the same when you view them from the
front?
6. Which of the objects look the same when you view them from
above?
130
Term 3
Platinum Maths Gr6_Term 3_CAPS.indd 130
02/02/13 1:46 AM
revision
1. Use rounding to the nearest 1 000 to estimate the answer to each calculation below:
a ) 1 321 854 + 1 187 988
b ) 2 135 756 + 1 909 121
(1)
(1)
2. Use the expanded column method to complete the following calculations:
a ) 9 538 054 + 3 348 201 + 1 498 768
b ) 202 800 344 + 340 693 545 + 122 293 187
(2)
(2)
3. Use the column method to complete the following calculations:
a ) 10 879 657 – 9 464 043
b ) 534 945 802 – 323 771 232
(2)
(2)
4. Complete the following number sentences:
a ) 343 214 + □ = 765 345, so 765 345 – □ = 343 214
b ) 785 340 – □ = 344 089, so 344 089 + □ = 785 340
(2)
(2)
5. The population of a certain country is 345 980 400. If 103 567 200 of the people are women
and 98 340 500 are men, how many are children?
(3)
6. Which solid object that you have studied has these views?
Top view:
(2)
Front view:
7. Look at the picture of a castle.
a ) Where was the person standing when he took this picture?
(1)
b ) Draw a top view of what this building might look like from a helicopter flying above it. (5)
Total marks: 25
Revision
Platinum Maths Gr6_Term 3_CAPS.indd 131
131
02/02/13 1:47 AM
Topic
Properties of 2D shapes
21
Maths ideas
• Identify, describe,
compare, sort and
name 2D shapes.
• Identify sides and
angles.
• Investigate the
properties of 2D
shapes.
• Draw 2D shapes
on grid paper.
• Investigate
drawing circles.
identify, describe and compare
2D shapes
Look at the picture of the Geometric tortoise.
This species of tortoise is only found in
Southern Africa and is named after the
patterns on its shell.
There are at least five different 2D shapes on
the tortoise shell. Try and find all five shapes
and identify them.
A regular polygon has all sides equal in length and all angles equal in
size. The only regular quadrilateral is a square. If the sides and angles
are not equal, the polygon is irregular.
Key words
ExErCiSE 21.1
• regular polygon
– a polygon with
all sides equal
in length and all
angles equal in size
1. Measure the straight sides of each shape shown below and complete
the table that follows. Shape A has been done as an example.
• irregular polygon
– a polygon with
sides and angles
that are not equal
AA
Shape
Length of sides
A
6 equal sides = 1 cm each
B
C
D
B
B
Classify angles
6 obtuse angles
C
C
D
D
Regular/ Irregular Name of shape
Regular
Regular hexagon
2. a ) Write down the names of all the quadrilaterals you have learnt
about.
b ) Explain why a rectangle is not a regular polygon.
132
Term 3
Platinum Maths Gr6_Term 3_CAPS.indd 132
02/02/13 1:47 AM
Draw 2D shapes on grid paper
You can use a pair of compasses to draw a triangle.
Example
To draw a triangle, you need to know the lengths of the three sides of
the triangle. Draw a triangle with sides of 8 cm, 7 cm and 5,5 cm.
Step 1: Draw a straight line exactly 8 cm long.
Step 2: Use a pair of compasses to measure 7 cm on a ruler.
Step 3: Place the metal point of your compasses at one end of your
straight line and draw an arc above the centre of the line. The arc
marks all the points that are 7 cm away from the line.
Step 4: Repeat Step 3 at the other end of your straight line, but this
time measure 5,5 cm and draw an arc above the line, cutting the
other arc. Ensure that the two arcs cut. If they do not, extend the
two arcs until they do cut.
Step 5: Draw a straight line from the point where the arcs cut each other
to one end of your line.
Step 6: Join the two ends of your lines to form a triangle from the point
where the arcs cut.
Step 1
Step 2
Step 3
Step 4
Step 6
Step 5
ExErCiSE 21.2
1. Draw a circle with a radius of 6 cm.
2. Draw triangles with these side measurements:
a ) 6 cm, 6 cm and 8,5 cm
b) each side = 7,5 cm
3. Practise drawing some of the patterns drawn below. Then use
these patterns to create an interesting full-page design.
a)
b)
c)
d)
e)
Key words
• radius – a line
from the centre
of a circle to the
outside of the
circle
Topic 21: Properties of 2D shapes
Platinum Maths Gr6_Term 3_CAPS.indd 133
133
02/02/13 1:47 AM
ExErCiSE 21.3
1. Use your compasses, set square and ruler to draw the following
shapes accurately. You may use grid paper
a ) a rectangle of length 7 cm and width 5 cm
b ) a triangle with sides measuring 6 cm, 8 cm and 10 cm
c ) a triangle with three sides of 7 cm each
2. Use your compasses, set square
and ruler to find where the treasure
is hidden on the map below. On
a sheet of 1 cm square grid paper
draw the camp in the bottom corner
as shown on the picture. The circle
marked on the corner of the camp
is the starting point for all your
measurements.
a) Mark the position of the store
room which is 5 cm to the right
(east) of the camp. Draw a line
to represent the road between
the camp and the store room.
b) Mark the position of the tree
which is 3 cm directly north of
the store room so that the line
joining the store room and the
tree is at right angles to the road
between the camp and the store
room. Draw the road between
the store room and the tree.
c ) Mark the position of the swimming pool which is 2 cm directly
north of the camp so that the line joining the swimming pool
and the camp is at right angles to the road between the camp
and the store room. Draw the road between the camp and the
swimming pool.
d ) The treasure lies 4,4 cm away from the swimming pool and
2,9 cm away from the tree. It is north of the line joining the
camp and the store room. Use a compass to mark the position
where the treasure can be found.
Camp
134
Term 3
Platinum Maths Gr6_Term 3_CAPS.indd 134
02/02/13 1:47 AM
To draw a circle, you need to know the length of the radius. You will
use square grid paper to practise drawing circles.
Did you know?
ExErCiSE 21.4
1. Practise drawing circles on grid paper.
First use a radius of four blocks,
then, keeping the metal point of the
compasses in the same place, increase
the radius to six blocks.
2. Let’s draw a Fibonacci spiral like the
spiral that we see in some sea shells.
a ) Fill in the next two numbers in the
Fibonacci sequence 1; 1; 2; 3; 5; 8; □; □
b ) Copy the squares and the markings and numbers, as shown in
the picture onto a piece of square grid paper.
13
21
F
E
Leonardo Fibonacci
Leonardo Fibonacci
was an Italian
mathematician who
lived in the 12th
century. People
considered him
to be one of
the most talented
mathematicians of
the Middle
Ages.
3 121 A
B
C
5
D
8
G
c ) Notice how the lengths of the sides of the blocks show the
Fibonacci sequence.
d ) Place the metal point of your compasses on position A as
marked on the sketch. Now with a radius of one block draw
the first half a circle through the blocks marked 1.
e ) Now change your radius to two blocks and move the position
of the metal point to position B and draw through a quarter of
a turn.
f ) Adjust your radius to three and move the metal point to
position C and continue the spiral. As you move from one
square to the next, keep adjusting your radius and the position
of the metal point to continue the spiral.
g ) Find a picture of a shell that has this shape and bring it to school.
The Nautilus shell
follows the
Fibonacci
sequence and
grows in a threedimensional
spiral.
Topic 21: Properties of 2D shapes
Platinum Maths Gr6_Term 3_CAPS.indd 135
135
02/02/13 1:47 AM
Topic
Transformations
22
Maths ideas
• Use
transformations to
describe patterns.
• Describe patterns
in the world
around us using
transformations.
Use transformations to describe
patterns
In Grade 5 you learnt about three types of transformations which can
change the position of a shape.
Reflection creates the mirror image of a shape. A line of symmetry
forms between the shape and the image.
A
B
line of symmetry of a
pentagon
Key words
• transformation
– a special way
to change the
position and/or
orientation of a
shape
• reflection – to
flip a shape over
to form a mirror
image of the shape
Rotation changes the position of
a shape by turning it around a
fixed point.
Top
Back to the top
• image – a copy or
likeness
• translation – to
shift a shape to
a new position
without turning it
136
Translation is copying or
sliding a shape into a new
position, without turning
it around.
translate the shape to the right two times
translate the shape down
• rotation – to turn
a shape around a
fixed point
4 units
4 units
Term 3
Platinum Maths Gr6_Term 3_CAPS.indd 136
02/02/13 1:47 AM
In tessellations, you use transformations to create regular tiling
patterns, which do not have any spaces between the shapes.
Key words
• tessellation – a
regular pattern of
identical shapes
with no overlaps
and no gaps
Example
This shape tessellates.
This shape does not tessellate.
There are spaces in between
the shapes.
There are no spaces in between the shapes.
ExErCiSE 22.1
1. Look at the pattern below:
Row 1
Challenge
Row 2
Row 3
a ) Identify the basic shape that is being tessellated.
b ) Explain how you would transform the basic shape to form this
pattern.
c ) Describe all the symmetries that you see in the pattern.
2. Look at the pattern below:
a ) Identify the basic shape that is being tessellated.
b ) Explain how you would transform the basic shape to form this
pattern.
c ) Describe any line symmetry in the pattern.
d ) Describe any rotational symmetry.
Find a picture to
show how we use
transformations to
form a pattern in
nature or in a cultural
context. Present your
picture to the class
and describe the
type(s) of
transformations used.
Patterns in an
Islamic building
Topic 22: Transformations
Platinum Maths Gr6_Term 3_CAPS.indd 137
137
02/02/13 1:47 AM
Describe patterns around us
You can identify transformations in the patterns that you see in nature,
in our cultural heritage and in objects around you.
ExErCiSE 22.2
1. Look the picture of the owl butterfly. This type of butterfly is
known as the owl butterfly because the two markings on the
wings look like an owl’s eyes. Rotate the picture to see if you agree
with the name this butterfly has been given.
a ) Identify the type of transformation on the wings of the butterfly.
b ) How many lines of symmetry does this picture have?
c ) Find a picture of an object or a shape with a pattern that uses
this type of transformation. Describe the shape that is being
transformed and explain how the pattern is made.
d ) Draw the picture that you found. Fill in the lines of symmetry.
2. Look at the picture of the medicine case that some people use to
remind them to take their daily medication.
a ) Identify the shape of the medicine case.
b ) Identify the shape of each pill compartment. (Remember that
this is a 3D object.)
c ) Describe the type of transformation used in this object.
3. Look at the African beadwork in the picture.
a ) There are many patterns in this colourful beadwork. Find one
pattern with one line of symmetry and redraw it.
b ) Describe the shape or shapes that have been tessellated and
explain the transformations that have been used.
c ) Draw any other pattern in the beadwork picture. Describe the
symmetry and the types of transformations.
4. Follow the instructions for each object shown below:
a ) Identify the shape that is being tessellated.
b ) Describe the tessellation that would give you
the same pattern as in each of the pictures.
c ) Identify any line of symmetry or point of
rotational symmetry.
138
Term 3
Platinum Maths Gr6_Term 3_CAPS.indd 138
02/02/13 1:47 AM
revision
1. Is it possible for the angle of a polygon to be a straight angle?
(1)
2. Say whether each of the following statements is true or false:
a ) A circle is a 2D shape with one vertex.
b ) A square has all angles equal.
c ) A parallelogram is a type of triangle.
d ) An octagon has seven sides.
(1)
(1)
(1)
(1)
3. Draw a circle with a radius of 8,5 cm.
(2)
4. Draw a triangle in which the length of each side is 8 cm.
(3)
5. Complete each of these statements:
a ) A rectangle is a quadrilateral with … .
b ) A rectangle is a parallelogram with … .
(2)
(2)
6. Look at the pattern below:
a ) Which shape is being tessellated to form the pattern?
b ) Is there any symmetry in the pattern? If so, describe the symmetry that you see
in the pattern.
c ) Describe all the transformations that you would use if you copied the pattern.
(1)
(2)
(3)
7. Look at the pattern below:
a ) Which shape is being tessellated?
b ) Is there any symmetry in the pattern? If so, describe the symmetry that you
see in the pattern.
c ) Describe all the transformations that you would use if you copied the pattern.
(1)
(2)
(2)
Total marks: 25
Revision
Platinum Maths Gr6_Term 3_CAPS.indd 139
139
02/02/13 1:47 AM
Topic
Percentages
23
Maths ideas
• Understand that
1
one percent is ___
100
or 1 in 100.
• Read and write
percentages.
• Understand the
equivalence
between decimals,
common fractions
and percentages.
• Convert between
common fractions,
decimals and
percentage.
What is a percentage?
The word ‘per-cent’ means ‘for each hundred’. A percentage is
a fraction that is measured out of a total of 100. The symbol for
percentage is %. One hundred percent (100%) is the whole amount,
100
= 1.
because ___
100
40
The fraction ___
is 40 percent, which you write as 40%.
100
Example
I have R100 to spend. I buy a shirt for R47, I give R30 to you, and I have
R23 left. This means that I have spent 47% of my money, I have given
you 30% and I have 23% left.
• Find percentages
of whole numbers
ExErCiSE 23.1
Key words
1. Write down five examples of how we use percentages in our
everyday lives.
• percentage –
‘per cent’ means
out of a 100; %
is the symbol for
percentage
Did you know?
We use percentages
in many different
areas of everyday life.
• Test marks
are given as
percentages.
• Workers may get
a percentage
increase in their
salary.
• Shops give a
percentage
discount during a
sale.
140
2. Below is a table showing the percentage of girls and boys in each class:
Class
Boys
Girls
Total
1
2
50%
3
4
40%
45%
5
6
16%
70%
7
8
58%
35%
52%
a ) Each learner is either a boy or a girl, so the percentages must
add up to 100% for each class. Copy and complete this table.
b ) For each class, write down whether there are more girls, more
boys or the same number of each.
c ) Which class has the highest percentage of boys?
d ) Which class has the highest percentage of girls?
3. Write down what percentage is left in each of the following situations:
a ) I spend 15% of my savings on a new fridge and 32% of my
savings on a holiday.
b ) You spend 20% of your homework time on Mathematics and
65% on English.
c ) You give 25% of your sweets to your sister and 15% to a friend.
Term 3
Platinum Maths Gr6_Term 3_CAPS.indd 140
02/02/13 1:47 AM
Convert fractions to percentages
Example
Challenge
What fraction of the whole is 25%?
25
= __1 , so 25% is one quarter of the whole.
25% = ___
100 4
Would you rather
have 45% of R3 600
or 37% of R5 200?
ExErCiSE 23.2
1. Copy the number line on the right.
a ) Show the following fractions on the left of the number line:
0
__ __
5 7 10
1 __
__
, 3 , __
, , .
10 10 10 10 10
20%
b ) Show the following percentages on the right of the number
line: 10%, 30%, 50%, 70%, 100%.
2. a ) What does your number line in question 1 show you?
b ) Can you see that the fraction one tenth is the same as 10%?
Use fractions to explain this.
40%
It is easy to write a percentage as a fraction, because it is always out of 100.
Example
11
55
55% = ___
= __
100 20
3
3% = ___
100
80%
75
75% = ___
= __3
100 4
To change a fraction to a percentage, first write it as a fraction with a
denominator of 100.
1
Example
3
30
__
= ___
= 30%
10 100
55
11 __
__
= 11 × __5 = ___
= 55%
20 20 5 100
ExErCiSE 23.3
1. Convert the following percentages to common fractions in their
simplest form:
a ) 29%
b ) 34%
c ) 30%
d) 90%
e) 45%
f ) 82%
g ) 100%
h) 50%
i ) 81%
j ) 16%
2. Write the following fractions as percentages. Remember to first
write it as a fraction with a denominator of 100.
3
a ) __
10
9
g ) __
20
9
b ) __
10
2
h ) __
50
14
c ) ___
100
1
i ) __
25
7
d ) __
10
j ) __14
77
e ) ___
100
k ) __34
66
f ) ___
100
l ) __55
Food companies sometimes
promote their products by
including a percentage more
food in the package.
Topic 23: Percentages
Platinum Maths Gr6_Term 3_CAPS.indd 141
141
02/02/13 1:48 AM
Decimal fractions and percentages
Decimal fractions and percentages are two different ways of saying
the same thing. Look at this number line:
uivalents
Useful eq
n Decimal
% Fractio
10%
20%
25%
50%
0
1
10
0,1
10%
1
5
1
4
0,2 0,25
20% 25%
0,1
_1
0,2
_1
Example
0,25
_1
67
67% = ___
= 0,67
100
0,5
_3
0,75
5
4
2
75%
4
3
4
0,5
50%
0,75
75%
100
It is easy to convert a percentage to a decimal, because it is a number
of hundredths.
_1_
10
1
2
70
70% = ___
= 0,70 or 0,7
100
3
3% = ___
= 0,03
100
ExErCiSE 23.4
Convert the following percentages to decimals:
1. 45%
2. 75%
3. 25%
4. 72%
6. 80%
7. 30%
8. 50%
9. 3%
5. 99%
10. 7%
To convert a decimal fraction to a percentage, first write the decimal
as a common fraction and then make sure the denominator is 100.
Example
6
60
0,6 = __
= ___
= 60%
10 100
ExErCiSE 23.5
Convert each of the following decimals to percentages:
1. 0,3
2. 0,95
3. 0,82
4. 0,7
5. 0,9
6. 0,4
7. 0,01
8. 0,07
9. 0,05
10. 0,55
Game
• Copy the table and play together in pairs.
• Choose one number in the table and tick off all the equivalent
fractions, decimals and percentages.
• Then do the same with another number, and another number
until all the numbers are ticked off.
• Time yourself, and see if you can both complete the table
together more quickly with practice.
142
_1
2
5
__
10
25%
20
___
100
_6
8
_2
4
_1
4
_2
5
0,4
0,5
1
__
10
0,05
50%
0,75
0,1
_1
4
1
___
100
10
___
200
1%
20%
4
__
20
75
___
100
_2
8
3
__
15
10
___
100
10%
0,2
25
___
100
75
___
150
4
__
10
0,01
40% 75%
_4
4
__
0,25
_1
5%
5
___
1
__
40
100
8
5
20
Term 3
Platinum Maths Gr6_Term 3_CAPS.indd 142
02/02/13 1:48 AM
Find the percentage of a whole
number
First change the percentage into a common fraction, and then use
your knowledge of fractions.
Example
25
Find 25% of R300. 25% = ___
= __1
100 4
How many times will 4 go into 300?
300 ÷ 4 = R 75, so 25% of R300 is R75.
Example
Find 3% of 1 500.
3
. How many times does 100 go into 1 500?
3% = ___
100
1
of 1 500 is 15.
1 500 ÷ 100 = 15, so ___
100
3
of 1 500 = 3 × 15 = 45.
So 3% of 1 500 = ___
100
ExErCiSE 23.6
1. Calculate:
a ) 10% of 240
d ) 50% of 670
g ) 70% of 9 090
2. Find 50% of:
a ) 680 g
Find 20% of:
e ) 245 g
Find 75% of:
i ) 444 kg
b ) 20% of 250
e ) 40% of 40
h ) 1% of 7 000
c ) 25% of 24
f ) 30% of 240
i ) 5% of 3 200
b ) 2 468 km
c ) R14
d ) 750 ml
f ) 45 km
g ) R555
h ) 2 085 ml
j ) R1 000
k ) 4 040 km
l ) 728 ml
3. In a class of 36 learners, 25% say that their favourite sport is tennis.
How many children does this represent?
4. A survey was conducted with 420 hotel guests.
a ) The finding was that 20% of the guests were not happy with
the standard of cleanliness in the hotel. How many guests
were not happy?
b ) The same survey found that 40% of the guests found the staff
unhelpful. How many guests were dissatisfied with the staff?
c ) The food was, however, highly rated by 70% of guests. How
many guests did not rate the food highly?
Challenge
Out of 240 people
50% said they liked
soccer the most.
Another 23% liked
cricket and 12% liked
basketball. The rest
liked rugby. How
many people liked
rugby?
A cricket player
Topic 23: Percentages
Platinum Maths Gr6_Term 3_CAPS.indd 143
143
02/02/13 1:48 AM
Topic
Temperature
24
Maths ideas
• Read temperature
from a
thermometer.
• Do calculations
involving
temperature.
• Solve problems
involving
temperature.
Did you know?
read, compare and measure
temperatures
Temperature is a measure of how hot or cold things are.
We use temperature to describe the weather conditions, and we
record temperature in degrees Celsius (°C).
The sun, frost, snow, rain and wind all affect the air temperature.
Weather maps provide clues to the future temperature, moisture level
and wind directions of an area. Meteorologists are scientists who study
weather conditions. They get measurements from many weather
stations and use satellite images to predict the weather on Earth.
ExErCiSE 24.1
Use the temperatures of these cities on one particular day to answer
the questions that follow:
Barometers measure
air pressure, which is
sometimes called
barometric pressure.
You can predict some
weather conditions
that are associated
with high and low air
pressure. High
pressure is normally
associated with fair
weather conditions.
Low pressure systems
often bring rain.
City
Johannesburg
Pretoria
Kimberley
Durban
Bloemfontein
Cape Town
Port Elizabeth
Minimum temperature Maximum temperature
18,9 °C
28,9 °C
19,1 °C
30,2 °C
10,2 °C
24,0 °C
22,8 °C
30,4 °C
15,5 °C
22,5 °C
17,6 °C
18,6 °C
20,1 °C
26,3 °C
1. Which city had the highest maximum temperature on that day?
2. Which city had the lowest minimum temperature on that day?
3. Order the maximum temperatures from the lowest to the highest.
4. Calculate the difference between the minimum and maximum
temperatures for each city. Name the city that had the highest
temperature difference during the day.
5. Round off the given temperatures to the nearest unit. Does this
change your answers to questions 1 or 2?
144
Term 3
Platinum Maths Gr6_Term 3_CAPS.indd 144
02/02/13 1:48 AM
We use different thermometers to measure different types of
temperatures. We use the Celsius (°C), Fahrenheit (°F) or Kelvin scales
to measure temperature. In South Africa, we use the Celsius scale to
record temperature. The freezing point of pure water is 0 °C and the
boiling point of pure water is 100 °C.
The analogue thermometer on the right shows a reading of 25 °C.
Laboratory thermometers are usually made of metal or glass, like
the ones shown below. The digital thermometer below shows a
temperature of 52,37 °C.
The normal body temperature of a healthy adult human is between
36,2°C and 37,0°C. You can measure body temperature with a
thermometer that looks like the one in the picture below.
ExErCiSE 24.2
1. Match these temperatures to the descriptions below:
180 °C
32 °C
7 °C
38,5 °C
a ) a hot summer’s day
b ) a sick person
c ) a cake baking in an oven
d ) a glass of cold fruit juice
2. Read the temperatures on the thermometers below. About how
many more degrees must each temperature rise to reach 100 °C?
a)
b)
c)
d)
e)
Topic 24: Temperature
Platinum Maths Gr6_Term 3_CAPS.indd 145
145
02/02/13 1:48 AM
Calculations with temperature
ExErCiSE 24.3
Did you know?
We show
temperatures below
freezing point by
using special
numbers with a ‘–’
sign in front of them.
These numbers are
called negative
integers.
1. From the temperatures given below, calculate how many degrees
the temperature must rise or drop to reach 100 °C:
a ) 25 °C
b ) 32,8 °C
c ) 148,7 °C
d ) 231 °C
e ) 69,1 °C
f ) 76,5 °C
g ) 195,4 °C
h ) 10,2 °C
2. Fill in the missing numbers in this temperature chain:
18,7 °C
–9,8 °C
…
+7,2 °C
…
6,2 °C
ExErCiSE 24.4
Answer the following questions without using a calculator:
An ice cube
1. The maximum temperature readings over four winter days in
London were 2,8 °C; 3,0 °C; 2,4 °C and 1,8 °C. Calculate the difference
between the highest and lowest temperatures for that period.
2. The minimum temperature for Cape Town was 15,8 °C and the
maximum temperature was 23,2 °C on a particular day. Calculate
the difference between the minimum and the maximum
temperatures.
3. Mom baked muffins, a cake, bread and a pie. For the muffins the
oven was set at 165 °C. To bake the cake, the oven was set at
220 °C. The bread required a temperature of 185 °C, while the pie
needed a temperature of 175 °C.
a ) What was the difference in oven temperature between the pie
and cake?
b ) If she baked the items in the order given in the question, write
down by how many degrees she had to increase or decrease
the temperature each time.
Challenge
Hurricanes are storms with high-speed winds and very heavy rains. A hurricane begins
over the ocean where air rising from warm seas creates a severe low-pressure zone.
If the hurricane is moving at an average speed of 80 km/h, calculate how long
it will take the hurricane to reach a city that is 1 540 km away from the zone.
146
Term 3
Platinum Maths Gr6_Term 3_CAPS.indd 146
02/02/13 1:48 AM
revision
1. Look at the weather report for ten different
capital cities of the world. Then answer the
questions that follow.
Capital City
Helsinki
London
Maputo
New Delhi
Ottawa
Paris
Prague
Pretoria
Tokyo
Washington DC
Max.
–5 °C
10 °C
27 °C
24 °C
2 °C
9 °C
2 °C
25 °C
8 °C
9 °C
Min.
–9 °C
8 °C
19 °C
9 °C
–5 °C
5 °C
–2 °C
19 °C
2 °C
6 °C
4. Convert the following decimal fractions into
percentages:
a ) 0,2
(1)
b ) 0,7
(1)
c ) 0,06
(1)
d ) 0,4
(1)
e ) 0,15
(1)
5. Four friends were comparing their test
6
17
, Lindiwe got __
,
marks. Linda got __
20
25
39
7
__
__
James got 50 and Mpho got 10 .
a ) Write each mark as a percentage.
b ) Whose mark was the highest
percentage?
c ) Whose mark was the lowest
percentage?
(4)
Revision
147
(1)
(1)
a ) Arrange the minimum temperatures
from lowest to highest.
(2) 6. Calculate the following amounts:
b ) Which city is the coldest?
(1)
a ) 25% of 700 ml
c ) Which city is the hottest?
(1)
b ) 50% of 1,4 kg
d ) How many cities have a temperature
c ) 20% of 2 345 g
below freezing point?
(1)
d ) 65% of 1 m
(4)
e ) Work out the diffence between the
Total marks: 30
minimum and maximum temperatures
for the following cities: Maputo, New
Delhi, Paris, Prague and Pretoria.
(4)
2. From the temperatures given below, work
out how many degrees the temperature
must rise or drop to reach 37 °C:
a ) 5 °C
b ) 10 °C
c ) 0 °C
d ) 74,5 °C
(4)
3. Convert the following common fractions
into percentages:
a ) __45
17
b ) __
20
Platinum Maths Gr6_Term 3_CAPS.indd 147
(1)
(1)
02/02/13 1:48 AM
Topic
Data handling
25
Collect, organise and represent data
Maths ideas
• Collect, organise
and record data.
• Draw pictographs,
bar graphs and
double bar
graphs, including
percentages.
• Order data and
find the mode and
median of a set of
data.
• Analyse and
interpret data
in tables,
pictographs, bar
graphs and pie
graphs.
In Term 1 you used questionnaires to collect data and then you
organised it into tables. You also drew pictographs, bar graphs and
double bar graphs to represent data. Read through the examples in
Topic 7 again if you have forgotten how to do these things.
ExErCiSE 25.1
1. Some Grade 6 learners want to investigate the following:
• How do learners get to school?
• Do learners and their families recycle old newspapers or not?
• How many people live in each household?
• What type of energy do these families use for cooking?
a ) For each question above, draw up a simple questionnaire that
the learners could use to collect the data they need.
b ) Choose one of the questions. Use your questionnaire to collect
data from at least 20 learners in your school.
c ) Draw a pictograph to show what you found out.
2. Mymoena did a survey in her class to find out how many learners
wear glasses. This is a table of her results:
Boys
Girls
Energy
% of
used for households
lighting (rounded)
Electricity
Gas
Paraffin
Candles
Solar
Other
148
70
0,3
10
20
0,3
0,4
Wear glasses
Don’t wear
7
14
4
11
a)
b)
c)
d)
How many boys wear glasses?
How many girls do not wear glasses?
How many learners did she survey?
Express the number of boys who wear glasses as a percentage
of the class.
e ) Express the number of girls who wear glasses as a percentage
of the class.
f ) Draw a double bar graph to show the data in the table.
3. This table from the National Census shows what type of energy
different households use for lighting their homes. Draw a
horizontal bar graph to show what percentage of homes use
electricity, paraffin or candles to light their homes. Use a scale of
1 cm to 10% on the horizontal axis.
Term 3
Platinum Maths Gr6_Term 3_CAPS.indd 148
02/02/13 1:48 AM
Find the mode and median of
data sets
Key words
The mode is the data value that occurs most often. When two values
occur most often, the data has two modes. We call this bi-modal data.
The median is the middle value in a data set. To find the median you
have to order the data from smallest to largest.
If there is an even number of data, the median is half-way between
the two middle values. To find the median, you add the two middle
values and divide the total by 2.
• bi-modal – a
data set with two
modes
Example
Find the mode and the median of this data set: 4; 3; 2; 3; 1; 1; 2; 3
First, we rewrite the data in ascending order: 1; 1; 2; 2; 3; 3; 3; 4
The most common value is 3, so the mode is 3.
The two middle values are 2 and 3.
The median is (2 + 3) ÷ 2 = 5 ÷ 2 = 2__12
ExErCiSE 25.2
1. Find the mode and the median of each set of data:
a)
c)
e)
g)
1; 1; 1; 2; 3; 4; 5; 5; 5; 5; 6
100; 100; 113; 115; 112; 111; 110
0; 3; 5; 3; 2; 1; 6; 4; 3; 4; 3
5; 3; 4; 5; 2; 1; 1; 1; 4
b ) 12; 13; 14; 12; 12; 12; 13; 15; 15
d ) 23; 25; 26; 25; 25; 26; 24; 26
f ) 5; 4; 2; 1; 5; 7; 8; 0; 3
h ) 5; 3; 2; 6; 0; 2; 1; 2; 3; 2; 2
2. Work in a group of four. Each learner must roll a die 12 times, and
keep a tally of how many of each number (1 to 6) they throw.
a ) Copy and complete this table to summarise how many of
each number each player threw. Write your names in the first
column.
Learner
1
2
3
4
5
6
b ) Who threw the most sixes?
c ) Who threw the most ones?
d ) Find the modal number for each learner.
Topic 25: Data handling
Platinum Maths Gr6_Term 3_CAPS.indd 149
149
02/02/13 1:48 AM
Key words
• pie chart – a
circle divided
into sections
that represent
percentages of data
interpret and analyse data
Data can be presented in many different ways – in words, tables and
in graphs. You have already learnt how to read and interpret data in
words, tables, pictographs, bar graphs, double bar graphs and pie
charts divided into fractions. You have also answered questions about
the data categories, the sources of data and the context in which it
was collected.
This term you will work with the above-mentioned graphs, some of
which will show the data categories as percentages.
Example
Northern Cape
Study the pie chart showing data
1,8%
from the 2001 National Census.
Mpumalanga
a ) What does this graph show?
North
7,0% Limpopo
11,8%
b) In which province does most
West
8,2%
of South Africa’s population
Western Cape
live? What percentage of the
10,1%
KwaZulu-Natal
population lives there?
21,0%
Eastern Cape
c ) Which provinces each
14,4%
have less than 10% of the
Gauteng
Free
population?
19,7%
State
d) Which provinces each
6,0%
have more than 15% of the
population?
e ) What percentage of the total population lives in KwazuluNatal, Gauteng and the Western Cape together?
f ) Order the provinces from the smallest percentage to the
greatest percentage of the population.
Answers
a ) The graph shows how the total population of South Africa is
divided up among the provinces.
b) KwaZulu-Natal; 21%
c ) North West, Northern Cape, Mpumalanga and Free State
d) KwaZulu-Natal and Gauteng
e ) 10,1% + 19,7% + 21% = 50,8%
f ) Northern Cape, Free State, Mpumalanga, North West, Western
Cape, Limpopo, Eastern Cape, Gauteng, KwaZulu-Natal.
150
Term 3
Platinum Maths Gr6_Term 3_CAPS.indd 150
02/02/13 1:48 AM
ExErCiSE 25.3
Key
2001
2007
Province
1. Study the graph on the right carefully.
Western Cape
a ) What does this graph show?
North West
Northern Cape
b ) What type of graph is this?
Mpumalanga
c ) What does the key tell you?
Limpopo
d ) Which province’s share of the
KwaZulu-Natal
population remained the same from
Gauteng
2001 to 2007? How can you tell this
Free State
from the graph?
Eastern Cape
e ) Which province’s share of the
0
5
10
15
20
population decreased from 2001 to
% of total population
2007?
f ) Where do you think these people went to? Suggest some
possible reasons for this movement.
g ) Which two provinces together have more than 40% of the
total population of South Africa?
h ) Complete these two sentences based on the graph:
Gauteng is the smallest province but it has the … share of the
population.
Northern Cape is the largest province but it has the … share of
the population.
i ) Use information from the graph to describe what happened to
Gauteng’s share of the total population from 2001 to 2007.
25
2. Study this double bar graph carefully:
Percentage of households that have
access to piped water by province
Key
2001
2007
South Africa
Western Cape
North West
Province
Northern Cape
Mpumalanga
Limpopo
KwaZulu-Natal
Gauteng
Free State
Eastern Cape
0
20
40
60
Percent
80
100
a ) What does the graph show?
Topic 25: Data handling
Platinum Maths Gr6_Term 3_CAPS.indd 151
151
02/02/13 1:48 AM
b ) Which three provinces have the highest percentage of
households with piped water?
c ) Which province has the lowest percentage of households with
piped water?
d ) The government says that they have worked hard to provide
more households with piped water. Does the graph support
this statement? Give a reason for your answer.
Households’ main source
of water supply (2001)
Borehole/ Other
spring/river 3,1%
Water
carrier
0,6%
11,8%
Public tap
23,2%
3. This pie chart on the left also gives data about where households
get their water.
a ) Write a paragraph summarising what this graph shows you.
Piped water
inside dwelling
b ) When was this data collected? How do you think the graph
or yard
may have changed since then? Why?
61,3%
4. Study the second pie graph carefully:
a ) What does the graph show?
Visitors by continent (SA Tourism)
b ) Where do most international tourists to South Africa come from?
S. America 4%
c ) What percentage of tourists to South Africa come from:
Asia 7%
the Middle East
Australasia 4%
Africa
Middle East 2%
South America
16%
North America
N. America
9%
Africa?
d ) From which two regions do we get the same number of
Europe
tourists?
58%
5. When tourists visit South Africa, they spend money on hotels,
meals, transport and entertainment. The table shows the average
amount of money spent each day by tourists from different places:
Origin
Africa
UK
Spend per day (R)
1 900
800
Rest of
Europe
1 100
North
America
1 450
South
America
1 300
Middle
East
900
Far East
Australasia
1 400
700
a ) Draw a pictograph to show this data. Use the symbol
to represent R250.
b ) Which tourists spend the most money per day?
c ) Which tourists spend the least money per day?
d ) Which tourists spend less than R1 000 per day when they visit
South Africa?
Challenge
Use the pie graph in question 4 and your own pictograph from question 5 to
suggest three things that the Department of Tourism could do to try to earn
more money from international tourists. Give reasons for your suggestions.
152
Term 3
Platinum Maths Gr6_Term 3_CAPS.indd 152
02/02/13 1:48 AM
Compare graphs on the same topic
Graphs that show the same data can look quite different. Some reasons for this are:
• The source of the data is different, for example:
ū The data was collected from different groups of people, so their answers were
different.
ū The data was collected in different places, so the results were different.
• The data was collected in different contexts, for example:
ū The data was collected at different times, so different things were happening.
ū The data was collected in different ways, so the results were different.
Day
Day
Day
Salma and Anita are in the same class. For a Science
Monday
project, they measured the temperature in their
classroom each day for a week. They drew these
Tuesday
graphs to display their data.
Wednesday
They each wrote a paragraph summarising their results.
Thursday
Salma’s paragraph:
Friday
Over the week the lowest temperature was 16 °C.
The temperature rose from 16 °C to reach a high of
10 12 14 16 18 20 22
Temperature (°C)
22 °C on Wednesday and Thursday. On Friday the
temperature dropped back down
16 °C.
Salma’sto
Graph
Anita’s Graph
Temperature in the classroom
Temperature in the classroom
Anita’s paragraph:
Monday
Monday
This week the temperature
did not go above 14°C.
Monday was the
coldest day with a temperature of
Tuesday
Tuesday
11 °C. Wednesday and Thursday were the hottest. On
Wednesday
Wednesday
those days the
temperature rose to a high of 14 °C. By
Friday the temperature
dropped again down to 12 °C.
Thursday
Thursday
Why do you think the two girls got such different results?
Friday
Friday
If you compare the graphs you can see that the pattern
10 12 14 16 18 20 22
10 12 14 16 18 20 22
for the week was the same. Both
graphs(°C)
show a low
Temperature
Temperature (°C)
temperature on Monday, getting hotter on Tuesday and
being the hottest on Wednesday and Thursday and then dropping down again on Friday.
However, Salma’s graph includes temperatures that are higher than Anita’s. The reason is that
Salma measured the temperature each day at lunchtime. Anita measured the temperature
when she got to school in the morning. This caused the girls to get different results because
the mornings are cool and then it gets warmer during the day. In other words, each set of
data was collected in a different context, one showing the minimum temperatures for each
day, the other showing the maximum temperature for each day.
Topic 25: Data handling
Platinum Maths Gr6_Term 3_CAPS.indd 153
153
02/02/13 1:48 AM
Day
Salma’s Graph
Temperature in the classroom
Example
Wed
T
ExErCiSE 25.4
1. The two pie graphs show what percentage of a particular day
Sammy and Nick spent on different activities:
Sammy’s day
Watch
movies
Help
dad in
garden
School
Eating
Visit
Granny
Homework
Nick’s day
Sleep
Homework
Sleeping
Sports
Church
Eat
a)
b)
c)
d)
Besides sleeping, what did Sammy do on that day?
Besides sleeping, what did Nick do on that day?
Write a paragraph summarising what each graph shows you.
Suggest one reason why these two graphs show different
results.
Heights of learners in Grade 6 and 7
171-180 cm
161-170 cm
151-160 cm
141-150 cm
131-140 cm
121-130 cm
111-120 cm
100-110 cm
8
10
12
Number of learners
Did you know?
Census at School is a
national project that
collects data from
2 500 schools across
the country.
154
14
2. Nicky collected data about the heights of learners in
Grade 6 and 7 at her school. She drew the graph on
the left to show her results.
a ) What is the modal height of Grade 6 and 7
learners?
b) How many learners are 120 cm or shorter?
c ) How many learners are taller than 170 cm?
d) Write a paragraph summarising Nicky’s results.
e ) If you collected the same data at your school,
would your graph look the same as Nicky’s or
not? Give a reason for your answer.
3. Compare Nicky’s graph with this
graph drawn from data collected
by Census at School:
a ) How are the graphs similar?
b ) How are they different?
c ) Why do you think Nicky got
different results to those of
Census at School?
Average heights of
learners by grade
Key
Male
Female
7
6
5
4
3
0
50
100
cm
150
200
Term 3
Platinum Maths Gr6_Term 3_CAPS.indd 154
02/02/13 1:48 AM
10+
9
Number of rooms
4. The graph to the right comes from the national
census data. It shows what percentage of households
in South Africa has homes with each number of
rooms.
a ) What is the modal number of rooms in a home?
b ) Approximately what percentage of households
lives in a home with four rooms or less?
c ) What percentage of households lives in a home
with eight or more rooms?
d ) If you draw this graph for your class would it have
a similar or different pattern to this one? Why?
8
7
6
5
4
3
2
1
0
5
10
15
Percent
20
25
30
5. a ) Draw up a table to collect data about the number of rooms in
different learners’ homes. Collect data from at least 15 learners.
b ) Draw a bar graph like the one above to show your results.
c ) Write a paragraph comparing your graph with the one from
the national census. Try to give reasons for any important
differences.
6. This pie chart shows what percentage of families had access to
telephones in 2001:
2001 Telephone facilities
Not nearby
3,4%
No access
to telephone
5,9%
At another location
nearby/work
3,2%
Did you know?
Public
telephone
38,5%
Telephone/
cellphone in
dwelling
42,4%
At a neighbour
6,6%
a ) What percentage of households had a telephone or cell phone
at home?
b ) What percentage of households only had access to a public
telephone?
c ) Predict how the graph would be different if you collected this
data today. Give a reason for your answers.
When cell phones
were first introduced
in South Africa in
1994 there were
200 000 cell phones
in use in South Africa.
In 2011 there were
42 300 000 cell
phones in use in
South Africa and
almost 90% of the
population had
access to a mobile
phone.
Topic 25: Data handling
Platinum Maths Gr6_Term 3_CAPS.indd 155
155
02/02/13 1:48 AM
Project
Data handling
You are going to work on your own to carry out a survey among
learners in your school. You will need to collect data, organise it in tally
tables, display it as a double bar graph and analyse and summarise
your findings.
Step 1: Choose your topic and start planning
Choose one of the following topics for your survey:
• Favourite music of boys and girls
• Favourite TV shows of boys and girls
• Favourite films of boys and girls
Write down what questions you will ask people in your
survey.
Decide how you will collect the data to answer the question.
How will you record your data?
Predict what your results will be. Write down your prediction.
Step 2: Conduct your survey
Design a simple questionnaire or table to collect your data.
Decide who you will include in your survey and how many
learners you will collect data from. Remember you need to
show the differences between boys and girls, so you need to
collect data from both.
Collect the data.
Make a list of the names of all the learners you survey.
(5)
Step 3: Organise your data
Draw up a neat tally table to summarise the data you have
collected. Include totals for each category.
(5)
156
Term 3
Platinum Maths Gr6_Term 3_CAPS.indd 156
02/02/13 1:48 AM
Step 4: Draw a double bar graph to represent the data
Draw one or more double bar graphs to represent and
summarise your data.
• Make sure you give each graph a heading a heading.
• Include a key to show what different bars represent.
(10)
Step 5: Summarise your findings and compare the data for boys
and girls
Write down what you found out during your survey. You
can make a list of your findings or you can write a short
paragraph summarising your findings.
(5)
Describe any differences between the results for boys
and girls. Try to give reasons for any differences.
(5)
Step 6: Make a poster to present your project
Your teacher will use your poster to assess your project,
so you need to make sure that your poster includes the
following:
• your topic and the questions you asked
• the questionnaire or table you used in your survey
• details of who you included in your survey
• a completed tally table
• your double bar graph(s)
• a paragraph summarising your findings.
You may decorate the poster in any way that you like.
Be as creative as possible.
(10)
Total marks: 40
Project
Platinum Maths Gr6_Term 3_CAPS.indd 157
157
02/02/13 1:48 AM
Topic
Numeric patterns
26
Maths ideas
• Recognise,
describe and
extend number
sequences.
• Construct number
sequences.
• Determine output
numbers for given
input numbers
using flow
diagrams.
• Determine the
rule that applies
to a given set of
input and output
numbers.
Special types of sequences
Some input-output rules create special types of sequences.
Example
Look at the following four sequences:
a) 8; 14; 20; 26; …
This is called a constant difference sequence because the
difference between two consecutive numbers is always 6.
b) 1; 3; 9; 27; …
This is a constant ratio sequence. The ratio is 3 because you
multiply by 3 to get the next number (81).
c ) 64; 16; 4; 1; …
This is also a constant ratio sequence, but you divide by 4 each
time so the ratio is __14 .
d) 1; 5; 2; 6; 3; 7; … There are two patterns here: 1; …; 2; …; 3; …
and 5; …; 6; …; 7; … . The next two numbers in the pattern are
4 and 8, but this is not a constant difference or a constant ratio
sequence.
ExErCiSE 26.1
Key words
• constant
difference
sequence – a
sequence of
numbers created
by adding or
subtracting by the
same number
• constant ratio
sequence – a
sequence of
numbers created
by multiplying or
dividing by the
same number
158
1. Write the next three numbers in each of the following sequences.
Say whether each sequence is a constant difference or a constant
ratio sequence or neither.
a ) 3; 5; 9; 15; 23; □; □; □
b ) 729; 243; 81; □; □; □
c ) 7; 11; 15; 19; □; □; □
d ) 1; 2; 4; 8; 16; □; □; □
2. Describe the pattern in each of the following number sequences.
State whether it is a constant difference or constant ratio sequence,
or neither. Then fill in the missing numbers to complete it.
a ) 18; 36; 54; 72; □; □; □
b ) 3; 9; 27; □; □; □
c ) 1,25; 2,5; 3,75; 5,0; □; □; □ d ) 122; 223; 324; 425; □; □; □
e ) 176; 88; 44; □; □; □
f ) 207; 198; 189; 180; □; □; □
1
1
__
__
h ) 93; 85; 77; 69; □; □; □
g ) 5 2 ; 7; 8 2 ; 10; □; □; □
Term 3
Platinum Maths Gr6_Term 3_CAPS.indd 158
02/02/13 1:48 AM
Find the rule
If you carefully study a given set of input numbers and the set of
output numbers that are formed using a rule, you can usually work
out the rule.
Example
In this flow diagram, you have been given part of the rule:
input number ? –3 output number. You can use inspection or trial
and improvement to find the rule.
Choose one pair of input and output numbers, say 5 and 32. Try an
operation:
We try × 5 : (5 × 5) – 3 = 22 Too small
We try × 8 : (5 × 8) – 3 = 37 Too big
We try × 7 : (5 × 7) – 3 = 32 correct
Test the rule on another input number: (11 × 7) – 3 = 74
So the rule is (input number × 7) – 3.
3. 1
2
3
4
5
6
7
–3
Calculator skills
For each set of input and output numbers:
• determine the rule
• find the missing output numbers.
1
2
3
4
5
6
?
14
28
42
56
70
Rule
Rule
+1
2.
3
6
9
12
15
18
4 4. 1
2
7
3
10
13
4
5
16
6
7
On some calculators,
if you want to keep
adding the number 2,
you first key in 2
and then + and
then = . Then if you
keep pressing the =
key, your calculator
will count in steps of
2 like this: 2, 4, 6, 8, …
1
2
3
4
5
Rule
Rule
–4
3
10
17
24
31
On another type of
calculator you may
have to key in
2 + 2 = 4. Then every
time you press the
= key, 2 more will
be added. Find this
short cut (also with
other operations) on
your own calculator.
Topic 26: Numeric patterns
Platinum Maths Gr6_Term 3_CAPS.indd 159
4
18
32
39
53
74
Did you know?
ExErCiSE 26.2
1.
1
3
5
6
8
11
159
02/02/13 1:48 AM
It is more difficult to find a rule with two parts if both parts are
unknown.
Example
Determine the rule for this set of input and output numbers:
Input
Output
1
2
3
4
5
3
3,5
4
4,5
5
6
7
1. It might help to see the pattern if you get rid of the fractions.
If you double each output number you get 6, 7, 8, 9, 10, … Each
of these numbers is 5 more than its input number. So the rule is
(input number + 5) ÷ 2.
2. You can get the next output number in the output row by
adding 0,5 to the previous output number. So the next two
output numbers are 5,5 and 6.
3. Apply the rule for inputs 6 and 7 to check:
(6 + 5) ÷ 2 = 11 ÷ 2 = 5,5
(7 + 5) ÷ 2 = 12 ÷ 2 = 6
4. The rule: (input number ÷ 2) + 2,5 can also be used for this set
of input and output numbers.
ExErCiSE 26.3
For each table below:
• find the rule for getting each output number from the input number
• fill in the missing input and output numbers
• state if there is a pattern for getting the numbers along the output
row, and use this pattern to check your new output numbers.
1.
2.
3.
4.
5.
160
Input
Output
2
4
6
8
10
3
4
5
6
7
Input
Output
1
2
3
4
5
2,5
3
3,5
4
Input
Output
2
4
6
8
7
21
35
49
Input
Output
1
2
3
4
8
10
12
14
Input
Output
4
8
16
24
3
4
6
8
12
14
16
32
84
6
7
8
17
29
5,5
10
12
10,5
14
16
44
84
7
8
19
45
77
5
6
20
28
32
40
48
12
Term 3
Platinum Maths Gr6_Term 3_CAPS.indd 160
02/02/13 1:48 AM
revision
1. Study the graph carefully:
Gender by grade
Key
Boys
Girls
7
6
5
4
3
0
a)
b)
c)
d)
e)
10
20
30
%
40
50
60
What percentage of Grade 3 learners are boys?
What percentage of Grade 6 learners are girls?
Which grade has the biggest difference between the number of boys and girls?
In which grade is the percentage of boys and girls more or less equal?
Write a short paragraph comparing the data on this graph with the situation
in your class.
(1)
(1)
(1)
(1)
(3)
2. Describe the pattern in each number sequence below. Write down the next
three numbers in each sequence.
a ) 7; 13; 20; 28; …
b ) 25; 22; 19; 16; …
c ) 100; 91; 83; 76; …
(4)
(4)
(4)
3. Use the given rule to find the output numbers in the table:
Rule: (Input number +3) ÷ 2
(3)
Input number
Output number
3
5
7
9
11
13
4. Start with the first number and complete each number sequence. Use the given
rule for each sequence.
a ) Rule: Subtract 13 from the previous number, then add 10:
260; …
b ) Rule: Multiply the previous number by 4 and add 4:
2; …
(3)
(3)
5. Write the rule that will give the output number for each input number:
3
6
9 12 15 18
Input number
Output number 7 13 19 25 31 37
(2)
Total marks: 30
Revision
Platinum Maths Gr6_Term 3_CAPS.indd 161
161
02/02/13 1:48 AM
Topic
Length
27
Maths ideas
• Round off lengths.
• Measure, compare,
order and estimate
lengths in metres,
centimetres,
millimetres and
kilometres.
• Compare
estimates to actual
measurements.
• Add, subtract,
multiply and divide
lengths given
in millimetres,
centimetres and
kilometres.
• Convert between
different units of
length.
• Solve multi-step
problems involving
length.
Did you know?
1 cm or 10 mm is
about the width of
your little finger;
5 cm or 50 mm is
about the length of
your little finger.
How long is your
shoe?
162
Estimate and measure lengths
You have already used
millimetres (mm),
centimetres (cm) and
metres (m) to
measure length, and
kilometres (km) to
measure distance. In
this topic you will
continue to learn
about these
measurements of
length.
Tape measure
Metre stick
Ruler
ExErCiSE 27.1
1. Estimate the following lengths and choose the closest answer in
each case. Discuss which instrument might be used to measure
each of these lengths:
a ) length of a classroom: 15 mm 15 cm
15 m
15 km
b ) length of your arm:
50 mm 50 cm
50 m
50 km
c ) length of a paper clip: 25 mm 25 cm
25 m
25 km
d ) width of a textbook:
20 mm 20 cm
200 cm 20 m
e ) a walk to school:
30 cm
3m
3 km
300 km
f ) length of a shoe:
20 mm 20 cm
2m
20 m
g ) height of a door:
20 mm 200 cm 20 m
200 m
h ) distance between Cape Town and Pretoria:
14,6 km
146 km
1 460 km
14 600 km
2. Write down the following in your book:
• length of your pencil
• length of your shoe
• width of your desk
• thickness of a 10c coin
• your height
• height of the classroom
• distance between school and home
a ) Which unit of measurement would you use to measure each of
the above?
b ) Estimate the measurement of each object.
c ) Carefully measure the object. If you cannot measure it, explain
how you could find out its measurement.
Term 3
Platinum Maths Gr6_Term 3_CAPS.indd 162
02/02/13 1:48 AM
Convert units of length
It is useful to convert units of measurement, especially if a problem
has mixed measurements. To compare 5,6 cm and 62 mm, we write
5,6 cm as 56 mm to see that 62 mm is longer.
Challenge
Rules for converting
greater to smaller:
cm to mm × 10
× 100
m to cm
smaller to greater:
mm to cm ÷ 10
÷ 100
cm to m
10 mm = 1 cm
100 cm = 1 m
1 000 m = 1 km
Example
Convert 5,6 cm to mm: cm to mm: × 10; 5,6 cm × 10 = 56 mm
Convert 276 cm to m: cm to m: ÷ 100; 276 cm ÷ 100 = 2,76 m
Convert 5,4 km to m:
km to m: × 1 000; 5,4 km × 1 000 = 5 400 m
Convert 3 680 m to km: m to km: ÷ 1 000; 3 680 m ÷ 1 000 = 3,68 km
1. Convert from
metres to
millimetres:
a) 4 m
b) 6,3 m
2. Convert from
centimetres to
kilometres:
a) 600 000 cm
b) 350 000 cm
Use the same rules for lengths with decimal places:
Convert 8 mm to cm:
mm to cm: ÷ 10;
8 mm ÷ 10 = 0,8 cm
Convert 0,75 m to cm:
m to cm: × 100;
0,75 m × 100 = 75 cm
Convert 40 m to km:
m to km: ÷ 1 000; 40 cm ÷ 1 000 = 0,04 km
ExErCiSE 27.2
1. Convert cm to mm:
a ) 8 cm
b ) 27 cm
c ) 3,8 cm
d ) 2,9 cm
2. Convert mm to cm:
a ) 40 mm
b ) 70 mm
c ) 123 mm
d ) 461 mm
3. Convert m to cm:
a) 4 m
b ) 2,8 m
c ) 3,75 m
d ) 2,96 m
4. Convert cm to m:
a ) 500 cm
b ) 900 cm
c ) 1 200 cm d ) 856 cm
5. Convert m to km:
a ) 33 000 m b ) 2 567 m
c ) 1 007 m
d ) 1 070 m
6. Convert km to m:
a ) 2,3 km
b ) 1,07 km
c ) 22,81 km d ) 2,13 km
7. Convert the following units:
a ) 0,9 cm to mm
b ) 5 mm to cm
c ) 0,16 km to m
d ) 0,03 m to cm
e ) 110 m to km
f ) 0,01 km to m
g ) 5 cm to m
h ) 20 m to km
i ) 0,05 km to m
The circumference of
the wheel of a
trundle wheel is
exactly one metre.
When a person
pushes a trundle
wheel and the wheel
makes a full rotation
(one metre), a
clicking device
sounds to allow the
user to count the
clicks and find an
approximate
distance. Can you
think of how you
could use a trundle
wheel at school?
Topic 27: Length
Platinum Maths Gr6_Term 3_CAPS.indd 163
163
02/02/13 1:49 AM
Order lengths
You already know how to compare whole numbers and decimal numbers.
To compare lengths, first convert the lengths to the same unit.
Example
Convert these measurements to the same unit. Then write them in
order from shortest to longest.
3,4 km
3 300 m
360 m
3,38 km ← Convert all to metres.
× 1 000
× 1 000
3 400 m
We use kilometres
to measure the
distances between
different places
Challenge
Order the following
lengths from shortest
to longest:
1. 2 500 cm;
2,5 km;
25 001 mm;
250 m
2. 0,041 km;
3 900 cm;
39 002 mm;
40 m
3. 347 cm;
3,4 m;
0,035 km;
3 460 mm
164
3 300 m
360 m
3 380 m
360 m is shortest ← 3 digits
3 400 m > 3 300 m and 3 380 m ← 4 hundreds > 3 hundreds
3 380 m > 3 300 m ← 8 tens > 0 tens
So from shortest to longest the measurements are:
360 m
3 300 m
3,38 km (3 380 m)
3,4 km (3 400 m)
Example
Convert these measurements to the same unit. Then write them in
order from longest to shortest.
19 mm
2,5 cm
24 mm
0,02 m ← Convert all to centimetres.
÷ 10
÷ 10
× 100
1,9 cm
2,5 cm
2,4 cm
2 cm
4th
1st
2nd
3rd
So from longest to shortest the measurements are:
2,5 cm
24 mm
0,02 m
19 mm
ExErCiSE 27.3
1. Write the following lengths in order from shortest to longest:
a ) 6,25 km
6 240 m
6,3 km
6 290 m
b ) 9,03 km
9 100 m
9 090 m
9,2 cm
2. Write the following lengths in order from longest to shortest:
a ) 52 mm
4,9 cm
5 cm
48 mm
b ) 5,7 m
591 cm
5,9 m
580 cm
3. Round off the following lengths as instructed and then order them
from the shortest to the longest:
1 002 mm; 876 mm; 95 mm; 874 mm; 1 099 mm
a ) to the nearest 5 mm
b ) to the nearest 10 mm
c ) to the nearest 100 mm
d ) to the nearest 1 000 mm
Term 3
Platinum Maths Gr6_Term 3_CAPS.indd 164
02/02/13 1:49 AM
Solve problems involving lengths
Read each problem carefully before you decide which operation
to use.
Challenge
2. In cross-country running practice, Ben ran 6,4 km and Amy ran
5 390 m. Calculate how much further Ben ran than Amy.
Erin travelled
1 546 km when she
was on holiday. She
travelled 120 km on
the first day of her
journey, 388 km on
each of the next two
days and 40% of the
remaining journey on
the second last day of
her travels. Which of
the following
number statements
can be used to find
out how far Erin
travelled on the last
day of her journey?
3. Unathi is 92 cm tall. Her brother, Mbali, is 14 cm taller than Unathi.
What is Mbali’s height in metres?
1. 1 546 – (120 + 2
× 388) × 40%
4. Omar must complete a journey of 1 500 km. He travels 150,6 km
on the first day, 138,95 km on the second day, and 210,45 km on
the third day. What fraction of the distance must he still travel?
2. 1 546 – (1 546 –
(120 + 2 × 388))
× 40%
5. A roll of material is 4,5 m long. If Sarah uses 1,2 m, Bongi uses 90 cm
and Cathy uses 105 cm, how much material is left on the roll?
3. 1 546 – 210 –
(2 × 388) –
(40% × 1 546)
ExErCiSE 27.4
Solve the following word problems involving length and distance.
Remember to first convert the measurements to the same units.
1. a ) I cut a piece of string measuring 254 cm from a ball of string
measuring 4 m. Calculate how much string is left on the ball.
b ) If I used 40% of the 4 m roll of string, how much string would
be left on the roll?
6. To make a model boat, you need a 90 mm pole to hold the sail.
How many poles can you make out of a 5 m length of pole? What
length will be left over?
7. A teacher wants to glue tape around the edges of four identical
display boards. Each roll holds 5 m of tape. Each display board is a
rectangle with length 125 cm and width 80 cm. How many rolls of
tape will she need?
8. Beryl records that she takes approximately 120 steps per minute
when she is walking.
a ) If Beryl walks for 1 hour, how many steps has she taken?
b ) Beryl wears a pedometer that records the number of steps
she has taken. If the pedometer reading is 10 080, how many
minutes has she walked for?
c ) If an average stride is 76 cm, how many metres has she
covered in 10 080 steps?
d ) Round off your answer to question c) to the nearest whole
kilometre.
A model boat
Topic 27: Length
Platinum Maths Gr6_Term 3_CAPS.indd 165
165
02/02/13 1:49 AM
4 Term 4
Thembi sent this SMS to Sipho
Nicole chats to her friend at school
Mr Jacobs phones his insurance company to
report a car accident
Mandy rudely speaks on her cellphone in a movie
theatre
166
Platinum Maths Gr6_Term 4_CAPS.indd 166
11/02/13 6:37 PM
Topics 28–38
Starting off
Cellphones have changed our lives. It has never
been so easy to contact each other any time
and anywhere.
1. Give an example of when cellphones are
useful and when they are a nuisance. Does
your school have a cellphone policy?
Zama speaks to her husband on her cellphone
2. Do this survey on the learners in your class:
a ) Count how many learners in your class
have access only to a cellphone at home
and do not have a landline telephone.
b ) Count how many learners in your class
have access only to a landline telephone
at home and not a cellphone.
c ) Count how many learners have access
to both a cellphone and a landline
telephone at home.
d ) Express each number in a) to c) as
a fraction of the class and then as a
percentage.
e ) Do you think that you can use these
results to talk about the cellphone
access of any Grade 6 learner anywhere
in South Africa? Why?
3. The abbreviation ‘SMS’ stands for ‘short
message service’. SMS language has become
popular and is a shorthand version of the
full message. Decode the message in the
picture on the top left of page 174 and write
it in full.
Content covered in Term 4
A ‘no cellphones’ sign
Topic 28: Whole numbers, Topic 29: Multiplication, Revision,
Topic 30: Common fractions, Assignment, Topic 31: Properties of 3D Objects,
Revision, Topic 32: The History of Measurement, Topic 33: Perimeter, area
and volume, Revision, Topic 34: Division, Assignment, Topic 35: Number
statements, Revision, Topic 36: Transformations, Topic 37: Position and
movement, Revision, Topic 38: Probability
167
Platinum Maths Gr6_Term 4_CAPS.indd 167
11/02/13 6:37 PM
Count, order, compare and represent
whole numbers
Topic
28
Maths ideas
• Represent, order
and compare large
numbers.
Read and write large numbers
Example
• Recognise place
value.
The number three hundred and sixty-three million, five hundred
and four thousand, six hundred and forty-two is written like this:
363 504 642.
• Round off to the
nearest 5, 10, 100
and 1 000.
ExERCiSE 28.1
1. Write down the following numbers in digits, and give the value of
the digit 5 in each number:
a ) ninety-five million, sixty-three thousand, two hundred and sixteen
b ) four hundred and eighteen million, three hundred and twentyfive thousand and ten
c ) nine hundred and six million, fifty thousand, eight hundred
and eight
2. Write the following numbers in words and give the value of the
digit 1 in each number:
a ) 315 090 450
b ) 130 230 430
c ) 999 107 455
If a large number is given in expanded form, you should first put the
parts in the correct order.
Example
number of digits →
Write the number in digits:
50 000 000 + 3 000 + 10 + 80 000 + 100 000 000 + 600 + 7 000 000
(8)
(4) (2)
(5)
(9)
(3)
(7)
100 000 000 + 50 000 000 + 7 000 000 + 80 000 + 3 000 + 600 + 10
So the number is 157 083 610.
ExERCiSE 28.2
Write the following expanded forms in digit form:
a ) 80 000 + 5 000 000 + 3 000 + 2 + 400 + 400 000
b ) 2 + 600 000 000 + 5 000 + 90 000 + 30 000 000 + 7 000 000 + 100 000
c ) 5 000 + 900 + 700 000 000 + 20 000 + 8 + 40 + 400 000
168
Term 4
Platinum Maths Gr6_Term 4_CAPS.indd 168
11/02/13 6:37 PM
Round off, compare and order
large numbers
You can round off large numbers in different ways, depending on how
accurate you want the number to be.
Remember the rules for rounding to the nearest 10, 100 or 1 000:
• Decide which place value you are rounding to.
• Increase it by one if the digit to the right is 5 or more. This is
called rounding up.
• Leave it the same if the digit to the right is 4 or less. This is called
rounding down.
ExERCiSE 28.3
Round each number to the nearest 5, 10, 100 and 1 000:
1. 345 908 635
2. 203 989 622
3. 23 755 908
When you work with whole numbers, the number with the most digits
is the larger number. If there are the same number of digits, then
compare the place value of each digit, working from left to right.
Example
Compare the numbers 567 309 989 and 567 311 213:
Both numbers start with the digits 567 3… . The first digits that are
different are 0 and 1, so the number 567 311 213 is larger.
ExERCiSE 28.4
1. Write < or > to make each number sentence true:
a ) 56 909 123 □ 56 912 345
b ) 230 341 256 □ 203 789 966
c ) 399 459 093 □ 398 098 432
Challenge
These are the
approximate distances
of four planets from
the sun.
Mercury – 57 910 000 km
Venus – 108 200 000 km
Earth – 149 600 000 km
Mars – 227 940 000 km
How do you think
these numbers have
been rounded off? To
the nearest … ?
Use rounding off to
decide which planet is
approximately three
times as far from the
sun as one other
planet.
Did you know?
• A decimal number
with fewer digits
can be larger than
a number with
more digits.
• The number 8,4
has two digits, and
the number 3,99
has three digits,
but 8,4 is the larger
number.
2. Write the following numbers in ascending order (from smallest to
largest). Remember to first write all numbers in the same units:
a ) 390 876 km; 390 885 098 m; 390 886 120 km
b ) 456 890 765 g; 456 890 kg, 456 809 145 g
c ) 546 908 870 m; 54 690 km; 54 699 km
3. Which number is larger, 244 985 or 245 320? What kind of
rounding will make the numbers equal?
Topic 28: Count, order, compare and represent whole numbers
Platinum Maths Gr6_Term 4_CAPS.indd 169
169
11/02/13 6:37 PM
Topic
29
Multiplication
Maths ideas
• Estimate answers
to multiplication.
• Work with factors.
• Use rounding and
compensating to
multiply numbers.
• Multiply a fourdigit number
by a three-digit
number.
• Solve problems
using
multiplication.
When working
with multiple
operations, you
should first solve
the brackets,
then do the
multiplication
and division and
lastly addition
and subtraction.
Estimate and calculate answers
You have learnt that you can use rounding off to estimate an answer.
You have also learnt that you can round off and compensate, to find
the exact answer.
Example
4 232 × 98 is almost the same as 4 232 × 100.
4 232 × 98 = 4 232 × (100 – 2)
= (4 232 × 100) – (4 232 × 2)
= 423 200 – 8 464
= 414 736 (by column subtraction)
This is an exact answer, because (-2) compensated for rounding up
98 to 100.
ExERCiSE 29.1
1. For each multiplication below:
• Estimate the answers by rounding the first number to the
nearest 1 000 and the second number to the nearest 100.
• Now use your calculator to find exact answers.
• Use the column method to find the difference between each
estimate and the exact answer.
a ) 4 569 × 399 b ) 3 998 × 160 c ) 2 259 × 576 d ) 6 986 × 675
2. Round one number and compensate to find the exact answer:
a ) 3 622 × 99
b ) 2 350 × 102
c ) 1 320 × 1 003
d ) 2 013 × 495
3. Use your calculator to multiply each answer in question 2 by 1. Then
multiply each answer by 0. Write down what you have noticed.
4. Calculate the answer to each number sentence.
a) 2 + 3 × 7
b ) 5 – 12 ÷ 4
c ) 2 × 15 + 30
d ) 18 ÷ 6 + 3
e ) 40 × 8 ÷ 2
f ) (11 + 4) ÷ (5 – 2)
g ) 18 ÷ (6 + 3)
h ) 3 × (4 + 1) – 10 i ) 3 × (4 × 1)
170
Term 4
Platinum Maths Gr6_Term 4_CAPS.indd 170
11/02/13 6:37 PM
Multiply using the column method
Earlier this year you used the column method to multiply two
numbers. You can also use this method to multiply large numbers.
First multiply by the units, then by the tens, then by the hundreds, and
add the results.
Example
First estimate the answer to 3 422 × 253.
Then use the column method to find the exact value of 3 422 × 253.
a ) An estimate is 3 400 × 250
= (34 × 5 × 5) × (100 × 10)
= (170 × 5) × (1 000)
= 850 000
b ) To use the column method, find (3 422 × 3), (3 422 × 50),
(3 422 × 200), and add the three results in columns.
3422
253
×
10266
(3 422 × 3)
171100
(3 422 × 50)
+684400
(3 422 × 200)
865766
The estimate in a) was quite close.
ExERCiSE 29.2
1. Use the column method to do the following calculations. Allow
your partner to check your answers with a calculator.
a ) 1 803 × 766
b ) 1 220 × 728
c ) 1 079 × 909
d ) 2 176 × 439
e ) 4 008 × 237
f ) 5 157 × 152
g ) 1 116 × 191
h ) 1 712 × 612
i ) 3 176 × 457
Challenge
2. In questions 1 a) and b), round the second number to the nearest
100 to find an estimate for the multiplication. Use subtraction to
find which estimate was more accurate.
Work out the missing
numbers.
3. Use an estimate and then compensate, to check your answers to
questions 1 c) and g).
×
4. Use column multiplication to check if these division answers are
correct. Correct any mistakes.
a ) 1 573 075 ÷ 623 = 2 525
b ) 188 238 ÷ 1 374 = 136
c ) 586 608 ÷ 2 424 = 242
d ) 918 090 ÷ 3 030 = 303
1 2 0 □
□ 1 □
2 4 0 2
□ 2 0 □ 0
+□ 6 0 3 0 0
□ 7 4 7 □ □
Topic 29: Multiplication
Platinum Maths Gr6_Term 4_CAPS.indd 171
171
11/02/13 6:37 PM
Solve problems with multiplication
You can use your knowledge of multiplication to solve many different
types of problems. Remember to estimate your answer first so that
you know more or less what your answer should be. Let your partner
check your answers with a calculator.
ExERCiSE 29.3
1. Phindile is preparing to plant a crop of potatoes this spring. The
field will have 1 212 rows of potatoes with 182 plants in each row.
a ) Estimate how many plants he must prepare for planting.
b ) Use column multiplication to calculate the exact number of
potato plants in total.
c ) The potatoes must be watered. Each row uses 525 litres per
month. How many kilolitres of water is used on the field each
month?
d ) Water costs R42,00 per kilolitre with an added service charge
of R220 per week. How much does it cost the farmer to water
the field for a month of 4 weeks?
2. During an endurance test, the cars race around a track at a speed
of 138 km/h for 16 hours.
a ) What is the total distance that each car travels?
b ) If 132 cars do the test, how many kilometres are travelled in
total by all the cars?
3. An English visitor to Africa has 1 284 British Pounds. One British
Pound can buy R13, or 156 Kenyan Shillings, or 34 Mozambican
Meticals.
a ) How many Kenyan Shillings can he buy?
b ) How many more Mozambican Meticals can he buy than
Rands?
Challenge
How many seconds
are there in a year of
365 days?
172
4. A fuel storage depot has 2 560 containers that hold 950 litres each.
Hector estimates the total number of litres by calculating
2 600 × 900, and Maria estimates by calculating 2 560 × 1 000.
a ) Work out both estimates.
b ) Use the column method to find the exact number of litres.
c ) Whose estimate was more accurate? How much closer was this
estimate to the exact answer?
5. a ) How many seconds are there in a day?
b ) Use an estimate and then compensate, to calculate how many
seconds there are in a month of 31 days.
Term 4
Platinum Maths Gr6_Term 4_CAPS.indd 172
11/02/13 6:37 PM
Revision
1. Write down the following numbers in digits and give the value of the digit 6 in each number:
a ) three hundred and sixteen million, two hundred thousand, nine hundred and fifty-six (2)
b ) 40 000 + 600 000 + 5 000 + 200 000 000 + 300 + 2
(2)
2. Round the following numbers to the nearest 1 000:
a ) 340 789 272
b ) 282 873 621
(1)
(1)
3. Write < or > to make each number sentence true:
a ) 108 529 898 □ 108 592 759
b ) 590 247 827 □ 595 392 939
(1)
(1)
4. Write each set of numbers in ascending order:
a ) 20 983 798 m; 209 830 km; 20 982 km
b ) 567 984 837; 567 988 437; 567 899 973
(2)
(2)
5. Round the following numbers to the nearest 1 000 to estimate each answer:
a ) 2 480 × 789
b ) 899 522 × 4 540
(2)
(2)
6. Use rounding and compensating to calculate 34 232 × 990.
(2)
7. Use multiplication and division to complete the following calculations:
a ) □ × 24 = 15 000
b ) □ ÷ 342 = 2 876
(2)
(2)
8. Use the column method to do the following multiplications:
a ) 116 × 2 098
b ) 5 567 × 754
(3)
(3)
9. A farmer will plant 326 tomato seedlings in each of 120 rows. How many
tomato seedlings will she plant?
(2)
Total marks: 30
Revision
Platinum Maths Gr6_Term 4_CAPS.indd 173
173
11/02/13 6:37 PM
Topic
30
Common fractions
Maths ideas
• Recognise and
use equivalent
forms of common
fractions.
• Recognise
equivalence
between common
fractions, decimal
fractions, and
percentage forms
of the same
number.
• Compare fractions.
• Add and subtract
fractions and
mixed numbers.
• Find fractions of
whole numbers.
• Solve problems
with fractions.
Find equivalent forms and
compare common fractions
You can multiply or divide the numerator and the denominator of a
fraction by the same whole number to find equivalent fractions.
Example
40
Here are three fractions that are equivalent to ___
:
100
40 ______
120 ______
10 _______
4
___
; 40 × 3 = ___
; 40 ÷ 4 = __
; 40 ÷ 10 = __
100 100 × 3 300 100 ÷ 4 25 100 ÷ 10 10
40
= __2
In its simplest form, ___
100 5
If you cannot easily see which fraction is larger, write them with the
same denominator.
Example
Which fraction is larger: __34 or __45 ?
Both 4 and 5 divide into 20, so use this as the new denominator for
both fractions:
3 __
15 __
16
__
× 5 = __
; 4 × __4 = __
4 5 20 5 4 20
16 is larger than 15, so __45 is larger than __34
ExERCiSE 30.1
1. Find four equivalent fractions for each fraction:
3
10
12
b) __
c ) __
d) __13
a ) __
15
15
16
2. Write each of these fractions in its simplest form:
16
15
56
36
42
b) __
c ) __
d) __
e ) __
a ) __
45
66
32
49
48
3. Which of these fractions are equivalent to __56 ?
76
f ) ___
100
55 __
25 __
51 __
35
__
; 60 ; __
; 45 ; __
; 95 ; __
66 50 30 54 61 96 36
4. Arrange the following fractions in ascending order:
2 __
1
; 3 ; __
a ) __
15 5 10
5
c ) __13 ; __34 ; __28 ; __
12
174
11
b ) __23 ; __49 ; __
27
5 __
d ) __
; 3 ; __1 ; __5
12 4 2 6
Term 4
Platinum Maths Gr6_Term 4_CAPS.indd 174
11/02/13 6:37 PM
Different fractions for different
contexts
The three different types of fractions are not really different things,
but are different ways of saying the same thing.
Example
The photo shows half of an apple. You can write this as __12 an apple.
You have 50% of the apple left. You can see 0,5 of the apple in the
picture. All these fractions are describing the same thing — half of
an apple.
In the above example, which is the best way to describe what you
see in the picture: __12 an apple, 50% of an apple, or 0,5 of an apple?
Although the three fractions all mean the same thing, some fraction
types are better for certain situations than other fraction types.
ExERCiSE 30.2
Which type of fraction would you use for each context? Give an
example of each.
1. To describe a mark you get for a test
2. To describe the time in which a race is run
3. To describe the ingredients of a recipe
4. To describe distance travelled in kilometres
Challenge
5. To describe the fraction of children in a class who play tennis
1
of __19
What is __
11
of __17 of __15 of __13 of
You need to be able to convert one fraction type to another.
135 135?
ExERCiSE 30.3
1. Below are fractions in all three forms. Fill in the missing fractions.
a) 0
b ) 0%
c ) 0,0
1
__
4
□
0,25
□
50%
□
3
__
4
□
0,75
1
□
□
2. Draw a picture to show each of the following as fractions of a whole:
a ) 25%
b ) 0,75
c ) 0,5
Topic 30: Common fractions
Platinum Maths Gr6_Term 4_CAPS.indd 175
175
11/02/13 6:37 PM
Convert decimals and common
fractions
Example
Converting a decimal to a common fraction is easy, because
decimals are in tenths, hundredths or thousandths.
6
in its simplest form is __35
• 0, 6 is 6 tenths and __
10
45
9
in its simplest form is 3__
• 3,45 is hundredths and 3___
100
20
Example
To convert a fraction to a decimal, make sure your denominator is
either ten or a hundred.
6
6
6
9
24
__
= 0,6 ; ___
= 0,09 ; 1__
= 1__
× __4 = 1___
= 1,24
10
100
25
25 4
100
ExERCiSE 30.4
1. Convert the following decimal fractions to common fractions or
mixed numbers in their simplest form:
a ) 0,4
b ) 0,7
c ) 0,55
d ) 0,32
e ) 0,21
f ) 0,03
g ) 0,01
h ) 0,02
i ) 1,1
j ) 2,06
k ) 11,89
l ) 21,07
m) 10,24
n ) 2,72
o ) 4, 75
p ) 9, 36
2. Convert these fractions to decimals:
61
9
2
b ) 2__
c ) ___
a ) __
10
10
100
55
e ) ___
100
i ) __35
7
m) 4__
25
3
g ) 4___
100
17
f ) 3___
100
1
j ) __
20
1
k ) 7__
25
3
o ) 3__
30
n ) 7__14
4
d ) ___
100
7
h ) 6__
10
l ) 6__12
p ) __72
Challenge
The clothing items in the table below are all made of a combination of cotton, polyester and wool. Work in
small groups and complete the table. Then convert each percentage to a decimal fraction and a common
fraction in its simplest form.
Jersey
Cotton
Polyester
Wool
176
Skirt
50%
30%
Socks
42%
45%
40%
Trousers Shorts
18%
68%
73%
57%
Scarf
38%
5%
62%
Clothing label
Term 4
Platinum Maths Gr6_Term 4_CAPS.indd 176
11/02/13 6:37 PM
Convert percentages and common
fractions
All percentages are out of 100, so to convert a percentage to a
common fraction you write the percentage over 100 and simplify.
To convert a common fraction to a percentage, make sure your
denominator is 100.
Example
15
3
15 % = ___
= __
100 20
40
___
= 40%
100
3 _____
60
__
= 3 × 20 = ___
= 60%
5 5 × 20 100
ExERCiSE 30.5
1. Convert the following percentages to common fractions in their
simplest form:
a ) 13% b) 59% c ) 12% d) 72% e ) 80% f ) 75%
g ) 4%
h) 20% i ) 16% j ) 35% k ) 92% l ) 48%
2. Convert the following common fractions to percentages:
65
3
7
1
b ) ___
c ) __
d ) __
a ) ___
100
100
10
10
3. Copy and complete the following table:
Common fraction
Decimal fraction
Percentage
0,25
22%
1,85
35%
Game
Play in pairs. Enlarge and cut out these fraction dominoes and lay them face down. Each player picks up 5
dominoes, and one domino is placed face up on the table. Take turns to lay down a domino to match an
equivalent fraction on one side. If you do not have an equivalent fraction, you miss a turn and pick up another
domino. The first player to lay down all his or her dominoes wins.
Fill in these 40 numbers on the 20 dominoes:
1
2
1
4
1
10
1
4
4
20
10%
25
100
4
10
5
10
2
4
0,05
1
100
75
100
0,2
75
150
0,01
25%
2
5
50%
10
200
2
8
40%
75%
4
8
20
100
0,4
0,75
1%
3
15
4
40
0,25
1
20
6
8
0,5
0,1
20%
10
100
5%
5
100
1
5
Topic 30: Common fractions
Platinum Maths Gr6_Term 4_CAPS.indd 177
177
11/02/13 6:37 PM
Add and subtract fractions
First write the fractions with the same denominator. Add and subtract
whole numbers first. Your answer should be a mixed number in its
simplest form.
Example
3 × 7 ____
4__35 + 2__47 = 6 + ____
+ 4 × 5 (5 and 7 both divide into 35)
5×7 7×5
21 + 20
= 6 + ______
35
41
= 6 + __
35
6
6
= 7__
= 6 + 1 + __
35
35
If you need to, use a whole unit to make more fractions.
Example
5
5 __
7
7__
– 3__
= 4 + __
– 7 (cannot subtract 7 from 5)
12
12
12 12
12 __
7
12
+ 5 – __
(4 = 3 + 1 = 3 + __
)
= 3 + __
12 12 12
12
10
= 3__
= 3__56
12
ExERCiSE 30.6
1. Calculate each of the following:
4
b ) 5__23 – 1__46
a ) 2__35 + 1__
15
7
c ) 3__34 + 2__
10
5
e ) 2__59 + 1__
18
g ) __37 + 6 __25
15
i ) __39 + 1 + 1__
18
d ) 3__18 – 1__14
3
f ) 7__12 – 2__
10
h ) 4 – 2__23
j ) 1__13 + 5__16 – 2__12
2. Double each quantity of this recipe:
2__14 cups of flour
5
__
of a teaspoon of baking powder
8
1__35 cups of syrup
2
__
cups of water
3
3
__
of a teaspoon of nutmeg
4
3
__
of a teaspoon of salt
10
3. Calculate the total length of the following cuts of wood:
4
m + 3__56 m + 2__13 m
1__34 m + __
12
178
Term 4
Platinum Maths Gr6_Term 4_CAPS.indd 178
11/02/13 6:37 PM
Solve problems with fractions
Example
To find __34 of 372, first divide the number into four equal groups:
372 ÷ 4 = 93
Now choose 3 of these groups: 3 × 93 = 279. So __34 of 372 = 279.
This example shows you that to calculate a fraction of a whole
number, you divide the number by the denominator of the
fraction. Then you multiply the answer by the numerator of the
fraction.
ExERCiSE 30.7
1. Calculate the following:
3
b ) __
of R90
a ) __23 of 36 kg
10
d ) __25 of 100 ml
7
of R840
g ) __
20
4
e ) __
of 900 cm
15
7
c ) __
of 72 mins
12
f ) __38 of 1000 ml
9
h ) __
of 1 km (in metres)
10
7
of 1 kg (in grams)
i ) ___
100
4
of an
2. James exercised for __25 of an hour, and Sandile exercised for __
12
hour? Who exercised for the longest time?
3. There are 36 learners in a Grade 6 class. If __59 are girls, how many
boys are there in the class?
4. At a birthday party, 9 pizzas were ordered for the guests. After
supper 1__58 of a pizza remained. How much pizza did the guests eat?
3
of the money, how much has she left?
5. Talia has R75. If she spends __
25
6. A water tank holds 600 kl of water.
a ) How much water is in the tank if it is __23 full?
b ) How much water must be added to completely fill the tank?
7. A school week is 26__14 hours long. After 5__38 hours have been used to
teach Mathematics, how many hours of the school week are left
for teaching the other subjects?
1
km
8. Aiden went on a hike of 24__12 km. On the first day he walked 9__
16
3
__
and on the second day he walked 8 4 km. How far did he still have
to walk on the last day?
Challenge
Which is greater: 1__14
of 400 or __45 of 500?
Topic 30: Common fractions
Platinum Maths Gr6_Term 4_CAPS.indd 179
179
11/02/13 6:37 PM
Assignment
Percentages, profit and loss
Percentages play a very important role in business and the economy
of the country. In this investigation you will learn about profit and
loss.
If a person buys something and then sells it for more than they paid
for it, we say the person has made a profit. If they sell it for less than
they paid for it, we say they have made a loss.
To f ind a prof it or loss, do the following:
10% prof it on R200:
• Find 10% of R200 = R20 prof it
• Selling price: add the prof it to the original amount:
R200 + R20 = R220
Key words
• economy – how a
country manages
its money and
goods
• profit – increase
in wealth; making
money
• loss – decrease
in wealth; losing
money
20% loss on R350:
• Find 20% of R350 = R70 loss ← Double 10% or ÷ 5
• Selling price: subtract the loss from the original amount:
R350 – R70 = R280
1. Find the profits and losses on these amounts:
a ) 25% profit on R400
(1)
b ) 10% loss on R320
(1)
c ) 20% profit on R5 000
(1)
d ) 25% loss on R48 000
(1)
e ) Lerato bought a bicycle for R480 and sold it 5 years later
at a loss of 10%. How much did she sell the bicycle for?
(2)
f ) Rajan bought eggs from the farmer and sold them at the
market. He paid R45 for the eggs and sold them at a 25%
profit. How much money did Rajan make? What was the new
selling price?
(3)
Identify the percentage profit or loss in this way:
Step 1: Is the new price more or less than the old price? More means
a profit, less means a loss.
Step 2: Find the difference between the two prices.
Step 3: Make a fraction, using the difference in price as the
numerator and the original price as the denominator.
Step 4: Find the equivalent of this fraction as a percentage.
180
Term 4
Platinum Maths Gr6_Term 4_CAPS.indd 180
11/02/13 6:38 PM
Penny bought a computer for R14 000 and sold it 2 years later for
R12 600.
What was her percentage prof it or percentage loss?
R12 600 < R14 000, so she made a loss.
14 000 – 12 600 = 1 400
1
or 0,1 = 10% loss
1 400 ÷ 14 000 = 10
—
2. Copy this table into your exercise book. Fill in whether each
situation is a profit or a loss. Calculate the percentage profit or
percentage loss.
(12)
Original
price
Selling price
R700
R630
R250
R300
R1 000
R750
R1 500
R750
R6 000
R7 500
R24 000
R28 800
Prof it or
loss?
Percentage
prof it or loss
3. Zane bought a horse for R5 000 but had to sell it for
R4 000. What percentage profit or percentage loss
did he make?
(2)
4. Jack bought a house for R1 100 000 and spent another
R50 000 on renovating the house. He sold the house
for R1 437 500, 5 years later. What percentage profit or
percentage loss did he make?
(3)
5. By putting her money in the bank, Sandy made a profit
on her money. She started with R2 000 and it grew to
R2 500 by the end of 4 years. What percentage profit did
she make?
(2)
6. A vegetable seller bought 100 sacks of onions for R400 and
sold them at R4,80 per sack. What percentage profit or
percentage loss did he make?
(2)
Total marks: 30
Assignment
Project
Platinum Maths Gr6_Term 4_CAPS.indd 181
181
11/02/13 6:38 PM
Topic
31
Properties of 3D objects
Maths ideas
Model 3D objects
• Identify and name
3D objects.
In Topic 11 you worked with different 3D objects. You will now explore
some of these 3D objects further by making models.
• Use features
of 3D objects
to distinguish,
describe, sort and
compare objects.
• Make models of 3D
objects.
• Interpret 3D
drawings.
• Interpret nets of
3D solids.
ExERCiSE 31.1
Make these models so that you can use them to answer questions
about solids later in this topic.
You will need: straws; thin string or cotton; paper glue; sticky tape;
tissue paper in different colours; paper to draw on.
1. Model a cube
Step 1: Cut twelve straws into 6 cm lengths.
Step 2: Thread cotton through four cut straws and tie the ends
tightly together to form a square. Repeat this step so that
you have two squares.
Step 3: Tie cotton to each vertex of one square.
Step 4: Thread a straw onto the cotton at each vertex and attach
the other side to the other square to form a cube.
Step 5: Use glue or sticky tape to cover each side with different
coloured tissue paper (if you have this).
2. Model a tetrahedron
Step 1: Cut six straws into 6 cm lengths.
Step 2: Thread cotton through three cut straws. Tie the ends of
the cotton tightly to form a triangle. This is the base.
Step 3: Tie cotton to each vertex of the triangle base.
Step 4: Thread cotton through the other three cut straws.
Step 5: Tie the cotton ends from each vertex together to form
a tetrahedron. Each straw represents an edge of the
tetrahedron.
Step 6: Use glue or sticky tape to cover each side of the
tetrahedron with different coloured tissue paper.
182
Term 4
Platinum Maths Gr6_Term 4_CAPS.indd 182
11/02/13 6:38 PM
3. Model a square-based pyramid
Step 1: Cut four straws to 6 cm lengths and four straws to 10 cm
lengths.
Step 2: Take the four shorter straws and thread them onto a
piece of cotton.
Step 3: Tie the ends of the cotton tightly to form a square.
Step 4: Tie cotton to each vertex of the square and thread a
longer straw onto each piece.
Step 5: Tie the ends of the cotton together to form a square
pyramid.
Step 6: Use glue or sticky tape to cover each side with different
coloured tissue paper.
4. Model a rectangular prism
Step 1: Cut four straws into 8 cm lengths, four straws to 6 cm
lengths and four straws to 5 cm lengths.
Step 2: Use cotton to tie the four 8 cm and the four 6 cm straws
together to form two rectangles of length 8 cm and width
6 cm. These will be the base and the top of your prism.
Now you need to join them with four vertical edges.
Step 3: Tie cotton to each vertex of each rectangle.
Step 4: Thread a 5 cm straw on to the cotton at each vertex
of one of the rectangles and tie the ends to the other
rectangle so that the new straws are all parallel to each
other.
Step 5: Use glue or sticky tape to cover each side with different
coloured tissue paper.
ExERCiSE 31.2
Here are diagrams of different solids:
A
B
Cone
C
Cylinder
D
Triangular prism
1. Write down the name of each solid.
Sphere
2. How many straight edges does each solid have?
3. How many curved edges does each solid have?
4. Which solid(s) could you form using only straight straws?
Topic 31: Properties of 3D objects
Platinum Maths Gr6_Term 4_CAPS.indd 183
183
11/02/13 6:38 PM
Sort and compare 3D objects
ExERCiSE 31.3
1. Use the models you constructed earlier to compare the following
pyramids. The tetrahedron has been filled in as an example.
Pyramids
Number of faces
4 faces
Tetrahedron
(triangular-based
pyramid)
Square-based
pyramid
Pentagonal pyramid
(5-sided base)
Hexagonal pyramid
(6-sided base)
Challenge
Here are three
number sentences
about the number of
faces (F), vertices (V)
and edges (E) of a
prism. Use your
models and 3D
diagrams to test
which is the correct
number sentence.
a) 2F + V = E
b) F + V – E = 2
c) F + V = E – 2
Prism
Shape of faces
1 triangular
base and 3 other
triangular faces
Number of
edges
6 edges
Number of
vertices
4 vertices (3 on the
base and 1 on top)
2. In small groups, discuss what you observed about the relationship
between the shape of the base in each pyramid and:
a ) the number of faces in each pyramid
b ) the number of edges in each pyramid
c ) the number of vertices in each pyramid.
3. Do you think the same rules would apply for working out the
number of faces, edges and vertices in a prism? Explain your
thinking.
4. Fill in the table below using your model of a cube and the diagram
of a triangular prism in Exercise 31.2 to check whether your answer
in question 3 was correct.
Number of faces
Shape of faces
Number of
edges
Number of
vertices
Triangular prism
Cube
5. a ) List the similarities between a tetrahedron and a rectangular
prism.
b ) What is the difference between a cone and a pyramid?
c ) In what ways are a cone and a pyramid similar?
184
Term 4
Platinum Maths Gr6_Term 4_CAPS.indd 184
11/02/13 6:38 PM
interpret 3D drawings
When you look at an object from different positions, its shape might
seem different.
ExERCiSE 31.4
1. Lay a rectangular piece of paper flat on your desk, and look down
at it from directly above. Draw a diagram to show what the paper
looks like. You do not have to draw it to scale, but show the size of
each corner angle correctly.
2. Hold the paper straight up in front of you, tilt it away from you at
the top, and then turn it in a clockwise direction. Which of these
diagrams shows what the piece of paper looks like to you now?
3. Use the corner of your piece of paper to measure the angles in the
diagram that you chose. Are the corner angles right angles in the
diagram?
a)
b)
c)
4. Explain what can happen when you draw a diagram of a shape
that is seen from an angle.
When you look at a solid object from an angle, the shape of its faces
might look different.
ExERCiSE 31.5
This picture shows everyday objects from
different angles:
1. How many of these 2D shapes can you see in
the diagram: triangles, squares, rectangles,
circles, parallelograms?
2. Write down the solid objects you can identify
in the picture.
3. Which faces of the solids look the same as in
real life?
4. Which faces of the solids have shapes that
are different to their shape in real life? In
what way are they different?
Topic 31: Properties of 3D objects
Platinum Maths Gr6_Term 4_CAPS.indd 185
185
11/02/13 6:38 PM
identify nets
Challenge
Nets are useful to make models of 3D objects. Making nets helps you
to see an object as a collection of related 2D shapes. You will now get
more practice in working with nets of solids.
Fill in two missing
faces to complete
this solid, and name
the solid.
ExERCiSE 31.6
Decide if each net can form a solid. If it can, name the solid. Copy and
cut out each net, and fold it up to check your thinking.
a)
b)
c)
d)
e)
f)
g)
h)
i)
ExERCiSE 31.7
Copy each net, and draw in the missing face to form a solid. Name the
solid. Cut out your nets, and fold them up to check your thinking.
a)
186
b)
c)
Term 4
Platinum Maths Gr6_Term 4_CAPS.indd 186
11/02/13 6:38 PM
Revision
8. Copy and complete the table below to
1. Convert the following decimal fractions to
compare the square-based pyramid and
common fractions in their simplest form:
the hexagonal pyramid:
(12)
a ) 0,35
(1)
b ) 0,07
(1)
Feature
SquareHexagonal
based
pyramid
2. Convert the following common fractions
pyramid
(six-sided
to decimal fractions:
7
__
base)
(1)
a ) 10
surfaces
(1)
b ) __34
number of
faces
3. Convert the following percentages to
common fractions in their simplest form:
shape of faces
a ) 55%
(1) edges
b ) 20%
(1) vertices
angles on
4. Convert these common fractions to
each face
percentages:
4
__
(1)
a) 5
9. Look at the picture below:
9
__
(1)
b ) 10
5. Create 2 equivalent fractions for each of
these common fractions:
25
(1)
a ) ___
100
b ) __49
6. Fill in the missing fraction:
1
=1
a ) □ + ___
10
b ) 2__14 + □ = 3__12
7. Calculate the following:
18
5
+ 3__
– 2__59
a ) __
36
12
b ) __23 of an hour (in minutes)
(1)
Triangular Prism
(2)
a ) How many vertices does the object
have?
(1)
b ) How many edges does the object
have?
(1)
c ) How many faces does the object
have?
(1)
(1)
Total marks: 30
(1)
(1)
Revision
Platinum Maths Gr6_Term 4_CAPS.indd 187
187
11/02/13 6:38 PM
Topic
32
The history of measurement
Maths ideas
• Early methods of
measurement.
• Early units of
measurement.
• Work with
early units of
measurement.
Where did measurement begin?
Thousands of years ago, people
needed to work with weights
and measurements in day-to-day
activities. Examples of such activities
were building homes and trading
for food or raw materials.
To find units for measurement,
people used their bodies and
natural surroundings.
Length was first measured using the
forearm, hand or finger. Time was
measured by the movements of the
sun, moon and stars. Capacity was
measured by filling clay or metal containers with plant seeds which
were then measured to find the volume of the contents. Seeds and
stones were also used as weights for early scales.
ExERCiSE 32.1
1. Work with a partner, using a ruler or measuring tape:
a ) Measure the length of your forearm, from your elbow down to
your wrist.
b ) Measure the length of your hand, from your wrist to the tip of
your middle finger.
2. Now that you know the length of your forearm in centimetres,
measure the following. You can use your ruler to help you.
a ) How many ‘forearms’ is the length of your desk?
b ) How many ‘forearms’ is the length of your body?
c ) How many ‘forearms’ is the height of the classroom door?
Did you know?
The height of a horse
is still measured in
hands.
188
3. Using the measurement for the length of your hand, measure the
following:
a ) How many ‘hands’ is the length of your desk?
b ) How many ‘hands’ is the length of your body?
c ) How many ‘hands’ is the width of the classroom door?
Term 4
Platinum Maths Gr6_Term 4_CAPS.indd 188
11/02/13 6:38 PM
How things have changed
The ancient measurements of ‘digit’, ‘palm’, ‘span’ and ‘cubic’ units of
length slowly changed to ‘inch’, ‘foot’ and ‘yard’. The ‘digit’ was the
length of a finger, while an ‘inch’ was the length of a thumb. A ‘span’
was an outstretched hand, which measured about 9 inches. The ‘foot’
was the length of an average foot, about a size-six shoe. A ‘yard’ was
the length of a belt.
ExERCiSE 32.2
1. If a foot is approximately 305 mm, calculate what the following
measurements will be:
a ) 2 feet in cm
b ) 7 feet in m
c ) 10 feet in m
d ) 50 feet in km
2. King Edward II of England ruled that 1 inch equalled 3 grains of
barley placed end to end lengthwise. How many grains of barley
would be equal to the following lengths?
a ) 3 inches
b ) 5 inches
c ) 9 inches
d ) 20 inches
Today all countries, except the USA, Liberia and Myanmar have
adopted the metric system of measurement. England still uses many
old, non-metric measurements, for example, speeds are still given in
miles per hour, as well as in km/h.
The metric system was simplified by 1960 and today we use the
following metric units:
• metre – for length
• kilogram – for mass
• second – for time
• litre – for capacity
• degrees Celsius – for temperature
ExERCiSE 32.3
What metric unit of measurement would you use to measure:
1. how hot water is?
2. how long a piece of wood is?
3. the mass of a bag of flour?
4. how much cool drink is in a bottle?
5. the distance travelled in a car?
Key words
• foot – a unit of
length which is
the length of an
average foot,
about a size-six
shoe.
• carats – the term
used to measure
the weight of
diamonds and
other precious
metals
Did you know?
How many carats?
Gems and precious
metals, such as gold,
were weighed using
the carob seed,
which comes from
the carob tree.
Diamonds were
weighed against a
quantity of carob
seeds. The ‘carat’ is
still used today as a
unit of mass for
measuring precious
gems, and is based
on the use of the
carob seed.
Challenge
There are 1,6 km in
one mile. How fast is
2 miles per minute in
kilometres per hour?
Topic 32: The history of measurement
Platinum Maths Gr6_Term 4_CAPS.indd 189
189
11/02/13 6:38 PM
Topic
33
Perimeter, area and volume
Maths ideas
Measure perimeter
• Measure and
calculate perimeter.
Perimeter is the total distance along the outside edges of a shape. You
can find the perimeter of a shape by adding the lengths of its sides.
Calculate the perimeter of this shape.
3,4 cm
3,4 cm →
39 mm →
4,7 cm →
56 mm →
6,8 cm →
61 mm →
34 mm
39 mm
47 mm
56 mm
68 mm
61 mm
305 mm
or 30,5 cm
61 mm
6,8 c
m
4,7
• Investigate the
relationship
between perimeter
and area.
Example
m
m
• Find the
volume/capacity
of containers and
objects.
39
• Find areas of shapes.
cm
• Find the surface
area of a
rectangular prism.
56 mm
ExERCiSE 33.1
1. Find the perimeter of each shape by measuring the outside
lengths and adding them up. Give your answers in millimetres.
a)
b)
c)
2. Calculate the perimeter of each of the following shapes. Change
the measurements to the same units if necessary.
5,2 cm
cm
6,2
2,4 cm
c)
cm
5,2 cm
b)
6,2
a)
6,2 cm
3,5 cm
1,8 cm
89 m
25 mm
m
,3 c
6,5 cm
12
m
m
3,9 cm
74 mm
f)
89 m
m
• perimeter – the
total distance
along the outside
edges of a shape
41
Key words
e)
1,4 cm
m
d)
60 mm
3. Find the third side of a triangular field if two sides are 750 m and
0,6 km long, and the perimeter is 1,86 km.
190
Term 4
Platinum Maths Gr6_Term 4_CAPS.indd 190
11/02/13 6:38 PM
Area
You already know how to find the area of a shape by counting the
number of square units that it covers. When there is not an exact
number of square units inside the shape you need to estimate.
To estimate the area of shapes:
Key words
• area – the amount
of surface that a
shape covers
• Count all whole squares.
• Combine half squares to make whole squares.
• Count any parts that are bigger than half a square as one square.
• Ignore any parts that are less than half a square.
ExERCiSE 33.2
1. Use counting to find the area of each shape in square units:
A
A
B
B
C
D
D
EE
F
C
G
G
2. Draw the following shapes on squared paper and give their
perimeters:
a ) three different rectangles each with an area of 18 blocks
b ) three different rectangles each with an area of 24 blocks.
3. In question 2 you saw that rectangles can have the same area
but different perimeters. Can two squares have the same area but
different perimeters? Explain.
Topic 33: Perimeter, area and volume
Platinum Maths Gr6_Term 4_CAPS.indd 191
191
11/02/13 6:38 PM
The area of rectangles
You already know how to find the area of a rectangle on grid paper by
counting the number of square units inside the shape. You can also
calculate the number of blocks inside a rectangle without counting
them.
Challenge
Look at this rectangle. If you count the blocks
4 units
you can see that the rectangle has an area of 8
square units.
The rectangle is four units long and two units
wide.
You have four units in each row and you have two rows.
Multiply the units per row by the number of rows: 4 × 2 = 8
This gives an area of 8 square units, the same as you got by counting.
This means that you can calculate the area of rectangle by using
this rule:
Number of blocks in a row × number of rows.
2 units
Find the width and
length of a rectangle
that has an area of
100 blocks and a
perimeter of 50 units.
Example
ExERCiSE 33.3
A
B
1. Count the number of square units to find the areas of rectangles
A and B. Check whether you get the same answer using the above
rule.
2. Nomsa has put three plastic rectangles on a grid, but she can’t
see through them to count the square units and find the area. Try
to work out the area of each rectangle. If you get stuck, draw the
shapes on squared paper and count the units to check the area.
A
B
3. Look at the two shapes on the left. Some of the grid blocks have
been rubbed out. Can you still work out the area? Tell your partner
how you did this.
4. Use the multiplying rule to give the length and width of as many
different rectangles as you can that have an area of 32 blocks.
192
Term 4
Platinum Maths Gr6_Term 4_CAPS.indd 192
11/02/13 6:38 PM
Volume
You can use multiplication to work out quickly how many cubes make
up the volume of a rectangular prism.
Example
In the container on the right, each layer is made up of 4 rows of
9 cubes, so there are 4 × 9 = 36 cubes in each layer.
There are five layers in the container, so the total number of cubes
is 5 × 36 = 180.
The volume of the container is 4 × 9 × 5 = 180 cubes.
ExERCiSE 33.4
1. Work out the volume of each stack of cubes:
a)
c)
b)
2 cm
1 cm
1 cm
2 cm
8 cm
2 cm
5 cm
4 cm
d)
2 cm
e)
5 cm
f)
8 cm
5 cm
5 cm
3 cm
5 cm
2 cm
8 cm
4 cm
Challenge
A box is in the shape
of a rectangular
prism, with a volume
of 2 000 cubes. If one
dimension is 25 units,
how many units can
the other sides be?
Find as many
possible answers as
you can. Would all of
these boxes be a
useful shape?
Explain.
2. How many cubes will it take to fill each of these containers?
a)
b)
c)
Topic 33: Perimeter, area and volume
Platinum Maths Gr6_Term 4_CAPS.indd 193
193
11/02/13 6:38 PM
Key words
• surface area – the
total area of all the
faces of the object
Remember, you
cannot see the
bottom, the back or
the right face of this
object, but these
faces still form part
of the total surface
area.
Surface area of an object
Area is the amount of space that a flat shape covers. When you work
with a solid object, each face has an area. The surface area of an
object is the total area of all the faces.
Example
The total surface area is:
Top + bottom + front + back + left side + right side
The top and bottom
= 24 + 24 + 12 + 12 + 8 + 8
faces each have the
= 48 + 24 + 16
same area: 24 blocks
= 88 square units.
The right and
left faces each
have the same
area: 8 blocks
The front and back faces each
have the same area: 12 blocks
Challenge
A cube has sides of 2
units. Find out what
happens to the
surface area and the
volume if you:
a) multiply the
sides by 2
b) multiply the
sides by 3.
194
ExERCiSE 33.5
1. Look at the rectangular prisms in Exercise 33.4 question 1 again.
a ) Calculate the area of each face of the solid.
b ) Work out the surface area of each solid.
2. Look at the rectangular prisms in Exercise 33.4 question 2 again.
a ) Calculate the area of each face of the solid.
b ) Work out the surface area of each solid.
Term 4
Platinum Maths Gr6_Term 4_CAPS.indd 194
11/02/13 6:38 PM
Revision
1. Work on grid paper in small groups to answer the following questions:
a ) Draw three possible rectangles with an area of 24 square units, if the sides are all whole
units.
(3)
b ) Calculate the perimeter of each rectangle you drew in question 1a).
(3)
c ) Draw three possible rectangles with a perimeter of 24 units.
(3)
d ) Calculate the area of each rectangle you drew in question 1c).
(3)
e ) Draw a square with a perimeter of 24 units.
(2)
2. Palesa draws the layout of a garden on grid paper. Use the diagram to answer the questions.
Shrubbery
Bench
Tree
Each square
represents 1 m2
Stone bed
F lower bed 1
Pond and
fountain
F lower bed 2
Paving stones in grey and white
Measure the areas of the shapes in blocks. Each paving stone is one block.
a ) What area does the tree cover?
b ) What is the area of the bench?
c ) What area do all the paving stones cover?
d ) Calculate the total area covered by the shrubbery, bench and tree.
e ) Calculate the perimeter of the path formed by the paving stones.
(1)
(1)
(1)
(1)
(1)
3. Write a few sentences about how people used their bodies as a measure
of length in ancient times.
(3)
4. What unit of measurement is used today to measure gems and precious metals?
(1)
5. If a foot is approximately 305 mm in length, write the following lengths in millimetres:
a ) 15 feet
b ) 30 feet
(1)
(1)
Total marks: 25
Revision
Platinum Maths Gr6_Term 4_CAPS.indd 195
195
11/02/13 6:38 PM
Topic
34
Division
Maths ideas
Use long division
• Divide by multiples
of 10, 100, and
1 000.
In this section you will practise using long division to divide
a four-digit number by a three-digit number.
• Do multiple
operations on
whole numbers.
• Estimate division
answers.
• Work with
multiplication and
division as inverses.
• Divide a four-digit
number by a
three-digit number.
• Solve problems
using rate and ratio.
Example
Find 4 983 ÷ 215. First estimate the answer:
5 000 ÷ 200 = 50 ÷ 2 = 25.
23 remainder 38
215 4 9 8 3
–
4300
← 215 × 20 = 4 300
683
–
645
← 215 × 3 = 645
38
20 + 3 rem 38 = 23 rem 38. The estimate was very close.
ExERCiSE 34.1
First estimate and then use the long division method to find the answers.
Let your partner use a calculator to check your answers.
1. 2 453 ÷ 312
2. 5 876 ÷ 424
3. 3 275 ÷ 255
4. 3 772 ÷ 323
5. 4 498 ÷ 634
6. 6 640 ÷ 305
7. 9 578 ÷ 162
8. 7 645 ÷ 502
9. 9 200 ÷ 834
You can also divide using the rules for multiples of 10:
the digits move 1 place to the right
÷10
the digits move 2 places to the right
÷100
the digits move 3 places to the right
÷1 000
Example
15 000 ÷ 300 = 15 000 ÷ 100 ÷ 3 = 150 ÷ 3 = 50
ExERCiSE 34.2
Do the following calculations by first breaking down the multiple of
100 or 1 000 into two factors:
1. 5 000 ÷ 200
2. 3 600 ÷ 900
3. 64 000 ÷ 16 000
196
Term 4
Platinum Maths Gr6_Term 4_CAPS.indd 196
11/02/13 6:38 PM
inverse operations
Remember that multiplication and division are inverse operations. You
can use one operation to check the answer of the other.
Example
Earlier, we used long division to find that 4 983 ÷ 215 = 23
remainder 38.
Use multiplication to check the answer: (215 × 23) + 38
= 215 × (20 + 3) + 38
= (215 × 2 × 10) + (215 × 3) + 38
= 4 300 + 645 + 38
= 4 938
ExERCiSE 34.3
1. Use division to calculate these missing numbers:
a ) 512 × ____ = 7 680
b ) 459 × ______ = 9 639
c ) 338 × _____ = 9 802
d ) 214 × ______ = 6 420
Challenge
A water tank holds
10 kl of water.
Calculate how many
750 ml bottles can be
f illed from this tank.
2. Use multiplication to check your answers to 1, 2 and 3 in
Exercise 34.1.
All the division methods that you have learnt can help you to solve
different types of problems.
ExERCiSE 34.4
First estimate the answer to each question, then find the exact answer,
and finally check your answer by using the inverse operation.
1. 8 550 kg of cargo has to be divided between 19 containers to be
shipped. How many kilograms will be in each container?
2. I have a wall with a surface area of 2 688 square cm. If my tiles are
24 square cm, how many tiles do I need to cover the wall?
3. A farmer wants to fence in some square feeding areas on his farm.
a ) If the square feeding areas each have a length of 120 metres,
how much fencing is needed to go around the perimeter of
each feeding area?
b ) If the farmer has 8 160 metres of fencing, how many feeding
areas can he fence in?
Topic 34: Division
Platinum Maths Gr6_Term 4_CAPS.indd 197
197
11/02/13 6:38 PM
Solve problems with division
You will now get some more practice in using division to compare two
different quantities.
Challenge
It takes Sabelo
30 minutes to walk
home from school. It
takes Sabelo’s mom
5 minutes to bring
him to school by car
travelling at an
average speed of
60 km per hour.
a) How far does
Sabelo live from
the school?
b) How fast does
Sabelo walk in
kilometres per
hour?
Example
You buy 5 kg of potatoes for R27.
a ) How much does each kilogram cost?
The price for one kg is R27 ÷ 5.
R27 ÷ 5 = R5 remainder R2
R2 = 200c ÷ 5 = 40c
The potatoes cost R5,40 per kilogram.
b ) How much will 8 kg cost at the same price?
8 kg will cost R5,40 × 8 = (5,0 × 8) + (40c × 8) = R40 × 320c
8 kg of potatoes will cost R43,20
ExERCiSE 34.5
1. If Pumzile cycles 5 km in 10 minutes, how many kilometres does
she cycle in 7 minutes?
2. If you need 20 ℓ of paint to cover 5 square metres of wall, how
many litres do you need to paint 15 square metres?
3. Ceiling fan A makes 10 revolutions every 2 minutes and ceiling fan
B makes 12 revolutions every 3 minutes. Which of the fans turns
the fastest (makes the most revolutions each minute)?
4. Car A travels 112 km on 8 litres of petrol. Car B travels 150 km on
12 litres. Which car is cheaper to run? (Find which car uses less
petrol to travel one kilometre.)
5. Feila sells pencils in boxes of 12 for R24 and in packs of 3 for R7.
Is it cheaper per pencil if you buy a box or a bag?
6. Bev sells apples in 2 kg bags for R24 and in 4 kg boxes for R46. Will
the 2 kg bags cost less per apple than the 4 kg box?
7. Which is a better buy: a 200 g slab of chocolate for R15 or an 80 g
bar of chocolate for R5?
8. Dumisani is making roast chicken for supper. He must cook the
chicken for 50 minutes for every kilogram. For how many hours
and minutes must he cook a 2,5 kg chicken?
198
Term 4
Platinum Maths Gr6_Term 4_CAPS.indd 198
11/02/13 6:38 PM
You can also solve problems by comparing quantities of the same type.
Example
In a class there are 3 girls for every 2 boys. If there are 35 learners
altogether in the class, how many are girls and how many are boys?
The total number of parts is 3 + 2 = 5.
35 ÷ 5 = 7
The number of girls is 3 × 7 = 21. The number of boys is 2 × 7 = 14.
Check that 21 + 14 = 35.
ExERCiSE 34.6
1. In an athletics club there are two boys for every three girls. If there
are twelve girls, how many boys are there?
2. A chef uses four tomatoes to make 500 ml of sauce. How much
sauce can he make from 20 tomatoes?
3. A recipe uses three cups of flour to make 48 cupcakes. Calculate
how many cups of flour you will need to make eight cupcakes, and
to make 80 cupcakes.
Challenge
A builder is making
concrete from 3 parts
sand, 2 parts cement
and 1 part gravel.
a) If he has 9 kg of
sand, how much
cement and
gravel does he
need?
b) If he wants to
make 36 kg of
concrete, how
much of each
material does he
need?
4. Liza’s recipe for fish pie uses 280 g of fish and 50 g of rice for two
people.
a ) How much fish and rice will she need to feed one person?
b ) How much fish and rice will she need to feed five people?
5. You make purple paint by mixing three parts red paint to one part
blue paint.
a ) If you use 400 ml of blue paint, how much red paint do you need?
b ) If the paint that you have mixed will cover two walls, how
much red paint will you need to cover ten walls? Give your
answer in litres.
6. To make 1 500 ml of green paint, Alex mixes twice as much yellow
paint as blue paint. How much yellow and how much blue paint
does he use?
7. Mulalo always makes two brown rolls for every three white rolls.
Today he made 360 bread rolls. How many brown and how many
white rolls did he make?
8. A builder mixes two parts cement to five parts sand to make
mortar. He needs to make 42 kg of mortar. How many kg of sand
and how many kg of cement does he need?
Mixing cement
Topic 34: Division
Platinum Maths Gr6_Term 4_CAPS.indd 199
199
11/02/13 6:38 PM
investigation
Use simple budgets and accounts
Key words
A budget is a plan that helps you to manage your money effectively.
It shows how much money you receive (income) and how you plan to
spend that money (expenses).
• budget – a
statement of
income and
expenses
Example
A budget is for a specific period. It has a section for the money that
you have received and a section for the plan of how you will spend
the money. The money you spend should not be more than the
money you receive.
Budget: Rael Taylor
September 2012
Income
Pocket money
Money from chores
Total
Expenses
Tuck shop
Charity
Gift – friend
Savings
Total
R50
R30
R80
• income – money
that you receive or
earn
• expenses – money
that you spend
• accounts – records
of money spent and
that you will repay
later
• limit – a maximum
value that you may
spend
R35
R5
R30
R10
R80
• repayments –
several payments
that you make to
settle an account
1. Try to draw up your own budget or your family’s budget for
one month.
(10)
2. Do you save any of the money you receive?
(1)
3. Explain why you should save.
(2)
4. If you do save, what could you do with your savings?
(1)
5. Look at the budget you have drawn up. What could you
possibly spend less money on in order to save more?
(2)
• transactions –
operations, such
as payments and
purchases, on an
account
• balance – the
amount that you
still owe on an
account
Did you know?
Every country has a budget based on the amount of money it raises
through tax collection and how they will spend the money, for
example, on building new schools and hospitals, and paying for grants
to help disabled people.
200
Term 4
Platinum Maths Gr6_Term 4_CAPS.indd 200
11/02/13 6:38 PM
Accounts allow people to buy items even when they do not have the
full amount of money needed to pay for them straight away. There is a
limit on the amount of money that people may spend on an account
and an agreement for when they must make repayments. Account
holders receive a statement of the account at the end of each month.
The statement details how money was spent
on the account, any repayments that have been
made, and the due date and amount for the next
repayment.
Rael Taylor
Tuckshop
Example
The account holder’s (Rael Taylor) details and
account number allow the shopkeeper to
identify the customer and keep track of any
transactions on the account. Amounts
credited or paid into the account are
subtracted from the balance.
6. On what date did the account holder pay
the account?
(1)
7. What is the account number?
8. What is the total amount owing on
24 September 2012?
(1)
36 Engel Crescent
Rondebosch
7440
Transaction details
31 Aug
Opening balance
03 Sept
Payment
05 Sept
Bagel
07 Sept
Sweets
10 Sept
Bottle of water
12 Sept
Bottle of water
14 Sept
Cheese roll
19 Sept
Meat roll
24 Sept
(1)
9. When is the payment due?
(1)
10. How many bottles of water did the account holder buy?
(1)
11. How much more expensive is a meat roll than a cheese roll?
(2)
12. If the account holder budgeted R35 for his tuckshop account,
how much money will he still need?
(2)
13. If the exact amount owing is paid, what is the fewest number
of coins and bank notes that he could use?
(2)
Statement date: 24 September
2012
Account number: 000213
Payment due by: 1 October 201
2
Transaction
amount
Balance
R28,31
– R28,31
R10,50
R 0,00
R10,50
R4,20
R14,70
R4,50
R4,50
R19,20
R23,70
R8,50
R10,25
Closing balance
R32,20
R42,45
R42,45
14. If he uses a R50 bank note to pay the full balance, calculate how
much change he will receive. What is the fewest number of coins
and bank notes that he could receive in change?
(4)
15. Draw up a statement of all your tuckshop expenses for one
month. Imagine that you had bought the items on account.
Use an opening balance of R0.
(9)
Total marks: 40
Investigation
Platinum Maths Gr6_Term 4_CAPS.indd 201
201
11/02/13 6:39 PM
Topic
35
Number sentences
Number sentences and rules
A number statement is a useful way to write a rule and to work with it.
Example
Maths ideas
• Write number
sentences
to describe
mathematical
problems.
• Solve and complete
number sentences.
• Identify equivalent
statements.
• Check answers by
substitution.
The rule “multiply the number by three and add 2” can be written as
□×3+2
Apply the rule to the number 7: 7 × 3 + 2 = 21 + 2 = 23
ExERCiSE 35.1
1. Write these rules as number sentences, and apply each rule to the
number 100:
a ) divide by 10 and add 8,4
b ) multiply by 200 and subtract 1 200
2. Substitute a number to show that these two rules are not the
same: “add 2 and multiply by 3”; “multiply by 3 and add 2”.
When there are mixed operations, you must be careful with the order
of the operations. Remember that after brackets are calculated, you
do multiplication and division before addition and subtraction.
ExERCiSE 35.2
1. A rule is “divide by 4 and add 3”.
a ) Apply the rule to the number 4.
b ) Which number statement(s) below shows this rule correctly?
Explain your choice.
□ ÷ 4 + 3 ; (□ ÷ 4) + 3 ; □ ÷ (4 + 3)
2. Can you write each of the following rules without brackets? Explain.
a ) multiply by 4 and add 2
b ) add 3 and multiply by 7
3. Write the following rules as number statements. Use brackets if
you need to.
a ) divide by 2 and add 9
b ) subtract 3 and multiply by 1
202
Term 4
Platinum Maths Gr6_Term 4_CAPS.indd 202
11/02/13 6:39 PM
Solve a number sentence
In Topic 2 you learnt how to find a missing number in a number statement.
You can either use trial and improvement, or you can use inspection.
Example
Find the missing number in the sentence (3 × □) + 2 = 32.
Use trial and improvement: substitute 2: (3 × 2) + 2 = 8 too small
substitute 8: (3 × 8) + 2 = 26 too small
substitute 12: (3 × 12) + 2 = 38 too large
substitute 10: (3 × 10) + 2 = 32 correct
Use inspection: What added to 2 makes 32? It must be 30.
So 3 × □ = 30, which tells us that the missing number is 10.
Check by substitution: 3 × 10 + 2 = 30 + 2 = 32
From your work with flow diagrams, you can also use inverse
operations to work out a missing number.
Example
Find the missing number in the sentence (□ × 3) + 2 = 17.
Rule
?
×3
+2
17
This rule first multiplies the number by 3 and then adds 2 to get 17.
To reverse the operations, you can first subtract 2 and then divide by 3.
Rule
17
–2
÷3
5
So, (17 – 2) ÷ 3 = 15 ÷ 3 = 5.
ExERCiSE 35.3
Use as many different methods as possible to find the missing
numbers. Substitute to check your answers.
1. (2 + □) × 4 = 48
2. (□ – 12 ) ÷ 3 = 6
3. □ – (12 ÷ 3) = 6
4. 18 + □ + 12 = 98
5. □ – 3 + 5 = 12
6. □ – (3 + 5) = 12
7. (16 – □) ÷ 3 = 4
8. 16 – (□ ÷ 3) = 4
Topic 35: Number sentences
Platinum Maths Gr6_Term 4_CAPS.indd 203
203
11/02/13 6:39 PM
Use number sentences when
solving problems
When you solve a word problem, it is often useful to write the
problem in the form of a number sentence. Then you can solve the
problem by solving the number sentence. You can use a symbol, like
□ or , to represent the unknown number in the sentence.
Example
Diego worked for 9 days and Marcelle worked for 13 days at a
supermarket during their school holidays. Together they were paid
R4 994. Calculate how much each should be paid, so that they are both
paid at the same rate.
There are three parts to this problem:
• First you must calculate how much they were paid per day.
• Then calculate how much Diego should receive.
• Then calculate how much Marcelle should receive.
You can write the parts of the problem as word sentences:
• Amount paid per day = total amount paid ÷ total number of days
worked.
• Amount that Diego should receive = number of days worked ×
amount paid per day.
• Amount that Marcelle should receive = number of days worked ×
amount paid per day.
Now you can write the word sentences as number sentences:
Use □ to represent the amount paid per day: □ = 4 994 ÷ (9 + 13)
Use to represent the amount that Diego should receive: = 9 × □
Use to represent the amount that Marcelle should receive: = 13 × □
Now solve the number sentences, to calculate the answer:
□ = 4 994 ÷ 22
= 227
= 9 × 227
= 2 043
= 13 × 227
= 2 951
Diego should therefore receive R2 043 and Marcelle should receive
R2 951.
Check your answer: R2 043 + R2 951 = R4 994
204
Term 4
Platinum Maths Gr6_Term 4_CAPS.indd 204
11/02/13 6:39 PM
ExERCiSE 35.4
For each of the following problems:
• First write a word sentence that describes how the problem can be
solved.
• Then write the number sentence(s) that describe(s) the problem.
Always write the numbers in the number sentence in the same
order as in the word sentence.
• Choose symbols to represent the unknown numbers.
• Then solve the sentence(s) to find the missing number(s).
• Finally, substitute to check that your answer does work in the
situation.
1. Farm workers are packing onions into bags. They have 727 onions
to pack. If they pack 11 onions in each bag, how many onions will
be left over?
2. In a school there are 105 learners in three classes. If two classes
have 34 learners each, how many learners are in the third class?
3. A father pays R80 in pocket money to five children. He gives the
eldest child R20 and divides the rest of the money equally among
the other children. Calculate how much money each of the other
children gets.
4. A dressmaker buys 150 cm of lace. A square cushion needs lace
trim on all four sides. The sides of the cushion are 28 cm in length.
Calculate how much lace will be left over after the dressmaker has
sewn lace around the whole cushion.
5. A recipe for 6 people needs 240 g of sugar. What mass of sugar
should be used for 8 people?
6. A car-wash company offers to pay workers at a rate of R35 per
hour. The manager paid R1 120 to three men – Frank, Fritz and
Fanie. Frank worked for 8 hours and Fritz worked for 11 hours. For
how many hours did Fanie work? How much was Fanie paid?
7. A medicine bottle holds 300 ml. A standard medicine measure is
5 ml. A patient has to take 2 measures twice a day. How many days
will the medicine last?
8. Farouk used 4 202 bricks to build a wall around his house. This was
11 times more than the number of bricks that his neighbour used
to build a braai. How many bricks did his neighbour use?
Topic 35: Number sentences
Platinum Maths Gr6_Term 4_CAPS.indd 205
205
11/02/13 6:39 PM
Multiple choice questions
Some tests have questions in multiple choice form. You are given
different possible answers to the question, and you must choose the
correct one.
ExERCiSE 35.5
Choose the correct answer(s) for each of the following questions:
1. 23 × 14 is less than 23 × 17. How much less?
a ) 23
b ) 14
c ) 69
d ) 51
2. 137 × 26 is less than 140 × 26. How much less?
a ) 78
b ) 140
c ) 137
d ) 420
3. 147 × 19 is greater than 147 × 17. How much greater?
a ) 294
b ) 147
c ) 38
d ) 34
4. For which pair(s) of numbers does the rule “multiply the first
number by 11 then subtract 5” apply?
a ) 11 ; 116
b ) 9 ; 84
c ) 116 ; 11
d ) 13 ; 138
5. Which one of the following statements is true? In each statement
□ represents the same number.
a ) 17 × □ = □ –17
b ) 17 × □ = □ × 17
c ) 17 + □ = □ – 17
d ) 17 × □ = 17 + □
6. Which of the statements below are equivalent to: 18 + (8 + 12)?
a ) 18 (8 + 12)
b ) (18 + 8) + 12
c ) (9 × 2) + 8 + (3 × 4)
d ) 8 + (18 + 12)
7. Which of the statements below is equivalent to: (14 × 18) + (14 × 2)?
a ) 14 × 20
b ) 14 × 36
c ) 14 + 20
d ) 14 + 36
8. Which of the statements below are equivalent to: 12 × ( 8 × 9)?
a ) (8 × 12) × 9
b ) (12 × 8) + (12 × 9)
c ) (10 – 2)(12 × 9)
d) 6 × 2 × 2 × 2 × 2 × 3 × 3
9. Which one of the following statements is true? □ represents the
same number in each statement.
a) □ ÷ 9 = 9 ÷ □
b) □ × 9 = 9 × □
c) □–9=9–□
d) □ + 9 = □ – 9
10. Which one of the following statements is true? □ represents the
same number in each statement.
a) □ + 1 = □
b) □ – 1 = □
c) □–□=1
d) □ + 0 = □
206
Term 4
Platinum Maths Gr6_Term 4_CAPS.indd 206
11/02/13 6:39 PM
Revision
1. Copy the table into your workbook. Use the tests for divisibility to check which factors are
factors of each number. Place a cross or a tick in each box.
(10)
2
3
4
Factors
5
6
8
9
10
Numbers 3 687 426
213 552
273 015
2. Use the long division method to calculate the following:
a ) 7 232 ÷ 422
b ) 3 564 ÷ 901
(3)
(3)
3. In a class of 45 Grade 10 learners, there are 2 girls to every 3 boys. How many are boys and
how many are girls?
(2)
4. A recipe for four people needs 260 g of butter. How much butter would be needed for six
people?
(2)
5. The single train fare from Wynberg to Cape Town is R8,20. Sonja makes the journey twice a
week for 8 weeks. She uses a R200 note to buy the tickets.
a ) Write a number sentence to calculate how much change she will get. Use a □ for the
missing number.
(2)
b ) Solve the number sentence.
(1)
6. A school has collected empty cool drink bottles for recycling. So far, 1 734 bottles have been
collected. Large plastic bags have also been collected for recycling. Learners decide to pack
the bottles into the bags. 47 bottles fit into each bag.
a ) Write a number sentence to calculate how many bags are needed for all the bottles.
(2)
b ) Solve the number sentence.
(1)
c ) One bag will not be completely full. How many bottles are there in this bag?
(1)
7. Find the missing number: (3 × □) – 12 = 30
8. Choose the correct answer for each of the following:
a ) 45 × 38 is less than 45 × 40. How much less?
B. 90
C. 38
A. 45
b ) (26 × 39) + (26 × 1) is equivalent to:
A. 26 × 27
B. 400
C. 26 × 4
(1)
(1)
D. 40
(1)
D. 26 × 40
Total marks: 30
Revision
Platinum Maths Gr6_Term 4_CAPS.indd 207
207
11/02/13 6:39 PM
Topic
36
Transformations
Maths ideas
• Use rotational and
line symmetry to
describe patterns.
• Find the factor of
enlargement and
reduction.
• Transform a shape
by enlarging and
reducing.
Recognise transformations
in patterns
In Topic 22 you explored different types of transformations. You will
now look at how transformations are used in patterns around us.
ExERCiSE 36.1
Below are pictures used in an advertisement for Quick Pave, a paving
company. Examine the different patterns and then answer the
questions that follow.
A
B
C
D
E
F
G
H
1. Identify the shape that is used to form these patterns.
2. Draw a rectangle on cardboard or paper and cut it out. Use the
rectangle to help you answer the following questions:
a ) Explain how rotation can be used to create pattern A. It may
help if you draw the pattern.
b ) Describe the transformations used in pattern B.
c ) Describe the symmetry in pattern B.
d ) Pattern C is similar to which other pattern in the
advertisement? Do these similar patterns have any symmetry?
e ) Look at patterns E and F. What do you notice about the
symmetry in both patterns?
f ) Describe the transformations used to create pattern E.
g ) Describe the transformations used to create pattern F.
h ) How are patterns E and G similar?
i ) How are patterns F and H similar?
208
Term 4
Platinum Maths Gr6_Term 4_CAPS.indd 208
11/02/13 6:39 PM
Enlarge a shape
One way to transform a shape is to make it larger without changing its
shape. This is called an enlargement.
Example
These dolls are exactly the same except for their size.
Doll D is three times the height of Doll A, so we say
that Doll A has been enlarged by a factor of 3. If you
enlarge Doll B by a factor of 2 you also get Doll D,
because 4__12 cm × 2 = 9 cm. Doll E is larger than Doll
C by a factor of 2 (6 cm × 2 = 12 cm), and larger than
Doll A by a factor of 4 (3 cm × 4 = 12 cm).
A
B
C
D
E
In the example, the dolls did not only get taller. All their dimensions
changed. In an enlargement, all lengths change by the same factor.
Example
All the sides of the two shapes have been doubled in length to
form new shapes. The original shapes have been enlarged by a
factor of 2. The original rectangle had a length 2 units and a
width 1 unit. The enlarged rectangle has a length 4 units and a
width 2 units. Notice that the ratio (relationship) of the length to
the width stays the same, because __21 is the same as __42.
When one enlarges a shape, the sides stay in the same proportion.
This means that the ratio of the sides stays the same.
Key words
ExERCiSE 36.2
• enlargement –
an increased size
of a shape, with
sides in the same
proportion as the
original shape
1. A triangle has sides of 8 cm, 10 cm and 15 cm.
a ) Find the lengths of the new triangle formed by enlargement
by a factor of 3.
b ) By what factor has the triangle been enlarged if the sides
become 0,8 m, 1 m and 1,5 m?
• proportion – the
same relative
amount or ratio
2. On grid paper, draw these shapes and transform them:
b)
a)
Enlarge the shape
by a factor of 2.
c)
Enlarge the shape
by a factor of 2.
Enlarge the shape
by a factor of 3.
Topic 36: Transformations
Platinum Maths Gr6_Term 4_CAPS.indd 209
209
11/02/13 6:39 PM
Key words
• reduction – a
decreased (smaller)
size of a shape with
sides in the same
proportion as in
the original shape
Reduce a shape
You can also transform a shape by making it smaller. Look at the dolls
on the previous page again. You can enlarge Doll A by a factor of 4 to
get Doll E, or you can reduce Doll E by a factor of 4 to get Doll A.
A reduction is a transformation that makes the size of a shape smaller
according to a given factor. To reduce a shape, you decrease all the
sides by the same factor, so that the sides stay in the same proportion.
Example
All the sides of the two shapes have been halved to form
new shapes. The new shape is half the size of the original
shape. The original shape has therefore been reduced by
a factor of 2. You can also say that the original shape has
been enlarged by a factor of __12 .
A shape is enlarged when the enlargement factor is larger
than one. A shape is reduced when the enlargement
factor is a fraction less than one.
ExERCiSE 36.3
1. On grid paper, draw the shapes below and transform them:
b)
c)
d)
a)
Reduce by a factor of 2.
Enlarge by a
Reduce by a factor of 3. factor of __1 .
2
a)
Enlarge by a factor of __14 .
2. Describe the type of transformation shown in each grid on the left.
3. A rectangular box has length of
base 12 cm, width of base 8 cm
and height 4 cm. It is reduced
and enlarged to form three other
rectangles. Copy and complete
the table for the new rectangles:
b)
Length
a)
b)
c)
210
Width
4 cm
12 cm
8 cm
Height
Enlargement
factor
3
6 cm
160 mm
Term 4
Platinum Maths Gr6_Term 4_CAPS.indd 210
11/02/13 6:39 PM
Compare the size and shape of
triangles and quadrilaterals
Example
Look at the picture on the right. The blue square has been enlarged to
form the red square. The length of the side of the blue square is 2 units
and the sides of the red square are 8 units long, so the enlargement
factor is 4. Even though the size of the shape has changed, the shape of
the image is the same as the original shape and the size of the angles
has not changed, as they are still right angles.
ExERCiSE 36.4
In the picture below, the original shapes are blue and the new shapes
are red:
a)
b)
c)
1. Which of the original shapes in the picture above have been
reduced and which have been enlarged?
2. Copy and complete the table below to compare the original
shapes with the new shapes:
Shapes
Enlargement
or reduction
What has stayed the
same?
What has changed?
a)
b)
c)
3. Transform each of the shapes on the right as instructed:
a)
• Transform a) by a factor of __12 .
• Transform b) by a factor of 3.
4. Transform shape 4 on the right by enlarging only the longer
sides of the shape by a factor of 2. What has stayed the same
4.
and what has changed between the original shape and its image?
5. Transform shape 5 on the right by reducing the size of two
opposite sides by a factor of 2. What has stayed the same and what
has changed between the original shape and the new shape?
b)
5.
Topic 36: Transformations
Platinum Maths Gr6_Term 4_CAPS.indd 211
211
11/02/13 6:39 PM
Topic
37
Position and movement
Maths ideas
• Locate the position
of symbols using
alpha-numeric
references on grids
and maps.
• Describe changing
positions on a map
or a grid.
• Give directions to
move positions on
grids.
Find positions on maps and grids
A grid is a reference system to help us find a point on a surface. You
have probably seen a map of an area like the map below. It uses
vertical and horizontal lines to divide a large area into smaller areas.
Example
On this alpha-numeric grid of the map, the columns are labelled
in letters from A to H, and the rows are labelled in numbers from
1 to 6. The position of a place is described by giving the column
letter and the row number. For example, B6 is the block where
Column B and Row 6 overlap. Cape Town has map reference B6.
Key words
• alpha-numeric
grid – a grid with
alphabet letters
for the columns
and numbers for
the rows, in which
every position
on the grid has
a particular
reference
• grid reference
position – refers
to a particular
cell in the grid
for example C2,
the cell where
column C and row
2 overlap
Nelspruit is found in Column H and in Row 2. Therefore the grid
reference position of Nelspruit on the map is H2.
ExERCiSE 37.1
1. Find the city or cities in each of the following blocks on the map:
a ) H3
b) G1
c ) C4
d) B6
e ) F5
f ) E6
2. Give the grid reference of each of the following places:
a ) East London
b ) Bloemfontein
c ) Pietermaritzburg
d ) Vereeniging
e ) Pretoria
f ) Bisho
3. Use another map to find the positions of two towns or cities that
are in the green areas of the map. Give their grid references.
212
Term 4
Platinum Maths Gr6_Term 4_CAPS.indd 212
11/02/13 6:39 PM
Describe how to change positions
on a grid
Often we need to plan a route or change our position. We can use grid
references to plan or to describe this movement.
Example
If you are given a starting point and a description of the
move that has been made, you can find the new position.
Look at the seating plan on the right.
To describe a move from C2 to F4, you can say the move is
3 blocks to the right and 2 blocks downwards.
The diagram also shows a change in position from seat G9
moving 4 blocks left and two blocks upwards on the grid.
The new position is C7.
A
B
C
D
E
F
G
H
I
J
1
2
3
4
5
6
7
8
9
10
ExERCiSE 37.2
Use the grid to answer the following questions:
1. You have moved two blocks left and three blocks up to reach
position C7. Give your starting position.
2. Practise changing positions by using the following descriptions.
Write down the final grid position.
a ) From A1 move 4 blocks right and 5 blocks downwards.
b ) From E6 move 3 blocks left and 2 blocks downwards.
c ) From G2 move 1 block right and 8 blocks downwards.
d ) From C3 move 2 blocks left and 2 blocks upwards.
e ) From A10 move 1 block right and 5 blocks upwards.
f ) From H1 move 6 blocks left.
g ) From D2 move 8 blocks downwards.
h ) From F10 move 1 block right and 4 blocks upwards.
3. Describe the movement from each position to the new position:
a ) From C2 to H2
b ) From I10 to F4
c ) From J5 to E7
d ) From F2 to I3
e ) From G7 to B6
f ) From H9 to J10
g ) From G3 to F7
h ) From A1 to J6
Topic 37: Position and movement
Platinum Maths Gr6_Term 4_CAPS.indd 213
213
11/02/13 6:39 PM
Work with street maps
Street maps give us a more detailed look at areas.
There are symbols on street maps that help you to find certain
features. From work that you have done in Grade 5, you already know
the symbols for schools, the library, municipal clinics and the police
station. Here are some more useful symbols:
Petrol
station
Place of
worship
Tennis
Car park
Hotel
Route
marker
ExERCiSE 37.3
Use the map of Parkview on the next page to answer the following
questions:
1. Give the grid reference of Zoo Lake Sports Club.
2. Name the sports that can be enjoyed at the Zoo Lake Sports Club.
3. Tennis can be enjoyed at four places on the map. Give the grid
reference of these places.
4. Give the grid references of the two entrances into the Zoo.
5. Which of the two entrances has a car park? Provide the grid
reference of this entrance.
6. Give the names of the roads around the zoo.
7. Provide the grid reference for the water feature in the zoo.
8. Give the position of the metro route marker on Jan Smuts Avenue.
9. Give the grid reference of the petrol station in Parkview.
10. What building is north-west of the petrol station in Tyrone
Avenue?
11. Give the grid reference of three places of worship in Parkview.
12. Identify the places that are located at the following grid reference
positions:
a ) C3
b ) B4
c ) H7
d ) D3
e ) D8
f ) C1
g ) F10
h ) H5
214
Term 4
Platinum Maths Gr6_Term 4_CAPS.indd 214
11/02/13 6:39 PM
1
2
3
4
5
6
7
8
9
10
A
B
C
D
E
F
G
H
I
Topic 37: Position and movement
Platinum Maths Gr6_Term 4_CAPS.indd 215
215
11/02/13 6:39 PM
Maps and compass directions
Refer to the map on page 215 for the example and the exercise below.
Example
N
NE
NW
E
W
SW
SE
S
Compass directions
Riaan lives in Kinross Road in the B2 grid position. He wants to go
to the swimming pool on A4. He is given the following directions.
Trace the path with your finger on the map to make sure the
directions are correct.
”Walk east to Donegal Avenue and turn right into Donegal Avenue.
Walk in a south-easterly direction until you reach Lower Park Drive.
Turn left into Lower Park Drive and walk in a northerly direction
until you see the swimming pool on your right.”
ExERCiSE 37.4
1. Thabo was dropped off at the Upper Park Drive entrance to the
Zoo in G8 and his friend Roy at the Jan Smuts entrance in F7. Write
down directions for Thabo to follow to find his friend at the other
entrance.
2. Hope Special Education School is at the end of Hope Road in
H5. Connor is at the end of the cul-de-sac in Pallinghurst Road
in Westcliff. Connor wants to visit Hope School. Write down
directions for him to get there from his house.
3. Follow the directions given and answer the following questions:
a ) Zita is visiting the Bernberg Fashion Museum with grid
reference H6 in Duncombe Road. She exits the building, turns
right and walks in a westerly direction until she reaches Jan
Smuts Road. She turns into Jan Smuts Road and walks towards
the Zoo entrance. Before she reaches the entrance she sees
her friend Leon at the school they attend. What school does
Zita go to?
b ) Zita and Leon exit the school and turn left, walking in a
northerly direction. They cross Dundalk Road and continue to
the traffic light opposite the zoo entrance where they turn left
into Lower Park Drive. They continue in Lower Park Drive until
they reach the traffic light and go for a picnic. Where are they
going to have their picnic?
216
Term 4
Platinum Maths Gr6_Term 4_CAPS.indd 216
11/02/13 6:39 PM
Revision
1. Copy the following shapes onto square grid paper and transform each shape as
instructed below:
a)
b)
d)
a)
b)
c)
d)
e)
c)
e)
Enlarge the shape by a factor of __13 .
Reflect the shape in the line of symmetry.
Enlarge the shape by a factor of 4.
Rotate the shape to complete one full turn.
Enlarge only the shorter sides of the shape by a factor of 2.
(2)
(2)
(2)
(2)
(2)
2. For each transformation in question 1, describe what has stayed the same and what has
changed for each shape.
(5)
3. a ) Design your bedroom. Draw a grid like this one to represent the layout of
your bedroom:
(4)
1
2
3
4
5
6
7
8
A B C D E F G H
door
bedside
table
television
cabinet
bed
chair
desk
bookshelf
b) Describe three different movements on the grid to take you from one point to another.
As an example, you enter the room and walk close to the wall to sit on the right of
the bed. From A1 you move 4 blocks to the right and 2 blocks downwards to E3.
(6)
Total marks: 25
Revision
Platinum Maths Gr6_Term 4_CAPS.indd 217
217
11/02/13 6:39 PM
Topic
38
Probability
Maths ideas
• Perform events
with coins, dice and
spinners.
• List the possible
outcomes of events.
• Make tally tables
to record actual
outcomes.
• Count and compare
frequencies of
outcomes.
Key words
• experiment –
something you do
to find out what
will happen
• trial – the activity
you do in an
experiment
• outcome – a result
of a trial
List possible outcomes of
experiments
You already know that when you roll a die a number
of times you are performing an experiment. Each
time you roll the die you are performing a trial to
see what the outcome will be. There are six possible
outcomes when you roll a normal die. You can roll the
numbers 1, 2, 3, 4, 5 or 6.
This year you are going to do more experiments with
coins, dice and spinners.
Before you perform any probability experiment, you
need to decide what the possible outcomes are and list them.
Example
What are the possible outcomes when you
spin this spinner?
The spinner can land on any of the five
colours, so the possible outcomes are:
red, blue, green, yellow or orange
ExERCiSE 38.1
Did you know?
One cube with dots
on each side is
actually called a ‘die’.
More than one die
are called ‘dice’.
1. What are the possible outcomes for each of these spinners?
a)
b)
c)
d)
e)
2. Draw and colour a circular spinner with the following possible
outcomes: red, black, white, green, blue, orange or yellow.
3. How many possible outcomes are there if you roll the following 3D
objects, which have a number on each face?
a ) tetrahedron
b ) square-based pyramid
218
Term 4
Platinum Maths Gr6_Term 4_CAPS.indd 218
11/02/13 6:40 PM
Record outcomes of experiments
When you have to record the outcomes of a number of trials, it makes
sense to first list the possible outcomes in a table. As you do each trial,
you make a tally mark to show what the outcome is. Once you have
finished, you add up the tally marks to find the frequency of each
outcome.
Key words
• frequency – how
often an outcome
occurs
Example
Safiya did an experiment with a spinner.
She did 50 trials in the experiment and recorded her results in a frequency table:
Possible outcomes
Red
Actual outcomes
Frequency
12
Blue
8
Green
11
Yellow
10
Orange
9
Number of trials conducted
50
Safiya listed the possible outcomes in the first column of the table.
In the second column she used tallies to record how often the spinner landed on each colour.
In the last column she added up the tallies to get the total number of times the spinner
landed on each colour. This is the frequency of each outcome.
The frequencies must add up to 50, as this was the number of trials she conducted.
ExERCiSE 38.2
1. Toss a coin 50 times and record the outcomes of your experiment.
a ) Draw up a table to record the outcomes of your experiment.
Your table should include a list of the possible outcomes and a
column to make tallies to record actual outcomes. It must also
include a column to record the frequency of each outcome.
b ) Do the experiment and record your results in your table.
2. a ) List the possible outcomes when you toss a normal die.
b ) Draw up a table to record the actual outcomes when you toss
a normal die 50 times.
c ) Perform the experiment and record your results.
d ) Compare your results with those of another learner. Are they
the same? Try to explain why or why not.
Topic 38: Probability
Platinum Maths Gr6_Term 4_CAPS.indd 219
219
11/02/13 6:40 PM
To make your own hexagonal spinner, you will need:
• a piece of paper and a piece of cardboard
• a pair of scissors and some glue
• a toothpick or small nail
• coloured pens or pencils.
Method
1. Trace this hexagon onto the piece of paper and cut it out.
2. Colour each section a different colour.
3. Glue the paper hexagon onto the cardboard and cut it out.
4. Push the toothpick or small nail through the dot in the centre of
the hexagon to make your spinner.
ExERCiSE 38.3
1. For this activity, use the spinner you made.
a ) List the possible outcomes when you spin your spinner.
b ) Choose one of the colours. Predict how many times your
spinner will land on that colour if you spin it 50 times.
c ) Draw up a table to record the outcome of 50 trials.
d ) Carry out the experiment by spinning your spinner 50 times
and recording the outcomes.
e ) How well did you predict the result?
WIN!
dia
e
44
ntr
Ce
R1 000 000
New
Pho
ne
220
Nothing
e
Me
R
R
rtim
Ai
100
e
irtim
A
100
Nothing
N
Ph ew
on
e
SPIN!
2. The picture shows a large wheel used in a TV show called Spin to
Win. The competitors spin the wheel to see what prize they can
win if they answer a question correctly.
a ) List the possible outcomes for this spinner.
b) Which prize is a competitor most likely to
Car
Big
win? Why?
Scre
0
0
e
n
c ) Which prize is going be the hardest to win?
R1 time
TV
r
i
Why?
A
Term 4
Platinum Maths Gr6_Term 4_CAPS.indd 220
11/02/13 6:40 PM
ExERCiSE 38.4
Challenge
1. When you toss a die, it can land on a number smaller than 4 (1,
2, or 3) or a number that is greater than 3 (4, 5, or 6). Copy the
following table into your book.
Outcome
A number smaller than 4
A number greater than 3
Total number of trials
Actual outcomes
Frequency
50
2. Work in a group of four. You will each need a die.
a ) Each person tosses their die 50 times. Record the actual
outcomes in your table.
b ) Each person adds up their own tallies to find the frequency of
each outcome.
c ) Write each frequency as a percentage.
Look at the dart
board. The value of a
dart depends on
where it lands, in any
region that is
enclosed by straight
or curved lines. How
many possible
outcomes are there
when you throw a
dart at the board?
You don’t need to list
the outcomes but
you do need to be
able to explain how
you worked out how
many there are.
3. Compare your results in question 2 with the other members of
your group.
a ) Who got the highest frequency of numbers smaller than 4?
b ) Who got the highest frequency of numbers greater than 3?
4. Combine your actual outcomes in question 2 with the outcomes
of a partner, to find the frequency of each outcome for 100 trials.
a ) What percentage of the total outcomes were smaller than 4?
b ) What percentage of the total outcomes were greater than 3?
c ) How does the total frequency compare with the frequency
that you got when you did the experiment on your own?
5. Nita has an eight-sided spinner. She
spins it 50 times and records her
results as shown on the right:
a ) Draw up a table to show the
possible outcomes for this
experiment.
b ) Organise Nita’s results into a table.
c ) Which outcome occurred most
frequently?
d ) Which outcome occurred least frequently?
Topic 38: Probability
Platinum Maths Gr6_Term 4_CAPS.indd 221
221
11/02/13 6:40 PM
Glossary
A
acute angle an angle smaller than a right angle
page 37
alpha-numeric grid a grid with alphabet
letters for the columns and numbers for the
rows, in which every position on the grid has a
particular reference
page 212
analogue clocks clocks with hands that point
to the numbers
page 30
angle the amount of turn around a fixed point
page 37
area the amount of surface that a shape covers
page 191
ascending order from the smallest to the
greatest
page 24
B
bar graph a graph with vertical or horizontal
bars each representing one set of data
page 47
bi-modal a data set with two modes
page 149
C
capacity the maximum amount a container
can hold
page 104
carats the term used to measure the weight of
diamonds and other precious metals
page 189
century 100 years
page 31
circle a 2D shape in which all points are the
same distance from the centre of the shape
page 36
column method writing numbers with the
same place value underneath each other to do
the calculation
page 14
222
composite number a number with more than
two factors
page 89
constant difference sequence a sequence of
numbers created by adding or subtracting by
the same number
page 158
constant ratio sequence a sequence of
numbers created by multiplying or dividing by
the same number
page 158
convert change from one unit into another
unit
page 106
D
data information that you collect
page 44
data cycle the process of asking a question,
collecting and organising data and
summarising the results
page 53
decimal fractions fractions written as decimal
numbers with a decimal comma
page 96
decimal places the digits after the decimal
comma
page 96
denominator the number below a fraction line
page 18
descending order from the greatest to the
smallest
page 24
difference the result of a subtraction operation
page 17
digital clocks clocks that show the time by
numbers only
page 30
dimensions the lengths of the sides of a shape
page 36
Glossary
Platinum Maths Gr6_Glossary_CAPS.indd 222
08/02/13 3:15 AM
G
double bar graph
per set of data
page 47
a bar graph with two bars
E
edge the line where two faces of a solid meet
page 73
enlargement an increased size of a shape, with
sides in the same proportion as in the original
shape
page 209
equivalent fractions fractions with the same
value
page 22
equivalent number statements number
statements that give the same answer
page 9
estimate to approximate the result of a
calculation
page 12
exchange break down larger units into smaller
units
page 15
experiment something you do to find out
what will happen
page 218
F
face a flat surface of an object
page 70
factor a number that divides exactly into
another number
page 64
flow diagram a diagram to show how a
number changes after a calculation
page 54
foot a unit of length which is the length of an
average foot, about a size-six shoe.
page 189
frequency how often an outcome occurs
page 219
grid reference position refers to a particular
cell in the grid for example C2, the cell where
column C and row 2 overlap
page 212
I
image a copy or likeness
page 136
infinite without a limit
page 84
input number the number that goes into a
calculation
page 54
inspection finding a missing number by
looking at each part of the number sentence
page 8
inverse operation opposite operation
page 16
irregular polygon a polygon with sides and
angles which are not equal
page 132
K
key explanation of what the symbols mean
page 46
kilolitre one thousand litres
page 106
L
line of symmetry a line that divides a shape
into two identical halves
page 84
line symmetry a shape has line symmetry if it
can be divided into two identical halves by a
straight line
page 84
long division a method of dividing using
multiplication facts
page 91
Glossary
Platinum Maths Gr6_Glossary_CAPS.indd 223
223
08/02/13 3:15 AM
M
mass the amount of matter contained in an object
page 114
median the middle number in an ordered data
set
page 49
million one thousand thousand
page 62
mixed number a number that is made up of a
whole number and a fraction
page 25
mode the number or the data item that
appears most often in a data set
page 49
multi-step you need more than one calculation
to solve a problem
page 17
multiple the result when you multiply whole
numbers
page 64
N
net a flat pattern that you can cut out, fold and
glue together to make a model of 3D object
page 71
numerator the number above a fraction line
page 18
O
obtuse angle an angle that is larger than a
right angle
page 37
order of rotational symmetry the number of
times that a shape fits onto itself when it turns
through a full revolution
page 86
outcome a result of a trial
page 218
output number the number that comes out of
a calculation
page 54
P
parallel lines straight lines that are always the
same distance from each other
page 38
224
Glossary
Platinum Maths Gr6_Glossary_CAPS.indd 224
parallelogram a quadrilateral with opposite
sides parallel and equal
page 38
percentage ‘per cent’ means out of a 100; % is
the symbol for percentage
page 140
perimeter the total distance along the outside
edges of a shape
page 190
pictograph a graph that uses symbols and a
key to represent data
page 46
pie chart a circle divided into sections that
represent percentages
page 150
polygon a 2D shape enclosed by three or more
straight sides
page 36
position the place where someone or
something can be found
prime number a number with only two factors,
1 and itself
page 89
prism a 3D object that has two identical, parallel
polygon faces and all its other faces are rectangles
page 70
profit increase in wealth; making money
proportion the same relative amount or ratio
page 209
pyramid a 3D object that has a polygon as a
base and all its other faces are triangles
page 70
Q
questionnaire a form for asking questions and
recording answers
page 44
R
radius a line from the centre of a circle to the
outside of the circle
page 133
reduction a decreased size of a shape with sides
in the same proportion as in the original shape
reflection to flip a shape over to form a mirror
image of the shape
page 136
08/02/13 3:15 AM
reflex angle an angle bigger than a straight
angle
page 37
regular polygon a polygon with all sides equal
in length and all angles equal in size
page 132
revolution a full turn
page 37
right angle an angle like the corner of a page
page 37
rotation to turn a shape around a fixed point
page 136
rotational symmetry a shape has rotational
symmetry if you can rotate (turn) it so that it
fits onto itself before it completes a full turn
page 86
rounding reducing or increasing a number
so that it is a multiple of the number you are
rounding to
page 12
rule a number calculation that changes an
input number to an output number
page 54
S
sequence a set of numbers or shapes in a
pattern
page 76
simplest form a fraction is in its simplest form
when no whole number can divide into both
the numerator and denominator
page 23
straight angle half a turn, forming a straight
line
page 37
surface area the total area of all the faces of
the object
page 194
T
table information arranged in rows and
columns
page 44
tessellation a regular pattern of identical or
combined shapes with no overlaps and no gaps
page 137
tetrahedron a pyramid made of four identical
triangles
page 70
time interval length of time that passes
page 35
three-dimensional (3D) objects objects
that have three dimensions: length, width (or
breadth) and height
page 70
transformation a special way to change the
position and/or orientation of a shape
page 136
translation to shift a shape to a new position
without turning it
page 136
trial the activity you do in an experiment
page 218
trial and improvement finding a missing
number by making more accurate guesses
based on your results
page 8
two-dimensional (2D) shapes shapes made
up of length and width only
page 36
V
vertex a corner point where two sides of a
polygon meet
page 39
vertices the plural of vertex
page 39
viewpoint the position from which you view
an object
page 126
volume the amount of space something takes up
page 104
W
whole numbers zero and all positive numbers
with no fractions
page 4
Glossary
Platinum Maths Gr6_Glossary_CAPS.indd 225
225
08/02/13 3:15 AM
Useful resources
Mathematical symbols
Word
Symbol
Meaning
sum
+
add
difference
–
subtract
product
×
multiply
quotient
÷
divide
Rules for rounding
If the digit to the right of the place you are rounding to is less than 5, round down.
If the digit to the right of the place you are rounding to is 5 or more, round up.
Hundred square
Percentages
Percentage Fraction
226
Percentage Fraction
Percentage Fraction
100%
1
50%
1
__
2
10%
1
__
10
75%
3
__
4
25%
1
__
4
5%
1
__
20
Useful resources
Platinum Maths Gr6_Glossary_CAPS.indd 226
08/02/13 3:15 AM
Useful resources
Fraction wall
Principal units
kilo – thousand (Greek)
centi – hundredth (Latin)
milli – thousandth (Latin)
metre – measure (Latin)
Distance
1 km = 1 000 m
1 m = 100 cm
1 m = 0,001 km
1 cm = 0,01 m
1 km = 100 000 cm
1 m = 1 000 mm
1 cm = 0,00001 km
1 mm = 0,001 m
1 km = 1 000 000 mm
1 cm = 10 mm
1 mm = 0,000001 km
1 mm = 0,1 cm
Capacity
1 litre = 1 000 millilitres
1 millilitre = 0,001 litre
1 kilolitre = 1 000 litres
1 litre = 0,001 kilolitre
or
or
or
or
1 ℓ = 1 000 ml
1 ml = 0,001 ℓ
1 kl = 1 000 ℓ
1 ℓ = 0,001 kl
Useful resources
Platinum Maths Gr6_Glossary_CAPS.indd 227
227
08/02/13 3:16 AM
Maskew Miller Longman (Pty) Ltd
Forest Drive, Pinelands, Cape Town
Offices in Johannesburg, Durban, King William’s Town, Polokwane, Bloemfontein, Mahikeng and
Nelspruit, and companies throughout southern and central Africa.
Website: www.mml.co.za
© Maskew Miller Longman (Pty) Ltd (2012)
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system,
or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or
otherwise, without the prior written permission of the copyright holder.
First published in 2012
ISBN 978 0 636 13535 2
Edited by Marga Vos and Careena Koch
Typesetting by Lizette Stuart
Book design by MML Studio
Artwork by Robert Hichens, Val Myburgh, Andre Plant, James Whitelaw and Adele Williams
DTP artwork by Tech-set Ltd
Cover design by Shereen Pearson
Cover photograph by David Pickett/DIS
Printed by
Acknowledgements
The author(s) and publisher wish to thank the following for granting permission to reproduce
photographs:
Bigstock (pp. 3, 20, 32, 34, 40, 60, 72, 75, 86, 93, 106, 115, 122, 123, 125, 128, 131, 135, 138, 144,
146, 172, 175, 180, 181, 197, 199, 200, 221); Corbis/Greatstock (pp. 3, 33, 110, 137, 146, 166);
David Pickett (pp. 41, 42, 49, 86, 102, 133, 162); Erika Gouws (pp. 110, 132, 138, 209, 223); Fotolia
(p. 92); Getty Images (pp. 135, 166); iStock (pp. 2, 18, 21, 40, 73, 115, 128, 138, 166, 167, 205);
Jeremy Howell (p. 74); Mike Carelse/LCA Studios (pp. 20, 104, 111, 117, 128, 133, 166, 179, 198);
PictureNET (pp. 45; 167 [Roger de la Harpe]); Science Photo Library (p. 163); The Bigger Picture (p. 89).
The author(s) and publisher wish to thank the following for permission to use copyrighted material:
Map Studio (p. 215).
Every attempt has been made to trace and contact copyright holders. Should any copyright
infringement have occurred, please inform the publisher so that the error can be rectified in the
next edition.
Platinum Maths Gr6_Glossary_CAPS.indd 228
08/02/13 3:16 AM