CAPS
Mathematics
a
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Le
5
rn
k
Grade
e r ’s B
L. Bowie • C. Gleeson-Baird • R. Jones
H. Morgan • K. Morrison • M. Smallbones
1 Term 1
The abacus helps us to do calculations.
2
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Topics 1–9
Starting off
Throughout the ages, people used different
devices to help them do calculations. One
of the best known calculating devices is the
abacus.
The abacus has been in existence for more than
4 000 years!
Many ancient cultures had their own versions
of the abacus, for example the Babylonians,
the Egyptians, the Greeks, the Romans and
many others. People in some parts of the world,
for example in China, Japan, India and some
African countries still use the abacus.
Abacus 1
Believe it or not, we can use an abacus for
adding, subtracting, multiplying and dividing.
We can use it to work with decimal numbers as
well as whole numbers. We can even use it to
calculate square roots and cubic roots!
1 Abacus 1 shows the number 2 459 when
you read it from left to right. See if you can
work out what the value of the different
beads on abacus 1 mean.
2. What number is represented on abacus 2?
Content covered in Term 1
Abacus 2
Topic 1: Count, order, compare and represent whole numbers, Topic 2: Number
sentences, Revision, Topic 3: Addition and subtraction, Topic 4: Numeric
patterns, Revision, Topic 5: Multiplication and division, Assignment, Topic
6: Time, Revision, Topic 7: Data handling, Topic 8: Properties of 2D shapes,
Revision, Topic 9: Capacity and volume
3
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Topic
1
Count, order, compare and
represent whole numbers
Maths ideas
• Count forwards
and backwards to
10 000.
• State the value of
any digit in whole
numbers up to
1 000.
• Read, write and
order numbers up
to 10 000.
• Round off to 10
and 100.
Key words
• place value − the
value of a digit
according to
its position in a
number
Challenge
What number am I?
• I am a four-digit
number.
• I am less than 5 000.
• My hundreds digit
is double my units
digit.
• My tens digit is half
of my units digit.
• My thousands digit
is the same as my
units digit.
4
Place value
The value of a digit in a number depends on the place that it has
in that number. You can also write a number in expanded form, by
adding the value of each digit.
Example
Use a place value table to find the value of the digit 4 in the
number 4 613.
Th
H
T
U
4
6
1
3
The value of the 4 is four thousand or 4 000.
The place value of the 4 is a thousands (Th).
When you write a four-digit number, remember to leave a space
between the thousands digit and the hundreds digit.
ExErCiSE 1.1
1. Write each number in expanded form.
a ) 534
b ) 368
c ) 7 899
d ) 5 464
2. Write down the value of the digit 3 in each number.
a ) 305
b ) 263
c ) 2 136
d ) 3 421
3. Underline the hundreds digit in the numbers below.
a ) 7 394
b) 421
c ) 6 385
d) 736
4. Fill in the missing numbers.
a ) 545 = □ hundreds, □ tens and □ units
b ) 6 491 = □ thousands, □ hundreds, □ tens and □ units
c ) 5 679 = □ thousands, □ hundreds, □ tens and □ units
d ) 5 000 = □ thousands, □ hundreds, □ tens and □ units
5. Write these numbers in digit form.
a ) 400 + 20 + 5
b ) 2 000 + 30 + 2
c ) 3 000 + 200 + 8
Term 1
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read, write and round off numbers
Do you remember how to read whole numbers?
Key words
Example
You read the number 398 as three hundred and ninety-eight.
You read the number 1 015 as one thousand and fifteen.
You read the number 3 240 as three thousand, two hundred and forty.
• round off −
a way of making a
number simpler to
use, according to a
given place value
ExErCiSE 1.2
1. Read these numbers with a partner.
a ) 234
b ) 7 676
c ) 5 034
d ) 4 706
2. Write these numbers out in words.
a ) 420
b ) 4 563
c ) 7 589
d ) 8 999
3. Write down the following numbers in digits.
a ) five thousand, six hundred and nineteen
b ) seven thousand and twenty-three
c ) nine thousand and five hundred
You already know how to round off numbers. Here is a reminder.
Example
324
200
300
578
400
500
600
700
324 → 300 and 578 → 600, rounded to the nearest 100
3 850
2 000
3 000
5 376
4 000
5 000
6 000
3 850 → 4 000 and 5 376 → 5 000, rounded to the nearest 1 000
ExErCiSE 1.3
Round off each number to the nearest 10 and the nearest 100.
1. 9 328
2. 782
3. 3 456
4. 6 081
Topic 1: Count, order, compare and represent whole numbers
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Count forwards and backwards
Now you will count forwards and backwards to find the missing numbers.
ExErCiSE 1.4
Fill in the missing numbers on these number lines.
Count forwards or backwards to find the answers.
1.
□
245
2.
445
□
5 150 5 200
3.
4.
□
□
745
□
□
5 450
□
2 550
7 984 7 981
□
5 300
2 675
2 650
□
2 600
□
7 996 7 993
□
□
□
845
□
5 550
□
7 975
□
2 500
□
7 969
ExErCiSE 1.5
1. Write down the numbers that are 2 more than:
a ) 99
b ) 999
2. Write down the numbers that are 10 less than:
a ) 189
b ) 1 899
Did you know?
The name of
this thumb is
‘Isithupha’ in
Zulu. ‘Isithupha’
is also ‘six’.
3
Seven is
‘Isikhombisa’
in Zulu. This
finger is used to
‘point’, which is
‘khomba’ in Zulu.
‘Shiyagalombili’
means ‘leave
out two fingers’
before ten.
4
7
‘Shiyagalolunye’
means ‘leave
out one finger’
before ten.
8
9
2
5
1
6
6
6
10
3. Write down the numbers that
are 50 more than:
a ) 7 789
b ) 7 899
4. Write down the numbers that
are 5 less than:
a ) 4 704
b ) 4 902
5. Write down the numbers that
are 100 more than:
a ) 994
b ) 8 946
6. Write down the numbers that
are 100 less than:
a ) 5 553
b ) 8 005
Term 1
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Compare and order numbers
To order whole numbers, follow these steps:
Step 1: Count the number of digits in each number. The whole
number with the most digits is the biggest number.
Step 2: What happens if the number of digits in each number is the
same? Compare the first digits of the numbers (starting from
the left).
Step 3: What happens if the first digits are the same?
Look at the next digit to the right until the digits are different.
Key words
• ascending order
− from smallest to
greatest
• descending order
− from greatest to
smallest
Example
Order these numbers from smallest to greatest (ascending order).
Th
H
T
U
8
8
3
4
9
4
3
9
5
4
9
The number 999 is the smallest number because it only has 3 digits.
The numbers 8 345 and 8 434 have the same first digit:
8(thousands). The next digits are 3 and 4 (hundred). Four is larger
than 3, so the number 8 434 is larger than 8 345.
The order from the smallest to the greatest is 999; 8 345; 8 434.
Example
Remember that < means smaller than and > means greater than.
9 464 < 9 471 because 6 is smaller than 7
563 > 521 because 6 is bigger than 2
ExErCiSE 1.6
1. Order these numbers from smallest to greatest.
a ) 653; 154; 874; 112; 551
b ) 6 134; 4 631; 6 431; 4 361; 699
c ) 1 345; 1 342; 1 543; 1 432; 1 600
2. Use the symbols < or > to show which number is greater.
a ) 334 □ 343
b ) 2 091 □ 2 019
c ) 3 245 □ 789
Challenge
Can you write all
the numbers in
Exercise 1.6 in
descending order?
Topic 1: Count, order, compare and represent whole numbers
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Topic
2
Number sentences
Maths ideas
• Complete and
solve number
sentences.
• Use inverse
operations.
• Recognise
equivalent number
sentences.
• Use order to
complete number
sentences.
• Use grouping to
complete number
sentences.
• Work with addition
and subtraction
facts.
Key words
• inverse −
opposite
• operation −
addition,
subtraction,
multiplication,
division
Addition and subtraction number
sentences
When you add or subtract, zero is a very special number as you will
see in the next exercise.
ExErCiSE 2.1
1. Complete each number sentence by filling in the missing number.
a ) 47 − 47 = □
b ) 682 − □ = 682 c ) □ − 451 = 0
d ) 244 + 0 = □
e ) □ + 589 = 589 f ) 499 + □ = 499
Write down what you notice about the number 0 in these number
sentences.
2. Now complete these number sentences by filling in the missing
numbers.
a ) 85 − 6 + 6 = □
b ) 426 − 8 + 8 = □
c ) 743 + 10 − 10 = □
d ) □ +4 − 4 = 656
e ) 59 + 8 − □ = 59
f ) 767 − □ + 3 = 767
Write down what you notice in these calculations.
You have found that when the same number is
+6
used, subtraction cancels addition and addition
cancels subtraction. Addition and subtraction are
38
44
inverse operations.
−6
If 30 + 70 = 100, then 100 − 70 = 30, and
100 − 30 = 70.
You can use one operation to check or to find the answer of the other
operation.
ExErCiSE 2.2
Complete these number sentences. Then check each answer by using
the inverse operation.
8
1. 68 − 14 = □
2. 312 + 145 = □
3. 486 − 264 = □
4. 73 − □ = 12
5. 213 + □ = 325
6. 98 − 16 = □
Term 1
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Division and multiplication number
sentences
Multiplication can undo division and division can undo multiplication
if the same number is used to multiply and divide.
Example
Look at this number sentence: 36 ÷ 4 = 9
You can get back to 36 by multiplying 9 by 4:
9 × 4 = 36
So, you can check the answer to a division
sentence by using multiplication.
×4
9
36
÷4
ExErCiSE 2.3
Complete these groups of equivalent number sentences.
1. 8 × 6 = □
2. 23 × 3 = □
3. 200 × 4 = □
□÷8=6
□ ÷ 23 = 3
□ ÷ 200 = 4
48 ÷ 6 = □
69 ÷ 3 = □
800 ÷ 4 = □
ExErCiSE 2.4
Complete these division number sentences. Then write a
multiplication sentence to check each answer.
1. 42 ÷ 7 = □
4. 64 ÷ 4 = □
2. 63 ÷ 9 = □
5. 1 000 ÷ 10 = □
3. 48 ÷ 2 = □
6. 60 ÷ 3 = □
The number 1 is very important when you multiply and divide, as you
will see in the next exercise.
ExErCiSE 2.5
1. Complete these number sentences.
b ) 12 ÷ 12 = □
c ) 172 ÷ 172 = □
a ) 37 × 1 = □
d ) 15 ÷ □ = 1
e ) 24 ÷ 6 × 6 = □ f ) □ ÷ 9 × 9 = 234
2. Write down three facts about the number 1 that you notice from
the number sentences in Question 1.
Topic 2: Number sentences
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The order in a number sentence
It is important to know when you can change the order of the
numbers in a number sentence.
ExErCiSE 2.6
Copy and complete these addition number sentences.
1. 33 + 12 = □;
12 + 33 = □
2. 121 + 300 = □;
300 + 121 = □
3. 705 + 23 = □;
23 + 705 = □
What happened when you changed the order of addition?
In the above exercise you saw that you can change the order of addition
and still get the same answer. Is the same true for subtraction?
ExErCiSE 2.7
1. If 35 − 15 = 20, can you also say that 15 − 35 = 20? Explain.
2. If 50 − 10 = 40, can you also say that 10 − 50 = 40? Explain.
3. Is it true that 28 − 12 = 12 − 28? Explain.
From your work so far, you can see that the answer stays the same
when you change the order of addition, but is not the same when you
change the order of subtraction except if both numbers are the same.
ExErCiSE 2.8
State whether each sentence is true or false.
10
1. 302 + 123 = 123 + 302
2. 302 − 123 = 123 − 302
3. 45 + 45 = 0
4. 45 − 45 = 0
5. 16 + 32 − 32 = 16
6. 100 + 5 + 15 = 100 + 15 + 5
Term 1
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Group numbers in different ways
When you add more than two numbers together, it is sometimes
easier if you change the order and group the numbers differently.
Look for additions that make multiples of 10 or 100.
Example
1. Look at this number sentence: (27 + 28) + 3 = □
You can add the numbers more easily if you change the grouping:
(27 + 3) + 28 = □ (30 + 28 = □, so the missing number is 58)
2. 489 + 23 = □ has the same answer as 489 + (11 + 12) = □
or (489 + 11) + 12 = □
This is 500 + 12 = □, so the missing number is 512.
ExErCiSE 2.9
Change the grouping in each of these number sentences to make
them easier to complete. Then write in the missing number.
1. (22 + 45) + 28 = □
2. (29 + 12) + 18 = □
3. (355 + 37) + 45 = □
4. (37 + 69) + 13 = □
Challenge
Get a friend to time
you to see how
quickly you can add
these columns of
numbers together:
You can also break down numbers and then group them to find a
missing number in a number sentence.
Example
Marie was asked to complete and solve this number sentence:
87 + 7 = □ + 5
Marie solved it like this: 7 = 2 + 5
So, 87 + 7 = 87 + (2 + 5) = (87 + 2) + 5 = 89 + 5
Check Marie’s answer: 87 + 7 = 94, and 89 + 5 = 94, so this is
correct.
ExErCiSE 2.10
Complete these number sentences.
1. 96 + 8 = □ + 5
2. 53 + 9 = □ + 4
3. 78 + □ − 5 = 80
4. 64 − 6 + □ = 66
5. 39 + 4 = □ + 6
6. 32 − 7 = □ + 7 − 7
Topic 2: Number sentences
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Addition and subtraction facts
It is very useful to know which numbers add up to 10, 100 and 1 000.
You will practise working with these in the next exercises.
Example
Challenge
This dartboard
shows multiples of
5 from 5 to 95.
55
85
70
35
45
5
50
60
95
10
40
80
15
20
25
30
65
90
50
75
1. Use the numbers
on the dartboard
to write as many
additions as you
can that add up to
100. For example,
45 + 55 = 100
1. 2 + 8 = 10. The inverse of this is: 10 − 8 = 2 or 10 − 2 = 8
2. 20 + 80 = 100. The inverse of this is: 100 − 80 = 20 or
100 − 20 = 80
Inverse operatons can be used to check addition and
subtraction answers.
3. To find 100 − 54, you can break down the second number.
100 − 50 − 4 = (100 – 50) – 4 = 50 − 4 = 46, so 100 − 54 = 46
4. To complete the number sentence □ + 32 = 100, write the
sentence as:
□ = 100 − 32 = 100 – 30 − 2 = (100 – 30) – 2 = 70 − 2 = 68
The missing number is 68.
Check by adding: 68 + 32 = 60 + 30 + 8 + 2 = 90 + 10 = 100
ExErCiSE 2.11
1. Fill in the missing numbers in these number statements.
a ) 10 = 4 + □; 10 − 6 = □
b ) 10 = □ + 3; □ − 7 = 3
c ) 100 = 40 + □; 100 − 60 = □
d ) 100 = □ + 30; □ − 70 = 30
To calculate each
addition, use one
number from the
outer ring and
one number from
the inner ring.
2. Fill in the missing numbers and then write each number sentence in
a different way. Check your answer by using an inverse operation.
a ) 100 − 56 = □
b ) 22 + □ = 100
c ) □ + 25 = 100
d ) 65 + 35 = □
2. Now write
as many
subtractions as
you can that start
with 100.
If you know number facts for 100, you can work out number facts for
1 000. For example, if 40 + 60 = 100, then 400 + 600 = 1 000.
For example:
100 − 55 = 45.
12
ExErCiSE 2.12
Fill in the missing numbers, and write each number sentence in a
different way. Check your answer by using an inverse operation.
1. 1 000 − 500 = □
2. 200 + □ = 1 000
3. □ + 300 = 1 000
4. 650 + 350 = □
Term 1
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revision
1. Write down the numbers that are 10 less than each of the following:
a ) 909
b ) 4 003
c ) 9 002
(1)
(1)
(1)
2. Write down the numbers that are 100 more than the following:
a ) 947
b ) 8 896
c ) 9 906
(1)
(1)
(1)
3. Write down the value of each digit in this number: 7 064.
(2)
4. Write this number in a place value table: 3 502.
(2)
5. Order these numbers from smallest to greatest:
1 629 1 926 1 296 1 692 1 269
(2)
6. Solve these number sentences. Show how you get your answer.
a ) 47 + 7 = □ + 3
b ) 63 + □ − 7 = 60
c ) 86 + 5 = □ + 1
(1)
(1)
(1)
7. Complete these number sentences, then use an inverse operation to check each one. Show all
your working.
a ) 576 − 98 = □
(2)
b ) 381 + 67 = □
(2)
8. Complete these division number sentences, then write a multiplication sentence to check
your answer. Show all your working.
a ) 96 ÷ 12 = □
(2)
b ) 138 ÷ 6 = □
(2)
9. Change the grouping in each of these number sentences to make it easier to calculate and
then complete the sentence:
a ) (16 + 137) + 24 = □
(1)
b ) 47 + (132 + 13) = □
(1)
Total marks: 25
Revision
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Topic
3
Addition and subtraction
Maths ideas
round off and estimate
• Estimate answers
by rounding
numbers.
When you round off a whole number, digits less than 5 round down to
zero, and digits 5 or larger round up to the next whole unit.
• Add and subtract
numbers with up
to 4 digits.
• Solve multi-step
problems that
involve addition
and subtraction.
5 or more − round up to the nearest 1, 10, 100 or 1 000
4 or less − round down to the nearest 1, 10, 100 or 1 000
Example
• Use breaking
down and
compensation to
add and subtract.
Round off the number 345.
• 345 rounded to the nearest 10 will be 350, because 5 rounds up.
• 345 rounded to the nearest 100 will be 300, because 4 tens is
less than 5 tens.
• Use inverse
operations to
check answers.
Example
Key words
1. Estimate the answer to 34 + 85 by rounding off to the nearest
10. Look at the units digits to round up or down.
• estimate − an
approximate
calculation
The approximate answer is 30 + 90 = 120.
2. Estimate the answer to 445 − 288 by rounding off to the
nearest 100. Look at the tens digits to round up or down.
The estimated answer is 400 - 300 = 100
ExErCiSE 3.1
1. Use rounding to the nearest 10 to estimate these answers.
a ) 29 + 44 + 12
b ) (28 + 62) − 18
c ) 43 − 11
2. Use rounding to the nearest 100 to estimate these answers.
a ) 765 − 335
b ) 688 + 367
c ) 221 + 245
d ) 655 + 211
e ) 612 + 783 + 389
f ) (765 + 923) − 239
g ) (956 − 323) − 440
h ) 543 + 841 + 999
14
Term 1
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Add whole numbers
You will use the same methods for adding as before. You will also use
brackets for larger numbers when you break them up for addition.
Remember that any calculation inside the brackets must be done first.
Example
Method 1: Add by breaking down all numbers according to place
value parts.
Calculate 1 362 + 7 486
= 1 000 + 300 + 60 + 2 + 7 000 + 400 + 80 + 6
= (1 000 + 7 000) + (300 + 400) + (60 + 80) + (2 + 6)
= 8 000 + 700 + 140 +8
= 8 848
Method 2: To calculate 5 362 + 2 486, break down the number to
be added.
Add the number 2 486 in parts: 2 000 then 400 then 80 then 6.
5 362 + 2 000 = 7 362 + 400 = 7 762 + 80 = 7 842 + 6 = 7 848
Method 3: Use rounding off and compensating to calculate
2 486 + 148. You add and subtract the same number when you
round off.
You can make 2 486 become 2 500 by adding 14, but you must also
subtract 14, because 14 − 14 = 0.
So, 2 486 + 148 = (2 486 + 14) + (148 − 14) = 2 500 + 134
= 2 500 + 100 + 34 = 2 634
Key words
• compensating −
to add or subtract
numbers after
rounding off one
of the numbers
when calculating
ExErCiSE 3.2
First round the numbers to the nearest 100 to estimate each answer.
Then use any of the above methods to do these additions.
1. 213 + 962
4. 5 076 + 8 104
7. 5 233 + 3 122
2. 467 + 964
5. 8 476 + 9 817
8. 9 002 + 898
3. 612 + 490
6. 2 380 + 7 999
9. 3 695 + 5 791
ExErCiSE 3.3
Double each number by filling in the missing numbers:
1. 361 = 300 + □ + 1 = 600 + 120 + □ = □
2. 582 = □ + 80 + □ = □ + □ + 4 = □
Topic 3: Addition and subtraction
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inverse operations
Addition and subtraction are related. Addition is the inverse operation
of subtraction. And subtraction is the inverse operation of addition.
This means that you can use addition facts that you know to find
subtraction facts that you do not know.
34
+ 17
51
17
+ 34
51
51
− 34
17
51
− 17
34
Example
Find related addition and subtraction facts.
34 + 17 = 51
The related addition and subtraction facts are:
17 + 34 = 51 and 51 − 34 = 17 and 51 − 17 = 34
This is a family of addition and subtraction facts of 34, 17 and 51.
ExErCiSE 3.4
Complete the family of four related facts.
1. 28 + 72 = 100
3. 249 + 167 = 416
5. 2 856 + 4 768 = 7 624
2. 100 − 56 = 44
4. 725 − 248 = 477
6. 9 375 − 2 618 = 6 757
You can find missing numbers by using families of related addition
and subtraction facts.
Example
Find the missing numbers. Use the family of related addition and
subtraction facts.
a) 78 − □ = 34, so 78 − 34 = □. So, □ = 44 ← Check 78 − 44 = 34
b) □ − 24 = 56, so 24 + 56 = □. So, □ = 80 ← Check 80 − 24 = 56
ExErCiSE 3.5
Find the missing numbers. Write down the addition or subtraction
fact that you used to check the calculation.
1. 23 + □ = 87
2. □ + 56 = 110
3. □ − 38 = 60
4. 125 − □ = 98
5. 68 − □ = 15
6. □ + 99 = 256
7. □ − 89 = 121
8. □ + 199 = 310
16
Term 1
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Subtract whole numbers
Here are two useful methods for subtracting whole numbers.
Example
Method 1: Breaking down both numbers according to place value
parts. If necessary, you can break down a number further to help
you subtract.
Calculate 8 743 − 5 684
8 743 − 5 684
= (8 000 + 700 + 40 + 3) − 5 000 − 600 − 80 − 4
= (8 000 + 600 + 130 + 13) − 5 000 − 600 − 80 − 4
(Break down 743 into 600 + 130 + 13)
= (8 000 − 5 000) + (600 − 600) + (130 − 80) + (13 − 4)
= 3 000 + 0 + 50 + 9
= 3 059
Method 2: Break down the number to be subtracted and then
subtract in parts.
Calculate 4 687 − 2 143. Subtract 2 143 by subtracting 2 000, then
100, then 40, then 3.
4 687 − 2 143 → 4 687 − 2 000 → 2 687 − 100 → 2 587 − 40
→ 2 547 − 3 = 2 544
ExErCiSE 3.6
First round each number to estimate the answer. Then do the
calculation and check the answer with an inverse calculation.
1.
2.
3.
4.
5.
6.
a ) 459 − 302
b ) 1 234 − 1 098
c ) 3 420 − 1 999
Find the sum of 7 098 and 1 754.
Find the difference between 8 472 and 4 097.
How much larger is 2 100 than 931?
By how much must you increase 1 618 to get 5 141?
In June, 3 580 people visited the museum. That was
86 more than the number of people who visited
the museum in April. How many people visited the
museum in April?
7. Double both numbers and then find the difference
between them. Finally halve that number to get to
the correct answer. 141 – 15 = □
The South African Museum
Topic 3: Addition and subtraction
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Topic
4
Numeric patterns
Maths ideas
• Investigate
numeric patterns.
• Extend numeric
patterns.
• Describe observed
rules in own
words.
• Determine input
and output values
in flow diagrams.
• Use flow diagrams
to describe rules
in mathematical
operations.
Key words
Patterns and flow diagrams
A number sequence is a group of numbers that follow each other in a
particular order. In some number sequences:
• The same number is added or subtracted to get the next number
• You multiply or divide by the same number to get the next number
Example
These sequences each follow a different rule. Fill in the missing
numbers in each sequence:
1. 3; 5; 7; 9; □; □; □
2. 2; 4; 8; 16; □; □; □
Answers:
1. Add 2 to get the next number. missing numbers are 11; 13; 15.
2. Multiply by 2 to get the next number. Missing numbers are 32;
64; 128
ExErCiSE 4.1
• flow diagram −
a diagram that
shows how
an operation
is applied to a
number to get an
answer
Find the rule then fill in the missing numbers:
1. 1; 4; 7; □; □; □
2. 20; 16; 12; □; □; □
3. 2; 6; 18; □; □; □
4. 128; 64; 32; □; □; □
• input number −
the number that
you start with in a
flow diagram
The flow diagrams in the next exercise will show an important rule
about multiplication and division. You can also use a table to show
input and output numbers.
• output number
− the answer that
you get in a flow
diagram
ExErCiSE 4.2
• number sequence
− a group of
numbers that
follow each other
in a particular
order
18
1. Draw these flow diagrams and then fill in each output number:
1
9
3
27
6
x9
8
54
÷9
72
9
81
Term 1
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2. Now complete the table below for the flow diagrams in Question 1.
Input number
Output number
Input number
Output number
Rule 1
3
6
8
9
27
54
72
81
×9
Rule 9
÷9
inverse operations
You have learnt that multiplication and division are inverse operations.
ExErCiSE 4.3
×9
8
1. Use what you have learnt so far to complete these flow diagrams.
Explain to a partner what method you used:
3
2
72
÷9
22
36
9
4
x6
44
x11
66
12
66
6
2. Complete these pairs of number sentences.
a ) 72 ÷ 6 = □ and □ × 6 = 72 b ) 54 ÷ 9 = □ and □ × 9 = 54
c ) 49 ÷ 7 = □ and □ × 7 = 49 d ) 60 ÷ 12 = □ and □ × 12 = 60
Order in multiplication
In the next exercise the order of the numbers has been changed.
ExErCiSE 4.4
1. Complete these flow diagrams and then write down what you
notice.
1
30
5
4
60 6
x5
7
x2
x2
x5
70
90 13
14
150
2. Now draw up two tables similar to those in Exercise 4.1. Enter the input
numbers and output numbers from the flow diagrams in Question 1.
Topic 4: Numeric patterns
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Discover more numeric patterns
Multiples of 100 or 1 000
You will discover an easy way to multiply by any multiple of 100, or
1 000 like 600, 800 and 3 000 in the next exercise.
ExErCiSE 4.5
1. First copy and complete these two flow diagrams:
3
1 200
3
2 000
5
x400
2 800 7
x4
x100
3 200
8
11
11
2. Write down, in your own words, an easy way to multiply by 400.
ExErCiSE 4.6
Use the method you discovered in Exercise 4.4 to complete these
number sentences:
1. 7 × 200 = □
2. 6 × 7 000 = □
3. 4 × 400 = □
4. 9 × 500 = □
5. 5 × 9 000 = □
6. 6 × □ = 2 400
7. 7 × □ = 63 000
8. 5 × □ = 2 000
9. 7 × □ = 4 200
Splitting other numbers
4
As you split 400 into 4 × 100 to make multiplication easier, you can split
other numbers. In this flow diagram, numbers are multiplied by 18.
72
ExErCiSE 4.7
6
x18
108
126
7
Complete this flow diagram and
explain how the flow diagram made
it easier for you to multiply by 18.
4
5
x9
6
x2
7
8
20
Term 1
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revision
1. Round off these numbers to the nearest 10.
a ) 77
b ) 125
(2)
2. Write down an inverse addition or subtraction operation that you can
use to check these calculations.
a ) 386 + 3 752 = 4 138
b ) 1 947 − 465 = 1 482
(1)
(1)
3. In a National Park, there are 856 elephants and 543 giraffes.
Estimate, by rounding off to the nearest 100, how many
elephants and giraffes there are altogether.
(2)
4. Use any method that you feel comfortable with to do the following calculations.
a ) 975 + 798
b ) 425 − 212
c ) 3 245 + 1 987
d ) 5 362 − 2 181
(1)
(1)
(1)
(1)
5. a ) Complete the following tables.
Rule: × 7
Input number
2
4
7
(2)
9
12
Output number
Rule: ÷ 7
Input number
Output number
(2)
14
28
49
63
84
b ) What conclusion you can draw from these tables?
(2)
6. Find the rule in each of the following. Write each rule in words.
a ) Input number
2
4
5
20 100
b)
Output number
6
12
15
60
300
Input number
Output number
1
6
8
40
96
5
10
12
44
100
(2)
(2)
Total marks: 20
Revision
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Topic
5
Multiplication and division
Maths ideas
Find factors and multiples
• Work with
multiples and
factors.
When you multiply two numbers, you get a multiple. A number that
can divide exactly into that number is called a factor.
• Round off to
estimate answers.
• Multiply two-digit
numbers by twodigit numbers.
• Divide three-digit
numbers by singledigit numbers.
• Check solutions.
• Solve problems
with multiplication
and division.
Key words
• multiple − the
answer when
you multiply two
numbers
• factor − a number
that divides exactly
into another
number
Challenge
Can you predict what
happens when you
multiply or divide by
10 000, 100 000 or
1 000 000? Make up a
rule related to the
numbers of zeros
when multiplying or
dividing.
Example
10 × 4 = 40, so 40 is a multiple of 10 and also a multiple of 4.
10 and 4 are called factors of 40.
10 = 5 × 2, so 5 and 2 are also factors of 40.
Check what all the factors of 40 are: 1, 2, 4, 5, 8, 10, 20 and 40.
You can write a number as a multiplication of a factor pair. Here are
all the factor pairs for 40: 1 × 20; 4 × 10; 5 × 8.
ExErCiSE 5.1
1. a ) Write down the first 5 multiples of 7.
b ) What are the multiples of 5 between 61 and 69?
2. Write down all the factors of each number.
a ) 10
b ) 25
c ) 36
d ) 50
e ) 100
3. Write down two factor pairs for each number.
a) 6
b ) 12
c ) 18
d ) 20
e ) 80
You can use factors to help you to multiply. Look for multiples of 10
because it is easier. You can change the order when you multiply.
Example
Find 24 × 20
Break down 20 into the factors 2 × 10
24 × 20 = 24 × 2 × 10 = 48 × 10 = 480
ExErCiSE 5.2
Multiply by breaking down the multiple of 10 into two factors.
1. 42 × 20
22
2. 21 × 60
3. 11 × 80
4. 32 × 30
Term 1
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Multiply two-digit numbers
Challenge
In Grade 4 you used two methods to multiply numbers. You can make
multiplication easier if you first break down one of the numbers.
Example
Here is a group of 15 coins.
You can move the coins to
make two groups of coins,
with 9 coins and 6 coins.
3×5
3×3
This means that you can
write 3 × 5 like this:
3 × 5 = 3 × (3 + 2) = (3 × 3) + (3 × 2) = 9 + 6 =15.
+ 3×2
1. What number
am I?
I am > 50 and < 70.
I am a multiple of 3.
I am not a multiple
of 6 or 9.
2. What number am I?
I am > 80 and
< 100.
I am a multiple of 4.
I am not a multiple
of 6.
?
Example
37 × 65
= 37 × (60 + 5)
(Break down one number.)
= (37 × 60) + (37 × 5)
= (37 × 2 × 3 × 10) + (30 × 5)+ (7 × 5)
= ( 74 × 3 x 10 ) + 150 + 35
= (222 × 10) + 185
= 2 220 + 185 = 2 405
Example
67 × 18 = 67 × 2 × 9 (Break down one of the numbers into factors.)
= (67 × 2 ) × 3 × 3
= (134 × 3 ) × 3
= (100 + 30 + 4) × 3 × 3
= (300 + 90 + 12) × 3
= (900 + 270 + 36) × 3
= 1 206
ExErCiSE 5.3
First estimate these answers and then use the method of breaking
down one of the numbers to find the exact answer. You can also use
factors to help you.
1. 57 × 34
2. 87 × 29 3. 45 × 66 4. 74 × 47 5. 23 × 35
Topic 5: Multiplication and division
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Key words
• inverse
operations −
reverse or opposite
operations
(for example,
multiplication
is the inverse of
division)
• whole numbers
− the numbers
we use to count,
including 0. 0, 1, 2,
3, …
Divide whole numbers
Remember that multiplication and division are inverse operations.
This means that you can use multiplication facts to help you divide
numbers.
Example
Estimate and then use multiplication facts to find 407 ÷ 9.
An estimate is 400 ÷ 10 = 40.
Now write down some simple multiplication facts for 9. This is
called a clue board.
Clue Board
9 × 10 = 90
9 × 20 = 180
9 × 40 = 360
9 × 5 = 45
The closest multiplication to 407 is 360, so start with 9 × 40.
Multiply
Subtract
9 × 40 = 360
407 − 360 = 47
9 × 5 = 45
47 − 45 = 2
407 ÷ 9 = 40 + 5 remainder 2 = 45 remainder 2.
The estimate was quite close! Finally check by multiplying:
9 × 45 plus remainder 2 = (9 × 40) + (9 × 5) + 2
= 360 + 45 + 2 = 407
ExErCiSE 5.4
Challenge
Which numbers, from
2 to 9, will divide into
these numbers with
remainders?
a) 2 445
b) 2 616
c) 2 128
24
1. Draw up your own multiplication clue boards to do these divisions.
a ) 654 ÷ 3
b ) 505 ÷ 4
c ) 722 ÷ 8
d ) 299 ÷ 7
e ) 190 ÷ 9
f ) 529 ÷ 6
2. Fill in the missing numbers.
a) □ ÷ 7 = 6
c ) 42 ÷ □ = 21
e) □ ÷ 9 = 6
g ) 72 ÷ □ = 8
b) □ ÷ 8 = 3
d ) 81 ÷ □ = 9
f ) □ ÷ 6 = 13
h ) 96 ÷ □ = 24
Term 1
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Solve multiplication or division
problems
To solve a word problem, first write the information as a number
sentence. You can then find the answers to the number sentence.
Example
A water tank holds 75 litres of water. How many litres can 16 tanks
hold?
This is a multiplication problem, because you have 75 litres of
water 16 times.
The number sentence is 75 × 16 = □
75 × 16 = 75 × 2 × 2 × 4 = 150 × 2 × 4 = 300 × 4
= 3 × 4 × 100 = 1 200
The tanks can hold 1 200 litres altogether. Use division to check
that 1 200 ÷ 16 = 1 200 ÷ 4 ÷ 2 ÷ 2 = 300 ÷ 2 ÷ 2
= 150 ÷ 2 = 75 litres.
Challenge
a) Find two fivedigit numbers
that are multiples
of 2 and 10.
b) Find two fivedigit numbers
that are multiples
of 5 and 4.
c) Find two fivedigit numbers
that are multiples
of 2, 5, 10 and 4.
ExErCiSE 5.5
1. A man is packing toys into boxes. He can fit 8 toys into each box.
He has 568 toys to pack. How many boxes will he fill?
2. One crate of pineapples has a mass of 45 kg. What will be the mass
of 18 crates of pineapples?
3. A minibus can carry 8 people. How many minibuses are needed to
take 152 people to a soccer match?
4. Look at these 3 containers of eggs. Which would be the
best buy? Show all your calculations.
5. Each bridesmaid needs 135 cm of ribbon for her pretty
dress. How much ribbon must the bride buy if she has
5 bridesmaids?
6. Forest Gate Railway Station is very busy. Every day
84 trains pass through the station. How many trains
will pass through the station in:
a ) 1 week?
b ) 25 days?
Topic 5: Multiplication and division
Platinum Maths Gr5_Term 1_CAPS.indd 25
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4
4
4
4
4
890123
67
4
012345
890123
345678
012345
4
9
67
890123
345678
012345
4
9
67
In this investigation, you will use a calculator to find number patterns.
Practice using your calculator before you begin this investigation.
Work out 25 × 12 on your calculator. To do this multiplication, press
these keys:
You should get an answer of 300.
1. Copy these multiplications into your exercise book. Then multiply
these numbers using your calculator.
b ) 25 × 22
a ) 25 × 12
d ) 25 × 42
(4)
c ) 25 × 32
Can you see the pattern? Find 25 × 52 and 25 × 62 without
multiplying the numbers.
2. Find the difference in these subtractions. Do you see the pattern?
a ) Subtract: 25 × 22 − 25 × 12. This is the same as saying 550 − 300.
b ) Subtract: 25 × 32 − 25 × 22
(4)
c ) Subtract: 25 × 42 − 25 × 32
3. Look for patterns in these sequences. Use your calculator to
work out the first three multiplications in each sequence.
Find the differences between the answers. Then use addition
to find the missing multiplications.
a ) 25 × 13; 25 × 23; 25 × 33; □; □; 25 × 63
b ) 25 × 14; 25 × 24; 25 × 34; □; □; 25 × 64
c ) 25 × 19; 25 × 29; 25 × 39; □; □; 25 × 69
What pattern do you see in the differences between the answers?
Find the difference between 25 × 37 and 25 × 47. Use your
calculator to check if you are correct.
(8)
4. Look at the pattern below.
2 × 9 = 18
2 × 99 = 198
2 × 999 = 1 998
3 × 9 = 27
3 × 99 = 297
3 × 999 = 2 997
4 × 9 = 36
4 × 99 = 396
4 × 999 = 3 996
In your own words, explain how to find the answers to
the second and third columns using the multiplication
facts in the first column.
(4)
890123
9
012345
4
345678
67
90
67
90
890123
345678
9
90
012345
90
67
90
890123
345678
9
90
012345
12
67
90
890123
345678
9
12
012345
12
67
90
890123
345678
9
12
012345
12
67
45678
9
12
012345
4
9
90
890123
345678
3
12
12
012345
12
Find patterns
345678
67
12
9
90
Assignment
26
Term 1
Platinum Maths Gr5_Term 1_CAPS.indd 26
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890123
5. Use the pattern in Question 4 to predict the answers to the
following divisions:
a ) 495 ÷ 99
b ) 594 ÷ 99
c ) 693 ÷ 99
d ) 4 995 ÷ 999
e ) 7 992 ÷ 999
f ) 8 991 ÷ 999
Use your calculator to check your predictions.
(6)
6. Look at the following pattern of
multiplications. How many
multiplications do you need to
do before you know the answer
without using your calculator?
(5)
1×1
11 × 11
111 × 111
1 111 × 1 111
11 111 × 11 111
7. Look at the following pattern of
multiplications. How many
25 × 11
25 × 111
multiplications do you need to
25 × 1 111
do before you know the answer
25 × 11 111
without using your calculator?
25 × 111 111
Explain the pattern in your own words.
(5)
8. Find patterns in these multiplications. Use your calculator to
help you.
b ) 35 × 1
c ) 72 × 1
a ) 42 × 1
35 × 11
72 × 11
42 × 11
35 × 111
72 × 111
42 × 111
Guess the answers to 42 × 1 111, 35 × 1 111 and 72 × 1 111.
What do you notice about the repeating number in the answers?
Look carefully at the digits of the number you are multiplying.
Guess what the repeating numbers are in 23 × 1 111 and
(9)
43 × 11 111. Use your calculator to check if you are correct.
Total marks: 45
Assignment
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Topic
6
Time
Maths ideas
• Time events
with a watch or
stopwatch.
• Read and write
time on analogue
and digital clocks.
• Use a.m. and p.m.
notation to show
time of day.
• Convert units of
time.
• Understand
the difference
between 12-hour
time and 24-hour
time.
• Read and use
calendars.
• Calculate durations
of time.
Key words
• analogue clock
− a clock that uses
hands to show the
time
• a.m. (ante
meridian) − the
time after midnight
but before noon
• p.m. (post
meridian) − the
time past noon up
to midnight
• digital clock − a
clock that shows
time using numbers
and a : separator
28
read and write time
There are 24 hours in a day. Midnight to midday (or noon) is called
‘before noon’. Midday to midnight is called ‘after noon’. You write
‘before noon’ as a.m. and ‘after noon’ as p.m.
Example
You can read or write the time as
5.14 a.m. or as 14 minutes past 5
in the morning (before noon) or
as 05:14.
10
9
8
11 12 1
7 6 5
2
3
4
Analogue clocks and watches have hands that turn to show the time.
Some analogue clocks show a second hand to indicate the seconds.
Digital clocks and watches use electronic digits (numbers) to show
the time. On these clocks hours, minutes and seconds are separated
by a :, for example: 18:22:10.
Example
The time on both clocks is
twenty-two minutes and
ten seconds past six o‘clock
in the evening
10
9
8
11 12 1
7 6 5
2
3
4
ExErCiSE 6.1
Read the times on these clocks and write down each time in words.
(Hint: Use hours, minutes and seconds.)
1.
4.
2.
10
9
8
11 12 1
7 6 5
2
3
4
5.
10
9
8
11 12 1
7 6 5
2
3
4
3.
6.
10
9
8
11 12 1
7 6 5
2
3
4
Term 1
Platinum Maths Gr5_Term 1_CAPS.indd 28
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The 12-hour time system can be confusing if you don‘t know whether
a time is showing a.m. or p.m. To avoid this, we use the 24-hour system
of telling time. In this system times are not repeated. Instead, the
hours are numbered from 00:00 hours (12 a.m.). Times before 12:00
(12 noon) are the same as a.m. times. Times after noon are numbered
from 12:00 onwards. So 1 pm = 12:00 + 1 hour = 13:00. We read this as
13 hundred hours.
Example
Write the following 12-hour times as 24-hour times.
1. 3.22 a.m.
2. 11 a.m.
3. 3.22 p.m.
4. 11.00:30 p.m.
Did you know?
The sundial is the
oldest known
instrument for
measuring time. As
the sun moves across
the sky, the shadow
on the sundial points
to different hours of
the day marked on
the sundial.
Remember to count on from 12 to show 24-hour times.
a) 3.22 a.m. is 03:22. ← Write the number of hours as a two-digit number.
b) 11 a.m. is 11:00. ← Write the number of minutes as a two-digit number.
c) 3.22 p.m. is 15:22. ← Remember to add 12 to the hour for p.m. time.
d) 11.00:30 p.m. is 23:00:30. ← Treat seconds the same way in 12-hour
time and 24-hour time.
ExErCiSE 6.2
1. Write the following 12-hour times as 24-hour times.
a ) 4.30 a.m.
b ) 12 a.m.
c ) 11.15 p.m.
d ) 12 p.m.
2. Write the following 24-hour times as 12-hour times.
a ) 16:45
b ) 20:48
c ) 00:15
d ) 12:27
Challenge
Betty starts reading.
11 12 1
10
2
9
3
8
4
7 6 5
Betty stops reading
11 12 1
10
2
9
3
8
4
7 6 5
For how long did Betty read? Give your answer in hours and minutes.
Topic 6: Time
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Key words
• leap year − the
name given to a
year that has 366
days in it and not
the usual 365
• decade − a period
of 10 years
• convert − to
change to
something else
Convert units of time
In this section, you will learn how to work with different units of time.
Example
1. How many seconds are there in
7 minutes?
7 × 60 s = 7 × 6 × 10 s = 420 s
There are 420 seconds in
7 minutes.
2. How many weeks in 84 days?
84 ÷ 7 = 12
There are 12 weeks in 84 days.
1 minute = 60 seconds
1 month = about 4 weeks
1 hour = 60 minutes
1 year = 12 months or
365 days
1 day = 24 hours
1 leap year = 366 days
1 week = 7 days
1 decade = 10 years
ExErCiSE 6.3
Did you know?
• It takes 365—14 days
for the Earth to
rotate once around
the sun every year.
• In normal years,
we do not count
the —14 of a day.
• So, every 4 years
we have an extra
day. These years
are leap years.
1. How many seconds are there in the following?
a ) 4 minutes
b ) 11 minutes
c ) 20 minutes
2. How many minutes are there in the following?
a ) 60 seconds
b ) 240 seconds
c ) 720 seconds
Months have 30 or 31 days, so they do not have exactly four weeks.
February is different because it has 28 days, but 29 days in leap years.
ExErCiSE 6.4
Use the examples in each question to help you.
1. 15 weeks = 15 × 7 days = 105 days
How many days are there in:
a ) 5 weeks
b ) 100 weeks c ) 35 weeks d ) 63 weeks?
2. 72 days = 72 ÷ 7 days = 10 weeks remainder 2 days
How many weeks are there in:
a ) 21 days
b ) 63 days
c ) 861 days
d ) 396 days?
3. 11 years = 11 × 12 months = 11 × 2 × 2 × 3 = 132 months
How many months are there in:
a ) 25 years
b ) 43 years
c ) 102 years d ) 86 years?
4. 5 decades = 5 × 10 years = 50 years
How many years are there in:
a ) 8 decades b ) 12 decades c ) 17 decades d ) 23 decades?
30
Term 1
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Measure and calculate time
You can use clocks and stopwatches to find out how long events take.
Example
For how long did Sonja run?
Sonja starts running.
Sonja stops running.
Calculate the seconds: From 30 seconds to 45 seconds = 15 seconds
Calculate the minutes: From 5 minutes to 8 minutes = 3 minutes
Calculate the hours: From 8 o’clock to 9 o’clock = 1 hour
Sonja ran for a total time of 1 hour 3 minutes and 15 seconds.
ExErCiSE 6.5
Key words
Use a stopwatch or watch with a second hand to answer these questions.
• stopwatch − an
instrument that
you use to time the
duration of events
very accurately
1. Find the number of steps you can take in 2 minutes.
2. Find the number of times you can hop on one foot for 1 minute.
3. Find the number of times your heart beats in 1 minute.
A calendar gives you the date, month and year of a particular day. For
example, 5 May 2011 is a calendar date. You read this date as ‘the fifth
of May 2011’.
Example
How many days are there from 16 August to 6 September?
August
S
M
T
W
T
September
F
S
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15
16 17 18 19 20 21 22
23 24 25 26 27 28 29
30 31
S
M
T W T
F
S
1
2
3
4
5
6
7
8
9 10 11 12
13 14 15 16 17 18 19
20 21 22 23 24 25 26
27 28 29 30
Did you know?
A sand timer is a very
old instrument for
measuring time.
Number of days in August (until 31 August) = 16 days
Number of days in September (until 6 September) = 6 days
Total number of days from 16 August to 6 September = 22 days
Topic 6: Time
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ExErCiSE 6.6
1. It is ten minutes past 11 a.m. What time was it three hours and
45 minutes ago?
2. Katlego went to sleep at 9.35 p.m. She woke up at 6.15 a.m. For
how long did Katlego sleep?
3. Maria’s race time was 1 min 15 seconds, and Mona’s time was 55
seconds. By how many seconds did Mona beat Maria?
4. Look at each pair of clocks. Calculate the amount of time that has
passed.
a)
b)
c)
12
12
12
10
9
8
10
9
8
Challenge
1. For how long are
you at school in:
a) 1 day?
b) 1 week?
c) February?
d) this year?
2. For how long will
you sleep in:
a) 1 night?
b) 1 week?
c) 1 month?
11
1
7 6 5
11 12 1
7 6 5
2
3
4
2
3
4
10
9
8
10
9
8
11
1
7 6 5
11 12 1
7 6 5
2
3
4
2
3
4
10
9
8
10
9
8
11
1
7 6 5
11 12 1
7 6 5
2
3
4
2
3
4
5. How many days are there from 12 October to 19 November? Use a
calendar to check how many days in each month.
6. Safwat and his class organised a ‘stay awake’ marathon to raise
money for books for the school library. The class started the
marathon at 6.30 a.m. on Saturday and they stayed awake until
10.30 p.m. on the same day.
a ) For how long did the class stay awake?
b ) Did the class reach their target of 24 hours?
c ) The class raised R10,00 for each hour that they stayed awake.
How much money did they raise?
7. Harry works in a factory. He starts work at 06:30. He works until
10:00. He has a break for 15 minutes and he then works until 13:00.
He has another break for 30 minutes and he then works until
16:30. How many hours does Harry work in 1 day?
d) 1 year?
32
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revision
1. Do the following multiplication and division calculations.
a ) 10 × 45
b ) 33 × 60
c ) 210 ÷ 30
d ) 720 ÷ 60
(1)
(1)
(1)
(1)
2. Use any method to do the following multiplications.
a ) 45 × 14
b ) 71 × 52
(1)
(1)
3. Use the clue board method to do the following divisions.
a ) 442 ÷ 7
b ) 717 ÷ 6
(1)
(1)
4. Write down all the multiples of 6 that lie between 50 and 70.
(2)
5. Write down all the factor pairs of 32.
(3)
6. One box holds six eggs. If the farmer has 594 eggs,
how many boxes of eggs does he have?
(2)
7. Arrange the following periods of time from the shortest to the longest.
day
leap year
second
decade
minute
week
hour
month
year
8. Read the times on these clocks, then
write down the times.
b)
a)
11 12 1
10
9
8
7 6 5
(2)
(3)
c)
2
3
4
10
9
8
11 12 1
7 6 5
2
3
4
9. Complete these sentences.
a ) There are □ decades in 60 years.
b ) There are □ seconds in 25 minutes.
(1)
(1)
10. How many days from 12 November to 29 November?
(1)
11. How much time has passed from 09:25 to 23:35?
(2)
Total marks: 25
Revision
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11/02/13 10:40 AM
Topic
7
Data handling
Maths ideas
• Collect, organise
and record data.
• Draw pictographs
and bar graphs to
show data.
• Order data and
find the mode of a
set of data.
• Analyse and
interpret data
in tables,
pictographs, bar
graphs and pie
graphs.
Key words
• data − a collection
of facts, numbers
or measurements
• tallies − marks
made to record
each item when
you are counting
• table −
information
arranged in rows
and columns
Collect, organise and display data
You already know how to collect data and use tallies and tables to
organise the data. You also know how to draw simple pictographs
and bar graphs to show the data.
Example
A teacher asked a Grade 5 class how they learn about the news.
These are the results:
• 14 learners said they watch the news on TV.
• 12 learners said they hear the news on the radio.
• 10 learners said they read newspapers.
• 6 learners said they read magazines.
a ) Organise this data into a tally table.
b ) Draw a pictograph to show the data. Use the symbol to
show 2 learners.
Source of
news
Number Source of
news
of
learners TV
Tally
TV
14
Radio
12
Newspapers
10
Magazines
6
Radio
Newspapers
Magazines
Key:
c ) Show this data as a bar graph.
• pictograph −
graph that uses
symbols (pictures)
to show data
Number of
learners
= 2 learners
Source of News
TV
Radio
Newspapers
Magazines
0
34
2
4
6
8
10
12
14
16
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When you draw bar graphs to show data you will often need to:
Show the axes which are the vertical (up−down) axis and horizontal
(left−right) axis fixed lines.
Use a scale to show the data. A scale is the ratio between the actual
measurements of the object and the measurements of the drawing.
ExErCiSE 7.1
1. Pete chooses the symbol
to show 4 learners. Draw the symbol
Pete would use to show:
a ) 1 learner
b ) 3 learners
c ) 2 learners.
to show 20 people. Draw the
2. Shamila chooses this symbol
symbol Shamila would use to show:
a ) 10 people
b ) 15 people
c ) 4 people.
3. This table shows information about the number of learners who
play different sports. Draw a pictograph to show this data. Use to
show two learners.
Sport
Number of learners
Soccer
Netball
Basketball
Cycling
10
6
7
5
4. Jabu collects 10c, 20c and 50c coins
in a jar. When Jabu counts the coins
he collected this is what he had:
a ) Draw up a tally table to
organise this data.
b ) Draw a pictograph to show
to
the data. Use a symbol of
show 3 coins.
c ) Draw a bar graph to show the same data. Use a scale of
1 cm = 5 coins on the vertical axis.
d ) Do your own investigation to find out what coins learners
have on them at school.
• Ask at least five learners.
• Record and organise the results.
• Draw a suitable graph to show the results.
Key words
• axis − one of
two fixed lines
in a graph; it can
be either vertical
(up−down) or
horizontal (left−
right)
Challenge
a) Plan and carry out
an investigation
to find out what
colour is most
common in a
pack of coloured
sweets.
b) Decide how you
will find out.
c) Record and
organise the data.
d) Draw a
pictograph to
show your results.
5. Write down the first names of 10 learners in your class.
a ) Make a tally table to show how many letters there are in
each name.
b ) Draw a bar graph to show your data.
Topic 7: Data handling
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Key words
Order data and find the mode
• mode − the data
value that occurs
most often
When you work with data it is sometimes useful to know which data
value is the smallest and which is the largest.
• modal − relating
to the mode
It is also useful to know the mode of the data. The mode is the data
value that occurs most often.
Example
This table shows the number of learners in six Grade 5 classes.
Class
Number of learners
A
B
C
D
E
F
28
34
29
34
30
34
Here are the numbers written in order from smallest to largest:
28; 29; 30; 34; 34; 34
• You can see that the smallest number of learners is 28.
• The largest number of learners is 34.
• There are 3 classes that have 34 learners.
• 34 is the number that occurs most often. It is the mode of
the data.
ExErCiSE 7.2
1. A teacher counted the number of different coloured crayons in the
art classroom. This is how she recorded her results:
a ) Draw up a table to organise the results.
b ) Write the results in order from smallest to largest.
c ) What is the mode of this data?
Challenge
What do you think
the most common
shoe size is among
your friends?
a) Collect data to
find out.
b) Order the data
and find the
mode.
c) How well did
you estimate the
mode?
36
2. Some Grade 5 learners did an experiment to see how many
counters they could pick up in one hand. These are their results:
Lilia 14
Paul 16
Annika 12
Bhusi 12
a)
b)
c)
d)
e)
Vusi 18
Tiny 16
Noni 15
Petrus 17
Pumla 18
Sihle 13
Kobus 15
Ziggy 18
Carol 17
Jess 18
Peter 14
Saul 16
Mandy 12
Josh 16
Simeon 18
Zanele 12
Palesa 11
Nina 18
Helene 17
Julius 12
Arrange the results in order from smallest to largest.
How many learners could hold less than 15 counters?
How many learners could hold 16 or more counters?
What is the modal number of counters?
Which learners picked up the modal number of counters?
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Work through a data cycle
Read this flow 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
5. Interpret and
summarise
the data
3. Record and
organise the
data
Key words
• data cycle
− process of
asking questions,
collecting and
organising data
and summarising
results
4. Represent
the data
graphically
ExErCiSE 7.3
You are going to find out which of the following colour
combinations is the favourite for the Grade 5 learners at
your school.
Black and White
1. Carry out your survey
Choose 20 Grade 5 learners to answer your questions.
Try to choose a mixture of boys and girls.
Blue and White
2. Organise and record your data
Use a tally sheet to record the answer to each question
in your survey. Your tally sheet could look something
like this:
Colour combination
Yes
No
3. Draw a bar graph to show how many learners chose
each colour combination.
Blue and Green
Red and Yellow
Blue and Yellow
4. Interpret your data
Write a short paragraph about what you found out during your
survey. Your paragraph should say which colour combination is
the favourite and which is the least favourite.
Topic 7: Data handling
Platinum Maths Gr5_Term 1_CAPS.indd 37
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11/02/13 10:41 AM
Did you know?
When you interpret
a graph you should:
• Read the heading.
• Check if there is a
key.
• Read the labels
and scale on the
axes.
interpret and analyse data
Data is shown in many different ways. You already know how to read
and interpret data in words, tables, pictographs, bar graphs and
pie charts.
Example
These three graphs show the same data in different ways.
Floyd’s Day
10
Floyd’s Day
• Look at the
patterns in the
data. Sleeping
Eating
7
School
6
Playing
5
Travelling
4
TV
3
Homework
Ot
he
r
ll
ve
Tra
Sleeping
8
Hours
• Summarise the
data in a sentence
Eating rk
School
ewo paragraph.
m
Ho or ia
ng
Floyd’s Day
9
TV
Playing
2
Other
1
0
Key:
Sleeping
Eating
School
Playing
Travelling
TV
Homework
= 2 hours
Other
Floyd’s Day
Activity
10
Floyd’s Day
9
8
7
Sleeping
School
5
4
3
Ot
he
r
Eating rk
ewo
Hom ling
l
ve
Tra
Hours
6
TV
Playing
2
1
0
Sleeping
Eating
School
Playing
Travelling
Activity
All the graphs show how many hours Floyd spent on different
activities:
• The bar graph has a vertical scale marked in hours. The length of
the bars shows how much time he spent on each activity.
• The pictograph uses a square to show two hours. Half a square
represents one hour, a quarter of a square is half an hour.
• The pie chart divides his day into fractions. The slices of the pie
show what fraction of the day he spent doing each activity.
From these graphs you can see that:
• Floyd spent most of the day sleeping.
• Floyd spent a quarter of his day at school (6 hours).
• Floyd spent the least time on ‘other’ activities.
38
TV
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Homework
ExErCiSE 7.4
1. Two game rangers were asked to count and record the number of
different types of animals they saw over a weekend. Sipho drew a
graph to show his data. Marius wrote a note.
Animals in a National Park
Type of animals seen
Buck
I went on five game drives this
weekend. We saw lots of buck, some
elephants and rhinos and a large
pride of lions.
I’d estimate we saw about 50 buck in
different locations, a herd of about
24 elephants, 12 rhino and four prides
of lions (26 in total).
Rhinoceros
Lion
Elephant
0
60
Which data is easier to interpret? Why?
How many buck did Sipho see?
Which type of animal was seen most often?
Which type of animal was seen least often?
Draw a pictograph to show Marius’ data. Use a key of one circle
for every four animals seen.
7
6
5
4
3
2
Topic 7: Data handling
Platinum Maths Gr5_Term 1_CAPS.indd 39
Eastern Cape
Northern Cape
North West
Mpumalanga
Limpopo
Kwazulu-Natal
Gauteng
0
Free State
1
Western Cape
2. Study this graph carefully.
a ) What type of graph is this?
b ) What does the graph show?
c ) What scale is used on the vertical axis?
d ) Which province has the most schools?
e ) Which province has the fewest schools?
f ) Which provinces have more or less the same
number of schools?
g ) Why do you think there are different
numbers of schools in different provinces?
h ) Is it easy to work out exactly how many
schools there are in each province from this
graph? Explain why or why not.
i ) Write a sentence describing how your
province compares with the other provinces
in terms of the number of schools.
Number of Schools
(Thousands)
a)
b)
c)
d)
e)
10 20 30 40 50
Number of animals seen
39
11/02/13 10:41 AM
ExErCiSE 7.5
Key words
• census − a
government count
of the whole
population
Challenge
Study this
pictograph.
Type of energy used for cooking
Electricity
Paraffin
Wood
Coal
Gas
Key:
= 12 families
b) Exchange
questions with
a partner. Try
to answer each
other’s questions.
c) Write a paragraph
summarising
what this graph
tells you.
Traditional
dwellings
Oth
er
Informal
dwellings
Formal dwellings
2. The data for the pie chart above
was collected by Census at
school. It included nearly 800 000
learners. Say whether these statements are true or false.
a ) About 600 000 learners live in formal dwellings.
b ) Around 400 000 learners live in traditional homes.
c ) Less than 100 000 learners live in informal settlements.
d ) More than 20 000 learners live in ‘other’ dwellings.
Type of energy used for light
3. This pie chart shows the main
energy source that South African
families use for light in their
homes.
a ) Complete these sentences
Candles
about the graph.
Electricity
• Almost □ of homes use
electricity for light.
• Nearly □ of homes use
candles for light.
• Less than □ of homes use
paraffin for light.
b ) This data was collected in
2007. Do you think the graph would be the same if the data
was collected today? Give a reason for your answer.
Paraffin
a) Make up five
questions that
can be answered
from the graph.
1. Study this pie chart carefully.
a ) What is the title of this graph?
b ) Approximately what fraction
of learners live in formal
dwellings?
c ) Approximately what fraction
of learners live in informal
dwellings?
4. The data in the pie chart in Question 3 is for the whole country.
Explain how and why the graph might be different if the data was
collected from the following sources:
a ) A large city
b ) A farming community in the North West
c ) An informal settlement in a poor area
d ) A very wealthy community
40
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Think about where data comes from
Key words
When you interpret data you have to think about:
• the source − where does the data come from?
• the context − where and how was the data collected?
• source − the
people or places
that data comes
from
Example
• context − where
and how data was
collected
Amina and David did a survey of 20 learners to find out how many
were sick at least once during the term.
• Amina’s results show that __14 of the learners were sick at least once.
• biased − skewed
or distorted
9
of the learners were sick at least once.
• David’s results show that __
10
Why are their results so different?
• Amina asked boys and girls from different Grade 5 classes.
Only 5 learners said that they had been sick during the last term.
• David only asked learners who had been absent (away) from
school during the term. The source of his data was biased because
learners who were absent are more likely to have been sick.
Example
Josh draws a graph that shows that __45 of the learners in his class love watching the school team play
soccer. Pume disagrees and she asks Josh what the source of his data was and how it was collected.
Josh says he asked five of his friends after the school soccer team won by 4 goals to 0. Four friends
said they love watching the school team play soccer, so Josh drew a graph to show this.
The source of Josh’s data was his five friends. This is a small group and he also asked his friends
after a match when the school team won. This means that the context and source of Josh’s data
were biased.
ExErCiSE 7.6
Study these two pictographs.
1. What is the source of the data for each graph?
2. How does the context in which each set of data
was collected change the results?
3. What do you think the results would be if
you did a similar survey in your community?
Why?
Diepsloot (Soweto)
Key:
= 4 households
Key:
= 4 households
House
Shack
Backyard
Rural family
compound
Lugangeni (Rural Eastern Cape)
House
Shack
Backyard
Rural family
compound
Topic 7: Data handling
Platinum Maths Gr5_Term 1_CAPS.indd 41
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11/02/13 10:41 AM
Topic
Properties of 2D shapes
8
Maths ideas
• Recognise and
name twodimensional (2D)
shapes.
• Describe and
compare twodimensional (2D)
shapes.
• Draw twodimensional (2D)
shapes on grid
paper.
• Name twodimensional (2D)
shapes.
Key words
• two-dimensional
(2D) − having two
dimensions: length
and width
• three-dimensional
(3D) − having
three dimensions:
length, width and
height
What is a 2D shape?
Two-dimensional (2D) shapes have two dimensions: length and
width (or breadth).
You live in a three-dimensional (3D) world, so everything that you
see around you has three dimensions: length, width (or breadth) and
height. However, if you were to take a photograph or draw a picture of
a 3D object, you would have a 2D view of that object.
You learn much about the world
around you from pictures, maps
and diagrams. These are all
This banner has some curved
examples of 2D representations
sides and some straight sides
of 3D objects.
• Some shapes have only curved
sides.
• Some shapes have curved and
straight sides.
• There are also shapes with only
straight sides. These shapes are A circle has only
curved sides
called polygons.
polygon
We use little lines like the ones on the picture to
show that the sides of 2D shapes are equal in
length. These little lines are very useful when we
have to identify 2D shapes.
ExErCiSE 8.1
3D
• polygon − a
2D shape that is
enclosed by three
or more straight
lines
42
Draw two examples of the following 2D shapes.
1. Shapes that have only curved sides, but are not circles.
2. Shapes with some curved sides and some straight sides.
3. Shapes that have only straight sides.
Term 1
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11/02/13 10:41 AM
identify polygons
Key words
When a 2D shape has only straight sides, you count the number of
sides to name the shape.
• triangle − a
polygon with three
sides
Naming polygons
Some examples
• quadrilateral − a
polygon with four
sides
Three-sided polygons are called
triangles
• square − a
rectangle with
all sides equal in
length
Four-sided polygons are called
quadrilaterals. Squares and
rectangles are examples of
quadrilaterals
• rectangle − a
quadrilateral
where all the
angles are right
angles
Five-sided polygons are called
pentagons
• pentagon − a
polygon with five
sides
Six-sided polygons are called
hexagons
• hexagon − a
polygon with six
sides
Seven-sided polygons are called
heptagons or septagons
• heptagon or
septagon − a
polygon with
seven sides
ExErCiSE 8.2
1. Study this picture with a friend.
a ) Identify and draw all the shapes
with only curved sides.
b ) Identify and draw all the shapes
with some curved and some
straight sides.
c ) Identify and draw all the
polygons in the picture.
d ) Name each polygon that you
drew.
2. Count how many polygons you can find in each of these sketches.
Which polygons can you name?
a)
b)
c)
Did you know?
A ten-sided polygon
is called a decagon
in the same way as
10 years is called a
decade.
Topic 8: Properties of 2D shapes
Platinum Maths Gr5_Term 1_CAPS.indd 43
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11/02/13 10:41 AM
Key words
• angle − the
amount of turn
between two
straight lines that
meet each other
Polygons and angles
Polygons do not only have sides, they also have angles. The angle
between two sides that join is the amount of turn between the two sides.
• right angle − a
quarter of a full
turn
angle
One important angle is a right angle. A right angle is one quarter of a
full turn. The corner of a page is a right angle.
A full turn
A right angle (a quarter of a full turn)
The little square is the symbol for
a right angle.
You will also work with angles that are smaller than one quarter of a
full turn, and angles that are larger than one quarter of a full turn.
This angle is larger than a right angle.
ExErCiSE 8.3
1. Write down four examples of where you can see a right angle in
your classroom. If you need to check the angle, see if it is the same
shape as the corner of a sheet of paper.
44
Term 1
Platinum Maths Gr5_Term 1_CAPS.indd 44
2. Open a cupboard door to an angle less than a right angle; then
open the door to a right angle and lastly open the door to an
angle larger than a right angle. Get a friend to check and see if
they agree with you.
11/02/13 10:42 AM
3. State which of the marked angles in these triangles are:
smaller than a right angle, a right angle, or larger than a right angle.
If you are unsure of the answer, use the corner of a page to
measure the size of the angle.
a)
b)
c)
A polygon can have all its sides equal and all its angles equal. Explore
this idea in the next exercise.
ExErCiSE 8.4
Matchsticks in a box of matches are all the same length.
1. Form a polygon with three matchsticks.
a ) What is this polygon called?
b ) Does this polygon have equal sides? Does it have equal angles?
Explain your answer.
c ) Classify the angles of this polygon as less than a right angle,
a right angle, or larger than a right angle.
2. Form a polygon with four matchsticks so that one angle is
a right angle.
a ) What is this polygon called?
b ) Does this polygon have equal sides and angles? Explain.
3. Form a quadrilateral with matchsticks, so that two of the angles
are smaller than right angles.
Key words
• composite − a 2D
shape that is made
up of two or more
2D shapes that are
joined together
4. Use five matchsticks to make a house shape.
a ) What is this polygon called?
b ) Does this polygon have equal sides? Does it have equal angles?
c ) Classify the angles in this shape.
d ) This polygon is a composite shape, which is made up of two
other shapes. Use one more matchstick to separate it into two
smaller shapes. Name the two smaller shapes.
e ) Rearrange your first five matchsticks into a shape where all the
angles are the same size. Classify the angles in the shape.
5. Use your polygon in Question 4e, and make new polygons by
adding in one more matchstick each time. When the polygon has
more and more sides, what is the name of this shape?
Topic 8: Properties of 2D shapes
Platinum Maths Gr5_Term 1_CAPS.indd 45
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11/02/13 10:42 AM
Describe and draw 2D shapes
It is important to be able to compare different shapes. You need to say
what is the same and what is different about each shape.
Example
Look at the diagrams below.
In which ways are these shapes
the same? Both are 2D shapes.
Both are polygons. Both have all
their sides of equal length.
How are these shapes different?
The triangle has three sides and three angles. The square has four sides
and four right angles.
ExErCiSE 8.5
1. a ) On a sheet of square dotted grid paper, draw two different
rectangles and one square.
b ) Compare your rectangles. How are they the same and how are
they different?
c ) Count the number of blocks that form each side of one of your
rectangles. What do you notice?
d ) Count the number of blocks that form each side of your other
rectangle. What do you notice?
e ) In which ways is the square the same as the rectangles? How is
it different?
A
B
46
2. Look at these two composite shapes.
a ) Which shapes make up both composite shapes?
b ) Compare the rectangles used in Shape A and Shape B. What
do you notice?
c ) Compare the triangles in Shape A and Shape B. How are they
the same and how are they different?
d ) Pretend that the composite shapes are real fields that you can
walk around. How many sides would each field have?
e ) On square dotted grid paper, draw any two composite
shapes made from the same basic shapes. Compare the two
composite shapes. How are they different and how are they
the same?
Term 1
Platinum Maths Gr5_Term 1_CAPS.indd 46
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revision
1. Match each term with the correct definition.
Term
(10)
Definition
a) Bar graph
A. A mark made to record one fact when counting
b) Table
B. The axis that runs across the page from left to right
c) Categories
C. A graph that uses bars to represent data
d) Data
D. A graph that uses pictures to represent data
e) Graph
E. Information arranged in rows and columns
f ) Horizontal axis
F. The axis that runs up and down the page
g) Pictograph
G. The different groups into which we can divide data
h) Tally
H. A collection of facts, numbers or measurements
i) Vertical axis
I. A diagram that represents data
2. a ) Represent the data in the table below in a bar graph.
b)
c)
d)
e)
School
Number of learners who cycle to school
A
B
C
D
E
F
40
25
50
30
25
50
(4)
What information is shown on the horizontal axis of your graph?
What information is shown on the vertical axis of your graph?
What is the mode of this data?
How many learners cycle to school altogether?
(1)
(1)
(1)
(1)
3. Which one is the odd one out in this list of 2D shapes: triangle; quadrilateral; circle;
square; polygon? Explain
(2)
4. I am a polygon with four angles, all of which are right angles. What am I?
(1)
5. A 2D shape has two dimensions. What do we call these dimensions?
(2)
6. I am a 2D shape enclosed by a single curved line. All the points on the line are the same
distance from my centre. What am I?
(1)
7. What fraction is a right angle of a full turn?
(1)
Total marks: 25
Revision
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Topic
9
Capacity and volume
Maths ideas
• Estimate, measure,
record, compare
and order
capacities of
different objects.
• Use measuring
instruments
correctly.
• Understand and
use different
measuring scales.
• Calculate and
solve problems
involving millilitres
and litres.
Key words
• capacity − the
maximum amount
an object can hold
• volume − how
much space an
object takes up
Estimate capacity
You worked with capacity and volume in Grade 4. Do you remember
the difference between them?
• Capacity is the maximum 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.
Remember that containers are not always filled
to the top. A bottle 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
cooldrink in the bottle is __12 litre.
The capacity of larger containers is given in
litres and the capacity of smaller containers is
given in millilitres. You should remember that
there are 1 000 ml in 1 ℓ.
ExErCiSE 9.1
1. State whether you would measure the following in litres
or millilitres.
a ) a full tank of petrol
b ) a baby’s bottle of milk
c ) the nail polish in a bottle
d ) water in a bath
e ) paraffin in a drum
f ) dish-washing liquid in a bottle
g ) medicine to give a baby
h ) water in a rainwater tank
2. Look at these containers.
Did you know?
You may hear people
talk about a capacity
crowd at a sports
match. This means
that every seat is
filled, as the capacity
of a stadium is the
number of seats it
has.
48
a ) Write the capacity of each container.
b ) Estimate the volume of liquid in each container.
Term 1
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Estimate,measure and record capacity
You can use a measuring jug to measure exact amounts. Do you
remember how to read the jug correctly?
The scale on measuring instruments is divided into spaces like a
number line. Each interval represents a fraction of the larger units.
0
500 ml
1ℓ
The above scale is divided into intervals of 100 ml. Only the 500 ml
mark and the 1 ℓ mark are labelled. When you use a measuring jug
you need to work out how much each interval represents.
Example
How much liquid is in each of these 1 ℓ measuring jugs?
On this scale each line represents 250 ml
or __14 of a litre.
On this scale each line represents 100 ml
1
or __
of a litre.
10
There is 500 ml in the jug.
There is 700 ml in the jug.
Step 1: Put the jug on a
flat surface.
Step 2: Put your eye level
with the top liquid in the
jug.
Step 3: Read the
numbered intervals
carefully.
ExErCiSE 9.2
1. Copy these scales into your book. Fill in all the missing measurements.
1ℓ
1 000 ml
1ℓ
ml
ml
ℓ
0 ml
1 000 ml
ml
ℓ
ml
ml
1
-2 ℓ
0ℓ
ℓ
ml
ml
250 ml
0ℓ
1 000 ml
ℓ
200 ml
0ℓ
0ℓ
2. a ) What volume of liquid does each jug contain? Read the scales
carefully as they are all different.
b ) Write the measurements in order from smallest to greatest.
Topic 9: Capacity and volume
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Compare, order and record capacities
Did you know?
Some of the
containers you will
work with may have
their capacity listed
in decimal form. For
example, you get
1,5 ℓ bottles of cold
drink. This means the
same as 1__12 litres.
The capacity of a container can be written in different ways:
• In litres, it can be written as 2__12 ℓ
• In millilitres, it can be written as 2 500 ml.
• In litres and millilitres, it can be written as 2 ℓ and 500 ml
It is easier to compare the capacities of containers when they are all
written in the same way.
Example
Look at these containers.
a ) Which has the greatest
capacity?
b ) Which holds the smallest
amount?
c ) List the capacities in order
from smallest to greatest.
Answer
a ) The oil drum has the greatest capacity. It can hold 220 litres.
b ) The teaspoon. It has a capacity of 5 ml.
c ) 5 ml; 400 ml; 1 ℓ 200 ml; 1,5 ℓ; 15 ℓ; 220 ℓ
ExErCiSE 9.3
1. Look at these containers.
F
E
D
B
A
100 ml
of eye
drops
C
500 ml of
juice
250 ml of
yoghurt
1 ℓ of
milk
1,5 ℓ of
cola
5 ℓ of
paraffin
a ) Which container has the greatest capacity?
b ) Which containers hold less than 1 ℓ?
c ) Rewrite the capacity of each container using different units.
2. Here are the capacities of eight different containers:
2 ℓ; 8 ℓ; 3 ℓ and 200 ml; 5__12 ℓ; 6__34 ℓ; 8__14 ℓ; 8 ℓ and 300 ml
a ) Write each measurement using millilitres only.
b ) List the capacities in order from greatest to smallest.
c ) Round each measurement to the nearest litre.
50
Term 1
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Convert units of capacity
When you do calculations and solve problems that involve capacity
you may need to covert measurements from one unit to another.
To convert from litres (the larger unit) to millilitres (the smaller unit)
you need to multiply by 1 000.
• 1 ℓ = 1 000 ml
1
• __ ℓ = 250 ml
4
Example
How many millilitres are there
in 2 ℓ?
1 ℓ = 1 000 ml
2 ℓ = 2 × 1 000 ml = 2 000 ml
• __12 ℓ = 500 ml
• __34 ℓ = 750 ml
A bucket contains 6__14 ℓ of water.
How many millilitres is this?
6__12 ℓ = 6 ℓ and 250 ml
6 ℓ = 6 × 1 000 ml = 6 000 ml
6 000 ml + 250 ml = 6 250 ml
To convert from millilitres (the smaller unit) to litres (the bigger unit)
you need to divide by 1 000.
Challenge
Example
A jug holds 4 000 ml. How many litres is this?
4 000 ml = 4 000 × 1 000 ml = 4 ℓ
You can also think like this: 4 000 ml = 4 × 1 000 ml = 4 litres
You should drink at
least 8 glasses of
liquid a day. One
glass holds about
250 ml of liquid.
ExErCiSE 9.4
1. How many litres
of liquid are there
in 8 glasses?
1. Convert each of these litre capacities to millilitres.
d ) __14 ℓ
a) 2 ℓ
b ) 7ℓ
c ) 3__12 ℓ
e ) 5__34 ℓ
2. Convert each of these millilitre capacities to litres.
a ) 2 000 ml
b ) 3 500 ml
c ) 5 250 ml
d ) 12 750 ml
3. Mrs Smit pours 25 cups of cold drink. Each cup holds 300 ml of
juice. How many litres of juice did she pour?
4. Kanye has 6__12 ℓ of juice. How many 250 ml containers can he fill
2. How many litres
of liquid would
you drink in a
month, if you
drank this amount
each day?
3. How many
millilitres is this?
with this?
Topic 9: Capacity and volume
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Calculate and solve problems that
involve capacity
Do you remember how to add, subtract, multiply and divide
capacities? Read through these examples to remind you.
• You can only add or subtract capacities if they are in the same
units. Convert the units before you start if they are different.
• Always write the correct units in your answer.
Example
Add: 4 ℓ and 450 ml + 3 ℓ and 750 ml
4 ℓ 450 ml
+ 3 ℓ 750 ml
7 ℓ 1 200 ml (1 200 ml = 1 ℓ and 200 ml, carry the 1 ℓ to
the litres column)
So, 4 ℓ 450 ml + 3 ℓ 750 ml = 8 ℓ and 200 ml
Example
Subtract: 12 ℓ 250 ml − 9 ℓ 780 ml
To subtract the millilitres, change 1 ℓ in the litre column to 1 000 ml
and move it to the millilitre column. This means that:
12 ℓ 250 ml = 11 ℓ 1 250 ml.
11 ℓ 1 250 ml
− 9 ℓ 780 ml
2 ℓ 470 ml
So, 12 ℓ 250 ml − 9 ℓ 780 ml = 2 ℓ and 470 ml
Example
Multiply: 25 ml × 15
25 ml × 15 = 2 500 ml + 125 ml = 2 ℓ and 675 ml
Example
Divide: 12 ℓ 450 ml ÷ 3
12 ℓ ÷ 3 = 4 ℓ
450 ml ÷ 3 = 150 ml
So, 12 ℓ 450 ml ÷ 3 = 4 ℓ
and 150 ml
52
Divide: 34 ℓ ÷ 4
34 ℓ ÷ 4 = 8 ℓ remainder 2 ℓ
2 ℓ = 2 000 ml
2 000 ml ÷ 4 = 500 ml
So, 34 ℓ ÷ 4 = 8 ℓ and 500 ml
Term 1
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ExErCiSE 9.5
1. Calculate.
a ) 65 ℓ 450 ml + 9 ℓ 550 ml
c ) 69 ℓ 978 ml + 45 ℓ 750 ml
e ) 56 ℓ 250 ml − 25 ℓ 750 ml
g ) 125 ℓ 607 ml − 98 ℓ 855 ml
b ) 355 ℓ 580 ml + 49 ℓ 890 ml
d ) 25 ℓ 700 ml + 4 900 ml
f ) 45 ℓ 205 ml − 12 ℓ 450 ml
h ) 79 ℓ 5 ml − 35 ℓ 760 ml
2. Calculate. Give your answers in litres and millilitres.
b ) 85 ml × 12
c ) 2 ℓ 100 ml × 9
a ) 31 ℓ × 17
f ) 508 ℓ ÷ 9
d ) 40 ℓ ÷ 4
e ) 380 ℓ ÷ 10
3. Solve these problems. Show all your working.
a ) Janet waters her garden three times a week. Last week she
used 39__12 ℓ, 56 ℓ and 300 ml and 38 000 ml. How much water
did Janet use altogether?
b ) Nomhle has 12__12 ℓ of milk and she sells 8__14 ℓ. How much does
Nomhle have left over?
c ) Solly has seven containers and he pours three 250 ml bottles
of juice into each. How much juice does Solly need for this?
d ) Josh has 4 litres of fruit juice to share equally between six
friends. How much can each friend have?
Challenge
These containers are
all full of milk. Henry
also has an empty
container that has a
capacity of 2,75 ℓ. He
wants to store the
milk in as few
containers as
possible. What
combination of
containers must
Henry use to store
the milk?
4. Round each answer in Question 3 to the nearest litre.
5. A shopkeeper sells fresh orange juice at R12 per litre. People bring
their own containers of different sizes to be filled. How much must
she charge each of these people to fill their containers?
a ) Mrs Ratsoma brings a plastic can that has a capacity of 2,5 ℓ.
b ) Ayeesha brings a glass jar that has a capacity of 400 ml.
c ) John brings two bottles. Each bottle has a capacity of 1,5 ℓ.
d ) Annie brings four plastic cups each with a capacity of 200 ml.
6. A baby must get 5 ml of medicine three times a day for seven days.
If a full bottle of medicine contains 150 ml, how much medicine
will be left over?
7. Nina wants to make a pudding by mixing the pudding powder
with milk. The recipe says use 250 ml of milk per tablespoon of
pudding powder. How much milk does she need if she uses:
a ) 4 tablespoons of pudding powder?
b ) 5__12 tablespoons of pudding powder?
400 ml
Topic 9: Capacity and volume
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2 Term 2
Buildings can be decorated using colourful tiles.
Many of Russia’s churches have beautiful spires
decorated in gold.
Traditional African buildings are decorated using
different natural materials.
54
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Topics 10–18
Starting off
People love to surround themselves with
decorative patterns. We wear clothes made
from patterned materials, we use patterned
cups, saucers and plates, and some of us even
decorate our buildings with different patterns.
In the photographs on the left, you will see
examples of buildings that people have
decorated in different styles.
1. Which pattern do you like most? Why?
2. Choose one of the photographs.
a ) What culture does it represent?
b ) Do you have a building like this in your
town or city?
c ) Would you expect to see a building like
Many traditional Chinese buildings are decorated
this elsewhere in South Africa?
using colourful designs.
d ) Does this building remind you of
another country? If so, which country?
3. Give an example of a building in the
photograph with the following:
a ) A 2D pattern
b ) A 3D pattern
Content covered in Term 2
Religious buildings often have domed roofs that
look like giant umbrellas.
Topic 10: Count, order, compare and represent whole numbers, Topic 11: Addition
and subtraction, Revision, Topic 12: Common fractions, Topic 13: Length,
Revision, Topic 14: Multiplication, Topic 15: Properties of 3D objects, Revision,
Topic 16: Geometric patterns, Topic 17: Symmetry, Revision,
Topic 18: Division
55
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Count, order, compare and
represent whole numbers
Topic
10
Maths ideas
• Read and write
numbers up to
six digits.
• Count forwards
and backwards to
10 000.
• Round whole
numbers to the
nearest 10, 100 and
1 000.
• Compare and
order numbers.
Count, read and write numbers
In this topic, you will practise working with larger numbers.
Example
You can show the number 321 456 in a place value table like this.
Hundred
Ten
Thousands Hundreds Tens Units
Thousands Thousands
3
2
1
4
5
6
In expanded form: 300 000 + 20 000 + 1 000 + 400 + 50 + 6
3 2 1 4 5 6
You read this in groups as:
three hundred and
four hundred and
twenty one thousand,
fifty-six.
ExErCiSE 10.1
1. Write these numbers in digits.
a ) five hundred and fifteen thousand, six hundred and eighteen
b ) eighteen thousand, six hundred and two
c ) eight hundred thousand, three hundred and fifty-two
2. Write these numbers in expanded form and then in words.
a ) 47 561
b ) 386 276
c) 129 560
d) 50 600
3. Count forwards or backwards to complete these number sequences.
Read the numbers first to decide what the intervals are.
a ) 2 500; □; 2 600; 2 650; □; 2 750; □
b ) □; 9 055; □; 9 255; □; 9 455
c ) 1 483; 1 478; □; 1 468; 1 463; □; □
d ) 1 260; 1 310; □; 1 410; □; 1 510; □
4. Write down the number that is 2 more than each of these:
a ) 4 009
b ) 9 998
5. Write down the number that is 25 less than each of these:
a ) 7 000
b ) 6 405
6. Write down the numbers that are 100 more and 100 less than
each of these:
a ) 2 978
b ) 9 019
56
Term 2
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round off and compare whole
numbers
You already know that rounding numbers is useful when you need to
estimate an answer. When you round a number, look at the digit to the
right of the place you are rounding to. If the digit is 5 or more, round
up. If it is less than 5, round down.
Example
Round off 46 185.
• To the nearest 10: 46 185 becomes 46 190, because 5 rounds up.
• To the nearest 100: 46 185 becomes 46 200, because 8 rounds
up.
• To the nearest 1 000: 46 185 becomes 46 000, because 1 rounds
down.
ExErCiSE 10.2
1. Round each number to the nearest 10, 100 and 1 000:
a ) 3 873
b ) 7 147
c ) 4 719
d ) 35 762
e ) 14 625
f ) 28 411
g ) 56 043
h ) 49 938
2. There are 29 371 people living in Makgabana. Round this number
to the nearest 1 000.
3. 50 708 people visited the aquarium last month. Round this
number to the nearest 1 000.
Remember that < means smaller than, and > means larger than.
ExErCiSE 10.3
Challenge
In 2005, the
population sizes of
some towns in South
Africa were as
follows:
1. Give the place value of each digit in the following numbers.
a ) 215 873
b ) 850 147
c ) 40 719
Ceres: 22 302
2. Write these numbers in ascending order.
Port Elizabeth: 737 658
456 043; 458 411; 514 625; 549 938
3. Use the symbols < and > to compare these numbers.
a ) 324 009 □ 329 999
b ) 189 000 □ 198 000
c ) 298 880 □ 298 980
d ) 76 000 □ 67 405
Richards Bay: 69 482
Welkom: 199 972
Round each number
to the nearest 100
and to the nearest
1 000.
Topic 10: Count, order, compare and represent whole numbers
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Topic
Addition and subtraction
11
Maths ideas
• Round numbers to
estimate answers.
• Use the inverse
relationship of
addition and
subtraction to
check answers
and find missing
numbers.
• Solve multi-step
problems that
involve addition
and subtraction,
with totals to
100 000.
Add whole numbers
In Topic 3 you used brackets to organise numbers by place value when
you add. In this topic, you will organise your numbers by writing them
in expanded form in rows or in columns. You will also round off the
numbers to estimate answers.
Example
Calculate 56 423 + 18 269 + 21 899
First estimate the answer by rounding off to the nearest 1 000.
Look at the hundreds digit to round up or down:
56 423 rounds down to 56 000 because 4 is less than 5.
18 269 rounds down to 18 000 because 2 is less than 5.
21 899 rounds up to 22 000 because 8 is more than 5.
The estimate is 56 000 + (18 000 + 22 000) = 56 000 + 40 000 = 96 000.
To find the exact answer then using place values.
(50 000 + 6 000 + 400 + 20 + 3) + (10 000 + 8 000 + 200 + 60 + 9)
+ (20 000 + 1 000 + 800 + 90 + 9)
= (50 000 + 10 000 + 20 000) + (6 000 + 8 000 + 1 000) +
(400 + 200 + 800) + (20 + 60 + 90) + (3 + 9 + 9)
= 80 000 + 15 000 + 1 400 + 170 + 21
= 95 000 + 1 000 + 500 + 70 + 20 + 1
= 96 000 + 500 + 90 + 1
= 96 591
The estimate was quite accurate!
ExErCiSE 11.1
Follow the method in the example to add these numbers.
58
1. 15 356 + 18 790
2. 18 233 + 21 876
3. 24 965 + 21 454
4. 22 879 + 31 489
5. 35 212 + 27 705
6. 38 462 + 19 360
Term 2
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To add numbers, you can also write the expanded forms in columns. It
is important to write the same place values below each other. This is a
useful way to add large numbers.
Example
26 235 + 15 469 + 33 502
First estimate the answer by rounding off the numbers to the nearest 1 000.
The estimate is 26 000 + 15 000 + 34 000 = 15 000 + (26 000 + 34 000) = 15 000 + 60 000 = 75 000.
Now expand the numbers, and write the same place values below each other in columns.
26 235 = 20 000 + 6 000 + 200 + 30 + 5
15 469 = 10 000 + 5 000 + 400 + 60 + 9
33 502 = 30 000 + 3 000 + 500 + 0 + 2
Total
60 000 + 14 000 + 1 100 + 90 + 16
= 60 000 + 10 000 + 4 000 + 1 000 + 100 + 90 + 10 + 6
= 70 000 + 5 000 + 200 + 6 = 75 206
You can also use the adding on method when you add only two
numbers. Break down the second number and add it in parts.
Example
Find 20 415 + 13 425
20 415 + 13 000 □ 33 415 + 400 □ 33 815 + 20 □ 33 835 + 5 □ 33
840
ExErCiSE 11.2
1. First round off the numbers to the nearest 1 000 to estimate each
answer. Then write the expanded numbers in columns and work
out the exact answers.
a ) 21 781 + 18 324
b ) 29 021 + 18 569
c ) 26 451 + 24 357
d ) 32 864 + 22 891
e ) 45 312 + 39 119 + 23 456 f ) 46 242 + 38 975 + 98 123
g ) 63 274 + 41 802 + 28 282 h ) 55 323 + 49 461 + 19 987
2. Use the adding on method to check your answers to Questions 1a)
to d). Break down the number that you are adding, and add it in
parts.
Challenge
Find the missing
numbers. Write down
three different
answers for each
number sentence.
The missing numbers
must be whole
numbers.
1. 120 − □ − □ = 56
2. 25 + □ + □ = 150
3. □ + □ − 45 = 200
3. Complete the following number sentences. Then write down what
you notice.
a ) 602 + 0 = □
b ) 0 + 4 119 = □
c ) □ + 786 = 786
Topic 11: Addition and subtraction
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Subtract whole numbers
You can also subtract large numbers by writing them in expanded form.
Example
845 − 376
= (700 + 130 + 15) − 300 − 70 − 6
= (700 − 300) + (130 − 70) + (15 − 6)
= 400 + 60 + 9
= 469
Example
48 534 − 37 833
48 534 − 37 833 = 40 000 + 7 000 + 1 500 + 30 + 4
(Write 8 500 as 7 000 + 1 500 to subtract 800.)
− 30 000 + 7 000 + 800 + 30 + 3
Difference
10 000 + 0 + 700 + 0 + 1 = 10 701
This method helps you to keep track of all of your numbers.
Example
88 743 − 54 684
Break down 743 into 600 + 130 + 13 so that you can do the
subtractions.
88 743 − 54 684 = 80 000 + 8 000 + 600 + 130 + 13
− 50 000 + 4 000 + 600 + 80 + 4
Difference
30 000 + 4 000 + 0 + 50 + 9 = 34 059
ExErCiSE 11.3
First estimate each answer by rounding off the numbers to the nearest
1 000. Then expand the numbers in a row to find the exact answer.
60
1. 28 564 − 17 341
2. 32 685 − 21 241
3. 43 680 − 21 520
4. 54 936 − 31 914
5. 46 286 − 24 165
6. 61 356 − 41 242
Term 2
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You can also break down the second number and subtract in parts.
Example
Notice how you write the brackets.
25 746 − 14 582
= 25 746 − 10 000 − 4 000 − 500 − 80 − 2
= (15 746 − 4 000) − 500 − 80 − 2
= (11 746 − 500) − 80 − 2
= (11 246 − 80) − 2 Break down the number 246.
= 11 000 + 100 + 140 + 6 − 80 − 2
= 11 000 + 100 + 60 + 6 − 2
= 11 164
ExErCiSE 11.4
First estimate each answer by rounding off the numbers to the nearest
1 000. Then subtract as the example above. Finally, use addition to
check your answer.
1. 26 403 − 18 351
2. 31 786 − 19 694
3. 43 970 − 25 884
4. 49 034 − 29 556
5. 53 486 − 25 696
6. 58 003 − 38 426
Remember that addition and subtraction are inverse operations. You can
use one operation to check the answer of the other. In the above example,
25 746 - 14 582 = 11 164. You can check this by doing two addition
calculations: 11 164 + 14 582 = 25 746, or 11 164 + 25 746 = 14 582.
ExErCiSE 11.5
1. Estimate your answer, then use any method to complete these
subtraction sums. Check your answers by addition calculation.
a ) 21 390 − 15 285 b ) 36 345 − 19 278 c ) 39 598 − 27 389
d ) 52 167 − 37 191 e ) 61 290 − 42 181 f ) 80 000 − 49 866
2. Find these missing numbers. Write down the inverse statement for
checking each answer.
a ) 1 786 + □ = 1 645
b ) □ + 3 983 = 11 376
c ) 12 659 − □ = 8 732
d ) 31 548 − □ = 2 736
e ) 73 975 − □ = 51 864
f ) 46 434 = □ + 33 281
Topic 11: Addition and subtraction
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Solve addition and subtraction
problems
Remember to show all your working when you solve a problem.
Example
Sipho is raising money for his school. He wants to raise a total of
R18 000. After four fundraising events, he has R9 725. How much
more money does he need to reach his target?
Find R18 000 − R9 725. Break down 18 000 until you can subtract
all the place values.
18 000 = 10 000 + 7 000 + 1 000 = 10 000 + 7 000 + 900 + 90 + 10
18 000 = 10 000 + 7 000 + 900 + 90 + 10
− 9 000 +
700 + 20 + 5
Answer 1 000 + 7 000 + 200 + 70 + 5
= R8 275
So, Sipho needs to raise R8 275 more.
Use addition to check your subtraction answer:
8 275 + 9 725 = (8 000 + 9 000) + (200 + 700) + (70 + 20) + (5 + 5)
= 17 000 + 900 + 90 + 10
= 17 000 + 1 000
= 18 000
Challenge
Make up some more
word problems of
your own. Use
five-digit whole
numbers. Work out
the answers to your
problems.
ExErCiSE 11.6
1. After reading three books during the holidays, I had read a total of
1 697 pages. If the first two books add up to a total of 948 pages,
how many pages does the third book have?
2. Find the difference between 28 404 cm and 19 212 cm.
3. Mpho’s father is saving up for a holiday overseas. The total amount of
money needed for the family of four is R58 950. He has managed to
save R31 920. How much more money does he need to save?
4. Kyle’s uncle is a farmer. He has collected 12 023 bags of wheat after
the harvest. If he sells 6 459 bags at the market and later another
2 772 bags to the bread mills, how many bags does he have left?
5. A tourist bus must make a journey of 25 500 km. In the first week it
travels 8 700 km and in the second week it travels 11 250 km. How
many kilometres must it still travel?
62
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revision
1. Write down the next four numbers in each of the following. Read through the numbers to
work out what the intervals are.
a ) 23 405; 23 410; 23 415; □; □; □; □
(2)
b ) 75 334; 75 324; 75 314; □; □; □; □
(2)
2. Write down the numbers that are 7 less and 7 more than each of these numbers.
a ) 33 556
b ) 102 378
(1)
(1)
3. Write down the following number in digits, then write down the number that is 100 more
than this number.
one hundred and twenty six thousand, nine hundred and forty-nine.
(2)
4. Complete the following table. Round off the numbers to the nearest 10, 100 and1 000.
Town
(3)
Population Nearest Nearest Nearest
size
10
100
1 000
Middlefort
65 785
Klawer
154 435
5. Find these totals, using any method.
a ) 18 456 + 23 154
b ) 37 344 + 25 467 + 30 452
(2)
(2)
6. Find the following differences using any method.
a ) 37 895 − 21 344
b ) 42 044 − 28 627
(1)
(1)
7. This year, 85 604 people visited Cape Point. This is 17 342 more than last year. How many
people visited Cape Point last year? Use an inverse operation to check your answer.
(3)
Total marks: 20
Revision
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Topic
Common fractions
12
Maths ideas
Name and recognise fractions
• Name and
recognise
fractions.
Fractions are numbers. They have a
specific place on the number line.
• Count in fractions.
• Recognise
equivalent
fractions.
• Compare and
order fractions.
• Make fractions
through grouping
and sharing.
3
is a common fraction. You can
The number __
4
3
__
see 4 of a cake on the left.
• Solve problems
using fractions.
1. Copy these diagrams.
number below a
fraction line which
shows how many
parts the whole
has been divided
into
• numerator − the
number above
a fraction line
showing the
number of parts of
the whole
64
0
1
2
3
__
4
1
__
4
ExErCiSE 12.1
• denominator − the
4
• The denominator in this fraction is 4. The
cake is divided into 4 equal parts.
• The numerator in this fraction is 3. There
are 3 of the 4 pieces left.
• Add fractions
with the same
denominator.
Key words
3
__
a ) What fraction is shaded in each diagram?
b ) Write the fractions in words.
c ) What fraction of each diagram is not shaded?
2. Fill in the missing words.
In the fraction __38 , the whole has been divided into □ equal parts.
The denominator is □ and the numerator is □.
3. Fill in the missing numbers.
□
a ) 1 banana = 6 of the whole bunch.
b ) □ bananas = __56 of the whole bunch.
□
c ) 6 bananas = 6 of the whole bunch.
d ) □ bananas = __12 of the whole bunch.
Term 2
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Equivalent fractions
When a fraction is equal to another fraction, we say the fractions are
equivalent. This means that they have the same value.
ExErCiSE 12.2
1. Use fraction strips to help you answer the following.
a ) How many sixths do you need to make up a third?
b ) How many sixths make two-thirds?
c ) How many tenths make a fifth?
d ) How many tenths make a half?
e) How many ninths do two-thirds make?
2. Use the fraction wall to complete these equivalent fractions.
1
Key words
1
__
2
1
__
2
1
__
3
1
__
3
1
__
4
1
__
3
1
__
4
1
__
5
1
__
4
1
__
5
1
__
6
1
__
7
1
__
8
1
__
8
1
__
9
1
__
9
1
__
9
1
___
10
1
___
10
1
___
10
1
___
11
1
___
11
1
___
11
1
___
12
1
___
12
1
___
12
1
__
8
1
__
9
1
___
10
1
___
12
1
__
9
1
___
12
1
___
10
1
___
11
1
___
12
□
b ) __15 = 10
1
___
12
1
__
7
1
__
7
1
__
8
1
__
8
1
__
9
1
__
9
1
___
10
1
___
10
1
___
11
1
___
11
1
___
11
1
___
12
1
___
12
1
___
12
1
__
8
1
__
9
1
___
10
1
___
11
1
__
6
1
__
7
1
__
8
1
___
11
1
__
6
1
__
7
1
__
8
1
__
5
1
__
6
1
__
7
a ) __12 = □
8
1
__
5
1
__
6
1
__
7
1
__
4
1
__
5
1
__
6
• equivalent
fractions −
fractions that have
the same value
1
__
9
1
___
10
1
___
11
1
___
10
1
___
11
1
___
12
1
___
12
□
c ) __23 = 9
□
d ) __34 = 12
□
□
□
□
5
3
4
__
__
e ) __46 = 3
f ) __
=
g
)
=
h
)
=
3
3
2
12
10
9
3. Look very carefully at the fraction wall. Try to find another fraction
1
. Explain your answer.
that is equivalent to __
11
Topic 12: Common fractions
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Count, compare and order fractions
Fractions have a specific place on a number line, so you can count in
fractions like you did with whole numbers.
ExErCiSE 12.3
1. Use this number chain to count in:
3
+ 12
+ 12
+1
2
+ 12
+ 12
+ -1
2
a ) halves
b ) thirds
1
+2
c ) quarters
2. Write down your answers to Question 1.
You should be able to see which fractions are larger than others when
you use your fraction strips or the fraction wall. When you compare
fractions use these symbols: >; < or =
Remember that < means less than and > means greater than.
ExErCiSE 12.4
1. Which fraction is larger?
b ) __34 or __12
c ) __15 or __35
d ) __26 or __14
a ) __12 or __13
2. Write these fractions in ascending order: __36 ; __14 ; __78 ; __23 ; __15
2 metres
1 metre
1
_
2
metre
1
_
2
metre
1 metre
1
_
2
metre
1
_
2
metre
3. You have 2 m pieces of wood that have been
cut into equal pieces in different ways.
a) Name all the different fractions that the
pieces of wood have been cut into.
b) Which piece in each pair is longer? Show
your answer by filling in < or >.
i.
ii.
5
1
__
□ __
5
10
5
iii. __
□ __24
10
iv. 1__15 □ __65
1 □ __65
vi. __78 □ __35
v.
66
1
1
__
□ __
2
10
Term 2
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Make fractions by sharing and
grouping
When you share something equally, you are actually dividing and
getting a fraction of the whole.
Example
If 4 learners share 12 sweets equally, how many sweets will each
learner get?
12 ÷ 4 = 3
So __14 of 12 = 3
Each learner will get 3 sweets
ExErCiSE 12.5
1. Look at this circle of balls.
a ) How many balls are there in the
circle altogether?
b ) What fraction of the whole do the
red balls represent?
c ) What fraction of the whole do the
blue balls represent?
d ) What fraction of the whole do the
green balls represent?
e ) What fraction of the whole do the
yellow balls represent?
2. Draw a picture to show how 3 friends can share
12 chocolate bars equally. How much chocolate
will they each get?
3. Draw a picture to show how 4 friends can share
20 apples equally. How many apples will they
each get?
Topic 12: Common fractions
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Add fractions
It is easy to add fractions that have the same denominator.
Example
A cake is cut into 8 equal slices. If you eat 3 slices and then you eat
1 more slice, what fraction of the cake have you eaten?
The cake is cut into eighths, so each slice is __18 of the cake. You have
eaten a total of four slices.
You have eaten __18 + __38 of the cake.
When the denominators are both eighths, you can add the fractions.
1
__
+ __38 = __48
8
You have eaten __48 of the cake. Can you see that this is equivalent to
half of the cake?
4 __
__
=1
8 2
ExErCiSE 12.6
1. Copy this fraction chain into your book. Then complete the chain.
1
+ 6-
4
+ 6
+ 5
6
2
+ 6
+ 36
+ -1
6
+ 5
6
4
+ -6
2
+ 6
2
+ 6
1
+6
+ 1
6
3
+ 6
+ 26
2
+6
16
2. Make your own fraction
chain in fifths for your friend to fill in.
- 3. Add these fractions.
a ) __25 + __25
b ) __29 + __59
3
9
1
e ) __
+ __
+ __
10
10
10
5
3
2
c ) __
+ __
+ __
11
11
11
d ) __46 + __56
4
of the pocket of oranges to make juice. The
4. Sipho’s dad used __
11
4
of the pocket of oranges.
next day he used another __
11
a ) How many elevenths has he used?
b ) How many elevenths were left of the pocket of oranges?
5. If __12 of a loaf of brown bread costs R6, how much will:
a ) 2 halves cost
b ) 4 halves cost?
68
Term 2
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Solve problems that involve fractions
When you have to solve word problems, read through each problem
very carefully. Then decide what the important facts are and how you
will solve the problem.
• Use the information in the word problem to write a number sentence.
• Do the calculation and finish with a sentence that explains your
answer.
Example
• Raeesa spends __15 of her homework time doing Maths, and __25 of the
time studying for a test. What fraction of her time has she used?
• She has used __15 + __25 = □
• She has used __35 of her time.
ExErCiSE 12.7
1. Mrs Rhada uses __27 of a block of cheese for sandwiches and __37 of the
cheese for making muffins. What fraction of the cheese has she used?
2. Mandla runs 7 km on Monday. If this is one-third of the distance he
ran on Sunday, what distance did he run on Sunday?
3. A painter paints __14 of a 300 metre wall. How much has he painted?
How much must he still paint?
4. After a class party, a teacher wants to know how much food is left
over. Work out what fractions of full packets, a jug and a plate are
left over.
a ) 3 packets of sweets each __14 full.
b ) 2 jugs of cool drink each __25 full.
c ) 4 plates of cake each __27 full.
1
full.
d ) 6 packets of chips each __
10
5. The Maho family uses a total of __13 of a litre of milk on their cereal
each morning. Answer these questions by drawing a picture to
work out your answers.
a ) How much milk will they use on their cereal in two mornings?
b ) How much milk will they use on their cereal in six mornings?
c ) If the Maho family use 5 ℓ of milk for breakfast, how many
mornings did they have cereal?
Topic 12: Common fractions
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Topic
Length
13
Maths ideas
• Estimate, measure,
compare and order
lengths.
• Check estimation
by measuring.
• Read and write
lengths in
kilometres, metres
and centimetres.
• Add, subtract,
multiply and
divide with units of
measurement.
• Convert between
different units of
measurement.
• Solve problems
that involve
length.
Estimate and measure length
To measure length, you use
different units of measurement
such as millimetres (mm),
centimetres (cm), metres (m)
and kilometres (km).
You also need to choose a
suitable measuring instrument.
Some examples of measuring
instruments can be seen on
the right.
a ruler
a measuring tape
a trundle wheel
a metre rule
10 mm = 1 cm
100 cm = 1 m
1 000 m = 1 km
You can use a ruler to measure short lengths. It is useful to estimate
the length first.
Example
Find the length of Peter’s line.
Peter estimates that his line is about
60 mm. Use your ruler to measure
the line.
Count in millimetres from 50 mm
to the end of the line. There are
8 millimetres.
So, the line measures 50 mm +
8 mm = 58 mm or 5 cm 8 mm
(10 mm = 1 cm, so 50 mm = 5 cm)
70
Term 2
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Example
Sandra is measuring a line. First she estimates that her line is about
2 m or 200 cm long. She then measures the line with her metre
rule to the 2 m mark. From that point, there are four 10 cm spaces
to the end of the line.
So, the length of the line is 2 m + 40 cm
= 2 m 40 cm or 200 cm + 40 cm = 240 cm.
ExErCiSE 13.1
1. Choose the best unit of measurement from the ones in brackets.
a ) I use (centimetres; metres) to measure the length of my
classroom.
b ) I use (centimetres; metres) to measure the width of my
classroom door.
c ) I use (centimetres; metres; kilometres) to measure the length
of the school’s soccer field.
d ) I use (centimetres; metres) to measure the length of my arm.
e ) I use (centimetres; metres; kilometres) to measure the distance
from Durban to Cape Town.
2. Look at these lines:
B
A
C
D
a ) Estimate the length of each line in millimetres.
b ) Measure the length of each line in centimetres and
millimetres.
c ) Order the lengths from the shortest to the longest.
3. a) Estimate the length of your classroom.
b ) Estimate the length of the board in your classroom.
c ) Estimate the length of the school building.
d ) Estimate the width of one of the windows in your classroom.
4. Use a metre stick or measuring tape to measure each object in
Question 3 in metres and centimetres. Order the lengths from
shortest to longest.
Topic 13: Length
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Convert units of length
When you solve problems, you will often work with different units
of measurement. This is why it is important that you know how to
convert units of measurement.
You can convert one unit of length to another unit of length
by multiplying or dividing by 10, 100 or 1 000.
× 1 000
km
× 100
m
÷ 1 000
× 10
cm
÷ 100
mm
÷ 10
Example
Mbulelo needs 5 km of wire to make a fence. How many metres of
wire is this?
1 km = 1 000 m
A metre is a smaller unit of measurement than a kilometre, so you
should multiply.
5 km = (5 × 1 000) m = 5 000 m
Mbulelo needs 5 000 m of wire.
Example
Tau travelled 18 000 m on his bicycle. How many kilometres did
he travel?
1 000 m = 1 km. A kilometre is a larger unit than a metre, so you
should divide.
18 000 m = (18 000 ÷ 1 000) km = 18 km
Tau travelled 18 km on his bicycle.
ExErCiSE 13.2
Convert these units of measurement.
72
1. 780 mm = □ cm
2. 45 cm = □ mm
3. 25 000 cm = □ m
4. 8 m = □ cm
5. 7 km = □ m
6. 135 000 m = □ km
7. 0,5 cm = □ mm
8. 0,5 km = □ m
9. 6,5 cm = □ mm
10. 2,5 km = □ m
Term 2
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Calculate with units of length
You will practise doing calculations with units of length. Remember to
convert to the same unit before you calculate.
Example
Add 850 m, 2,5 km and 3 121 m. Write your answer in kilometres
and metres.
850 m = 800m + 50m
+ 2 500 m = 2 000m + 500m
+ 3 000 m = 3 000m + 100m + 20m + 1m
Total
= 5 000m + 1 400m + 70m + 1m
The total distance is 6 471 m.
6 471 m = 6 000 m + 471 m = (6 000 ÷ 1 000) km + 471 m
= 6 km 471 m
ExErCiSE 13.3
1. Round off these lengths to the nearest 10 m and 100 m.
a ) 34 566 m
b ) 135m
c ) 2 408 m
2. Round off these lengths to the nearest 1 000 m.
a ) 3 500m
b ) 34 567 m
c ) 1 423 m
3. Do these calculations.
a ) 32 456 m + 12 123 m
c ) 34__12 cm + 12 cm + 500 mm
4. Do these calculations.
a ) 340 m × 12
c ) 440 m ÷ 20
b ) 9 km − 7 367 m
d ) __18 m + __28 m
b ) 224 km × 15
d ) 345 cm ÷ 5
5. How many quarters of a metre are there in 4__34 m?
6. Erin recorded the following distances from the odometers of the
cars of her aunts and uncles.
45 674 km; 25 135 km; 60 901 km; 78 345 km; 11 961 km
a ) Order the distances the cars have travelled from the shortest
to the longest distance.
b ) Estimate and then calculate the total distance that all the cars
have travelled.
Topic 13: Length
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Solve problems that involve length
Study the examples of how to solve problems that involve length.
Then do the exercise on your own.
Example
Ayesha has four pieces of material in these lengths: 400 cm;
700 cm; 13 m and 9 m. She needs 50 m of this material to make
shirts. How many more metres of material does Ayesha need?
First convert these lengths to the same units of measurement.
400 cm = (400 ÷ 100) m = 4 m
700 cm = (700 ÷ 100) m = 7 m
Ayesha has 4 m + 7 m + 13 m + 9 m = 33 m of material.
She still needs 50 m − 33 m = 17 m of material.
Example
Vusi is a salesman and he usually travels 95 km for his job each day.
How many kilometres does he travel in five working days?
95 × 5 = (100 − 5) × 5 = (100 × 5) − (5 × 5) = 500 − 25 = 475 km
ExErCiSE 13.4
1. The distance from Fatima’s house to her school is 2__12 km. She walks
to and from school each day.
a ) How many kilometres does she walk each day?
b ) How many kilometres does she walk in five school days?
2. Rhada must cut a string of 6__23 metres into lengths of __13 metre. How
many pieces of string will she have?
3. To make his fencing, a farmer uses 8 times more plain wire than
barbed wire. If he uses 320 m of plain wire, how much barbed wire
will he use?
4. Cephus is training to take part in a cycle race. On Day 1, he cycles
10 km 750m. On Day 2, he cycles 12,5 km. On Day 3, he cycles 9
km 250m. On Day 4, he cycles twice the distance he travelled on
Day 2.
a ) How far did Cephus cycle on Day 4?
b ) Calculate the total distance he cycled while in training.
c ) How much more must Cephus cycle to cover a total distance
of 50 km?
74
Term 2
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revision
1. Write the fractions shown by the shaded areas.
b)
c)
a)
(3)
2. Draw diagrams and shade the following fractions of your diagram
a ) __14
(1)
7
b ) __
10
(1)
3. Use the fraction wall on page 65 to complete these equivalent fractions.
4
=□
a ) __
10
(1)
5
□
2
__
b) 4 = 2
(1)
3
full. The blue tin
4. A painter has some paint left over in tins of the same size. The red tin is __
11
3
4
__
__
(2)
is 11 full and the yellow tin is 11 full. What fraction of a paint tin is left over in total?
5. a ) Mark off a 10 m measurement in the school grounds. Use a metre stick,
measuring tape or trundle wheel to do this.
b ) Use this 10 m length to make a 10 m long piece of string. Use this piece of string
to mark off a distance of 10 times the 10 m. What will this new length be?
c ) Walk 10 times the 100 m length. What is the total distance that you walked?
6. Mashadi needs 25 cm, 347 mm and 64 cm of string to finish her project.
a ) How many millimetres of string does she need?
b ) She buys a ball of string that is 2 m long. How much string will she have left over?
(1)
(2)
(2)
(2)
(1)
7. Below is a diagram of Asanda’s farm.
1 km
m
946 m
3k
m
28
65
m
2k
m
54
13
5m
a ) Calculate the total distance around her farm. Write your answer in metres.
b ) Write your answer to Question 7. a) in kilometres.
(2)
(1)
Total marks: 20
Revision
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Topic
Multiplication
14
Maths ideas
Work with factors and multiples
• Find multiples and
factors of whole
numbers up to
100.
Factors and multiples have an important relationship with each other.
For example, if a number is a multiple of 5, then 5 is also a factor of
that number.
• Multiply two-digit
numbers by twodigit numbers.
Example
• Check solutions.
1. You can divide 644 by 2 without remainder. You can say that:
• 2 is a factor of 644, because 644 ÷ 2 = 322. You can also say that
• 644 is a multiple of 2, because 322 × 2 = 644.
• Solve problems
that involve
multiplication.
2. The first three multiples of 7 are 7 × 1 = 7, 7 × 2 = 14, and
7 × 3 = 21. This means that 7 is a factor of 7, 14 and 21.
• Round off to
estimate answers.
• Work with rates.
To help you find factors, here are some useful rules of division.
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.
Challenge
If a number has 6 as a
factor, it also has 2
and 3 as factors. This
is because 2 × 3 = 6.
If a number has the
following factors,
what other factors do
you know it will
have? Do not look at
the tests for factors.
a) 10
b) 9
c) 8
d) 12
e) 20
f) 24
76
Example
Is the number 4 014 divisible by 3, 9 or 4?
Add the digits: 4 + 0 + 1 + 4 = 9. This is divisible by 3 and by 9, so
3 and 9 are both factors of 4 014.
The last two digits form the number 14 which is not divisible by 4,
so 4 is not a factor of 4 014.
ExErCiSE 14.1
Make up a two-digit number that is a multiple of:
1. 3 and 6
2. 9
3. 3, 6 and 9
4. 2 and 3
5. Will the number in Question 2 also be a multiple of 6? Explain.
Term 2
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6. Can you share R745 equally among
9 people? Explain your answer.
70
71
72
73
74
7. Copy this grid and cross off:
75 76 77 78 79
a ) Multiples of 3 between 70 and 99.
80 81 82 83 84
b ) Multiples of 6 between 80 and 90.
c ) Multiples of 9 between 80 and 99.
85 86 87 88 89
d ) All even numbers.
90 91 92 93 94
e ) Any remaining numbers with 10 as
a factor.
95 96 97 98 99
f ) Any remaining numbers with 5 as
a factor.
g ) Find all the factors of all the numbers that are not crossed off
on the grid.
Challenge
Find the greatest and
smallest product
possible. Use all the
digits 2, 3, 5 and 8
only once in each
calculation that you
try. Try □ □ □ × □ or
□ □ × □ □.
For example, try
385 × 2 or 38 × 52
You can use factors to help you multiply numbers. Look for the
factors 10 and 100, because it is easy to multiply by these numbers.
Remember that you can multiply numbers in any order. First estimate
the answer.
Example
a ) Find 211 × 6
An estimate is 200 × 6 = 2 × 6 × 100 = 12 × 100 = 1 200.
211 × 6 = (211 × 2) × 3 = 422 × 3 = 1 266
b ) Find 65 × 80
An estimate is 70 × 80 = 5 600
65 × 80 = 65 × (2 × 2 × 2) × 10 = 130 × 2 × 2 × 10
= 260 × 2 × 10 =5 200
ExErCiSE 14.2
1. Fill in the missing numbers.
a ) 36 × 14 = 36 × □ × 2
b ) 24 × 18 = 24 × □ × □ × □
c ) 40 × 15 = □ × 2 × □ × 3 × 10
2. First estimate each answer. Then use factors to find the exact answer.
a ) 12 × 45
b ) 21 × 80
c ) 11 × 14
d ) 31 × 21
3. There are 30 eggs in 1 tray. How many eggs are there in 25 trays?
4. There are 35 buttons in 1 box. How many buttons are there in
16 boxes?
Topic 14: Multiplication
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Multiply three-digit numbers by
two-digit numbers
You can break down a number to form different groups.
Example
Challenge
Find the missing
numbers. Then, work
out the answer using
any method you
choose.
a) 28 × 56
= (7 × □)
× (4 × □ × □)
b) 240 × 48
= ( 60 × □ )
× (□ × 8)
Imagine a group of 150 coins
in thirty rows and five columns.
You can move the coins to
make two groups of coins, with
90 coins and 60 coins.
…
…
…
30 × 5
30 × 3
30 × 2
This means that you can write 30 × 5 like this:
30 × 5 = 30 × (3 + 2) = (30 × 3) + (30 × 2) = 90 + 60 =150
ExErCiSE 14.3
1. Draw your own diagram to show that 3 × 6 = (3 × 4) + (3 × 2).
2. Fill in the missing numbers:
a ) 27 × 35 = (27 × □) + (□ × 5)
b ) 34 × □ = (34 × 50) + (34 × 6)
c ) □ × 19 = (16 × 10) + (16 × □)
d ) 240 × 48 = (240 × □) + (240 × 8)
e ) 345 × 82 = (345 × □) + (345 × □)
f ) 134 × □ = (134 × 70) + (□ × 2)
When you break down a number, you can multiply in steps.
Remember to estimate the answer first.
Example
Find 211 × 15
An estimate is 200 × 15 = 2 × 15 × 100 = 3 000.
211 × 15 = 211 × (10 + 5) = (211 × 10) + (211 × 5)
= 2 110 + 1 055 = 3 165
ExErCiSE 14.4
First estimate each answer. Then multiply by breaking down one number.
1. 312 × 13
78
2. 121 × 42
3. 250 × 23
4. 101 × 36
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You may use all the methods that you know to multiply numbers.
Look for different ways to break down numbers to make the work
easier. Work through these examples carefully to help you with the
next exercise.
Example
Find 123 × 17
An estimate is 120 × 20 = 12 × 2 × 10 × 10 = 24 × 100 = 2 400.
123 × 17 = 123 × (20 − 3) (This is easier than writing 17 as 10 + 7.)
= (123 × 2 × 10) − (123 × 3) (Notice the subtraction sign.)
= 2 460 − 369
= 2 000 + 300 + 160 − 300 − 69
= 2 000 + 160 − 70 + 1 = 2 091
Example
Find 642 × 56.
An estimate is 600 × 60 = 36 000.
642 × (50 + 6)
= (642 × 50) + (642 × 6)
= (642 × 10 × 5) + (600 × 6) + (40 × 6) + (2 × 6)
= 32 100 + 3 600 + 240 + 12
= 35 952
ExErCiSE 14.5
1. Use any methods to work out the answers to these multiplication
sums. First estimate the answer.
a ) 142 × 16 b ) 261 × 25 c ) 328 × 43 d ) 308 × 66
e ) 578 × 73 f ) 478 × 32 g ) 521 × 54 h ) 858 × 92
2. Zubair thinks you need 570 chairs for 16 classes of 35 children
each. Mark thinks you need 560 chairs. Who is correct?
3. A new school library has been built at Langa Primary School. The
builders have put in 121 shelves. Each shelf holds 32 books. How
many books does the school need to fill the library?
4. Doubling is the inverse of halving. Double the following numbers.
a ) 410
b ) 808
c ) 4 143
d ) 2 755
5. Fill in the missing numbers. Write down what you notice.
a ) 12 × 1 = □
b ) 73 × 1 = □
c ) 1 × 852 = □
d ) 1 × 5 000 = □
Topic 14: Multiplication
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Division is the inverse of multiplication
Multiplication and division are inverse operations. This means that you
can use division facts to check multiplication facts.
Example
Find the missing number: 50 × □ = 3 500
Use division: 3 500 ÷ 50 = □
3 500 ÷ 50 = 3 500 ÷ 10 ÷ 5 = 350 × 5 = 70
The missing number is 70.
Check: 50 × 70 = 5 × 7 × 10 × 10 = 35 × 100 = 3 500
Challenge
ExErCiSE 14.6
1. You multiply an
even number by
an even number.
Is the product
always an even
number?
1. Write down two related division facts for each multiplication.
a ) 333 × 87 = 28 971
b ) 419 × 35 = 14 665
c ) 563 × 72 = 40 536
2. You multiply an
odd number by
an odd number.
Is the product
also an odd
number?
3. You multiply an
odd number by
an even number.
Is the product an
odd number or
an even number?
4. Explain the
pattern you
found.
2. Find the missing numbers.
a ) 15 × □ = 45, so 45 ÷ 15 = □
b ) 220 × 4 = □, so □ ÷ 4 = 220
c ) 360 ÷ □ = 5, so 5 × □ = 360
d ) □ ÷ 30 = 5, so 30 × 5 = □
3. Check these divisions by doing multiplication. Are all the answers
correct? If not, then correct the largest number.
a ) 9 720 ÷ 45 = 216
b ) 30 195 ÷ 99 = 305
c ) 12 584 ÷ 242 = 62
d ) 6 632 ÷ 510 = 13 remainder 2 (Hint: find 13 × 510 first)
4. Calculate the missing number.
a ) □ ÷ 341 = 23
b ) □ ÷ 17 = 771
c ) □ ÷ 42 = 333
d ) □ ÷ 441 = 36
5. Halving is the inverse of doubling. Halve these numbers.
a ) 508
b ) 992
c ) 6 018
d ) 7 314
6. Explain why the following numbers may be difficult to halve.
a ) 885
b ) 4 131
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Solve multiplication problems
Many word problems involve multiplication.
Example
a ) Lerato is paid R12 an hour at the local shop. This means that
she is paid R12 for each hour that she works. If she works for
4 hours, she will earn 4 × R12 = R48.
b ) A car is travelling at a speed of 80 km an hour. If the car travels
at this speed for 30 minutes (half an hour), then it will travel
1
__
of 80 km = 80 km ÷ 2 = 40 km.
2
c ) A bag of potatoes costs R8 a kilogram. If you buy __14 of a kg,
you will pay __14 × R8 = R8 ÷ 4 = R2.
Key words
• average speed
− total distance
divided by total
time; the rate at
which the distance
is changing over
time. We normally
write speed in
km/hr. This means
kilometres per
hour
ExErCiSE 14.7
1. A shop sells boxes of pens at R22 a box. How much will a teacher
pay if he buys 18 boxes?
2. Look at these pictures of fruit. Find the cost of:
R9 per kg
R8 per sack
a ) 150 apples?
c ) 215 sacks of oranges?
R26 per kg
R3 each
b ) 2 kg of grapes?
d ) 6 kg of bananas?
3. An ice machine makes 128 ice cubes in 1 hour. The machine only
operates from 5 p.m. to 11 p.m.
a ) How many ice cubes can the ice machine make in 1 day?
b ) How many ice cubes can the ice machine make if it works on
all 31 days in January?
4. A family left for a holiday from Cape Town to the Kruger Park. The
total journey was 1 938 km.
a ) On the first day, they travelled 24 × 42 km. How far do they
still have to go?
b ) On the second day, they reached the Kruger Park gate after
travelling for 10 hours. How many kilometres an hour did the
family travel on average? (This is their average speed.)
Topic 14: Multiplication
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Topic
Properties of 3D objects
15
Maths ideas
• Recognise and
name 3D objects.
• Describe, sort
and compare 3D
objects.
• Make 3D models.
• Interpret diagrams
of 3D objects.
recognise and name 3D objects
Solid objects are objects that you can see
around you, like books and tables. They are
height
also called three-dimensional (3D) objects.
face
width
An object has three dimensions when it has
length
length, height and width. A flat surface of a
solid is called a face of the solid.
There are many different types of 3D objects. If you want to know
what type of 3D object you have, look at the number of faces and the
shapes of the faces.
Types of 3D objects
Key words
• face − a flat surface
of an object
• prism − a 3D
object with two
identical end faces
(bases)
• pyramid − a 3D
object that has a
polygon base and
all its other faces as
triangles
• base − the face on
which the object
rests
• identical −
exactly the same
A prism has two opposite faces identical (equal) polygon, with exactly
the same size and shape. All the other faces of a prism are rectangles.
triangular prism
cube
All the faces of a cube are
equal squares.
rectangular prism
The two identical triangular
The two identical, parallel
faces are parallel to each other. faces of a rectangular prism
are rectangles.
A pyramid has a polygon as its base. The other faces of a pyramid
are triangles, because the edges of a pyramid come to a single point
above the base. The shape of the polygon base gives you the name of
the pyramid.
square-based
pyramid
A square-based pyramid has its base in the
shape of a square. Its other faces are triangles.
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Some 3D objects do not have polygons as bases, such as cones,
cylinders and spheres. These objects have curved surfaces.
Key words
• curved surface
− a surface of a 3D
object that is not
flat
cone
cylinder
sphere
The base of a cone
is a circle.
The base of a cylinder
is a circle.
A sphere does not have
a base.
Did you know?
ExErCiSE 15.1
1. Draw and name each 3D object from the description. Use the
drawings of 3D objects on the previous page to help you.
a ) I am box-shaped and have rectangular faces.
b ) I am a prism. Two of my faces are triangles and three of my
faces are rectangles.
c ) I have a round base and come to a point at the top.
d ) I have the shape of a tube with circles at both ends.
e ) I have six equal square faces.
f ) I am the shape of a round ball.
g ) I have a square base and four of my faces are triangles.
The Egyptians built
pyramids as tombs
for their kings and
queens. They
believed that the
pyramid would allow
the king or queen’s
soul to live forever.
In 1100 AD, the Shona
people of Zimbabwe
built stone structures.
The whole civilisation
lived in these
structures. The ruins
of these structures are
known as Great
Zimbabwe.
2. Look at the objects in the
photograph on the right.
a ) Write down the name of two
of the objects.
b ) What is the same about the
two objects?
c ) What is different about the
two objects?
3. Match the two halves that will complete the 3D objects below.
A
E
B
F
D
C
G
H
Topic 15: Properties of 3D objects
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Construct models of 3D objects
Key words
• dimension − a
measurement of
length, breadth or
height
To make a model of a 3D object, you first cut out the correct shape of
each face. The dimensions of each face must be measured carefully,
so that the sides of the faces match each other exactly in length when
you tape them together to make the solid.
Steps to construct a cube
Step 1: Count all the faces of a cube. There are six identical squares.
Step 2: Glue some square grid paper onto cardboard. Draw and cut
out six identical squares.
Step 3: Tape the squares together to form a cube.
Steps to construct a square-based pyramid
Step 1: Count all the faces of a square-based pyramid. There is one
square base, and four identical triangles.
Step 2: Glue some square grid paper onto cardboard. Copy the
triangle in the picture on the left onto your paper, and also
put in the writing.
Step 3: Draw three more triangles that are the same as the first one
and cut them out.
Step 4: Draw a square with the dimensions shown in the picture on
the left.
Bottom
84
Step 5: Each triangle has one side marked with the word ‘bottom’.
Tape this side of each triangle to one side of the square. Now
tape the sides of the triangles together to make the pyramid.
Notice that the other two sides of the triangles are equal in
length so that they can be taped together.
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Steps to construct a triangular prism
Step 1: Draw three identical rectangles using the dimensions shown
in the picture. Cut them out.
These three rectangles will form the side/faces of the prism.
The shape on the end of this prism is a triangle.
Tape the longer sides of your rectangles together to form a
triangular shape at the two ends.
Step 2: Measure the length of the shorter side of your rectangle. This
is the length of each side of the base triangle.
On your grid paper, draw a line that is the exact length of the
base triangle, down the left of the page.
Step 3: Draw a line of symmetry from left to right, across the middle
of your first line.
Now place your ruler on one end of the first line you drew.
Measure the correct distance to the line of symmetry.
Draw the second side of the triangle.
Repeat this instruction from the other end of the line.
Step 4: Repeat Steps 2 and 3 to draw another equal triangle. Then
cut out your two triangles.
Step 5: Use sticky tape to construct your triangular-based prism.
ExErCiSE 15.2
Challenge
1. Use the steps for making a cube to construct a rectangular prism.
First draw six faces of the correct shapes. Think carefully about
which lengths must be equal in each face so that you can tape
them together.
2. Look carefully at your models of the cube, the rectangular prism
and the triangular prism.
a ) What do you see that is the same in these solids?
b ) What do you see that is different in these solids?
Could you construct
a cylinder in the
same way by using
only grid paper, a
ruler, a pair of
scissors, a pencil and
sticky tape? Explain
in your own words
how you would
do this.
Topic 15: Properties of 3D objects
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Work with diagrams of 3D objects
You can view a 3D object from different sides. You can look at the
object from the top, bottom, front, back or sides. Look at the different
views of a chair shown below.
The side view
The top view
The front view
The view from the bottom
ExErCiSE 15.3
1. Look at photographs A to E.
Identify all the 3D objects that
you can see in each photograph.
How many faces are there in each
3D object?
2. What shapes are the top and
the side view of the house in
Photograph B?
3. What shapes are the top and
the side view of the tin in
Photograph A?
A
C
D
86
B
E
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revision
1. Which one of these numbers is a factor
of 27?
a) 2
b) 3
c) 4
d) 5
2. Which of these numbers are multiples
of 3 and also of 6?
45; 54; 99; 24
4. Use any method that you like to do these
multiplications.
a ) 421 × 23
(1)
b ) 588 × 36
(1)
(1)
5. Calculate the missing costs, if cheddar
cheese costs R65 for one kg.
(2)
3. Write down a two-digit number that is a
multiple of 9.
(1)
Number of
kilograms
Cost
1
2
3
5
10
a)
Description
45
R65
6. Copy and complete the table below:
Name of a 3D object
(5)
(9)
Drawing of the object
a 3D object with a flat
circular base and one
curved face that comes to a
point
b) cube
c)
a 3D object with two
opposite circles as bases
and one curved face
d)
e)
sphere
Total marks: 20
Revision
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Topic
Geometric patterns
16
Maths ideas
• Identify geometric
patterns.
• Extend geometric
patterns.
• Identify and
extend number
patterns.
• Identify a rule
that applies to a
pattern.
• Describe a rule in
different ways.
• Create your own
geometric pattern.
Key words
• patterns
− repeated
arrangements of
shapes, numbers,
colours or lines
• geometric
patterns
− repeated
arrangements of
shapes
Describe and extend geometric patterns
In this topic, you will look at different geometric shapes and try to find
patterns made by the shapes. You will also try to find rules that will
help you to draw the next shape in a pattern.
Example
Here is a geometric pattern made with matchsticks.
1
2
3
1. You can describe this pattern in different ways:
• It is a pattern of squares, with each square bigger than the
one before.
• It is a pattern of squares, with each square using more
matchsticks than the one before.
2. You can explain how to get from one diagram to the next,
by saying:
• I add four matchsticks.
• I add one match to each side of the square.
3. Draw the next
two squares in
the pattern:
4
5
4. In a table, write the number of matchsticks used in each square.
1
Diagram number
Number of matchsticks 4
88
2
3
4
5
8
12 16 20
6
8
12 35
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5. If the diagram number is the input, what rule can you use to
get the number of matchsticks as the output?
Rule: The number of matchsticks = 4 × the diagram number
of the square.
6. Draw a flow diagram with 1; 2; 3; 4 and 5 as input numbers.
Show the output numbers.
1
4
2
Rule
8
3
?
12
4
16
5
20
7. How many matchsticks will you need for Diagrams 6, 8, 12 and 35?
Use the rule: 6 × 4 = 24; 8 × 4 = 32; 12 × 4 = 48 and
35 × 4 = 140
Now try these exercises yourself.
ExErCiSE 16.1
Look at this pattern of matchsticks:
1
2
3
Challenge
1. Draw the next three diagrams in the pattern.
2. In your own words, describe how you get from one diagram in the
pattern to the next diagram.
3. Count the number of matchsticks in each diagram and then
complete this table:
Diagram number
Number of matchsticks
1
2
3
4
7
11
Can you see 7
triangles in this
matchstick diagram?
If you remove only 3
matchsticks you can
change the diagram
to have only 3
triangles. How do
you do this?
Topic 16: Geometric patterns
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ExErCiSE 16.2
Look at this pattern of squares:
1
2
3
1. Draw the next two diagrams in the pattern.
2. In your own words, describe how you get from one diagram in the
pattern to the next diagram.
3. Count the number of squares in each diagram. Then complete this
table for the first five diagrams.
Diagram number
Number of squares
1
2
3
4
5
7
9
10
4. If the diagram number is the input, what rule can you use to get
the number of squares as the output?
5. Complete this flow diagram. In the box write the rule that will
change the input numbers to output numbers.
1
2
Rule
3
?
4
5
6. Use the rule to calculate the number of squares in Diagrams 7, 9
and 10. Write these in the table.
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investigate geometric patterns
Key words
You can write a number pattern as a sequence, or list, of numbers. The
pattern of a number sequence can help you to find rules and missing
numbers.
Example
• sequence − a
group of numbers
or shapes that
follow each other
in a particular
order
Describe this number sequence. Then find the next three numbers
in the pattern. 1; 4; 7; 10; …
This sequence follows the rule: Add 3 to the previous number.
+3 +3 +3
1; 4; 7; 10;
…
So, the next three numbers in the sequence are 13; 16 and 19.
Here is a geometric pattern that might match the number sequence:
Example
Describe this number sequence. Then find the next three numbers.
101; 99; 96; 92; …
This sequence follows this rule: Subtract the next counting number
from the previous number in the sequence.
−2
−3
−4
101; 99; 96; 92;
…
So, the next three numbers in the sequence are: 92 − 5 = 87;
87 − 6 = 81 and 81 − 7 = 74
ExErCiSE 16.3
Describe the pattern in each number sequence. Then fill in the missing
numbers to complete the sequence.
1.
1; 6; 11; 16; □; □; □
2.
17; 27; 37; □; □; □
3.
117; 219; 321; □; □; □
4.
1 005; 955; 905; □; □; □
5.
270 000; 27 000; □; □; □
6.
23; 29; 36; 44; □; □; □
7.
89; 82; □; 68; □; □; 47
8.
63; 51; □; □; 15; □
9.
3; 6; 12; □; □; □; 192
10. 16; 8; □; □; □; __12
11. 2; 10; 50; □; □; □; 31 250
12. 4; 16; 64; □; □; □; 16 384
Challenge
1. Create your
own geometric
pattern.
2. Swop your
patterns with
your partner.
3. See if you can
identify the rule
in the pattern
and then extend
the pattern.
Topic 16: Geometric patterns
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Topic
Symmetry
17
Maths ideas
• Recognise lines of
symmetry.
• Draw and describe
lines of symmetry.
• Recognise
rotational
symmetry.
• Describe order
of rotational
symmetry.
Line symmetry
You already know that some shapes have line symmetry.
If a shape has line symmetry, this means that there is a line that
divides the shape into two identical halves that are mirror images of
one another. This line is called a line of symmetry.
Some shapes have no lines of symmetry; some have one line of
symmetry and other shapes have more than one line of symmetry.
ExErCiSE 17.1
1. Look at these shapes:
b)
a)
e)
c)
d)
f)
g)
Key words
• line symmetry
− a shape has line
symmetry if it can
be folded in such a
way that one half
lies exactly on the
other half
• line of symmetry
− a line that
divides a shape
into two identical
halves
92
How many lines of symmetry does each shape have?
2. Here is a fun way to make your own shape that has line symmetry.
a ) Take a piece of dotted grid paper.
b ) Draw a vertical line down the middle of the piece of paper,
along one of the lines of dots.
c ) Put the point of a pencil on the line that you have drawn.
d ) Without lifting your pencil, draw any shape on the left hand
side of the piece of paper. Stop only when you have reached
the line again.
e ) Now draw the mirror image of your shape on the right hand
side of the paper. Try to make sure that both halves of the
shape are exactly the same. Use the dots to help you.
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rotational symmetry
Another kind of symmetry is rotational symmetry.
If a shape has rotational symmetry, this means that the shape can be
rotated (turned) so that it looks exactly like the original shape at least
once before completing a full turn.
The number of times that a shape looks like the original while
completing a full turn is called the order of rotational symmetry of
the shape.
Example
1. Look at this square:
Key words
• rotational
symmetry
− a shape
has rotational
symmetry if it can
be rotated so that
it looks exactly
like the original
shape at least once
before completing
a full turn
• order of
rotational
symmetry − the
number of times
that a shape looks
like the original
while completing a
full turn
Do you see that when you rotate a square through a full turn, it
looks exactly like the original four times? We say that a square
has rotational symmetry of order 4.
2. Look at this rectangle:
Do you see that when you rotate a rectangle through a full
turn, it looks exactly like the original only twice? We say that a
rectangle has rotational symmetry of order 2.
Did you know?
• A circle is the most perfect 2D shape that there is. It has an
infinite number of lines of symmetry, because any diameter
that you can draw in a circle is a line of symmetry of the circle.
Topic 17: Symmetry
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Key words
ExErCiSE 17.2
• infinite − never
ending
Look at these shapes:
a)
b)
c)
d)
e)
What is the order of rotational symmetry of each shape?
Some shapes have only line symmetry or only rotational symmetry.
Some shapes have both, and other shapes have neither.
ExErCiSE 17.3
Look at these shapes:
a)
b)
c)
d)
e)
f)
g)
h)
1. Which shapes have line symmetry and rotational symmetry?
2. Which shapes have line symmetry, but not rotational symmetry?
3. Which shapes have rotational symmetry, but not line symmetry?
4. Which shapes do not have line symmetry or rotational symmetry?
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revision
1. Look at this pattern that uses squares:
□□□
□
□□□
□
□□□
1
□□□□
□
□□□□
□
□□□□
2
□□□□□
□
□□□□□
□
□□□□□
3
a ) Draw the next diagram in the pattern.
b ) In your own words, describe how you get from one diagram to the next diagram.
c ) Count the number of squares in each diagram.
Copy this table and complete it for the first 4 diagrams.
(1)
(2)
(2)
Diagram
1 2 3 4 5 9 10 21
number
Number
of squares
d ) If the ‘diagram number’ is the input, what rule can you use to get the ‘number
of squares’ as the output?
e ) Use the rule to calculate the number of squares in diagrams 5, 9, 10 and 21.
Write these in the table.
2. Describe the pattern in these number sequences. Then fill in the missing numbers
to complete them.
a ) 1; 8; 15; 22; □; □; □
b ) 89; 83; □; 71; □; □; 53
c ) 4 500 000; 450 000; 45 000; □; □; □
d ) 448; 224; 112; □; □; □
3. Write down the missing word in each of the following sentences about symmetry:
a ) A shape has _____ symmetry if it can be folded so that one half lies exactly on
the other half.
b ) A line of symmetry divides a shape into ___ identical halves.
c ) The _____ of rotational symmetry is the number of times that a shape looks like the
original while completing a full turn.
d ) A regular _____ has six lines of symmetry.
e ) A square has rotational symmetry of order _____.
f ) A circle has an _____ number of lines of symmetry.
(2)
(4)
(2)
(2)
(2)
(2)
(1)
(1)
(1)
(1)
(1)
(1)
Total marks: 25
Revision
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Topic
Division
18
Maths ideas
Work with factors and multiples
• Find multiples and
factors of whole
numbers.
Factors are very useful for dividing numbers. You can look back at the
division rules in Topic 14, to help you find factors.
• Round off to
estimate answers.
When you find factors of a number, you can use factor pairs to help
you find all the factors.
• Divide a threedigit number by a
two-digit number.
• Check answers
by inverse
operations.
• Express two
quantities with
different units as a
rate.
• Solve problems
using division.
Example
The factors pairs of 12 are:
The factors pairs of 15 are:
12 = 1 × 12
15 = 1 × 15
12 = 2 × 6
15 = 3 × 5
12 = 3 × 4
15 is a multiple of 1, 3, 5 and 15.
12 is a multiple of 1, 2, 3, 4, 6 and 12.
ExErCiSE 18.1
1. Decide whether each number is a multiple of 3; 4; 5 or 10.
a ) 24
b ) 95
c ) 42
d ) 204
e ) 730
f ) 420
g ) Is every multiple of 2 also divisible by 4? Explain your answer.
2. Write down a:
a ) three-digit number that is a multiple of 2
b ) two-digit number that is a multiple of 5 and 10
c ) two-digit number that is a factor of 100 and a multiple of 4.
3. Write down the multiples of:
a ) 3 from 21 to 30
b ) 9 between 35 and 55
c ) 6 that are smaller than 20.
4. Find all the factor pairs for these numbers:
a ) 20
b ) 32
d ) 42
e ) 80
c ) 36
f ) 100
5. Find a number between 50 and 60 that has:
a ) 3 as a factor
b ) 4 as a factor
c ) 6 and 9 as factors.
96
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Use multiplication facts to divide
In Topic 5 you used multiplication facts and a clue board to divide
numbers. Do you remember how to use a clue board to help with
division?
Example
Challenge
Estimate and then use multiplication facts to find 479 ÷ 8.
An estimate is 500 ÷ 10 = 50
Now write down some simple multiplication facts for 8 in a
clue board.
8 × 10 = 80
8 × 100 = 800
8 × 50 = 400
8 × 9 = 72
I want to share a bag
of sweets equally
among some
children.
The closest number to 479 is 400, so start with 8 × 50.
Multiply
Subtract
50 × 8 = 400
479 − 400 = 79
9 × 8 = 72
79 − 72 = 7
479 ÷ 8 = 50 + 9 remainder 7 = 59 remainder 7.
Check by multiplying:
8 × 59 plus remainder 7 = (8 × 50) + (8 × 9) + 7
= 400 + 72 + 7 = 479
If I share the sweets
between 2 children,
there are no sweets
left over.
If I share the sweets
between 5 or 6
children, there are no
sweets left over.
ExErCiSE 18.2
1. Find the missing numbers. Use multiplication and division facts to
help you.
a ) 28 ÷ □ = 7
b ) 49 ÷ □ = 7
c ) 54 ÷ □ = 9
d ) □ ÷ 8 = 72
e ) □ ÷ 4 = 16
f ) □ ÷ 9 = 30
How many sweets
could there be in
the bag?
2. Use your own clue boards to divide these numbers. First estimate
the answer, and multiply to check your answer.
a ) 390 ÷ 5
b ) 810 ÷ 8
c ) 736 ÷ 7
d ) 468 ÷ 9
e ) 756 ÷ 6
f ) 928 ÷ 9
Topic 18: Division
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Division by a two-digit number
You can also use a clue board to divide by bigger numbers.
Example
Challenge
In how many ways
can you make this
number sentence
correct?
120 ÷
∙=∙
One answer can be
120 ÷ 2 = 60.
Use a clue board to divide 665 by 19.
An estimate is 700 ÷ 20 = 70 ÷ 2 = 35.
First write down some easy multiplication facts for 19. Start with
10 × 19, and then multiply and divide to find some more facts.
Clue Board
10 × 19 = 190
20 × 19 = 380
30 × 19 = 570
5 × 19 = 95
2 × 19 = 38
4 × 19 = 76
The closest number to 665 is 570, so start with 30 × 19.
Multiply
Subtract
30 × 19 = 570
665 − 570 = 95
4 × 19 = 76
1 × 19 = 19
95 − 76 = 19
So 665 ÷ 19 = 30 + 4 +1 = 35
Now check by multiplying: 19 × 35 = 19 × 5 × 7
= (20 − 1) × 5 × 7
= (100 − 5) × 7
= 95 × 7
= (90 × 7) + (5 × 7)
= 630 + 35 = 665
ExErCiSE 18.3
1. Find the missing numbers. Use multiplication and division facts to
help you.
a ) 280 ÷ □ = 7
b ) 490 ÷ □ = 70
c) 540 ÷ □ = 90
d ) □ ÷ 6 = 72
e ) □ ÷ 4 = 96
f ) □ ÷ 9 = 32
2. Use your own clue boards to divide these numbers. First estimate
the answer, and multiply to check your answer.
a ) 390 ÷ 15
b ) 810 ÷ 18
c ) 736 ÷ 23
d ) 468 ÷ 36
e ) 756 ÷ 42
f ) 928 ÷ 29
98
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Division with remainders
You can use division to share amounts equally. Sometimes there may
be a remainder when you share objects.
Example
The Grade 5 learners in a school are digging rows of earth to plant
vegetables for the school kitchen. Together they must dig 830
metres. If there are 36 learners, how many metres must each one dig?
First find an estimate: 800 m ÷ 40 = 80 ÷ 4 = 20 metres
Use the clue board method with multiplication facts for 36.
10 × 36 = 360
20 × 36 = 720 (360 × 2)
5 × 36 =180 (360 ÷ 2)
2 × 36 = 72
4 × 36 = 144
The closest number to 830 is 720, so start with 20 × 36
Multiply
Subtract
20 × 36 = 720
830 − 720 = 110
2 × 36 = 72
1 × 36 = 36
110 − 72 = 38
38 − 36 = 2
So 830 ÷ 36 = 20 + 2 +1 remainder 2 = 23 remainder 2
Each learner must dig 23 metres and there will be 2 m left over.
Multiply to check: 23 × 36 = 23 × 2 × 18 = 46 × 18
= 46 × (20 − 2)
= (46 × 2 × 10) − (46 × 2)
= 920 − 92 = 828
So 23 × 36 + remainder 2 = 828 + 2 = 830
ExErCiSE 18.4
1. Use your own clue boards to divide these numbers and find the
remainders. First estimate the answer and then multiply to check.
a ) 392 ÷ 15 b ) 815 ÷ 18 c ) 336 ÷ 23 d ) 469 ÷ 29
2. One school bus can carry 22 learners. How many school buses
do you need to carry 600 learners to a soccer game? How many
learners will be in the bus that is not full?
3. A teacher bought 805 coloured pens for 33 learners. How many
pens will each learner get? How many pens will be left over?
Topic 18: Division
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Challenge
Lerato can read
100 words in
1 minute.
Martin can read
2 pages in
360 seconds.
Mikateko can read
128 pages in 6 hours
and 24 minutes.
If there are 300 words
on 1 page, who is the
fastest reader?
Compare two quantities by dividing
You can use division to compare two quantities.
Example
Tebogo is paid R72 for working an 8 hour day in the local shop.
a ) His hourly rate of pay is R72 ÷ 8 = R72 ÷ 2 ÷ 4 = R9 an hour.
b) In three hours he earns R9 × 3 = R27.
Example
A 3 kg bag of apples costs R24.
a ) The price for one kilogram is R24 ÷ 3 = R8.
b) Five kilograms of apples costs R8 × 5 = R40.
ExErCiSE 18.5
R42 for 2 kg
1. A teacher buys 18 boxes of pens for R396.
a ) What is the price of one box of pens?
b ) How much will she pay for 16 boxes of pens?
2. Look at these pictures of fruit. Find the price for 1 kg of each fruit
R96 for 8 kg
3. Dan earns R20 for delivering 400 newspapers.
a ) How much does he earn for delivering one newspaper?
b ) How much does he earn for delivering 200 newspapers?
4. What is the cost of one cabbage if 5 cabbages cost R25,50?
R33 for 6 kg
5. An aeroplane travels a distance of 600 km in 2 hours.
a ) What is its speed in km per hour?
b ) If it continues to fly at this speed, how far will it travel in
1__12 hours?
6. Some bamboo plants can grow at the rate of 96 cm in one day.
a ) How many cm will they grow in one hour at this rate?
b ) How many cm will they grow in four hours at this rate?
7. A craft market sells candles in different gift packs. If a pack of
3 candles cost R12 and a pack of 5 candles costs R15,50, which
pack is the best value for money?
8. The mass of 12 ℓ of cold drink is 24 kg. What is the mass of 5 ℓ?
100
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Solve problems that involve division
Use your knowledge of division methods to answer these questions.
Remember to first estimate your answer and then to check your
answer by multiplication.
ExErCiSE 18.6
1. Thelma has handed her work to you to be marked. If
Thelma has made a mistake, correct the answer for her.
a ) 240 ÷ 10 = 24
b ) 2 is a factor of 233
c ) 176 ÷ 21 = 3 remainder 8 d ) 24 × 16 = 384
e ) 32 × 23 = 436
f ) 80 × 90 = 720
What was Thelma’s final score out of 6?
2. Thiathu uses 25 beads to make one necklace.
a ) If Thiathu has 792 beads, how many necklaces can he make,
and how many beads will be left over?
b ) If Thiathu sells each necklace for R15 each, how much money
can he make?
3. A Grade 5 class of 32 learners plans an outing to Table Mountain. If
the bus hire costs R672, how much will they each have to pay?
4. An aeroplane flew a distance of 900 km in 180 minutes.
a ) How far did it fly in 1 minute?
b ) How far can the aeroplane fly in 2 hours at this speed?
5. If 16 boxes of shampoo cost R720, how much will 124 boxes of the
same shampoo cost?
6. 6 golf balls cost R125. What will 24 golf balls cost?
Topic 18: Division
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3 Term 3
The opening ceremony of the Olympic Games in 2008 in Beijing, China.
Drummers perform during the opening ceremony.
102
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Starting off
The summer Olympic Games in Beijing were
held in August 2008. The opening ceremony
was a great show of fireworks and artistic
performances, such as dancing and singing.
More than 2 000 artists took part.
1. How many interlocking rings make up the
Olympic symbol?
2. What are the colours of the Olympic rings?
What do these colours represent?
Children use their hands to form Olympic rings in
front of a mirror.
3. Look at the picture of the drummers and
then answer these questions.
a ) Your friend says that the pattern is
formed by repeating a single picture
of a drummer. The drummer faces a
different direction in every second row.
Do you agree with this description?
Explain your answer.
b ) Another friend says that every second
row of drummers is a copy of the other.
Do you agree with this description?
Explain.
Content covered in Term 3
Each ring represents one of the five parts of the
world that were joined together in the Olympic
movement: America, Africa, Asia, Australia
and Europe.
Platinum Maths Gr5_Term 3_CAPS.indd 103
Topic 19: Common fractions, Topic 20: Mass, Revision, Topic 21: Count, order,
compare and represent whole numbers, Topic 22: Addition and subtraction,
Revision, Topic 23: Viewing objects, Topic 24: Properties of 2D shapes,
Revision, Topic 25: Transformations, Topic 26: Temperature, Revision,
Topic 27: Data handling, Project, Topic 28: Numeric patterns, Revision,
Topic 29: Multiplication
:
103
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Topic
Common fractions
19
Maths ideas
Add and subtract common fractions
• Add and subtract
fractions with
the same
denominators.
In Topic 12, you learnt how to add fractions with the same
denominator:
3
8
8
4
1
__
__
+ __
+ __
= __
is eqivalent to __45
10
10
10
10
10
• Add and subtract
mixed numbers
with the same
denominators.
• Solve problems
with fractions.
You can also subtract fractions when they have the same
denominator.
Example
1. Peter has an orange that he has cut
into fifths. He decides to give __25 to his
friend. How much orange does Peter
have left?
5 __
__
– 2 = __35
5 5
2. Shaye has __78 of his pizza left. For lunch he eats another __48 of his
pizza. How much of his pizza is left?
7 __
__
– 4 = __38
8 8
ExErCiSE 19.1
1. Draw a picture that shows a birthday cake that has been cut into
12 equal pieces. Alan has 7 friends at his birthday party. Everyone
gets 1 piece of cake. Show how much of the cake is left over after
the party if Alan also has a piece of cake.
Did you know?
More than
680 000 000
(680 million) people
live in Africa. This is a
very big number, but
this number makes
up only one-tenth of
the world’s
population.
104
2. Draw diagrams to show the answers to these sums.
a ) __78 – __48 = □ b ) __25 + __35 = □ c ) __34 – __14 = □ d ) __16 + __46 = □
3. Do these calculations without drawing a picture.
2
1
b ) __
+ __
=□
a ) __47 – __27 = □
11
11
7 __
–1=□
c ) __
12 2
4. Fill in the missing fractions.
a ) __78 – □ = __18
□
□
+ __14 = __24
c)
□
3
1
d ) __
+ __
=□
10
10
b ) __45 – □ = __15
□
6
2
□
d)
– __
= __
11
11
□
Term 3
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Mixed numbers
Key words
• mixed number – a
whole number and
a fraction making
one number
Look at this number line. Each section is divided into thirds.
0
1
0
_
3
1
_
3
2
3
_
3
2
_
3
5
_
3
4
_
3
3
6
_
3
8
_
3
7
_
3
4
10
__
3
9
_
3
11
__
3
12
__
3
On this line you are able to count up from 0 in thirds: __03 , __13 , __23 and __33 .
You know that __03 = 0; __33 = 1 whole; __63 = 2 wholes; __93 = 3 wholes and
1__23 = 4 wholes.
If you redraw the number line, you can use whole numbers as well as
fractions.
1
_
3
{
2
_
3
0
5
_
3
4
_
3
1
8
_
3
7
_
3
2
10
__
3
3
11
__
3
4
Now look at the fraction __43 .
This is written as __43 = __33 + __13 = 1 __13 .
This new number is called a mixed number because it has a whole
number part and a fraction part. If you use these mixed numbers on
the number line, then the line will look like this:
0
1
_
3
2
_
3
1
1_13
1_23
2
2_13
2_23
3
3_13
3_23
4
4_13
4_23
5
ExErCiSE 19.2
Did you know?
• The ancient
Egyptians used
unit fractions
to write all their
fractions. A unit
fraction always has
1 as its numerator.
• They put a mouth
picture above a
number to make it
into a unit fraction.
• The mouth picture
means ‘part’.
• They wrote other
fractions as the
sum of different
unit fractions.
1. Write in the missing numbers to show that you can count from 1
to 3 in thirds. 1; 1__13 ; □; 2; □; 2__23 ; □; □; □; 3
2. Fill in the missing numbers to show how you can count backwards
from 3 to 0 in halves. 3; □; 2; □; □; □; 0
3. Draw your own number line to show how you can count from:
a ) 0 to 4 in fifths
b ) 4 to 7 in quarters.
1
__
5
Challenge
Write these fractions as the sum of unit fractions, which have a numerator
of 1. Use as few unit fractions as possible. For example:
3
1
1
1
__
= __
+ __
+ __
20
20
20
20
2
1
+ __
= __
20
20
1
1
+ __
= __
10
20
13
a) __
20
b)
99
__
10
7
c) __
17
41
d) __
45
Topic 19: Common fractions
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Add and subtract mixed numbers
You can also do calculations with mixed numbers. Add or subtract the
whole numbers first.
Example
a) Calculate 1__13 + 5__13 .
1__13 + 5__13
1
1
2
+ __
) = 6 + __
= 6__23
= (1 + 5) + (__
3
3
3
b) Calculate 9__45 – 3__25 .
4 __
2
– 2 = __
, so the answer is 6__25 .
9 – 3 = 6 and __
5 5
5
c) Calculate 6__34 + 4__24 .
3
2
+ __
)
(6 + 4) + (__
4
4
5
4
1
= 10 + __
+ __
= 10 + 1 + __14 = 11__14
= 10 + __
4
4
4
d) Calculate 12__16 – 8__56 .
5
__
from __16 , so you should break down the number 12 to help you
6
subtract them.
12 __16 – 8 __56
6
1
1
– 8__56 = 11 + __
+ __
– 8__56
= 11 + 1 + __
6
6
6
= 11__76 – 8__56
Now you can subtract in the usual way:
7 __
2
1
– 5 = __
. So, 12__16 – 8__56 = 3__26 = 3__
.
11 – 8 = 3 and __
6 6
6
3
ExErCiSE 19.3
Calculate these mixed number sums.
3
5
8
1
+ 10__
2. 2__
+ 9__
1. 6__
10
10
12
12
3. 1__16 + 2__26 + 3__16 4. 4__28 + 4__58 + 4__38
6. 13__15 – 11__45
5. 8__34 – 2__14
+
6
10
7. 7__37 – 5__47
3
__
10
5 __
4
8. 5__
+ 1__
– 3
11
11 11
1
__
10
106
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Solve problems that involve fractions
In this exercise, you will practise what you have learnt in this topic by
solving more fraction word problems.
ExErCiSE 19.4
1. Sandy finished __38 of her homework before supper and
another __48 after supper.
a ) What fraction of her homework did Sandy finish?
b ) What fraction of her homework will Sandy still have to do
before school the next day?
2. Hlengiwe has 2__58 bags of sweets and Craig has 3__78 bags of
sweets. If they combine their sweets, how many bags of sweets
will they have together? Write your answer as a mixed number.
5
6
of his salary on rent and another __
of his salary
3. Lucas spent __
12
12
on food.
a ) What fraction of his salary did he use?
b ) What fraction of his salary was left over after these expenses?
4. A gardener collected 12__14 bags of raked leaves. After lunch, he
found that the dog had emptied out 5__34 of the bags. How many
bags were still full of leaves?
5. Moyo added 3__13 kg of mince to 2__23 kg of spaghetti. Later he also
added 1__13 kg of onion and tomato mix. What was the total mass of
all the ingredients for this meal?
6. Two planks of wood are laid out end to end and nailed together.
Their lengths are 4__24 m and 3__14 of a metre. The builder decides that
the planks are too long, so he saws off 2__14 m. How much wood is
left nailed together?
Topic 19: Common fractions
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Topic
Mass
20
Maths ideas
• Measure, compare,
order and estimate
mass.
• Calculate using
units of mass.
• Convert grams and
kilograms.
• Solve problems
that involve mass.
Key words
• mass – the
amount of matter
in an object
Estimate, measure and compare
masses
You measure the mass of an object to find out how much matter is in
the object. An object with more matter packed closely together will
have a larger mass.
The two units of mass that you will use are grams (g) and kilograms
(kg). You use grams for the mass of lighter objects, and kilograms for
the mass of heavier objects.
It is useful to estimate the mass of everyday objects. To estimate the mass
of an object, hold the object to see how heavy or light it is. Compare its
‘lightness’ or ‘heaviness’ with the mass of an object you do know.
A paperclip has a mass of about 1 g.
Small masses are measured in grams.
A bunch of six bananas has a mass of
about 1 kg. There are 1 000 g in 1 kg.
It is important to choose the correct unit of mass when you estimate
the mass of an object.
Example
Choose the most suitable unit of mass for each object below.
1. A television: The mass of a television is much greater than the
mass of a paperclip. The appropriate unit of mass is kilograms.
2. A tennis ball: The mass of a tennis ball is much less than the mass
of a bunch of six bananas. The correct unit of mass is grams.
ExErCiSE 20.1
Choose the most suitable unit of mass to measure these objects.
1. a pencil
3. your empty school bag
5. your desk
108
2. your school bag with books
4. your own mass
6. a large bag of flour
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You often need to measure the mass of an object very accurately. To
do this, you use a scale.
A bathroom scale A kitchen scale
A balance scale
Key words
• scale – an
instrument used to
measure mass
A digital scale
Example
What mass is shown on the scale?
The pointer is between 100 g and 200 g.
There are 4 equal spaces between 100 g and 200 g. Each space
represents 100 g ÷ 4 = 25 g.
The pointer is after the 1st space, so the mass = 100 g + 25 g = 125 g.
ExErCiSE 20.2
1. Look at the readings on the three scales below.
A
B
C
a ) Write down the mass shown on each scale in grams and kilograms.
b ) Order these masses from lightest to heaviest.
2. Complete this table. Use a scale to find the mass of each item.
Then order each actual mass from lightest to heaviest.
Object
Unit of
mass to use
Type of
scale
Estimated
mass
Actual
mass
Difference
in mass
A brick
A 500 ml bottle of water
1 litre of water
A soccer ball
Topic 20: Mass
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Key words
Convert units of mass
• kilo – means
one thousand,
therefore kilogram
means 1 000 g
Sometimes measurements are given in units not suitable for you to
work with. You can convert the measurement into units that you can
work with.
To convert from a larger unit to a smaller unit, you need to multiply.
This means that to convert kilograms to grams you multiply by 1 000.
Example
5 kg = □ g
(5 × 1 000) g = 5 000 g
← (larger unit to a smaller unit)
To convert from a smaller unit to a larger unit, you need to divide. This
means that to convert grams to kilograms you divide by 1 000.
Example
33 400 g = □ kg
(33 000 ÷ 1 000) kg = 33 kg ← (smaller unit to a larger unit), and there are
There are 1 000 grams in
1 kilogram.
1 kg = 1 000 g
Challenge
1. Find a partner
to discuss this
question, ‘Are
bigger objects
always heavier
than smaller
objects?’
2. Find five small
objects and
five big objects.
Compare their
masses.
3. Do you think
that a small
object can be
heavier than a
large object?
Explain.
110
400 grams left over, so 33 400 g = 33 kg 400g
Example
Convert __34 kg to grams.
1
__
of 1 kg = 1 000 g ÷ 4 = 250 g
4
3
__
of 1 kg = 3 × 250 g = 750 g
4
Have you seen a bag of sugar in the shops labelled ‘2,5 kg’? This is
the same as 2__12 kg. When you see 0,5 kg remember that this means
1
__
kg or 500 g.
2
ExErCiSE 20.3
1. Convert each kilogram measurement to grams.
a ) 3 kg = □ g
b) 10 kg = □ g
c) 25 kg = □ g
1
__
e) 2,5 kg = □ g
f ) 83 kg = □ g
d ) 12 4 kg = □ g
2. Convert each gram measurement to kilograms.
a ) 9 000 g = □ kg
b) 18 000 g = □ kg
c ) 50 000 g = □ kg
d) 8 500 g = □ kg □ g
e ) 12 700 g = □ kg □ g
f ) 45 000 g = □ kg
3. How many grams are there in these amounts?
b ) 6__12 kg
c ) 10 kg
a ) __18 kg
d ) 15__34 kg
4. How many kilograms and grams are there in these amounts?
a ) 2 500 g
b ) 5 500 g
c ) 7 250 g
d ) 12 500 g
Term 3
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Calculate with units of mass
You can only add and subtract masses if they are in the same units.
Example
Calculate: 3__12 kg + 500 g – 2__14 kg
1
__
of 1 kg is __12 of 1 000 g = 500 g and __14 of 1 kg is __14 of 1 000 g = 250 g
2
3__12 kg + 500 g – 2__14 kg
= 3 500 g + 500 g − 2 250 g
= 4 000 g − 2 250 g
= 1 750 g
Did you know?
The average mass of
a human brain is
about 1,35 kg. The
average mass of an
elephant’s brain is
5,4 kg and that of a
sperm whale is 9 kg.
If you have to multiply or divide a mass, use the methods you know.
Remember to write the unit of measurement in the answer.
Example
1. Calculate 515 g × 4.
515 g × 4 = 2 060 g = 2 000g + 60 g or 2 kg 60 g
2. Calculate 165 kg 33 g ÷ 11.
165 kg 33 g ÷ 11 = 165 kg ÷11 and 33 g ÷ 11 = 15 kg 3 g
ExErCiSE 20.4
1. Arrange these masses in descending order.
a ) 5 kg; 12,5 kg; 2__14 kg; 3 000 g; 10 000 g
b ) 150 g; 1 kg; 7 500 g; 2 kg; __12 kg
2. Round off each mass to the nearest 10 kg.
a ) 123 kg
b ) 101 kg
c ) 97 kg
3. Round off each mass to the nearest 1 000 g.
a ) 998 g
b ) 1 201 g
c ) 2 580 g.
4. Do these calculations.
a ) 34,5 kg + 32 kg – 4 kg 500 g
b ) 250 g × 6 (give your answer in kg and g)
c ) __23 of 1,5 kg (give your answer in g)
d ) 456 g × 24 (give your answer in kg)
e ) 495 kg ÷ 15 (give your answer in g)
f ) Find the difference between 1 kg 35 g and 5 kg 40 g.
g ) How much mass is needed to make up 9 kg from 5 kg 40 g?
Topic 20: Mass
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Did you know?
• Throughout
history, people
have needed to
measure the mass
of objects.
• One of the earliest
units of mass was
a grain of wheat
or a grain of barley
corn.
• They used this unit
to measure the
masses of precious
metals, such as
silver and gold.
Challenge
Sammy needs to buy
rice. The shop
advertises rice as
follows.
100 g
300 g
R1,50
R9,00
Solve problems that involve mass
Example
Riaaz brought 340 g of clay to school for a project. He gave 110 g
of the clay to Simon and 100 g of the clay to Vuyo. How many
grams of clay did Riaaz keep?
Find how much Riaaz gave away altogether: 110 g + 100 g = 210 g
Subtract this from what he started with: 340 g – 210 g = 130 g
Riaaz kept 130 g of clay.
Example
Mrs Ndloku wants to put the same amount of food in eight chicken
cages. She has 2 kg 600 g of food. How many grams of food will
she put in each cage?
First convert 2 kg 600 g to grams: (2 × 1 000) g + 600 g = 2 600 g.
Divide 2 600 g by 8.
Use multiplication facts to help you:
8 × 100 = 800, 8 × 200 = 1 600, 8 × 300 = 2 400
8 × 50 = 400, 8 × 25 = 200
The closest multiplication to 2 600 is 2 400, so start with 8 × 300.
Multiply
Subtract
300 × 8 = 2 400
2 600 – 2 400 = 200
25 × 8 = 200
200 – 200 = 0
She will put 325 g in each cage.
ExErCiSE 20.5
1 kg
1. A recipe for a chocolate cake uses 250 g of flour for every 3 eggs. If
9 eggs are used, how much flour must be used?
R20
2. An elephant eats 283 kg of plants each day. How many kilograms
of plants does an elephant eat in one week?
1. Which packet
of rice should
Sammy buy to
get the best value
for his money?
2. Give a reason for
your answer.
112
3. A lion has a mass of 248 kg, an elephant has a mass of 3 596 kg
and a rhinoceros has a mass of 1 298 kg.
a ) What is the combined mass of these three animals?
b ) What is the difference in mass between the heaviest and the
lightest animal?
4. Calculate the total cost of buying 1__12 kg of nuts at R100 per
kilogram, 2__14 kg of sweets at R36 per kilogram and 500 g of
potatoes at R2 per 100 grams.
Term 3
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revision
1. Do these calculations.
a ) __16 + __36 + __16
5. Convert these units of mass.
(1)
a ) 2__12 kg to grams
(1)
b ) 32 000 grams to kg
(1)
(1)
b ) __47 + __27 − __37
6. Five friends have the following masses:
(1)
c ) 3__13 + 2__13
35 kg; 41 kg; 33 kg; 42 kg and 39 kg.
A sign in the lift says that it can carry a
(1)
d ) 4__14 − 2__34
maximum of 11 people and a maximum
2
2
1
__
__
__
(2)
e ) 43 + 3 − 13
mass of 600 kg.
a ) How many more people are allowed
2. Write down one equivalent fraction for each
to get into the lift with the five
fraction.
friends?
(1)
(1)
a ) __23
b ) What is the maximum combined
1
__
(1)
b) 2
mass that the people in Question a)
may have?
(1)
(1)
c ) __6
9
Total marks: 20
3. James plants __38 of his tomato seeds
on Thursday and __58 of the seeds
on Friday. What fraction of the seeds
does he still have to plant
on Saturday?
(2)
4. Complete the table by writing in the
unit that you would use to measure
the mass of each object.
(5)
Object
Unit of mass
your pen
a cow
ruler
your friend
a potato
Revision
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Count, order, compare and
represent whole numbers
Topic
21
Maths ideas
• Round whole
numbers to the
nearest 5, 10 , 100
and 1 000.
• Compare and
order whole
numbers to
6 digits using <
and >.
• Work with place
value.
round off numbers
You already know that rounding numbers is useful when you need to
estimate an answer. Up to now you have rounded off numbers to the
nearest 10 or 100 or 1 000. Now you will also learn how to round off to
the nearest 5.
When you round a number to the nearest 5, you must find which
multiple of 5 is closest to your number.
Example
Round 18 to the nearest 5.
The multiples of 5 are 5, 10, 15, 20 and so on.
The multiples of 5 that are closest to 18 are 15 and 20.
18 – 15 = 3
20 – 18 = 2
18 is closest to 20.
So, 18 rounded to the nearest 5 is 20.
ExErCiSE 21.1
At a school sale, the prices are rounded to the nearest R5 so that no
one has to give change in smaller coins.
1. Round each price to the nearest R5.
a ) R3
b ) R36
c ) R29
e ) R9
f ) R19
g ) R11
d ) R21
h ) R12
2. If the stalls sell one of each item, will the school make more money
or less money by rounding the prices to the nearest R5?
ExErCiSE 21.2
1. Round the following numbers to the nearest 10 and then to the
nearest 100:
a ) 174
b ) 264
c ) 219
d ) 305
e ) 392
f ) 436
g ) 598
h ) 627
2. Round the following numbers to the nearest 1 000:
a ) 1 342
b ) 2 895
c ) 2 387
d ) 4 921
e ) 25 298
f ) 19 967
g ) 56 388
h ) 99 784
114
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Expand and compare whole numbers
You write large numbers in expanded notation. This can help you to
compare large numbers.
ExErCiSE 21.3
1. Write these numbers in expanded notation.
a ) 5 719
b ) 37 926
c ) 213 040
d ) 928 008
2. Write these expanded additions as whole numbers.
a ) 70 000 + 5 000 + 400 + 30 + 6
b ) 300 000 + 60 000 + 800 + 50 + 6
When you compare numbers, compare the digits starting on the left.
Remember that < means smaller than, and > means larger than.
76 574 > 76 386 means that 6 574 is bigger than 6 386. The hundreds
digit tells you which is the bigger number.
Did you know?
Over 4 000 years ago,
the ancient Egyptians
had their own way of
recording numbers.
They used these
symbols:
1
2
3
4
5
6
7
8
9
10
20
100
200
ExErCiSE 21.4
1. Compare each pair of numbers by writing in < or >.
a ) 7 142 □ 7 264
b ) 68 236 □ 68 223
c ) 174 679 □ 174 769
d ) 399 989 □ 399 998
2. Five schools have been raising money for a charity. The board
shows how much money each school raised.
Order the list of schools starting with the school that raised the
most money to the school that raised the least money.
Challenge
1. How do you think Egyptians wrote these numbers?
a ) 12
b) 17
c ) 24
d) 33
f ) 102
g) 122
h) 351
i ) 409
e ) 56
2. With a partner, make up a useful symbol for 1 000.
3. Use your symbol with the ancient Egyptian symbols to write:
a ) 1 206
b ) 4 587
c ) 5 005
d) 7 612
Topic 21: Count, order, compare and represent whole numbers
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Topic
Addition and subtraction
22
Maths ideas
• Add and subtract
whole numbers
using the column
method.
• Solve multi-step
problems that
involve addition
and subtraction,
with totals to
100 000.
• Round off to
estimate answers.
• Check answers
using inverse
operations.
Game
Play in pairs. Cut out
squares of paper. On
each write down one
number between
10 000 and 50 000.
Now each player
writes down the
number 10 000 on a
separate piece of
paper. Take turns to
pick up a piece of
paper and add that to
your number until all
the papers have been
used. The player with
highest number at the
end wins.
Key words
• column method –
add and subtract by
writing digits with
the same value
below each other in
the same column
116
Add numbers in columns
You know that you can add numbers by expanding the numbers and
writing them in columns.
In the column method you add in the same way, but to save time
you only write down the digits and not the zeros. Write digits with
the same place value in the same column, one under the other. Add
each column separately starting with the units column. If any column
adds up to 10 or more, write down that number in two parts: the ten
goes to the next column on the left. This is because in the place value
system you multiply by 10 to get the next place value on the left.
Example
Add: 42 345 + 24 478
TTh Th
4
2
4
+ 2
6
6
H
3
4
8
1
Add: 45 678 + 7 680
T
4
7
2
1
U
5
8
3
TTh
1
4
+
5
Th
1
5
7
3
5 + 8 = 13
4 + 7 + 1 = 12
H
6
6
3
1
T
7
8
5
U
8
0
8
7 + 8 = 15
6 + 6 + 1 = 13
5 + 7 + 1 = 13
ExErCiSE 22.1
1. Add these numbers using the column method. First round off the
numbers to the nearest 1 000 and estimate the answer.
a ) 8 476 + 9 817
b ) 2 390 + 9 999
c ) 60 089 + 32 987
d ) 4 056 + 29 476 e ) 19 099 + 79 011 f ) 12 323 + 7 890
2. Fill in the missing numbers. What do you notice?
a ) 2 436 + 4 523 = 4 523 + □
b ) 11 246 + □ = 10 978 + 11 246
c ) □ + 39 813 = □ + 52 169
3. Complete the following.
12 369 + (2 500 + 17 101) = (12 369 + □) + 17 101
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Subtract numbers in columns
You can also use the column method to subtract numbers. First revise
how to subtract in expanded form.
Example
Find 68 237 − 54 684.
Break down 8 237 so that you can do all the subtractions.
8 237 = 7 000 + 1 000 + 237 = 7 000 + 1 100 + 130 + 7.
68 237 − 54 684 = 60 000 + 7 000 + 1 100 + 130 + 7
50 000 + 4 000 + 600 + 80 + 4
Difference
10 000 + 3 000 + 500 + 50 + 3 = 13 553
Now you will do the same subtraction in columns without writing the
zeros. Remember that in the place value system, each place value is
10 times the place value on the right: 1 000 = 10 × 100, 100
= 10 × 10 and so on. This means that when you cannot subtract, you
can take a number from the column on the left and write it as 10 in
the column where you are subtracting.
Example
−
6
5
8
4
2
6
?
1
3
8
5
1
7 ← Write 200 as 100 + 100, or 100 + 10 tens.
4 ← Now you have 13 tens.
3
Next take 1 from the thousands column and give it to the
hundreds column.
−
6
5
1
8
4
3
7
2
6
5
11
3
8
5
1
7 ← Write 8 000 as 7 000 + 10 hundreds.
4 ← Now you have 11 hundreds.
3
ExErCiSE 22.2
First round off to estimate these answers and then subtract the numbers
in columns without expanding. Use addition to check your answers.
1. 6 742 – 3 458
2. 7 219 – 4 733
3. 3 856 – 1 976
4. 23 107 – 19 872
5. 56 122 – 28 001
6. 60 671 – 3 940
Topic 22: Count, order, compare and represent whole numbers
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Solve addition and subtraction
problems
In this section, you will solve addition and subtraction problems. You
can use these ideas to help you:
• Write down the calculation that you need to do.
• Estimate the answer.
• Use the column method of addition to find totals.
• Use the column method of subtraction to find differences.
• Check your answer by using an inverse operation.
ExErCiSE 22.3
1. a ) In November, 21 946 people visited the National Park. In
December, there were 24 187 visitors. How many people visited
the National Park altogether in November and December?
b ) In January, 23 576 people visited the National Park. In
February, there were 19 289 visitors. How many more people
visited the National Park in January than in February?
c ) The total number of visitors in January, February and March
was 68 092. How many people visited the park in March?
National
Park
entrance
fees
Adults:
R28
Children:
R14,50
2. The population of Higher Town is 80 936. The population of Lower
Town is 43 486 less than the population of Higher Town. What is
the population of Lower Town?
3. Michelle’s mother has saved R41 550 for a house. Sindi’s mother
has saved R12 950 more than this.
a ) How much money has Sindi’s mother saved?
b ) If they put their money together, how much would they have
in total?
c ) How much more must each mother save to have R80 000?
Challenge
Make up some more
word problems of
your own. Use
five-digit whole
numbers. Work out
the answers to your
problems.
118
4. On Saturday and Wednesday, 64 916 people went to soccer
matches. On Wednesday, 26 947 people went to the soccer match.
How many people went to the soccer match on Saturday?
5. An explorer plans to travel a total distance of 85 000 km. In the first
three weeks he travels 12 350 km, 18 048 km and 10 990 km.
a ) How many kilometres did he travel in the first three weeks?
b ) How many kilometres must he still travel?
Term 3
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revision
1. Write each of these numbers as expanded addition.
a ) 3 457
b ) 29 416
(1)
(1)
2. Write these expanded additions as whole numbers.
a ) 40 000 + 5 000 + 300 + 60 + 2
b ) 700 + 5 000 + 3 + 40 + 20 000
(1)
(1)
3. Compare these pairs of numbers by writing < or >.
a ) 23 978 □ 23 767
b ) 570 835 □ 571 893
(1)
(1)
4. Round these number to the nearest 5.
a ) 78
b ) 54
c ) 712
(1)
(1)
(1)
5. Last Saturday, 18 578 people went to watch City play a soccer match, On Wednesday, City
played another match and 16 442 people went to watch. Approximately how many people
watched the two matches altogether? Round off your answer to the nearest 100.
(3)
6. Use any method to do these calculations.
a ) 53 456 + 12 943
b ) 64 129 + 11 978
c ) 24 160 + 42 974
d ) 17 935 + 28 887
(1)
(1)
(1)
(1)
7. Use the column method to do these calculations.
a ) 56 473 – 31 594
b ) 60 070 – 12 598
(1)
(1)
8. In June, 73 580 visited the Waterfront in Cape Town. This is 2 579 more than the number of
people who visited in April. How many people visited the Waterfront in April?
(2)
Total marks: 20
Revision
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Topic
Viewing objects
23
Maths ideas
• Identify objects
from different
viewpoints
• Match the views
of objects with
the position of the
viewer.
This photograph
has been taken by
a photographer at
the entrance into the
stadium.
identify objects from different
viewpoints
Objects look different depending on
where you are standing when you
look at them.
Look at these photographs of the
Nelson Mandela Stadium in Port
Elizabeth. Where do you think the
photographer was positioned for
each picture?
In the next exercise, you will match
the view of an object with the
position of the person viewing the
object.
This photograph has been
taken by a photographer in
a helicopter flying over the
stadium.
ExErCiSE 23.1
Write down the position the photographer was standing in to take the
following photographs.
1. a )
b)
c)
This photograph
has been taken by a
photographer inside
the stadium.
120
2. a )
b)
3. a )
b)
Term 3
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Match the view with the position
of the viewer
Key words
• evacuation plan –
a plan of a building
that shows the
quickest way out
of the building to a
place of safety
In the next exercise, you will need to identify the viewpoint of
different objects.
ExErCiSE 23.2
• viewpoint –
position from
which you view an
object
1. Look at this picture of a
garden shed.
The pictures below show the
view if you:
• are standing on a ladder
looking down at the shed
• are standing in front of the
shed, or
• are standing at the side of
the shed.
Decide where you would be standing in each of these cases.
a)
b)
c)
2. Most big buildings that have many people working or living in
them will have an evacuation plan in every room. This plan shows
the quickest route out of the building to a place of safety.
a ) Check if your classroom has an evacuation plan. Make sure you
know your school’s evacuation plan.
b ) Draw a plan of your school building showing the quickest
route to the school’s sports field or playground from your
classroom.
Topic 23: Viewing objects
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Topic
Properties of 2D shapes
24
Maths ideas
• Recognise and
name twodimensional (2D)
shapes.
• Describe and
compare twodimensional (2D)
shapes.
• Draw twodimensional (2D)
shapes.
identify, describe and compare
2D shapes
You can use sides and angles to name 2D shapes.
ExErCiSE 24.1
1. Study the 2D shapes below and answer the questions that follow.
A
B
C
D
E
F
G
H
I
J
K
L
• Identify twodimensional (2D)
shapes around us.
Challenge
Samir says that he
has discovered a new
definition for a
rectangle:
A rectangle is a
quadrilateral that has
two pairs of equal
sides.
1. Do you agree
with Samir?
Explain your
answer
2. If you do not
agree with Samir,
try to prove him
wrong. Draw
one or more
quadrilaterals that
are not rectangles
but have two
pairs of equal
sides.
122
a)
b)
c)
d)
e)
f)
g)
h)
Which shapes have only curved sides?
Which shapes are polygons?
Name the polygons that you see.
Which polygons have all equal sides?
Which polygons have all equal angles?
Which polygons have all angles smaller than a right angle?
Which polygons have only some right angles?
Which polygons have all their angles as right angles?
2. Draw an example of:
a ) a quadrilateral with exactly two right angles
b ) a pentagon with only one right angle.
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ExErCiSE 24.2
1. Study this picture. Then answer the questions that follow.
a ) How many sides does Shape A have?
b ) How many sides does Shape B have?
c ) What is the name given to shapes that have the same number
of sides that Shape B has?
d ) Look at the corner of a page. What type of angle is formed by
the corner?
e ) Use the corner of the piece of paper to test the size of the
angles of both Shape A and Shape B. What do you notice?
A
B
2. Find the lengths of the sides of Shape A and Shape B, using the
square dotted paper. Mark the equal sides using markers as you
have been shown before.
3. Copy and complete each statement below using one of these
words: square; rectangle; quadrilaterals; right angles; four.
a ) Shape A and Shape B both have _________ sides. They can
therefore both be called _________.
b ) All the angles of both Shape A and Shape B are equal in size
and are called _________.
c ) Shape B has two pairs of opposite sides equal and is called a
_________.
d ) Shape A has all sides equal in length and all angles equal in
size. It is called a _________.
A
4. Here are some more 2D shapes.
a ) Which shapes, labelled A, B, C, D and E, have the same number
of sides?
B
b ) Which shapes have equal sides and equal angles? A
c ) Which shapes have one or more right angles?
d ) Which shapes are quadrilaterals?
A
B
C
A
B
B
C
D
C
D
E
E
Challenge
1. Take a 10 cm piece of string and tie the ends together.
A
B
C
2. What shape can you form with your loop of string? Remember, you
can also form shapes with curved sides.
D
E
3. Can you form a circle?
Topic 24: Properties of 2D shapes
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D
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09/02/13 12:41 PM
ExErCiSE 24.3
1. What shape is a piece of A4 paper?
• Place the paper horizontally, so that the longer side is on top.
• Fold the bottom left corner up to the top of the sheet so that
the edges of the paper line up. Press the fold with your thumb.
• Now cut off the extra piece of paper that does not form a part
of your triangle.
2. What is the shape of the piece that you cut off?
3. Unfold your triangle. What shape have you formed?
• Fold this shape to find all four lines of symmetry and press the
folds.
• Open up the shape and put it straight in front of you.
• Fold the top two corners inwards to the middle of the shape
and press the folds.
4. What shape do you have?
5. Describe the sizes of all the angles in the shape.
6. This shape has one line of symmetry. Fold the shape closed on its
line of symmetry. How many sides does your new shape have?
7. Name your new shape.
8. Think of your new shape as a composite shape. Which two shapes
is your composite shape made of?
124
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Draw 2D shapes
ExErCiSE 24.4
Look at this picture of a pot plant. It is made up of 2D shapes.
1. Identify all the 2D shapes in the picture. Remember that some of
the shapes may be part of a larger shape that you can
also name.
2. Copy the picture onto square dotted grid paper.
3. Is the picture symmetrical?
4. What would you move to make the picture symmetrical?
5. Cut out each of the 2D shapes in your copied picture.
6. Put the parts back together in a different way to make the picture
symmetrical.
7. Now choose only the parts that form the leaves and the pot of the
pot plant.
a ) What shape is the complete leaf?
b ) What two shapes make up the leaf?
c ) What shape is the complete pot?
8. Remove the rectangle that forms the lip of the pot. What shape are
you left with?
9. Which shapes make up the pot?
10. Create your own picture on square dotted grid paper made of
2D shapes.
Topic 24: Properties of 2D shapes
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09/02/13 12:41 PM
2D shapes in African art
Geometric (2D) shapes are used in many African patterns and art
works. These patterns often decorate buildings, beadwork and
basket work.
You can use these steps to create a small part of an African pattern on
square dotted grid paper.
Key words
• midpoint – the
point that divides a
line into two equal
parts
Step 1: Draw a six block by four block rectangle.
Step 2: Find the middle of each side.
Step 3: Draw a line joining the middle of each side to the middle of
the next side.
Step 4: Draw a line through the midpoints of the longer sides.
Step 5: Repeat these rectangles three times next to each other to
create a pattern.
Step 6: Colour each different shape in a different colour.
ExErCiSE 24.5
1. Use square dotted grid paper to create your own pattern with
geometric shapes.
2. Identify the shapes you used in the pattern.
3. Colour each different shape in a different colour.
4. Look at a partner‘s pattern and copy it.
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revision
1. Work in groups of 3 for this question.
a ) As a group, choose an object in your classroom. Each member of the group draws a
different view of the object.
b ) Exchange your three pictures with each other. Work out where the person who
drew each view was standing in relation to the object that they drew.
(2)
(3)
2. Write down the position the photographer was standing in to take each of these
photographs.
a)
b)
c)
(3)
3. Is a square also a rectangle? Explain your answer.
(2)
4. Is a rectangle also a square? Explain your answer.
(2)
5. Draw each of the following shapes on square dotted grid paper.
a ) A quadrilateral with four right angles and all sides equal.
b ) A five-sided shape with two right angles.
c ) A six-sided shape with at least one line of symmetry.
d ) A seven-sided shape.
(1)
(1)
(1)
(1)
6. Name each of the shapes described in Question 4.
(4)
Total marks: 20
Revision
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09/02/13 12:42 PM
Topic
Transformations
25
Maths ideas
• Investigate
patterns in nature.
• Use
transformations
to describe the
movement of
shapes.
• Make composite
shapes from
two-dimensional
shapes.
Key words
• transformation
– change in the
position and/or
the direction of a
shape
• translate – to
transform a shape
by sliding it
• reflect – when you
flip a shape (as in a
mirror image)
• rotate – to
transform a shape
by turning it
Use transformations to describe
the movement of shapes
Translation, reflection and rotation are transformations that describe
the movement of shapes.
• You can translate (slide) a shape.
• You can reflect (flip) a shape.
• You can rotate (turn) a shape.
Example
Look at the shape on the right.
We can translate (slide) it.
We can reflect (flip) it.
We can rotate (turn) it.
ExErCiSE 25.1
Look at the following pairs of shapes. In each case, say what you
would do to Shape A to get Shape B. Would you slide, turn or flip
Shape A? Give the correct mathematical name for the transformation
that you have chosen.
A
B
A
B
A B
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09/02/13 12:42 PM
Use transformations to make
composite shapes
Key words
A tangram is an ancient Chinese puzzle consisting of geometrical
shapes that can be combined to form different shapes. You will now
make a tangram like you did in Grade 4.
Step 1: Take the sleeve of a match box and flatten it.
Step 2: Use a ruler to draw a line diagonally across the flattened box.
Step 3: Cut the line through both sides of the box to form two triangles.
Step 4: Cut on every fold to form the tangram pieces.
• tangram – an
Ancient Chinese
puzzle consisting
of geometrical
pieces that fit
together to make a
square
Example
Choose one of your tangram pieces to start your investigation with.
Trace around the shape and then perform the following steps.
Step 1: Shape A is translated.
A
Step 2: Shape A is flipped over to form a reflection.
B
C
Step 3: Shape A is turned to form a rotation. Turning
the shape four times completes the rotation.
The pictures on the right show of each type of
transformation.
ExErCiSE 25.2
D
E
F
G
H
1. a) Use Shape A from the example to practise translating a shape
in a straight line at least four times.
b ) Use Shape A to translate the shape in a different way.
c ) Use Shape A to reflect the shape four times.
d ) Draw in any lines of symmetry that you see.
2. Look at this pattern.
a ) Find the tangram pieces that were used to create this pattern.
b ) Use your own tangram pieces to explain how the shapes were
transformed to create the pattern.
3. a ) Use any of your tangram pieces to create a pattern. Trace
around a shape to form a pattern.
b ) Describe the transformations you used in your pattern.
Topic 25: Transformations
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Topic
Temperature
26
Maths ideas
• Read temperatures
from different
thermometers.
• Estimate
temperature.
• Order
temperatures.
• Solve problems
involving
temperature.
Estimate temperature
When you decide if you should wear a jacket, or if your food is cool
enough to eat, you are estimating temperature. You are deciding how
hot or how cold something is.
Water freezes at 0 °C and pure water boils at 100 °C.
These two temperatures describe the freezing point and the
boiling point of water. Hot water and hot food are around
60 °C to 70 °C.
ExErCiSE 26.1
Key words
• freezing point
– temperature
at which liquid
turns to solid, for
example, water
freezes to form ice
• boiling point –
the temperature
at which liquid
turns to gas, for
example, water
boils to form
vapour
Did you know?
To check if bath
water is the right
temperature for a
baby, put your elbow
in the water. The
water must not feel
hot or cold to you.
It should be around
37 °C .
130
1. Use the temperatures given above to estimate the temperatures in
the following pictures.
a)
b)
c)
d)
2. Look at the picture on the right and
answer the following questions:
a ) Estimate the air temperature from the
picture.
b ) Why is it useful to know what the
temperature will be each day before
you go to school?
3. Gary’s mom baked a cake. The recipe
stated that the temperature of the oven
should be 180 °C, but she set the oven at
225 °C. What was the difference between
the actual temperature and what the
recipe stated?
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Measure temperature
Key words
• thermometer – an
instrument you
use to measure
temperature
You use a thermometer to measure temperature in degrees Celsius
(°C). You can measure the temperature of water, food, the air, the
human body, an oven or a refrigerator. There are different kinds of
thermometers, and you need to choose the correct thermometer for
its purpose.
The temperature of a healthy human being is 36,9 °C, which is
about 37 °C.
ExErCiSE 26.2
Study the weather forecasts and answer the following questions:
1. Study the world temperature chart and name three places that are
A medical
experiencing cold temperatures.
2. What is the difference between the minimum temperature
and the maximum temperature in Johannesburg?
thermometer
An oven
thermometer
3. Find the difference between the minimum and maximum
temperatures in Geneva and in Durban.
4. According to this weather forecast, which city is the
coldest in South Africa?
5. Do you think it is summer or winter in South Africa?
A digital
thermometer
An outdoor
thermometer
World temperatures
City
Platinum Maths Gr5_Term 3_CAPS.indd 131
Temperature
Weather
Min °C
Max °C
Amsterdam
5
7
Rain
Beirut
16
23
P/Cloudy
Cairo
14
26
P/Cloudy
Geneva
0
3
Snow
London
5
11
Rain
New York
6
17
P/Cloudy
Perth
15
31
Clear
Tokyo
12
18
Clear
Topic 26: Temperature
131
09/02/13 12:43 PM
Key words
read and order temperatures
• maximum
temperature
– the highest
temperature
reading recorded
On page 131 you saw pictures of different types of thermometers. In
this section, you will work with an outdoor thermometer.
• minimum
temperature
– the lowest
temperature on
one day
The outdoor thermometer on the left shows the temperature as
25 degrees Celsius or 25 °C. This was the maximum temperature
(highest temperature) one day in Cape Town. A minimum
temperature is the lowest temperature reading on one day.
Example
ExErCiSE 26.3
1. Show these temperatures on a drawing of an outdoor
thermometer.
c ) 23 °C
d ) 24 °C
a ) 18 °C
b ) 31__12 °C
2. Read and write down the temperatures shown on the
thermometers below.
a)
b)
c)
3. Read the temperatures shown on the digital thermometers.
a)
b)
Did you know?
Temperatures below
0 °C are shown as
negative numbers.
These numbers have
a ‘–’ sign in front of
them and are used to
keep track of values
below a certain mark.
132
c)
4. Order the temperatures in Questions 3 from lowest to highest.
Calculate the difference between the highest and lowest
temperatures.
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revision
1. Look at the pattern that has been created
from 2D shapes.
H
c ) Which method is healthier? Why?
d ) How long is it going to take to bake
the fish?
e ) What temperature do you need
to make sure that the fish is well
cooked in an oven?
f ) How long is it going to take to
pan-fry the fish?
g ) What would happen if you tried
to bake the fish at a temperature
of 300°C?
(1)
(2)
(1)
(2)
(1)
a ) Are there any lines of symmetry in this
pattern?
(1) 4. Choose the most likely temperature
b ) If all the pentagons in the centre of the
in each case.
pattern were in the same colour, would
a ) your body temperature: 12 ºC, 36 ºC
the pattern have line symmetry?
(1)
or 56 ºC
(1)
c ) Identify the type of transformation that
b ) a pie taken out of the oven: 30 ºC,
is happening with the pentagons in
80 ºC or 300 ºC
(1)
the middle of the pattern.
(1)
c ) iced water: 4 ºC, 28 ºC or 43 ºC
(1)
d ) Find three examples of translation in
d ) the air on a hot day: 18 ºC, 32 ºC or
the pattern and describe them.
(3)
65 ºC
(1)
e ) boiling water: 56 ºC, 98 ºC or 180 ºC (1)
2. a ) Draw a pattern that uses at least two
different kinds of transformations.
(2) 5. Read these temperatures on a digital
b ) Describe the kinds of transformations
thermometer.
that you used to make your pattern. (2)
a)
(1)
3. Study the instructions on the packet in the
picture on how to cook the fish.
b)
(1)
c)
(1)
6. Order the temperatures in Question 5
from lowest to highest. Find the
difference between the highest and
lowest temperatures.
a ) What are the two methods that you can
use to cook this fish?
(2)
b ) Which method is quicker?
(1)
Total marks: 30
Revision
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(2)
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09/02/13 12:44 PM
Topic
Data handling
27
Maths ideas
Collect, organise and display data
• Collect, organise
and record data.
In Term 1 you collected data and organised it into tally tables. You also
drew pictographs and bar graphs to show the data.
• Draw pictographs
and bar graphs to
show data.
• Order data and
find the mode of a
dataset.
ExErCiSE 27.1
1. A group of learners was given this set of shapes by their teacher.
• Analyse and
interpret data.
• Answer questions
about data
and how it was
collected.
• Compare different
graphs on the
same topic.
a ) Copy and complete this tally table to show the number of
each type of shape.
Shape
Tally
Total
Square
Triangle
Circle
Oval
Star
Total number of shapes
b ) Draw a bar graph to compare the number of ovals and circles
in this set of shapes. Label the vertical scale in intervals of 2.
c ) Draw a pictograph to compare the number of shapes with
straight sides with the number of shapes with curved sides.
Use the symbol ⊕ to show four shapes.
2. Ask 10 learners in your Maths class how many brothers and sisters
they have.
a ) Draw up a tally table to collect and record your results.
b ) Draw a bar graph to show the data you collected.
3. Use the table and the graph you drew in Question 2 to answer
these questions.
a ) What is the most common number of brothers and sisters?
b ) How many learners have three or more brothers and sisters?
c ) How many learners have no brothers and sisters?
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ExErCiSE 27.2
1. This table shows the number of goals scored by
30 different soccer teams during a tournament.
a ) Draw up a tally table to organise the data.
b ) Draw a bar graph to show the data.
0
4
0
3
3
5
2
2
2
1
2
2
0
2
5
0
1
1
1
1
3
2
0
1
1
1
0
1
2
0
2. Use your graph from Question 1 to answer these questions.
a ) What was the highest number of goals scored?
b ) What was the lowest number of goals scored?
c ) How many teams scored 1 goal?
d ) How many teams did not score any goals at all?
e ) How many teams scored 3 or more goals?
f ) How many teams scored 1 or 2 goals?
3. Lillian and Abu did a survey to find out what flavour of chips was most
popular at the school tuck shop. These are their results for one week.
45 children chose plain
23 children chose salt and vinegar
19 children chose cheese puffs
24 children chose hot chilli
18 children chose other flavours
a ) Draw a pictograph to show the data.
= 4 children.
Use a scale of
b ) How could this graph be useful to
the people who own the tuck shop?
4. Carry out a short survey among the learners at your school to find
out what flavour of chips they like best.
a ) Draw up a table to collect and organise your data.
b ) Survey at least 12 learners.
c ) Combine your results with a partner.
d ) Decide which type of graph would be best to show the data
and draw it.
e ) Write a short paragraph explaining what your data shows and
how it compares to the data from Question 3.
5. Predict what you think the weather will be like for the next two weeks.
a ) Use a table like this to record your predictions.
Sunny days
Overcast days
Rainy days
b ) Draw a pictograph to show your weather predictions.
c ) Keep a record of the actual weather for the next two weeks.
How well did you predict the weather?
Topic 27: Data handling
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Key words
Find the mode
• bimodal – a
data set with two
modes
The mode is the data value that occurs most often. When two values
occur most often, the data has two modes and we say it is bimodal.
ExErCiSE 27.3
1. Here is a list of the number of goals scored by different netball
players in one year.
Player
Busi
Anna
Nomsa
Parvati
Jessica
Marie
Goals scored
4; 2; 0; 1; 1; 3; 4; 2; 3; 0; 1; 0
2; 1; 1; 3; 4; 4; 2; 3; 0; 1; 1; 0
1; 1; 2; 1; 3; 4; 1; 1; 0; 2; 1; 1
0; 2; 3; 1; 1; 2; 2; 4; 2; 2; 1; 0
0; 1; 0; 0; 0; 3; 3; 1; 0; 0; 3; 0
5; 3; 4; 3; 3; 1; 1; 1; 3; 1; 3; 0; 2
a ) Write each player’s scores in order from smallest to largest.
b ) Find the modal number of goals scored by each player.
c ) Which data set has two modes?
2. Three learners each toss a dice 15 times. These are their results:
a ) Complete this table to organise the scores.
Learner
1
2
3
4
5
6
Laila
Josh
Priya
b ) Who threw the most 6s?
c ) Who threw the most 3s?
d ) Find the modal score for each learner.
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ExErCiSE 27.4
1. A school nurse recorded the heights of 10 boys and 10 girls in
Grade 5 in centimetres. These are her results rounded to the
nearest 10.
Boys 120
Girls 110
a)
b)
c)
d)
140
130
120
130
110
120
150
140
140
100
110
130
130
140
110
120
130
120
Write each set of heights in order from shortest to tallest.
What height is the mode for boys?
What height is the mode for girls?
Why do you think the mode for boys is different to the mode
for girls?
2. A shop that sells children’s shoes kept a record of the shoe sizes
sold on a Saturday morning. These are the results.
2
3
3__12
4
3
4__12
3
2__12
3
2__12
3
3__12
4__12
4
3
2__12
4
3
2__12
3
4__12
3
2
3__12
2__12
4
4__12
3
2__12
4
3__12
4
4__12
a ) Complete this table to organise the data.
Shoe size
2
2__12
3
Number of pairs sold
b ) What is the modal shoe size?
c ) What do you think the modal size would be in a shop that sells
adult women’s shoes? Why?
d ) What do you think the modal size would be in a shop that sells
adult men’s shoes? Why?
3. A farmer recorded the maximum temperature in
Sutherland each day for a month. These are her
results.
a ) List the temperature in order from lowest to
highest.
b ) What was the lowest temperature recorded?
c ) What was the highest temperature recorded?
d ) What is the mode of this data?
e ) In which month do you think this data was
recorded? Why?
Topic 27: Data handling
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09/02/13 12:44 PM
Work through a data cycle
In Topic 7 you learnt that a data cycle is a process of asking questions,
collecting and organising data and summarising results.
In this exercise you are going to find out the shoe size of girls and
boys in Grade 5 at your school. Follow the steps in the exercise to carry
out your investigation.
ExErCiSE 27.5
1. Carry out your survey.
Choose at least 10 girls and 10 boys in Grade 5 to answer your
questions.
2. Organise and record your data
Use a table to record the data. Your table could look something
like this but you will need to think about what the smallest and
largest shoe sizes are likely to be.
Shoe size
Challenge
A friend says that
taller learners have
larger feet than
shorter learners.
Write down what you
would do to find out
if this is true or not
for learners in your
class.
138
Girls
Boys
3. Draw bar graphs to show your data
a ) Draw two separate bar graphs to show the data for boys
and girls.
b ) Put the shoe sizes on the horizontal axis and the number of
learners in intervals of 2 on the vertical axis.
c ) Don’t forget to give each graph a heading.
4. Interpret your data
Write a short paragraph about what you learnt from your survey.
Your paragraph should include
information about the modal
size for girls and the modal size
for boys. You must also talk
about the differences between
girls’ and boys’
shoe sizes.
Term 3
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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 and pie charts. You have also learnt
how to find the answers to questions by using the data you are given.
1. Study this graph carefully.
a ) What type of graph is this?
b ) What does this graph show you?
c ) How many people live alone?
d ) What is the largest number of people living in one home?
e ) How many homes have five or more people living there?
f ) What is the modal number of people per home?
g ) How many homes were surveyed to find this
information? Explain how you got your answer.
h ) Do you think this data was collected in a city or in a
rural area? Why?
Number of people living
in the home
ExErCiSE 27.6
9
8
7
6
5
4
3
2
1
1 2 3 4 5 6 7 8 9
Number of homes
2. Redraw the graph above as a pictograph. Use a scale of
= 3 homes.
3. Vusi lives in a rural area in Limpopo. He collected data about
the number of people per household in his community. This
is Vusi’s data.
Number of people
Number of households
3
4
5
6
7
8
9
10
11
12 or more
2
2
4
7
7
5
12
11
9
6
a ) What is the fewest number of people in a
household in this community?
b ) What is the greatest number of people in
a household in this community?
c ) What is the modal number of people per
household?
4. Vusi drew this pictograph to show his data.
Compare Vusi’s pictograph with the one you
drew in Question 2. Write a short paragraph
summarising the main differences between
the two graphs.
Number of persons in each household
3
4
5
Key
= 3 households
= 2 households
= 1 household
6
7
8
9
10
11
>12
Topic 27: Data handling
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Compare graphs
Sometimes graphs that show the same data can look quite different.
Some reasons for this are that the data was collected:
• from different groups of people so their answers were different
• in different places so the results were different
• at different times so different things were happening
• in different ways so the results were different.
Example
Mandla and Elton counted how many cars, taxis and buses passed the gate of their school
during a 15 minute period. They drew these bar graphs to show their results.
Number of different types of
vehicles passing the school
Number of different types of
vehicles passing the school
Taxis
Taxis
Buses
Buses
Cars
Cars
4
8
12
16
20
Number of vehicles
24
28
4
8
12
16
20
Number of vehicles
24
28
If you compare the graphs you can see that Mandla recorded a much higher number of
vehicles than Elton.
• Mandla collected his data at 07:30 in the morning just before school started. This means
that there were lots of vehicles passing the school to drop off learners.
• Elton collected his data at 10:30 during first break. At that time the street outside the
school was not busy. So he counted far fewer vehicles than Mandla.
Example
Nina and Maria investigated how much time learners spent on homework each week. They
drew these pictographs to show their results.
Number of hours spent on homework each week
3
2
6
5
1
Key
0
= 2 Learners
140
Number of hours spent on homework each week
4
3
Key
= 2 Learners
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• If you look at Nina’s graph it seems like most learners spend 1 or 2 hours on homework
each week.
• If you look at Maria’s graph it seems like most learners spend 5 or 6 hours on homework
each week.
Looking at their data sources:
• Nina did her survey of 30 Grade 2 and 3 learners.
• Maria did her survey of 30 Grade 7 learners.
Grade 2s and 3s don’t get much homework but Grade 7s do. They asked different groups of
people and that is why they got different results.
ExErCiSE 27.7
1. Work with a partner. Discuss how the following investigations into
the same topic could give different results.
a ) Jabu and Peter want to know if people go to church regularly.
Jabu did his survey by asking people at a church on a Sunday
morning. Peter did his survey by asking people at a shopping
mall on a Sunday morning.
b ) Samira and Hussein measure amount of rain that falls in their
town each day in July.
Samira lives in Cape Town.
Hussein lives in Durban.
c ) Nino and Themba want to know what type of car is most
popular.
Nino collected his data by observing the cars that passed his
school.
Themba collected his data by observing cars parked at a local
factory.
d ) Jessica and Simone want to know what TV programme is most
popular with the learners in their class.
Jessica asks 10 girls.
Simone asks 8 boys and 2 girls.
e ) Data about the facilities available at local schools is collected by:
Grade 5s at a school in a rural area.
Census at School from 2 500 schools spread across South Africa.
Challenge
1. Combine the
data for boys
and girls shown
on the graph in
Question 2 of
Exercise 27.2.
2. Use the
combined data
to draw one
graph that
shows the data
for all learners in
the survey.
3. Give your graph
a heading.
Topic 27: Data handling
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Favourite sports of boys
Favourite sports of girls
2. These two bar graphs show the
favourite sports of learners. The
data was collected from 2 500
No favourite sport
schools during the 2009 Census
Netball
at School.
a ) How are the graphs similar?
Athletics
b ) How are the graphs
Rugby
different?
c ) Which sport is the favourite
Cricket
among boys?
Tennis
d ) Which sport is the favourite
among girls?
Vollyball
0
1
2
3
__
__
__
e ) Which sports are not
5
5
5
Fraction of learners
popular with boys or girls?
f ) Why do you think these two graphs look different even
though they show the same data and the data was collected
at the same time?
Soccer
Soccer
No favourite sport
Netball
Athletics
Rugby
Cricket
Tennis
Vollyball
1
__
5
0
2
__
5
Fraction of learners
3
__
5
ExErCiSE 27.8
1. Study these two graphs carefully.
Fraction of learners above a certain age (2001)
Fraction of learners above a certain age (2009)
Grade 3
10yrs & older
Grade 3
10yrs & older
Grade 7
14yrs & older
Grade 7
14yrs & older
0
1
__
10
2
__
10
3
__
10
Data collected by Census at School
a)
b)
c)
d)
e)
142
4
__
10
0
1
__
10
2
__
10
Data collected by Census at School
What is the source of the data for these graphs?
Write a few sentences summarising what each graph tells you.
When was the data for the first graph collected?
When was the data for the second graph collected?
How has the number of older children in each grade changed
over time?
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2. These two graphs show the
amount of rainfall (in mm) that
fell each month in Springbok
(Northern Cape) and Bloemfontein
(Free State) during one year.
a ) Describe the rainfall pattern
for the year in Springbok.
b ) Describe the rainfall pattern
for the year in Bloemfontein.
c ) Which three months have the
highest rainfall in each place?
d ) Which three months have the
lowest rainfall in each place?
e ) What do the graphs tell you
about the rainy season in each
place?
f ) The horizontal scales on the
graphs are different. What
does this tell you about the
amount of rain that falls in
each place?
g ) Estimate the total annual
rainfall for each town.
Monthly precipitation in
Springbok
Monthly precipitation in
Bloemfontein
Dec
Dec
Nov
Nov
Oct
Oct
Sep
Sep
Aug
Aug
Jul
Jul
Jun
Jun
May
May
Apr
Apr
Mar
Mar
Feb
Feb
Jan
Jan
0
15 mm 30 mm
0
20 mm 40 mm 60 mm 80 mm
3. Durban in KwaZulu-Natal is in a summer rainfall region and it has
quite a high rainfall. Predict what the bar graph for Durban would
look like. Draw a rough sketch to show what you think the graph
would show.
4. Here is the rainfall data for each month of one year in Durban.
J
Month
Rainfall (mm) 160
F
M
A
M
J
J
A
S
O
N
D
140
110
60
20
15
20
20
40
90
115
125
a ) Use the data to draw a bar graph like the ones in Question 2.
b ) Write a short paragraph comparing the rainfall pattern of
Durban and Springbok.
c ) How is the rainfall pattern in Durban similar to that in
Bloemfontein? How is it different?
Topic 27: Data handling
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Project
Sources of water in our country
and our community
Everyone needs a supply of clean fresh water to be healthy and to
live well. However in our country, many people do not have access to
clean piped water in their homes.
In this project you are going to find out more about where people in
South Africa get their water. You are also going to do a local survey to
find out where people in your community get their water.
Part 1 – Sources of water in South Africa
This data was collected during a national census. It
includes all households in South Africa. The numbers
have been rounded off to make it easier for you to
work with them.
1. Answer these questions.
a ) Which type of water supply is used by:
i. the greatest number of households?
ii. the least number of households
b ) How many households get water from
a piped source in the community?
2. Copy and complete this pictograph to show
households who have a supply of piped water.
(2)
(2)
(6)
Number of households with access to a piped water supply
In home
In yard
Less than 200 m away
More than 200 m away
Water supply in
South Africa
Number of
households
Piped water in home
Piped water inside yard
Piped water in community
• Less than 200 m away
• More than 200 m away
Borehole
Spring
Rain-water tank
Dam/pool/standing water
River/stream
Water vendor
Other
3 600 000
3 300 000
1 200 000
1 400 000
300 000
200 000
75 000
100 000
700 000
80 000
300 000
Key
= 200 000 households
Your teacher will collect and mark your answers to Part 1.
Total marks for Part 1: 10
Part 2
You are going to work on your own to carry out a survey in your
community to find out where people get their water. You will need to
collect data, organise it in tally tables, display it as a pictograph and
analyse and summarise your findings.
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Step 1: Choose your questions and then plan
• Write down which 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 in order to collect data.
• Decide who you will include in your survey and how many people you will collect data from.
• Collect the data.
• Make a list of the names of all the people you survey.
Step 3: Organise your data
• Draw up a neat tally table to summarise the data you have collected. Include totals
for each category.
Step 4: Draw a pictograph to represent the data
Draw one or more pictographs to represent and summarise your data.
Remember to:
• make sure you give each pictograph a heading
• include a key to show what the symbols mean.
Step 5: Summarise your findings and compare the data from your survey with the data for the
whole country
• 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.
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:
• 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 pictograph(s)
• a paragraph summarising your findings.
You may decorate the poster in any way that you like. Be as creative as possible.
(15)
Total marks for Part 2: 15
Total marks for project: 25
Project
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Topic
Numeric patterns
28
Maths ideas
Create number patterns
• Recognise,
describe and
continue number
sequences.
Numeric patterns may have a rule that you can use to find the next
number in the pattern. The pattern may also have a rule that helps you to
find a number much further along the pattern, such as the 100th number.
• Construct number
sequences.
Example
• Determine output
numbers for given
input numbers
using flow
diagrams.
• Determine the
rule that applies
to a given set of
input and output
numbers.
Use the given rule to complete the row of output numbers:
Rule: (Input number × 3) – 2
Input
1
3
5
7
9
11
number
Output (1 × 3) – 2 (3 × 3) – 2 (5 × 3) – 2 (7 × 3) – 2 (9 × 3) – 2 (11 × 3) – 2
number
=1
=7
= 13
= 19
= 25
= 31
• You can describe the input number sequence in this way: To get the
next number you add 2 to the previous number. So, the next three
input numbers are: 11 + 2 = 13; 13 + 2 = 15 and 15 + 2 = 17.
• Now look at the output numbers: 1; 7; 13; 19; 25; 31. To get the
next number you add 6 to the previous number. So, the next
three output numbers are: 37; 43 and 49.
ExErCiSE 28.1
1
2
3
4
5
6
1. Use the given rule to complete the row of output numbers:
a ) Rule: (Input number + 2) × 4
Input number
Output number
1
2
3
4
5
6
8
11
14
17
20
b ) Rule: (Input number – 2) ÷ 3
Input number
Output number
5
2. Look at the input numbers in Question 1.
a ) Describe each number sequence.
b ) Write down the next three numbers in each sequence.
3. Look at the output numbers in Question 1.
a ) Describe each number sequence.
b ) Write down the next three numbers.
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Find the rule for a number pattern
If you look at a given set of input numbers and output numbers that
are formed using a rule, you can find the rule.
Example
Output
A special rule was used to form the pattern of output Input
1
9
numbers from the input numbers in the flow diagram. 3
19
5
You have been given part of the rule:
5
29
15
25
?
+4
6
34
30
input number → ? → number → +4 → output number.
40
8
44
55
The number in the middle helps you to find the rule for 11
59
the first box.
To find the first rule, choose a pair of input and output numbers, for example, 3 and 19. Now
try an operation:
Try ×3: (3 × 3) + 4 = 9 + 4 = 13 but this is too small.
Try ×6: (3 × 6) + 4 = 18 + 4 = 22 but this is too big.
Try ×5: (3 × 5) + 4 = 15 + 4 = 19.
Test on another input number: (5 × 5) + 4 = 25 + 4 =29.
You can also subtract 4 from the output number to get the middle number, for example, 59 –
4 = 55. Therefore, 55 ÷ 5 = 11 which is the input number.
So, the rule is: (input number × 5) + 4.
ExErCiSE 28.2
1. Look at these two flow diagramsand see if you can find the rule.
Input
6
12
14
16
Input
Output
4
5
2
4
÷?
6
9
10
+3
Output
8
12
1
3
5
8
×?
+3
10
12
11
16
22
26
30
2. Look at these sets of input and output numbers. There are two
operations in each rule. Find the rule for each set.
Input numbers
Output numbers
Input numbers
Output numbers
1
3
5
7
9
11
10
26
42
58
74
90
2
4
6
8
10
12
9
15
21
27
33
39
Topic 28: Numeric patterns
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investigate number sequences
You can work out what the next number in the sequence will be.
Some number sequences are easy to predict, for example:
• When you add or subtract the same number to get the next number.
• When you multiply or divide by the same number to get the next
number.
Other number sequences are more difficult to predict.
Example
a) Find the rule for this number sequence: 97; 96; 94; 91; 87; ___.
b) Find the next three numbers.
You will get the next number in the sequence by subtracting the
next counting number from the previous number in the sequence:
−1 −2 −3 −4
97; 96; 94; 91; 87; □
So, the next three numbers in the sequence must be: 87 – 5 = 82;
82 – 6 = 76 and 76 – 7 = 69.
ExErCiSE 28.3
For each of these questions describe the pattern in each number
sequence. Then fill in the missing numbers to complete the sequence.
1. a) 2; 4; 6; 8; □; □; □
b ) 1; 5; 9; 13; □; □; □
c) 27; 36; 45; 54; □; □; □
d ) 122; 223; 324; 425; □; □; □
e) 207; 198; 189; 190; □; □; □ f ) 105; 91; 77; 63; □ □ □
g) 11 777; 11 770; 11 763; 11 756; □; □; □
148
2. a) 67; 71; □; 79; 83; □; □
c) 90; 83; □; 69; □; □; 48
b ) 84; 72; □; □; 36; □; 12
d ) 990; 880; 770; □; □; □
3. a ) 3; 6; 12; 24; □; □; □
c ) 972; 324; 108; □; □; □
b ) 64; 32; 16; 8; □; □; □
d ) 20 000; 4 000; 800; □; □; □
4. a ) 4; 6; 9; 13; □; □; □
c ) 2; 12; 21; 29; □; □; □
e ) 3; 7; 12; 18; □; □; □
b ) 3; 5; 9; 15; □; □; □
d ) 100; 98; 94; 88; □; □; □
f ) 0; 2; 6; 14; 30; □; □; □
Term 3
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revision
1. During the 2012 Olympic Games in London, Pete did a survey to see which track and field
events the learners in his school liked to watch. He drew this graph to show his results:
Key
Type of event
Number of learners
= 4 learners
High jump
Sprinting
Javelin
Marathon
What type of graph is this?
How many learners does one stick person represent?
What is the most popular event?
What is the least popular event?
How many learners chose javelin?
How many learners chose sprinting?
Which event is the mode?
3. Describe the pattern in each of these
number sequences. Write the next three
numbers in each sequence.
a ) 7; 13; 19; 25; □; □; □
b ) 100; 91; 83; 76; □; □; □
(1)
(1)
(1)
(1)
(1)
(1)
(1)
Graph 1
16
14
12
10
8
6
4
2
0
Number of boys
2. The learners in Lima’s class want to raise
money. The teacher asks the boys and the
girls in the class for their opinion about
how to raise money. The following graphs
represent the results of her survey:
a ) What is the name for this type of graph?
b ) Explain what the graphs show you
about the opinions of the learners.
Number of girls
a)
b)
c)
d)
e)
f)
g)
Talent Soccer Bazaar
contest
Event
18
16
14
12
10
8
6
4
2
0
Graph 2
(1)
(2)
Talent Soccer Bazaar
contest
Event
(3)
(3)
4. Use the given rule to complete each number sequence.
a ) Rule: Subtract 13 from the previous number.
260; □; □; □
b ) Rule: Multiply the previous number by 2 and add 4.
2; □; □; □
(2)
(2)
Total marks: 20
Revision
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Topic
Multiplication
29
Maths ideas
Multiples and factors
• Revise factors and
multiples.
Remember that a multiple is the answer you get when you multiply
two numbers. A factor divides exactly into another number.
• Multiply by
multiples of 1 000.
• Revise
multiplication by a
2-digit number.
• Solve problems
using
multiplication –
including rate,
length and mass.
ExErCiSE 29.1
1. Say whether the following are true or false? Explain how you
know this.
a ) 102 is a multiple of 4.
b ) 7 935 is a multiple of 5.
c ) 999 is a multiple of 3.
d ) 999 is a multiple of 6.
e ) 10 000 is a multiple of 2, 5 and 10.
f ) 645 is a multiple of 9.
2. What number am I?
a ) I am a multiple of 9. I have 5 as a factor. I am smaller than 50.
b ) I am an even number. I am a multiple of 6 and 9. I am bigger
than 50 but smaller than 60.
Do you remember what happens to numbers when you multiply
by 1 000? The number that you are multiplying becomes a 1 000
(thousand) times bigger. For example, 25 × 1 000 = 25 000. The 5 was
in the units column, but in the answer it is in the thousands column.
In 25, 2 was in the tens column, now it is in ten thousands column.
You5 can50use factors
when you multiply
by multiples of 1 000:
500
5 000
12 ×x103 000 x10
= 12 × 3 × x10
1 000 = 36 × 1 000 = 36 000.
5
50
x10
2
500
x10
20
x10
2
x10
200
x10
20
x10
5 000
2 000
x10
200
x10
20 000
x10
2 000
x10
20 000
x10
ExErCiSE 29.2
1. To multiply these numbers, first break down the multiple of 1 000
into two factors.
a ) 44 × 1 000
b ) 345 × 2 000
c ) 106 × 5 000
d ) 3 000 × 26
e ) 9 000 × 41
f ) 121 × 7 000
2. Fill in the missing numbers.
a ) 9 000 × □ = 18 000
150
b ) □ × 5 000 = 250 000
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Multiply three-digit numbers by
two-digit numbers
In Topic 14, you learnt how to multiply three-digit by two-digit
numbers. You will now practise using those methods. Round off
numbers first to estimate your answer.
Example
Find 253 × 88.
An estimate is 250 × 100 = 25 000.
You can use factors to multiply: 253 × (2 × 2 × 2 × 11).
Or you can break down one of the numbers:
253 × (100 – 12) = (253 × 100) – (253 × 12)
= 25 300 – (253 × 10) – (253 × 2)
= 25 300 – 2 530 – 506
= 22 770 – 506 = 22 264
You can check your answer by showing that 22 264 × 88 = 253.
ExErCiSE 29.3
1. Fill in the missing numbers. Then calculate the answers.
a ) 561 × 67 = (561 × 70) – (561 × □)
b ) 236 × 79 = (236 × 80) – (236 × □)
c ) 452 × 58 = (452 × □) – (452 × □)
d ) 809 × 45 = (809 × □) + (809 × □)
2. Before you calculate each of these answers, round off and write
down an estimated answer.
a ) 634 × 73
b ) 711 × 94
c ) 808 × 41
d ) 532 × 69
e ) 74 × 812
f ) 17 × 619
g ) 91 × 353
h ) 274 × 82
i ) 321 × 29
3. Use your calculator and multiply each answer in number 2 by 1.
Write down what you notice. Compare your answers to your
partner’s answers.
4. Use multiplication as an inverse operation to check if these
calculations are correct. Correct any mistakes you find.
a ) 4 112 ÷ 36 = 114
b ) 989 ÷ 52 = 19
c ) 782 ÷ 45 = 17
d ) 5 005 ÷ 38 = 131 remainder 27
e ) 3 333 ÷ 99 = 33 remainder 70
Topic 29: Multiplication
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Compare quantities
In Topic 14, you compared two quantities with different units of
measurement.
Example
a) If sweets cost R28 per kilogram, this means that you pay R28 for
one kg of the sweets. You can write this as R28/kg. If you buy
15 kg of these sweets, you will pay 15 × 28 = R240.
b) If you drive in a car at 60 km per hour, you can write this speed as
60 km/h. If you drive for __13 hour, you will travel __13 of 60 = 20 km.
ExErCiSE 29.4
Show how you work out the answers to these problems.
1. A recycling business pays you 34c for each newspaper that you
bring them. How much will you be paid if you bring them 669
newspapers?
2. An aeroplane is travelling at 595 km/h. How far will it fly in 13 hours?
3. A laser printer can print 26 pages per minute. How many pages
will it print in 1 hour and 15 minutes?
4. You are selling cabbages for your aunt. If she pays you 72 c for
each cabbage that you sell, how much will you earn if you sell
6 dozen cabbages? (Give your answer in rand, and remember
that 1 dozen = 12.)
5. At a busy airport, 145 planes take off each day. How many planes
will take off in 16 days?
6. In a sprint an athlete runs at the speed of 12 m/sec. How many
metres will he run in 2 minutes and 45 seconds if he keeps to
this speed?
Challenge
Fill in the squares on the sides of a triangle with even numbers, so that when
the numbers on each side are multiplied together the answer is 96.
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Solve multiplication problems
In this exercise, you will practise what you learnt in this topic by
solving word problems.
ExErCiSE 29.5
1. To train for the Comrades Marathon, an athlete has to run
112 km/week.
a ) If he is in training for 6 months before the marathon,
how many kilometres does he run during training?
b ) The same athlete runs at the rate of about one
kilometre in 5 minutes. How much time does the
athlete spend running per week? Give your answer in
hours and minutes.
2. On the day of the marathon, the organisers must have
enough food and drink for the athletes.
a ) 784 bags of oranges are ordered. If there are
33 oranges in each bag, how many oranges are there
altogether?
b ) 600 boxes of bananas are ordered. What is the total number of
bananas if there are 55 in each box?
c ) 1 200 kg of chocolate is supplied. If each kilogram is made up
of 40 squares, how many squares of chocolate are available for
the runners?
3. At a festival a pizza hut sells large pieces of pizza for R11 each.
How much money do they make if they sell 320 pieces?
4. Super-Duper cleaning services pay their cleaners R23/hr. How
much can a cleaner earn if they work:
a ) an 8 hour day
b ) a five-day week c ) a 21-day month?
5. Jonah wants to put a new carpet in his lounge. Each metre of
carpet costs R85. He will need 14 metres of carpet. How much will
it cost Jonah for the carpet?
6. A manager wants to repaint the walls of her small hotel. She
buys 34 large tins of paint which cost R125 each, and 16 paint
brushes at R34 each. How much will she spend in total at the
hardware shop?
Challenge
I am ...
smaller than 173 × 2
bigger than
(14 × 20) + (14 × 2)
a multiple of 5 and 2.
Choose the answer
from the box:
345
340
426
315
500
311
Topic 29: Multiplication
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4 Term 4
Trains are a great way to travel between cities.
You can get to most destinations in South Africa by bus.
Johannesburg has the biggest airport in South Africa.
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Topics 30 – 40
Starting off
The photographs on the left show different
kinds of transport that people use to get from
one place to another.
1. Write down the names of all the different
kinds of transport in the photographs.
2. Arrange your answers to Question 1 from
the slowest to the fastest.
Bicycle rides are fun and an affordable way of getting around.
3. Each of the different kinds of transport
shown in the photographs has a common,
very important feature. Use the clues below
to find out what this feature is.
a ) The first letter of the word is the 23rd
letter of the alphabet.
b ) The last letter of the word is the 12th
letter of the alphabet.
c ) The second letter of the word comes
just after G in the alphabet.
d ) The third and fourth letters of the word
are the same.
4. Think of at least three other kinds of
transport that do not have the feature in
Question 3. Write them down.
Content covered in Term 4
Cars provide a convenient way of getting from one place
to another.
Topic 30: Count, order, compare and represent whole numbers, Topic 31: Addition
and subtraction, Revision, Topic 32: Properties of 3D objects, Topic 33: Common
fractions, Revision, Assignment, Topic 34: Division, Topic 35: Perimeter,
area and volume, Revision, Topic 36: Position and movement,
Topic 37: Transformations, Revision, Investigation, Topic 38: Geometric
patterns, Topic 39: Number sentences, Revision, Topic 40: Probability
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Topic
30
Count, order, compare and represent
whole numbers
Maths ideas
Round off large numbers
• Round off whole
numbers.
You can round off large numbers to estimate answers.
• Compare and
order whole
numbers.
Example
Look at the picture of United’s soccer stadium on the left. The seats
are colour-coded to show different prices.
• There are 11 942 red seats in the stadium.
• The number of yellow seats is three times the number of red seats.
• The number of blue seats is half the number of red seats.
• The number of green seats is double the number of red seats.
• Work with place
value.
United’s soccer stadium
Estimate how many seats there are in total. Round 11 942 to
12 000. So, there are about 12 000 red seats.
• Yellow seats: 12 000 × 3 = 36 000
• Blue seats: 12 000 ÷ 2 = 6 000
• Green seats: 12 000 × 2 = 24 000
Estimated total number of seats:
12 000 + 36 000 + 6 000 + 24 000 = 78 000
Challenge
ExERCiSE 30.1
The total number of
seats in a soccer
stadium is 60 000.
The seats are colourcoded in the same
way as above. Find
how many seats of
each colour there
could be. Use the
numbers in the table
to help you.
Copy the table below into your exercise book.
Soccer
stadium
2. Use the information in the top row of the table to fill in the missing
estimated numbers of seats.
Red seats
United
City
Rovers
12 000
Town
9 000
156
1. Round off these numbers for three other stadiums, and write them
in the table.
• City: 253 blue seats (round to the nearest 10)
• Rovers: 15 146 yellow seats (round to the nearest 1 000)
• Town: 17 787 green seats (round to the nearest 1 000)
Yellow seats
(3 × red)
Blue seats
(Half of red)
Green seats
(2 × red)
36 000
1 500
6 000
24 000
Total seats
78 000
10 000
Term 4
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More practice with whole numbers
Place value and expanded form
Example
1. Write 145 648 in expanded form.
145 648 = 100 000 + 40 000 + 5 000 + 600 + 40 + 8
2. Write the number below in digits.
100 000 + 70 000 + 9 000 + 400 + 90 + 5 = 179 495
ExERCiSE 30.2
1. Write these numbers in expanded form. Give the place value of
the digit 8 in each number.
a ) 132 853
b ) 198 233
c ) 236 187
d ) 385 349
e ) 866 127
f ) 670 853
g ) 720 085
h ) 822 010
2. Write these expanded additions as whole numbers in digit form.
a ) 100 000 + 20 000 + 3 000 + 900 + 80 + 7
b ) 30 + 2 + 600 000 + 5 000 + 600 + 70 000
c ) 60 000 + 800 000 + 5
Order and compare numbers
ExERCiSE 30.3
1. Write the following numbers in ascending order.
a ) 132 498, 123 498, 312 984, 149 832, 314 849; 182 439
b ) 254 763, 245 763, 254 673, 254 367, 245 367, 234 567
2. Write the following numbers in descending order.
a ) 126 753, 175 623, 132 675, 125 753, 135 672, 152 673
b ) 486 291, 468 219, 486 912, 426 189, 412 896, 468 129
3. Use the symbols < and > to compare these pairs of numbers.
a ) 123 465 □ 123 654
b ) 167 930 □ 167 903
c ) 303 567 □ 330 657
Topic 30: Count, order, compare and represent whole numbers
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Topic
31
Addition and subtraction
Maths ideas
• Add and subtract
two or more whole
five-digit numbers
using the column
method.
• Solve multi-step
problems that
involve addition
and subtraction.
Add numbers in columns
In Topic 22 you learnt how to add numbers using the column method.
Look at the example box on page 116 of Topic 22 to revise how to set
this out.
Remember to write the correct digits under each other according to
their place value. If you do not write the digits in the correct place, it is
difficult to get the correct answer.
ExERCiSE 31.1
1. Use the column method to do these calculations.
a ) 36 027 + 50 378
b ) 46 366 + 72 472
c ) 30 432 + 58 930
d ) 57 038 + 70 484
e ) 79 000 + 10 091
f ) 60 072 + 42 519
2. Do these calculations involving three numbers.
a ) 37 037 + 13 587 + 19 997
b ) 25 589 + 13 706 + 38 896
c ) 21 456 + 19 786 + 33 874
d ) 20 307 + 57 463 + 17 564
e ) 68 237 + 27 855 + 14 099
3. Complete these number sentences to show inverse operations.
a ) 23 157 + □ = 55 788
So, 55 788 − 23 157 = □
b ) 36 424 + □ = 68 338
So, 68 338 − 36 424 = □
c ) 47 551 + □ = 77 990
So, 77 990 − 47 551 = □
d ) 51 467 + □ = 80 300
So, 80 300 − 51 467 = □
e ) 49 577 + □ = 90 009
So, 90 009 − 49 577 = □
Challenge
Add these sets of three
numbers. Use any method
of addition.
1. 38 673 + 38 562 + 28 564
2. 3 089 + 56 925 + 28 560
3. 609 + 28 565 + 27 599
4. 27 507 + 6 920 + 540
158
4. Use rounding and compensating (Method 3 on page 15)
to complete the following.
a ) 17 394 + 15 352
b ) 25 575 + 21 739
c ) 46 782 + 37 115
d ) 80 369 + 13 571
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Subtract numbers in columns
Key words
When you subtract a bigger number from a smaller number in a
column, you must break down a number from the column on the left.
We say that you exchange a larger number for 10 smaller numbers.
For example, you exchange 1 ten for 10 units, or you exchange
1 hundred for 10 tens.
• exchange − break
down a larger
number into
smaller units: 100
is 10 tens; 1 000 is
10 hundreds and
so on
Example
Subtract: 7 228 − 4 346
First approximate your answer: 7 200 − 4 300 = 2 900
Step 1:
−
Th
7
4
H
2
3
T
2
4
U
8
6
2
8−6=2
Step 3:
Th H
7 112
4 3
8
6
−
T
2
4
8
12
U
8
6
2
Step 2:
Th
7
4
−
H
2
3
1
T
2
4
8
1
U
8
6
2
To subtract 4 tens from 2 tens, exchange
1 hundred for 10 tens. You now have 10
+ 2 = 12 tens. 12 − 4 = 8
Step 4:
To subtract 3 hundreds from
1 hundred, exchange 1 thousand for
10 hundreds. 10 + 1 = 11 hundreds.
11 − 3 = 8
−
Th H
6
7 112
4 3
2 8
T
2
4
8
1
U
8
6
2
6−4=2
2 882 is very close to the
approximation of 2 900.
ExERCiSE 31.2
1. Find the difference between each pair of numbers.
a ) 13 542 − 11 971 b ) 27 998 − 15 078 c ) 61 100 − 28 964
d ) 44 000 − 18 346 e ) 50 348 − 24 771 f ) 78 001 − 45 882
2. Rewrite these subtractions in columns and fill in the missing digits.
a ) 16 531 − 11 □□9 = □ 55□
c ) 3□ 718 − 21 □□1 = □8 71□
b ) 29 652 − □□ 493 = 10 □5□
d ) 63 □□4 − □□ 824 = 25 40□
3. Complete these money calculations.
a ) R2 345 − R1 156 b ) R4 237 − R2 399 c ) R5 899 − R3 635
d ) R4 995 − R2 745 e ) R6 733 − R3 850 f ) R7 805 − R4 222
Topic 31: Addition and subtraction
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Solve addition and subtraction
problems
When solving problems, remember to set your work out carefully and
to show all your working. First estimate the answer and then check
your answer with an inverse operation.
Example
A shop has 12 950 magazines to sell. They sell 2 585 magazines
in the first week, 3 250 in the second week and 1 200 in the third
week. How many magazines are left?
This number sentence describes how to answer the question:
12 950 − (2 585 + 3 250 + 1 200) = □
An estimate is 13 000 − (2 600 + 3 300 + 1 200)
= 13 000 − 7 100 = 5 900
2
3
+ 1
7
5
2
2
0
8
5
0
3
5
0
0
5
1 2 9 5 0
7 0 3 5
−
5 9 1 5
Answer: There are 5 915 magazines left to sell.
Check: 5 915 + 7 035 = 12 950
ExERCiSE 31.3
1. A factory produced 2 340 cell phones on the first day, 2 885 on the
second day, 8 046 on the third and 9 556 on the fourth day. On the
fifth day 6 900 cell phones were produced, but 780 of these were
damaged. How many undamaged cell phones were produced?
2. Jabu’s mother saves R17 850 towards buying a house. Jabu’s dad
has saved R15 350. How much more must they save together to
have R50 000 deposit for a house?
3. The Animal Anti-Cruelty Society held a number of collections to
raise money. They collected R3 450 on Monday, R6 590 on Tuesday,
R8 540 on Friday and R7 548 on Saturday.
a ) What is the difference between Friday’s amount and Monday’s
amount?
b ) What was the total amount raised?
c ) How much more must they raise to meet the target of R50 000?
160
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Revision
1. Round off these numbers to the nearest 10.
a ) 26
b ) 834
c ) 1 301
d ) 45 938
(1)
(1)
(1)
(1)
2. Round off these numbers to the nearest 100.
a ) 257
b ) 981
c ) 8 965
d ) 13 581
(1)
(1)
(1)
(1)
3. In each row, round off the bold number to the nearest thousand and then choose
the correct answer from the four options offered:
(2)
Number
Option A
Option B
Option C
Option D
5 318
5 000
5 300
5 400
6 000
10 555
10 000
10 500
10 600
11 000
4. Write the following numbers in expanded notation.
a ) 64 891
b ) 37 023
(1)
(1)
5. Write the following separate values as whole numbers.
a ) 100 000 + 6 000 + 400 + 80 + 1
b ) 20 000 + 5 000 + 100 000 + 7
(1)
(1)
6. Use the column method to do the following calculations.
a ) 19 087 + 23 435
b ) 25 788 + 26 224
c ) 32 665 − 18 254
d ) 46 897 − 29 017
(1)
(1)
(1)
(1)
7. Complete the money calculations below.
a ) R4327 + R2756
b ) 7 560 c − 3 356 c
(1)
(1)
8. Find the sum of 13 546 m and 27 499 m.
(2)
9. Add the difference between 29 355 and 17 496 to the sum of 43 220 and 18 080.
(3)
Total marks: 25
Revision
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Topic
32
Properties of 3D objects
Maths ideas
• Identify and name
3D objects.
• Cut open boxes to
form nets.
• Describe, sort
and compare 3D
objects.
• Interpret diagrams
of 3D objects.
Recognise and name 3D objects
In Topic 15 you learnt about the properties of some 3D objects. You
also learnt how to construct models of some 3D objects. Now you will
do more exercises on the properties of 3D objects.
ExERCiSE 32.1
1. Below are pictures of 3D objects that are found in our
environment. For each object do the following:
• Draw a picture of the front view.
• Draw a picture of the top view.
• Name all the faces of this 3D object.
• What type of 3D object is this?
a)
b)
Table Mountain in Cape Town
c)
Conical trees like the ones
you decorate at Christmas.
162
The Giant’s Causeway in
Northern Ireland
d)
The Earth
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2. Below are pictures of some 3D objects. For each object do
the following:
• Draw a picture of the front view.
• Draw a picture of the top view.
• Name all the faces of this 3D object.
• What type of 3D object is this?
a)
b)
Balls used in sport
Houses
c)
d)
Structures like The FNB Stadium in Soweto
Structures like The Louvre pyramid in Paris
Topic 32: Properties of 3D objects
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Sort and compare 3D objects
Some 3D objects may be new to you, but try to imagine what they
look like when you read about them.
Example
Which shape do you think is described below?
Description
Flat or curved
It only has flat faces.
Faces
Number of faces It has seven faces.
pentagonal prism
Shapes of faces
Two opposite faces (the end faces) are identical
pentagons. The other five faces are rectangles.
Type of 3D
object
It is a prism with two identical end faces that are
pentagons.
The object described here is called a pentagonal prism because
two end faces are identical pentagons.
When you compare objects, you look at the faces of each object. You
then explain how the objects are the same or different.
rectangular prism
Example
Rectangular prism
Flat or curved
It has only flat faces.
faces
Number of faces It has six faces.
Shapes of faces
cube
164
Type of 3D
object
Cube
It has only flat faces.
It has six faces.
All six faces are rectangles.
The opposite faces are the Six equal square faces.
same.
It is a rectangular prism,
It is a prism with identical
because the opposite
square faces.
faces are identical
rectangles.
What is the same?
What is different?
• Both objects have only flat faces.
• Both objects have six faces.
• Both are prisms.
• All the faces of rectangular prisms
are rectangles.
• All the faces of cubes are squares.
• Only the opposite faces of
rectangular prisms are identical.
• All the faces of cubes are identical.
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ExERCiSE 32.2
Look at the square-based pyramid and triangular prism on the right.
1. Copy and complete the table for both pyramids.
Square-based
pyramid
Flat or curved
faces
It has only flat faces
Shapes of faces
It has one □ face and
□ triangular faces
Triangular prism
Number of faces It has □ faces
It is a pyramid, because it
has a square base and all
other faces are □
Type of 3D
object
2. Explain in which ways these two objects are the same.
3. Explain in which ways these two objects are different.
Key words
• net − a flat
arrangement of
the 2D shapes
that make up a 3D
object
In Grade 4 you made nets of 3D objects. Do you remember how to do this?
ExERCiSE 32.3
1. Look at these objects. Which of these objects have:
a ) curved faces
b ) triangular faces
c ) six faces
d ) five faces?
A
D
B
C
G
E
F
2. Cut open a box and trace its net onto paper.
3. Find two other boxes with different shapes. Cut the boxes open
and trace their nets.
Net of cone
Net of a square-based pyramid
Topic 32: Properties of 3D objects
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Topic
33
Common fractions
Maths ideas
• Compare and
order fractions.
• Add and subtract
mixed fractions
with the same
denominator.
• Find fractions of
whole numbers
which result in
whole numbers.
• Solve problems
with fractions.
Compare and order fractions
When you compare fractions, you can use the fraction wall on page 65
of the Learner’s Book. If the numerator is 1, you can compare the
denominators. A larger denominator makes a smaller fraction. For
example, __14 is smaller than __12 .
ExERCiSE 33.1
1. Write these fractions in descending order.
1 __
1 __
__
; 1 ; __
; 1 ; __1 ; __1
8 5 12 10 2 3
2. Use the fraction wall on page 65 to find three fractions that are
larger than each fraction below.
a ) __18
3
b ) __
10
2
c ) __
11
4
d ) __
12
3. Use the fraction wall on page 65 to find three fractions that are
smaller than each fraction below.
a ) __34
b ) __12
c ) __56
8
d ) __
10
4. Use the fraction wall on page 65 to compare these fractions. Then
fill in <, > or =.
a ) __13 □ __12
b ) __12 □ __23
c ) __14 □ __26
4
d ) __25 □ __
10
e ) __28 □ __24
f ) __13 □ __39
g ) __52 □ __12
h ) __46 □ __23
5. Use the fraction wall to convert these fractions to twelfths. Then
order the fractions from smallest to largest.
1 __
__
, 2 , __3 , __2
2 3 4 6
Challenge
Pumle has a recipe for 8 pancakes. The recipe asks for: 2__12 cups of flour, 1__14
cups of milk, __13 teaspoon of salt and 2 eggs. How much of each item does
Pumle need to make 24 pancakes for her party?
166
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Calculate fractions of whole numbers
You can use division to find a fraction of a whole number.
Example
Maria divides 20 sweets into four groups of five sweets each.
Each group is __14 of 20.
1
__
of 20 = 20 ÷ 4 = 5.
4
She then eats three of the groups.
She eats __34 of 20.
3
__
of 20 is three times __14 of 20.
4
3
__
of 20 = 3 × 5 = 15.
4
So, __34 of 20 = 15.
Challenge
What is the answer to
this sum: __12 of (__21 of
(__12 of 3 000))?
Calculate __23 of 30 m.
Divide 30 by 3 to find __13 of 30 m
30 m ÷ 3 = 10 m.
2
__
of 30 is double __13 of 30.
3
2
__
of 30 m = 10 × 2 = 20 m.
3
So, __23 of 30 m = 20 m.
ExERCiSE 33.2
1. Calculate these amounts.
b ) __15 of 30
a ) __13 of 15
d ) __19 of 27
g ) __38 of 24
e ) __27 of 14
h ) __46 of 12
2. Calculate the fraction of each quantity.
b ) __16 of 18 ml
a ) __34 of 20 g
4
of 200 km
d ) __
10
4
of R220
g ) __
11
1
e ) __
of 600 g
12
7
h ) __
of 96 kg
12
c ) __12 of 18
f ) __25 of 25
6
i ) __
of 50
10
c ) __37 of 35 cm
9
f ) __
of 500 mm
10
i ) __78 of 248 m
3. 265 people watched a school play and __35 were learners. How many
learners were there?
Topic 33: Common fractions
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Solve problems that involve fractions
Use your knowledge of fractions to answer these questions.
ExERCiSE 33.3
1. __13 of a bag of 24 oranges is rotten.
a ) How many oranges are rotten?
b ) How many oranges are not rotten?
7
of 48 pieces of chocolate. How many pieces
2. Pumi’s brother ate __
12
were left?
3
of a pocket of potatoes during the
3. Solomon’s mother cooked __
10
2
__
week. She then gave 10 of the pocket to her friend.
a ) What fraction of the pocket of potatoes was left over?
b ) The pocket of potatoes had a mass of 20 kg when she bought
it. How many kilograms were left over?
4. Lukas earns R640 per week. If he spends __38 of his money on food
each week, how much money does he have left over?
5. David divides a packet of apples into three equal groups to share
with his friends. If each person gets 6 apples, how many apples are
there altogether?
6. For your birthday you get R50 as a present. If the R50 is equal
to __14 of the money that you had saved, how much money had you
already saved?
7. There are 36 learners in a Grade 5 class and __49 of the learners take
part in sport.
a ) How many learners take part in sport?
b ) How many learners do not take part in sport?
8. Thembi plays soccer for 2__14 hours on Monday, __34 of an hour on
1
hours on Wednesday.
Tuesday and 1__
4
a ) How many hours does Thembi play each week?
b ) Thembi wants to play for 8__14 hours each week. How many
more hours will she have to play over the weekend?
9. James bought shelving in the following lengths: 1__58 m; __48 m; 2__38 m
and 1__78 m. What length of wood did he buy altogether?
10. At a Grade 5 class party, __78 of the class of 40 were there. How many
learners did not go to the party?
168
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Revision
1. Write down the name of each of these 3D objects.
C
B
A
E
(6)
F
D
2. Use the nets below and grid paper to construct models of the 3D objects.
A
(2)
B
3. Look at the models you constructed in Question 2.
a ) Write down the name of each 3D object.
b ) How many faces does each object have?
c ) Describe the shapes of the faces of each object.
(1)
(1)
(1)
4. Calculate each of the following:
a ) __49 of 360
(1)
7
of 120
b ) __
12
(1)
c ) __34 of 320 ml
(1)
d ) __23 of R99
(1)
5. There are 45 biscuits on the baking tray. Joe eats __49 of them.
a ) What fraction of the biscuits did Joe leave?
b ) How many biscuits did Joe eat?
c ) How many biscuits are left?
(1)
(1)
(1)
6. I have a packet of sweets. If __12 of the sweets is 6 sweets, how many sweets do I have
altogether?
(1)
7. If __14 of Jabu’s money is R10, how much money does he have altogether?
(1)
Total marks: 20
Revision
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Assignment
Travelling times and distances
In this assignment, you will learn to work
with maps and tables. Use the map,
the table on this page and the table
on the next page, to answer
the questions.
1. What do the zeros in the
distance table mean?
Cairo
(1)
Abuja
2. a ) Which city is furthest from
Windhoek?
b ) Write down the distance from Windhoek
to the city furthest from it.
c ) Write down the place value of each digit
in your answer to Question 2b).
d ) Round your answer to Question 2b) to
the nearest 100 km.
Accra
Kampala
Nairobi
Dar es Salaam
Lusaka
(4)
Harare
Windhoek
Gaborone
Johannesburg
This table shows that the distance from
Harare to Gaborone is 962 km.
Abuja
Accra
Cairo
Dar es Salaam
Gaborone
Harare
Kampala
Lusaka
Maseru
Nairobi
Johannesburg
Windhoek
Maseru
Abuja
0
869
3 409
3 977
4 287
4 021
2 963
3 591
4 804
3 483
4 494
3 694
Accra
869
0
4 240
4 548
4 369
4 307
3 622
3 872
4 849
4 135
4 662
3 626
Cairo
3 409
4 240
0
4 187
6 120
5 325
3 309
5 064
6 614
3 534
6 229
6 043
Dar es Salaam 3 977
4 548
4 187
0
2 451
1 489
1 099
1 535
2 793
672
2 453
2 953
Gaborone
4 287
4 369
6 120
2 451
0
962
2 874
1 067
531
2 860
278
930
Harare
4 021
4 307
5 325
1 489
962
0
2 021
436
1 341
1 939
977
1 591
Kampala
2 963
3 622
3 309
1 099
2 874
2 021
0
1 808
3 337
521
2 970
3 037
Lusaka
3 591
3 872
5 064
1 535
1 067
436
1 808
0
1 548
1 829
1 196
1 418
Maseru
4 804
4 849
6 614
2 793
531
1 341
3337
1 548
0
3 270
354
1 280
Nairobi
3 483
4 135
3 534
672
2 860
1 939
521
1 829
3 270
0
2 912
3 184
Johannesburg 4 494
4 662
6 229
2 453
278
977
2 970
1 196
354
2 912
0
1 186
3 626
6 043
2 953
930
1 591
3 037
1 418
1 280
3 184
1 186
0
Windhoek
170
3 694
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3. a ) What is the distance in kilometres from Kampala to Lusaka?
b ) What is the distance in kilometres from Windhoek to Accra?
c ) You travel from Nairobi to Gaborone but you stop off at Dar es Salaam on the way. What is
the total distance of your trip?
(3)
This table shows the average speed of four different types of transport.
Aeroplane
Bus
Bicycle
Walking
750 km/h
80 km/h
20 km/h
6 km/h
4. a ) You travel in an aeroplane for 3 hours. What distance do you travel?
b ) How far can you travel in 4_12 hours on each type of transport?
c ) A bus travelled 2 240 km. For how long did it travel?
(3)
5. You have an ‘around Africa’ air ticket. Your ticket allows you up to five flights, but you
cannot fly more than 10 000 km in total. Plan a trip that starts in Johannesburg
and finishes in Cairo. Try to use all five flights and as many of the 10 000 km as you can.
Calculate the duration of each flight to the nearest 30 minutes. Then, calculate your
total flying time.
(9)
6. Plan three different bus trips. Each bus trip must be less than 1 000 km. Calculate the
duration of each bus trip to the nearest 30 minutes.
(6)
7. A hiker must walk from Johannesburg to Gaborone. The hiker has one full week
to complete the trip. Plan how many kilometres the hiker should walk each day
to complete the trip in one week. The hiker should not walk more than 60 km
on any one day. Remember to plan times for him to rest, eat and sleep. Draw
up a timetable for each day of his trip.
(7)
8. How many days would it take someone to cycle the same trip given in Question 6?
Draw up a new timetable to show this.
(4)
9. Plan your own trip. Your trip must include at least one flight
and one bus ride. How far can you get in 12 hours?
(3)
Total marks: 40
Assignment
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Topic
34
Division
Maths ideas
• Find multiples and
factors of whole
numbers up to 100.
• Divide three-digit
numbers by twodigit numbers.
• Use inverses to
check answers.
• Solve problems
using rate, ratio
and financial
settings.
Factors and multiples
In Topic 18 you learnt how to use factors and multiples to help you to
divide three-digit numbers by two-digit numbers. You can use factor
pairs to break down a number into its smallest factors.
Example
Write 36 as a multiplication of its smallest factors, excluding 1.
Start with one factor pair and then break down each factor into
smaller factors.
36 = 9 × 4 = (3 × 3) × (2 × 2), so 36 = 2 × 2 × 3 × 3
Or: 36 = 2 × 18 = 2 × (9 × 2) = 2 × (3 × 3 × 2) = 2 × 2 × 3 × 3
ExERCiSE 34.1
1. Find all of the factor pairs for each number.
a ) 45
b ) 62
c ) 39
d ) 78
f ) 88
g ) 42
h ) 50
i ) 66
k ) 36
l ) 56
e ) 100
j ) 96
2. Write each number in Question 1 as a multiplication of its smallest
factors.
3. List the factors for each number. Then circle the factors that are in
both numbers.
a ) 8 and 16
b ) 5 and 10
c ) 3 and 6
d ) 12 and 16
e ) 12 and 18
f ) 10 and 15
g ) 24 and 36
h ) 20 and 24
i ) 40 and 16
4. Find a multiple of 8 between 41 and 50.
5. Find a multiple of 9 between 60 and 70.
6. Find a number between 80 and 90 that has 7 as a factor.
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inverse operations
Remember that multiplication and division are inverse operations.
This means that you can use one operation to check your answer to
the other operation.
Example
You can use division to check your answer when you multiply
9 × 5 = 45.
You can write the statement 50 × 4 = 200 in two different ways:
200 ÷ 50 = 4
200 ÷ 4 = 50
You can also use multiplication to check your answer when you divide.
Example
Divide 5 000 ÷ 250 and check your answer.
Remember how to divide by 10:
5 000 ÷ 250 = 5 000 ÷ 10 ÷ 25 = 500 ÷ 25 = 20
Check: 250 × 20 = (25 × 10) × (2 × 10) = (25 × 2) × (10 × 10)
= 50 × 100 = 5 000
ExERCiSE 34.2
1. Find the missing numbers.
a ) 25 × □ = 75 so 75 ÷ 25 = □
b ) 120 ÷ 10 = □ so □ × 10 = 120
c ) 110 × 3 = □ so □ ÷ 3 = □
d ) 440 ÷ 4 = □ so □ × 110 = 440
e ) 60 × 30 = □ so □ ÷ □ = 30
f ) □ ÷ 5 = □ so 12 × □ = 60
2. Find the missing number and use an inverse operation to check
your answer.
a ) 350 ÷ 50 = □
b ) 880 ÷ □ = 44
c ) 720 ÷ □ = 6
d ) □ ÷ 70 = 60
e ) □ ÷ 9 = 200
f ) □ ÷ 30 = 40
Topic 34: Division
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Use a clue board for division
In this section you will practise using a clue board to help you with
division of bigger numbers.
Round off the numbers to estimate the answer before you begin. Then
use multiplication to check your answer when you have finished.
Example
Estimate and then use multiplication facts to find 766 ÷ 24.
An estimate is 800 ÷ 20 = 40.
Write down some simple multiplication facts for 24 on a clue board.
Clue Board
24 × 10 = 240
24 × 20 = 480
24 × 30 = 720 (480 + 240)
The closest multiplication to 766 is 720, so start with 24 × 30.
Challenge
Make up two word
problems that
involve division and
give it to a friend to
solve. You must also
work out the answer!
Multiply
Subtract
30 × 24 = 720
766 − 720 = 46
1 × 24 = 24
46 − 24 = 22
766 ÷ 24 = 30 + 1 remainder 22 = 31 remainder 22
Check by multiplying:
24 × 31 plus remainder 22 = (24 × 30) + (24 × 1) + 22
= 720 + 24 + 22 = 766
ExERCiSE 34.3
1. Write down two division facts for each statement.
a ) 21 × 43 = 903
b ) 15 × 62 = 930 c ) 72 × 11 = 792
2. Check these multiplications by doing division.
a ) 12 × 34 = 408
b ) 18 × 26 = 468 c ) 22 × 17 = 374
3. First estimate each answer, and then use a clue board to do these
divisions. Show how you would check your answer.
a ) 735 ÷ 15
b ) 536 ÷ 53
c ) 710 ÷ 72
d ) 614 ÷ 61
e ) 439 ÷ 32
f ) 257 ÷ 47
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Solve division problems
Use a clue board to help you find the answers in this exercise.
Remember to estimate the answer first, and to check the answer by
multiplying.
ExERCiSE 34.4
1. There are 44 learners in Grade 5C and each learner has 15 coins.
Sonto thinks that the learners have 660 coins altogether. Use
division to check if Sonto is correct.
2. There are 18 shelves in the library and each shelf has 28 books on
it. Kyle thinks that there are 502 books altogether. Use division to
check if Kyle is correct.
3. A car travels 693 km in 7 hours. What is the speed of the car per
hour? Write your answer in km/h.
4. The total mass of 14 bags of cement is 588 kg. What is the mass of
each bag?
5. Lillian collects stamps and sticks them into a scrapbook. She has
800 stamps and she sticks 32 on each page. How many pages of
her scrapbook will she use?
6. A builder mixes 4 kg of gravel with 1 kg of cement. How many
kilograms of cement must he put into the cement mixer together
with 156 kg of gravel?
7. Sam works at the local shop and gets paid R784. If he worked
seven times as long as his friend, how much was his friend paid?
8. Melanie buys 16 boxes of beads for R416 to make necklaces.
a ) What is the price of 1 box of beads?
b ) How much will she pay for 13 boxes of beads?
Challenge
1. How can you arrange 960 chairs in rows of equal length?
2. What would be the best layout for a school hall?
Topic 34: Division
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Topic
Perimeter, area and volume
35
Maths ideas
Measure perimeter
• Measure and
calculate perimeter
in standard units.
Perimeter is the distance around a shape. You find the perimeter of a
shape by measuring the lengths of its sides.
• Find area of shapes
using squares on a
grid.
Example
Use a ruler to measure the perimeter
of your Maths textbook.
Give your answer in mm.
275 mm + 210 mm + 275 mm + 210 mm
= 970 mm
• Find volume/
capacity of
containers and
objects by counting
cubes or blocks.
Key words
• perimeter − the
total distance
around the outside
of a shape
It is not possible to measure the perimeter of a curved shape
accurately using your ruler, but you can use string or wool to help you
do this.
Example
Measure the perimeter of these two shapes using string.
Place a piece of string around the
edge of the circle.
Place a piece of string around the
edge of the irregular shape.
Measure the pieces of string with a ruler.
ExERCiSE 35.1
Find the perimeter of each shape by measuring the sides. Give your
answers in millimetres.
F
A
176
B
C
D
E
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Calculate perimeter
You can calculate perimeter when you are given the lengths of the
sides. To do this, you add all the lengths together. Your answer should
be in the same units as those given in the diagram.
Example
Calculate the perimeter of this soccer field.
90 m
Did you know?
50 m
The word ‘perimeter’
comes from the
Greek words ‘peri’
meaning ‘around’
and ‘meter’ meaning
‘measure’. So
’perimeter‘ means ‘to
measure around.’
50 m + 90 m + 50 m + 90 m = 100 m + 180 m = 280 m
ExERCiSE 35.2
1. Calculate the perimeter of each shape.
2 cm
27 mm
19 mm
3,8 m
2,5 cm
2 cm
19 mm 6 cm
A
B
5,1 m
C
4,9 m 6,9 cm
12,4 cm
D
3,5 cm
27 mm
68,2 cm
4 cm
2. Look at each shape below.
3m
5m
A
10 m
4 cm 4,5 cm
3,5 m
2 m 700 cm
270 cm
3m
3,5 cm
3,5 cm
1,9 m
B
8 cm
90 mm
C
2,6 m
310 cm
5 cm
2m
12 cm
a ) Find the measurement that is not in the same units as the
others. Convert it to make it the same.
b ) Calculate the perimeter of each shape.
Topic 35: Perimeter, area and volume
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Key words
• area − the
amount of surface
that a shape covers
Area
The area of a shape can be measured by counting or working out the
number of square units needed to cover it. You can find the area of a
shape by counting the number of square units inside it. When there
are not an exact number of square units inside the shape you need
to estimate.
To estimate the area of shapes:
• Count all whole squares.
• Combine half squares to make whole squares.
• Count any parts that are bigger than half a square.
• Ignore any parts that are less than half a square.
Example
Estimate the area of this shape.
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✗
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✗
✗
✗
✗
✗
• Count the whole squares. There are 14 (red ticks).
• Count the parts that are half a square or bigger. There are
12 (green ticks).
• Ignore the parts that are less than half a square (the crosses).
• The area is approximately 14 + 12 = 26 square units.
ExERCiSE 35.3
Find the area of each shape in square units.
B
A
F
178
C
G
E
D
H
I
J
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ExERCiSE 35.4
1. Look at these shapes.
A
E
B
C
D
a ) Estimate the area of each shape by counting the squares on
the grid.
b ) Order the shapes from the one with the smallest area to the
one with the greatest area.
2. Look at these shapes.
C
D
E
B
A
a ) Estimate the area of each shape by counting the squares on
the grid.
b ) Order the shapes from the one with the smallest area to the
one with the greatest area.
Challenge
Some of the blocks on these shapes have been accidentally rubbed out.
1. Find the area of each shape in square units.
2. Write down how you worked out how many square units were missing.
Topic 35: Perimeter, area and volume
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Volume
During Term 1 you worked with capacity of containers and the volume
of liquids they contained. Now you are going to find the volume of
solids which you will measure in cubes. Remember, volume is the
amount of space that an object takes up.
Look at the two different arrangements of cubes in the picture below.
Each arrangement uses 12 cubes. We can say each
shape has a volume of 12 cubes.
Sometimes you cannot see all the cubes in an arrangement. But, you
can still work out the volume of the object.
Example
What is the volume of this stack of cubes?
You can think of this as three layers of 8
cubes: 3 × 8 = 24.
Or, you can think of this as a front half
and a back half. The front half has
12 cubes and the back half has
12 cubes: 12 + 12 = 24.
The volume of this stack of cubes is 24 cubes.
ExERCiSE 35.5
Work out the volume of each of these stacks of cubes.
1.
180
2.
3.
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ExERCiSE 35.6
1. What is the volume of each of these stacks of cubes?
a)
b)
c)
e)
f)
d)
2. What is the volume of each of these stacks of cubes?
3. Three Grade 5 learners built these shapes with blocks. Find the
volume of each shape. Give your answer in blocks.
Topic 35: Perimeter, area and volume
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Find the volume of a container
Sometimes it is useful to know how
many objects can fit into a container.
You can work out how many cubes or
blocks can fit inside a box by finding
its volume. For example, you might
need to know how many big dice can
fit into a small purple box like the one
in the photograph.
Example
How many blocks will it take to fill this container?
Challenge
Angelique has a
number of small
boxes like this. Each
box has a volume of
eight cubes.
This box is 4 blocks long and it is 3 blocks
wide. It will take 12 blocks to fill one layer.
The box is 2 blocks high. So there will be
two layers of blocks. It will take 12 × 2 = 24
blocks to fill the container.
ExERCiSE 35.7
Angelique wants to
pack the smaller
boxes into a big box
like the one below.
1. Work out how
many boxes will
fit in the big box.
2. Tell a partner how
you worked out
the answer.
182
1. Work out how many blocks you can fit into each of these
containers. Build models with blocks if you need to.
2. How many blocks with the
same volume as A could
you pack into a box with the
same volume as B? Build
models with blocks if you
need to.
B
A
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Revision
1. Which of these numbers has 2, 5 and 10 as factors?
(1)
2 334; 40 090; 299; 3 205
2. Find all the factor pairs for each of the following numbers.
a ) 35
b ) 99
(2)
(2)
3. Find these numbers.
a ) a multiple of 7 between14 and 24
b ) a factor of 18 less than 10
(1)
(1)
4. Find two numbers between 75 and 85 that have 6 as a factor.
(1)
5. Estimate the area of each shape. Give your answers in square units.
(4)
A
B
C
D
6. Measure the sides of shapes A and B on the grid above, then calculate the perimeter
of each shape. Give your answers in millimetres.
(4)
7. Give the volume of each object below in cubes.
(4)
a)
b)
d)
c)
Total marks: 20
Revision
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Topic
36
Position and movement
Locate positions on a grid
Maths ideas
• Locate the
position of objects,
drawings and
symbols using
alpha-numeric
references on grids
and maps.
When you use a grid to location position of objects, drawings or
symbols, you will use two new terms:
• Learners will be
able to trace a path
between positions
on a map.
The grid reference position refers to a particular cell or coordinate in the
grid, for example, C2 is the position where Column C and Row 2 overlap.
An alpha-numeric grid is a grid with alphabet letters for the columns
and numbers for the rows. This means that every position on the grid
has a particular reference.
Brian drew the grid below that shows part of the town in which he lives.
A
Key words
• alpha-numeric
grid − a grid with
alphabet letters for
the columns and
numbers for the
rows
• grid reference
position − refers
to a particular cell
or coordinate in
the grid
• coordinate − a
reference that is
used to show the
exact location of
objects
1
B
C
Mrs
Tambara’s
house
Seth’s
house
Jyoti’s Unathi’s
house house
2
Bev’s
house
4
Kurt’s
house
Ken’s
House
5
Mr
Nxawe’s
house
Dr
Jeftha’s
house
6
7
8
Brian’s
house
spaza
shop
doctor’s
surgery
clinic
town
post
hall
office
public
library
police station
9
fire station
E
Mr
Julia’s
Stewart’s
house
house
Shafik’s
house
Thembi’s
house
3
D
Cathy’s
house
Mpho’s
house
Pete’s
house
10
sports
fields
church
shop
Lungi’s
house
Mr Vos’s
house
Mrs Sogiba’s Carla’s
house house
doctor’s
surgery
dentist’s
surgery
Dr
Moore’s
house
church hall
botanical gardens
garden
church-yard
rose
garden
swings
market
Spaza
Dr
Green’s
house
clinic
school
G
Mr Meyer’s
house
rose
taxi
rank
F
craft market
jungle
gym
campsite
park
pool
Example
Brian’s house is in Row 3 of Column B. You say that the grid position
of Brian’s house is B3.
To get from Brian’s house to Seth’s house, you can go one grid
block left and two grid blocks up, or two grid blocks up and one
grid block left. The distance is three grid blocks either way.
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ExERCiSE 36.1
Study Brian’s grid carefully and then answer these questions:
1. What do you find at the following grid positions?
a ) C2
b) F4
c ) F1
d) B8
e) D6
f ) A8
2. Write down the grid positions of each of the following:
a ) the pool
b ) the taxi rank
c ) the post office
d ) the jungle gym e ) the town hall
f ) the church hall.
3. What are the grid positions of the spaza shops?
4. What do the school, the botanical gardens and the market have in
common?
5. The dentist lives right next door to his surgery. What is his name?
6. Who lives closest to:
a ) the swings
c ) the public library
b ) the park
d ) the pool?
Challenge
Use the grid below to
answer the questions.
1
2
3
4
5
7. Trace the shortest route and count the grid blocks from:
a ) Lungi’s house to the school
b ) Mr Nxawe’s house to the pool
c ) Julia’s house to the public library
d ) The taxi rank to the dentist’s surgery
e ) Bev’s house to Carla’s house
f ) Mrs Tambara’s house to the swings
A
y
s
m
g
a
B
z
t
n
h
b
C D E F
u v w x
o p q r
i j k l
c d e f
1. Decode this word:
A4; E5; B3; C4; C2;
A2.
2. Write your name
using the code.
3. Decode the
sentence below.
Game
Divide into small groups.
• Take turns to call out a coordinate reference while the learner to your right
finds the position on the grid.
• If the learner finds the position correctly, they get to call the next grid
reference.
• If the learner is incorrect, the learner to their right gets a turn to find the
position.
F5; F3; C2; C4; B2/
B4; E5; F4; D3; A2/
A1; C3; C2/B2; C3/
A2; B2; A5; A1/B4;
E5; A5; F4; B2; B4;
A1.
• Mark off the grid references as they are called.
• The learner at the end of the game who marks off the last position wins the
game.
Topic 36: Position and movement
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Locate positions on a map
Not all towns are neatly laid out in rectangular blocks. Some town
planners have to work around natural features such as dams, forests,
hills and streams. Look at the map of Ratanda in Gauteng.
A
B
C
D
E
F
G
H
I
1
2
3
4
5
School
6
Example
Grid positions make it easy to locate a position.
If you were just told to find Walter Sisulu Drive, you would have to
look at all the streets on the map, but if you know its grid position
you would know where to look. Walter Sisulu Drive is in B4 and C4.
Symbols are used to show different places of interest, schools,
clinics, libraries, police stations and places of worship on maps.
Here are some of the symbols used on the map above.
School
186
Police station
L
Library
+
Medical facility
Cemetry
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Example
You can use the compass points that you
have been taught in Social Studies to
describe direction.
N
NW
NE
W
E
SE
SW
S
ExERCiSE 36.2
Use the map of Ratanda to answer these questions:
1. What is the name of the school in I5?
2. Give the grid position of the library.
3. Give the direction of the sports ground from the library.
4. Give the grid position of the police station.
5. Calum lives in Tshungu Street in E3. He walks east to Malebane
Street in E3 and then turns north-east and continues walking until
he reaches his school.
a ) What is the name of Calum’s school?
b ) Calum’s school is going to play soccer against Boneha Primary
School in G3. Write directions for the players who need to walk
to Boneha Primary school. Use compass points as well as street
names.
6. Lindiwe lives in Dithako Street and attends Fountain Five Primary
school.
a ) Find the grid position of Lindiwe’s school.
b ) Next year Lindiwe will be going to secondary school, what is
the name of her closest secondary school?
7. Find the medical facility to the east of the library.
a ) What is the grid position of this facility?
b ) Which school is closest to this medical facility?
8. a ) Find the grid position for the cemetery.
b ) Which roads surround the cemetery?
9. a ) Do you see any shopping facilities or markets on the map?
b ) How do you think the people here buy food?
Topic 36: Position and movement
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Topic
37
Transformations
Maths ideas
• Build composite
shapes from
2D shapes,
including
some with line
symmetry.
• Pack out
2D shapes to
make tessellating
patterns, including
some with line
symmetry.
• Describe patterns
in terms of the
lines of symmetry,
2D shapes,
3D objects from
real life.
Use transformations to create
tessellations
You form a tiling or tessellating pattern by reflecting, rotating or
translating one shape without leaving any gaps.
Example
• The shape below tessellates because there are no spaces in
between the shapes.
• The shape below does not tessellate because there are spaces
between the shapes.
Example
Describe how this tessellating pattern was made.
Row 1
Row 2
The pattern was made by translating the hexagon to the right
to form the first row of the pattern. The first row is then reflected
downwards to form the second row.
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ExERCiSE 37.1
1. Look at the pattern on the right.
a ) Describe the transformation that moved the triangle from
Position 1 to Position 2.
b ) Describe the transformation that moved the triangle from
Position 1 to Position 3.
c ) Describe the transformation that moved the triangle from
Position 4 to Position 5.
d ) Can the movement from Position 3 to Position 4 to Position 5
be described as a rotation? Explain your answer.
e ) Does the pattern that was formed have any lines of symmetry?
How many?
1
3
4
5
2
2. Look at the pattern on the right.
a ) Identify the shape that is tessellated.
b ) There are two possible transformations to move the shape
from Position 1 to Position 2. List both types of transformation.
c ) Describe the transformation that moves the shape from
Position 1 to Position 3.
d ) Look at the pattern from Position 1 to Position 6. Does this part
of the pattern have symmetry?
1
5
3
6
2
4
3. Look at these patterns.
a ) What transformations were used in Pattern A?
b ) What type of shape is transformed?
c ) Does Pattern A have any lines of symmetry?
d ) Look at Pattern B. The different colours show the different
layers of the tessellation. What order were the different layers
created in?
A
4. Use grid paper to draw a 2 cm by 3 cm rectangle; cut out your shape
and paste it on cardboard. Then cut out the shape on the cardboard.
a ) Use your shape to create these patterns on square grid
paper. The numbers on each pattern show the position of the
rectangle that you must draw at each
5
level of the pattern.
6
7
1
1
3
2
b ) Explain the type or types of
2
3
9
10
8
8
4
transformation you used to create
9
10
each pattern.
c ) Which of the patterns have straightA
B
line symmetry?
B
5
4
7
5
6
8
1
2
4
3
7
C
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Describe patterns around us
Patterns occur in nature, in art and in everyday life. Patterns can be
made from both 3D objects and 2D shapes.
ExERCiSE 37.2
1. Investigate the patterns made by these 3D objects, A to D.
A
B
C
a)
b)
c)
d)
e)
f)
Which of the pictures show examples of rotation?
Which of the pictures show examples of reflection?
Which of the pictures show examples of translation?
Identify the 3D object that is used in Picture B and Picture C.
Identify the 3D object that is used in Picture A and Picture D.
Look at a honeycomb and say which 3D object is used and
how is the pattern formed?
2. Investigate these patterns.
A
D
B
C
a ) Draw the shape that is transformed.
b ) Draw in any lines of symmetry for your shape.
c ) Describe the transformation that is shown.
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Revision
1. The grid in the picture below shows a piece
of parkland that has a stream running
through it. The grid is a 6 cm by 6 cm
square.
A
B
C
D
E
F
1
2
3
4
b ) Does your pattern have
line symmetry?
(1)
c ) Use your pattern from Question a)
and reflect it downwards.
(2)
d ) Does your new pattern have line
symmetry?
(1)
e ) Start a new pattern by rotating your
rectangle four times to complete one
full turn.
(2)
3. On 1 cm square grid paper draw the kite
as shown in the picture below, paste it on
cardboard and cut out the shape. Use the
cardboard kite to answer the following.
5
6
a ) Estimate the length of the stream in
centimetres.
(1)
b ) Use a piece of string to measure the
length of the stream in centimetres. (1)
c ) Sally walks along the path. Through
which grid positions will she walk? (2)
d ) There is a tree at A2. Draw three more
trees in the grid so that they fall in the
middle of an empty square. What are
the grid positions of your trees?
(2)
e ) If you are in grid position F4, what is the
grid position of the tree that is closest
to you?
(1)
2. On 1 cm square grid paper draw a rectangle
4 cm by 2 cm, paste it on cardboard and cut
it out. Use the cardboard rectangle to create
the following patterns on 1 cm square grid
paper.
a ) Translate the rectangle six times in a
straight line.
(2)
a ) Use your cardboard kite to recreate
the pattern below.
(2)
Row 1
Row 2
Row 3
b ) Describe this pattern using translation. (1)
c ) Describe the pattern using reflection. (1)
d ) Would it be possible to create this
pattern using rotation?
(1)
Total marks: 20
Revision
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investigation
Palindromes
A palindrome is a number, a word, a phrase or even a sentence that reads the
same backwards as it does forwards. In other words, it reads the same from
right to left as it does from left to right.
Here are some examples of palindromes in the English language.
civic; deed; madam; radar; rotor
Here are some examples of palindromes in some of the other languages
spoken in South Africa.
Afrikaans: daad (deed); ewe (even); lepel (spoon); ses (six); soos (like)
Sesotho: ebe (then); ee (of ); efe (which); eme (stood)
isiXhosa: inani (numeral); uku (to)
isiZulu: inani (numeral)
Here are some examples of numbers that are
palindromes.
99; 121; 12 321; 601 106
The Madoko Dam in Zimbabwe is a real-life
example of a palindrome. This palindrome
consists of two words that together form the
palindrome. See if you can read it backwards.
Note: It is true, but not very interesting, to say
that any word consisting of a single letter is a
palindrome. The same goes for any number
that consists of a single digit. We will ignore
these basic palindromes in this investigation, and concentrate on more interesting examples.
1. Write down all the palindromes in the following sentence, excluding the word ‘a’.
Anna, Otto, Hannah, and Bob travelled by kayak on a level stretch of water in 1991.
(7)
2. Write down all the palindromic years between the year 1 000 and the year 2 000.
(10)
3. The palindromes 11, 22 and 33 are all multiples of 11. What is the smallest multiple
of 11 that is not a palindrome?
(1)
4. Use as few digits as possible to make each of the following a palindrome.
a ) 567□
b) 34□
c) □867
d) 5□495
(4)
5. How many different three-digit palindromes can you make
using only the digits 1 and 2? Write them all down.
(3)
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6. How many different three-digit palindromes can you make
using only the digits 1, 2 and 3? Write them all down.
(5)
7. Use your answers to Questions 5 and 6 to complete the following.
a ) Using two different digits, you can make 2 × 2 = □
different three-digit palindromes.
(1)
b ) Using three different digits, you can make □ × □ = □
different three-digit palindromes.
(2)
8. Pam says that, using 100 different digits, she can make
100 × 100 = 10 000 different three-digit palindromes.
Zareena says that this does not make any sense.
a ) Who is correct, Pam or Zareena? Why?
(3)
b ) What is the maximum number of different three-digit
palindromes that you can make using any digits that you like? (2)
c ) What is the maximum number of different three-letter
palindromes that you can make using any letters of the
alphabet that you like?
(3)
9. Zandi has to choose a pin code for his cellphone. The pin code
must consist of any three digits or letters of the alphabet. If Zandi
wants his code to be a palindrome, how many different choices
does he have?
(4)
Note: When you solve a number of related mathematical
problems, it is often very useful to look for a pattern. In this
investigation, your answers to Questions 6 and 7 led to a
pattern that helped you to solve related questions without
having to write out, and then count, the different options
for each one. This is one of the reasons that patterns are so
important!
Total marks: 45
Investigation
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Topic
Geometric patterns
38
Maths ideas
• Extend geometric
patterns.
• Identify a
sequence in a
geometric pattern.
• Find a rule in a
number sequence.
Extend a geometric pattern
In Topic 16 you described geometric patterns and learnt how to draw
the next diagrams to extend a pattern. In this topic you will be given
more practice in working with geometric patterns.
ExERCiSE 38.1
Here is a geometric pattern that uses matchsticks.
• Use flow diagrams
to describe a
geometric pattern.
• Design a
geometric pattern
of your own.
1
3
2
1. Describe this geometric pattern in your own words.
2. Draw the next diagram in the pattern.
3. Count the number of matchsticks in each diagram. Then complete
this table for the first four diagrams.
Diagram number
Number of matchsticks
1
Rule
2
?
3
1
2
3
4
5
7
47
134
4. Look at the sequence of numbers in the bottom row of the table.
How do you get from one number to the next? Use this sequence
rule to fill in the table for diagrams 5 and 7.
5. If the diagram number is the input, what rule can you use to get
the number of matchsticks as the output?
6. Complete this flow diagram. In the box write your rule that will
change the input numbers into output numbers.
4
7. Use the rule to calculate the number of squares in diagrams 47
and 134. Write these in the table.
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Change the shape of a geometric
pattern
In Exercise 38.1, you made a shape larger by adding more matchsticks.
The rule was easy to find because in each diagram the number of
matchsticks was 8× the number of its diagram position.
In the next exercise, you will also add matchsticks to make a shape
larger, but the rule is not so easy to find.
ExERCiSE 38.2
Use matchsticks to make this pattern, and then answer the questions
that follow:
1
3
2
1. In your own words, describe this geometric pattern.
2. In your own words, describe how you get from Diagram 1 in
the pattern to Diagram 2, and how you get from Diagram 2 to
Diagram 3.
3. How many more matchsticks do you need to make Diagram 4?
Draw Diagram 4.
4. Count the number of matchsticks in each diagram. Then complete
this table for the first four diagrams.
Diagram number
Number of matchsticks
1
2
3
4
5
7
27
150
5. How do you get from one number to the next in this sequence?
Fill in the table for diagram numbers 5 and 7.
1
6. If the diagram number is the input, what rule can you use to get
the number of matchsticks as the output?
2
7. Complete this flow diagram. In the box write your rule that will
change the input numbers into output numbers.
3
8. Use the rule to calculate the number of matchsticks in Diagrams
27 and 150. Write these in the table.
Rule
?
4
Topic 38: Geometric patterns
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A different type of geometric pattern
See if you can find the rule for the next geometric pattern in this
exercise.
ExERCiSE 38.3
Look at this pattern made with matchsticks:
1
2
3
1. Copy Diagram 1 with matchsticks.
2. Now add matchsticks to copy Diagrams 2 and 3.
3. Now make Diagram 4 in the pattern.
4. Count the number of matchsticks in each diagram. Then complete
this table for the first four diagrams.
Diagram number
Number of matchsticks
1
2
3
4
5
6
7
5. Look at the sequence of numbers in the bottom row of the table.
How do you get from one number to the next? What rule can you
use to get the next number from the previous number?
6. Use the rule to calculate the number of matchsticks in Diagrams 5,
6 and 7. Write these in the table.
Challenge
Create your own geometric pattern.
a) Use matchsticks or counters to create your own geometric pattern.
b) Write questions (similar to those above) for your pattern.
c) Give your pattern to a partner to answer.
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Another type of pattern
This next pattern is different to the ones that you worked with above.
ExERCiSE 38.4
Copy the pattern of dots into your workbook.
1
2
3
5
4
1. In your own words, describe this geometric pattern.
2. Count the number of dots in each diagram. Then complete this
table for the first five diagrams.
Diagram number
Number of dots
1
2
3
4
5
6
7
8
3. Look at the sequence of numbers in the bottom row of the table.
How do you get from one number to the next? What rule can you
use to get the next number from the previous number?
4. Use the rule to calculate the number of matchsticks in Diagrams 6,
7 and 8. Write these in the table.
Did you know?
• Leonardo Fibonacci was an Italian mathematician who lived between 1170 AD and 1240 AD. He found that
certain numbers appear again and again in patterns in nature. The pattern he found was: 1; 1; 2; 3; 5; 8; 13;
21; 34; …
• To get the next number in the pattern, you add the two previous numbers together. This pattern is called
the Fibonacci sequence.
• You can find these numbers in the way leaves and petals of flowers are arranged. You can also see these
numbers in the way sunflower seeds are arranged or inside the fruit of many plants.
This flower has 3 petals.
This flower has 13 petals.
This flower has 21 petals.
Topic 38: Geometric patterns
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Topic
39
Number sentences
Maths ideas
• Express ‘word
problems’ in the
form of number
sentences.
• Solve and
complete number
sentences.
Use number sentences to solve
problems
When you solve word problems, remember to write the problem in
the form of a number sentence. Then solve it by solving the number
sentence.
Example
Mr and Mrs Samsodien went camping in the Kruger National Park.
They are both senior citizens, so they paid a reduced rate. They
paid R72 less than the normal rate of R180 per night. How much
did they pay for the four nights in the Kruger National Park?
Answer
This problem is in two parts. You need to:
1. Calculate how much they paid per night.
2. Calculate how much they paid for 4 nights.
You can write these as word sentences:
1. Normal rate (R180) − reduction (R72) = rate paid per night
2. Rate paid per night (□) × number of nights (4) = total amount
paid (◇)
You can write these as number sentences:
1. 180 − 72 = □
2. □ × 4 = ◇
You can solve the number sentences to find the missing numbers:
1. 180 − 72 = 108
2. 108 × 4 = 432
So, they paid R432 for four nights.
ExERCiSE 39.1
For each of these problems, write a number sentence(s). Use □ or ◇
for the missing number(s). Solve the number sentences by finding the
missing number(s).
1. A school collected toys for children in an orphanage.
During August 172 toys were collected and during September
237 toys were collected. How many more toys were collected
during September than during August?
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2. Tania has worked as a mechanical engineer for 4 years and
has saved enough money to buy a flat. She buys the flat for
R73 550 less than the price advertised in the newspaper. Tania
pays R450 700 for the flat. What was the price advertised in the
newspaper?
3. A truck delivers boxes of tomatoes to a supermarket. Each box
has 37 tomatoes in it. If there are 6 364 tomatoes on the truck
altogether, how many boxes are on the truck?
4. A shopkeeper sells 27 slabs of chocolate in January for a total of
R351. What is the cost of one slab of chocolate? If the shopkeeper
sells a further 395 slabs during the year for the same price, how
much money does he get for the whole year?
5. Sarah and Toni each collect stickers. Sarah has 472 stickers in her
collection. If Toni were to collect another 187 stickers she will have
the same number as Sarah. How many stickers does Toni have?
6. Hlahla used 9 120 bricks to build a wall around his house. This was
12 times more than the number of bricks that his neighbour used
for building a braai. How many bricks did his neighbour use?
7. Feroza and Margie both work at the local car wash over weekends.
Feroza works for 13 hours in March. Margie works for 11 hours in
March. If the car wash pays a total of R1 128 for their wages, what
is the hourly rate? How much do they each get?
8. Sophie is the shooter in her school
netball team. Gloria has scored a third
of the number of goals that Sophie has
scored. If Sophie has scored 411 goals
altogether, how many has goals has
Gloria scored?
9. Pumela uses __23 of an apple to bake one
apple pie. If she has 18 apples, how many
apple pies can she bake?
10. A recipe for making shortbread says that
for every 2 cups of sugar, __14 cup of butter
must be used. If a baker uses 40 cups of
sugar, how many cups of butter should
he use?
Topic 39: Number sentences
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Key words
Multiple choice questions
• multiple choice
− a question in
which you must
choose the correct
answer from a
number of possible
answers
The questions in this exercise are multiple choice questions. You
will be given a number of answers and usually only one is correct.
However, sometimes there can be more than one correct answer.
Always read the question carefully to see how many correct answers
you can find.
Example
How much greater is 27 × 18 than 27 × 16?
a) 27
b ) 18
c ) 54
d ) 32
Answer
• There is no need to do the calculation. The number that
is the same in both expressions is 27. The first expression
has 18 twenty-sevens and the second expression has only
16 twenty-sevens. This means that the first expression is
2 twenty-sevens more than the second expression.
• The answer is 54 (2 × 27). So, choose answer c).
ExERCiSE 39.2
1. By how much is 15 × 14 less than 15 × 17?
a ) 15
b ) 14
c ) 45
d ) 51
2. Which of these pairs of numbers has the rule, ‘add three to the first
number, then multiply by 7 to get the second number’?
a ) 11 and 80 b ) 9 and 84
c ) 84 and 9
d ) 14 and 119
3. Which one of these statements is always true? In each statement
□ represents the same number.
a) 7 × □ = □ − 7
b) 7 × □ = □ × 7
c) 7+□=□−7
d) 7 × □ = 7 + □
4. Which of these statements are equivalent to 18 + (8 + 12)?
a ) 18(8 + 12)
b ) (18 + 8) + 12
c ) (9 × 2) + 8 + (3 × 4)
d ) 8 + (18 + 12)
5. Which of these statements is equivalent to (14 × 18) + (14 × 2)?
a ) 14 × 20
b ) 14 × 36
c ) 14 + 20
d ) 14 + 36
6. Which of these statements 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
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Revision
1. Look at this geometric pattern:
1
3
2
a ) Count the squares and complete the table below for the first three diagrams.
Diagram number
1
2
3
4
5
6
29
(1)
78
Number of squares
b ) For the sequence of numbers in the bottom row, write down how the next
number can be obtained from the previous number.
c ) Use the rule that you found in Question b) to write the number of squares in
diagrams 4, 5 and 6 in the pattern.
d ) If the ‘diagram number’ is the input and ‘number of squares’ is the output, write
down the rule that enables you to get the output for any input number.
e ) Complete the table for diagrams 29 and 78.
2. Write the rule that was used to find these sequences of output numbers and then use
the rule to find the missing output number.
a ) Input number
1
2
3
4
5
6
27
b)
Output number
6
12
18
24
30
36
Input number
1
2
3
4
5
6
Output number
13
33
53
73
93
113
(1)
(3)
(2)
(2)
(2)
(2)
17
3. A school has collected empty cool drink bottles for recycling. 1 734 bottles have been
collected. Large plastic bags have also been collected for recycling. Learners 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.
b ) Solve the number sentence.
c ) One bag will not be completely full. How many bottles are there in this bag?
(2)
(3)
(2)
Total marks: 20
Revision
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Topic
40
Probability
Maths ideas
• Perform simple
repeated
experiments with
coins, dice and
spinners.
• List the possible
outcomes
of events or
experiments.
• Make tally tables
to record actual
outcomes.
• Count and
compare
frequency 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
When you toss a coin a number of times you are performing an
experiment. Each time you toss the coin you are doing a trial to see
what the outcome will be.
There are only two possible outcomes when you toss a coin:
• The coin can land heads up.
• The coin can land tails up.
tails
The picture shows you heads and tails on a R2
coin. The face with the value of the money is
called heads.
Before you perform an experiment you need to
head
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 four colours. So the possible
outcomes are:
• Red
• Blue
• Green
• Yellow
ExERCiSE 40.1
Here are four different spinners. List all the possible outcomes if you
spin each one.
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Record outcomes of experiments
When you record the outcomes of a number of trials, it makes sense to
list the possible outcomes in a table.
As you do each trial make a tally mark to show what the outcome was.
Then add the tally marks to find the frequency of each outcome.
Key words
• frequency − how
often an outcome
occurs
Example
April did an experiment with this spinner. She did 20 trials and
recorded his results in this frequency table.
Possible outcomes
Tallies
Frequency
Red
5
Blue
4
Green
6
Yellow
5
Number of trials conducted
20
• Notice that April has listed the possible outcomes.
• Then April used tallies to record how often the spinner landed
on each colour.
• Finally, April added up the tallies to get the total number of times it
landed on each colour. This is the frequency of each colour.
When you add up the frequencies the total should come to 20, as
this was the total number of trials.
ExERCiSE 40.2
1. In pairs, toss a coin 20 times and record the outcomes.
a ) Draw up a table to record the outcomes of your experiment.
b ) Do the experiment and record your results in your table.
2. a ) List the possible outcomes when you toss a normal dice.
b ) Draw up a table to record the actual outcomes when you toss
a normal dice 20 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 40: Probability
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Make your own spinner
You will need:
• a piece of paper and a piece of cardboard
• a round lid or a pair of compasses to draw
a circle
• a pair of scissors and some glue
• a toothpick or small nail
• coloured pens or pencils.
Step 1
Step 2
Step 3
Step 4
Step 4
ExERCiSE 40.3
1. Use the spinner you made for this activity.
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 20 times.
c ) Draw up a table to record the outcome of 20 trials.
d ) Carry out the experiment by spinning your spinner 20 times
and recording the outcomes.
e ) How well did you predict the outcomes?
Step 5
Step 6
A
Method
Step 1: Draw a small circle on the piece of paper and cut it out. Your
circle should not be wider than 4 cm.
Step 2: Fold the paper in half. Then fold the half circle in half again to
make 4 equal parts.
Step 3: Open up the paper and make a dot in the centre of the circle.
Step 4: Colour each section a different colour.
Step 5: Stick the paper circle onto the cardboard and cut it out.
Step 6: Push the toothpick or small nail through the dot in the centre
of the circle to make your spinner.
2. Look at these spinners.
B
C
D
E
F
G
H
a ) List the possible outcomes for each spinner.
b ) Which spinner is most likely to land on blue? Why?
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ExERCiSE 40.4
1. When you toss a dice, it can land on an odd number (1, 3 or 5) or
and even number (2, 4 or 6). Copy this table into your book.
Possible outcomes
Tallies
Frequency
An even number
An odd number
Number of trials conducted
20
2. Work in a group of four. You will each need a dice.
a ) Toss your dice 20 times. Record the actual outcomes in the table.
b ) Add up the tallies to find the frequency of odd and even numbers.
c ) Write each frequency as a fraction with a denominator of 20.
1
3. Compare your results with the other members of your group.
a ) Who got the highest frequency of odd numbers?
b ) Who got the highest frequency of even numbers?
3
4. Combine your totals to find the frequency of odd and even
numbers for 80 trials.
a ) What fraction of the total outcomes were odd?
b ) What fraction of the total outcomes were even?
c ) Write each total frequency as a fraction with a denominator
of 80.
d ) How does the total frequency compare with the frequency
that you got when you did the experiment on your own?
7
5. Angie has a six-sided dice with shapes on it. She tosses it 20 times
and records her results like this.
a ) Draw up a table to show the possible outcomes for this dice.
b ) Organise Angie’s results into the table.
c ) Which outcome occurred most frequently?
d ) Which outcome occurred least frequently?
3
2
Challenge
Conduct your own experiment to find the outcomes when you toss two coins at
the same time. Record your results in a table like this one. Perform 20 trials of
the experiment.
Possible outcomes
Tallies
Frequency
4
Number of trials conducted
Topic 40: Probability
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Glossary
A
a.m. (ante meridian) – the time after midnight
but before noon
page 28
alpha numeric grid – a grid with alphabet letters
for the columns and numbers for the rows
page 184
analogue clock – a clock that uses hands to
show the time
page 28
angle – the amount of turn between two straight
lines that meet each other
page 44
area – the amount of space that a shape covers
page 178
ascending order – from smallest to greatest
page 7
average speed – total distance divided by
total time; the rate at which the distance is
changing over time. We normally write speed
in km/hr. This means kilometres per hour
page 81
axis – one of two fixed lines in a graph; it can be
either vertical (up–down) or horizontal (left–
right)
page 35
B
base – the face on which the object rests
page 82
biased – skewed or distorted
page 41
bimodal – a data set with two modes
page 136
boiling point – temperature at which liquid turns
to gas for example, water boils to form vapour
page 130
C
capacity – the maximum amount an object can
hold C
page 48
206
census – a government count of the whole
population
page 40
column method – add and subtract by writing
digits with the same place value below each
other in the same column
page 116
compensating – to add or subtract numbers
after rounding off one of the numbers when
calculating
page 15
composite – a 2D shape that is made up of two
or more 2D shapes that are joined together
page 45
context – where and how data was collected
page 41
convert – to change to something else
page 30
coordinate – a reference that is used to show the
exact location of objects
page 184
curved surface – a surface of a 3D object that is
not flat
page 83
D
data – a collection of facts, numbers or
measurements
page 34
data cycle – process of asking questions,
collecting and organising data and
summarising results
page 37
decade – a period of 10 years
page 30
denominator – the number below a fraction line
which shows how many parts that the whole
has been divided into
page 64
descending order – from greatest to smallest
page 7
Glossary
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digital clock – a clock that shows time using
numbers and a : separator
page 28
dimension – a measurement of length, breadth
or height
page 84
E
equivalent fractions – fractions that have the
same value
page 65
estimate – an approximate answer
page 14
evacuation plan – a plan of a building that
shows the quickest way out of the building to
a place of safety
page 121
exchange – break down a large number into
smaller units – 100 is 10 tens; 1 000 is 10
hundreds and so on
page 159
experiment – something you do to find out what
will happen
page 202
F
face – a surface of an object
page 82
factor – a number that divides exactly into
another number
page 22
flow diagram – a diagram that shows how an
operation (for example, addition) is applied to
numbers to get an answer
page 18
freezing point – temperature at which liquid
turns to solid, for example, water freezes to
form ice
page 130
frequency – how often an event or outcome
occurs
page 203
G
geometric patterns – repeated arrangements of
shapes
page 88
grid reference position – refers to a particular
cell or coordinate in the grid
page 184
H
heptagon or septagon – a polygon with seven
sides
page 43
hexagon – a polygon with six sides
page 43
I
identical – exactly the same
page 82
infinite – never ending
page 94
input number – the number that you start with
in a flow diagram
page 18
inverse – opposite
page 8
inverse operations – reverse or opposite
operations (for example, multiplication is the
inverse of division)
page 24
K
kilo – means one thousand, therefore kilogram
means 1 000 g
page 110
L
leap year – the name given to a year that has 366
days in it and not the usual 365
page 30
line of symmetry – a line that divides a shape
into two identical halves
page 92
Glossary
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line symmetry – a shape has line symmetry if it
can be folded in such a way that one half lies
exactly on the other half
page 92
M
mass – the amount of matter an object has
page 108
maximum temperature – the highest
temperature reading recorded
page 132
midpoint – the point that divides a line into two
equal parts
page 126
minimum temperature – the lowest
temperature reading on one day
page 132
mixed number – a whole number and a fraction
making one number
page 105
modal – relating to the mode
page 36
mode – the data value that occurs most often
page 36
multiple – the answer when you multiply two
numbers
page 22
multiple choice – a question in which you must
choose the correct answer from a number of
possible answers
page 200
N
net – a flat arrangement of the 2D shapes that
make up a 3D object
page 165
number sequence – a group of numbers that
follow each other in a particular order
page 18
numerator – the number above the fraction line
showing the number of parts of the whole.
page 64
O
operation – addition, subtraction, multiplication,
division
page 8
208
order of rotational symmetry – the number of
times that a shape looks like the original while
completing a full turn
page 93
outcome – a result of a trial
page 202
output number – the answer that you get in a
flow diagram
page 18
P
p.m. (post meridian) – the time past noon up to
midnight
page 28
patterns – repeated arrangements of shapes,
numbers, colours or lines
page 88
pentagon – a shape with five sides
page 43
perimeter – the total distance around the
outside of a shape
page 176
pictograph – graph that uses symbols (pictures)
to show data
page 34
place value – the value of a digit in a number
according to the position of the digit within
the number
page 4
polygon – a 2D shape that is enclosed by three
or more straight sides
page 42
prism – a 3D object with two identical, parallel
end faces (bases)
page 82
pyramid – a 3D object that has a polygon base
and all its other faces as triangles
page 82
Q
quadrilateral – a polygon with four sides
page 43
R
rectangle – a quadrilateral where all the angles
are right angles
page 43
Glossary
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reflect – when you flip a shape ( as in a mirror
image)
page 128
right angle – a quarter of a full turn
page 44
rotate – to transform a shape by turning it
page 128
rotational symmetry – a shape has rotational
symmetry if it can be rotated so that it looks
exactly like the original shape at least once
before completing a full turn
page 93
round off – a way of making a number simpler to
use, according to a given place value
page 5
S
scale – an instrument used to measure mass
page 109
sequence – a group of numbers or shapes that
follow each other in a particular order
page 91
source – the people or places that data comes
from
page 41
square – a rectangle with all the sides equal
page 43
stopwatch – an instrument that you use to time
the duration of events accurately
page 31
T
table – information arranged in rows and
columns
page 34
tallies – marks made to record each item when
you are counting
page 34
tangram – an Ancient Chinese puzzle consisting
of geometrical pieces that fit together to make
a square
page 129
thermometer – an instrument we use to
measure temperature
page 131
three-dimensional (3D) – having three
dimensions: length, width and height
page 42
transformation – change in the position and/or
the direction of a shape
page 128
translate – transform a shape by sliding it
page 128
trial – the activity you do in an experiment
page 202
triangle – a polygon with three sides
page 43
two-dimensional (2D) – having two dimensions:
length and width
page 42
V
viewpoint – position from which you view an
object
page 121
volume – how much space an object takes up
page 48
W
whole numbers – the numbers we use to count,
including 0. 0, 1, 2, 3, …
page 24
Glossary
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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
210
Useful resources
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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
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Maskew Miller Longman (Pty) Ltd
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