Greg Byrd, Lynn Byrd and Chris Pearce
Cambridge Checkpoint
Mathematics Skills Builder Workbook
9
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Contents Introduction 1 Integers, powers and roots
5 7
1.1 Directed numbers 1.2 Square roots and cube roots 1.3 Indices
7 9 12
2 Sequences and functions
13
2.1 Generating sequences 2.2 Finding the n th term 2.3 Finding the inverse of a function
13 15 17
3 Place value, ordering and rounding
20
3.1 Multiplying and dividing decimals mentally 3.2 Multiplying and dividing by powers of 10 3.3 Rounding 3.4 Order of operations 4 Length, mass, capacity and time
20 22 24 25 27
4.1 Solving problems involving measurements 27 4.2 Solving problems involving average speed 29 4.3 Using compound measures 32
7 Fractions
47
7.1 Writing a fraction in its simplest form 7.2 Adding and subtracting fractions 7.3 Multiplying fractions 7.4 Dividing fractions
47 49 51 53
8 Constructions and Pythagoras’ theorem
55
8.1 Using Pythagoras’ theorem
55
9 Expressions and formulae
59
9.1 9.2 9.3 9.4 9.5 9.6
59 61 63 65 67
Simplifying algebraic expressions Constructing algebraic expressions Substituting into expressions Deriving and using formulae Factorising Adding and subtracting algebraic fractions
69
10 Processing and presenting data
71
10.1 Calculating statistics 10.2 Using statistics
71 75
11 Percentages
78
11.1 Using mental methods 11.2 Percentage changes
78 81
5 Shapes
34
5.1 Regular polygons 5.2 Plans and elevations
34 37
12 Tessellations, transformations and loci
84
6 Planning and collecting data
40
6.1 Designing data collection sheets 6.2 Collecting data
40 44
12.1 Tessellating shapes 12.2 Solving transformation problems 12.3 Enlarging shapes
84 85 88
3
13 Equations and inequalities
90
18 Graphs
13.1 Solving linear equations 13.2 Trial and improvement
90 93
18.1 Gradient of a graph 120 18.2 The graph of y = mx + c 124
14 Ratio and proportion
97
14.1 Comparing and using ratios 14.2 Solving problems
97 99
19 Interpreting and discussing results 127
15 Area, perimeter and volume
101
15.1 Solving circle problems 101 15.2 Calculating with prisms and cylinders 104 16 Probability
108
16.1 Calculating probabilities 16.2 Sample space diagrams
108 111
17 Bearings and scale drawings
114
17.1 Using bearings 17.2 Making scale drawings
114 117
120
19.1 Interpreting and drawing frequency diagrams 127 19.2 Interpreting and drawing line graphs 128 19.3 Interpreting and drawing scatter graphs 131 19.4 Interpreting and drawing stem-and-leaf diagrams 134 Glossary 136
Introduction Welcome to Cambridge Checkpoint Mathematics Skills Builder Workbook 9. The Cambridge Checkpoint Mathematics course covers the Cambridge Secondary 1 Mathematics curriculum framework. The course is divided into three stages: 7, 8 and 9. You can use this Skills Builder Workbook with Coursebook 9 and Practice Book 9. It gives you extra practice in all the topics, focusing on those that are the most important, to improve your understanding and confidence. Like the Coursebook and the Practice Book, this Workbook is divided into 19 units. In each unit there are exercises on each topic. There are introductory explanations and either worked examples or guided questions. These explain the skills you need to master and use to solve more complex problems. This Workbook also gives you a chance to try further questions on your own. This will improve your understanding of the units. It will also help you feel confident about working on your own when there is no teacher to help you. At the end of each unit is a link to exercises to attempt in the Coursebook. If you get stuck with a task: • Read the question again. • L ook back at the introductory explanations and worked examples or guided questions. • Read through the matching section in the Coursebook.
5
1 1.1
Integers, powers and roots
Directed numbers To subtract a negative number, you add the inverse. Look at these examples: 3 – –2 = 3 + 2 = 5
–2 – –5 = –2 + 5 = 3
–6 – –4 = –6 + 4 = –2
1
Fill in the missing numbers.
a
5 – –1 = 5 + . . . . . . = . . . . . .
b
2 – –4 = 2 + . . . . . . = . . . . . .
c
8 – –1 = 8 + . . . . . . = . . . . . .
d
–3 – –1 = –3 + . . . . . . = . . . . . .
e
–1 – –5 = –1 + . . . . . . = . . . . . .
f
–7 – –5 = –7 + . . . . . . = . . . . . .
The numbers can be decimals: 2.5 – –1.3 = 2.5 + 1.3 = 3.8
–1.2 – –3.4 = –1.2 + 3.4 = 2.2 +3.4
+1.3 0
1
2.5
3.8
–2 –1.2
5
0
2.2
2
Fill in the missing numbers.
a
3.5 – –1.2 = 3.5 + . . . . . . = . . . . . .
b
2.2 – –3.4 = 2.2 + . . . . . . = . . . . . .
c
2.7 – –2.2 = 2.7 + . . . . . . = . . . . . .
d
–4.6 – –2 = –4.6 + . . . . . . = . . . . . .
e
–2 – –3.5 = –2 + . . . . . . = . . . . . .
f
–6 – –2.3 = –6 + . . . . . . = . . . . . .
4
7
Unit 1 Integers, powers and roots
3
Fill in the missing outputs. Input Subtract –4 2.5 – –2.5 3.1 –1.6 –5.5
Output –1.5 ...... ...... ...... ......
The rule for multiplication is: Same signs 3 × 4 = 12 positive answer –2 × –7 = 14 Different signs 3 × –4 = –12 negative answer –2 × 7 = –14
4
a 3 × –4 = . . . . . .
b –2 × –3 = . . . . . .
c –5 × 4 = . . . . . .
d 2.4 × –2 = . . . . . .
e –3.2 × 3 = . . . . . .
f –4.1 × –5 = . . . . . .
5
Complete these multiplications.
Fill in the missing outputs. Input 3 –4.2 2.5 –1.2 –6.1
Multiply
× –3
Output –9 ...... ...... ...... ......
Now try Exercise 1.1 on page 8 of Coursebook 9.
8
Unit 1 Integers, powers and roots
1.2
Square roots and cube roots This number line shows the squares of whole numbers from 1 to 7. 12 22 1 4
0
32 9
10
The square of 3 is 9
42 16
52 25
20
62 36
30
72 49
40
32 = 9
50 _ √ 9 = 3
The square root of 9 is 3
1
Fill in the missing numbers.
a
The square of 6 is . . . . . .
62 = . . . . . .
The square root of . . . . . . is 6
_ √ . . . . . . = 6
b
The square of . . . . . . is 25
. . . . . . ² = 25
The square root of 25 is . . . . . .
_ √ 25 = . . . . . .
2
Fill in the missing numbers.
a
_ √ 49 = . . . . . .
b
_
c
√ 4 = . . . . . .
_ √ 16 = . . . . . .
d
_
√ 1 = . . . . . .
_ What is √ 30 ? Look at the number line above. 30 is between 25 and 36. _ So √ 30 is between 5 and 6.
3
Fill in the missing numbers.
a
_ √ 20 is between . . . . . . and . . . . . .
c
√ 14 is between . . . . . . and . . . . . .
e
√ 27 is between . . . . . . and . . . . . .
_ _
25 is 52 and 36 is 62.
b
_ √ 45 is between . . . . . . and . . . . . .
d
_ √ 6 is between . . . . . . and . . . . . .
f
_ √ 39 is between . . . . . . and . . . . . .
9
Unit 1 Integers, powers and roots
4
Mark 72, 82, 92 and 102 on this number line.
50
5
70
80
90
100
140
150
Mark 102, 112 and 122 on this number line.
100
60
110
120
130
6
10
Fill in the missing numbers. _ a √ 60 is between . . . . . . and . . . . . .
_ b √ 93 is between . . . . . . and . . . . . .
_ c √ 129 is between . . . . . . and . . . . . .
_ d √ 108 is between . . . . . . and . . . . . .
_ e √ 52 is between . . . . . . and . . . . . .
_ f √ 77 is between . . . . . . and . . . . . .
Unit 1 Integers, powers and roots
This number line shows the first five cube numbers. 13 23 1 8
0
33 27
20
43 64
40
The cube of 4 is 64
60
43 = 64 _ 3 √ 64 = 4
The cube root of 64 is 4
7
Fill in the missing numbers.
a
3
8
Fill in the missing numbers.
a
3
_
c
3
_
e
3
_
_
√ 27 = . . . . . .
b
53 125
3
80
_
√ 121 is between . . . . . . and . . . . . . √ 5 is between . . . . . . and . . . . . .
120
43 = 4 × 4 × 4.
c
√ 8 = . . . . . .
√ 40 is between . . . . . . and . . . . . .
100
b
3
_
d
3
_
f
3
_
3
_
√ 125 = . . . . . .
d
3
_
√ 1 = . . . . . .
√ 85 is between . . . . . . and . . . . . . √ 20 is between . . . . . . and . . . . . . √ 100 is between . . . . . . and . . . . . .
Now try Exercise 1.2 on page 10 of Coursebook 9.
11
Unit 1 Integers, powers and roots
1.3
Indices 1 3–1 is another way of writing the fraction _ 3
When you write 3–1, the –1 is called an index. The plural is indices.
1 7–1 is another way of writing the fraction _ 7 1
Write these numbers as fractions.
a
2–1 . . . . . .
2
Write these fractions using indices.
a
1 _ = . . . . . . 5
b
b
6–1 . . . . . .
1 _ = . . . . . . 12
c
8–1 . . . . . .
d
10–1 . . . . . .
c
1 _ = . . . . . . 9
d
1 _ = . . . . . . 4
52 = 25
1 You can write _ = 5–2 25 3
Fill in the missing numbers. The first one has been done for you.
a
1 42 = 16 so _ = 4–2 16
b
1 62 = 36 so _ = . . . . . . 36
c
102 = 100 so . . . . . .
d
92 = 81 so . . . . . .
e
72 = 49 so . . . . . .
f
82 = 64 so . . . . . .
Now try Exercise 1.3 on page 11 of Coursebook 9.
12
2 2.1
Sequences and functions
Generating sequences This sequence of numbers is called a linear sequence: 2, 5, 8, 11, 14, . . . In a linear sequence, the terms increase or decrease by the same amount each time. The term-to-term rule for this sequence is ‘Add 3’.
1
2 + 3 = 5, 5 + 3 = 8, 8 + 3 = 11, etc.
Write down the first five terms of each linear sequence. Some of them have been started for you. term-to-term rule: ‘Add 2’.
a
1st term: 5
5
+2
7
+2
9
+2
......
+2
......
term-to-term rule: ‘Add 3’. b
1st term: 0
0
+3
3
+3
......
+3
......
+3
......
term-to-term rule: ‘Subtract 2’. c
1st term: 11
11
–2
......
–2
......
–2
......
–2
......
term-to-term rule: ‘Subtract 20’.
d
1st term: 210
210
......
......
......
......
term-to-term rule: ‘Add 0.5’. e
1st term: 2
2
+0.5
...... ...... ...... ...... ...... +0.5 +0.5 +0.5 +0.5
13
Unit 2 Sequences and functions
This sequence of numbers is called a non-linear sequence: 2, 5, 9, 14, 20, . . . In a non-linear sequence, the terms increase or decrease by a different amount each time. The term-to-term rule for this sequence is 2 + 3 = 5, 5 + 4 = 9, 9 + 5 = 14, . . .
‘Add 3, add 4, add 5, add 6, . . .’.
2
Write down the first five terms of each non-linear sequence. Some of them have been started for you. term-to-term rule: ‘Add 1, add 2, add 3, …’.
a
1st term: 4
4
+1
5
7
+2
+3
......
+4
......
term-to-term rule: ‘Add 2, add 4, add 6, …’. b
1st term: 5
5
+2
7
+4
......
+6
......
+8
......
term-to-term rule: ‘Subtract 3, subtract 4, subtract 5, …’. c
1st term: 20
3
Look at these sequences. Two of them are linear and two are non-linear.
20
–3
......
–4
......
–5
......
–6
......
Write ‘Linear’ or ‘Non-linear’ next to each one. a
0, 3, 8, 15, 24
.......................
b
18, 20, 22, 24, 26 . . . . . . . . . . . . . . . . . . . . . . .
c
30, 25, 20, 15, 10 . . . . . . . . . . . . . . . . . . . . . . .
d
50, 49, 47, 44, 40 . . . . . . . . . . . . . . . . . . . . . . .
Now try Exercise 2.1 on page 16 of Coursebook 9.
14
Unit 2 Sequences and functions
2.2
Finding the nth term
A linear sequence is also called an arithmetic sequence. You can write the position-to-term rule of an arithmetic sequence as an expression called the nth term. Look at this sequence: 2, 4, 6, 8, 10, . . .
1st term: 2 × 1 = 2
The position-to-term rule is: term = 2 × position number
2nd term: 2 × 2 = 4
The n th term expression is: 2n (which means 2 × n)
3rd term: 2 × 3 = 6
You can see that ‘position number’ and n are the same thing. position number n
1
The n th term of a sequence is 5n. Complete the workings to find the first four terms of the sequence. The first one has been done for you.
1st term: 5 × n = 5 × 1 = 5
2nd term: 5 × n = 5 × 2 = . . . . . .
3rd term: 5 × n = 5 × 3 = . . . . . .
4th term: 5 × n = 5 × . . . . . . = . . . . . .
2
The n th term of a sequence is 8n. Work out the first four terms of the sequence.
1st term: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2nd term: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3rd term: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4th term: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
The n th term of a sequence is n + 3. Complete the workings to find the first four terms of the sequence.
1st term: n + 3 = 1 + 3 = 4
2nd term: n + 3 = 2 + 3 = . . . . . .
3rd term: n + 3 = . . . . . . + 3 = . . . . . .
4th term: n + 3 = . . . . . . + 3 = . . . . . .
15
Unit 2 Sequences and functions
4
The n th term of a sequence is n + 10. Work out the first four terms of the sequence. 1st term: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2nd term: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3rd term: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4th term: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
Complete the workings to find the 10th term of each of these sequences.
a
n th term = 3n
10th term = 3 × 10 = . . . . . .
b
n th term = 8n
10th term = 8 × . . . . . . = . . . . . .
c
n th term = n + 4
10th term = . . . . . . + . . . . . . = . . . . . .
d
n th term = n + 9
10th term = . . . . . . + . . . . . . = . . . . . .
6
Draw a line connecting each n th term expression with the correct value for the 10th term. The first one has been done for you. 6n
16
n+6
20
n+1
60
n–5
21
2n
11
2n + 1
5
Now try Exercise 2.2 on page 19 of Coursebook 9.
16
In each of the nth term expressions, swap n for 10.
Unit 2 Sequences and functions
2.3
Finding the inverse of a function
Here is a function machine. Input 4 7 ......
To work out the output values: 4+3=7 7 + 3 = 10
Output ...... ...... 15
+3
To work out the input value, reverse the function machine: 15 – 3 = 12
When you reverse the function machine you find the inverse function. You can write a function and its inverse as a function machine like this:
Function:
x
+3
x+3
You can also write the function and its inverse as an equation: Function: y = x + 3 Inverse: y = x – 3
Inverse:
x–3
–3
x
Or you can write it as a mapping: Function: y → x + 3 Inverse: y → x – 3
1
Work out the missing input and output numbers from these function machines.
a
c
Input 1 3 ...... Input 4 6 ......
+5
×2
Output ...... ...... 15 Output ...... ...... 20
b
d
Input 30 25 ...... Input 10 25 ......
–9
÷5
Output ...... ...... 10 Output ...... ...... 8
17
Unit 2 Sequences and functions
2
Complete the workings to find the inverse of each of these functions. Use your answers to Question 1 to help you.
a
1
+5
......
1
–5
6
b 30
–9
......
......
+9
21
4
×2
......
......
÷2
8
d 10
÷2
......
......
×2
5
c
3
x
+5
x+5
x – ......
–5
x
x
–9
x–9
x + ......
+9
x
x
×2
2x
x ......
÷2
x
x
÷2
x 2
...... x
×2
x 2
Find the inverse for each of these functions. Write each one as a machine, an equation and as a mapping. The first one has been done for you.
a Machine Function Inverse 18
Equation
Mapping
x
+4
x+4
y = x + 4 x → x + 4
x–4
–4
x
y = x – 4 x → x – 4
Unit 2 Sequences and functions
b Machine Function
Equation x
Inverse . . . . . .
+7
......
Mapping
x+7
y = . . . . . . . . . . . . . x →. . . . . . . . . . .
x
y = . . . . . . . . . . . . . . x →. . . . . . . . . . .
c Machine
Function
x
Inverse . . . . . .
–6
......
Equation
Mapping
x–6
................
................
x
................
................
Equation
Mapping
d Machine
Function
x
Inverse . . . . . .
×9
9x
................
................
......
x
................
................
Equation
Mapping
e Machine
Function
x
Inverse . . . . . .
÷4
x 4
................
................
......
x
................
................
Now try Exercise 2.3 on page 20 of Coursebook 9.
19
3 3.1
Place value, ordering and rounding
Multiplying and dividing decimals mentally When you multiply and divide by decimals mentally, you can use jottings to help you. Here is an example of how you can work out 8 × 0.3: 8 x 3 = 24 8 x 0.3 = 2.4
20
This method works because you change the 0.3 to 3 by multiplying by 10:
0.3 × 10 = 3
Then you divide the answer of 24 by 10:
24 ÷ 10 = 2.4
1
Fill in the missing numbers to work out these mental multiplications.
a
4 × 0.3
4 × 3 = 12
so
4 × 0.3 = . . . . . .
b
7 × 0.4
7 × 4 = ......
so
7 × 0.4 = . . . . . .
c
9 × 0.1
9 × 1 = ......
so
9 × 0.1 = . . . . . .
d
15 × 0.2
15 × 2 = . . . . . .
so
15 × 0.2 = . . . . . .
e
8 × 0.02
8 × 2 = 16
so
8 × 0.02 = . . . . . .
f
5 × 0.04
5 × 4 = ......
so
5 × 0.04 = . . . . . .
g
11 × 0.07
11 × 7 = . . . . . .
so
11 × 0.07 = . . . . . .
Unit 3 Place value, ordering and rounding
Here is an example of a method you can use to work out 12 ÷ 0.3:
12 x 10 = 120 0.3 x 10 = 3 120 ÷ 3 = 40
You multiply both numbers by 10 so that the 0.3 becomes a whole number instead of a decimal.
This method works because when you are dividing two numbers, if you multiply both numbers by 10, 100 or any other number, the answer to the division will remain the same.
2
Fill in the missing numbers to work out these mental divisions.
a
6 ÷ 0.3
6 × 10 = 60
0.3 × 10 = 3
60 ÷ 3 = . . . . . .
b
8 ÷ 0.2
8 × 10 = 80
0.2 × 10 = . . . . . .
80 ÷ . . . . . . = . . . . . .
c
9 ÷ 0.1
9 × 10 = . . . . . .
0.1 × 10 = . . . . . .
...... ÷ ...... = ......
d
12 ÷ 0.4
12 × 10 = . . . . . .
0.4 × 10 = . . . . . .
...... ÷ ...... = ......
e
6 ÷ 0.02
6 × 100 = 600
0.02 × 100 = 2
600 ÷ 2 = . . . . . .
f
8 ÷ 0.04
8 × 100 = . . . . . .
0.04 × 100 = . . . . . .
...... ÷ ...... = ......
g
16 ÷ 0.08
16 × 100 = . . . . . .
0.08 × 100 = . . . . . .
...... ÷ ...... = ......
Now try Exercise 3.1 on page 24 of Coursebook 9.
21
Unit 3 Place value, ordering and rounding
3.2
Multiplying and dividing by powers of 10 When you multiply and divide numbers by powers of 10, it is easiest to start by changing the power of 10 into a normal number. For example, × 103 means × 1000, because 103 means 10 x 10 x 10 = 1000.
1
Draw a line from each rectangular card to its matching oval card. One has been done for you. ×100 1
×1000
×10
×1
×102
×10000
×103
×100
4
×100000
5
×10
×10 ×10
When you multiply, use this method of moving the decimal point: 5.8 × 102 = 5.8 × 100 = 580
22
Move the decimal point two places to the right as 100 has two zeros.
2
Complete the workings for each of these multiplications.
a
3.4 × 102 = 3.4 × . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
b
4.8 × 103 = 4.8 × . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
c
12.5 × 101 = 12.5 × . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
d
5 × 105 = 5 × . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
e
14 × 103 = 14 × . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.8.0.
After you have moved the decimal point, remember to fill in any spaces in the answers with zeros.
Hint: 5 = 5.0
Unit 3 Place value, ordering and rounding
3
Draw a line from each rectangular card to its matching oval card. ×10–1
÷10 000
×10–2
÷10
×10–3
÷100 000
×10–4
÷1000
×10–5
÷100
When you divide, use this method of moving the decimal point: 5.8 × 10 = 5.8 ÷ 100 = 0.058 –2
Move the decimal point two places to the left as 100 has two zeros.
4
Complete the workings for each of these divisions.
a
3.4 × 10–2 = 3.4 ÷ . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
b
8 × 10–3 = 8 ÷ . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
c
15 × 10–4 = 15 ÷ . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
d
12 × 10–1 = 12 ÷ . . . . . . = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
0.0.5.8
After you have moved the decimal point, remember to fill in any spaces in the answers with zeros.
Now try Exercise 3.2 on page 27 of Coursebook 9.
23
Unit 3 Place value, ordering and rounding
3.3
Rounding Use this method to round a decimal number to one decimal place (1 d.p.). Round 5.623 to 1 d.p. Underline the digit in first place after the point:
5.623
Circle the next digit:
5.623
The number in the circle is less than 5, so leave the underlined number as it is:
5.6
Round 5.693 to 1 d.p. Underline the digit in first place after the point:
5.693
Circle the next digit:
5.693
The number in the circle is 5 or more, so add 1 to the underlined number:
5.7
1
Round each of these numbers to 1 d.p.
a
4.12
......
b
3.85
......
c
6.337 . . . . . .
d
8.164 . . . . . .
e
12.77 . . . . . .
f
43.05 . . . . . .
2
Round each of these numbers to 2 d.p.
a
5.168 . . . . . .
b
c
0.7769 . . . . . .
d
0.0444 . . . . . .
e
12.005 . . . . . .
f
25.118 . . . . . .
3
All the following numbers have been rounded to 3 d.p.
5.223 . . . . . .
This time, underline the second digit and put a circle round the third, then follow the same method as above.
Put a tick (✓) if they are correct and a cross (✗) if they are wrong. If they are wrong, write the correct rounded answer. a
5.6647 correct to 3 d.p. is 5.665 . . . . . .
b
3.1212 correct to 3 d.p. is 3.121
c
0.0079 correct to 3 d.p. is 0.08 . . . . . .
d
12.0454 correct to 3 d.p. is 12.045 . . . . . .
Now try Exercise 3.3 on page 29 of Coursebook 9. 24
......
Unit 3 Place value, ordering and rounding
3.4
Order of operations When you carry out calculations, you must use the correct order of operations. B Brackets
I Indices (powers)
D Division
M Multiplication
A Addition
S Subtraction
Write down the operations you have to do for each calculation, then do them in the right order. For example: 5+3×6
1
5+3×6 A M = 5 + 18 = 23
M comes before A in the list, so Multiply and then Add M: 3 x 6 = 18 A: 5 + 18 = 23
Complete the table showing the workings for each of these calculations. The first one has been done for you. Calculation
Operations
Order of operations
1st step
2nd step
a
20 – 4 × 3
S and M
M then S
4 × 3 = 12
20 – 12 = 8
b
8×2+4
M and A
M then A
c
8 3 + _ 2
A and D
d
30 _ – 1 6
D and S
e
7 + 22
A and I
f
2 × (9 + 11)
M and B
g
50 – (30 – 10)
S and B
h
52 – 15
I and S
25
Unit 3 Place value, ordering and rounding
2
Complete the workings for these calculations.
a 3 × 2 + 4 × 2
3 × 2 = 6
4 × 2 = ...... 6 + ...... = ......
I then M then A
22 = . . . . . .
5 × 3 = ...... ...... + ...... = ......
I then D then S
32 = . . . . . .
20 ÷ 2 = . . . . . . . . . . . . – . . . . . . = . . . . . .
M A M
b 22 + 5 × 3
Ms then A
I A M
20 c _ – 32 2 D S I
Now try Exercise 3.4 on page 30 of Coursebook 9.
26
4 4.1
Length, mass, capacity and time
Solving problems involving measurements When you solve problems involving measurements, you need to be able to convert metric units.
1
Complete these conversion tables. Length
Mass
1 km = . . . . . . m
Capacity
1 kg = . . . . . . g
1 l = . . . . . . ml
1 m = . . . . . . cm 1 cm = . . . . . . mm
2 km
Complete these conversions of length. Use the flow charts to help you. × 1000
m
× 100
cm
× 10
mm
a
5 km = . . . . . . m
b
4.5 m = . . . . . . cm
c
3.2 cm = . . . . . . mm
d
0.25 km = . . . . . . m
km
÷ 1000
m
÷ 100
cm
÷ 10
mm
e
700 cm = . . . . . . m
f
450 mm = . . . . . . cm
g
3200 m = . . . . . . km
h
1250 cm = . . . . . . m
27
Unit 4 Length, mass, capacity and time
3
Complete these conversions of mass. Use the flow charts to help you. kg kg
× 1000 ÷ 1000
g g
a 5 kg = . . . . . . g b 3.2 kg = . . . . . . g
c 0.5 kg = . . . . . . g
d 2600 g = . . . . . . kg
f 14 000 g = . . . . . . kg
4
e 250 g = . . . . . . kg
Complete these conversions of capacity. Use the flow charts to help you. l l
× 1000 ÷ 1000
ml ml
a 8 l = . . . . . . ml b 12 l = . . . . . . ml
c 2.4 l = . . . . . . ml
d 9000 ml = . . . . . . l
f 750 ml = . . . . . . l
e 3500 ml = . . . . . . l
Now try Exercise 4.1 on page 35 of Coursebook 9.
28