Step 3
Maths Frameworking Intervention Workbook Chris Pearce
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How to use this book Organise your learning The Contents table at the start of the Workbook shows the topics you are going to cover. Your teacher or tutor can set a date for you to complete each topic by. You can give a traffic light colour for each topic to show how you feel it went. Y You, your teacher and your parent or carer can write comments.
Work through each topic step by W step For each topic, there are: Clear learning objectives Worked examples to show you how to answer the questions Practice questions to help you consolidate what you have learnt. A glossary and answers are available on the Collins website. At the end of each chapter, there’s a comments box for your teacher or tutor to fill in on how you did.
Practise your mental maths Try the mental maths questions at the end of the Workbook to see what you have learned.
Celebrate your progress When you finish the Workbook, your teacher or tutor can fill in the Record of achievement certificate for you to keep.
How to use this book
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Step 3 Contents Strand/topic
Page
Hours Due date
1 Number 1.1
Multiplying and dividing by 10, 100 and 1000
Feedback Student
6
1
1.2 Rounding
9
1
1.3 Fractions
11
1
1.4 Fractions and decimals
13
1
1.5 Calculations
15
1
1.6 Fractions and percentages
16
1
1.7 Calculations without a calculator
18
1
1.8 Negative numbers
20
1
1.9 Approximations
22
1
1.10 Units of measurement
24
1
1.11 Number relationships
27
1
1.12 Number patterns
29
1
1.13 Squares, cubes and roots
31
1
2.1 Formulae
34
1
2.2 Coordinates
37
1
Teacher/ tutor/mentor
Parent/carer
2 Algebra
4
Contents
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Strand/topic
Page
Hours Due date
3 Ratio, proportion and rates of change
Feedback Student
3.1 Ratio
40
1
3.2 Proportion
43
1
4.1 Symmetry
46
1
4.2 3D shapes
50
1
4.3 Measuring angles
52
1
4.4 Calculating angles
54
1
4.5 Area and perimeter
57
1
5.1 Probability scales
61
1
5.2 Equally likely outcomes
64
1
6.1 Statistical measures
67
1
6.2 Statistical diagrams
69
1
6.3
73
1
4 Geometry and measures
Teacher/ tutor/mentor
5 Probability
Parent/carer
6 Statistics
Line graphs
Mental maths warm-ups Certificate of achievement
Contents
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1
Number
1.1 Multiplying and dividing by 10, 100 and 1000 I can t multiply whole numbers and decimals by 10, 100 and 1000 t divide whole numbers and decimals by 10, 100 and 1000
Example Work out the following: a 5.7 Ă— 100
b 570 á 1000
Solution a Write 5.7 in columns headed units (U), tens (T), hundreds (H), tenths (t), and so on. It is easier to do this neatly on square paper. To multiply by 100, move each digit two places to the left. Th H
T
U . t
h
th
5 . 7
5
7
.
The answer is 570. You must put a 0 in the units column. b Write 570 in columns. To divide by 1000, move each digit three places to the left. Th H
T
U . t
5
7
0 .
. 5
h
th
7
0
The answer is 0.57. You do not need a 0 in the thousandths column.
6
1 Number
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Practice questions 1
2
3
4
Multiply these numbers by 10. a 7
b 7.4
c 7.9
d 66
e 6.6
f 0.66
g 40
h 0.45
i 9.87
j 0.023
k 807
l 6.052
a 5
b 5.2
c 5.4
d 28
e 2.8
f 0.28
g 0.9
h 800
i 20.99
j 0.03
k 569
l 75.63
Divide these numbers by 10.
Multiply these numbers by 100. a 3
b 3.1
c 3.14
d 90
e 2
f 9.25
g 0.54
h 290
i 0.025
j 7.654
k 30.5
l 1.26
a 400
b 430
c 438
d 60
e 69
f 5
g 0.3
h 8.7
i 42.3
j 90.2
k 8000
l 6250
Divide these numbers by 100.
1.1 Multiplying and dividing by 10, 100 and 1000
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5
6
7
8
8
Multiply these numbers by 1000. a 4
b 4.4
c 44
d 0.44
e 6.7
f 3.75
g 0.5
h 2.64
i 0.075
j 0.106
k 17.2
l 4.93
Divide these numbers by 1000. a 7000
b 7300
c 7125
d 350
e 35
f 8
g 505
h 8040
i 3.5
j 0.9
k 7800
l 69.2
Fill in each of the missing numbers using 10, 100 or 1000. a 3.8 ×
= 38
b 0.65 ×
= 650
c 7.45 ×
= 7450
d 15 ×
= 1500
e 0.03 ×
= 30
f 1.03 ×
= 103
c 5460 ÷
= 546
f 303 ÷
= 0.303
Fill in each of the missing numbers using 10, 100 or 1000. a 45 ÷
= 0.45
b 320 ÷
d 7.9 ÷
= 0.79
e 15 ÷
= 0.32 = 0.15
1 Number
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1.2 Rounding I can t round decimals to one, two or three decimal places
Example A calculator display shows that 52 á 7 = 7.4285714. Round this answer to:
a one decimal place
b two decimal places.
Solution a The digit in the first decimal place is 4. 7 . 4 1st decimal place
2
8
5
7
2nd decimal place
1
4
3rd decimal place
You either leave the answer as 7.4 or round it up to 7.5. Look at the digit in the next (second) decimal place. It is 2. As 2 is less than 5, the answer is 7.4. 7.4285714 = 7.4 to 1 d.p.
1 d.p. is short for ‘one decimal place’.
b The digit in the second decimal place is 2. The answer is either 7.42 or 7.43. Look at the digit in the next (third) decimal place. It is 8. As 8 is more than 5, you round up to 7.43. 7.4285714 = 7.43 to 2 d.p.
Practice questions 1
Round these numbers to one decimal place. a 4.444 =
to 1 d.p.
b 7.777 =
c 3.535 =
to 1 d.p.
d 23.618 =
to 1 d.p.
f 0.7612 =
to 1 d.p.
e 75.907 =
to 1 d.p.
to 1 d.p.
1.2 Rounding
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2
3
Round these calculator answers to two decimal places. a 2 รท 3 = 0.6666666
b 1 รท 11 = 0.09090909
c 19 รท 11 = 1.727272
d 100 รท 7 = 14.285714
Round these numbers to three decimal places. a 0.3333 = c 0.181818 = e 4.56789 =
4
to 3 d.p. to 3 d.p.
to 3 d.p.
d 9.87654 =
to 3 d.p.
f 3.45962 =
to 3 d.p.
Round these numbers to the given number of decimal places. a 6.329 = c 64.8181 = e 28.972 =
to 1 d.p. to 1 d.p.
Here is a number: 4.29631.
6
Round it
b 0.0364 =
to 2 d.p.
d 9.9029 =
to 2 d.p.
f 0.00691 =
to 1 d.p.
5
a to 1 d.p.
10
b 0.6666 =
to 3 d.p.
b to 2 d.p.
to 3 d.p.
c to 3 d.p.
1 Number
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1.3 Fractions I can t reduce fractions to their simplest form
Example What fraction of this shape is a red?
b blue?
c yellow?
Put your answer in the simplest form. Solution 9 . a There are 24 squares and 9 are red. The red fraction is 24
Both 9 and 24 divide by 3. We say that 3 is a factor of 9 and 24. Divide both numbers by 3 to simplify the fraction. The fraction is 3 . 8
b The blue fraction is 8 = 4 = 2 = 1 . 24 12 6 3 You can divide by 3 three times or divide by 8 and get the answer straightaway. c The yellow fraction is 7 . 24 7 and 24 have no factor in common (apart from 1), so the fraction cannot be simplified.
Practice questions Write all your answers as fractions in the simplest possible form. 1
What fraction of this shape is a red?
2
b blue?
c yellow?
What fraction of this shape is a red?
b yellow?
c green?
d blue?
1.3 Fractions
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3
What fraction of this shape is a red?
4
6 = 24
6 c 12 =
d 6 = 15
b 8 = 24
c 9 = 24
d 15 = 24
b 12 = 20
c 15 = 18
d 10 = 45
Complete these equivalent fractions. a 3 = 4 16
12
b 6 = 10
Simplify these fractions as much as possible. a 14 = 21
7
d blue?
Simplify these fractions as much as possible. a
6
c green?
Simplify these fractions as much as possible. a 6 = 9
5
b yellow?
b 2= 3 18
c 5 = 8 40
d 1= 5 30
e 4 = 5 60
1 Number
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1.4 Fractions and decimals I can t use the equivalence between fractions t put fractions and decimals in order of size
Example Put these fractions and decimals in order of size, from smallest to largest. 5 8
0.8
5 6
7 10
0.72
Solution Write the fractions as decimals. 5 = 5 8
7 = 07 10
8 = 0.625
5 = 5 6
6 = 0.833
Compare these with 0.8 and 0.72. The order is
0.625 5 8
That is
0.7 7 10
0.72
0.8
0.72
0.8
0.833 5 6
Practice questions 1
Look at these fractions. 3 8
5 12
1 3
1 4
a The smallest is
2
b The largest is
Look at these fractions. 2 3
5 6
3 4
7 8
a The smallest is
3
b The largest is
Write these numbers in order, from smallest to largest. 1 4
1 5
0.24
0.3
0.19
1.4 Fractions and decimals
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4
Write these numbers in order, from largest to smallest. 3 4
5
7 10
0.8
2 3
0.55
5 8
0.6
Write these numbers in order, from largest to smallest.
132 7
0.65
Write these numbers in order, from smallest to largest. 1 2
6
5 8
3 14
1.7
Here are two fractions:
15 8
1.65
1 1 and . 6 3
Circle the fraction that is between them. 5 12
8
4 9
Here are two fractions:
1 4
1 2
3 1 and . 5 2
Circle the fraction that is between them. 7 10
9
7 15
3 4
11 20
Here are two decimals: 4.285 and 4.3. Circle the decimal that is between them. 4.28
14
4.29
4.2809
4.2828
1 Number
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1.5 Calculations I can t do calculations that involve several steps, including brackets
Example Calculate 3.5 + 2.6 × 3. Solution Do the multiplication before the addition.
26 × 3
7.8
Then add 3.5 + 7.8 = 11.3.
Practice questions Do not use a calculator for these questions. 1
Work out the following. a 24 – 4 × 2 =
2
b 4 × 9 – 2× 3 =
c 20 ÷ 5 + 5 =
b 6 × ( 4 + 7) =
c (6.8 – 3.8) – 13 . + 2.4 =
Work out the following. a 6 + 14 = 4
6
c 15 . ×8–5=
Work out the following. (Hint: work out the bracket first.) a ( 3 + 8) × 5 =
5
b 17 . + 2.5 × 4 =
Work out the following. a 3×3+ 4× 4 =
4
c 14 + 6 × 3 =
Work out the following. a 3 × 6.2 – 4 =
3
b 2×8 – 5 =
b
36 = 3+9
c 19 − 5 = 12 − 5
Work out the following. a 7.75 + 2.25 – 4.75 – 125 . =
b 7.75 + 2.25 – ( 4.75 – 125 . )= 1.5 Calculations
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1.6 Fractions and percentages I can t calculate fractions and percentages of quantities, using a calculator where appropriate
Example Work out the following: a 75% of £48
b 32% of £48
Solution a 75% = 3 4 So 75% off £48 = 3 of £48 4 = £48 ÷ 4 × 3 = £12 × 3 = £36 b 32% is not a simple fraction so it is appropriate to use a calculator. 32% off £48 = 0.32 .32 3
£48
= £15.36.
Practice questions 1
Work out these fractions without a calculator. a 1 of 45 cm = 5
2
16
b 3 of 45 kg = 5
c 4 of 45 ml = 5
Work out these fractions without a calculator. a 2 of 12 = 3
b 3 of 24 = 4
c 2 of 30 = 3
d 3 of 32 = 4
e 2 of £25 = 5
f 4 of £25 = 5
g 3 of 40 cm = 10
h 7 of 40 cm = 10
i 3 of 15 ml = 5
3 j 4 of 36 kg =
k 5 of 30 cm = 6
l 3 of $32 = 8
1 Number
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3
4
Work out these percentages without a calculator. a 50% of £13 =
b 25% of 60 g =
c 20% of 35 km =
d 75% of 14 =
e 80% of 50 =
f 30% of 400 =
g 30% of £6.00 =
h 70% of 120 m =
i 90% of £3.10 =
j 30% of £3.30 =
Work out these percentages. You can use a calculator. a 23% of £16.00 =
b 42% of 65 =
c 86% of 24 m =
d 9% of £67 =
e 17% of 17 kg =
f 4% of 7900 =
g 72% of 8.5 g =
h 36% of 7300 people =
i 2.5% of £3000 =
j 6.5% of £370 =
k 44% of 680 =
l 39% of 6.2 km =
1.6 Fractions and percentages
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1.7 Calculations without a calculator I can t multiply a three-digit number by a two-digit number t divide a three-digit number by a two-digit number
Example Work out the following: a 383 × 17
b 383 ÷ 17
Solution a You could use the column method or the grid method. 383 x 1 7 3830 2 65 82 1 65 1 1 1
or
300 80 10 3000 800 7 2100 560
3 30 21
1
3000 800 30 2 1 00 560 + 2 1 65 1 1 1
1
The answer is 6511. b You could set out the division like this. 2 2 r 9 17 3 8 43
17 does not go into 3. 17 goes into 38 twice, because 17 × 2 = 34. The remainder is 4. 17 goes into 43 twice with a remainder of 9. The answer is 22 remainder 9.
Practice questions Do not use a calculator for these questions. Show your working. 1
Work out these multiplications. a 142 × 32 =
18
b 163 × 15 =
c 309 × 45 =
1 Number
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2
d 427 × 33 =
e 530 × 18 =
f 612 × 71 =
g 23 × 254 =
h 52 × 381 =
i 802 × 19 =
a 325 ÷ 13 =
b 602 ÷ 14 =
c 960 ÷ 15 =
d 851 ÷ 23 =
e 930 ÷ 21 =
f 840 ÷ 32 =
g 720 ÷ 54 =
h 900 ÷ 16 =
i 666 ÷ 41 =
Work out these divisions.
1.7 Calculations without a calculator
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1.8 Negative numbers I can t put negative numbers in order t add and subtract negative numbers
Example Work out the following: a –3 + 5
b –3 – 5
c –3
–5
d –3 – –5
Solution a A number line can be useful. +5
–4 –3 –2 –1
0
1
2
3
4
–3 + 5
To add 5 to −3, move 5 to the right. b
2
–5
–8 –7 –6 –5 –4 –3 –2 –1
0
1
To subtract 5 from −3, move 5 to the left.
–3 – 5 = –8
c To add a negative number (−5), subtract the inverse (5). The diagram is the same as part b. –3
–5 = –3 – 5 = –8
d To subtract a negative number (−5), add the inverse (5). The diagram is the same as part a. –3 – –5 = –3 3
5= 2
Practice questions 1
Write these numbers in order, from smallest to largest. 8
20
−10
7
−3
0
−5
1 Number
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2
Write these numbers in order, from largest to smallest. −4
3.5
0
−2
−2.5
6
3
Write down the number that is 10 more than −5.
4
Write down the number that is 4 less than −8.
5
Work out
6
7
8
9
a 6 + –4 =
b 6 – –4 =
c –6 + –4 =
d –6 – –4 =
e 8 + –10 =
f 8 – –10 =
a –5 + –4 =
b 3 + –6 =
c –3 + –2 =
d –5 + –5 =
e –15 + 5 =
f –5 + –15 =
a –2 – 7 =
b 5 – –3 =
c –3 – 12 =
d –2 – –10 =
e 10 – –20 =
f –20 – 10 =
a 3 + –8 =
b –7 – 1 =
c 0 – –11 =
d –8 + –8 =
e 2.5 + –4 =
f –6 – –15 . =
g –20 + –26 =
h –15 – –19 =
i –4.5 + 6 =
Work out
Work out
Work out
Fill in the missing numbers. a 10 + d –30 +
= 3 = –20
b –4 + e
= 4 +6= 4
c 5+ f
= –10 – –2 = 5
1.8 Negative numbers
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1.9 Approximations I can t use approximations to check answers t check that answers are of the correct order of magnitude
Example 1 Here is a multiplication.
38 × 22.5
Without using a calculator, say which one of these is the correct answer. a 8.55
b 85.5
c 855
d 8550
Solution 38 × 22.5 is approximately 40 40
20.
20 = 800
C is the correct answer because it is closest to 800.
Example 2 Here is a division.
3135 ÷ 9.5 = 69
Without using a calculator, give a reason why this could be correct or must be incorrect. Solution 3135 ÷ 9.5 is approximately 3000 ÷ 10 which is 300. The answer must be incorrect because 69 is not close to 300. The correct answer is 330.
Practice questions Do not use a calculator for these questions. 1
22
Here are some multiplications. Estimate the answer and circle the one that is correct. a 495 × 32
A 1584
B 15840
C 158400
D 1584000
b 5.6 × 12.5
A 0.7
B 7
C 70
D 700
1 Number
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2
3
c 0.18 × 9.5
A 171
B 17.1
C 1.71
D 0.171
d 0.35 × 0.64
A 0.0224
B 0.224
C 2.24
D 22.4
e 980 × 0.32
A 0.3136
B 3.136
C 31.36
D 313.6
Here are some divisions. Estimate the answer and circle one that is correct. a 98.7 ÷ 23.5
A 0.42
B 4.2
C 420
D 4200
b 7449 ÷ 38.2
A 1.95
B 19.5
C 195
D 1950
c 93.5 ÷ 425
A 0.22
B 2.2
C 4.4
D 44
d 172.8 ÷ 72
A 0.24
B 0.42
C 2.4
D 4.2
e 621 ÷ 0.23
A 2.7
B 27
C 270
D 2700
Estimate whether each of these calculations could be correct or must be incorrect. Circle your answer and give an approximation to justify it. a 234 × 415 = 17110 Could be correct
Must be incorrect
Approximation: . ÷ 8.15 = 53 b 43195 Could be correct
Must be incorrect
Approximation:
1.9 Approximations
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1.10 Units of measurement I can t convert between units t estimate measurements
Example a How many millimetres is 25 centimetres? b What is the most suitable unit to use to measure the length of a football pitch? Solution a 1 cm
10 mm
25 cm = 25 Ă— 10 mm m
25 50 mm
b You can measure lengths in millimetres, centimetres, metres and kilometres. The best unit for measuring a football pitch is the metre. A centimetre is too short. A kilometre is over half a mile and is too long.
Practice questions 1
Here are metric units of length. m
cm
km
mm
Choose the most sensible unit to measure a the distance between two towns b the length of a fingernail c the length of a room d the height of a table
24
1 Number
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2
Here are some metric units. g
m
l
kg
km
mg
ml
Choose the most suitable unit to measure a the capacity of a spoon
b the mass of a dog
c the height of a building
d the quantity of water in a swimming pool
e the amount of salt used in a recipe
3
Change these lengths to metres. a 200 cm =
4
m
b 3 km =
m
c 5000 mm =
m
Change these lengths to centimetres. a 2.5 m =
cm
b 340 mm =
cm c 7 mm = 1.10 Units of measurement
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cm
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5
Draw a line between the two capacities that are the same. Circle the odd one out.
6
Convert these masses into grams. a 4 kg = d 9.5 kg =
7
b 13 . kg =
g
e 2000 mg =
g
g
c 0.25 kg =
g
f 500 mg =
g
Convert these units. a 300 mm =
cm
d 4250 g = g 2.5 l =
26
g
kg ml
b 300 cm =
m
c 300 m =
e 0.4 kg =
g
f 6500 mg =
h 100 ml =
l
i 350 ml =
km g l
1 Number
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1.11 Number relationships I can t work out common factors and common multiples
Example a Find all the common factors of 12 and 15. b Find two common multiples of 12 and 15. Solution a 12 = 1 12 12 = 2
6
12 = 3
4
15 = 1 15 15 = 3
The factors of 12 are 1, 2, 3, 4, 6 and 12.
The factors of 15 are 1, 3, 5 and 15.
5
The common factors are factors of both numbers. They are 1 and 3. b The multiples of 12 are 12, 24, 36, 48, 60, 72, ‌ The multiples of 15 are 15, 30, 45, 60, 75, ‌ The common multiples are the numbers in both lists. The first common multiple is 60. Further common multiples are 120, 180, 240, and so on.
Practice questions 1
a Work out the factors of 16. b Work out the factors of 28. c Write down the common factors of 16 and 28.
2
a Work out the factors of 18. b Work out the factors of 27. c Write down the common factors of 18 and 27.
1.11 Number relationships
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3
a Work out the factors of 45. b Work out the factors of 63. c Write down the common factors of 45 and 63.
4
Work out the common factors of 30 and 40.
5
Work out the common factors of 50 and 75.
6
a Write down the first six multiples of 3. b Write down the first six multiples of 5. c Work out two common multiples of 3 and 5.
7
a Write down the first five multiples of 10. b Write down the first five multiples of 15. c Work out three common multiples of 10 and 15.
28
8
Work out two common multiples of 20 and 25.
9
Work out two common multiples of 9 and 12.
1 Number
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1.12 Number patterns I can t recognise number patterns
Example Here is a sequence of numbers.
1
2
4
7
11
16
Work out the next two numbers in the sequence. Solution Look at the differences between the numbers. 1
2 +1
4 +2
7 +3
11 +4
16 +5
+6
+7
The differences increase by one each time. The next number is 16 + 6
22.
Then the one after that is 22 + 7
29.
Practice questions 1
Work out the next two numbers in each of these sequences. a 17
20
23
26
29
32
3
4
6
9
13
18
c 10
12
16
22
30
40
1
2
5
10
17
e 80
76
72
68
f 50
49
47
44
40
b
d
g
3
4
7
12
19
28
h
9
9.5
11
13.5
17
21.5
1.12 Number patterns
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2
Here is a sequence of patterns.
1+2=3
1+2+3=6
1 + 2 + 3 + 4 = 10
Work out the next two sums in the sequence.
3
Here is a different sequence of patterns.
1+3=4
1+3+5=9
1 + 3 + 5 + 7 = 16
Work out the next two sums in the sequence.
4
30
Fill in the missing numbers in these sequences. a
9
14
19
b
40
37
34
31
c
4
5
7
10
19
d
1
2
5
10
26
29
44 19 32 37
65
1 Number
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1.13 Squares, cubes and roots I can t work out the square or the cube of a number t work out the square root or the cube root of a number
Example Work out a 52 b 53
c
64
d
3
64
Solution a 52 is ‘5 squared’.
52
b 53 is ‘5 cubed’.
53 = 5
c
64 is the square root of 64.
82
64 is the cube root of 64.
43 = 4
d
3
5×5 5
8×8 4
25 5 = 125 64 so
64 = 8
4 = 64 so
3
64 = 4
Practice questions Do not use a calculator for these questions. 1
Work out a 32 =
2
c 92 =
d 122 =
b 33 =
c 63 =
d 103 =
Work out a 23 =
3
b 62 =
Work out a 12 + 22 =
b 32 + 4 2 =
c 82 + 10 2 =
d 152 + 20 2 =
1.13 Squares, cubes and roots
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4
5
6
7
Work out a 32 – 23 =
b 33 – 52 =
c 53 – 43 =
d 103 – 10 2 =
Work out a
9 =
b
49 =
c
81 =
d
121 =
e
100 =
f
1=
g
196 =
h
400 =
i
0 =
Work out a
3
1=
b
3
8 =
c
3
125 =
d
3
1000 =
e
3
216 =
f
3
729 =
2
= 100
Fill in the missing numbers. a
8
= 36
b
3
c
= 27
3
d
= 64
Fill in the missing numbers. a
32
2
= 5
b
= 9
c
3
= 4
d
3
=6
1 Number
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Comments, next steps, misconceptions
1.13 Squares, cubes and roots
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