Maths Frameworking Step 5 Intervention Workbook

Step 5

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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. It can be used by: your teacher or tutor can set a date for you to complete each topic by you can give a tra ffic light colour for each topic to show how you feel it went yYou, your teacher and your parent or carer to write comments

Work through each topic step by step For each topic, there are: clear learning objectives worked exam ples 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 Collins Connect.

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 5 Contents Strand/topic

Page Hours Due date

Feedback

1 Number 1.1 Multiplying and dividing decimals

6

1

1.2 Multiplying and dividing fractions

10

1

1.3 Rounding and estimates

14

1

1.4 Errors

16

1

2.1 Expanding brackets

20

1

2.2 Sequences

22

1

2.3 Quadratic graphs

26

1

2.4 Solving simultaneous equations

32

1

2.5 Formulae

38

1

3.1 Inverse proportion

41

1

3.2 Proportional change

43

1

3.3 Compound units

45

1

2 Algebra

3 Ratio, proportion and rates of change

4

Contents

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Strand/topic

Page Hours Due date

Feedback

4 Geometry and measures 4.1 Pythagoras’ theorem

48

1

4.2 Area

52

1

4.3 Prisms and cylinders

55

1

5.1 Scatter graphs

59

1

5.2 Grouped data

63

1

5.3 Comparing distributions

66

1

5 Statistics

Mental maths warm-ups Record of achievement certificate

Contents

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1

Number

1.1 Multiplying and dividing decimals I can • use place value to help multiply and divide decimals

Example Work these out, without using a calculator. a 75 × 0.3

b 1.3 × 0.6

Solution a If you ignore the decimal point, you have: 75 × 3 = 225 3 ÷ 10 = 0.3 You are multiplying two numbers together. If you divide one of them by 10, you divide the answer by 10.

75 × 0.3 = 225 ÷ 10 = 22.5 b Ignore the decimal points.

13 × 6 = 78 In the question, both numbers have been divided by 10.

So divide the answer by 10 × 10 = 100.

1.3 × 0.6 = 78 ÷ 100 = 0.78

6

1 Number

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Example Work these out, without using a calculator. a 75 ÷ 0.3

b 3 ÷ 0.04

Solution a Writing the question without the decimal point you have: 75 ÷ 3 = 25 If you divide the divisor by 10, you multiply the answer by 10.

Hint

75 ÷ 0.3 = 25 × 10

The divisor is the number you are dividing by.

= 250 b Writing the question without the decimal point you have: 3 ÷ 4 = 0.75 If you divide the divisor by 100, you multiply the answer by 100.

3 ÷ 0.04 = 0.75 × 100 = 75

Practice questions Do not use a calculator for these questions.

1

2

Work these out. a 12 × 0.6 = __________

b 12 × 0.06 = __________

c 1.2 × 6 = __________

d 1.2 × 0.6 = __________

Work these out. a 3.5 × 4 = __________

b 3.5 × 0.4 = __________

c 0.4 × 0.35 = __________

d 0.04 × 35 = __________

1.1 Multiplying and dividing decimals

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3

4

5

Work these out. a 0.8 × 0.5 = __________

b 11 × 0.03 = __________

c 2.5 × 0.6 = __________

d 4.1 × 0.3 = __________

Work these out. a 0.7 × 0.7 = __________

b 30 × 0.3 = __________

c 5.2 × 0.5 = __________

d 5.5 × 0.2 = __________

48 × 85 = 4080 Use this fact to work these out.

6

7

8

a 48 × 8.5 = __________

b 4.8 × 8.5 = __________

c 0.48 × 85 = __________

d 48 × 0.85 = __________

Work these out. a 24 ÷ 4 = __________

b 24 ÷ 0.4 = __________

c 24 ÷ 0.04 = __________

d 2.4 ÷ 4 = __________

Work these out. a 36 ÷ 0.6 = __________

b 4.2 ÷ 6 = __________

c 5.4 ÷ 0.6 = __________

d 30 ÷ 0.06 = __________

1 Number

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9

10

Work these out. a 20 ÷ 0.4 = __________

b 40 ÷ 0.2 = __________

c 7 ÷ 0.5 = __________

d 7 ÷ 0.2 = __________

Work these out. a 150 ÷ 0.5 = __________

b 80 ÷ 0.2 = __________

c 6 ÷ 0.4 = __________

d 10 ÷ 0.04 = __________

648 ÷ 24 = 27 Use this fact to work these out.

11

12

a 648 ÷ 2.4 = __________

b 648 ÷ 0.24 = __________

c 64.8 ÷ 2.4 = __________

d 6.48 ÷ 2.4 = __________

Work these out. a 12 × 0.8 = __________

b 12 ÷ 0.8 = __________

c 1.2 × 0.8 = __________

d 1.2 ÷ 0.8 = __________

Work these out. a 44 × 0.4 = __________

b 44 ÷ 0.4 = __________

c 4.4 × 0.4 = __________

d 4.4 ÷ 0.4 = __________

1.1 Multiplying and dividing decimals

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1.2 Multiplying and dividing fractions I can • multiply and divide fractions or mixed numbers

Example Work these out. a

2 3

×

4 5

b

2 3

÷

4 5

c 132 × 1 12

d 132 ÷ 1 12

Solution a To multiply two fractions, multiply the numerators and multiply the denominators.

2 4 2× 4 × = 3 5 3× 5 8 = 15 b To divide by a fraction, multiply by the inverse. The inverse of 2÷ 4 = 2× 5 3 5 3 4 2× 5 = 3× 4 10 = 12 5 = 6 Look at the expression

4 5

is 45 .

Simplify by dividing each number by 2.

2 × 5 3 × 4

above.

The 2 and the 4 have a common factor of 2 so you can divide them both by this before you multiply.

1

2×5 5 = 3 × 42 6

In this case you get the answer directly. It cannot be simplified, like c 132 × 1 12 =

5 31

×

1

3 2

=

5 2

10 12

can.

= 2 12

Notice that the two 3s have been divided by 3.

If you do not do this you will get the answer

15 6

for the multiplication.

You then need to simplify this to 2 12. d Change the divisor to an improper fraction and then multiply by its inverse.

132 ÷ 1 12 = = = = =

10

5 ÷ 32 3 5 × 32 3 5 × 2 3 × 3 10 9 191

1 Number

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Practice questions 1

2

3

Work these out. a 1 × 2 = _________________________ 3 3

b 2 × 2 = _________________________ 5 3

c 3 × 1 = _________________________ 5 2

d 3 × 3 = _________________________ 4 4

Work these out. Write each answer as simply as possible. a 1 × 3 = _________________________ 3 4

b 2 × 1 = _________________________ 5 4

c 2 × 5 = _________________________ 3 8

d 5 × 3 = _________________________ 6 4

Work these out. Write each answer as simply as possible. a 3 × 5 = _________________________ 5 6

b 2 × 5 = _________________________ 5 8

c 7 × 4 = _________________________ 8 5

d

9 × 2 = _________________________ 10 3

1.2 Multiplying and dividing fractions

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4

5

6

7

Work these out. a 1 12 × 1 12 = _________________________

b 1 12 × 2 12 = _________________________

c 131 × 2 12 = _________________________

d 132 × 132 = _________________________

Work these out. a 1 ÷ 1 = _________________________ 3 4

b 2 ÷ 3 = _________________________ 3 4

c 3 ÷ 1 = _________________________ 8 3

d 5 ÷ 2 = _________________________ 6 3

Work these out. a 1 ÷ 2 = _________________________ 5 3

b 2 ÷ 1 = _________________________ 3 5

c 3 ÷ 3 = _________________________ 8 5

d 3 ÷ 3 = _________________________ 5 8

Work these out. a 131 ÷

c

12

3 4

1 2

= _________________________

÷ 1 12 = _________________________

b 1 12 ÷

d

3 5

1 3

= _________________________

÷ 2 12 = _________________________

1 Number

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8

9

10

Work these out. a 1 12 ÷ 143 = _________________________

b 143 ÷ 141 = _________________________

c 2 12 ÷ 1 12 = _________________________

d 3 31 ÷ 2 12 = _________________________

Work these out. Write each answer as simply as possible. a 2 × 1 = _________________________ 3 2

b 2 ÷ 1 = _________________________ 3 2

c 1 × 2 = _________________________ 2 3

d 1 ÷ 2 = _________________________ 2 3

Work these out. Write each answer as simply as possible. a 3 12 × 1 12 = _________________________

b 3 12 ÷ 1 12 = _________________________

c 1 12 × 3 12 = _________________________

d 1 12 ÷ 3 12 = _________________________

1.2 Multiplying and dividing fractions

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1.3 Rounding and estimates I can • round numbers to make estimates of calculations

Example Round the numbers to one significant figure (1 sf) to estimate the answers to these calculations. a 32.7 × 19.8

b 5750 ÷ 2.86

Solution a 32.7 is 30, correct to 1 sf and 19.8 is 20, correct to 1 sf.

32.7 × 19.8 ≈ 30 × 20 = 600 ≈ means ‘approximately equals’. b 5750 is 6000, correct to 1 sf and 2.86 is 3, correct to 1 sf.

5750 ÷ 2.86 ≈ 6000 ÷ 3 = 2000

Practice questions Do not use a calculator for these questions.

1

2

Round each number to one significant figure. a 329 ≈ _________

b 27.32 ≈ _________

c 0.716 ≈ _________

d 5678 ≈ _________

Round each number to one significant figure to find an estimate of the answer. a 32.5 × 61.8 ≈ 30 × ______ = ______

b 491 × 2.15 ≈ ______ × ______ = ______

c 72.9 × 3.14 ≈ ______ × ______ = ______ d 0.471 × 8.36 ≈ ______ × ______ = ______ e 7230 × 37 ≈ ______ × ______ = ______

14

f 0.81 × 0.792 ≈ ______ × ______ = ______

1 Number

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3

Round each number to one significant figure to find an estimate of the answer. a 432 ÷ 38 ≈ ______ ÷ ______ = ______

b 7.93 ÷ 4.09 ≈ ______ ÷ ______ = ______

c 691 ÷ 0.97 ≈ ______ ÷ ______ = ______ d 6.14 ÷ 0.518 ≈ ______ ÷ ______ = ______ e 82.3 ÷ 97.2 ≈ ______ ÷ ______ = ______ f 2700 ÷ 30.6 ≈ ______ ÷ ______ = ______

4

Estimate the answer to each calculation. a 27.2 × 41.2 ≈ _____________________ 5.91

b

8.2 × 113 ≈ _____________________ 22.6

5 58.4 cm

a The circumference of a circle with a diameter of 58.4 cm is 3.14159 × 58.4 cm.

Estimate this length.

_____________________ cm

b The area of a circle with a diameter of 58.4 cm is 3.14159 × 29.2 × 29.2 cm2.

Estimate this area.

6

_______________________ cm2

The Body Mass Index of a man with a mass of 81.4 kg and a height of 1.91 m is Estimate the value of the Body Mass Index.

81.4 .

1.91 × 1.91

_______________________

1.3 Rounding and estimates

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1.4 Errors I can • use interval notation to express possible errors

Example The lengths of the sides of a rectangle are 14 cm and 20 cm, to the nearest centimetre.

20 cm

14 cm

Work out an interval estimate for: a the perimeter of the rectangle b the area of the rectangle.

Solution a The width is between 13.5 cm and 14.5 cm so you can write: 13.5 cm  width < 14.5 cm.

Notice the signs: 13.5 rounds up to 14 and 14.5 rounds up to 15.

Similarly: 19.5 cm  length < 20.5 cm The perimeter = 2(width + length) so: 2(13.5 + 19.5)  perimeter < 2(14.5 + 20.5) So 66 cm  perimeter < 70 cm. b The area = width × length so: 13.5 × 19.5  area < 14.5 × 20.5 So 263.25 cm2  area < 297.25 cm2.

16

1 Number

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Practice questions 1

2

Write interval estimates for each of these measurements. a A length of 35 cm to the nearest centimetre

__________  length < __________

b A length of 620 m to the nearest ten metres

__________  length < __________

c A length of 7.4 cm to the nearest millimetre

__________  length < __________

d A mass of 70 kg to the nearest kilogram

__________  mass < __________

e A mass of 70 kg to the nearest ten kilograms

__________  mass < __________

f A capacity of 50 ml to the nearest millilitre

__________  capacity < __________

g A capacity of 50 ml to the nearest ten millilitres

__________  capacity < __________

The sides of these regular polygons are given to the accuracy shown.

Work out an interval estimate for the perimeter of each one. a

b

7 cm to the nearest centimetre

7120 cm to cmthe to the nearest nearest centimetre 10 centimetres

120 cm to the nearest 10 centimetres

__________  perimeter < __________ c

__________  perimeter < __________ d

6.2 cm to the nearest millimetre

14 to m the to the 6.2 cm nearest metre nearest millimetre

__________  perimeter < __________

14 m to the nearest metre

__________  perimeter < __________

1.4 Errors

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3

a The side of a square is 8 cm long, to the nearest centimetre.

Work out an interval estimate for the area of the square. ______________cm2  area < ______________cm2 b The side of a square is 8.0 cm long, to the nearest millimetre.

Work out an interval estimate for the area of the square. ______________cm2  area < ______________cm2

4

This is a plan of a football pitch. The lengths of the sides are given to the nearest metre.

90 m

50 m

a Work out an interval estimate for the perimeter of the pitch. ______________m  perimeter < _____________m b Work out an interval estimate for the area of the pitch. ______________m2  area < ______________m2

18

1 Number

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Comments, next steps, misconceptions

1.4 Errors

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