Step 5
37709_P01_03.indd 1 7537709 Book 5 Intervention TITLE.indd 1
7/22/14 9:48 AM 30/06/2014 09:30
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
37709_P01_03.indd 3
3
22/07/14 3:50 PM
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
37709_P04_19.indd 4
7/22/14 9:35 AM
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
37709_P04_19.indd 5
5
7/22/14 9:35 AM
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
37709_P04_19.indd 6
7/22/14 9:35 AM
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
37709_P04_19.indd 7
7
7/22/14 9:35 AM
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
37709_P04_19.indd 8
7/22/14 9:35 AM
8
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
37709_P04_19.indd 9
9
7/22/14 9:35 AM
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
37709_P04_19.indd 10
7/22/14 9:35 AM
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
37709_P04_19.indd 11
11
7/22/14 9:35 AM
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
37709_P04_19.indd 12
7/22/14 9:36 AM
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
37709_P04_19.indd 13
13
7/22/14 9:36 AM
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
37709_P04_19.indd 14
7/22/14 9:36 AM
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
37709_P04_19.indd 15
15
7/22/14 9:36 AM
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
37709_P04_19.indd 16
7/22/14 9:36 AM
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
37709_P04_19.indd 17
17
7/22/14 9:36 AM
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
37709_P04_19.indd 18
7/22/14 9:36 AM
Comments, next steps, misconceptions
1.4 Errors
37709_P04_19.indd 19
19
7/22/14 9:36 AM