9X1 and 9Y1

p

Key Stage 3 Mathematics

Level by Level

Pack D: Level 7

Stafford Burndred

ISBN 1 899603 25 5

Published by Pearson Publishing Limited 1997

Pearson Publishing 1996

Revised February 1997

A licence to copy the material in this pack is granted to the purchaser strictly within theirschool, college or organisation. The material must not be reproduced in any other formwithout the express written permission of Pearson Publishing.

Pearson Publishing, Chesterton Mill, Frenchs Road, Cambridge CB4 3NP Tel 01223 350555 Fax 01223 356484Web site http://www.pearson.co.uk/education/

Pembrokeshire e-Portal Licence exp 31Aug10

Estimating, multiplying and dividing by numbersbetween 0 and 1

Rounding to one significant figure (1 sig. fig.)

3725 4000

28.63 30

421.3 400

Note: 421.3 does not become 400.0

0.038 0.04

0.724 0.7

0.0306 0.03

Multiplying by numbers larger than 1

The answer is larger. Example: 3 x 5 = 15

Multiplying by numbers smaller than 1

The answer is smaller. Example: 3 x 0.05 = 0.15

Dividing by numbers larger than 1

The answer is smaller. Example: 800 20 = 40

Dividing by numbers smaller than 1

The answer is larger. Example: 800 0.2 = 4000

Questions

Estimate: 1 3127 x 493 __________ 2 3814 x 0.019 __________

3 3957 813 __________ 4 911 0.0301 __________

Answers1 3000 x 500 = 1 500 000 2 4000 x 0.02 = 80

3 4000 800 = 5 4 900 0.03 = 30 000

Pearson Publishing, Chesterton Mill, Frenchs Road, Cambridge CB4 3NP Tel 01223 350555 Fax 01223 356484 3

KS3 Mathematics D: Level 7 Number and Algebra


Round to one figure, then add noughts tothe decimal point. Do not add noughts afterthe decimal point.

Significant figures are counted from the firstnon-zero number.

C = / x7 8 9 +4 5 6 -1 2 3

.0 =

Estimating, multiplying and dividing by numbersbetween 0 and 1

Exercises

1 3000 x 40 2 1200 20 3 40 x 50

4 30 0.3 5 600 x 0.6 6 8000 0.4

7 40 x 0.07 8 5 0.01 9 600 x 300

10 20 0.004 11 7 x 0.04 12 6000 30

13 30 x 0.6 14 40 0.08 15 500 x 0.002

16 8 0.02 17 700 x 0.04 18 800 000 200

19 60 x 0.5 20 400 0.04

Estimate the answers to the following questions. (Show your working.)

21 3928 x 7081 22 8024 21 23 6927 x 49

24 89 3.01 25 39 x 0.099 26 3.98 0.193

27 6.01 x 0.099 28 91 0.029 29 7120 x 0.099

30 6.014 0.031

31 A man worked for 52 weeks and earned 391 per week. Estimate his annual salary.

32 A lorry can carry 39 tonnes in one load. Estimate how many loads will be required to transport 159 000 tonnes of sand.

33 A garage sells 492 000 litres of petrol at 59.8 pence per litre. Estimate the total value of the sale.

34 Water flows through a pipe at a rate of 39.8 cc per second. Estimate how long it will take for 361 412 cc to flow through the pipe.

35 There are 49 483 people in a town. Each person uses 19.2 litres of water each day. Estimate how long 803 280 000 litres of water will last.

36 A jam jar contains 398 grams of jam. Estimate how many jars can be filled from 80 kilograms of jam.




C = / x7 8 9 +4 5 6 -1 2 3

.0 =

Efficient use of a calculator 1

Use of brackets

3 ( 6 + 8 ) this means 3 x ( 6 + 8 )

Calculator keys: Answer 42

( 8 - 5 ) 3 this means ( 8 - 5 ) x 3


Questions involving division

Method A: Using brackets

3.86 - 4.23 Place brackets at the start and end of the top line (3.86 - 4.23)7.25 x 3.68 Place brackets at the start and end of the bottom line (7.25 x 3.68)

Calculator keys:

Answer - 0.013868065

Method B: Using the memory

First work out the answer to the bottom line (remember to press =).Place this number in memory.Clear your calculator.Work out the answer to the top line.Divide by memory recall.

Calculator keys: (look at your calculator instruction booklet if you do not know how touse the memory)

Answer -0.013868065




your clear keycould say

your memorykey could say


Exercises

Use a calculator to answer the following questions.

Show your calculator keys. Try to use an efficient method.

1 2.8 (3.65 - 1.72)

2 4.6 (2.7 + 1.35)

3 0.7 (3.6 + 2.8 + 4.3 - 1.9)

4 (7.38 - 5.6) 2.7

5 (9.32 + 2.6) 0.17

6 A = 12 (a + b) h Find the value of A when a = 3.6, b = 7.4 and h = 1.32.

7 Y = 3 (c - d) + c (4d + 8) Find the value of Y when c = 6.82 and d = 4.71

8 A = B C (D - E) - D (B + C) Find the value of A when B = 0.78, C = 3.62, D = 8.4 and E = 7.34.

9 5.72 + 6.358.24 x 3.2

10 4.832.86 - 1.31

11 8.63 - 2.94 8.23.8 x 4.2 3.7 - 1.64

12 6.82 (5.74 + 3.82) 4.82 (5.3 - 2.91)6.87 x 3.91 6.87 x 3.2

13 B (C + D) C (C - D) C D B C

14 A B (C - D) C (B + D)C D B + C




-

+

-A =

Y = +

Find the value of A when B = 7.32, C = 5.74 and D = 11.31.

Find the value of Y when A = 3.81, B = 4.26, C = 3.14 and D = 2.87.


Use of power and root keys

Use of memory

If you need to use a number more than once it may help reduce the calculation bysaving the number in memory.

But remember: When you put a number into memory, you will lose the previousnumber in memory.

Questions

1 Calculate 82 + 52

2 What is the value of 45?

3 A square has an area of 81 cm2. What is the length of each side?

4 A cube has a volume of 64 cm3. What is the length of each side?

5 y = 3x3 + 4x2 + 2x. Calculate the value of y when x = 2.974.

Answers

2 2Calculator keys Answer 891

Calculator keys Answer 10242

Calculator keys Answer 9 cm3

Calculator keys

Calculator keys

Answer 120.24

Answer 4 cm4

First put 2.974 into memory. Calculator keys5

y

1

3

9

y 2

Most calculators: to put into memory. to recall what is in memory.

2

xy

y

y

This is used to square a number, eg 82 = 64

This is used to calculate the square root of a number, eg 36 = 6

3 This is used to calculate the cube root of a number, eg

or


This is used to calculate powers, eg 34




3125 = 5


Exercises

Use a calculator to answer the following questions.

Show your calculator keys. Try to use an efficient method.

1 6.32

2 5.822

3 4.622 + 3.842

4 8.25

5 9.34

6 8.77 19.328 38729 A square has an area of 300 cm2. Find the length of one side.

10 A square has an area of 831 cm2. Find the length of one side.

11 38.4 cm3 of gold is used to make a gold cube. Calculate: a the length of one sideb the total surface area.

12 (8.74)3 + (3.6)2

8.4 - 2.6

13 7.9 + 32.3(5.4)2 - (1.3)3

14 8.63 (2.94 -(1.1)3)(9.8)3 + 8.4

15 y = 4 x3 - 3 x2 + 2x - 4. Calculate the value of y when:a x = 3.6b x = 5.2c x = 3.74




Solving problems

Before you attempt the question you must be clear about what you are doing.

Plan your calculation before you start.

If units are different, eg grams and kilograms, it is wise to change everything to thesame unit before you start eg change everything to grams.

In a long calculation it is essential to show all of your working; if you make a carelesserror you will still gain some marks for your working.

Question

A garage buys 428 286 litres of petrol at 41.6 pence per litre. The garage sells thepetrol at 2.84 per gallon. Calculate the profit made by the garage. Show all of yourworking. (1 gallon = 4.5 litres)

AnswerThere are several ways of solving this problem. If your answer is correct then you can assume that yourmethod is also correct.

Notice the traps

1 Prices are given in s and pence

2 Capacity is given in gallons and litres

The calculation

The petrol is sold for 2.84 per gallon

1 gallon = 4.5 litres

Therefore petrol is sold for 2.84 for 4.5 litres

2.84 4.5 = 0.63111111 for litre (you must not approximate to 0.63)

Petrol is bought for 41.6p per litre. Change the price into s 0.416

Profit per litre is 0.631111 0.416 = 0.2151111 (you must not approximate to 0.215)

Total profit: 0.2151111 x 428 286 = 92 129.0773

= 92 129.08 correct to the nearest penny




put this number into yourcalculator memory

obtain from calculatormemory

Solving problems

Exercises

1 A motor car manufacturer in Singapore exports cars to Britain. Her total costs (costof production, labour cost, shipping costs, selling costs, etc) are $17 364 for eachcar. She wishes to make a profit of 32% of her total costs on each car. What will bethe selling price of each car in Britain? Give your answer to the nearest . The exchange rate is 1 = $3.95.

2 Mr King bought four bottles of cola. He paid with a 10 note and received 7.24change. Each bottle contained 5/8 of a litre. Work out the cost of cola per litre.

3 A long playing record rotates at 33 revolutions per minute. It rotates 842 times.How long does the record last? Give your answer to the nearest second.

4 A shop buys 2780 shirts for 10 000. 70% of the shirts are sold at a price whichproduces a 30% profit. At what price must each of the remaining shirts be sold toproduce an overall profit of 15%? Give your answer correct to the nearest penny.

5 An empty tank is filled with water at the rate of 436 cc per second. a How much water does the tank hold after two minutes? b The tank holds 80 litres when full. How long does it take to fill?

6 A lorry delivered sacks of grain to Mr Whites chicken farm. The lorry contained3885 kilograms of grain packed into 15 kg sacks. Each chicken ate 120 grams ofgrain per day. The total load lasted Mr Whites chickens for 175 days. How manychickens did he have?

7 A measuring jug contains water and a metal cube of side length 3.87 cm. Themarking on the measuring jug is 983 ml. What will be the reading on the jug whenthe metal cube is removed? Give your answer to the nearest ml.

8 In a factory 30% of the workers receive a 5% pay rise, a quarter of the workersreceive an 8% pay rise and the remainder receive a 9% pay rise. What is the meanaverage percentage increase per worker?

9 Mr Smith and Mrs Jones share the profits in a business in the ratio 5:4 respectively.Mr Smith shares his part with his wife in the ratio 3:2. Mrs Jones gives all of hershare to her three children, which they share in proportion to their ages of 18, 15and 12. The youngest child receives 9600. How much does Mr Smiths wifereceive?




Proportional change

Fractions

The original number is considered to be 1 whole.

Increase by 1/3 means 1 + 1/3 we find 11/3Decrease by 1/4 means 1 - 1/4 we find 3/4

Percentages

The original number is considered to be 100%.

Increase by 30% means 100% + 30% we find 130%Decrease by 15% means 100% - 15% we find 85%

Ratio

Increase in the ratio 5:3 means multiply by 5/3Decrease in the ratio 2:7 means multiply by 2/7

Questions

1 Increase 38 by 1/5

2 Decrease 48 by 1/3

3 Increase 46 by 20%

4 Decrease 72 by 30%

5 Increase 20 in the ratio 5:4

6 Decrease 8 in the ratio 5:16

Answers1 (1 + 1/5 = 11/5) 38 x 11/5 = 453/5

2 (1 - 1/3 = 2/3) 48 x 2/3 = 32

3 (100% + 20% = 120%) 46 x 120% = 55.2

4 (100% - 30% = 70%) 72 x 70% = 50.4

5 20 x 5/4 = 25

6 8 x 5/16 = 2.5




Proportional change

Exercises

1 A plank of wood is 12 m long. 2/5 is cut off. What is the length of the remainingpiece?

2 A stadium holds 20 000 people. The capacity is increased by 3/8. How manypeople does the stadium hold after the increase?

3 A woman receives a wage of 280 per week. She receives a 3% increase. What isher new wage?

4 A television manufacturer sold 38 000 televisions in August. In September salesfell by 16%. How many televisions were sold in September?

5 This is a photograph.

It is enlarged in the ratio 8:5.

What are the new dimensions?

6 Decrease 60 in the ratio 2:3.

7 A train journey normally takes 90 minutes. A new engine was introduced and thisreduced the journey time by 3/20. How long does the new engine take for thejourney?

8 Newspaper sales in 1994 averaged 14 000 000 per day. Sales increased by 1/8 in1995. What were the average sales per day in 1995?

9 A football ground used to hold 18 500 spectators. Safety regulations were thenintroduced and capacity was reduced by 12%. How many spectators does theground now hold?

10 A reservoir normally holds 800 million litres of water. Heavy rain in Januaryincreased the amount of water by 17.5%. How much water was in the reservoirafter the heavy rain?

11 The weights of these packets are in the ratio 8:5.

What is the weight of the small packet?

12 Decrease 18 in the ratio 5:6.




15 cm

20 cm

1000 g

largesmall

The nth term for quadratics

Finding the formula for the nth term

This is very useful for coursework projects.

Find the rule to produce the sequence 6, 17, 34, 57, 86

We now know that the rule begins 3n2.

We now know that the rule begins 3n2 + 2n.

The required sequence is 6, 17, 34, 57, 86

Using n = 1, n = 2, n = 3, etc 3n2 + 2n produces 5, 16, 33, 56, 85

This is 1 less than the required sequence.

Therefore the nth term rule to produce 6, 17, 34, 57, 86 must be 3n2 + 2n + 1.

To find the 8th term:

Let n = 8 3 x 82 + 2 x 8 + 1 = 209

2 2 2 2Breaknumber

Look at the break number 1 means n2 means 2n3 means 3nNote: -5 means -5n

A break number on the first line meansthe rule is linear, ie x or n

63 (ie 3x12)3

1st term

1712 (ie 3x22)5

2nd term

3427 (ie 3x32)7

3rd term

5748 (ie 3x42)9

4th term

867511

5th term

SequenceSubtract 3n2New sequence

6 17 34 57 86

11 17 23 29

Breaknumber

6 6 6

Look at the break number 2 means n24 means 2n26 means 3n2etc

A break number on the second line meansthe rule is quadratic, ie x2 on n2




The nth term for quadratics

Exercises

Fill in the missing numbers:

Look at these sequences. Find:

a An expression for the nth term.

b The 10th term.

c The 21st term.

1 1, 4, 9, 16, 25

2 2, 8, 18, 32, 50

3 4, 13, 28, 49, 76

4 3, 8, 15, 24, 35

5 8, 14, 22, 32, 44

6 5, 15, 31, 53, 81

7 9, 24, 47, 78, 117

8 2, 13, 30, 53, 82

9 2, 7, 18, 35, 58

10 7, 17, 35, 61, 95

8 23 46 77 116

23




Solving simultaneous equations by algebraic methods

Question

Solve the simultaneous equations: 4x - 5y = 2 3x - 2y = 5

Answer

Substitute y = 2 into the original equation

4x - 5y = 2

4x - 5 x 2 = 2

4x - 10 = 2

4x = 2 + 10

4x = 12

x = 124

x = 3

Now check by substituting x = 3 and y = 2 in the other original equation

3x - 2y = 5

3 x 3 - 2 x 2 = 5

9 - 4 = 5

Answer: x = 3

y = 2

4 x - 5 y = 2

3 x - 2 y = 5

multiply by 3

multiply by 4

12x - 15y = 6

12x - 8y = 20

-7y = -14

-14

- 7

y = 2

Multiply the top line by 3

Multiply the bottom line by 4

Subtract

-15y- -8y = -7y 6 - 20 = -14

y =




Solving simultaneous equations by algebraic methods

Exercises

Solve the following simultaneous equations by algebraic methods.

1 2x + 3y = 21 2 5x + 3y = 194x + 2y = 22 2x + 4y = 16

3 5x + 2y = 13 4 3a + 2c = 233x + 3y = 15 4a + 3c = 32

5 a - 3c = 1 6 3a + 4x = 292a + 4c = 22 2a - 3x = -9

7 x - 3d = -5 8 2x - 5y = -193x - 2d = 6 3x + 2y = 0

9 3c - 5d = 17 10 3x - 2y = 04c + 3d = 13 4x - 2y = 2

11 5a + 2d = 4 12 x - 3y = -13a - d = -2 2x - 2y = -6

13 4c + 3d = 10 14 6a - 3d = -122c - 4d = -11.5 4a - d = -10

15 10a + 3c = 4 16 4x - 3y = -0.55a - 2c = -8.5 3x + 7y = 32

17 A man bought eight cakes at x pence and four rolls at y pence. The total cost was1.12. The following day he bought five cakes and three rolls from the same shop.The total cost was 0.74.

Form two equations to find the value of x and y.

18 Two tables and four chairs cost 84. Three tables and eight chairs cost 156.

What is the cost of a table?

What is the cost of a chair? Show your working.




Solving simultaneous equations by graphical methods

Question

Solve the following pair of simultaneous equations by drawing a graph.

y - 2x = 1 and 2y - x = 8

AnswerWrite y - 2x =1 as y = 2x + 1

(ie y on the left, everything else on the right of the equals sign)

Choose three simple values of x:

eg When x = 0 When x = 1 When x = 3

y = 2x + 1 y = 2x + 1 y = 2x + 1y = 2 x 0 + 1 y = 2 x 1 + 1 y = 2 x 3 + 1y = 1 y = 3 y = 7

(0,1) (1,3) (3,7)

Write 2y - x = 8 as 2y = x + 8

Choose three simple values of x:

When x = 0 When x = 1 When x = 3

2y = x + 8 2y = x + 8 2y = x + 82y = 0 + 8 2y = 1 + 8 2y = 3 + 82y = 8 2y = 9 2y = 11y = 4 y = 4.5 y = 5.5

(0,4) (1,4.5) (3,5.5)

Plot the values on a graph.

Where the lines cross draw dotted lines.

The solution is x = 2, y = 5.




01

1

2

3

4

5

6

7

2 3 4 5 6 7

x

x

x

x

x

xy

x

y =

2x +

1

2y = x +

8

Solving simultaneous equations by graphical methods

Exercises

Solve the following simultaneous equations by drawing suitable graphs. You will needvalues from -10 to +10 on the x and y axes.

1 2x + y = 4 2 3x + y = 5x + y = 3 x + y = 3

3 y - x = -2 4 x + 3y = 6x + y = 8 2x - y = 5

5 x + 2y = 6 6 3x - y = -12x + y = 3 2x + y = 6

7 2x - 2y = -6 8 x + y = 53x - y = -1 x - y = -1

9 2x - 2y = -4 10 2x + y = 53x + y = 6 x + y = 1

11 x = y 12 2x + y = -62x + y = 9 x - y = -3

13 2x + 4y = 2 14 2x - y = -33x + 5y = 5 x + y = -3

15 2x - y = -4 16 x + 2y = -1x + 2y = 3 2x + 3y = 0

17 3x = 6 18 3x + 3y = 6x + y = 3 x + 2y = 1

19 3x - 2y = -5 20 2x - y = -102x + y = -1 x + 2y = 0




Inequalities

> means greater than

< means less than

means greater than or equal to

means less than or equal to

Questions

1 Solve these inequalities:

a 5x > 20 b x - 7 < 10 c -2x > 8

2 Draw and indicate the following regions by shading:

a x > 4 b y 2

Answers1 Inequalities are very similar to equations:

a 5x > 20 b x - 7 < 10 c -2 x > 8x > 20/5 x < 10 + 7 x < 8/-2x > 4 x < 17 x < - 4

Note: When we have a negative multiplication or division the inequality sign reverses.

This causes many difficulties. If you are not certain which way the inequality sign should point, try acheck. The solution shows x is less than -4.

Choose a value less than -4, eg -5:

Is it true that -2x > 8?ie -2 x -5 > 8

10 > 8 Yes, it is true. So x < -4 is correct.

2a b




Note: The symbol always points to thesmaller number.

10-1

1

2

3

4

5

2 3 4 5 6x

x

yy

-1

-2

-1

-2

1 2 3 4 5-1-2

-3

-4

0

1

2

3

x = 4

The shaded region is x > 4

The shaded region is y 2

Note: We use a solid line when it is or

Note: We use a dotted linewhen it is < or >

y = 2

Inequalities

Exercises

Solve these inequalities:

1 3x < 15 2 x + 3 5

3 4x + 3 > 11 4 3x - 5 7

5 -4x > 12 6 -3x -24

7 -2x + 7 -3 8 -4x - 3 < 5

9 8x -3 < 1 10 2 - 5x -18

Draw and indicate the following regions by shading:

11 x 3 12 y < 4

13 x < 1 14 x -2

15 x -1 16 y -3

17 x + 3 5 18 y - 2 -5

19 y + 4 < 3 20 4y > 12

21 3 x 6 22 2x + 4 2

23 3x - 5 > 1 24 2y + 4 < -2

Describe the shaded regions:




4

3

2

1

0-1 1 2 3

-1

x

y2

1

0-1-2-3 1 2

-1

-2

-3

x

y3

2

1

0-1-2 1 2

-1

-2

x

y

25 26 27

Pythagoras theorem

Pythagoras theorem can be used when a triangle has a right angle.

Pythagoras theorem:

a2 + b2 = c2

(where c is the longest side)

Note: The longest side is always opposite the right angle.

Example 1

Find x

Example 2

Find y

Question

Find the height of this isosceles triangle:

Answer

An isosceles triangle can be split into two right-angled triangles.

h2 + 42 = 102

h2 = 102 - 42

h2 = 100 - 16h2 = 84h = 84h = 9.165 cm

To find the long side

Square both numbersAdd

Square root

To find either short side

Square both numbersSubtract

Square root

Short side

Short side

Long s

ide


KS3 Mathematics D: Level 7 Shape, Space and Measures


a

b

c

5 cm

12 cm

x

10 cm

8 cm

y

52 + 122 = x2

25 + 144 = x2

169 = x2

169 = x13 cm = x

y2 + 82 = 102

y2 = 102 - 82

y2 = 100 - 64y2 = 36y = 36y = 6 cm

10 cm 10 cm

8 cm

10 cm h

4 cm

Pythagoras theorem

Exercises

Find the value of x. Write answers correct to three significant figures.

Find the length of the diagonals in these rectangles.

Calculate the area of these triangles. (Area of a triangle = 1/2 base x height)

18 A square has a diagonal of 8 cm. Calculate the length of each side.

16 17

4 cm

8 cm

8 cm 12 m

12 m

7 m

13 14 15

7 cm

10 cm 6 cm12 m

4 m7 cm

5 cm 15 cm

15 cm 20 m

15 cm

15 m11 cm

17 cm

12 cm

10 cm6 cm

x

x

x

xx x

x

x

x

x

8 cm

8 m

7 m

8 cm

3.4 cm

1.4 m

3.9 cm

3.2 cm

8.4 cm1.7 m

8.2 cm

2.7 cm

8 cm

x

x

1 2 3 4

5 6 7 8

9 10 11 12




8 cm

Calculating lengths, areas and volumes 1

You are expected to know how to use these formulae:

Area of a triangle = 1/2 x base x perpendicular height (P.H.)

Area of a parallelogram = base x perpendicular height

Area of a trapezium = 1/2 (a + b) x perpendicular height

Volume of a cuboid = length x width x height

4 m5 m

3 m = 5 m x 4 m x 3 m

= 60 m3

P.H. 8 cm

12 cm

20 cm

a

b

Area = 1/2 x (12 + 20) x 81/2 x 32 x 8128 cm2

==

5 m

12 m

7 m

Area = 12 x 5 = 60 m2

(Note: 7 m is not used)

Base

P.H.6 cm

7 cm

1/2 x 7 x 6 = 21 cm2

or





Exercises

Calculate the areas of these shapes.

3 cm

8 cm 12 m

84 cm

12 cm

14 cm

5 m 5 cm

5 cm

12 cm

4 m

5 m

7 m

1.3 cm

3.52 cm

5.4 cm8.2 mm

4 cm

3 cm

30 cm

8 cm

18 cm

11 cm

92 cm1.6 m

30 cm

0.5 m0.4 m

Calculate the volumes of these cuboids

2.1 m

88 cm72 cm

109

87

5 64

2 31

11 12





Example

Find the volume of this prism:

Volume = cross-sectional area x length

First find the cross-sectional area

(Area of a triangle = 1/2 base x height = 1/2 x 5 x 3 = 7.5 cm2)

Note: The length is 1.5 m. This must be changed into centimetres, ie 150 cm.

Volume = 7.5 cm2 x 150 cm = 1125 cm3

Question

Find the area and perimeter of this shape:

AnswerThe formula to find the area of a trapezium is 1/2 (a+b) x perpendicular height.

Area = 1/2 (4 + 10) x 6

= 1/2 (14) x 6

= 7 x 6

= 42 cm2

To find the perimeter we must use Pythagoras theorem to find the missing side.

x2 = 62 + 62

x2 = 36 + 36

x2 = 72

x = 72x = 8.49 cm

Perimeter = 4 + 6 + 10 + 8.49 = 28.49 cm

P.H.

a

b




5 cm

1.5 m3 cm

4 cm

6 cm

10 cm

6

6x


Exercises

1 Square tiles, side length 15 cm, are usedto tile this floor. How many tiles arerequired?

2 Calculate the area of this shape.

3 AOB is a semi-circle.

ABCD is a square of side 6 m.

a What is the perimeter of the shaded shape?

b What is the area of the shaded shape?

Find the volumes of these shapes:




1.35 m

2.85 m1.8 m

4.05 m

17 m

10 m

8 m

A B

CD

O

40 cm

1.5 m

15 cm5 cm

4 cm

7 cm

8 cm 3 cm

8 cm3 cm

12 cm

8 cm

10 cm

7 cm

2 cm

3 cm

This cylinder has a radius of 8 cm and a length of 12 cm.

4 5

6

8

7

Enlargement by a fractional scale factor

Before attempting this work revise Level 6 enlargement.

Question

Enlarge the triangle by a scale factor of 2/3. Centre of enlargement is the point (1,1).

Answer

Count the distance from the centre of enlargement to each point.

9 along

3 up

6 along

2 up

Scale Factor

Point A x 2/3 =

12 along

3 up

8 along

2 upPoint B x 2/3 =

12 along

4.5 up

8 along

3 upPoint C x 2/3 =

0

1

1

2

2

3

3

4

4

5

6

5 6 7 8 9 10 11 12 13 14

A

A1

C1

B1

B

C

x6

9

23

0

1

1

2

2

3

3

4

4

5

6

5 6 7 8 9 10 11 12 13 14

A B

C




Enlargement by a fractional scale factor

Exercises

1 Enlarge A by a scale factor of 1/4, centre of enlargement (2,23)

2 Enlarge B by a scale factor of 1/2, centre of enlargement (20,19)

3 Enlarge C by a scale factor of 1/3, centre of enlargement (9,6)

4 Enlarge D by a scale factor of 2/3, centre of enlargement (7,1)

0

2

2

4

4

6

6

8

8

10

12

14

16

18

20

22

24

26

28

30

10 12 14 16 18 20 22 24 26 28 30

D

A

B

C




Locus

Questions

1 Draw the locus of a point which is always 1 cm from the line AB.

A B

2 Draw the locus of a point which is always an equal distance from two points P andQ which are 6 cm apart.

3 Draw the locus of a point which is always an equal distance from the lines BA andBC.

Answers

1

MethodJoin P and Q

a Place a pair of compasses on P

b Open the compasses over halfway

c Draw an arc above and below the line (1 and 2)

d Keep the compasses the same distance apart

e Place the pair of compasses on Q

f Draw an arc above and below the line (3 and 4)

g Join the intersections of both arcs. This is the locus.

B

A

C




A B

Methoda Open a pair of compasses. Keep them the

same distance apart.

b Place the pair of compasses on B

c Draw an arc on AB (1)

d Draw an arc on BC (2)

e Place the pair of compasses at 1 wherethe arc crosses the line AB

f Draw an arc (3)

g Place the pair of compasses at 2 wherethe arc crosses the line BC

h Draw an arc (4)

i Join B to the intersection of arcs 3 and 4

1

24

3

P Q

This is the locusof the point whichis always an equaldistance from P and Q

2

B

A

C

1

3

4

2

This is the locus of thepoint which is alwaysan equal distance fromBA and BC

3

Locus

Exercises

1 Construct the locus of the point Psuch that the locus is always 2 cm Paway from the point P.

2 A goat is tethered to the L shaped rail and moves so that it is always 2 m from therail. Using a scale of 1 cm represents 1 m construct the locus of the goats path.

3 A helicopter flies between two roads AB andAC so that it is always an equal distance fromeach road. Draw the path of the helicopter.Show all construction lines.

4 A ladder 5 m high is placed against a vertical wall. Gradually it slips until it lieshorizontally along the ground. Using a scale of 2 cm represents 1 m draw thelocus of this midpoint of the ladder as it slips down the wall.

5 Two points A and B are 10 m apart. Draw the locus of the point which is always anequal distance from A and B. Use a scale of 1 cm represents 1 m.




B

A

C

Accuracy of measurement

When a measurement, such as a length, mass or capacity is given, it may be inaccurate.

Example

If the length of a desk is given as 1.3 m this indicates that the length is approximately1.3 m. The measurement may be inaccurate by up to one half in either direction.

To calculate the minimum possible value

To calculate the maximum possible value

Therefore if the length is given as 1.3 m this means the actual length lies between 1.25 m and 1.35 m inclusive.

Questions

1 A book has a mass of 2.18 kilograms. What are the minimum and maximumpossible masses of the book?

2 The length of a blackboard is given as 2.80 m. What are the minimum andmaximum possible lengths of the blackboard?

Answers1

2

1.3 1.35 mAdd a 5

1.3 1.2 1.25 mAdd a 5Reduce the last digit by 1




2.18 2.17 2.175 kgAdd a 5Reduce the last digit by 1

2.18 2.185 kgAdd a 5

Minimum value

Maximum value

2.80 2.79 2.795 mAdd a 5Reduce the last digit by 1

2.80 2.805 mAdd a 5

Minimum value

Maximum value

Accuracy of measurement

Exercises

1 What are the minimum and maximum possible values of the followingmeasurements?

a 8.5 m b 6.28 cm c 4.82 g

d 12.3 seconds e 3.84 cl f 1.36 litres

g 8.17 km h 3.20 kg i 5.201 litres

j 3.026 cc k 5.87 mm l 0.030 g

2 What is the maximum error in each of the following measurements?

a 3.7 cm b 4.23 g c 7.82 ml

d 4.327 km e 3.21 mm f 5.2 kg

g 3.78 m h 4.20 ml i 0.230 g

3 The length of a room was measured to the nearest centimetre. The length was 628 centimetres.

a Could the room be longer than 628 centimetres?

b Explain your answer.

4 Three weighing scales were used to find the weight of a package.

The results were:

scale 1 3.8 kilograms

scale 2 3820 grams

scale 3 3.823 kilograms

Which scale gives the weight to the greatest accuracy?

Explain your answer.




Compound measures

Compound measures involve two measurements, for example:

metres per second, kilometres per hour

The following formulae must be memorised:

Questions

1 A car takes 8 hours 10 minutes to travel 343 kilometres. Calculate the averagespeed.

2 A man walks at a speed of 24 metres in 10 seconds. Calculate his speed inkilometres per hour.

Answers1 Decide if you require the answer in kilometres per hour or kilometres per minute.

If you choose kilometres per hour change 8 hours 10 minutes into hours.

10 minutes is 10/60 of an hour = 1/6 of an hour. Therefore 8 hours 10 minutes = 81/6 hours.

Speed = Distance = 343 = 42 kilometres per hour.Time 81/6

2 24 metres in 10 seconds

(multiply by 6) 144 metres in 1 minute

(multiply by 60) 8640 metres in 60 minutes (ie 1 hour)

(divide by 1000) 8.64 kilometres in 1 hour

The speed is 8.64 kilometres per hour.

Distance

Speed Time

Speed =Distance

Time

Time =DistanceSpeed

Distance = Speed x Time

Mass

Density Volume

Density =Mass

Volume

Volume =Mass

Density

Mass = Density x Volume




Compound measures

Exercises

Calculate the speed of the following:

1 A car travels 368 kilometres in 4 hours.

2 A train travels 728 kilometres in 5 hours.

3 A man runs 1000 metres in 184 seconds (calculate his speed in kilometres per hour).

4 A bird travels 185 kilometres in 6 hours 23 minutes.

5 A greyhound runs 100 metres in 9.2 seconds.

Calculate the time taken for the following:

6 A bus travels 180 miles at 40 miles per hour.

7 A cyclist travels 80 kilometres at a speed of 25 kilometres per hour.

8 A plane travels 2200 kilometres at a speed of 540 kilometres per hour.

9 A man walks 8.3 kilometres at a speed of 1.5 metres per second.

10 A bird flies 82 kilometres at a speed of 35 kilometres per hour.

Calculate the distance travelled:

11 A train travels for 6 hours at a speed of 145 kilometres per hour.

12 A horse travels at a speed of 52 kilometres per hour for 6 minutes.

13 A car travels for 3 hours 22 minutes at a speed of 85 kilometres per hour.

14 A snail travels at a speed of 1.4 centimetres per minute for two weeks.

15 A spaceship travels for 10 weeks at a speed of 3.4 x 104 kilometres per hour.

16 A block of wood has a volume of 180 cm3. Find the mass given that the density is 0.95 g/cm3.

17 Calculate the density of snow given that the mass is 617.5 g and the volume is 950 cm3.

18 Find the mass of a block of metal given that the volume is 370 cm3 and thedensity is 18.4 g/cm3.

19 Convert a speed of 36 metres per second to kilometres per hour.

20 Convert a speed of 162 kilometres per hour to metres per second.




Designing questionnaires

1 Design your questions to obtain information you can present and analyse in avariety of ways.

2 Make your questions easy to understand.

3 Do not ask embarrassing questions, eg How many boyfriends do you have?

4 Provide a choice of answer eg, Do you do a lot of homework? will produceanswers such as yes, sometimes, only in Maths. These responses are difficult topresent and analyse. A better question would be:

How much time did you spend doing homework last night? Tick the box nearestto the amount of time.

0 hours n1 hours n2 hours n3 hours n

Types of question

Your questionnaire should contain one or two questions of each of the following types:

1 Questions with yes/no responses, eg Do you own a bicycle? Yes n No n

Try to avoid questions to which everyone will answer yes or everyone will answerno. Your results can be shown as a percentage, in a bar graph, pictogram, piechart, etc.

2 Questions with numerical answers, eg How many televisions do you have inyour house?

Your results can be presented in graphs, tables, etc.

You can calculate the mean, median and mode of the data.

3 Questions you can compare, eg What was your percentage mark in the Englishexam? and What was your percentage mark in the Maths exam?

These questions will allow you to draw a scatter diagram to test a hypothesis suchas Pupils who obtain high marks in English also obtain high marks in Maths.

How many people to ask

Twenty is a good number. Each person represents 5% of the total and each personcan be represented by 18 on a pie chart.

Forty is a good number. Each person represents 2.5% of the total and each personcan be represented by 9 on a pie chart.

How many questions to ask

A maximum of ten.


KS3 Mathematics D: Level 7 Handling Data


Designing questionnaires

Exercises

Look at the following questions. Which questions are good? Which questions are bad?Criticise the bad questions. Remember to refer to the difficulty of responding to thequestion, the difficulty of analysing the responses, etc. Rewrite the bad questions.

1 Do you often travel in a car?

2 Which is your favourite television channel?

BBC1 n BBC2 n ITV n Channel 4 n

3 Do you own a CD player? Yes n No n

4 What do you think of Maths?

5 What is your favourite type of music?

6 How many packets of crisps do you eat in a week?

Design and use a questionnaire to collect data which is useful in proving or disprovingone of the following hypotheses:

1 Most pupils have school meals.

2 Most boys like heavy rock.

3 The majority of pupils do not smoke cigarettes.

4 Girls receive less pocket money than boys.




Specifying and testing hypotheses

Task

Choose a hypothesis. Decide how to test it. Collect data. Present the data in a varietyof ways. Analyse the data. Draw conclusions from the results. Was the hypothesis proved?

What to do

1 Think of a hypothesis. A hypothesis is a statement or observation which may betrue, eg More men than women drive cars. A drawing-pin lands point upwardsmore than point downwards, Girls favourite television channel is BBC1.

2 Decide how to test your hypothesis. How will you collect your data? The above hypotheses could be tested in these ways:

More men than women drive cars (observation).A drawing-pin lands point upwards more than point downwards (experiment).Girls favourite television channel is BBC1 (questionnaire).

3 How will you analyse and present your data? The following should be included:

Tables eg percentages Graphs pictograms, bar charts, line graphs Pie charts including your calculations Frequency polygons Averages mean, median, mode Range Scatter diagrams positive correlation, negative correlation, line of best fit Cumulative frequency (National Curriculum Level 8) upper quartile, lower

quartile, inter-quartile range Bias are the results honest? For example, a coin could be weighted to give

more heads than tails.

If you can use a computer you could include spreadsheets, etc.

Remember to make your graphs neat; try to use colour.

Do not produce dozens of one type of graph. It is far better to draw three or fourpie charts than 20 pie charts.

Remember to state your hypothesis at the start.

Remember to analyse your findings. Draw conclusions from your results. Justifyyour conclusions is your hypothesis proved?

If your hypothesis does not allow you to analyse and present your data in a variety ofways it is far wiser to choose a different hypothesis immediately. Do not waste time ona hypothesis which will not allow you to demonstrate your mathematical ability.




Specifying and testing hypotheses

Exercise

Use this worksheet to check that your hypothesis will allow you to test, present andanalyse your data in a variety of ways.

If it will not you should choose another hypothesis.

Plan

My hypothesis is:

I shall test my hypothesis by using the following methods:

I shall present my data using the following methods:

I shall analyse my data using the following methods:

My conclusion could be:




Grouped data

Questions

This table shows the number of cars using a car park over a period of 100 days:

1 What is the modal class?

2 Estimate the median.

3 Estimate the mean.

4 Estimate the range.

Answers1 The modal class is the class with the highest number. In this question it is 200 299 cars.

2 There are 100 days. The median is the middle day when arranged in order of size. The question asksfor an estimate, therefore we can assume that the median is the 50th day.

5 + 18 = 23. Therefore there are 23 days with less than 200 cars.

5 + 18 + 30 = 53. Therefore there are 53 days with less than 300 cars.

The 50th day is towards the high end of the 200 299 class.

A good estimate of the median would be about 290 cars.

3 The mean is found by first multiplying the mid-value of each class by the frequency. The question asksfor an estimate, therefore we can use 50, 150, 250, 350 and 450 as the mid-values.

(5 x 50) + (18 x 150) + (30 x 250) + (27 x 350) + (20 x 450)100

= 250 + 2700 + 7500 + 9450 + 9000100

= 28900100

The mean number of cars is about 289.

4 The range is from 0 to 500, ie a range of 500.

Number of cars

Frequency

0 99

5

100 199

18

200 299

30

300 399

27

400 500

20




= 289

Grouped data

Exercises

In each of the following questions:

a Find the modal class.

b Estimate the median.

c Estimate the mean.

d Estimate the range.

1 This table shows the masses (in kilograms) of 100 sailors on a ship.

2 This table shows the times (in seconds) taken by 200 pupils to thread a needle.

3 This table show the heights (in centimetres) of 100 men.

4 This table shows the number of nails in 500 boxes of nails.

Number of nails

Frequency

500 519

320

520 539

142

540 559

27

560 580

11

Height (cm)

Frequency

150 159

22

160 169

26

170 179

38

180 189

12

190 200

2

Time (seconds)

Frequency

0 9

30

10 19

25

20 29

20

30 39

30

40 49

80

50 60

15

Mass (kg)

Frequency

50 59

3

60 69

17

70 79

24

80 89

30

90 99

16

100 110

10




Comparison of data

Question

1 The heights of 20 boys and 20 girls aged 16 are shown in this table.

a Present the data in a frequency polygon.

b Compare the distributions and comment on your findings.

Answersa

b The frequency polygon shows that boys aged 16 are generally taller than girls of the same age.

0

144.5

1

2

3

4

5

6

7

8

150

154.5

160164.5

170174.5

180184.5

190194.5

200

Freq

uenc

y

Height in centimetres

GirlsBoys

144.5 is the mid point for the class interval 140 149

Height (cm)

140 149

150 159

160 169

170 179

180 189

190 199

1

6

8

4

1

1

3

8

6

2

Number of boys Number of girls




Comparison of data

Exercises

Draw frequency polygons to illustrate the following data.

Compare the distributions and comment on your findings.

1 This table shows the price of detached houses on two housing estates.

2 This table shows the Maths GCSE grades obtained by two classes of pupils.

3 This table shows the hours of sunshine in a town in Scotland and a town in Wales.

Month

October

November

December

January

February

March

48

35

18

22

36

54

57

46

27

30

42

59

Town in Scotland Town in Wales

Grade

A

B

C

D

E

3

5

7

8

5

6

6

7

5

4

Class 1 Class 2

Price ()

70 000 under 80 000

80 000 under 90 000

90 000 under 100 000

100 000 under 110 000

110 000 under 120 000

3

8

6

2

1

0

2

6

8

4

Estate A Estate B




Scatter diagrams (line of best fit by inspection)

A line of best fit is drawn by looking at the crosses on a scatter diagram and thendrawing a line. Normally there would be a similar number of crosses above the line asbelow the line.

Question

Draw a line of best fit on this scatter diagram. This scatter diagram shows the masses of18 pupils against their ages.

Answer

The line of best fit should be in a similar position to the line shown.

30

40

50

60

70

80

98 1110 1312 1514

Mas

s (k

g)

Age (years)

xxx

xx x

x xx

xx

x x xx x

x

x

30

40

50

60

70

80

98 1110 1312 1514

Mas

s (k

g)

Age (years)

xxx

xx x

x xx

xx

x x xx x

x

x




Scatter diagrams (line of best fit by inspection)

Exercises

Draw a line of best fit on each of the following scatter diagrams.

1 This scatter diagram shows the examinations passed and hours spent watchingtelevision of 20 students.

a John spent five hours each day watching television. How many examinationpasses would you expect him to achieve?

b Sandy passed eight examinations. Estimate the number of hours she watchedeach week.

2 This scatter diagram shows the heights and masses of 20 people.

Complete this table.

Name

Jayne

Paul

Marie

Dave

60

75

190

165

Mass (kg) Height (cm)

150

160

170

180

190

200

504540 6055 7065 80 85 90 95 100 105 11075

Hei

ght

in c

entim

etre

s

Mass in kilograms

xx x

xx

xx

x

x

x

xx x x x

x xx

xx

1

0

2

3

4

5

6

7

8

9

10

21 43 65 8 9 10 11 127

Num

ber

of e

xam

inat

ion

pas

ses

Average number of hours per day spent watching TV

x

x

x

x

x

x

x

x

x

xx

x

x x x

x x

x

x

x




Probability

Estimation of probability by experiment

The more times an experiment is carried out the more likely the data obtained isaccurate.

Example

A six-sided die is thrown. Here are the results.

John threw the die 30 times.Andrea threw the die 300 times.

Andrea is more likely to obtain the better estimate because she has thrown the diemore times than John.

Question

1 A die is thrown 600 times. These results are obtained.

a Do the results indicate the die is biased?

b Justify your answer.

c Use the data to work out the probability of the die landing on:

i 1, ii 3, iii 4, iv 6

d If the die were fair how many times would you expect it to land on each number if it were thrown 600 times?

Answers1 a The die seems to be biased.

b More 3s were obtained than would be expected by chance. Less 4s were obtained than would be expected by chance.

c i 102/600 = 51/300 = 17/100 ii 181/600 iii 31/600 iv 92/600 = 23/150

d We would expect the die to land on each number a similar amount of times. The chance of each number is 1/6. Therefore we would expect each number to occur about 100 times.

1 2 3 4 5 6

102 112 181 31 82 92

1

2

2

7

3

3

4

8

5

4

6

6

46

Side of die

John

Andrea 51 53 47 46 57




Probability

Exercises

1 This table shows the shoe sizes of 200 women.

A woman is chosen at random. Use the information in the table to decide theprobability that she takes:

a Size 4 b Size 2 c More than size 5 d Less than size 5

e A week later the shoe sizes of 1000 woman were recorded. Fill in this table to show the numbers of each shoe size you would expect.

f Explain how you decided what numbers to write in the table.

2 2000 cartons of six eggs were accidentally dropped during transportation. 100 cartons were examined at random and the following numbers of eggs werebroken in each carton:

a If an unchecked box is opened at random what is the probability that it will contain: i one broken eggii three broken eggsiii less than five broken eggs?

b Estimate the number of cartons with exactly four broken eggs.

c Estimate the total number of broken eggs. Show your working.

Eggs broken

Frequency

0 1 2 3 4 5 6

2 17 35 22 12 8 4

Shoe size

Number of women

2 3 4 5 6 7

Shoe size

Number of women

2

10

3

41

4

72

5

36

6

24

7

17




Investigation

Arrange 16 chairs as shown.

Boys and girls can move left, right, up or down, onto an empty space next to them.No one may move diagonally.

What is the least number of moves to move the boy to the empty space?

Advice

1 First look at simple cases. Use counters instead of people.

2 Draw diagrams to show what is happening. Draw tables to show results.

Look for repeating moves or patterns. Describe what you find. Record themoves. Try to find a rule. Can you predict the next answer? Check yourprediction. You now have a rule for 2 rows and any number of columns.

3 Now find rules for other shapes. Be systematic.

Investigate 3 rows. Repeat points 1 and 2 above. Is the solution for 2 rows and 3columns the same as 3 rows and 2 columns? Investigate 4 rows, 5 rows. Can youpredict 6 rows? Check your prediction.

4 Now look for a general case. Can you predict the number of moves for 7 rows and8 columns? Explain your findings in words and using formulae.

G G G G G G

G G G G G GB B BTry then then etc

G G G

G G G G

G

G

G G G

B G G G

B = Boy

= Girl

= Empty space


KS3 Mathematics D: Level 7 Investigation


R O W S

COLUMNS

Nine Mens Morris

You need: 9 black counters 9 white counters

Rules

1 Take it in turns to place a counter on an empty dot.

2 When all counters are placed take it in turns to move one counter along a line toan empty dot.

3 Each time you have three counters in a straight line you can remove one of youropponents counters. It cannot be used again.

4 The player who loses all of their counters from the board loses.


KS3 Mathematics D: Level 7 Game


Number and AlgebraEstimating, multiplying and dividing by numbers between 0 and 1Efficient use of a calculator 1Efficient use of a calculator 2Solving problemsProportional changeThe nth term for quadraticsSolving simultaneous equations by algebraic methodsSolving simultaneous equations by graphical methodsInequalities

Shape, Space and MeasuresPythagoras theoremCalculating lengths, areas and volumes 1Calculating lengths, areas and volumes 2Enlargement by a fractional scale factorLocusAccuracy of measurementCompound measures

Handling DataDesigning questionnairesSpecifying and testing hypothesesGrouped dataComparison of dataScatter diagrams (line of best fit by inspection)Probability

Activity and InvestigationInvestigationNine Mens Morris

Documents

9X1 and 9Y1