key stage 3 - succeed in maths

8/2/2019 Key Stage 3 - Succeed in Maths

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p

Key Stage 3 Mathematics

Level by Level

Pack C: Level 6

Stafford Burndred

ISBN 1 899603 24 7

Published by Pearson Publishing Limited 1997 Pearson Publishing 1995

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 permission of Pearson Publishing.

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

Pembrokeshire e-Portal Licence exp 31Aug10


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Trial and improvement

You should draw four columns as shown below.

In the first column write down your guess.In the second column work out the answer using your guess.

If your answer is too big write your guess in the too big column.

If your answer is too small write your guess in the too small column.

Question

Find the value of x correct to 1 decimal place using trial and improvement methods.

You must not use the key on your calculator x2 = 78.

Answer

You may have used different values in your calculations.

8.8 is too small, 8.85 is too big.

The answer must be between 8.8 and 8.85.

Therefore the value of x correct to 1 decimal place is 88.

Note: If the question was Find the square root of 78 without using the square root key

() on your calculator you would use exactly the same method.

Guess x Answer x2 Too big Too small

8

9

8.5

8.8

8.9

8.85

64

81

72.25

77.44

79.21

78.3225

9

8.9

8.85

8

8.5

8.8

8 is too small. Guess higher

8 is too small. 9 is too big

Guess between 8 and 9

8.5 is too small. 9 is too big

Guess between 8.8 and 9

8.8 is too small. 8.9 is too big

Guess between 8.8 and 8.9

Guess Answer Too big Too small

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

KS3 Mathematics C: Level 6 Number and Algebra



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Trial and improvement

Exercises

Use Trial and Improvement methods to calculate the following. You must show all ofyour working. You may use a calculator but you must not use the square root key () or

the cube root key (3).

1 Find the value of x correct to 2 decimal places given x2 = 5 _________

2 Find the square root of 20 correct to 2 decimal places. _________

3 Find the square root of 17 correct to 2 decimal places. _________

4 Find the value of y correct to 2 decimal places given y2 = 45. _________

5 Find the cube root of 35 correct to 2 decimal places. _________

6 Find the value of x correct to 1 decimal place given x3 = 15. _________

7 Find the value of x correct to 1 decimal place given x3 = 48. _________

8 Find the exact value of x in each of the following questions:

a x2 + x = 20 b x2 + 3x = 54

c x3 + 5x = 18 d x2 - 4x = 21

e x3 - 2x = 980 f x2 + 4x - 3 = 282

g x2 - 2x + 8 = 296 h x3 - x2 = 294

i x3 + x2 = 8400 j x3 - x2 + 2x = 920





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Calculating fractions and percentages

To calculate one number as a fraction of another number.

Example 10 people out of 25 went to work by bus.Write this as a fraction in its lowest terms.

Without a calculator

10/25 Divide top and bottom by 5 2/5

With a calculator. The calculator should have a fraction key .

Calculator keys: Answer 2/5

To calculate one number as a percentage of another number.

Example 284 people out of 800 wore glasses.

Write this as a percentage.

Without a calculator.

284/800 x 100 = 35.5%

With a calculator.

Calculator keys: Answer 35.5%

With some calculators you may have to press at the end.

Questions

1 Find 3/8 of 12

2 A man earns 250 per week. He receives a 4 increase.

What percentage increase is this?

Answers1 Of means multiply 3/8 x 12 = 4.50

2 = 16 %4 52 0 %

=

82 4 8 0 0 %

01 2 5 =ab c/

ab c/





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Exercises

Write the following as percentages.

1 3 out of 20 men have beards. _________

2 5 people out of 40 wear hats. _________

3 There are 360 girls in a school with 600 pupils.

What percentage are boys? _________

4 28 people out of 50 people liked pop music. _________

5 Find 3/8 of these numbers:a 18 b 30 c 12

d 27 e 16 f 360

g 420 h 480 i 630

6 A shop had a sale in which all goods were 1/3 off the normal price. What were the

sale prices of these goods with the following normal prices?

a Tie 6 b Shirt 12 c Handkerchief 4.50

d Blouse 7.50 e Skirt 6.30 f Scarf 7.80

g Sweat shirt 12.60 h Trousers 15.90 i Dress 2130

7 The following list gives the marks obtained by a class of pupils in a history test. The

marks are out of 80. Express each mark as a percentage correct to the nearest

whole number.

a 40 b 20 c 36

d 48 e 72 f 68

g 27 h 29 i 33j 52

8 A car dealer bought a car for 3000 and sold it for 3210.

What was his percentage profit? _________





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Notes and Examples

When working with percentages the total is 100%.

If we are working with price increases or decreases the cost price or original price is

100%.

Example

A man earns 350 per week. He then receives a wage increase. His new wage is 371.

Calculate the percentage wage increase.

Calculator keys

Your calculator should show . If it does not, try pressing the key.

Answer 6% Note: without a calculator 21/350 x 100

Questions

1 A golf club had 1200 members. 840 of the members were males.

What percentage of the members were female?

2 A car was bought for 8000 and sold for 5000.

Calculate the percentage loss.

Answers1 The question asks for the percentage of females 1200 - 840 = 360

360/1200 Calculator keys: 30%

2 The loss is 3000. 3000/8000 Calculator keys 37.5%03 0 0 8 0 0 0 %

63 0 1 2 0 0 %

=6

12 3 5 0 %

His wage increase is 371 - 350 = 21

His original wage is 350

21

350The cost price, original priceor total goes on the bottom line





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Exercises

1 In a sale all goods are sold at 20% off the normal price. Calculate the sale price ofgoods normally sold at these prices.

a 30 b 400 c 380

d 5 e 3 f 16

g 4.50 h 3.60 i 7.20

j 5.35 k 7.65 l 20.05

2 An estate agent calculates his fee for selling a house by the following rules.

Houses with a selling price under 50,000, selling fee 3% of the house price.

House with a selling price over 50,000, selling fee 2.75% of the house price.

a Calculate the estate agents fees for the following houses:

i Selling price 28,000 ii Selling price 58,000

iii Selling price 37,000 iv Selling price 125,000

b The estate agent received the following fees. Calculate the house prices.

i 1200 ii 2200 iii 960

iv 3025 v 870 vi 2695

3 Two shops are selling identical television sets. The normal selling price is 690.

In AA Electrics there is a sale with 1/3 off everything.

In Hardys Video and TV store there is a discount of 30%.

a What is the cost in AA Electrics? _________

b What is the cost in Hardys Video and TV? _________

c Where should you buy the television and how much cheaper is it in this shop

than in the other shop? _______________________________________________





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Equivalences between decimals and percentages

Converting percentages to decimals

Move the decimal point two places to the left.

Converting decimals to percentagesMove the decimal point two places to the right.

Questions

1 Convert the following percentages to decimals:

a 74% b 6% c 42.2%

2 Change these decimals to percentages:

a 0.52 b 0.08 c 0.026

Answers1 a 0.74 b 0.06 c 0.422

2 a 52% b 8% c 2.6%

0.52 0.5 2 = 52%

0.7 0.7 = 70%

0.03 0.0 3 = 3%

0.365 0.3 6 5 = 36.5%

38% 3 8. = 0.38

30% 3 0. = 0.30

5% 5. = 0.05

27.4% 2 7. 4 = 0.274





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Equivalences between decimals and percentages

Exercise

1 Write the following percentages as decimals:

a 38% b 23% c 6%

d 26% e 80% f 5%

g 45% h 3% i 60%

j 7% k 2.32% l 25%

m 48.5% n 40% o 0.5%

p 50% q 3.6% r 75%

s 27.32% t 36.8%

2 Write the following decimals as percentages:

a 0.27 b 0.72 c 0.47

d 0.54 e 0.63 f 0.6

g 0.03 h 0.9 i 0.272

j 0.453 k 0.1 l 0.01

m 0.02 n 0.24 o 0.4

p 3.72 q 2.01 r 4.1

s 0.0072 t 0.104

3 Complete the table and then memorise the following information if you do not

already know these equivalences.

Fraction Decimal Percentage

a 1/2 = 0.5 =

b 1/4 = 0.25 =

c 3/4 = 0.75 =

d 1/3 = 0.333 =

e 2/3 = = 66.7%

f 1/8 = = 12.5%

g 1/10 = = 10%

h 1/100 = = 1%





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Equivalences between decimals, fractions andpercentages

Converting percentages to fractionsFirst convert the percentage to a decimal and then proceed as below.

Converting decimals to fractions

Converting fractions to decimals

Divide the top number by the bottom number.

3/4 means 34 = 0.75

17/20 means 1720 = 0.85

3/40 means 340 = 0.075

Converting fractions to percentages

First convert the fraction to a decimal, then convert the decimal to a percentage.

Questions

1 Convert the following decimals to fractions:

a 0.4 b 0.24 c 0.02 d 0.027

2 Write these fractions as decimals:

a 3/5 b 17/25 c 5/8

Answers

1 a 4/10 =2/5 b 24/100 = 6/25 c 2/100 = 1/50 d 27/10002 a 0.6 b 0.68 c 0.625

0.3 = 310

One number afterthe decimal point

One nought

0.3 7 = 37100

Two numbers afterthe decimal point

Two noughts

0.3 7 1 = 3711000

Three numbers afterthe decimal point

Three noughts

0.0 3 = 3100

Two numbers afterthe decimal point

Two noughts





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Equivalences between decimals, fractions andpercentages

Exercise1 Write the following decimals as fractions:

a 0.78 b 0.93 c 0.47

d 0.6 e 0.8 f 0.05

g 0.07 h 0.98 i 0.63

2 Convert the following fractions to decimals:

a 1/4 b 3/5 c 7/8d 5/8 e 3/10 f 28/40

g 29/50 h 87/100 i 23/80

3 Complete this table:

Fraction Decimal Percentage

a 9/16 = =

b = 0.34 =

c = = 25.5%

d = 0.224 =

e 7/8 = =

f = 0.04 =

4 Write the following fractions as percentages:

a 3/4 b 1/2 c 3/8

d 4/5 e 1/10 f 1/5

g 15/16 h 5/8 i 17/20

5 Convert the following percentages to fractions:

a 25% b 47% c 28%

d 37% e 39% f 42.7%

g 60% h 30% i 8%





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Ratio 1

Questions

1 This is a recipe for soup for four people.

800 cc water

2 tomatoes

100 g beef

8 g salt

How much of each ingredient should you use for:

a two people

b six people

2 Simplify these ratios:

a 4:18 b 30:45

3 The scale of a map is 1:1,000,000

a The distance between Longton and Hilton is 18 cm on the map.

What is the actual distance?

b The distance between Bursley and Higham is 142 km.

What is the distance on the map?

Answers1 a Two people will need half the ingredients: 400 cc water, 1 tomato, 50 g beef and 4 g salt

b Six people will need one and a half times the ingredients: 1200 cc water, 3 tomatoes, 150 g beef,

12 g salt

2 a 4:18, divide both sides by 2 2:9

b 30:45, divide both sides by 15 2:3

3 1:1,000,000 means 1 cm on the map represents 1,000,000 cm on the ground

1,000,000 cm = 10,000 m = 10 km

Therefore 1 cm on the map represents 10 km on the ground

a 18 cm on the map means (18x10) km on the ground 180 km

b 142 km is represented by (142 10) cm on the map 14.2 cm





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Ratio 1

Exercises

1 This is a recipe for Yorkshire pudding for four people:

120 g of flour

480 ml of milk

2 eggs


a 2 people b 6 people c 10 people

2 This is a recipe to make hot pot for four people:

440 g of beef steak

50 g of flour

2 onions

600 g of potatoes

350 g of cube stock


a 2 people b 6 people c 10 people

3 Simplify these ratios:

a 3:12 b 4:8 c 15:9

d 10:4 e 18:12:9 f 150:250:350

4 The scale of a map is 1:100,000. What are the actual distances between the

following towns? Give your answer in kilometres.

a Ayton is 8 cm from Beeton on the map

b Beeton is 12 cm from Ceeton on the map

c Ceeton is 3 cm from Deeham on the map

d Deeham is 4.5 cm from Exford on the map

e Exford is 7.6 cm from Effingham on the map

5 These are the actual distances between the following towns.

What are the distances on the map?

a Kayham is 5 km from Elton

b Elton is 8.2 km from Emton

c Emton is 13.5 km from Newtown.





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Ratio 2

You use ratio every day of your life. A simple example is making a glass of orange

squash. You use undiluted orange and water in the ratio

Example

How many litres of squash can be made with a three litre bottle of undiluted orange?

The ratio is undiluted orange water squash

1 : 4 5

one part four parts five parts

One part is 3 litres

Therefore five parts is 5 x 3 = 15 litres

Question

A man leaves 5000 in his will. The money is to be divided between his three sons

Adam, Ben and Carl in this ratio 2:3:5. How much does each son receive?

AnswerAdam receives 2 parts

Ben receives 3 parts

Carl receives 5 parts

10 parts

10 parts is 5000

Therefore 1 part is 500

Adam receives 2 parts 1000

Ben receives 3 parts 1500

Carl receives 5 parts 2500




1 : 4

1 part 4 parts

4 parts water1 part undiluted orange

Produces 5 parts squash


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Ratio 2

Exercise

1 Express the following as ratios in their simplest form:

a A school has 200 boys and 300 girls

b A tennis club has 250 female members and 300 male members

c A factory has 900 men and 600 women

d A ship has 800 passengers and 160 crew

2 Express these scales as ratios in their simplest form:

a The scale of a map is 1 cm represents 50 cmb The scale of a map is 1 mm represents 1 m

c The scale of a map is 1 cm represents 20 m

d The scale of a map is 5 cm represents 10 m

e The scale of a map is 5 cm represents 80 m

f The scale of a map is 2 cm represents 15 m

3 In a will, money is left to three daughters, Angela, Barbara and Carolyn in the ratio3:4:5. If the total amount of money is 4500, how much will each daughter receive?

4 A sum of money was left to three sons Adam, Ben and Calvin. The money was

divided in the ratio 3:5:6. If Adam received 2100, how much did the other two

sons receive?

5 Sweets were divided between Paul, Sarah and Tony in the ratio 2:4:5. Sarah

received 40 sweets less than Tony.

a What was the total number of sweets?

b How many sweets did each person receive?

6 A model of a sailing ship was made. The model of the sailing ship was built to a

scale of 1:400. Complete this table.

LengthBreadthHeight of main sail

Length of rudderWidth of main sail

Model (centimetres)7.5 centimetres

3 centimetres

Full-sized ship (metres)200 metres10 metres

2 metres

abc

de





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Explore number patterns

Example

a Describe how to find each term in the pattern 5, 8, 11, 14, 17.

b What is the tenth term?

Method:

The difference is 3. This is what you multiply by:

The rule is multiply the term by 3, then add 2.

b The tenth term is 3 x 10 + 2 = 32

Question

a Find the rule to produce this pattern: 2, 9, 16, 23, 30

b What is the 20th term?

c What is the 362nd term?

Answer

Multiply each term by 7.

The rule is multiply the term by 7, then subtract 5

b The 20th term is 7 x 20 - 5 = 135 c The 362nd term is 7 x 362 - 5 = 2529

1st term is

2nd term is

3rd term is

7 x 1 = 3

7 x 2 = 6

7 x 3 = 9

What do you haveto do to find theanswer?Subtract 5

7 - 5 = 2

14 - 5 = 9

21 - 5 = 16

1st2

2nd9

3rd16

4th23

5th30

7 7 7 7

a Term

Find the difference

1st term is

2nd term is

3rd term is

3 x 1 = 3

3 x 2 = 6

3 x 3 = 9

What do you haveto do to find theanswer?Add 2

3 + 2 = 5

6 + 2 = 8

9 + 2 = 11

1st5

2nd8

3rd11

4th14

5th17

3 3 3 3

a Term

Find the difference





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Explore number patterns

Exercise

a Find the rule to produce each pattern

b Find the 15th term

c Find the 127th term

1 2, 4, 6, 8, 10

2 6, 12, 18, 24, 30

3 3, 6, 9, 12, 15

4 2, 6, 10, 14, 18

5 3, 8, 13, 18, 23

6 5, 7, 9, 11, 13

7 4, 7, 10, 13, 16

8 -3, 1, 5, 9, 13

9 -5, -2, 1, 4, 7

10 27, 32, 37, 42, 47

11 6, 11, 16, 21

12 3, 11, 19, 27, 35, 43

The rules below will produce sequences. Produce the first five terms for each

sequence.

13 Multiply the term by 3 then add 2



16 Multiply the term by 6 then subtract 1







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Solving linear equations

Rules for solving equations

1 3a means 3 x a

2 The sign in front of a number is attached to that number.

eg -3 + 6a The - is attached to the 3, the + is attached to 6a

3 Always keep the equals signs in straight columns. Work down the page not across

4 When you take a number from one side of the equals to the other.

+ becomes -

- becomes +

x becomes

becomes x

5 Do the addition and subtraction parts before the multiplication and division.

Question

1 a + 5 = 8 2 a - 2 = -7 3 -7y = 28

4 y/3 = 6 5 5a + 7 = 27 6 a/3 - 5 = 1

Answer

a + 5 = 8aa

= 8 -5= 3

a - 2 = -7aa

= -7+2= -5

1 2

Keep equals signsin straight columns

-7y = 28yy

= 28/-7= -4

y/3 = 6yy

= 6 x 3= 18

3 4

5a + 7 = 275a5aaa

= 27 -7= 20= 20/5= 4

a/3 - 5 = 1a/3a/3aa

= 1 + 5= 6= 6 x 3= 18

5 6

Deal with theadd first

Now deal withthe multiplication

+ is the opposite of -

- is the opposite of +

x is the opposite of

is the opposite of x





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Solving linear equations

Solve these equations:

1 x + 4 = 7 2 x - 5 = 8 3 x - 5 = -3

4 x - 8 = -13 5 x + 3 = -8 6 x + 10 = -4

7 3x = 12 8 5x = 35 9 7x = 42

10 8x = 16 11 6x = 3 12 10x = 2

13 8x = -8 14 5x = -30 15 7x = -21

16 -3x = -12 17 -4x = -20 18 -8x = 32

19 -5 x = 30 20 -2x = 1 21 3y + 1 = 13

22 4a + 3 = 23 23 5c - 3 = 7 24 8d -1 = 31

25 6e + 3 = 15 26 5a + 12 = 3a + 20 27 7a + 2 = 4a + 20

28 8a + 25 = 3a +10 29 5a - 3 = 2a - 12 30 4a - 8 = 3a - 3

31 d/3 = 5 32 a/8 = 4 33 c/7 = 2

34 y/4 = 3 35 6a/3 = 4 36 5a/2 = 20

37 3a/4 = 12 38 5x/3 = 15 39 3x/4 - 3 = 3

40 5x/2 + 1 = 11





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Formulating linear equations

You must understand a problem before you can write an equation to solve it. Try

putting numbers in for the letters. This will help you to understand what the question is

asking.

Question

1 A man buys a apples at 8p each. The total cost is 96p.

a Form an equation to show this.

b Solve the equation.

2 I think of a number N, I double it and add 15. The answer is 31.

a Form an equation to show this.

b Solve the equation.

Answer1 a Try putting numbers in for the letters.

How would you work out the cost of:

5 apples 8 x 5 = 40

6 apples 8 x 6 = 48

7 apples 8 x 7 = 56

a apples 8 x a = 96

The equation is 8a = 96

b 8a = 96

8a = 96/8

8a = 12

2 a Choose numbers. See what happens:

if N = 3 3 x 2 + 15 = 21

if N = 4 4 x 2 + 15 = 23

if N = 5 5 x 2 + 15 = 25

Try N N x 2 + 15 = 31

The equation is N x 2 + 15 = 31 or 2N + 15 = 31

b N x 2 + 15 = 31

N x 2 = 31 - 15

N x 2 = 16

N = 16/2

N = 8





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Formulating linear equations

Exercise

Form an equation for each question and solve it.

1 Jayne buys x kilograms of sugar at 75p per kilogram she pays 5.25. How many

kilograms did she buy?

2 Mr Adams works for x hours per week at 4 per hour. How many hours does he

work if he earns 144?

3 A woman bought Y apples at 8p each and 12 oranges at 15p each. She spent

2.52. How many apples did she buy?

4 John has x sweets, David has five more than John, Paul has twice as many asDavid. They have 51 sweets altogether. How many sweets does John have?

5 Here are some instructions: Start with a number, double the number, then add 3.

What is the start number if the result is 55?

6 The cost of hiring a car is 45 plus 8p per mile. Mrs Johnson hires a car and the

cost is 65. How many miles did she travel?

7 Mrs Shaw walks 2K kilometres and runs 5K kilometres. She travels a total of 56

kilometres. How far did she walk?

8 Mr Davis is four times as old as his daughter. Six years ago he was ten times as

old. How old is Mr Davis now?

9 A number N is chosen. Five times the number minus 4 is equal to three times the

number plus 12. What is the number

10 Paul and Mark worked as waiters. Paul worked for 2H hours and Mark worked for

7H hours. The wage rate was 5 per hour. Mark earned 100 more than Paul.

How much did Paul earn?





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Graphical representation

Question

Complete this table of values and draw the graph of y = -x2

+ 4Note: Sometimes the question states draw the function f(x) = -x2 + 4

AnswerIf the question asks for the function f (x) = -x2 + 4 the table and graph will be the same with f(x) instead

of y. When x = -3 y = -(-3)2 + 4 = -5

-4

4

-3

-3

3

3

-2

-2

2

2

-1

-1

1

1xx

x

x

x

x

y

x -3 -2 -1 0 1 2 3

-5 0 3 4 3 0 -5y

x -3 -2 -1 0 1 2 3

y

-4

-4-5

4

4 5

-3

-3

3

3

-2

-2

2

2

-1

-1

1

1

This is the y axisand the line x=0

This is the x axisand the line y=0

y=-x

y=-x+

3

y=x

y=x

-2

x=

-5

y = 2





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Graphical representation

Exercise

1 You will need graph paper. Draw the x-axis from x = -10 to x = 10

Draw the y-axis from y = -10 to y = 10

Draw the following lines on your graph and label the lines.

a y = 0 b x = 0 c x = 4

d y = 3 e x = -5 f y = -7

g y = x h y = -x i y = x + 3

j y = -x + 4 k y = 1/2 x l y 3 x

2 Complete the following tables of values and then draw the graphs:

a y = x2 - 4

b y = 2 - x2

c y = 1/2 x2 - 6

3 Complete the following tables of values and then draw the graphs of the following

functions:

a f(x) = -2x + 3

b f(x) = 3x2 - 5

c f(x) = x2

+ 3

d f(x) = x2 + 2x + 3 x -3 -2 -1 0 1 2 3

y

x -3 -2 -1 0 1 2 3

y

x -3 -2 -1 0 1 2 3

y

x -3 -2 -1 0 1 2 3

y

x -3 -2 -1 0 1 2 3

y

x -3 -2 -1 0 1 2 3

y

x -3 -2 -1 0 1 2 3

y




2


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2-D Representations of 3-D Shapes

2-D and 3-D shapes

Question Answer

1 cm

1 cm 2 cm

Draw an accurate 2-D netof this cuboid.

Cube

Cuboid

Square basedpyramid

Triangular prism

This net folds to makea cube

This net folds to makea cuboid

This net folds to makea square based pyramid

This net folds to makea triangular prism


KS3 Mathematics C: Level 6 Shape, Space and Measures



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2-D Representation of 3-D shapes

Exercise

1 What 3-D shape will this net form?

2 Which of these nets will fold to form a cube?

3 Draw an accurate 2-D net of this cuboid.

4 Draw an accurate 2-D net of this triangular prism.

2 cm

2 cm2 cm

2 cm

3 cm60 60

60

2 cm

3 cm

4 cm

a b c d





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Properties of quadrilaterals

Example A quadrilateral is a four sided shape. The angles add up to 360.

Opposite sides are paralleland the same length.Opposite angles are equal.

Diagonals bisect each other.Rotational symmetry order 2.

A parallelogram with all anglesequal (ie 90).Rotational symmetry order 2.

A quadrilateral with one pair ofparallel sides.No rotational symmetry.

This is a parallelogram with four equal sides.Diagonals bisect each other.Rotational symmetry

order 2.

A rectangle with all sidesequal length.

Rotational symmetry order 4.

Two pairs of equal length sides adjacentto each other.

Diagonals cross at right angles.One diagonal bisects the other. No rotational symmetry.

Parallelogram

Rectangle Square

Trapezium Kite

Rhombus





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Axes of symmetry

Parallelogram

Usually none.

Trapezium

Usually none.

Square Kite

Rhombus Rectangle





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Exercise

1 Give the name of the quadrilateral best described by the following statements:

a All angles equal, opposite sides parallel

b All sides equal, diagonals bisect each other at right angles

c One pair of parallel sides

d Two pairs of parallel sides

e All angles equal, all sides equal

2 Name each quadrilateral.

3 Find the sizes of the following angles.

a

c

bx 110

62 70

a

b 50

c

30

d

e

50

a b c

d e f





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The Quadrilateral Game

Rules: This is a game for two players. You need two dice. Each player starts with six

rhombus, five kites, four trapeziums, three parallelograms, two rectangles and one

square.

Cut out the shapes at the bottom of the page.

Choose one different number from 2 to 12 for each shape. Suppose you choose 8 for

the kite. Each time you throw an 8 you can get rid of one kite. The first player to get rid

of all of their shapes wins. Use a pencil to complete the table, then you can change

your numbers for the next game.

Example Player A Player B

Player A Player B

Rhombus Rhombus Rhombus Rhombus

Rhombus Rhombus Kite Kite Kite

Kite Kite

Parallelogram

Parallelogram

Parallelogram

Trapezium Trapezium Trapezium

Trapezium

Square

Rectangle Rectangle

Rhombus Rhombus Rhombus Rhombus

Rhombus Rhombus Kite Kite Kite

Kite Kite

Parallelogram

Parallelogram

Parallelogram

Trapezium Trapezium Trapezium

Trapezium

Square

Rectangle Rectangle

Quadrilateral

Rhombus

Kite

Trapezium

Parallelogram

Rectangle

Square

Dice totalQuadrilateral

Rhombus

Kite

Trapezium

Parallelogram

Rectangle

Square

Dice totalQuadrilateral

Rhombus

Kite

Trapezium

Parallelogram

Rectangle

Square

Dice total

5

8

3

11

9

7





30/49

Regular polygons

A regular polygon has all of its sides the same length and all of its angles the same

size.

Questions

1 Find the size of an exterior and an interior angle of a regular octagon.

2 Find the size of an exterior and an interior angle of a regular hexagon.

Answers1

2 This question can be solved using the above method.

An alternative method is to split the shape into triangles.

4 triangles are formed

Therefore the sum of the interior angles is

4 x 180 = 720

6 interior angles = 720

1 interior angle = 120

Interior + exterior = 180

120 + exterior = 180

Exterior = 60

E

EE

E

EI

I

I I

I

I = Interior angles

E = Exterior angles

The sum of the exterior angles of a polygon is 360

Interior angle + exterior angle = 180




An octagon has 8 sides, 8 exterior angles, 8 interior angles

8 exterior angles = 360

Therefore 1 exterior angle = = 45

Interior angle + exterior angle = 180

Interior angle + 45 = 180

Interior angle = 135

II

I

I

I I

I

I

E

E

E

EE

E

E

E

3608


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Intersecting and parallel lines

Intersecting lines

Parallel lines

Look for shapes. Angles at corner of shapes are equal.

Questions

1 Find the missing angles:

2 Find x: 3 Find y:

Answers1 a = 140, b = 40, c = 140, d = 40, e = 140, f = 40, g = 140.

2 It often helps to extend the parallel lines to produce Z shapes. 3 Try adding an extra parallel line.

50

50

20

y = 70

20

70

70110

x = 70

50

20

y110

x

40 abc

d efg

or

or

a + b = 180

b + c = 180

c + d = 180

d + a = 180

Angles on a straight line add up to 180

Vertically opposite angles are equal

a = c

b = d

a bcd





32/49

Regular polygons, intersecting and parallel lines

Exercise

1 Find the size of each exterior and interior angle of the following:

a A regular pentagon

b A regular hexagon

c A regular nonagon (9 sides)

d A regular decagon (10 sides)

e A regular 12 sided polygon

2 How many sides does a regular polygon have if the size of each exterior angle is?

a 45 b 24 c 18

3 How many sides does a regular polygon have if the size of each interior angle is?

a 170 b 168 c 160

4 Find the angles indicated.

60

y

y

z

b

a

45

120

x

x a

110

a b

c d

e f

f g

dea b

c50

c

100

10

x

140





33/49

Circumference and area of a circle, areasand volumes

You must learn these formulae.

Circumference of a circle = 2 x x radius

= x diameter

Area of a circle = x radius x radius

Volume of a cuboid = length x width x height

Volume of a cuboid = 6 cm x 3 cm x 4 cm

= 72 cm3

Note: Area is in units2 eg cm2, m2

Volume is in units3 eg cm3, m3

Questions

1 Find the circumference and area of a circle radius 8 cm.

2 Find the area.

3 This is a diagram of a garden with a lawn and a

path around the edge.

The path is 2 m wide.

Answers1 Circumference = 2 x x r Area = x r x r

= 2 x 3.14 x 8 = 3.14 x 8 x 8

= 50.24 cm = 200.96 cm2

2 Split the shape into three parts.

Area = 32 m2

3 Find the area of the large rectangle = 10 x 16 = 160 m2

Find the area of the small rectangle = 6 x 12 = 72 m2

Take away = 88 m2




Circum

ference

Diameter

Radius

4 cm

6 cm

3 cm

1 m

2 m

3 m

5 m

8 m

Lawn

Path

16 m

10 m

1mx

5m=

5m2

4 m x 3 m= 12 m2 5 m x 3 m

= 15 m2


34/49

Circumference and area of a circle, areaand volumes

Exercise1 Find the circumference and areas of the following circles:

a radius 5 cm b radius 12 m c radius 7 m

d radius 3.2 cm e diameter 12 cm f diameter 20 m

g diameter 9 cm h diameter 4.8 m

2 Find the area of a circular path 2 m wide which goes allthe way around a pond radius 20 m.

3 a Find the distance around this cycle track.

b Find the area of the cycle track

4 This is a diagram of a garden.

a What is the perimeter?

b What is the area?

5 This is a garden. It has a lawn with a path, 3 m wide around the outside. What is the

area of the path?




Path

20 m

80 m

12 m

18 m

5 m

2 m

4 m

4 m

5 m

20 m

12 m


35/49

Enlargement

Questions

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

2 R1 is an enlargement of R.

a What are the coordinates of the centre of enlargement?

b What is the scale factor of the enlargement?

0

2

2

4

4

6

6

8

8

10

10

R

R1

0

1

1

2

2

3

3

4

4

5

5

a b

c





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Enlargement

Answers1

2

0

2

2

4

4

6

6

8

8

10

10

R

R1

Use a ruler to join the corners.The dotted lines cross at (1,2). Therefore the centre ofenlargement is the point (1,2)

Scale factor =Scale factor = = 3

new lengthoriginal length

62

a

b

0

1

1

1

2

2

2

2

3

3

4

4

4

5

6

7

8

5 6 7 8

a b

c

x

a1 b1

c1

Point a

scale factor

Count the distance from the centre of enlargement toeach point

2 along

1 up

4 along

2 upx 2

Point b3 along

1 up

6 along

2 upx 2

Point c 4 along

3 up

8 along

6 upx 2





37/49

Enlargement

Exercise

1 Enlarge L by a scale factor of 2, centre of enlargement (9,18). Label this A.

2 Enlarge L by a scale factor of 3, centre of enlargement (15,16). Label this B.

3 Enlarge L by a scale factor of 4, centre of enlargement (12,10). Label this C.

4 D is an enlargement of L. Find the centre of enlargement and the scale factor.

5 E is an enlargement of L. Find the centre of enlargement and the scale factor.

20

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

4 6 8 10 12 14 16 18 20 22 24 26 28 30

L

D

E





38/49

Collect and record continuous data in frequencytables and frequency diagrams

Continuous data is data which can have any value eg distance between two places,

height of a person. The height of a person can be measured to any degree of accuracy.

A person could be 1.783642 m tall.

Discrete data is data which can only have certain values eg the number of people in a

room can only have whole number values. You cannot have 3.2 people in a room.

If you are asked to collect data you must choose an appropriate method. Usually a

survey or an experiment. You must record your data and then present it in tables,

diagrams and graphs.

Questions

The following are the times taken by 20 people to complete a jigsaw. The times are in

minutes.

8.62, 28.4, 48.13, 30.1, 26.03, 47.42, 36.01, 25.23, 22.6, 29.97, 18.63, 30.00, 42.73,

38.62, 20.01, 19.99, 27.6, 16.32, 8.7, 12.58

a Record the information in a frequency table. Choose suitable equal class intervals.

b Show this information in a frequency diagram.

Answersa

b

0

1

2

3

4

5

6

7

10 20 30 40 50

Time in minutes

Frequency

0 - under 1010 - under 2020 - under 3030 - under 4040 - under 50

0 - 1010 - 2020 - 30

A common error is:

Where would you record 20?In the 10-20 or 20-30?

I II I I II I I I I II I I II I I

24743

MinutesMinutes Tally Frequency


KS3 Mathematics C: Level 6 Handling Data



39/49

Collect and record continuous data in frequencytables and frequency diagrams

Exercise1 The data below shows the height of 20 children in a class. Height is in centimetres.

a Choose four suitable equal class intervals.

b Record the information in a frequency table.

c Show the information in a frequency diagram.

137 148 164 150 136 156 148 139 156 159

168 157 162 154 146 149 153 167 139 140

2 The data below shows the mass of 20 adults. Mass is in kilograms.

a Choose suitable equal class intervals.



80 62 58 72 49 63 74 68 82 63

58 54 72 60 58 71 63 61 59 71

3 The data below shows the distance 12 pupils travel to school. The distance is in

kilometres.

a Choose suitable equal class intervals.



078 032 183 224 168 132

013 124 173 164 113 087

4 Collect data for the following:

a The height of each pupil in your class.

b The mass of each pupil in your class.

c The circumference of each pupils wrist.

5 A task of your own choosing eg the time taken to complete a task.

a Record the information.

b Present the information in a frequency table.






40/49

Constructing pie charts

Question

Thirty people were asked what sort of holiday they would choose. 5 said a mountainresort, 10 said a beach holiday, 7 said an activity holiday and 8 said a cruise. Show this

information in a pie chart.

AnswerThere are 360 in a circle. The pie chart must represent 30 people.

360 30 = 12. Therefore 12 represents 1 person.

How to draw the pie chart

1 Draw a circle.

Draw a line from the centre to the edge.

2 Place the protractor on the circle.

Place the centre of the protractor on the centre of

the circle.

Make sure 0 is on the line.

Measure the angle, 60.

3 Draw a line from the centre to the edge at 60.

Label the sector mountain resort write 60.

4 Move the protractor as shown.

Measure 120.

Draw a line from the centre to the edge.

5 Repeat for 84.

Check the remaining angle is 96.

Label each sector.

Mountain resortBeach holidayActivity holidayCruise

51078

Holiday choice Frequency

601208496

x 12x 12x 12x 12

Angle at the centre of the pie chartMultiply

by 12




0

180

60

60

0

Mountainresort

120

Mountain

resort60120

Beachholiday

84 96

Activityholiday Cruise


41/49

Constructing pie charts

Exercise

1 Draw a pie chart to show this information:




Pupils transport to school Frequency Angle at the centre of the circle

BusCarCycleWalk

15867

Vehicles passing school Frequency Angle at the centre of the circle

CarsVansLorriesBuses

421884

Shoe size34567

Frequency36876

Angle at the centre of the circle

Favourite type of musicPopClassicalCountrySoul

Frequency

288

2529

Angle at the centre of the circle





42/49

Scatter diagrams

A scatter diagram is used to compare two sets of results.

A positive correlation indicates that as one quantity increases so does the other

quantity. The diagram shows that in general taller people are heavier.

A negative correlation indicates that as one quantity increases the other quantity

decreases. The diagram shows that in general the more time a person spends at work,

the less time he spends at home.

No correlation indicates that there is no relationship between the two quantities. The

diagram shows that a pupils house number has no connection with the pupilsclassroom number.

Question

a Describe the type of correlation shown by this

scatter diagram.

b Explain the reason for this correlation.

Answersa Negative correlation.

b Sweets can cause harm to teeth. Therefore in general the more sweets a person eats, the more fillings

will be required.

Height

Mass

This diagram shows a

positive correlation


negative correlation

Timespentatwork

Time spent at home


no correlation

Housenumber

Classroom number

x

xx x

x x

x xx

xx

xx x

xx

xx

x

x

x

x x xx

x

x

xx

xx

x x x

xx

x

x x

xx

x

xx

x

x

x




Sweetseaten

Number of fillings

x

xx

xx

xx

xx

x


43/49

Scatter diagrams

Exercise

1 a Describe the type of correlation shownby this scatter diagram.

b What can you say about pupils marks in

English and Maths?

2 a Describe the type of correlation shown

by this scatter diagram.


3 a Describe the type of correlation shownby this scatter diagram.


4 a Place some crosses on this scatter

diagram to show the type of correlation

you would expect.

b Does this scatter diagram show positive,

negative or no correlation?

c Explain the reason for this correlation.




Englishmark

Maths mark

x

x

x x xx

xx

x

x

xx

xx

xx

xx

x

x

x

x x

x xx

x

x

x x

x

x

xxx x

x xx x

xx

xx

0

100

Height

Last digit of house number

Cigarettessmokedperday

Age at death

Temperature

Ice-creams sold


44/49

Probability

Questions

1 Draw a tree diagram to show all of the possible outcomes when two coins aretossed.

2 a Complete this table to show all of the possible outcomes when throwing two

dice.

1 2 3 4 5 6

1

2

3

4

5

6

b How many different ways can two dice land?

c What is the probability of a double?

3 The probability of a new light bulb not working is 0.03. What is the probability of a

new light bulb working?

Answers1

2 a

b 36 ways

c There are 6 doubles

There are 36 different ways

Probability = 6/36 = 1/6

3 A light bulb can either work or not work total probability is 1.

Probability of working + probability of not working = 1.

? + 0.03 = 1

Probability of working = 1 - 0.03

Probability of working = 0.97




First coin

Second coinHTHT

OutcomesHHHTTHTT

H

T

12

34567

1

23456

23

45678

34

56789

45

678910

56

7891011

67

89101112


45/49

Probability

Exercises

1 In Scotland a jury can find a defendant not guilty, not proven or guilty. Two casesare held. Draw a tree diagram to show all of the possible outcomes.

2 Traffic lights can show red or green. Each day Mrs Sims drives through two sets of

traffic lights. Draw a tree diagram to show the outcome.

3 A drawing pin can land point up or point down. Two drawing pins are dropped onto

the floor. List all of the possible outcomes.

4 David can afford to buy one can of drink and one bag of crisps. List all of his

possible choices.

5 Andrea has two dice. One is six sided and one is four sided.

a Complete this table to show all of

the possible totals.

b How many different ways can the

dice land?

c What is the probability of scoring

i 6, ii 7?

6 This is a bag of counters.

a What is the probability of choosing

a blue counter?

b What is the probability of not

choosing a blue counter?

7 The probability of a car breaking down is 0.02.

What is the probability of it not breaking down?

Explain how you worked out the answer.

Bacon

flavour

crisps

Plain

crisps




1

1

2

3

4

3

8

7

2 3 4 5 6

y

b

b p

p p

yy

b

py

y = yellowb = bluep = pink


46/49

Blocks

Building blocks or cubes will be useful for making the shapes.

a Shape 1 Shape 2 Shape 3

How many blocks for a height of 10, 20, 100?

Try to find a rule for a height of H.

b Now try [draw a results table]


Find a rule for a height of H.

c Now try


Find a rule for a height of H.

d Investigate other shapes such as:

Now try pyramids.

Numberof

blocks1 3 6


KS3 Mathematics C: Level 6 Activity and Investigation


Results table

Height

1

2

3

Blocks

1

3

6

Continue the shapes.Record you results

in a table


47/49

Diamonds

Rules: You will need squared paper or graph paper.

Stage 1 Start with 1 square in the centre of your paper.

Stage 2 Add new squares to the first square.

Each new square must touch the previous square along one side only.

1 This is how the pattern starts. At stage 4 a diamond is produced.

Make a copy of this and fill it in as far as stage 9.

2 Now you are ready to draw some diamond patterns. You will need some graph

paper. Size A4 with 2 mm squares is best.

Use four or eight different colours eg red, blue, yellow, green. Colour each stage

eg stage 1 red, 2 blue, 3 yellow, 4 green, 5 red, 6 blue, etc.

4

5

3

2

1 2 3 4 523

4

44

4

4

44

4

45

2

3

4

5

1 1

23

2

2

22 2

2

2

3Correctonly oneside only

WrongTwo sides

are touching





48/49

3 You can make more interesting patterns by changing the colours after each

diamond shape. It is best to use four or eight different colour.

4 Now you must investigate the patterns. Look at what happens for the first few

stages.

Stage Squares

1 1

2 4

3 4

4 12

5 4

6 12

etc

Can you predict the stages when diamond shapes appear?

Can you find rules for each stage?





49/49

The Symmetry Puzzle

Cut out the strips below.

Place them on the picture above so that the picture has rotational symmetry order 4.


key stage 3 - succeed in maths

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