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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.
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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
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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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Calculating fractions and percentages
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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Calculating fractions and percentages
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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Calculating fractions and percentages
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
How much of each ingredient should you use for:
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
How much of each ingredient should you use for:
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
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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
14 Multiply the term by 4 then add 7
15 Multiply the term by 2 then add 1
16 Multiply the term by 6 then subtract 1
17 Multiply the term by 3 then subtract 4
18 Multiply the term by 2 then subtract 10
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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
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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
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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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Properties of quadrilaterals
Axes of symmetry
Parallelogram
Usually none.
Trapezium
Usually none.
Square Kite
Rhombus Rectangle
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Properties of quadrilaterals
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
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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
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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
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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
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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
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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
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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?
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Path
20 m
80 m
12 m
18 m
5 m
2 m
4 m
4 m
5 m
20 m
12 m
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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
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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
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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
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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.
b Record the information in a frequency table.
c Show the information in a frequency diagram.
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.
b Record the information in a frequency table.
c Show the information in a frequency diagram.
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.
c Show the information in a frequency diagram.
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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
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0
180
60
60
0
Mountainresort
120
Mountain
resort60120
Beachholiday
84 96
Activityholiday Cruise
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Constructing pie charts
Exercise
1 Draw a pie chart to show this information:
2 Draw a pie chart to show this information:
3 Draw a pie chart to show this information:
4 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
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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
This diagram shows a
negative correlation
Timespentatwork
Time spent at home
This diagram shows a
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
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Sweetseaten
Number of fillings
x
xx
xx
xx
xx
x
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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.
b Explain the reason for this correlation.
3 a Describe the type of correlation shownby this scatter diagram.
b Explain the reason for this correlation.
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.
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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
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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
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First coin
Second coinHTHT
OutcomesHHHTTHTT
H
T
12
34567
1
23456
23
45678
34
56789
45
678910
56
7891011
67
89101112
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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
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KS3 Mathematics C: Level 6 Handling Data
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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
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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]
How many blocks for a height of 16, 30, 200?
Find a rule for a height of H.
c Now try
How many blocks for a height of 10, 15, 100?
Find a rule for a height of H.
d Investigate other shapes such as:
Now try pyramids.
Numberof
blocks1 3 6
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Results table
Height
1
2
3
Blocks
1
3
6
Continue the shapes.Record you results
in a table
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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
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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?
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The Symmetry Puzzle
Cut out the strips below.
Place them on the picture above so that the picture has rotational symmetry order 4.
KS3 Mathematics C: Level 6 Activity and Investigation
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