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Term pack

Year 9 Maths

Name: ________________________

Skills in this pack

  • Properties of rational numbers
  • Squares and square roots
  • Cubes and cube roots
  • Algebraic identities
  • Linear equations with the unknown on both sides
  • Factorisation
  • Introduction to graphs
  • Direct and inverse proportion
  • Discount and tax
  • Why constructions work
  • Mensuration
  • Understanding quadrilaterals
  • Mean, median and mode
  • Grouped data, histograms and pie charts
  • Chance and probability

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Year 9 Maths · 1 of 15

Properties of rational numbers

Add, subtract, multiply and divide rational numbers, and use the properties that make the work shorter.

Before you start

Wander along this line in quarter steps. Nothing here is right or wrong — see how the numbers crowd together.

-2.00-1.000.001.002.00

The line only shows quarters, but between any two of them sits another rational number — an eighth, a sixteenth, and so on for ever. There is no 'next' rational number.

  1. 1.5/7 × 3/4 gives the same answer as 3/4 × 5/7. Which property is that?

    • a) Multiplication of rational numbers is associative
    • b) One is the multiplicative identity
    • c) Multiplication of rational numbers is commutative
    • d) Multiplication is distributive over addition

  2. 2.What is the reciprocal of -7/9? Write your answer as a fraction.

  3. 3.2/3 × 9/8 = ? Write your answer as a fraction in its simplest form.

  4. 4.Sort each statement by whether it is true for every choice of rational numbers a, b and c.

    Groups: Always true · Not always true

    • (a - b) - c = a - (b - c)
    • a - b = b - a
    • a + b = b + a
    • a ÷ b = b ÷ a
    • (a + b) + c = a + (b + c)
    • a × b = b × a

  5. 5.Write the rational number that lies exactly halfway between 1/3 and 1/2.

  6. 6.4/9 × 7/5 + 4/9 × 3/5 = ? Write your answer as a fraction.

  7. 7.Match each rational number to its additive inverse — the number you add to it to get zero.

    • 2/7
    • -5/9
    • 3/4
    • -11/6
    • 8/3
    • -2/7
    • -3/4
    • -8/3
    • 5/9
    • 11/6

  8. 8.-3/5 + 3/5 = ?

Properties of rational numbers · Year 9 Maths · www.arenapublications.com/learn

Year 9 Maths · 2 of 15

Squares and square roots

Square whole numbers and find the square roots of perfect squares.

Before you start

Every whole number from zero to twenty-five sits on this line. Find 1, 4, 9, 16 and 25 on it, and look at the gaps between them.

0510152025

The gaps go 3, then 5, then 7, then 9 — the perfect squares spread out as you go, which is why so few numbers have a whole square root.

  1. 1.√225 = ?

  2. 2.Which of these is a perfect square?

    • a) 200
    • b) 250
    • c) 196
    • d) 150

  3. 3.√144 = ?

  4. 4.Match each number to its square root.

    • 36
    • 100
    • 121
    • 400
    • 625
    • 10
    • 20
    • 25
    • 11
    • 6

  5. 5.17² = ?

  6. 6.Sort each number by whether it is a perfect square.

    Groups: Perfect square · Not a perfect square

    • 49
    • 81
    • 64
    • 50
    • 99
    • 70

  7. 7.A square garden has an area of 169 square metres. How long is each side, in metres?

  8. 8.√0.25 = ?

Squares and square roots · Year 9 Maths · www.arenapublications.com/learn

Year 9 Maths · 3 of 15

Cubes and cube roots

Cube a number, recognise a perfect cube, and find a cube root by prime factorisation.

  1. 1.Which of these is a perfect cube?

    • a) 512
    • b) 500
    • c) 1200
    • d) 600

  2. 2.What is 12 cubed?

  3. 3.What happens when a negative number is cubed?

    • a) A negative number cannot be cubed
    • b) It becomes zero
    • c) It turns positive
    • d) It stays negative

  4. 4.What is 7 cubed?

  5. 5.Sort each number by whether it is a perfect cube.

    Groups: A perfect cube · Not a perfect cube

    • 125
    • 500
    • 343
    • 100
    • 300
    • 216
    • 64

  6. 6.What is the cube root of 3375?

  7. 7.392 is 2 × 2 × 2 × 7 × 7. What is the smallest number, apart from zero, it must be multiplied by to make a perfect cube?

  8. 8.What is the cube root of -125?

Cubes and cube roots · Year 9 Maths · www.arenapublications.com/learn

Year 9 Maths · 4 of 15

Algebraic identities

Use (a + b)², (a - b)² and a² - b² to expand brackets and to multiply quickly.

  1. 1.Factorise a² - b².

    • a) (a + b)(a - b)
    • b) a(a - b)
    • c) (a - b)(a - b)
    • d) (a + b)(a + b)

  2. 2.Expand (a - b)².

    • a) a² + b²
    • b) a² - b²
    • c) a² + 2ab + b²
    • d) a² - 2ab + b²

  3. 3.Use (50 + 1)² to work out 51².

  4. 4.Use (100 + 2)² to work out 102².

  5. 5.Use (100 - 2)² to work out 98².

  6. 6.Match each product to its expansion.

    • (x + 1)²
    • (x - 1)²
    • (x + 2)(x - 2)
    • (x + 4)²
    • x² + 2x + 1
    • x² + 8x + 16
    • x² - 2x + 1
    • x² - 4

  7. 7.If a + b = 10 and ab = 21, what is a² + b²?

  8. 8.Expand (x + 3)².

    • a) x² + 6x + 6
    • b) x² + 3x + 9
    • c) x² + 6x + 9
    • d) x² + 9

Algebraic identities · Year 9 Maths · www.arenapublications.com/learn

Year 9 Maths · 5 of 15

Linear equations with the unknown on both sides

Solve equations that have the unknown on both sides, and use them to answer word problems.

Before you start

The scale is level, and there is a mystery box on each side. Move the weights about and find out what keeps it level and what tips it over.

37

Lift one box off each side and the beam does not move at all. That is what “take the unknown off both sides” means, and it is why the boxes on the two sides can be gathered into one.

  1. 1.Solve 4(x - 3) = 2x + 6. x = ?

  2. 2.Solve 5x - 4 = 3x + 10. x = ?

  3. 3.Solve (x + 4)/3 = 2. x = ?

  4. 4.Solve 7 - 2x = x + 1. x = ?

  5. 5.A rectangle is 3 cm longer than it is wide, and its perimeter is 26 cm. How wide is it, in cm? Give just the number.

  6. 6.Which value of x makes 6x - 5 = 4x + 3 true?

    • a) 1
    • b) -4
    • c) 2
    • d) 4

  7. 7.Put these equations in order by their solution, smallest x first.

    • 3x = x + 6
    • 5x + 2 = 3x + 18
    • x - 4 = 6
    • 2x + 1 = x + 2
    • 4x - 5 = 2x + 5

  8. 8.Solve x/5 + 2 = 5. x = ?

Linear equations with the unknown on both sides · Year 9 Maths · www.arenapublications.com/learn

Year 9 Maths · 6 of 15

Factorisation

Break an algebraic expression into factors, and use those factors to divide one expression by another.

  1. 1.Which of these is 4x² - 9 written in factorised form?

    • a) (2x - 9)(2x + 1)
    • b) (4x - 3)(x + 3)
    • c) (2x - 3)(2x - 3)
    • d) (2x - 3)(2x + 3)

  2. 2.x² - 49 factorises as (x - 7)(x + ?). What number belongs in place of the question mark?

  3. 3.Divide x² + 5x + 6 by x + 2. Write the answer.

  4. 4.Put these expressions in order by the highest common factor of their terms, smallest first.

    • 6x + 18
    • 3x + 9
    • 14x + 21
    • 10x + 25
    • 4x + 8

  5. 5.Which of these is x² + 10x + 25 written in factorised form?

    • a) (x + 25)(x + 1)
    • b) (x + 5)²
    • c) (x + 10)²
    • d) (x + 5)(x - 5)

  6. 6.What is the highest common factor of the two terms in 12x + 18?

  7. 7.Take a out of a² + ab as a common factor. What is left inside the bracket?

  8. 8.Divide 15x⁴ by 5x². Write the answer.

Factorisation · Year 9 Maths · www.arenapublications.com/learn

Year 9 Maths · 7 of 15

Introduction to graphs

Plot and read points on a grid, read a line graph, and work out values from a simple linear relation.

Before you start

This graph follows one journey from start to finish, with time along the bottom and distance up the side. Find out what each stretch of the line says about it.

1234
  1. 1. Rising — moving away
  2. 2. Flat — standing still
  3. 3. Steeper — faster
  4. 4. The origin

Flat does not mean time has stopped — the clock still runs along the bottom. It means the distance stopped changing, and standing still is exactly what that looks like on a graph.

  1. 1.What is the x-coordinate of the point (5, 3)?

  2. 2.On a distance-time graph, what does a flat horizontal part of the line mean?

    • a) The object is going backwards
    • b) The object is speeding up
    • c) The object is not moving
    • d) The graph has been drawn wrongly

  3. 3.A line graph shows a plant 4 cm tall on day 1 and 16 cm tall on day 7. How much did it grow over those six days, in cm? Give just the number.

  4. 4.Which of these points lies on the x-axis?

    • a) (6, 6)
    • b) (-6, 6)
    • c) (6, 0)
    • d) (0, 6)

  5. 5.For the relation y = 3x, what is y when x is 6?

  6. 6.This distance-time graph shows one journey in three parts. Tap the part where the car was not moving.

    Write the number of the part.

    123

  7. 7.For the relation y = 2x + 1, what is y when x is 5?

  8. 8.A shop sells notebooks at £20 each. On a graph of cost against number bought, what is the cost of 7 notebooks? Give just the number.

Introduction to graphs · Year 9 Maths · www.arenapublications.com/learn

Year 9 Maths · 8 of 15

Direct and inverse proportion

Tell direct proportion from inverse proportion, and use each to find a missing quantity.

  1. 1.If 5 pens cost £60, what do 8 pens cost? Give just the number.

  2. 2.12 pipes fill a tank in 20 minutes. How many minutes would 15 pipes take?

  3. 3.A job takes 6 people 12 days. Put these teams in order by how long they would take, shortest first.

    • 4 people
    • 18 people
    • 9 people
    • 12 people
    • 6 people

  4. 4.6 workers build a wall in 12 days. How many days would 8 workers take, working at the same rate?

  5. 5.y is in inverse proportion to x. When x is 4, y is 15. What is y when x is 10?

  6. 6.Two quantities are in inverse proportion. Which of these stays the same as they change?

    • a) Their product
    • b) Their difference
    • c) Their ratio
    • d) Their sum

  7. 7.A car travels 180 km in 3 hours at a steady speed. How far does it go in 5 hours, in km? Give just the number.

  8. 8.6 workers take 12 days to finish a job. Match each number of workers to the days they would take at the same rate.

    • 4 workers
    • 8 workers
    • 9 workers
    • 12 workers
    • 18 workers
    • 9 days
    • 6 days
    • 4 days
    • 8 days
    • 18 days

Direct and inverse proportion · Year 9 Maths · www.arenapublications.com/learn

Year 9 Maths · 9 of 15

Discount and tax

Work out a discount, a sale price, and the price once tax has been added.

  1. 1.A bag is marked £1200 and is sold at a discount of 25 per cent. What is the sale price? Give just the number.

  2. 2.A toy costs £600 before tax. With tax of 5 per cent added, what is the total to pay? Give just the number.

  3. 3.A shirt is marked £800 and is sold at a discount of 15 per cent. How much is taken off the marked price? Give just the number.

  4. 4.Which of these offers takes the most off an item marked £1000?

    • a) A discount of 30 per cent
    • b) 10 per cent off, and then 10 per cent off again
    • c) £250 off
    • d) A discount of one fifth

  5. 5.Match each marked price to its sale price after a discount of 20 per cent.

    • £150
    • £250
    • £400
    • £600
    • £900
    • £720
    • £320
    • £200
    • £120
    • £480

  6. 6.A shop takes 10 per cent off an item marked £500, then adds tax of 10 per cent to the reduced price. What is the final price? Give just the number.

  7. 7.A book costs £250 and tax of 8 per cent is added. How much is the tax? Give just the number.

  8. 8.Each offer is on an item marked £2000. Put them in order by what you end up paying, cheapest first.

    • A quarter off
    • Half price
    • A discount of 10 per cent
    • A discount of 35 per cent
    • £600 off

Discount and tax · Year 9 Maths · www.arenapublications.com/learn

Year 9 Maths · 10 of 15

Why constructions work

Explain what each compass step guarantees, and decide when a triangle can be constructed at all.

  1. 1.Put these steps in order to build the perpendicular bisector of a segment AB.

    • Join the two crossing points with a straight line.
    • Open the compasses to more than half of AB.
    • Keeping that same opening, put the point on B and draw two more arcs, crossing the first two.
    • With the point on A, draw an arc above AB and another below it.

  2. 2.An angle of 60° is bisected, and one of the halves is then bisected again. How big is the smallest angle now, in degrees?

  3. 3.You draw two arcs of the same radius, one from each end of a segment. What is true of the point where they cross?

    • a) It is one radius from the mid-point
    • b) It is the mid-point of the segment
    • c) It lies on the segment itself
    • d) It is the same distance from both ends

  4. 4.A triangle is to be built on a base of 8 cm, with the other two sides adding up to 12 cm. What would its perimeter be, in centimetres?

  5. 5.Sort each set of measurements by whether it pins down exactly one triangle.

    Groups: Fixes exactly one triangle · Does not fix one triangle

    • One side on its own
    • Three angles: 40°, 60° and 80°
    • Three angles: 60°, 60° and 60°
    • Three sides: 5, 6 and 7
    • Two sides and the angle between them
    • Two angles and the side between them

  6. 6.Match each set of compass moves to what it produces.

    • Equal arcs above and below, swung from each end of a segment
    • Equal arcs swung from two marks on the arms of an angle
    • An arc of the same radius stepped once round from a point
    • A perpendicular raised on a line, then bisected
    • The perpendicular bisector of that segment
    • An angle of 60°
    • An angle of 45°
    • The bisector of that angle

  7. 7.Swing an arc from a point, then from where it cuts the arm swing another arc with the compasses opened exactly as before, and join up. All three lengths come out equal, so the triangle is equilateral. What angle have you just built, in degrees?

  8. 8.A triangle can be built only if any two of its sides add up to more than the third. Two sides are 6 and 9 cm. What is the largest whole number the third side could be, in centimetres?

Why constructions work · Year 9 Maths · www.arenapublications.com/learn

Year 9 Maths · 11 of 15

Mensuration

Find areas of trapeziums and circles, and volumes and surface areas of cuboids and cylinders.

  1. 1.A cube has edges of 5 cm. What is its total surface area, in square centimetres?

  2. 2.A cylinder has a radius of 7 cm and a height of 10 cm. Take π = 22/7. What is its total surface area, in square centimetres?

  3. 3.Which of these gives the volume of a cylinder?

    • a) 2 × π × r
    • b) π × r × r
    • c) π × r × r × h
    • d) 2 × π × r × h

  4. 4.A cuboid is 12 cm long, 8 cm wide and 5 cm tall. What is its total surface area, in square centimetres?

  5. 5.A trapezium has parallel sides of 10 cm and 6 cm, and a height of 4 cm. What is its area, in square centimetres?

  6. 6.A cylinder has a radius of 7 cm and a height of 10 cm. Take π = 22/7. What is its volume, in cubic centimetres?

  7. 7.A circle has a radius of 14 cm. Take π = 22/7. What is its circumference, in centimetres?

  8. 8.A circle has a radius of 14 cm. Take π = 22/7. What is its area, in square centimetres?

Mensuration · Year 9 Maths · www.arenapublications.com/learn

Year 9 Maths · 12 of 15

Understanding quadrilaterals

Work out the angles in a polygon and use the properties of parallelograms, rhombuses, rectangles and squares.

Before you start

The four corners of this parallelogram hold four angles. Find out how big each one is — nothing here is right or wrong.

1234
  1. 1. Angle A — 45°
  2. 2. Angle B — 135°
  3. 3. Angle C — 45°
  4. 4. Angle D — 135°

Opposite corners match. Any two corners next to each other add up to 180°, which is why one known angle gives you all four.

  1. 1.Put these polygons in order by what their interior angles add up to, smallest first.

    • Octagon
    • Hexagon
    • Pentagon
    • Quadrilateral
    • Triangle

  2. 2.Each exterior angle of a regular polygon is 45 degrees. How many sides has it?

  3. 3.Which of these is true of every rhombus but not of every rectangle?

    • a) The angles add up to 360 degrees
    • b) Opposite sides are parallel
    • c) All four sides are the same length
    • d) Opposite angles are equal

  4. 4.What do the interior angles of a pentagon add up to, in degrees? Give just the number.

  5. 5.One angle of a parallelogram is 65 degrees. What is the angle next to it, in degrees? Give just the number.

  6. 6.What do the interior angles of a hexagon add up to, in degrees? Give just the number.

  7. 7.Sort each property by whether it is true of every parallelogram.

    Groups: True of every parallelogram · Not true of every parallelogram

    • All four sides are equal
    • The diagonals are equal in length
    • Opposite angles are equal
    • Opposite sides are equal
    • Every angle is a right angle
    • The diagonals cut each other in half

  8. 8.Three angles of a quadrilateral are 80, 100 and 65 degrees. What is the fourth, in degrees? Give just the number.

Understanding quadrilaterals · Year 9 Maths · www.arenapublications.com/learn

Year 9 Maths · 13 of 15

Mean, median and mode

Find the mean, median, mode and range of a small set of numbers.

  1. 1.Find the median of 3, 7, 9, 11 and 15.

  2. 2.Find the mode of 5, 8, 8, 2, 9, 8 and 1.

  3. 3.Find the mean of 4, 8, 6, 10 and 2.

  4. 4.Find the median of 12, 5, 9, 3 and 7.

  5. 5.Which average is the middle value once the data has been put in order?

    • a) The median
    • b) The mean
    • c) The mode
    • d) The range

  6. 6.Find the mode of 2, 3, 3, 5, 7 and 3.

  7. 7.The mean of four numbers is 12. What do the four numbers add up to?

  8. 8.Match each word to what it tells you about a set of numbers.

    • Mean
    • Median
    • Mode
    • Range
    • The middle value, once in order
    • The total shared out equally
    • The value that turns up most often
    • The gap between the largest and the smallest

Mean, median and mode · Year 9 Maths · www.arenapublications.com/learn

Year 9 Maths · 14 of 15

Grouped data, histograms and pie charts

Group data into class intervals, read a histogram, and work out the angles in a pie chart.

Before you start

This pie chart shows what a class of 40 children picked for their club. Find out what each slice is and how wide its angle is — nothing here is right or wrong.

1234
  1. 1. Reading — 90°, so 10 children
  2. 2. Sport — 90°, so 10 children
  3. 3. Music — 108°, so 12 children
  4. 4. Art — 72°, so 8 children

The four angles add up to 360°, one whole turn. Music is the biggest slice because 12 out of 40 is the biggest share — and 12/40 of 360° really is 108°.

  1. 1.Put these pie-chart sectors in order by the size of their angle, smallest first.

    • 1/3 of the whole
    • 1/8 of the whole
    • 1/12 of the whole
    • 1/4 of the whole
    • 1/6 of the whole

  2. 2.A pie chart shows 200 people. One sector measures 72 degrees. How many people does that sector stand for?

  3. 3.What is the main difference between a bar graph and a histogram?

    • a) A histogram has no gaps between its bars, because its classes run on from each other
    • b) A bar graph cannot show how many there are of anything
    • c) A histogram always has more bars than a bar graph
    • d) A histogram has to be drawn in colour

  4. 4.A class interval runs from 15 to 25. What is its midpoint?

  5. 5.A class interval runs from 20 to 30. What is its class size?

  6. 6.These marks are grouped in intervals of 10 starting at 0: 5, 12, 18, 23, 27, 31, 34, 38, 45. How many marks fall in the interval 30 to 40?

  7. 7.Match each share of the whole to the angle of its sector in a pie chart, in degrees.

    • Half
    • A quarter
    • A third
    • A fifth
    • A tenth
    • 120
    • 36
    • 72
    • 180
    • 90

  8. 8.A pie chart is split between 4 subjects with an equal share each. What angle does each sector take, in degrees? Give just the number.

Grouped data, histograms and pie charts · Year 9 Maths · www.arenapublications.com/learn

Year 9 Maths · 15 of 15

Chance and probability

Say how likely an event is, and write its probability as a fraction.

Before you start

Every chance there is sits somewhere on this line, from impossible at one end to certain at the other. Move along it.

0.00.51.0

0.5 in the middle is an even chance — as likely to happen as not. To the left of it things are unlikely and to the right they are likely, and nothing at all lives past the two ends.

  1. 1.A bag holds 5 red balls and 3 blue balls. One is drawn without looking. What is the probability that it is red? Write your answer as a fraction.

  2. 2.A fair die numbered 1 to 6 is rolled. What is the probability of getting a 4? Write your answer as a fraction.

  3. 3.Sort each event by how likely it is.

    Groups: Impossible · Possible but not certain · Certain

    • Rolling a 7 on a die numbered 1 to 6
    • Drawing a blue ball from a bag of only red balls
    • Drawing a red ball from a bag of only red balls
    • Rolling a number below 7 on a die numbered 1 to 6
    • Tossing a fair coin and getting a tail
    • Rolling an even number on a die numbered 1 to 6

  4. 4.The probability that it rains tomorrow is 3/10. What is the probability that it does not rain? Write your answer as a fraction.

  5. 5.A fair die numbered 1 to 6 is rolled. What is the probability of getting an even number? Write your answer as a fraction.

  6. 6.An event that is certain to happen has which probability?

    • a) 1/2
    • b) 0
    • c) 100
    • d) 1

  7. 7.A fair die numbered 1 to 6 is rolled. Which of these is impossible?

    • a) Getting a 6
    • b) Getting a number less than 3
    • c) Getting an odd number
    • d) Getting a 7

  8. 8.A fair coin is tossed. What is the probability of getting a head? Write your answer as a fraction.

Chance and probability · Year 9 Maths · www.arenapublications.com/learn

Answer keys — Year 9 Maths

In the same order as the worksheets.

Properties of rational numbers

  1. 1. c) Multiplication of rational numbers is commutative
  2. 2. -9/7
  3. 3. 3/4
  4. 4. a + b = b + a → Always true; a - b = b - a → Not always true; a × b = b × a → Always true; a ÷ b = b ÷ a → Not always true; (a + b) + c = a + (b + c) → Always true; (a - b) - c = a - (b - c) → Not always true
  5. 5. 5/12
  6. 6. 8/9
  7. 7. 2/7 → -2/7; -5/9 → 5/9; 3/4 → -3/4; -11/6 → 11/6; 8/3 → -8/3
  8. 8. 0

Squares and square roots

  1. 1. 15
  2. 2. c) 196
  3. 3. 12
  4. 4. 36 → 6; 100 → 10; 121 → 11; 400 → 20; 625 → 25
  5. 5. 289
  6. 6. 49 → Perfect square; 64 → Perfect square; 81 → Perfect square; 50 → Not a perfect square; 70 → Not a perfect square; 99 → Not a perfect square
  7. 7. 13
  8. 8. 0.5

Cubes and cube roots

  1. 1. a) 512
  2. 2. 1728
  3. 3. d) It stays negative
  4. 4. 343
  5. 5. 64 → A perfect cube; 100 → Not a perfect cube; 125 → A perfect cube; 216 → A perfect cube; 300 → Not a perfect cube; 343 → A perfect cube; 500 → Not a perfect cube
  6. 6. 15
  7. 7. 7
  8. 8. -5

Algebraic identities

  1. 1. a) (a + b)(a - b)
  2. 2. d) a² - 2ab + b²
  3. 3. 2601
  4. 4. 10404
  5. 5. 9604
  6. 6. (x + 1)² → x² + 2x + 1; (x - 1)² → x² - 2x + 1; (x + 2)(x - 2) → x² - 4; (x + 4)² → x² + 8x + 16
  7. 7. 58
  8. 8. c) x² + 6x + 9

Linear equations with the unknown on both sides

  1. 1. 9
  2. 2. 7
  3. 3. 2
  4. 4. 2
  5. 5. 5
  6. 6. d) 4
  7. 7. 1. 2x + 1 = x + 2 2. 3x = x + 6 3. 4x - 5 = 2x + 5 4. 5x + 2 = 3x + 18 5. x - 4 = 6
  8. 8. 15

Factorisation

  1. 1. d) (2x - 3)(2x + 3)
  2. 2. 7
  3. 3. x + 3
  4. 4. 1. 3x + 9 2. 4x + 8 3. 10x + 25 4. 6x + 18 5. 14x + 21
  5. 5. b) (x + 5)²
  6. 6. 6
  7. 7. a + b
  8. 8. 3x²

Introduction to graphs

  1. 1. 5
  2. 2. c) The object is not moving
  3. 3. 12
  4. 4. c) (6, 0)
  5. 5. 18
  6. 6. 2 — Stopped by the roadside
  7. 7. 11
  8. 8. 140

Direct and inverse proportion

  1. 1. 96
  2. 2. 16
  3. 3. 1. 18 people 2. 12 people 3. 9 people 4. 6 people 5. 4 people
  4. 4. 9
  5. 5. 6
  6. 6. a) Their product
  7. 7. 300
  8. 8. 4 workers → 18 days; 8 workers → 9 days; 9 workers → 8 days; 12 workers → 6 days; 18 workers → 4 days

Discount and tax

  1. 1. 900
  2. 2. 630
  3. 3. 120
  4. 4. a) A discount of 30 per cent
  5. 5. £150 → £120; £250 → £200; £400 → £320; £600 → £480; £900 → £720
  6. 6. 495
  7. 7. 20
  8. 8. 1. Half price 2. A discount of 35 per cent 3. £600 off 4. A quarter off 5. A discount of 10 per cent

Why constructions work

  1. 1. 1. Open the compasses to more than half of AB. 2. With the point on A, draw an arc above AB and another below it. 3. Keeping that same opening, put the point on B and draw two more arcs, crossing the first two. 4. Join the two crossing points with a straight line.
  2. 2. 15
  3. 3. d) It is the same distance from both ends
  4. 4. 20
  5. 5. Three sides: 5, 6 and 7 → Fixes exactly one triangle; Two sides and the angle between them → Fixes exactly one triangle; Two angles and the side between them → Fixes exactly one triangle; Three angles: 60°, 60° and 60° → Does not fix one triangle; Three angles: 40°, 60° and 80° → Does not fix one triangle; One side on its own → Does not fix one triangle
  6. 6. Equal arcs above and below, swung from each end of a segment → The perpendicular bisector of that segment; Equal arcs swung from two marks on the arms of an angle → The bisector of that angle; An arc of the same radius stepped once round from a point → An angle of 60°; A perpendicular raised on a line, then bisected → An angle of 45°
  7. 7. 60
  8. 8. 14

Mensuration

  1. 1. 150
  2. 2. 748
  3. 3. c) π × r × r × h
  4. 4. 392
  5. 5. 32
  6. 6. 1540
  7. 7. 88
  8. 8. 616

Understanding quadrilaterals

  1. 1. 1. Triangle 2. Quadrilateral 3. Pentagon 4. Hexagon 5. Octagon
  2. 2. 8
  3. 3. c) All four sides are the same length
  4. 4. 540
  5. 5. 115
  6. 6. 720
  7. 7. Opposite sides are equal → True of every parallelogram; Opposite angles are equal → True of every parallelogram; The diagonals cut each other in half → True of every parallelogram; All four sides are equal → Not true of every parallelogram; The diagonals are equal in length → Not true of every parallelogram; Every angle is a right angle → Not true of every parallelogram
  8. 8. 115

Mean, median and mode

  1. 1. 9
  2. 2. 8
  3. 3. 6
  4. 4. 7
  5. 5. a) The median
  6. 6. 3
  7. 7. 48
  8. 8. Mean → The total shared out equally; Median → The middle value, once in order; Mode → The value that turns up most often; Range → The gap between the largest and the smallest

Grouped data, histograms and pie charts

  1. 1. 1. 1/12 of the whole 2. 1/8 of the whole 3. 1/6 of the whole 4. 1/4 of the whole 5. 1/3 of the whole
  2. 2. 40
  3. 3. a) A histogram has no gaps between its bars, because its classes run on from each other
  4. 4. 20
  5. 5. 10
  6. 6. 3
  7. 7. Half → 180; A quarter → 90; A third → 120; A fifth → 72; A tenth → 36
  8. 8. 90

Chance and probability

  1. 1. 5/8
  2. 2. 1/6
  3. 3. Rolling a 7 on a die numbered 1 to 6 → Impossible; Tossing a fair coin and getting a tail → Possible but not certain; Rolling a number below 7 on a die numbered 1 to 6 → Certain; Drawing a red ball from a bag of only red balls → Certain; Drawing a blue ball from a bag of only red balls → Impossible; Rolling an even number on a die numbered 1 to 6 → Possible but not certain
  4. 4. 7/10
  5. 5. 1/2
  6. 6. d) 1
  7. 7. d) Getting a 7
  8. 8. 1/2

Year 9 Maths · www.arenapublications.com/learn