← Back to Grade 8 Math

Term pack

Grade 8 Math

Name: ________________________

Skills in this pack

  • Direct and inverse proportion
  • Exponents and powers
  • Squares and square roots
  • Cubes and cube roots
  • Linear equations with the unknown on both sides
  • Pair of linear equations
  • Introduction to graphs
  • Properties of triangles
  • Surface area and volume
  • Congruence

Free at www.arenapublications.com/learn — no account, no ads, ever.

Grade 8 Math · 1 of 10

Direct and inverse proportion

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

  1. 1.6 workers build a wall in 12 days. How many days would 8 workers take, working at the same rate?

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

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

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

  4. 4.Sort each pair of quantities by how they vary together.

    Groups: Direct proportion · Inverse proportion

    • The amount of fuel bought and the price paid
    • The side of a square and its perimeter
    • Number of taps and the time to fill a tank
    • Speed and the time taken for a fixed journey
    • Number of identical tickets and the total cost
    • Number of workers and the days to finish a job

  5. 5.Which of these pairs is in direct proportion?

    • a) The number of workers and the days needed to finish a job
    • b) The speed of a car and the time it takes for a fixed journey
    • c) The number of identical books bought and the total cost
    • d) The number of taps and the time to fill a tank

  6. 6.6 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
    • 6 days
    • 18 days
    • 9 days
    • 8 days
    • 4 days

  7. 7.A journey takes 4 hours at 60 miles per hour. How many hours would it take at 80 miles per hour?

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

Direct and inverse proportion · Grade 8 Math · www.arenapublications.com/learn

Grade 8 Math · 2 of 10

Exponents and powers

Work out powers and use the laws of exponents to simplify them.

  1. 1.What is 3⁰?

  2. 2.Work out 2⁵ ÷ 2³ as an ordinary number.

  3. 3.What is 5²?

  4. 4.Match each power to its value.

    • 2⁴
    • 3³
    • 5²
    • 10³
    • 7²
    • 27
    • 25
    • 1000
    • 49
    • 16

  5. 5.64 can be written as 2 to some power. What is that power?

  6. 6.Simplify (2³)².

    • a) 2⁵
    • b) 2⁹
    • c) 2¹
    • d) 2⁶

  7. 7.Simplify a⁵ × a³.

    • a) a⁸
    • b) a¹⁵
    • c) 2a⁸
    • d) a²

  8. 8.What is 2³ (two to the power 3)?

Exponents and powers · Grade 8 Math · www.arenapublications.com/learn

Grade 8 Math · 3 of 10

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. 1.13² = ?

  2. 2.√0.25 = ?

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

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

  4. 4.√576 = ?

  5. 5.17² = ?

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

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

    Groups: Perfect square · Not a perfect square

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

  8. 8.√225 = ?

Squares and square roots · Grade 8 Math · www.arenapublications.com/learn

Grade 8 Math · 4 of 10

Cubes and cube roots

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

  1. 1.The prime factors of 5832 are 2 × 2 × 2 × 3 × 3 × 3 × 3 × 3 × 3. What is the cube root of 5832?

  2. 2.Put these in order by value, smallest first.

    • The cube of 3
    • The cube root of 8000
    • The cube root of 8
    • The cube root of 64
    • The cube of 2

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

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

  4. 4.Match each number to its cube.

    • 2
    • 3
    • 5
    • 9
    • 11
    • 125
    • 27
    • 1331
    • 729
    • 8

  5. 5.What is 7 cubed?

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

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

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

  8. 8.What is the cube root of 1000?

Cubes and cube roots · Grade 8 Math · www.arenapublications.com/learn

Grade 8 Math · 5 of 10

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.Match each equation to its solution.

    • 2x + 3 = 11
    • 3x - 1 = 14
    • 5x + 4 = 4x + 10
    • 7x = 4x + 21
    • 2x - 9 = x - 1
    • 7
    • 8
    • 4
    • 6
    • 5

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

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

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

  5. 5.Why may the same number be taken away from both sides of an equation?

    • a) Because taking away always makes an equation simpler
    • b) Because the two sides stay equal if you do the same thing to each
    • c) Because the left side is always the more important one
    • d) Because the unknown must end up on the left

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

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

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

  8. 8.Solve 3x + 5 = 2x + 11. x = ?

Linear equations with the unknown on both sides · Grade 8 Math · www.arenapublications.com/learn

Grade 8 Math · 6 of 10

Pair of linear equations

Solve a pair of linear equations in two variables by substitution and elimination, and say how many solutions a pair has.

Before you start

One scale, and every box on it hides the same weight. Move things about and find out what a single scale can and cannot tell you.

113

A scale can pin down one unknown, never two at once — which is exactly why a pair of equations is needed. Elimination is the step that turns the pair into a single scale like this one.

  1. 1.Two adult tickets and one child ticket cost 900 $. One adult ticket and one child ticket cost 500 $. What does one child ticket cost? Give just the number.

  2. 2.Solve the pair 5x - 2y = 16 and x + 2y = 8. What is x?

  3. 3.For which of these pairs do the two lines cross at exactly one point?

    • a) x + 2y = 5 and 2x + y = 4
    • b) x + y = 5 and 2x + 2y = 10
    • c) 2x + 3y = 6 and 4x + 6y = 5
    • d) x + y = 5 and x + y = 8

  4. 4.Two adult tickets and one child ticket cost 900 $. One adult ticket and one child ticket cost 500 $. What does one adult ticket cost? Give just the number.

  5. 5.Match each pair of equations to the value of x that solves it.

    • x + y = 6 and x - y = 2
    • x + y = 9 and x - y = 1
    • x + y = 12 and x - y = 4
    • x + y = 14 and x - y = 8
    • 4
    • 8
    • 11
    • 5

  6. 6.The sum of two numbers is 25 and their difference is 7. What is the smaller number?

  7. 7.Solve the pair 2x + y = 11 and x - y = 1. What is x?

  8. 8.Solve the pair x + y = 10 and x - y = 4. What is y?

Pair of linear equations · Grade 8 Math · www.arenapublications.com/learn

Grade 8 Math · 7 of 10

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.A point sits 4 units to the right of the origin and 7 units up. Write its y-coordinate.

  2. 2.Sort each point by where it sits on the grid.

    Groups: On the x-axis · On the y-axis · On neither axis

    • (0, -1)
    • (-4, 0)
    • (6, 6)
    • (0, 5)
    • (3, 0)
    • (2, 7)

  3. 3.A car travels at a steady 50 miles per hour. On a distance-time graph, how far has it gone after 3 hours, in miles? Give just the number.

  4. 4.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

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

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

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

  7. 7.For the relation y = 4x - 1, match each value of x to its value of y.

    • x = 1
    • x = 2
    • x = 3
    • x = 5
    • x = 6
    • 11
    • 19
    • 23
    • 7
    • 3

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

Introduction to graphs · Grade 8 Math · www.arenapublications.com/learn

Grade 8 Math · 8 of 10

Properties of triangles

Use the angle sum, the exterior angle rule and Pythagoras' theorem in a triangle.

Before you start

A right-angled triangle, with the corner that makes it one marked. Find out what each of its three sides is called.

1234
  1. 1. A short side
  2. 2. The other short side
  3. 3. Hypotenuse — the longest
  4. 4. The right angle

Square the two short sides, add them, and you land on the square of the hypotenuse — 60² + 80² = 100². Swapping which short side you call which changes nothing, because addition does not care about order.

  1. 1.Each row gives the three side lengths of a triangle. Sort them by whether the triangle is right-angled.

    Groups: Right-angled · Not right-angled

    • 4, 5, 6
    • 9, 12, 15
    • 8, 15, 17
    • 5, 12, 13
    • 2, 3, 4
    • 3, 4, 5

  2. 2.Every angle of an equilateral triangle is the same size. How big is each one, in degrees?

  3. 3.A right-angled triangle has a hypotenuse of 13 inches and one shorter side of 5 inches. How long is the other short side, in inches?

  4. 4.A right-angled triangle has shorter sides of 6 inches and 8 inches. How long is the hypotenuse, in inches?

  5. 5.In an isosceles triangle the two base angles are always...

    • a) right angles
    • b) 60° each
    • c) supplementary
    • d) equal

  6. 6.An isosceles triangle has an apex angle of 40°. How big is each base angle, in degrees?

  7. 7.Can a triangle have sides of 2 inches, 3 inches and 7 inches?

    • a) Only if it is isosceles
    • b) Only if it is right-angled
    • c) Yes
    • d) No — 2 and 3 together are shorter than 7

  8. 8.An exterior angle of a triangle is 120°. One of the two opposite interior angles is 45°. What is the other, in degrees?

Properties of triangles · Grade 8 Math · www.arenapublications.com/learn

Grade 8 Math · 9 of 10

Surface area and volume

Find surface areas and volumes of cones, spheres and hemispheres.

  1. 1.A sphere has a radius of 14 inches. Take π = 22/7. What is its surface area, in square inches?

  2. 2.A hemisphere has a radius of 7 inches. Take π = 22/7. What is its total surface area, in square inches?

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

    • a) (1/3) × π × r × r × h
    • b) 4 × π × r × r
    • c) (4/3) × π × r × r × r
    • d) 2 × π × r × r

  4. 4.A cone and a cylinder have the same radius and the same height. How many coneloads of water would fill the cylinder?

  5. 5.A sphere has a radius of 21 inches. Take π = 22/7. What is its volume, in cubic inches?

  6. 6.A cone has a radius of 7 inches and a height of 24 inches. What is its slant height, in inches?

  7. 7.A cone has a radius of 7 inches and a slant height of 25 inches. Take π = 22/7. What is its curved surface area, in square inches?

  8. 8.A cone has a radius of 7 inches and a height of 12 inches. Take π = 22/7. What is its volume, in cubic inches?

Surface area and volume · Grade 8 Math · www.arenapublications.com/learn

Grade 8 Math · 10 of 10

Congruence

Say when two figures are congruent, read a congruence statement in the right order, and use it to find matching sides and angles.

  1. 1.In the statement 'triangle ABC is congruent to triangle PQR', which side matches AB?

    • a) QR
    • b) PR
    • c) AP
    • d) PQ

  2. 2.Triangle LMN is congruent to triangle XYZ. Match each part of LMN to the part of XYZ that matches it.

    • side LM
    • side MN
    • side LN
    • angle L
    • angle N
    • side XY
    • side XZ
    • angle X
    • side YZ
    • angle Z

  3. 3.Two figures are congruent when they...

    • a) have the same area as each other
    • b) have the same shape, but may be different sizes
    • c) have exactly the same shape and exactly the same size
    • d) have the same number of sides as each other

  4. 4.Triangle ABC is congruent to triangle PQR, with A matching P, B matching Q and C matching R. Tap the side of triangle PQR that matches side AB.

    Write the number of the part.

    123

  5. 5.Sort each pair of figures by whether they have to be congruent.

    Groups: Must be congruent · Need not be congruent

    • Two equilateral triangles with sides of 5 inches
    • Two rectangles with the same perimeter
    • Two rectangles with the same area
    • Two squares with sides of 4 inches
    • Two triangles with the same three angles
    • Two line segments 7 inches long

  6. 6.Triangle ABC is congruent to triangle DEF. BC is 9 inches long. How long is EF, in inches?

  7. 7.Triangle ABC is congruent to triangle DEF. Angle C is 72°. How big is angle F, in degrees?

  8. 8.A square tile is lifted up and put back down turned a quarter turn. Is it congruent to the way it was before?

    • a) Yes — turning a shape changes neither its shape nor its size
    • b) No, because it is facing a different way
    • c) Only if it is turned back first
    • d) Only if it was the right way up to begin with

Congruence · Grade 8 Math · www.arenapublications.com/learn

Answer keys — Grade 8 Math

In the same order as the worksheets.

Direct and inverse proportion

  1. 1. 9
  2. 2. 1. 18 people 2. 12 people 3. 9 people 4. 6 people 5. 4 people
  3. 3. 300
  4. 4. The amount of fuel bought and the price paid → Direct proportion; Speed and the time taken for a fixed journey → Inverse proportion; Number of workers and the days to finish a job → Inverse proportion; Number of identical tickets and the total cost → Direct proportion; Number of taps and the time to fill a tank → Inverse proportion; The side of a square and its perimeter → Direct proportion
  5. 5. c) The number of identical books bought and the total cost
  6. 6. 4 workers → 18 days; 8 workers → 9 days; 9 workers → 8 days; 12 workers → 6 days; 18 workers → 4 days
  7. 7. 3
  8. 8. 96

Exponents and powers

  1. 1. 1
  2. 2. 4
  3. 3. 25
  4. 4. 2⁴ → 16; 3³ → 27; 5² → 25; 10³ → 1000; 7² → 49
  5. 5. 6
  6. 6. d) 2⁶
  7. 7. a) a⁸
  8. 8. 8

Squares and square roots

  1. 1. 169
  2. 2. 0.5
  3. 3. b) 196
  4. 4. 24
  5. 5. 289
  6. 6. 13
  7. 7. 49 → Perfect square; 64 → Perfect square; 81 → Perfect square; 50 → Not a perfect square; 70 → Not a perfect square; 99 → Not a perfect square
  8. 8. 15

Cubes and cube roots

  1. 1. 18
  2. 2. 1. The cube root of 8 2. The cube root of 64 3. The cube of 2 4. The cube root of 8000 5. The cube of 3
  3. 3. b) It stays negative
  4. 4. 2 → 8; 3 → 27; 5 → 125; 9 → 729; 11 → 1331
  5. 5. 343
  6. 6. -5
  7. 7. b) 512
  8. 8. 10

Linear equations with the unknown on both sides

  1. 1. 2x + 3 = 11 → 4; 3x - 1 = 14 → 5; 5x + 4 = 4x + 10 → 6; 7x = 4x + 21 → 7; 2x - 9 = x - 1 → 8
  2. 2. 5
  3. 3. 15
  4. 4. 2
  5. 5. b) Because the two sides stay equal if you do the same thing to each
  6. 6. 1. 2x + 1 = x + 2 2. 3x = x + 6 3. 4x - 5 = 2x + 5 4. 5x + 2 = 3x + 18 5. x - 4 = 6
  7. 7. 9
  8. 8. 6

Pair of linear equations

  1. 1. 100
  2. 2. 4
  3. 3. a) x + 2y = 5 and 2x + y = 4
  4. 4. 400
  5. 5. x + y = 6 and x - y = 2 → 4; x + y = 9 and x - y = 1 → 5; x + y = 12 and x - y = 4 → 8; x + y = 14 and x - y = 8 → 11
  6. 6. 9
  7. 7. 4
  8. 8. 3

Introduction to graphs

  1. 1. 7
  2. 2. (3, 0) → On the x-axis; (0, 5) → On the y-axis; (2, 7) → On neither axis; (-4, 0) → On the x-axis; (0, -1) → On the y-axis; (6, 6) → On neither axis
  3. 3. 150
  4. 4. 2 — Stopped by the roadside
  5. 5. 11
  6. 6. a) The object is not moving
  7. 7. x = 1 → 3; x = 2 → 7; x = 3 → 11; x = 5 → 19; x = 6 → 23
  8. 8. 5

Properties of triangles

  1. 1. 3, 4, 5 → Right-angled; 5, 12, 13 → Right-angled; 9, 12, 15 → Right-angled; 8, 15, 17 → Right-angled; 2, 3, 4 → Not right-angled; 4, 5, 6 → Not right-angled
  2. 2. 60
  3. 3. 12
  4. 4. 10
  5. 5. d) equal
  6. 6. 70
  7. 7. d) No — 2 and 3 together are shorter than 7
  8. 8. 75

Surface area and volume

  1. 1. 2464
  2. 2. 462
  3. 3. c) (4/3) × π × r × r × r
  4. 4. 3
  5. 5. 38808
  6. 6. 25
  7. 7. 550
  8. 8. 616

Congruence

  1. 1. d) PQ
  2. 2. side LM → side XY; side MN → side YZ; side LN → side XZ; angle L → angle X; angle N → angle Z
  3. 3. c) have exactly the same shape and exactly the same size
  4. 4. 1 — PQ
  5. 5. Two squares with sides of 4 inches → Must be congruent; Two rectangles with the same area → Need not be congruent; Two equilateral triangles with sides of 5 inches → Must be congruent; Two triangles with the same three angles → Need not be congruent; Two line segments 7 inches long → Must be congruent; Two rectangles with the same perimeter → Need not be congruent
  6. 6. 9
  7. 7. 72
  8. 8. a) Yes — turning a shape changes neither its shape nor its size

Grade 8 Math · www.arenapublications.com/learn