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Term pack

Year 9 Maths

Name: ________________________

Skills in this pack

  • Number systems
  • Linear equations in two variables
  • Quadratic equations
  • Distance and section formulas
  • Algebraic identities
  • Introduction to trigonometry
  • Similar triangles
  • Probability from experiments

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Year 9 Maths · 1 of 8

Number systems

Tell a rational number from an irrational one, and place any real number on the number line.

Before you start

Every tick on this line is a rational number. Move along it and watch where √2 (about 1.41), √3 (about 1.73) and π (about 3.14) would have to squeeze in.

0.00.51.01.52.02.53.03.54.0

No tick will ever land exactly on √2, however finely you chop the line up. That is what irrational means — not that the number is missing, but that no fraction can name it.

  1. 1.√25 turns out to be rational. Which whole number is it equal to?

  2. 2.Write the recurring decimal 0.777… as a fraction in simplest form.

  3. 3.Write 3/8 as a decimal.

  4. 4.Every square root of a perfect square is rational. Tap where -√4 sits.

    Mark the line with an X.

    -5-4-3-2-1012345

  5. 5.√2 is 1.414 to three decimal places. Tap the tick it is nearest to.

    Mark the line with an X.

    1.01.52.0

  6. 6.Which of these is irrational?

    • a) 0.6
    • b) √36
    • c) 2/9
    • d) √11

  7. 7.Is every whole number also a rational number?

    • a) Only when it is positive
    • b) Only when it is even
    • c) No — whole numbers are a separate kind of number
    • d) Yes — a whole number n can be written as n/1

  8. 8.Match each number to the smallest family it belongs to.

    • 12
    • -7
    • 5/6
    • √11
    • Rational, but not an integer
    • Irrational
    • Whole number
    • Integer, but not a whole number

Number systems · Year 9 Maths · www.arenapublications.com/learn

Year 9 Maths · 2 of 8

Linear equations in two variables

Find solutions of an equation like 2x + y = 7 and read its graph as a straight line.

  1. 1.The graph of every linear equation in two variables is...

    • a) a curve
    • b) a straight line
    • c) a single point
    • d) a circle

  2. 2.Written in the form ax + by + c = 0, the equation 3x = 5 has a = 3 and c = -5. What is b?

  3. 3.In the equation 2x + y = 7, what is y when x = 3?

  4. 4.Each pair of coordinates solves exactly one of these equations. Match them up.

    • (1, 4)
    • (2, 6)
    • (0, 7)
    • (3, 3)
    • x = y
    • x + y = 5
    • y = 3x
    • y = 7

  5. 5.Is (2, 3) a solution of x + y = 5?

    • a) No — it solves x - y = 5 instead
    • b) Yes — 2 + 3 comes to 5
    • c) Only if x and y are swapped over
    • d) No — no pair of numbers can solve it

  6. 6.How many solutions does a linear equation in two variables have?

    • a) Exactly one
    • b) Infinitely many
    • c) Exactly two
    • d) None

  7. 7.Two numbers add up to 12. Calling them x and y, the equation is x + y = c. What is c?

  8. 8.The line 2x + y = 6 crosses the y-axis. Give the y-coordinate of that crossing point.

Linear equations in two variables · Year 9 Maths · www.arenapublications.com/learn

Year 9 Maths · 3 of 8

Quadratic equations

Solve quadratic equations by factorising, and use the discriminant to describe the roots.

  1. 1.For x² - 7x + 10 = 0, what is the sum of the two roots?

  2. 2.Solve x² + 7x + 12 = 0. Give the smaller root.

  3. 3.What is the discriminant, b² - 4ac, of x² - 3x + 5 = 0?

  4. 4.Solve x² - 5x + 6 = 0. Give the smaller root.

  5. 5.For x² - 7x + 10 = 0, what is the product of the two roots?

  6. 6.Solve x² - 4x - 5 = 0. Give the positive root.

  7. 7.Match each quadratic to its factorised form.

    • x² + 5x + 6
    • x² - 5x + 6
    • x² - 4
    • x² + 4x + 4
    • (x + 2)(x + 3)
    • (x + 2)(x + 2)
    • (x + 2)(x - 2)
    • (x - 2)(x - 3)

  8. 8.Solve x² - 5x + 6 = 0. Give the larger root.

Quadratic equations · Year 9 Maths · www.arenapublications.com/learn

Year 9 Maths · 4 of 8

Distance and section formulas

Find the distance between two points, the point that divides a segment in a given ratio, and the area of a triangle from its vertices.

  1. 1.Find the distance between the points (1, 2) and (4, 6). Give the answer in units.

  2. 2.The point P divides the segment joining (1, 2) and (7, 11) in the ratio 2 : 1, measured from (1, 2). What is the x-coordinate of P?

  3. 3.Which of these points is the same distance from the origin as (3, 4)?

    • a) (4, 4)
    • b) (5, 0)
    • c) (2, 3)
    • d) (1, 6)

  4. 4.The point P divides the segment joining (1, 2) and (7, 11) in the ratio 2 : 1, measured from (1, 2). What is the y-coordinate of P?

  5. 5.Find the area of the triangle whose vertices are (2, 1), (8, 1) and (2, 9). Give the answer in square units.

  6. 6.Find the distance between the points (0, 0) and (6, 8). Give the answer in units.

  7. 7.Find the area of the triangle whose vertices are (0, 0), (4, 0) and (0, 6). Give the answer in square units.

  8. 8.Find the midpoint of the segment joining (2, 3) and (8, 11). What is its x-coordinate?

Distance and section formulas · Year 9 Maths · www.arenapublications.com/learn

Year 9 Maths · 5 of 8

Algebraic identities

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

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

  2. 2.Expand (x + 3)².

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

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

  4. 4.Use (50 + 1)² to work out 51².

  5. 5.Expand (x - 5)².

    • a) x² - 10x + 25
    • b) x² - 10x - 25
    • c) x² - 25
    • d) x² + 10x + 25

  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.Expand (a - b)².

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

  8. 8.Use a² - b² = (a + b)(a - b) to work out 43² - 37².

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

Year 9 Maths · 6 of 8

Introduction to trigonometry

Find sine, cosine and tangent in a right-angled triangle and recall the standard angles.

Before you start

One angle of this right-angled triangle has been marked. Find out what that angle and each of the three sides are called.

1234
  1. 1. Opposite
  2. 2. Adjacent
  3. 3. Hypotenuse
  4. 4. The marked angle

Mark the other corner instead and opposite and adjacent swap over. The hypotenuse never moves, because it is fixed by the right angle rather than by the angle you chose.

  1. 1.Sort each ratio by its value.

    Groups: Equals 0 · Equals 1

    • sin 0°
    • tan 45°
    • sin 90°
    • cos 90°
    • cos 0°
    • tan 0°

  2. 2.In a right-angled triangle the side next to an angle is 4 cm and the hypotenuse is 5 cm. What is the cosine of that angle? Give your answer as a fraction.

  3. 3.What is sin 60°?

    • a) 1/√3
    • b) 1/2
    • c) √3
    • d) √3/2

  4. 4.Match each trigonometric ratio to its value.

    • sin 30°
    • cos 30°
    • tan 45°
    • sin 0°
    • tan 60°
    • 0
    • 1/2
    • √3
    • √3/2
    • 1

  5. 5.What is cos 60°? Give your answer as a fraction.

  6. 6.In a right-angled triangle the side opposite an angle is 3 cm and the hypotenuse is 5 cm. What is the sine of that angle? Give your answer as a fraction.

  7. 7.What is tan 30°?

    • a) 1/√3
    • b) √3/2
    • c) √3
    • d) 1/2

  8. 8.What is cos 0°?

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

Year 9 Maths · 7 of 8

Similar triangles

Use equal angles and side ratios in similar triangles, including the ratio of their areas.

  1. 1.Which of these is true of two similar triangles?

    • a) They must be the same size
    • b) They must be right-angled
    • c) Matching sides are in the same ratio
    • d) Matching sides are equal

  2. 2.Two similar triangles have sides in the ratio 2:3. The area of the smaller is 16 square centimetres. What is the area of the larger, in square centimetres?

  3. 3.Two similar triangles have sides in the ratio on the left. Match it to the ratio of their areas.

    • 1:2
    • 1:3
    • 2:3
    • 3:4
    • 1:5
    • 9:16
    • 4:9
    • 1:9
    • 1:25
    • 1:4

  4. 4.Two similar triangles have sides in the ratio 3:5. The perimeter of the smaller is 18 cm. What is the perimeter of the larger, in centimetres?

  5. 5.Triangle ABC is similar to triangle PQR. AB is 4 cm and PQ is 12 cm. BC is 5 cm. How long is QR, in centimetres?

  6. 6.Two triangles have all three pairs of angles equal. The triangles are...

    • a) congruent
    • b) always right-angled
    • c) similar
    • d) unrelated

  7. 7.A 2 m pole casts a shadow 3 m long. At the same moment a tree casts a shadow 18 m long. How tall is the tree, in metres?

  8. 8.Sort each pair of shapes by whether they must always be similar.

    Groups: Always similar · Not always similar

    • Two rectangles
    • Two isosceles triangles
    • Two rhombuses
    • Two equilateral triangles
    • Two squares
    • Two circles

Similar triangles · Year 9 Maths · www.arenapublications.com/learn

Year 9 Maths · 8 of 8

Probability from experiments

Work out the empirical probability of an event from the results a trial actually gave.

Before you start

Every probability there has ever been lives somewhere on this line. Move along it and read off the values.

0.00.51.0

Zero means it never happened and one means it happened every single time. Nothing sits outside the two ends — a probability of 1.4 or of -0.2 is not unlikely, it is impossible.

  1. 1.A die is rolled 60 times and shows a six on 12 of them. What is the empirical probability of a six? Give your answer as a fraction.

  2. 2.A coin is tossed 100 times and lands heads 54 times. What is the empirical probability of a head? Give your answer as a fraction.

  3. 3.An event has an empirical probability of 0.7. Tap where that sits on the probability line.

    Mark the line with an X.

    0.00.51.0

  4. 4.A batch of 50 bulbs was tested and 3 of them were faulty. What is the empirical probability that a bulb is faulty? Give your answer as a decimal.

  5. 5.In a trial, seeds germinated with an empirical probability of 0.8. Out of 500 more seeds, how many would you expect to germinate?

  6. 6.Sort each number by whether it could possibly be a probability.

    Groups: Could be a probability · Could not be a probability

    • 0.25
    • 7/5
    • -0.2
    • 1.4
    • 1
    • 0

  7. 7.A coin gives 7 heads in 10 tosses, and 503 heads in 1000 tosses. Which empirical probability is closer to the theoretical one half?

    • a) They are exactly as close as each other
    • b) Neither is anywhere near
    • c) The one from 10 tosses
    • d) The one from 1000 tosses

  8. 8.A coin is tossed 100 times and lands heads 54 times. What is the empirical probability of a tail? Give your answer as a fraction.

Probability from experiments · Year 9 Maths · www.arenapublications.com/learn

Answer keys — Year 9 Maths

In the same order as the worksheets.

Number systems

  1. 1. 5
  2. 2. 7/9
  3. 3.
    • 0.375
    • .375
  4. 4. -2
  5. 5. 1.4
  6. 6. d) √11
  7. 7. d) Yes — a whole number n can be written as n/1
  8. 8. 12 → Whole number; -7 → Integer, but not a whole number; 5/6 → Rational, but not an integer; √11 → Irrational

Linear equations in two variables

  1. 1. b) a straight line
  2. 2. 0
  3. 3. 1
  4. 4. (1, 4) → x + y = 5; (2, 6) → y = 3x; (0, 7) → y = 7; (3, 3) → x = y
  5. 5. b) Yes — 2 + 3 comes to 5
  6. 6. b) Infinitely many
  7. 7. 12
  8. 8. 6

Quadratic equations

  1. 1. 7
  2. 2. -4
  3. 3. -11
  4. 4. 2
  5. 5. 10
  6. 6. 5
  7. 7. x² + 5x + 6 → (x + 2)(x + 3); x² - 5x + 6 → (x - 2)(x - 3); x² - 4 → (x + 2)(x - 2); x² + 4x + 4 → (x + 2)(x + 2)
  8. 8. 3

Distance and section formulas

  1. 1. 5
  2. 2. 5
  3. 3. b) (5, 0)
  4. 4. 8
  5. 5. 24
  6. 6. 10
  7. 7. 12
  8. 8. 5

Algebraic identities

  1. 1. 9604
  2. 2. c) x² + 6x + 9
  3. 3. 58
  4. 4. 2601
  5. 5. a) x² - 10x + 25
  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. a) a² - 2ab + b²
  8. 8. 480

Introduction to trigonometry

  1. 1. sin 0° → Equals 0; cos 90° → Equals 0; tan 0° → Equals 0; cos 0° → Equals 1; sin 90° → Equals 1; tan 45° → Equals 1
  2. 2. 4/5
  3. 3. d) √3/2
  4. 4. sin 30° → 1/2; cos 30° → √3/2; tan 45° → 1; sin 0° → 0; tan 60° → √3
  5. 5. 1/2
  6. 6. 3/5
  7. 7. a) 1/√3
  8. 8. 1

Similar triangles

  1. 1. c) Matching sides are in the same ratio
  2. 2. 36
  3. 3. 1:2 → 1:4; 1:3 → 1:9; 2:3 → 4:9; 3:4 → 9:16; 1:5 → 1:25
  4. 4. 30
  5. 5. 15
  6. 6. c) similar
  7. 7. 12
  8. 8. Two squares → Always similar; Two circles → Always similar; Two equilateral triangles → Always similar; Two rectangles → Not always similar; Two isosceles triangles → Not always similar; Two rhombuses → Not always similar

Probability from experiments

  1. 1. 12/60
  2. 2. 54/100
  3. 3. 0.7
  4. 4. 0.06
  5. 5. 400
  6. 6. 0.25 → Could be a probability; 1.4 → Could not be a probability; 0 → Could be a probability; -0.2 → Could not be a probability; 1 → Could be a probability; 7/5 → Could not be a probability
  7. 7. d) The one from 1000 tosses
  8. 8. 46/100

Year 9 Maths · www.arenapublications.com/learn