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Cambridge Lower Secondary Maths: Stage 9

Stage 9 of the Cambridge Lower Secondary Mathematics (0862) curriculum framework, usually Year 9 (ages 13 to 14). Here is what each part asks, a practice question for each, and the mistakes that cost the most marks.

Practise Stage 9 as a game

The 53 parts of Stage 9

Each part with one practice question, made by the same engine your child plays.

Integers, powers and roots

Rational and irrational numbers

Which of these is RATIONAL?

  1. √5
  2. √81
  3. √7
  4. √2
Show the answer

√81. √81 can be written as a fraction of integers (it is a whole number: 9), so it is rational.

Positive, negative and zero indices; the index laws

Evaluate 2⁻².

  1. 1/8
  2. −4
  3. 1/4
  4. 1/2
Show the answer

1/4. A negative power means "one over": 2⁻² = 1/2² = 1/4.

Standard form for large and small numbers

Write 66 in standard form.

  1. 6.6 × 10¹
  2. 6.6 × 10²
  3. 6.6 × 10⁰
  4. 6.6 × 10⁻¹
Show the answer

6.6 × 10¹. Move the point 1 place so the number is between 1 and 10: 6.6 × 10¹.

Estimate surds from square and cube roots

∛57 lies between which two whole numbers?

  1. 4 and 5
  2. 2 and 3
  3. 9 and 10
  4. 3 and 4
Show the answer

3 and 4. 3³ = 27 and 4³ = 64. 57 is between them, so ∛57 is between 3 and 4.

Place value, ordering and rounding

Multiply and divide by 10 to any positive or negative power

Work out, giving your answer in standard form.
(9 × 10⁶) ÷ (3 × 10¹)

  1. 3 × 10⁵
  2. 3 × 10⁶
  3. 6 × 10⁵
  4. 3 × 10⁷
Show the answer

3 × 10⁵. 9 ÷ 3 = 3, and 10⁶ ÷ 10¹ = 10⁵ (take away the powers).

A rounded number has an upper and a lower limit

A length is 23 cm, to the nearest cm.
What is the lower bound?

  1. 22.6 cm
  2. 23 cm
  3. 22 cm
  4. 22.5 cm
Show the answer

22.5 cm. Half a unit either way: 23 − 0.5 = 22.5 cm.

Fractions, decimals, percentages, ratio and proportion

Will a fraction recur or terminate as a decimal?

Will 4/11 be a terminating or a recurring decimal?

  1. it cannot be told without dividing
  2. neither: it is a whole number
  3. recurring
  4. terminating
Show the answer

recurring. Look at the denominator in lowest terms, 11. It has a prime factor other than 2 or 5 (11), so the decimal recurs.

Add and subtract proper and improper fractions and mixed numbers

1/5 + 2/3 = ?
(simplest form)

  1. 3/8
  2. 1/5
  3. 13/15
  4. 2/15
Show the answer

13/15. Use 15 as the denominator: 3/15 + 10/15 = 13/15.

Multiply and divide fractions, cancelling first

4/5 ÷ 2/5 = ?

  1. 3/5
  2. 8/25
  3. 2
  4. 1/2
Show the answer

2. Flip the second fraction and multiply: 4/5 × 5/2 = 2.

Simplify calculations with decimals and fractions, including brackets and indices

Work out 3/5 × 29 × 5/3

  1. 145
  2. 29
  3. 17
  4. 87
Show the answer

29. 3/5 × 5/3 = 1, whatever order you do it in. So the answer is 29.

Compound percentages

$2,000 is invested at 10% compound interest per year.
What is it worth after 2 years?

  1. $420
  2. $2,400
  3. $2,200
  4. $2,420
Show the answer

$2,420. Multiply by 1.1 each year: 2,000 × 1.1 × 1.1 = 2,420.

Multiply and divide decimals by integers and decimals

4.2 × 0.5 = ?

  1. 2.08
  2. 21
  3. 0.21
  4. 2.1
Show the answer

2.1. 42 × 5 = 210. There are two decimal places altogether, so 2.1.

Direct and inverse proportion

y is directly proportional to x.
When x = 4, y = 12.
Find y when x = 3.

  1. 3
  2. 16
  3. 9
  4. 11
Show the answer

9. y = kx. From the first pair, k = 12 ÷ 4 = 3. So y = 3 × 3 = 9.

Ratios and equivalence in a range of contexts

$287 is shared in the ratio 3 : 4.
How much is the larger share?

  1. $123
  2. $41
  3. $71.8
  4. $164
Show the answer

$164. 3 + 4 = 7 parts. One part is 287 ÷ 7 = 41. The larger share is 4 × 41 = 164.

Expressions, equations and formulae

The laws and the order of operations in algebra, with integer powers

Which is the same as x × 5 × y?

  1. 5x + 5y
  2. 5x + y
  3. 5 + xy
  4. 5xy
Show the answer

5xy. Multiplication can be done in any order (commutative): x × 5 × y = 5xy.

Expand two brackets, apply the laws of indices, simplify algebraic fractions

Expand and simplify
(x + 4)(x + 1)

  1. x² + 5x − 4
  2. x² + 4
  3. x² + 5x + 4
  4. x² + 4x + 5
Show the answer

x² + 5x + 4. x × x = x². The x terms: 4x and 1x make 5x. The numbers: 4 × 1 = 4.

A situation in words or as an expression with squares, cubes and roots

A cube has edge n cm.
Which expression is the total volume of 3 of these cubes?

  1. 3n³
  2. 3n²
  3. 9n
  4. (3n)³
Show the answer

3n³. One cube is n × n × n = n³. 3 of them: 3n³. Only the n is cubed.

A situation in words or as a formula with squares and cubes; change the subject

Make x the subject.
y = 7x − 9

  1. x = (y + 9) ÷ 7
  2. x = y ÷ 7 + 9
  3. x = (y − 9) ÷ 7
  4. x = 7(y + 9)
Show the answer

x = (y + 9) ÷ 7. Add 9 first: y + 9 = 7x. Then divide everything by 7.

Equations from situations, including an unknown in the denominator

Solve
8x + 3 = 3x − 2

  1. x = −1
  2. x = 0
  3. x = 0.2
  4. x = 1
Show the answer

x = −1. Collect x on one side: 5x = −5, so x = −1.

Simultaneous linear equations: by elimination and by graph

x + y = 5
x − y = −1
Find x.

  1. x = −1
  2. x = 2
  3. x = 4
  4. x = 5
Show the answer

x = 2. Add the equations: 2x = 4, so x = 2. Then y = 3.

A situation in words or as an inequality; solve linear inequalities

Solve
−5x + 7 > 22

  1. x < −3
  2. x > −3
  3. x > 3
  4. x < 3
Show the answer

x < −3. Dividing by a negative number (−5) REVERSES the inequality sign.

Sequences, functions and graphs

Generate linear and quadratic sequences from a term-to-term rule

The nth term of a sequence is 3n + 3.
Which term of the sequence is 21?

  1. 5
  2. 6
  3. 18
  4. 7
Show the answer

6. 3n + 3 = 21, so 3n = 18 and n = 6.

nth term rules: an ± b, n², n³ and n² ± a

Find the nth term.
31, 49, 67, 85, …

  1. 31n + 18
  2. 18n + 13
  3. n + 18
  4. 18n + 31
Show the answer

18n + 13. It goes up in 18s, so it starts 18n. When n = 1, 18n = 18, and the first term is 31: adjust by 13.

Functions: outputs, and inputs by inverse operations, including indices

f(x) = 6x − 9
Find f(−4).

  1. −7
  2. 55
  3. −33
  4. 15
Show the answer

−33. Put −4 where x is: 6 × −4 − 9 = −33.

A situation in words or as y = mx + c or ax + by = c

A phone plan costs $40 a month plus $45 for every gigabyte used.
h is the number of gigabytes and C is the bill in dollars.
Which formula is right?

  1. C = 45h + 40
  2. C = 85h
  3. C = 45(h + 40)
  4. C = 40h + 45
Show the answer

C = 45h + 40. Every one of the gigabytes adds 45, which is 45h. The 40 is paid once, whatever h is: C = 45h + 40.

Tables of values and graphs of ax + by = c and y = x² ± a

The questions for this part come with a picture or are played as a game, so they are best tried in Huh Lab.

Find the equation of a straight-line graph, any gradient

Find the gradient of the line through
(−1, −1) and (3, −6).

  1. 5/4
  2. −5/4
  3. 4/5
  4. −4/5
Show the answer

−5/4. Change in y ÷ change in x = −5 ÷ 4 = −5/4.

Read, draw and interpret graphs; compare them with compound measures

A car travels at 48 km/h for 40 minutes.
How far does it go?

  1. 1920 km
  2. 1.2 km
  3. 32 km
  4. 19.2 km
Show the answer

32 km. 40 minutes is 2/3 of an hour. 48 × 2/3 = 32 km.

Geometrical reasoning, shapes and measurements

Area and circumference of a circle

A circle has radius 4 cm.
Find its circumference, in terms of π.

  1. 10π cm
  2. 16π cm
  3. 4π cm
  4. 8π cm
Show the answer

8π cm. Circumference = 2πr = 2 × π × 4 = 8π cm.

Very small and very large units of length, capacity and mass

Which unit is the most sensible for the distance from the Earth to the Moon?

  1. micrometres
  2. millimetres
  3. kilometres
  4. millilitres
Show the answer

kilometres. The distance from the Earth to the Moon is measured in kilometres. The other units would give a number that is absurdly large or absurdly small.

Area of compound shapes made from rectangles, triangles and circles

The questions for this part come with a picture or are played as a game, so they are best tried in Huh Lab.

Volume of prisms and cylinders

A cylinder has radius 6 cm and height 6 cm.
Find its volume, in terms of π.

  1. 36π cm³
  2. 216π cm³
  3. 432π cm³
  4. 72π cm³
Show the answer

216π cm³. V = πr²h = π × 6² × 6 = 216π cm³.

Surface area, now including cylinders

Find the total surface area of a 6 cm by 4 cm by 5 cm cuboid.

  1. 74 cm²
  2. 120 cm²
  3. 148 cm²
  4. 60 cm²
Show the answer

148 cm². Three different faces: 24, 30 and 20 cm². There are two of each: 2 × 74 = 148 cm².

Reflective symmetry in 3D shapes

The questions for this part come with a picture or are played as a game, so they are best tried in Huh Lab.

The sum of the interior angles of any polygon

What is the sum of the interior angles
of a polygon with 9 sides?

  1. 1620°
  2. 1440°
  3. 360°
  4. 1260°
Show the answer

1260°. (n − 2) × 180 = 7 × 180 = 1260°.

The exterior angles of any polygon add to 360°

What is each exterior angle of a
regular polygon with 9 sides?

  1. 140°
  2. 1260°
  3. 20°
  4. 40°
Show the answer

40°. Exterior angles add to 360°. 360 ÷ 9 = 40°.

Missing angles from parallel lines, triangles and quadrilaterals together

Two parallel lines are crossed by a third line.
One angle is 50°. What is its corresponding angle?

  1. 310°
  2. 50°
  3. 130°
  4. 40°
Show the answer

50°. Corresponding angles (the F shape) are equal: 50°.

Pythagoras' theorem

The questions for this part come with a picture or are played as a game, so they are best tried in Huh Lab.

Position and transformation

Bearings and scale on maps and plans

The bearing of B from A is 185°.
What is the bearing of A from B?

  1. 003°
  2. 185°
  3. 005°
  4. 175°
Show the answer

005°. Turn round: add 180° if the bearing is under 180°, take 180° away if it is over. 185° − 180° = 005°.

Find points on a line segment from coordinates

Find the midpoint of
(5, −3) and (11, −5).

  1. (16, −8)
  2. (3, −1)
  3. (8, −4)
  4. (−4, 8)
Show the answer

(8, −4). Average the x values and the y values: (8, −4).

Combine reflections, translations and rotations

The point (1, 5) is reflected in the x-axis,
then translated by (1, −2).
Where does it end up?

  1. (2, −7)
  2. (−7, 2)
  3. (0, 3)
  4. (2, 3)
Show the answer

(2, −7). Reflecting in the x-axis flips the y: (1, −5). Then the translation adds 1 to x and −2 to y: (2, −7).

Identify and describe a transformation from an object and its image

The questions for this part come with a picture or are played as a game, so they are best tried in Huh Lab.

Any mix of reflections, translations and rotations leaves the image congruent

A shape is reflected, then rotated, then translated.
Which of these is TRUE about the final image?

  1. It is congruent to the original: the same shape and size.
  2. Its angles have changed.
  3. It is smaller than the original.
  4. It is bigger than the original.
Show the answer

It is congruent to the original: the same shape and size.. Reflections, rotations and translations move a shape without stretching it. Any combination of them leaves the image congruent to the object. Only an enlargement changes the size.

Enlarge from any centre; identify the enlargement, its centre and scale factor

The point (6, 4) is enlarged with centre (0, 0)
and scale factor 3. Where does it go?

  1. (6, 12)
  2. (18, 4)
  3. (18, 12)
  4. (9, 7)
Show the answer

(18, 12). From the origin, multiply BOTH coordinates by 3: (18, 12).

How perimeter and area change when a rectangle is enlarged

A 6 cm by 2 cm rectangle is enlarged by scale factor 2.
What is the perimeter of the enlarged rectangle?

  1. 64 cm
  2. 32 cm
  3. 24 cm
  4. 18 cm
Show the answer

32 cm. Every length is multiplied by 2: the new rectangle is 12 cm by 4 cm, so its perimeter is 2 × (12 + 4) = 32 cm, which is 2 times the old perimeter of 16 cm.

Statistics

Bias in data collection and sampling

Which question is LEADING (it pushes people towards one answer)?

  1. Most people love our new logo. Do you?
  2. What do you think of the new logo?
  3. How do you travel to school?
  4. How old are you?
Show the answer

Most people love our new logo. Do you?. "Most people love our new logo. Do you?" tells people what to think before they answer. "What do you think of the new logo?" leaves the answer open.

Record and show data, now including frequency polygons and back-to-back stem-and-leaf

Score: 1, 2, 3, 4
Frequency: 2, 1, 2, 5
Find the mean score.

  1. 30
  2. 7.5
  3. 3
  4. 2.5
Show the answer

3. Total of score × frequency = 30. Total frequency = 10. 30 ÷ 10 = 3.

Compare two distributions, including grouped data

Test scores.
Class A: mean 58, range 34.
Class B: mean 64, range 21.
Which class had the more CONSISTENT scores?

  1. Class B
  2. Class A
  3. They are the same
  4. It is impossible to tell
Show the answer

Class B. Consistent means close together, so compare the ranges: 21 is smaller, so Class B. The mean says nothing about how spread out the scores were.

Interpret trends and relationships; spot wrong or misleading information

The questions for this part come with a picture or are played as a game, so they are best tried in Huh Lab.

Probability

Probabilities of mutually exclusive events add, and total 1

A spinner lands on red (0.1), blue (0.2), green (0.55) or yellow.
Find the probability of red OR blue.

  1. 0.3
  2. 0.02
  3. 0.15
  4. 0.7
Show the answer

0.3. It cannot land on red and blue at once (mutually exclusive), so add: 0.1 + 0.2 = 0.3.

When successive and combined events are independent, and when they are not

The probability that Ana is late is 1/3.
The probability that it rains is 1/3.
The events are independent. Find P(Ana is late AND it rains).

  1. 1/6
  2. 1/3
  3. 2/3
  4. 1/9
Show the answer

1/9. For independent events, AND means multiply: 1/3 × 1/3 = 1/9.

Theoretical probability of combined events

The probability that Ana is late is 4/6.
The probability that it rains is 1/3.
The events are independent. Find P(Ana is late AND it rains).

  1. 2/9
  2. 4/9
  3. 5/9
  4. 1
Show the answer

2/9. For independent events, AND means multiply: 4/6 × 1/3 = 2/9.

Chance experiments; expected frequency against observed

The probability of winning a game is 0.9.
The game is played 460 times.
How many wins would you expect?

  1. 46
  2. 230
  3. 414
  4. 23
Show the answer

414. 0.9 × 460 = 414.

The mistakes that cost the most marks

Huh Lab checks every wrong answer against known mistakes like these, and goes back to the step behind the one your child keeps making.

Adds the denominators too

Quarters plus quarters are still quarters. Only the tops are added.

Gives the y-coordinate first

Along the corridor, THEN up the stairs: x comes before y.

Forgets the first one was not put back

After one counter is taken there is one fewer in the bag. Both numbers change.

Loses the negative sign

Use a number line: below zero the numbers keep counting, in the other direction.

Forgets the final square root

a² + b² gives c², not c. One more step.

Leaves out some of the faces

Count the faces before adding: a cuboid has 6, a closed cylinder has two circles and the curved part.

Stops one step before the end

After removing the number, there is still the multiplying to undo: divide as well.

Adds or subtracts the sides instead of their squares

Pythagoras works on the SQUARES of the sides.

Other Cambridge Lower Secondary Maths stages

Stage 7Year 7, ages 11 to 12 Stage 8Year 8, ages 12 to 13

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