Spring Year 9 - Home - Oasis Academy Lister Park · 2020-03-26 · (constructing triangles, perpendicular bisectors and angle bisectors) Throughout this section, make sure you show

Workbook

Spring

Year 9

2 Copyright © Mathematics Mastery 2018-19

Copyright © Mathematics Mastery 2018-19 3

Contents Unit 8: Construction .................................................................................................................................................. 4

8.1: Constructing triangles (review of Year 8)........................................................................................... 4

8.2: Constructing perpendicular bisectors and angle bisectors .......................................................... 7

8.3: Mixed questions (constructing triangles, perpendicular bisectors and angle bisectors)

.................................................................................................................................................................................... 10

8.4: Mixed problems ........................................................................................................................................... 12

Unit 9: Congruence .................................................................................................................................................. 16

9.1: Congruent figures ....................................................................................................................................... 16

9.2: Congruence conditions for triangles ................................................................................................... 20

9.3: Mixed problems ........................................................................................................................................... 25

Unit 10: Pythagoras’ Theorem ............................................................................................................................ 30

10.1: Pythagoras’ Theorem.............................................................................................................................. 30

10.2: Calculating the missing length of the hypotenuse ....................................................................... 32

10.3: Calculating the lengths of other sides .............................................................................................. 35

10.4: Mixed problems ........................................................................................................................................ 39

Unit 11: Angles in polygons .................................................................................................................................. 44

11.1: Sum of interior angles in polygons .................................................................................................... 44

11.2: Sum of exterior angles of polygons ................................................................................................... 47

11.3: Mixed questions ........................................................................................................................................ 51

Unit 12 Linear equations and inequalities ..................................................................................................... 56

12.1: Writing Linear equations and inequalities ..................................................................................... 56

12.2: Solving equations and inequalities with the unknown on only one side ........................... 58

Concept corner: Properties of equality ...................................................................................................... 62

12.3: Solving equations with the unknown on both sides of the equality ..................................... 63

Unit 13: Graphical solutions ................................................................................................................................ 70

13.1: Simultaneous equations ........................................................................................................................ 70

13.2: Quadratic graphs ...................................................................................................................................... 80

13.3: Further quadratic graphs ...................................................................................................................... 85

13.4: Exponential, reciprocal and piecewise linear graphs ................................................................ 90


Unit 8: Construction

8.1: Constructing triangles (review of Year 8)

Throughout this section, make sure you show all your construction lines.

1. Draw the following triangles accurately and measure the sides and angles not given in the

diagram.

a)

Concept corner

In order to construct a triangle, you need to have some information:

All 3 sides: Side Side Side (SSS)

Two side lengths and the angle between them: Side Angle Side (SAS)

Two angles and the length of the side between them: Angle Side Angle (ASA)

Note: AAS does not give a unique triangle.

For example:

7 cm 7 cm

56°

56° 72°

72°

A A

B B

C C


b)

2. Draw ∆CED, if ED = 6 cm, EC = 4 cm and ∠𝐶𝐸𝐷 = 40°.

Measure the sides and angles not given in the diagram.


3. Draw an isosceles triangle that has two sides of length 7 cm and an angle between

them of 40°.

a) Measure the length of a base and the corresponding height of the triangle to the nearest

millimetre.

base = …………………………. cm

height = ……………………….. cm

b) What is the area of the triangle?


8.2: Constructing perpendicular bisectors and angle bisectors


1. Construct the perpendicular bisector of the line segment AB.

Concept corner

Angle bisector

Perpendicular bisector


2. Measure the size of DEF ……………………°

Construct the angle bisector of DEF.

3. Construct the line which passes through M and is perpendicular to the line segment PQ.


4. Construct a right angle at Y using only a pencil, a pair of compasses and a ruler.

5. In the space below, construct an equilateral triangle of side length 6 cm using only a pencil,

a pair of compasses and a ruler.


8.3: Mixed questions

(constructing triangles, perpendicular bisectors and angle bisectors)


1. Construct triangle 𝑃𝑄𝑅 where PQ = 4 cm, PR = 4.5 cm and QR = 4.9 cm.

Measure and write down the size of ∠𝑃𝑅𝑄.

Construct the perpendicular bisector of PR.

2. Construct triangle XYZ where XY = 4.4 cm, XZ =4.6 cm and YZ =5.2 cm.

Measure and write down the size of ∠𝑌𝑋𝑍.

Construct the angle bisector of ∠𝑋𝑌𝑍.

Concept corner

When constructing geometrical figures, use a sharp pencil to draw points and lines

clearly.

All construction lines must be clearly shown.

Do not erase the construction lines you have drawn.


3. In the space below, construct triangle PQR with PQ = 4.8 cm, QR = 6.5 cm and

∠PQR=90°using only a pencil, a pair of compasses and a ruler.

Measure and write down the size of ∠PQR giving your answer correct to one decimal

place.

4. Construct △XYZ with XY = 9.4 cm, XZ = 8.8 cm and ∠Y= 60° using only a pencil, a pair of

compasses and a ruler.

Measure and write down the length of YZ.


8.4: Mixed problems


1. Explain what is meant by, ‘equidistant from two points’.

2. Here is a plan of some land.

There will be a fence that is always the same distance from tree A as from tree B, going all

the way from one road to the other road,

Use compasses and a ruler to accurately show, on the plan, where the fence will be.

You must leave in your construction lines.


3.

a) Use a ruler and a pair of compasses to draw a triangle that has these side lengths:

5 cm, 5 cm, 8 cm

b) Fin says it is possible to draw a triangle with these side lengths:

5cm, 5cm, 12cm

Is she correct?

Yes No

Explain how you know.

……………………………………………………………………………………………………………………………

4. Construct the angle bisector of angle ABC.


5. In the space below, construct and label the following angles using only a pencil, a pair of

compasses and a ruler.

a) Construct an angle of 30˚.

b) Construct an angle of 45˚.


c) Construct an angle of 330˚.

d) Construct an angle of 345˚.


Unit 9: Congruence

9.1: Congruent figures

1. In the diagram below, which triangles can be moved to overlap ∆ABC completely?

Concept corner

Congruent shapes are identical but can have different orientations.

For example, these five shapes are all congruent

Properties of congruence

Quadrilateral 𝐴𝐵𝐶𝐷 and quadrilateral 𝐴’𝐵’𝐶’𝐷’ are congruent.

This can be written as 𝐴𝐵𝐶𝐷 ≡ 𝐴’𝐵’𝐶’𝐷’, where the symbol ‘≡’ means ‘congruent’.

The angle marked in the diagram below can be written as 𝐴�̂�𝐶 or ∠𝐴𝐵𝐶.

Complete the statements below:

A and A’ are corresponding v…………………

AB and A’B’ are corresponding s……………..

∠𝐴𝐵𝐶 and ∠𝐴′𝐵′𝐶′ are corresponding a……………..

a

b

c

d


2. The two pentagons below are congruent.

a) Name two pairs of corresponding vertices

……………………………………………………….……………………………………………………………………..

b) Name two pairs of corresponding sides

……………………………………………………….……………………………………………………………………..

c) Name two pairs of corresponding angles

……………………………………………………….……………………………………………………………………..

3. Go back to question 1, name the triangles that are congruent to ∆ABC using the symbol ≡.

………………………………………………………………………………………………………………………………

4. Quadrilateral 𝑃𝑄𝑅𝑆 is congruent to 𝑊𝑋𝑌𝑍 .

Name the corresponding sides and angles.


5. Given that 𝑀𝑁𝑂𝑃 ≡ 𝑊𝑋𝑌𝑍, complete the following:

a) 𝑍𝑌 = 𝑃𝑂 = …………….cm

b) WZ = ………….. = …………….cm

c) XY = ………….. = …………….cm

d) 𝑊𝑋 = ………………… = ……………. cm

e) 𝑋�̂�𝑍 =………………… = ……………. °

6. Given that 𝐴𝐵𝐶𝐷 ≡ 𝑊𝑋𝑌𝑍, complete the following:

a) 𝐴�̂�𝐶 = 𝑊�̂�𝑌 = …………..°

b) ………….. = 𝑋�̂�𝑍 = …………..°

c) 𝐴𝐷 = ………………… = ……………. cm

d) ………………….. = 𝑊𝑋 = …………… cm


7. Use the diagrams to help you decide if the following statements are true or false:

a) The two quadrilaterals below are congruent. True/False

b) The two regular pentagons below are congruent. True/False

c) In congruent figures, corresponding line segments have equal lengths.

True/False

In the figures below 𝑨𝑩𝑪𝑫 ≡ 𝑨′𝑩′𝑪′𝑫′.

d) CD = C’D’ = 7 cm True/False

e) 𝐴′𝐵′̂𝐶′ = 59° True/False

f) In congruent figures, corresponding angles are equal. True/False


9.2: Congruence conditions for triangles

Concept corner

Two triangles are congruent if any of the following conditions hold true.

1. All three pairs of corresponding sides are equal (Side-Side-Side).

2. Two pairs of corresponding sides are equal and the angles between them are

equal (Side-Angle-Side).

3. Two pairs of corresponding angles and the corresponding sides between them

are equal (Angle-Side-Angle).

AB = A′B′

BC = B′C′

CA = C′A′

AB = A′B

∠ABC = ∠A′B′C′

BC = B′C′

∠ABC = ∠A′B′C′

BC = B′C′

∠ACB = ∠A′C′𝐵′


1. These two triangles are congruent:

a) What is the size of ∠𝑍𝑋𝑌 ? …………………

b) What is the length of XY? …………………

2. Which of the triangles below are congruent to the ∆ABC, and why?

∆𝐴𝐵𝐶 ≡ …………………… reason ……………………………………………………………………………

∆𝐴𝐵𝐶 ≡ …………………… reason ……………………………………………………………………………

Not drawn to

scale.

Not drawn to

scale.


3. Identify the triangles below which are congruent and give the reasons why.

∆𝐴𝐵𝐶 ≡ …………………… reason ……………………………………………………………………….

……………… ≡ ……………… reason ………………………………………..………………………………

……………… ≡ ……………… reason ………………………………………..………………………………

Not drawn to

scale.


4. In each diagram below, identify a pair of congruent triangles and give reasons for your

answers.

a)

b)

c)

How do you know

∠𝐴𝑂𝐵 = ∠𝐶𝑂𝐷?


5. In the diagram, ABX and ACX are congruent.

Given that AC = 13 cm, CX = 5 cm and 𝐴�̂�𝐶 = 67°.

Work out:

a) The length of BC.

b) 𝐵�̂�𝐶

6. Two triangles have sides of lengths 4 cm and 3 cm, and contain an angle of 30°.

Show that it is possible to draw 4 different triangles, none of which are congruent, using

this information.


9.3: Mixed problems

1. Two students each drew a triangle with one side of 5 cm, one angle of 20° and one angle of

60°

Must their triangles be congruent? Why?

2. In the diagram, the lines AC and BD

intersect at E.

AB and DC are parallel and AB = DC.

Prove that triangles ABE and CDE are

congruent.

3. If O is the centre of both circles, prove that

the triangles OAB and ODC are congruent.

Show your working, giving reasons for the

statements you make.


4. If O is the centre of the circle, prove that

the triangles OPQ and OQR are congruent.

5. ABCD and DXYZ are squares.

ADZ and XDC are straight lines.

Prove that triangles BDX and BDZ are

congruent.

Show your working and give reasons for

the statements you make.


6. ABC is an isosceles triangle in which AB = AC.

X and Y are points on AB and AYC such that ∠ XCB = ∠YBC.

Prove that triangles XCB and YBC are congruent.

7. PQRS is a square and QX = SY.

Identify the pairs of congruent triangles in the

diagram.

Give reasons for your answers.


8. A rectangular sheet of paper was folded as shown below.

Determine the measurement of ∠𝑎.

Show your working, giving reasons for the statements you make.


Reflections

This space is for you to write your reflections on the work you have covered so far.

You may wish to write about:

Things you’ve learnt (key concepts or formulae, new skills)

Things you found difficult (how will you address your difficulties?)

Other areas of maths you used in this topic

Topics you need to revisit/revise in the future


Unit 10: Pythagoras’ Theorem

10.1: Pythagoras’ Theorem

1. Mark the hypotenuse on each of the following right angled triangles:

2. On a separate piece of paper, construct the three triangles below accurately, measure the

missing length for each triangle and fill in the following table.

𝒃 𝒄 𝒂 𝒃𝟐 𝒄𝟐 𝒂𝟐 𝒃𝟐 + 𝒄𝟐

a) 8 6 64 36

b) 5 12

c) 8 15

Concept corner

This is a right angled triangle with 𝐴�̂�𝐶 = 90°.

The side opposite the right angle B is called the hypotenuse.

The hypotenuse is the longest side of a right-angled triangle.


3. The integers 3, 4 and 5 are called a Pythagorean triple because 32 + 42 = 52.

A triangle with sides of length 3 cm, 4 cm and 5 cm is right-angled.

Use Pythagoras’ Theorem to determine which of the sets of numbers below are

Pythagorean triples:

a) 5, 12, 13

b) 11, 22, 33

c) 10, 24, 26

d) 6, 8, 9

e) 7, 24, 25

Concept corner

If we use the letters 𝑎, 𝑏 and 𝑐 for the sides of a right-angled

triangle, then Pythagoras’ Theorem states that

𝑎2 + 𝑏2 = 𝑐2

where 𝑐 is the length of the hypotenuse.


10.2: Calculating the missing length of the hypotenuse

1. Calculate the length of the hypotenuse of each of these triangles:

a)

b)

Example

Calculate the length of the hypotenuse of a triangle in

which the other two sides are 8 m and 5 m.

Solution

𝑎2 = 82 + 52

𝑎2 = 64 + 25

𝑎2 = 89

𝑎 = √89 metres

𝑎 = 9.4 m, correct to 1 decimal place

m


2. Calculate the length of the hypotenuse of each of the following triangles, giving your

answer correct to one decimal place.

a)

b)

3. Calculate the length of the diagonals of the rectangle below:


4. Which of the rectangles below has the longer diagonal?

5.

a) Use Pythagoras’ Theorem to show that the length of AC is 12.0 cm correct to one

decimal place.

b) Sam says:

Is Sam correct? Why?

a b

This triangle is isosceles because

two of the sides are equal in

length.


10.3: Calculating the lengths of other sides

1. Calculate the length of the side marked 𝑥 in each of these triangles:

a)

b)

2. Calculate the length of the side marked 𝑥 in each of the following triangles, giving your

answers correct to one decimal place.

a)

b)


3. Calculate the perpendicular height of this equilateral

triangle, giving your answer correct to one decimal place.

4. Calculate the perpendicular height of this isosceles

triangle, giving your answer correct to 1 decimal place.

5. The width of a rectangle is 12 cm and the length of the diagonal is 13 cm.

a) How long is the other side of the rectangle?

b) What is the area of the rectangle?


6. Calculate the length of the diagonals of the square below:

7. Calculate the length of the side marked 𝑥 in each of the following triangles, giving your

answer correct to one decimal place.

a)

b)


8. Calculate the length of the sides marked with letters in each of the following, giving your

answers correct to two decimal places.

a)

b)

c)


10.4: Mixed problems

1. a) In which triangle below does 𝑎2 + 𝑏2 = 𝑐2?

b) For the other triangle, write an equation linking a, b and c

…………………………………………………..

c) In which triangle below does 𝑎2 + 𝑏2 = 𝑐2?

d) For the other triangle, explain why 𝑎2 + 𝑏2 does not equal 𝑐2.

…………………………………………………..

2.

Is the triangle a right-angled triangle?

Justify your answer.


3. Use Pythagoras’ theorem to find the value of 𝑥 in each of the trapezia below.

a)

b)

4.

a) Explain why the length XZ is 13 cm.

b) Calculate the length WZ.


5. A hiker walks 600 m due south and then 800 m due east.

How far is the hiker from her starting point?

6. A ladder of length 4 m leans against a wall so that the top of the ladder is 3.2 m above

ground level.

How far is the bottom of the ladder from the wall?

7. Alice draws a square with sides of length 6 cm.

She then measures a diagonal as 8.2 cm.

Use Pythagoras’ Theorem to decide if Alice has

drawn the square accurately.


8. The diagonals of a rhombus are of lengths 16 cm and 12 cm.

Find the length of its sides.


Reflections








Unit 11: Angles in polygons

11.1: Sum of interior angles in polygons

1. Name each of the polygons below.

How many non-overlapping triangles can be drawn from the vertices of each polygon?

Name of polygon: …………..…………………….. Name of polygon: …………..……………………..

Number of triangles:…………… Number of triangles:……………

2. What is the sum of the interior angles of the above polygons?

Concept Corner

This is a regular polygon with five equal sides and five equal angles.

The name of this regular polygon is a ……………………………

Two non-overlapping lines can be drawn to form three triangles.

The sum of the interior angles of each triangle is equal to ……………

So the sum of the interior angles of the pentagon is equal to 3 × …………… = ……………..


3. Repeat questions 1 and 2 for the following polygons:

a)

b)

c)

d)

Complete the table below:

Name of polygon: …………..……………………..

Number of triangles: ……………………

Sum of the interior angles: ………………………………………….

Name of polygon: …………..……………………..


Sum of the interior angles: ………………………………………….

Name of polygon: …………..…………………….. O……………………………………………….

Number of triangles: …………………… Number of triangles ……………………

Sum of the interior angles: ……………………………………. Sum of the interior angles………………………………………….

Name of polygon: …………..……………………..


Sum of the interior angles: …………………………………….


Name of Polygon Number of

sides

Number of non-

overlapping triangles

Sum of interior angles (in terms of 180°)

Quadrilateral 4 2 𝟐 × 180° = (𝟒 − 𝟐) × 180°

Pentagon 5 3 𝟑 × 180° = (𝟓 − 𝟐) × 180°

Hexagon 6

7

8

9

10

𝑛-gon 𝑛

e) From your completed table, what can you say about the number of non-overlapping

triangles formed within a polygon, in relation to its number of sides?

Concept corner

In general, we have the following conclusion:

The sum of the interior angles of an 𝑛-sided polygon is (𝒏 − 𝟐) × 𝟏𝟖𝟎°.


11.2: Sum of exterior angles of polygons

1. Calculate the size of the exterior angles of a regular polygon which has interior angles of:

a) 150°

b) 162°

c) 175°

Concept corner

The following diagram shows a regular pentagon:

In a regular polygon the sides are the same length and the interior angles are all the

same size.

For any regular polygon: interior angle + exterior angle = 180°

The exterior angles of any polygon add up to 360°

In this diagram,

the angles marked

are the interior angles of the

pentagon.

The angles marked

are the exterior angles of the

pentagon.


2. Calculate the sizes of the interior and exterior angles of:

a) a regular hexagon,

b) a regular octagon.


3. Calculate the size of an exterior angle of a regular heptagon.

4. The size of the exterior angle of a regular polygon is 24°.

How many sides does the polygon have?

5. a) Calculate the size of the interior angle of a 10-sided polygon.

b) What is the sum of the interior angles of a regular 10-sided polygon?


6. a) Complete the table below for regular polygons:

Number of sides

Size of each exterior angle

Size of each interior angle

Sum of interior angles

3 120°

4

5

6

7

8

9

10

12

b) Describe how to use the exterior angles to find the size of each of the interior angles of

a regular polygon.

c) Would your method work for irregular polygons too?


11.3: Mixed questions

1. Work out the size of each interior angle of the following regular polygons with:

a) 6 sides

b) 20 sides

2. Work out the number of sides of a regular polygon with an interior angle of:

a) 140°

b) 170°

3. An isosceles triangle has an angle of 50°

What are the other two angles? Give the two possible pairs of answers.


4. P and Q are vertices of a regular polygon.

The exterior angle at Q is 45°


5. A and B are vertices of a regular polygon. The interior angles are 168°


6. The £1 coin is a regular dodecagon, it has 12 sides. Work out the exact size of the interior

angle of a regular dodecagon.


7. The regular tridecagon is used as the shape of the Czech 20 Korun coin.

Work out the exact size of an interior angle of a regular tridecagon.

8. For each of the following polygons (not drawn to scale), work out the value of 𝑥:

a)

https://en.wikipedia.org/wiki/Coins_of_the_Czech_koruna


b)

c)

9. The interior angles in a quadrilateral are 𝑥, 2𝑥, 3𝑥 and 3𝑥.

Work out the value of 𝑥.

What are the values of each of the angles in degrees?


Reflections








Unit 12 Linear equations and inequalities

12.1: Writing Linear equations and inequalities

1. In a school tuck shop, packets of crisps cost 𝑔 pence and drinks cost ℎ pence.

If 3𝑔 + 5ℎ > 200, are the following statements true or false?

True False The cost of three packets of crisps and five drinks is more than £2.00

The cost of three drinks and five packets of crisps is less than £2.00

£2.00 is less than the cost of three packets of crisps and five drinks.

2. Tickets to a theme park cost 𝑥 pounds per adult and 𝑦 pounds per child.

What does each equality or inequality represent?

a) 𝑥 + 𝑦 = 100

b) 2𝑥 + 3𝑦 < 250

c) 𝑥 − 𝑦 < 15


3. In the school shop, pencils cost 𝑎 pounds, pens cost 𝑏 pounds and rulers cost 𝑐 pounds.

Match each situation to the correct equality or inequality.

The cost of five pencils and three pens is less than three pounds.

5𝑎 + 3𝑏 > 3

Three pounds is less than the cost of five pencils and three pens.

3(5𝑎 + 3𝑏) = 𝑐

The cost of five pencils and three pens is three times the cost of a ruler

5𝑎 + 3𝑏 < 3

The cost of five pencils and three pens together is the same as a third of the cost of a ruler.

5𝑎 + 3𝑏 = 3𝑐

4. Andrea buys five puzzle books at 𝑤 pounds each, and four magazines, at 𝑟 pounds each.

The total cost is less than £10.

Write an inequality to represent this.

5. When 𝑠 is subtracted from 𝑡, the difference is 9.

Write an equation to represent this.

6. Samuel is 𝑡 years’ old, and his mother is 𝑞 years old.

Samuel’s mother is more than three times Samuel’s age.

Write an inequality to represent this.


12.2: Solving equations and inequalities with the unknown on only one side

1. Solve each of these equations.

a) 2𝑥 + 5.2 = 9

b) 3𝑝 − 2.9 = 7

c) 12.2 − 2𝑥 = 3.8



a) 8 + 2𝑥 = 19

b) 8 − 2𝑥 = 19

c) 24 = 6 + 5𝑥

d) 24 = 6 − 5𝑥

e) 3𝑥 + 5 = 24

f) 63 − 7𝑥 = 13

3. Solve the equations to work out the unknown values.

a) 3(𝑥 + 7) = 102

b) 3(𝑥 − 7) = 102

c) 102 = 3𝑥 − 7


4. Solve the equations.

a) 5(2𝑚 + 1) + 3(4𝑚 − 2) = 99

b) 5(2𝑚 + 1) − 3(4𝑚 − 2) = 99

c) 5(2𝑚 − 1) − 3(−4𝑚 − 2) = 99


a) 11 =𝑠

4.5 b) 8 =

𝑡

3.1− 2

c) 12 =2

3(𝑤 − 2) d) 6 =

2

3(2 − 𝑥)


6. Solve these equations and inequalities.

a) 2𝑥 − 3 = 12 b) 2𝑥 + 3 < 12 c) −2𝑥 + 3 < 12

d) 5𝑥 + 3 = 19 e) 5𝑥 + 3 > 19 f) −5𝑥 − 3 > 19

g) 24 − 8𝑥 = −12 h) 24 − 8𝑥 > −12 i) 8𝑥 − 24 > −12

j) 34 = 9 − 5𝑥 k) 34 < 9 + 5𝑥 l) 34 < 9 − 5𝑥

m) 17 = 35 − 3𝑥 n) 17 > 35 − 3𝑥 o) 17 > 3𝑥 − 35


Concept corner: Properties of equality

1. If the same expression is added to two equal quantities, then the two resulting expressions

will be equal.

For example, if then

𝐴 = 𝐵 𝐴 + 3 = 𝐵 + 3

2. If the same expression is subtracted from two equal quantities, then the two resulting

expressions will be equal.


𝐴 = 𝐵 𝐴 − 3 = 𝐵 − 3

3. If two equal expressions are both multiplied by the same quantity, then the two resulting



𝐴 = 𝐵 3𝐴 = 3𝐵

4. If two equal expressions are both divided by the same quantity, then the two resulting



𝐴 = 𝐵 𝐴

3=

𝐵

3


12.3: Solving equations with the unknown on both sides of the equality

1. Solve these equations.

a) 7𝑥 + 5 = 9𝑥

b) 3𝑥 + 18 = 7𝑥 + 7

c) 19 − 4𝑥 = 𝑥 + 10

d) 7𝑥 + 2 = 32 − 3𝑥

e) 16𝑥 − 11 = 14 − 4𝑥



a) 0.8𝑥 + 1.1 = 0.3𝑥 + 4.1

b) 0.6𝑥 + 3.2 = 4.7 + 2.1𝑥

c) 23.4 − 3.7𝑥 = 1.3𝑥 − 1.6

3. Solve the following equations.

a) 6(2𝑥 − 4) = 5𝑥 + 28.5

b) 5(3𝑥 − 1.5) = 10𝑥 + 4

c) 6𝑥 + 169 = 8(1 − 5𝑥)


d) 33 − 2𝑥 = 3(2𝑥 − 19)

e) 6(𝑥 − 1.5) = 7(𝑥 − 2.5)

f) 3(2𝑥 + 1) = 2(4𝑥 − 11)

g) 8(8 − 3𝑥) = 7(10 − 6𝑥)

h) 9(2 − 𝑥) = −7(𝑥 + 14)

4. Expand brackets, and gather like terms before solving the equations.

a) −5𝑥 = 7 + 3(4𝑥 + 9)


b) 4𝑥 = 13 − 3(2𝑥 − 9)

c) 8 − 3(4 − 2𝑥) = 5(𝑥 − 2)

d) 10 − 2(5𝑥 − 3) = 4(𝑥 − 17)


a) 2

3𝑥 = 10 −

1

6

b) 18

𝑥= 4.5

c) 4.2 =9

𝑥


d) 2

3𝑥 +

4

5=

1

5−

1

3𝑥

e) 5

9𝑥 − 1 =

1

3𝑥 − 3

f) 𝑥+4

5=

𝑥+6

6

6. Consider the following diagrams. In each diagram, write and solve an equation to work

out the values represented by letters.

a)


b)

c)

7. The area of the rectangle and the triangle are equal. Work out the value of 𝑥.

8𝑥 + 15


8. Half of the sum of three consecutive integers, is 7 greater than the first integer.

What are the three integers?

9. I am thinking of a number, 𝑎.

Four more than my number is four less than twice my number.

a) Write an equation in 𝑎, to represent this.

b) Solve the equation.


Unit 13: Graphical solutions

13.1: Simultaneous equations

1. The cost of a mobile phone contract is £8 per month, plus £1 for every gigabyte of data

used.

a) Complete the table.

Gigabytes data used (𝑥 Gb)

0 1 2 3 4

Total cost (£ 𝑦)

£10

b) Write an equation for 𝑦 in terms of 𝑥.

𝑦 =

2. Alanna and Danni work at the same company.

a) Alanna gets paid £120 for working a six-hour day.

For every additional half-hour that she works, Allana earns an additional £11.

Complete the table.

Additional hours worked (𝑥 hours)

0 0.5 1 1.5 2

Total pay (£ 𝑦1)

£120

b) Write an equation for 𝑦1 in terms of 𝑥.

𝑦1 =


c) Danni gets paid £138 for working a six-hour day.

For every additional half-hour that she works, Danni gets paid a further £9.

Complete the table.

Additional hours worked (𝑥 hours)

0 0.5 1 1.5 2

Total pay (£ 𝑦2)

£138

d) Write an equation for 𝑦2 in terms of 𝑥.

𝑦2 =

Plot the graphs of 𝑦1 and 𝑦2 on the same axes.

e) How many additional hours do they each need to work if they are to earn the same

amount?


3. Tina buys a laptop on a credit arrangement.

Tina pays an initial payment of £80, and then pays £35 per month for 12 months.

Asif also buys a laptop. He pays an initial payment of £30, and then £45 each month for 12

months.

a) Complete the table.

Number of months (𝑥) 0 1 2 3 4

Amount paid by Tina (£ 𝑦1)

Amount paid by Asif (£ 𝑦2)

b) Express the amount paid by Tina as a formula in terms of 𝑥.

𝑦1 =

c) Express the amount paid by Asif as a formula in terms of 𝑥.

𝑦2 =

d) By plotting the graphs of 𝑦1 and 𝑦2, work out when Tina and Asif will have paid the

same amount of money.


e) How much do they each pay by the end of the 12-month payment period?

4. Lucy is doing a sponsored bike ride.

Lillie-Mae has sponsored her £20, plus £3 per additional mile.

Jenny has sponsored her £30, plus £2 per additional mile.

By drawing a graph, identify the number of miles that Lucy will need to cycle so that both

Lillie-Mae and Jenny donate exactly the same amount.


5. Liz takes a taxi. It costs £9, plus 50p per mile.

Tyrone also hires a taxi. His taxi charges £3, plus £1.50 per mile.

By drawing a graph, identify the distance for which they both charge the same.


6. On the grid below, plot the two equations and so find the solution.

𝑦 = 5𝑥 − 2

𝑦 = 4 − 𝑥


7. On the graph paper below, plot these two equations and so find the solution.

𝑦 =1

2𝑥 + 4

𝑦 = 9 − 2𝑥


8. On the graph paper below, plot these two equations and so find the solution.

𝑥 + 2𝑦 = 7

2𝑥 + 𝑦 = 8


9. Solve the following pair of simultaneous equations by drawing a graph.

𝑦 = 2𝑥 − 6

𝑦 = −𝑥 + 6


10. Solve the following pair of simultaneous equations.

2𝑦 = 5𝑥 + 3

𝑦 = 5 − 𝑥


13.2: Quadratic graphs

1. Fill in the missing coordinate for each of the functions below:

a)

b)

c)

𝑦 = 𝑥2 + 3

( 1 , … ) ( −1 , … )

(0 , … )

(… , 12)

𝑦 = 2𝑥2 + 1

( 1 , _ ) ( −1 , _ )

(0 , _ )

(… ,19)

𝑦 = 𝑥2 + 2𝑥

( 1 , _ ) ( −2 , _ )

(0 , _ )

(… ,3)

(… , 12)

(… ,19)

(… ,3)


d)

e)

f)

𝑦 = −𝑥2

( 10 , _ ) ( −4 , _ )

(−3 , _ )

( _ , −25)

𝑦 = 3 − 𝑥2

( 1 , _ ) ( −1 , _ )

(0 , _ )

( _ , −6)

𝑦 = 2𝑥2 − 3𝑥

( 1 , _ ) ( 5 , _ )

(−3 , _ )

( _ , 0)

( _ , −25)

( _ , −6)


2. Complete the tables for each of the function below:

a) 𝑦 = 𝑥2 − 5

b) 𝑦 = 𝑥2 + 5

c) 𝑦 = 2𝑥2 − 5

d) 𝑦 = 2𝑥2 + 5𝑥

b) 𝑦 = −𝑥2 − 5

e) 𝑦 = −𝑥2 − 5𝑥

f) 𝑦 = −2𝑥2 + 5

𝑥 −1 0 1 2 10

𝑦

𝑥 −1 0 1 2 10

𝑦

𝑥 −1 0 1 2 10

𝑦

𝑥 −1 0 1 2 10

𝑦

𝑥 −1 0 1 2 10

𝑦

𝑥 −1 0 1 2 10

𝑦

𝑥 −1 0 1 2 10

𝑦


3.

a) Complete the table and then plot the graphs.

𝒙 −𝟐 −𝟏. 𝟓 −𝟏 −𝟎. 𝟓 𝟎 𝟎. 𝟓 𝟏 𝟏. 𝟓 𝟐

𝑦 = 𝑥2 4 2.25 1 0.25 0 0.25 1 2.25 4

𝑦 = 2𝑥2

b) Compare the graphs of 𝑦 = 𝑥2 and 𝑦 = 2𝑥2.

Describe what the graph of 𝑦 = 3𝑥2 would look like.


Reflections








13.3: Further quadratic graphs

1. Complete the table, and then draw the graph.

𝒙

𝟎 𝟏 𝟐 3 4 5

𝑦 = 𝑥2 − 2


2.

a) Complete the table below.

𝒙

−𝟏 0 𝟏 2 3 4

𝑦 = 𝑥2 − 3𝑥 −2

b) Plot the graph of 𝑦 = 𝑥2 − 3𝑥


3.

a) Complete the table below.

𝒙

−𝟐 −𝟏 𝟎 1 2 3 4 5

𝑦 = 𝑥2 − 3𝑥 − 4 0 −4

b) Draw the graph of 𝑦 = 𝑥2 − 3𝑥 − 4.


4.

a) Complete the two tables.

𝒙 −𝟓 −𝟒 −𝟑 −𝟐 −𝟏 𝟎 1 2

𝑦 = 2𝑥2 − 3 15

𝒙 −𝟓 −𝟒 −𝟑 −𝟐 −𝟏 𝟎 1 2

𝑦 = 2𝑥2 + 5𝑥 − 3 0

b) Plot both graphs on the axes below.


5. Plot each of these graphs on the grid provided.

𝒙 −𝟑 −𝟐 −𝟏 𝟎 𝟏 𝟐 𝟑 𝟒 𝟓 𝟔

a) 𝑦 = 2𝑥2 − 7𝑥 − 4 18 −7

b) 𝑦 = 2𝑥2 − 17𝑥 + 30 72 −3

c) 𝑦 = 3𝑥2 − 7𝑥 − 6 20 0


13.4: Exponential, reciprocal and piecewise linear graphs

1. A van driver charges a flat fee of £5 to deliver a parcel up to 2 miles.

For deliveries over 2 miles, additional costs are calculated at the rate of £2 per mile.

a) Draw a graph to represent this.

b) What is the distance that the van driver will travel for a payment of £21?


2. The number of bacteria on a Petri dish double every hour.

There are 120 bacteria to begin with.

a) Complete the table and plot the resulting graph.

Number of hours

0 1 2 3 4 5 6

Number of bacteria

120

b) Use your graph to estimate when there be 5000 bacteria on the Petri dish.


3. An investment of £250,000 starts to lose 15% of its value each month.

a) If this continues without any changes, draw a graph of the value of the investment over ten

months. Use the table below to help you.

Time (months) Value (nearest £) Time (months) Value (nearest £)

0 250,000 6

1 212,500 7

2 8

3 153,531 9

4 10

5

b) After how many months will the value of the investment fall below one quarter of its

original value?


4. The current passing through a wire (measured in Amperes), is inversely proportional to

the resistance (measured in Ohms).

a) Complete the table and plot the graph.

Current, 𝑰 (Amperes)

1 2 4 6 8 10 12

Resistance, 𝑹 (Ohms)

6

𝑰 × 𝑹

12

b) What is the resistance when the current is 10 Amperes?


5. The pressure, P Pascals, of a fixed mass of gas at constant temperature is inversely

proportional to its volume, V m3.

Complete the table and plot the graph.

Pressure, 𝑷 (Pascals)

5 10 20 50 100 200

Volume, 𝑽 (m3)

50

𝑷 × 𝑽


Reflections








Reflections


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Spring Year 9 - Home - Oasis Academy Lister Park · 2020-03-26 · (constructing triangles, perpendicular bisectors and angle bisectors) Throughout this section, make sure you show