Scale drawings and bearings - Teacher Slides.pptx

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Scale drawings and bearings

Transformations and constructions

Lesson 5 of 8

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Warm-up

Transformations and constructions

Lesson 5 of 8

Answer each one, then check.

1

How many centimetres are in a kilometre?

2

What do the angles round a point add up to?

3

What do co-interior angles add up to?

4

Work out 3.2 × 50 000.

5

What is the compass direction opposite to north-east?

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Warm-up - Answers

Transformations and constructions

Lesson 5 of 8

1

How many centimetres are in a kilometre?

100 000

2

What do the angles round a point add up to?

360°

3

What do co-interior angles add up to?

180°

4

Work out 3.2 × 50 000.

160 000

5

What is the compass direction opposite to north-east?

South-west

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Learning Objectives

Transformations and constructions

Lesson 5 of 8

1

Use a scale in the form 1 : n to change between drawing and real lengths.

2

Measure and write three-figure bearings.

3

Find the bearing back from B to A.

4

Solve problems using bearings and right-angled triangles.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Map Scales

Transformations and constructions

Lesson 5 of 8

A scale of 1 : 50 000 means 1 cm on the map is 50 000 cm in real life.

Map to real

Multiply by the scale, then change units. 3.2 cm on a 1 : 50 000 map is 160 000 cm, which is 1.6 km.

Real to map

Divide by the scale. 4.5 km is 450 000 cm, so on a 1 : 25 000 map it is 450 000 ÷ 25 000 = 18 cm.

Units

100 000 cm = 1 km and 100 cm = 1 m.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Transformations and constructions

Lesson 5 of 8

BEARINGS

A bearing is an angle measured clockwise from north, always written with three figures.

North is 000°, east is 090°, south is 180° and west is 270°.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Reading a Bearing

Transformations and constructions

Lesson 5 of 8

Measure clockwise from north, at the point you are travelling from.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Measuring a Bearing

Transformations and constructions

Lesson 5 of 8

Follow the same steps every time.

1

Find the starting point

The bearing of B from A starts at A.

2

Draw or imagine a north line at A

It points straight up the page.

3

Measure clockwise

From the north line round to the line AB.

4

Write three figures

Add zeros at the front, e.g. 065°.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

A Back Bearing

Transformations and constructions

Lesson 5 of 8

The bearing of B from A is 065°. Work out the bearing of A from B.

1

The north lines at A and B are parallel

Co-interior angles add up to 180°

2

Add 180° to a bearing under 180°

065 + 180

3

Work it out

245

ANSWER

245°

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

A Journey with Bearings

Transformations and constructions

Lesson 5 of 8

A ship sails 8 km on a bearing of 070° and then 6 km on a bearing of 160°. How far is it from its starting point?

1

The two bearings differ by

160 − 70 = 90, so the path turns through a right angle

2

Use Pythagoras

8² + 6² = 64 + 36 = 100

3

Distance

√100 = 10

ANSWER

10 km

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Using a Map Scale

Transformations and constructions

Lesson 5 of 8

Two towns are 3.2 cm apart on a map with scale 1 : 50 000. How far apart are they in real life?

1

Real length in cm

3.2 × 50 000 = 160 000

2

Change to kilometres

160 000 ÷ 100 000

ANSWER

1.6 km

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Back Bearings

Transformations and constructions

Lesson 5 of 8

Add or subtract 180 so the answer stays between 0 and 360.

Bearing of B from A

Rule

Bearing of A from B

065°

+ 180°

245°

130°

+ 180°

310°

250°

− 180°

070°

305°

− 180°

125°

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Key Terms

Transformations and constructions

Lesson 5 of 8

Bearing

An angle measured clockwise from north, written with three figures.

Scale

The ratio of a length on a drawing to the real length.

Back bearing

The bearing of A from B when you know the bearing of B from A.

Clockwise

The direction the hands of a clock turn.

Three-figure bearing

A bearing written with hundreds, tens and units, such as 045°.

Compass point

One of N, E, S, W and their combinations.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Transformations and constructions

Lesson 5 of 8

YOUR TASK

Treasure Trail

15 minutes

Start at a point marked A. Draw the route: 5 cm on a bearing of 050°, then 4 cm on a bearing of 140°, then 6 cm on a bearing of 230°. Measure the final distance and bearing back to A. If the scale is 1 cm to 100 m, work out the real distance.

1

Draw a north line at each point.

2

Use a protractor from north.

3

Convert with the scale.

WHAT A GOOD ANSWER SHOWS

Students should end near the start; the route back to A is roughly 3 to 4 cm on a bearing near 300°. Real distance is about 300 to 400 m.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Can I...?

Transformations and constructions

Lesson 5 of 8

Use a map scale 1 : n.

Convert between cm, m and km.

Measure a bearing.

Write a bearing with three figures.

Find a back bearing.

Draw a bearing accurately.

Use bearings with Pythagoras.

Explain why the north lines are parallel.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Summary & Exam Focus

Bearings are measured clockwise from north, with three figures.

Back bearing: ± 180°.

Scale 1 : n: multiply to get the real length; divide to get the drawing length.

Draw a sketch and mark north lines.

EXAM FOCUS

The bearing of B from A is 072°. Work out the bearing of A from B. (3 marks)

Add 180 when the bearing is less than 180, and subtract 180 when it is more. Draw a north line at both points and mark the co-interior angles.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Exam Practice: Scale Drawings and Bearings

Transformations and constructions

Lesson 5 of 8

Answer all questions. Show your working. · 30 minutes

Question 1 · 2 marks · Non-calculator

The scale on a map is 1 : 50 000. Two towns are 3.2 cm apart on the map. Work out the real distance between the towns, in kilometres.

Question 2 · 2 marks · Non-calculator

Two villages are 4.5 km apart. Work out the distance between them on a map with scale 1 : 25 000. Give your answer in centimetres.

Question 3 · 3 marks · Non-calculator

The diagram shows two points, A and B. The bearing of B from A is 072°. Work out the bearing of A from B.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Exam Practice: Scale Drawings and Bearings (cont.)

Transformations and constructions

Lesson 5 of 8

Question 4 · 4 marks · Calculator

A ship sails 8 km on a bearing of 070°. It then sails 6 km on a bearing of 160°. Work out the distance of the ship from its starting point.

Question 5 · 2 marks · Non-calculator

A lighthouse is due south-west of a harbour. Write down the bearing of the lighthouse from the harbour.

Question 6 · 2 marks · Non-calculator

The bearing of Q from P is 250°. Work out the bearing of P from Q.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 1 · 2 marks · Non-calculator

Transformations and constructions

Lesson 5 of 8

“

The scale on a map is 1 : 50 000. Two towns are 3.2 cm apart on the map. Work out the real distance between the towns, in kilometres.

HOW TO ANSWER IT

Command word: Non-calculator. Worth 2 marks, so plan before writing.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 1 · mark scheme

Transformations and constructions

Lesson 5 of 8

2 marks available. Award a mark for each point made.

3.2 × 50 000

M1

1.6

A1

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 2 · 2 marks · Non-calculator

Transformations and constructions

Lesson 5 of 8

“

Two villages are 4.5 km apart. Work out the distance between them on a map with scale 1 : 25 000. Give your answer in centimetres.

HOW TO ANSWER IT

Command word: Non-calculator. Worth 2 marks, so plan before writing.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 2 · mark scheme

Transformations and constructions

Lesson 5 of 8

2 marks available. Award a mark for each point made.

450 000 seen

M1

18

A1

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 3 · 3 marks · Non-calculator

Transformations and constructions

Lesson 5 of 8

The diagram shows two points, A and B. The bearing of B from A is 072°. Work out the bearing of A from B. (3 marks)

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 3 · mark scheme

Transformations and constructions

Lesson 5 of 8

3 marks available. Award a mark for each point made.

180 + 72 or 360 − (180 − 72)

M1

252

A1

A reason: parallel lines or back bearing

C1

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 4 · 4 marks · Calculator

Transformations and constructions

Lesson 5 of 8

“

A ship sails 8 km on a bearing of 070°. It then sails 6 km on a bearing of 160°. Work out the distance of the ship from its starting point.

HOW TO ANSWER IT

Command word: Calculator. Worth 4 marks, so plan before writing.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 4 · mark scheme

Transformations and constructions

Lesson 5 of 8

4 marks available. Award a mark for each point made.

Right angle found

M1

8² + 6²

M1

100

A1

10

A1

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 5 · 2 marks · Non-calculator

Transformations and constructions

Lesson 5 of 8

“

A lighthouse is due south-west of a harbour. Write down the bearing of the lighthouse from the harbour.

HOW TO ANSWER IT

Command word: Non-calculator. Worth 2 marks, so plan before writing.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 5 · mark scheme

Transformations and constructions

Lesson 5 of 8

2 marks available. Award a mark for each point made.

225

B2

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 6 · 2 marks · Non-calculator

Transformations and constructions

Lesson 5 of 8

“

The bearing of Q from P is 250°. Work out the bearing of P from Q.

HOW TO ANSWER IT

Command word: Non-calculator. Worth 2 marks, so plan before writing.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 6 · mark scheme

Transformations and constructions

Lesson 5 of 8

2 marks available. Award a mark for each point made.

250 − 180

M1

070

A1

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