Graphing rates of change.pptx

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Graphing rates of change

Graphs

Lesson 3 of 8

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Last Lesson and Before

Graphs

Lesson 3 of 8

Answer each one, then check.

1

Last lesson: what is the gradient between (0, 0) and (2, 60)?

2

A car travels 120 km in 2 hours. What is its average speed?

3

How many minutes are in 0.75 hours?

4

Area of a triangle with base 4 and height 12?

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Last Lesson and Before - Answers

Graphs

Lesson 3 of 8

1

Last lesson: what is the gradient between (0, 0) and (2, 60)?

30

2

A car travels 120 km in 2 hours. What is its average speed?

60 km/h

3

How many minutes are in 0.75 hours?

45 minutes

4

Area of a triangle with base 4 and height 12?

24

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Learning Objectives

Graphs

Lesson 3 of 8

1

Read and draw distance-time graphs.

2

Work out speed from the gradient of a distance-time graph.

3

Work out average speed for a whole journey.

4

(Higher) Use the gradient of a velocity-time graph for acceleration, and the area under it for distance.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Reading a Distance-Time Graph

Graphs

Lesson 3 of 8

Steeper means faster; flat means stopped; going down means coming back.

On a distance-time graph the gradient is the speed. The first section rises 30 km in 1 hour: 30 km/h. The flat section is a 30-minute stop. The return covers 30 km in 45 minutes: 30 ÷ 0.75 = 40 km/h.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Distance-Time Graphs

Graphs

Lesson 3 of 8

Distance goes up the side, time along the bottom.

Gradient is speed

Speed = distance/time - the change up over the change across.

Horizontal line

Not moving: the distance from the start is not changing.

Steeper

Faster.

Minutes to hours

For km/h, change minutes to hours first: 45 minutes = 0.75 hours.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Speed from the Graph

Graphs

Lesson 3 of 8

On the journey in the diagram, work out (a) the speed on the way home, and (b) the average speed for the whole journey, including the stop.

1

(a) Distance and time on the way home

30 km in 45 minutes = 0.75 hours

2

Speed

30 ÷ 0.75 = 40 km/h

3

(b) Total distance and total time

60 km in 2 hours 15 minutes = 2.25 hours

4

Average speed

60 ÷ 2.25 = 26.66…

ANSWER

(a) 40 km/h (b) 26.7 km/h

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Other Rates of Change

Graphs

Lesson 3 of 8

Any graph of a quantity against time has a gradient that is a rate.

Water level against time

Gradient = how fast the level rises (cm per minute).

Cost against number

Gradient = cost of one more item.

Units of the gradient

Units up the side "per" units along the bottom: km per hour, litres per minute.

Curved graphs

A curve means the rate is changing: steeper parts change faster.

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PART TWO · HIGHER

Velocity-Time Graphs

Gradient is acceleration; area is distance.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Velocity-Time Graphs

Graphs

Lesson 3 of 8

Now velocity (speed) is up the side.

Gradient is acceleration

Acceleration = (change in velocity)/time, in m/s².

Negative gradient

Deceleration: slowing down.

Horizontal line

Constant speed - NOT stopped.

Area is distance

The area under the graph is the distance travelled. Split it into triangles, rectangles and trapeziums.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

A Car Journey

Graphs

Lesson 3 of 8

A car accelerates uniformly from rest to 12 m/s in 4 seconds, travels at 12 m/s for 10 seconds, then slows uniformly to rest in 6 seconds. Work out (a) the acceleration in the first 4 seconds, and (b) the total distance travelled.

1

(a) Gradient of the first section

12 ÷ 4 = 3 m/s²

2

(b) Area: first triangle

½ × 4 × 12 = 24

3

Rectangle

10 × 12 = 120

4

Last triangle

½ × 6 × 12 = 36

5

Add

24 + 120 + 36 = 180

ANSWER

(a) 3 m/s² (b) 180 m

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Don't Mix Them Up

Graphs

Lesson 3 of 8

DISTANCE-TIME

• Gradient = speed

• Horizontal = stopped

• Going down = coming back

• Area means nothing useful

VELOCITY-TIME (HIGHER)

• Gradient = acceleration

• Horizontal = constant speed

• Going down = slowing down

• Area = distance travelled

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Key Terms

Graphs

Lesson 3 of 8

Speed

Distance travelled per unit of time.

Average speed

Total distance divided by total time, including stops.

Rate of change

How fast one quantity changes compared with another; the gradient.

Velocity

Speed in a given direction.

Acceleration

Rate of change of velocity, in m/s².

Deceleration

Slowing down; a negative acceleration.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Graphs

Lesson 3 of 8

YOUR TASK

Tell the Story

10 minutes

Draw a distance-time graph for this journey: Priya walks 2 km to a friend's house at 4 km/h, stays for 20 minutes, then they both cycle 6 km to the park in 20 minutes. Then write your own journey story for a partner to graph.

1

Work out each time first.

2

Plot the corners of the journey.

3

Join with straight lines.

WHAT A GOOD ANSWER SHOWS

Walk: 2 km in 30 minutes. Flat for 20 minutes (to 50 minutes). Cycle: from 2 km to 8 km between 50 and 70 minutes - a steeper line, 18 km/h.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Can I...?

Graphs

Lesson 3 of 8

Read a distance-time graph.

Draw a distance-time graph.

Work out speed from a gradient.

Work out an average speed.

(Higher) Work out acceleration from a velocity-time graph.

(Higher) Work out distance from the area under a velocity-time graph.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Summary & Exam Focus

Distance-time: gradient = speed; flat = stopped.

Average speed = total distance ÷ total time.

(Higher) Velocity-time: gradient = acceleration; area = distance.

EXAM FOCUS

The distance-time graph shows Tom's cycle ride. Work out Tom's speed on his way home, in km/h. (2 marks)

Read the scales carefully - small squares are often 5 or 10 minutes, not 1. Convert minutes to hours before dividing to get km/h.

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