**Gradient and Rates of Change**

Starter

Example

Example

**L.O. - Understand the gradient as indicating the rate of change of a linear function.**

Explain why the red line shows a proportional relationship and the blue line doesn't.

Given the red line and blue line are parallel, what is true about both the blue line and the red line.

Starter

Explain why the red line shows a proportional relationship and the blue line doesn't.

Given the red line and blue line are parallel, what is true about both the blue line and the red line.

The red line passes through the point (0, 0) and the blue line doesn't.

They both have the same gradient.

The graph shows how the cost of a ticket to the Super Bowl has changed over time.

(a) Work out the rate of change from 1995 to 1996, giving the units of your answer.

(b) State the year that showed the greatest change and work out the rate of change in that year.

Example

The graph shows how the cost of a ticket to the Super Bowl has changed over time.

(a) Work out the rate of change from 1995 to 1996, giving the units of your answer.

75 $/year

(b) State the year that showed the greatest change and work out the rate of change in that year.

2005 to 2006, $100/year.

This graph shows how the speed of an object changes over time.

(a) What does the gradient of each line represent in this graph?

(b) Work out the rate of change of speed over the first 10 seconds.

(c) Work out the rate of change of speed over the last 5 seconds.

Example

This graph shows how the speed of an object changes over time.

(a) What does the gradient of each line represent in this graph?

Acceleration.

(b) Work out the rate of change of speed over the first 10 seconds.

1.6 m/s

(c) Work out the rate of change of speed over the last 5 seconds.

-3.2 m/s

2

2

Activity

Complete the 'Gradients and Rates of Change' worksheet.

Activity

Complete the 'Gradients and Rates of Change' worksheet.

1) a) £1 b) £28

2)

Mass (Kg) 1 2 3 4 5

Volume (cm ) 1.08 2.15 3.23 4.30 5.38

Gradient = 0.93

3)

0 1 2 3 4 5 6 7 8 9 10

1.6 2.4 3.2 4 4.8 5.6 6.4 7.2 8 8.8 9.6

(ii) 1.6 means that at birth piglets weigh 1.6 kg

(ii) 0.8 indicates an increase of 0.8Kg per day

.

4) Fred 2hr at 4m/hr stops 2 hrs and then walks home in 2hrs at 4m/hr

Arthur 3 hrs at 1 1/3 mph, stops for 1hr, then walks home in 2hrs at 2m/hr

Activity

Complete the 'Gradients and Rates of Change' worksheet.

5)

0 1 2 3 4 5 6 7 8 9 10

1.2 4.2 7.2 10.2 13.2 16.2 19.2 22.2 25.2 28.2 31.2

(i) 1.2 A basic charge regardless of the length of advert e.g. an administration fee.

(ii) 3 indicates an Increase of £3 per additional line.

6) Car 60mph to Southley. It leaves Southley at 10.20 and arrives in Lawntown 11.00.

Coach arrives in Winchester at 9 20 leaves at 10.20 and travels at 40mph.

The car arrives first by 5 minutes.

Activity

Complete the 'Gradients and Rates of Change' worksheet.

7 a) 1 m/s2 b) 13m/s , – 0.8 m/s2

8) -10 l/min. The tank is emptying at a rate of 10 litres per minute.

9) The rate of increase was £20 a month for 3 months, then £35 a month for two months, then there was a decrease of £40 in December and finally a rate of increase of £20 per month for the last 6 months.

10) OA accelerates at 2m/s2, AB constant speed (10 m/s) for 5 seconds, BC decelerates at 3 1/3 m/s2. CD at rest for 3seconds, DE accelerate 1 m/s2 and finally EF constant speed 5m/s for 4 seconds.

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