1. © Nuffield Foundation 2011 Free-Standing Mathematics Activity Speed and distance

2. © Nuffield Foundation 2011 How can we model the speed of a car? How will the model show the distance the car travels?

3. © Nuffield Foundation 2011 Using a graph Car travelling at 70 mph Area = 2 × 70 = 140 This is the distance travelled, 140 miles v mph t hours 0 2 70 Think about How far will it travel in 2 hours? Think about How far will it travel in 2 hours?

4. © Nuffield Foundation 2011 Car accelerating steadily from 0 to 72 kph in 10 seconds Distance travelled = 100 metres = 100 v kph t seconds 0 10 72 72 kph = = 20 metres per second 20 ms -1 Area of triangle Think about What was the car’s average speed? What is the connection with the graph? Think about What was the car’s average speed? What is the connection with the graph?

5. © Nuffield Foundation 2011 Car accelerating steadily from 18 ms -1 to 30 ms -1 in 5 seconds Distance travelled = 120 metres = 24 × 5 Area of a trapezium Area = v ms -1 t seconds 0 18 30 5 = 120 a b h Think about What was the car’s average speed? What is the connection with the graph? Think about What was the car’s average speed? What is the connection with the graph?

6. © Nuffield Foundation 2011 Car travelling between 2 sets of traffic lights Area of A t (s) v (ms -1 ) 0 0 2 5 4 8 8 8 10 5 12 0 6 9 0 v ms -1 t seconds 12 624810 A A C C B B = 70 Area of C = 17 Area of B = 13 Total area = 5 Distance travelled = 70 metres 8 5 9 Think about Why are the strips labelled A, B & C? How will this help to find the area? Think about Is this a good estimate? How can it be improved? Is the graph realistic? Think about Is this a good estimate? How can it be improved? Is the graph realistic?

7. © Nuffield Foundation 2011 Area = Car travelling with speed v = 0.5 t 3 – 3 t 2 + 16 t (s) v (ms -1 ) 01243 1613.5 8 0 2.5 v t 0 16 4 2 1 3 13.5 8 2.5 = 14.75 + 10.75 + 5.25 + 1.25 Distance travelled = 32 metres Think about What did this car do? Think about What did this car do? Think about How could this estimate be improved? Think about How could this estimate be improved?

8. © Nuffield Foundation 2011 At the end of the activity Explain why using triangles and trapezia can only give an estimate of the area under a curve When is an estimate smaller than the actual value? When is it larger? How can you improve the estimate? How well do you think the graphs and functions you have studied model the actual speed of real cars? In what way would graphs showing actual speeds differ from those used in this activity?