# How fast is fast?

Purpose

This unit is designed to provide students with a measurement context in which rates are used. In calculating speed from distance and time, they will apply number strategies as they convert between units.

Achievement Objectives
GM4-1: Use appropriate scales, devices, and metric units for length, area, volume and capacity, weight (mass), temperature, angle, and time.
GM4-2: Convert between metric units, using whole numbers and commonly used decimals.
NA4-1: Use a range of multiplicative strategies when operating on whole numbers.
Specific Learning Outcomes
• Calculate speed from measured distance and time
• Solve multiplication problems using doubling and halving strategies
• Use known multiplication facts to solve multiplication problems
Description of Mathematics

This unit explores the concept of speed that is a rate connecting two measurements, distance travelled, and elapsed time. Given two measurements speed can be calculated, e.g., 120 kilometres in 2 hours equates to a rate of 120 ÷ 2 = 60 kilometres per hour (km/h). Rates can be converted to different units to meet a purpose. For example, at 60 km/h what is the speed using minutes as the unit of time (useful in predicting time to get to a destination)? 60 kilometres per hour equals 60 kilometres per 60 minutes. Since 60 ÷ 60 = 1 the speed can also be expressed as 1 kilometre per minute.

The learning opportunities in this unit can be differentiated by providing or removing support to students and by varying the task requirements. Ways to differentiate include:

• Acting out the relationship between distance and time using lengths and times that are easily visible to establish how the attribute of speed works, e.g., Mark out ten metres and time walking the length at different speeds.
• Use structured recording methods like double number lines and ratio tables to support student thinking (notes within the unit).
• Use calculators to ease the burden of calculation while retaining an emphasis on understanding speed as a rate.
• Working in collaborative groups so students can leverage off the ideas of others.

The topic for this unit is speed. Choose contexts like those in the unit that reflect the interests of your students. While most students while enjoy the personal aspect of measuring speed there might also be topical situations that capture their interest. For example, Are cars observing the 40 km/h speed limit when driving past the school? An America’s cup yacht sails at 50 knots (93 km/h). How long will it take to sail between marks? What does it mean for a jet to fly faster than the speed of sound?

Students might also enjoy historical investigations into speed, e.g., Who was first person to measure the speed of sound? How did they do it? Investigate the pony express in USA. How fast did the horses travel?

Required Resource Materials
• Bike
• Measuring tapes,,metre rulers, trundle wheels
• Note paper and pencils
• Calculators
• Online access
• Stopwatches (optional) or mobile phones
Activity

#### Session 1

In this session we discuss speed. What does it mean? How fast is fast?

1. Ask students the key question:
How fast is fast?

Clarify that you are referring to objects travelling at high speed. List some objects that travel fast.
2. Record some answers on the board. Expect responses to come in a range of forms, e.g., 100 km an hour, 100 miles an hour, a cheetah running, the speed of light, 100 ‘kay’(meaning kilometres per hour), 10 metres per second, etc. You may wish to record the speed of light and sound under these examples if they are suggested, possibly after all other suggestions have been collected. The speed of light is about 300,000,000 metres per second and the speed of sound is about 340 metres per second (in air). Early attempts to measure the speed of sound using cannon shots is a fascinating story as is how you can tell how far from you a lightning strike is by the time it takes for thunder to be heard (Look online).
3. Ask students to tell you how they would measure the speed of a moving object, like a person, a ball, or a car. Discuss their suggestions.
What devices do we use to measure speed? (Note that a stopwatch measures time, not speed, but a speedometer on a car and a radar gun measure speed)
4. Refer back to the list of speeds that students said were fast.  Ask:
Which of these are actual speeds?
What do all the actual speeds have in common?

Support and guide the discussion to reach the conclusion that all have both a distance unit and a time unit involved in them. A rate also includes the word “per” which, in this case means “for every.” So 20 kilometres per hour means 20 kilometres are travelled for every hour that elapses.
5. Explain to students that speed is usually described by how far an object travels in a given amount of time. We usually use kilometres per hour or metres per second.
6. Pose the question:
How fast am I travelling, on average, if I travel 100 kilometres in 2 hours?
7. As the students give their answers encourage them to explain the strategy they used to find them:
I went half of 100 which is 50. So I was travelling 50 kilometres per hour.
I divided 100 by 2 to work out how far for each hour. 100 divided by 2 is 50.
8. Pose further questions of the same form:
How fast am I travelling if I travel 40 kilometres in half an hour?
How fast am I travelling if I travel 10 metres in 2 seconds?
9. Ask the students to work online to find out the speeds of some example that were listed in the first discussion.
How fast does a Cheetah run? How fast does a kārearea (falcon) fly?
How fast does a tennis ball travel when served?
How fast does a satellite travel?
How fast does the Earth travel?
10. Make a class display of speeds in the real world.

#### Session 2

In this session students find their average speed over a 1km run in kilometres per hour.

1. Ask students to estimate how fast they think they can run for 1 kilometre. Record students’ predictions.
2. Ask students how far they think a kilometre is. Use a unit like ‘lengths of the field’, or ‘laps of the tennis court’.
3. Explain to students that you are going to find out how fast they can actually run, by running a kilometre and timing how long it takes.
4. Move outside and measure either the length of the field or the distance round the tennis court and agree on a course that is 1 kilometre long. You could have an existing track for athletics or use Google Maps to mark out a distance of 1km using physical education cones or bottles of water.
5. Time students running around the course. Give each student their time in minutes and seconds. Have small pieces of paper and pencils so times are not forgotten.
6. Return to the classroom and challenge students to work out their speed in kilometres per hour. Give students access to calculators and an appropriate amount of time to work out their speed on their own, then return together as a class to discuss strategies used. Use a double number line to support students to visualise the relative size of the measures.
I took over 6 minutes, and there are 60 minutes in an hour. I know 6 x 10 is 60, so if I ran that speed for an hour I would go 10 kilometres.That means my speed is 10 kilometres per hour (10 kph or 10 km/h)

I took 4 minutes and 52 seconds to run 1 kilometre, and I know that that is nearly 5 minutes. There are 60 minutes in an hour, which is 12 lots of 5, so I ran at about 12 kilometres per hour.
7. Help students work out some ‘benchmarks’ as above. 1km in 5 min = 12km per hour, 1km in 6 min = 10km per hour. This will make it easier for them to approximate their own speeds.
Link working out the number of times a time fits into 60 minutes to division. For example, suppose a student takes 8 minutes to run 1kilometre.
60 ÷ 8 = 7.5 gives how many times 8 minutes fits into 60 minutes. That student is running at an average speed of 7.5 km/h.
8. Discuss how students could work out their speeds if they are between the ‘benchmarks'. List a few examples in a table:
 Time taken to run 1 kilometre Average speed 4 minutes 15 km/h 5 minutes 12 km/h 6 minutes 10 km/hr 8 minutes 7.5 km/h 12 minutes 5 km/h
9. If students are using mental strategies the nearest half or quarter kilometre per hour is reasonable. If students are using calculators, then you will need to discuss how many decimal places are appropriate.
How many minutes did you take?
How many minutes are there in an hour?
How many times will …go into…?
How many seconds did you take?
How many seconds are there in an hour?
How many times will …go into…?
10. Ensure that all students have found their speed in kilometres per hour, then compare that speed with their predictions from the start of the lesson.
11. Ask students how fast they think the fastest runners in the world can run 1 kilometre.
12. The world records for the 1000m are:
Men: 2:11.96 by Noah Ngeny (1999)
Women: 2:28.98 by Svetlana Masterkova (1996)

These records are amazing in that they have stood for over 20 years. Discuss the meaning of the numbers in the times.
2:11.96 means 2 minutes, 11.96 seconds.
How far around the 1 kilometre circuit were you when these athletes finished?
13. Ask students to work out the approximate speeds of the world record holders in kilometres per hour. Doing so accurately is quite difficult because time measurement is based on sixty, that is there are 60 seconds in a minute. You need to calculate as follows:
11.96 ÷ 60 = 0.199 which is close to 0.2. That means Ngeny took 2.2 minutes to run 1 kilometre. 60 ÷ 2.2 = 27.27 kilometres per hour was his speed.
28.98 ÷ 60 = 0.483 which is close to 0.5. That means Masterkova took about 2.5 minutes to run 1 kilometre. 60 ÷ 2.5 = 24 kilometres per hour was her speed.

#### Session 3

In this session students find their speed over 100m, both in metres per second and in kilometres per hour.

1. Ask students to estimate how fast they think they can run for 100 metres. Record students’ predictions. Discuss whether they think they can run faster for 100 metres than they did for a kilometre. Why or why not?
2. Ask students to describe how far they think 100 metres is.  They should be able to use their experience from the previous session to make a reasonable estimate.
3. Explain to students that today you are going to find out how fast they can actually run, by running 100 metres and timing how long it takes.
4. Move outside and agree on a course which is 100 metres long, preferably straight.
5. Time students running the course. Give each student their time in seconds. Have note paper and pencil available for students to record the time.
6. Back in the classroom challenge students to work out their speed in metres per second. First give students a couple of minutes to try to work out their speed on their own, then return together as a class to discuss strategies used. Use double number lines to support calculation where needed.
I took 20 seconds and went 100 metres. I know that 20 goes into 100 5 times so I must have run 5 metres each second.

I took 25 seconds to run 100 metres. That means I ran 20 metres in 5 seconds. Every second, I ran 4 metres because 5 x 4 = 20. My speed was 4 metres per second (4 m/s).
7. Discuss how students could work out their speeds if the numbers don’t work out evenly. If students are using mental strategies, then rounding to the nearest half (0.5) of a metre per second is probably reasonable. If students are using calculators, then you will need to discuss how many decimal places are appropriate.
How many seconds did you take? e.g., 16 seconds
How many metres would that be for each second? e.g. 100 ÷ 16 = 6.66…
If you share 100 metres among … seconds, how far did you go each second? e.g., 6.67 metres per second.

Ratio tables can support the progressive calculations of students. For example:
 Metres Seconds 100 20 50 10 5 1
8. Ensure that all students have found their speed in metres per second, then challenge them to convert the speed to kilometres per hour. They may want to start again from their time for 100 metres or work it out from their speed in metres per second. Discuss the different methods:
Let’s say you took 22 seconds to run 100 metres. At that speed, how many seconds would I take you to run 1 kilometre? (10 x 22 = 220 seconds).
How many seconds are in 1 hour? (60 x 60 = 3600 seconds)
How many lots of 220 seconds go into 3600 seconds? (3600 ÷ 220 = 16.36)
What is your speed? (16.36 kilometre per 3600 seconds or 16.36 kilometres per hour).
In a ratio table the calculations look like this:
 Metres Seconds 100 22 1000 220 1636.36 3600
9. Share other ways to convert the metres per second speed to kilometres per hour.
There are 60 seconds in a minute, so 6 metres per second is the same as 360 metres per minute, and there are 60 minutes per hour so 360 metres per minute is 60 x 360 metres per hour. 60 x 360 = 21 600 metres in 1 hour. My speed is 21600 kilometres per hour. I was running more than 6 metres per second, so that is more than 21.6 kilometres per hour.
16 seconds for 100 metres is the same as 160 seconds for 1 kilometre. That’s 2 minutes and 40 seconds. Two and a half minutes goes into an hour 24 times, so I was running less than 24 kilometres per hour.
10. Compare students’ speeds with their predictions from the start of the lesson.
11. Ask students how fast they think the fastest runners in the world can run 100 metres.
12. The world records for the 100m are:
Men:              9.58 seconds                Usain Bolt (2009)
Women:           10.49 seconds              Florence Griffith-Joyner (1998)
13. Ask students to work out the approximate speeds of the world record holders in kilometres per hour. They might use a ratio table to support their calculation steps:
 Metres Seconds 100 9.58 1000 95.8 37 578.28 3600

Usain Bolt averaged 37.58 kilometres per hour when he set the world record.

 Metres Seconds 100 10.49 1000 104.9 34 318.4 3600

Florence Griffith-Joyner averaged 34.32 kilometres per hour when she set the world record.

How far would you have been from the start when Usain and Florence crossed the finish line?
Students might be interested in seeing a video of each record-breaking run.

#### Session 4

In this session students will attempt to calculate the speed of student on a bike, both in metres per second and in kilometres per hour.

1. Ask students how fast they think a student on a bike can travel.
Can you ride twice as fast as you run?
How many times faster is your riding speed compared to your running speed?
Students might comment that the ratio of speed will change as the distances get longer.
2. Provide a student with a bike and allow them to carry out the experiment using both courses (100m and 1km) from the previous sessions.
3. Provide assistance, as required, with calculations as outlined in the previous session.
4. Encourage the students to share their strategies used in the calculations.
5. Discuss results as a class.
How much faster did you go on the bike?
Did you think it would be that much faster?
6. The world records for the 1 kilometre time trial are:
Men: 56.303 seconds by Francois Pervis
Women: 66.144 seconds by
Anastasiia Voinova
7. Ask students to work out the approximate speeds of the world record holders in kilometres per hour.

#### Session 5

In this session students are challenged to work out the speeds travelled by athletes in their world record performances over various distances.

1. Ask students what distances they think people can run the fastest.
2. Discuss why we can run faster over shorter distances.
3. Work out the speed travelled by the fastest 200m runners
The world records for the 200m are:
Men: 19.19 seconds by Usain Bolt (2009)
Women: 21.34  seconds by Florence Griffith-Joyner (1988)
4. Are these speeds slower or faster than for the 100 metres? Why or why not?
5. Challenge students to find the times for other races (either running, riding, or driving) and calculate their speeds. The IAAF (International Association of Athletic Federations) website is a good source of records. http://www.iaaf.org/statistics/records/