How fast is fast?
This unit is designed to provide students with a measurement context in which rates are used. Students will calculate speed from distance and time and apply number strategies as they convert between units.
About this resource
Specific learning outcomes:
- Calculate speed from measured distance and time.
- Solve multiplication problems using appropriate strategies (e.g., doubling and halving).
- Use known multiplication facts to solve multiplication problems.
How fast is fast?
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.
Description of mathematics
This unit explores the concept of speed, meaning a rate connecting two measurements: distance travelled and elapsed time. With these 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 the 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.
Opportunities for adaptation and differentiation
The learning opportunities in this unit can be differentiated by providing or removing support for 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
- using structured recording methods like double number lines and ratio tables to support student thinking (see the notes within the unit)
- using calculators to ease the burden of calculation while retaining an emphasis on understanding speed as a rate
- valuing the multiplication strategies demonstrated by students and strengthening and refining these while moving students towards the use of more efficient mental strategies
- providing opportunities for students to work in collaborative groups with students with different levels of mathematical knowledge and confidence.
The topic for this unit is speed. Choose contexts, like those in the unit, that reflect the interests of your students. While most students will 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 the first person to measure the speed of sound? How did they do it? Investigate the Pony Express in the USA. How fast did the horses travel? The context of running 1 km, presented in Session 2, could also present a context for exploring fitness or speed in different sports.
Te reo Māori kupu such as tere (speed, velocity, fast), pāpātanga (rate) could be introduced in this unit and used throughout other mathematical learning.
Required materials
- bike
- measuring tapes, metre rulers, and trundle wheels
- note paper and pencils
- calculators
- online access
- stopwatches (optional) or mobile phones
Activity
In this session, we discuss speed. What does it mean? How fast is fast?
1.
Ask students the question:
- How fast is fast?
2.
Clarify that you are referring to objects travelling at high speed. Discuss and record students' ideas. List some objects that travel fast. Expect responses to come in a range of forms, e.g., 100km 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 are a fascinating story, as is how you can tell how far away a lightning strike is from you by the time thunder is heard. This could be investigated as part of a wider, integrated unit.
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?
5.
Support and guide the discussion to reach the conclusion that all of these speeds have both a distance unit and a time unit. 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.
6.
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.
7.
Pose the question:
- How fast am I travelling, on average, if I travel 100 kilometres in 2 hours?
8.
As the students give their answers, encourage them to explain the strategy they used to find them.
- I went at 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.
9.
Pose further questions of the same type:
- 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?
10.
Ask the students to work online to find out the speeds of some examples 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?
11.
Make a class display of speeds in the real world. You might choose to structure this or provide guidelines around the way students present their information.
In this session, students find their average speed over a 1-kilometre run in kilometres per hour.
1.
Ask students to estimate how fast they think they can run 1 kilometre. Record students’ predictions.
2.
Ask students how far they think a kilometre is. Encourage students to refer to benchmarks 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 around 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 1 km using cones.
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. This could also be done in pairs, with half of the class running and half of the class timing, before switching roles.
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 the strategies used. Use a double number line to help students 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. 1 km in 5 min = 12 km per hour; 1 km in 6 min = 10 km 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 1 kilometre.
60 ÷ 8 = 7.5 gives how many times 8 minutes fit 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 (1000 m).
12.
The world records for the 1000 m are:
- Men: 2:11.96 by Noah Ngeny (1999)
- Women: 2:28.98 by Svetlana Masterkova (1996)
One of the most amazing facts about these records is that they have remained unbeaten. 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.
14.
As an extension, students could investigate the world record 1000 m speeds for other sports (e.g., swimming, rowing, cycling). During this time, you can check in with students and provide additional support and direct teaching to those who need it.
In this session, students find their speed over 100 metres, both in metres per second and in kilometres per hour.
1.
Ask students to estimate how fast they think they can run 100 metres. Record students’ predictions. Discuss whether they think they can run 100 metres faster than they did 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 that is 100 metres long, preferably straight.
5.
Time students running the course. Give each student their time in seconds. Have notepaper and pencils 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 calculations where needed.
- I took 20 seconds and went 100 metres. I know that 20 goes into 100 five times, so I must have run 5 metres each second.
It took me 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 it 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 kilometres 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.
14.
As a challenge, ask:
- 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.
15.
As a further extension, students could investigate the world record 100m speeds for other sports (e.g., swimming, rowing, cycling). During this time, you can check in with students and provide additional support and direct teaching to those who need it.
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 (100 m and 1 km) 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 the 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 (2013)
- Women: 66.144 seconds by Anastasiia Voinova (2019)
7.
Ask students to work out the approximate speeds of the world record holders in kilometres per hour.
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 200 m runners.
The world records for the 200 m 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.
6.
Return together and discuss the findings.
7.
Look up the speeds of animals.
- Do you run faster than a bull? …a cat? ...a dog? ...a horse? …a hedgehog? …a snail?
8.
As in the previous sessions, encourage students to share their calculation strategies.
9.
Discuss: How fast is fast?
Home link
Parents and whānau,
This week we have been investigating speed and measuring our speed for different activities. As part of this, we are doing some research, reading in maths, and presenting our findings. Ask your child to tell you about what they have found out and how they are going to present their information to the class. Your child has selected a research topic from the following:
- Compare the speed of human world record holders to several different animals using a graph.
- Graph the men's and women's 100 m sprint times for the past 10 Olympic Games.
- Research an animal with a long migration route (for example, godwits or eels) and calculate the speed at which they migrate.
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