This unit is about making calculations in the real life situations of athletics. The main task is to construct a scale plan of an athletics track.
Specific Learning Outcomes
- Draw a plan to scale, of an object based on a rectangle and two semi-circles.
- Understand the relation of length on the plan to actual length.
- Find lengths and areas use these for costing purposes.
- Be able to link speed, distance and time (given two find the third).
- Estimate distances and times.
Description of Mathematics
In this unit proportional reasoning is required to solve realistic problems. Students work with scale drawing. That means that distances shown on the drawing are in proportion to those in the real space of an athletics track.
An appropriate scale for representing the track on an A3 sized piece of paper is 1cm:5m (1 centimetre per 5 metres). This means that one centimetre on the plan drawing is equivalent to 5 metres in real life. Students also need to deal with speed as a rate connecting distance with time. A speed of 6 metres per second means that six metres are covered for every second. There is an assumption that the rate applies uniformly, that is, to every second. In the case of both scale and speed, calculations involve multiplication and division. For example, to draw the distance of the straight section of track students might find that it is about 92 metres in real life. Since 92 ÷ 5 = 18.4 students need to make that length 18.4 centimetres on their scale drawing.
Required Resource Materials
- PowerPoint One
- A3 sized paper
- Rulers, compass, and other drawing tools
- A list of World and Olympic records
- Trundle wheels, tape measures, stop watches (if outside work is done)
In session 1 we ask the students to construct simplified athletics track that has the following properties:
- 400m for the inside perimeter
- 100m straight along one side
- Eight lanes
- Staggered starting lines for the 200m and the 400m across the lanes
- A finish line.
Athletics tracks usually consist of two semi-circular ends joined by straight sections.
In session 2 the students need to find the area and perimeters of the various shapes on their tracks. They calculate the area of the inner field to establish a realistic estimate of cost for re-surfacing.
For session 3 we look at different events and the speeds that runners achieve over different distances. We also look at how records would change if, say, a 100m runner could keep that pace up for 200m or 10,000m.
To help with this session you will need to find Olympic and World records as well as New Zealand national records. These can be done easily by searching on the internet. It would be preferable for your students to look these things up for themselves.
Sessions 1 and 2
In these sessions, students will construct the plan of an athletics track on a piece of A3 sized paper and use their plan to measure some distances.
- If possible, take the class to a track before you start this session. By pacing out, or by using a trundle wheel, students can measure the length of the straight sections and the radius of the semi-circles.
Alternatively use PowerPoint One to discuss important features of the track. As the questions appear invite students to share their ideas.
Do they identify that the track is a composite of two straight sections and two part-circles?
- Get the class to work in pairs to make a scale drawing on A3 paper of an athletics track. Provide Copymaster One. You may need prompts like:
What tools will be useful for the task?
How will you work out the length of the straight parts of the track?
How will you work out the length of the curved parts of the track? (Note that the part circles are at a radius of 36.5 metres.)
How will you work out the staggered starts? (Using the formula for the circumference of a circle, c = 2πr (two x pi x radius). Alternatively, students might use string and relate the length of each arc to the scale.)
What scale will you use for your track so that it takes as much of the page as possible?
What will you need to do before you get started? (Measure the paper; do several calculations; find a scale that will fit your track on your paper.)
Allow time for questions. (It might be useful to make a smaller draft drawing first to iron out any problems. It would probably be good to use pencil so that any errors can be erased easily.)
- While the students are working move around to give help and advice. Check their scales. Bring the class together if there are some common problems.
- After an appropriate time bring the class together for reporting back. Ask:
What would you do differently if you were to start again?
What was the most difficult part of the exercise?
Could you fit a hockey or soccer field inside your track? (Students will need to look up dimension tolerances for soccer fields. They may know from experience that soccer fields are often located within athletic tracks).
- The ‘throwing’ field events are usually held on a grassed area inside the runnoing track. You may need to discuss each event and show an online video so students know what is involved.
The world records for the events were as follows in 2019:
Shot put (23.12m); Discus (74.08m); Javelin (98.48m)
All events are safe, in that the missile will not hit the track if thrown straight. You might also work out the permissible angle of throw. For the shot put and discus the angle is about 35° and for the javelin the measure is about 29°.
Find locations on your diagram to place the throwing areas for each of these events so that the missiles will never hit the track if thrown within the permissable angle of throw.
Why is the angle for the javelin stricter than for the shot put and discus? Does it make much difference to the safety of a throw?
In this session students perform some calculations relating to the dimensions of the track.
- Suppose the local council decides to replace the grass with a new free draining playing surface. That involves laying a new gravel pad, putting in irrigation system, and putting in a new surface suitable for multi-sport events and concerts.
- A similar project at Napier’s McLean Park cost about $2million.
What is the cost per square metre?
Students will need to find the surface area of the interior. Let them investigate the problem in pairs. Look for them to:
Partition the field into three parts; two semi-circles and a rectangle.
Combine the semi-circles into one circle.
Apply the formulas for area of a circle and a rectangle.
- After a suitable time bring the class back together to process the solutions. A suitable answer might be:
Semi-circles/single circle; area = πr2 = 3.14 x 36.52 = 4 182.265 m2.
Rectangle; area = 84.4m x 73m = 6 132 m2.
The combined area is about 10 314 m2.
The cost per square metre is 2 000 000 ÷ 10 314 ≈ $194 per m2.
Sessions 4 and 5
In this session, we look at the relative speeds of Olympic and World record runners on the track.
- Discuss the relation between speed, distance and time. Students will know that speed of a car is usually measured in kilometres per hour (e.g. 60 kph)
What does 60 kilometres per hour mean?
- Discuss speed as a rate, a relationship of two measures (different attributes).
What does this equation mean? s = d/t
Students are usually not aware of the division meaning of the vinculum (horizontal line in a fraction). Aim at getting a word form of the formula, “Speed equals distance divided by time.”
- Give students some examples about travelling by car between two cities. Make sure that all three types of problem below are covered, that is one of the measures, speed, time or distance, is unknown. Let the students attempt the problems in small groups and compare strategies. Let students use calculators but expect that they check their answers for reasonableness.
If the distance between Dunedin and Christchurch is 360 km and it takes Grandpa 4.5 hours to do the trip, what is his average speed?
If the distance between Dunedin and Christchurch is 360 km and Grandpa’s average speed is 80 kph, how long does he take to do the trip?
If Grandpa’s average speed between Dunedin and Christchurch is 95 kph and it takes 3.75 hours to do the trip, how far did he travel?
- Tell the students that they will need to calculate speed in metres per second rather than kilometres per hour. That is because the distances are in metres and the times are in seconds.
Let each group choose three women’s or men’s running events, e.g. 100m, 400m, 5 000m.
What was the average speed of the holders of each of these records?
- Choose three events where the World, Olympic and New Zealand national records are different. Assume these runners were running against each other. Ask
By how much would the World record runner be ahead of the other two runners at the finish?
- Look at the women’s or men’s national records in the 100m, 400m, 1500m, and 10,000m events.
If the 100m runner could keep going at that pace the other distances, what would the records for each of the other events be?
If they competed in the same race, how far ahead of the other runners would the 100m runner be at the finish?
- You might time your students sprinting over 100m.
Which world record distance is your speed closest to?
For example, the world record for men over 10 000 m is 27 minutes and 1.17 seconds (1621.17 seconds). That is an average speed of 10 000 ÷ 1621.17 ≈ 6.17 metres per second (m/s).
A student who runs 100 metres in 15 seconds has an average speed of 100 ÷ 15 ≈ 6.67 m/s which is close.
- Discuss the results.
Was there anything that the students found surprising about these results?
- If you are giving this unit before an Olympics or Commonwealth Games, you might like to speculate on which records will be broken and by how much.
- Suppose that a woman runner broke the existing National record for the 100m by 0.1s. Estimate by how much distance she would beat the current record holder.
Now calculate this distance.
Repeat with the men’s 800m World record and the women’s 1500m Olympic record.
- Suppose that a woman runner beats the existing Olympic record runner in the 100m by 1m. Estimate by how many seconds she would beat the current record holder.
- Now calculate their time.
Work out your percentage error.
- Repeat the last exercise for the women’s National 800m record and the men’s World record 400m.
- So far, the men’s records are faster than the women’s records. A match race is planned between the women’s Olympic 100m record holder and the men’s 100m record holder.
What handicap would you give the woman in order to produce a dead heat?
- Calculate the handicap distance.
- Repeat the last exercise for the National 800m record and the National 5,000m record.
- Allow the students to set their own problems.
- Summarise the unit.