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Level Five > Geometry and Measurement

# Round the Track

Purpose:

This unit is about making calculations in the real life situations of athletics.

Achievement Objectives:

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

Calculate practical percentages

Description of mathematics:

In this unit relatively simple calculations are required to solve real problems.

The main task of the unit is to construct a scale plan of an athletics tack. There is no right magnitude for the dimensions involved. However, common sense says that there are some wrong ones. No track would have straight of length 1m or a curve section of length 3m. So the students need to be aware of actual tracks and the purposes for which they are used, and take these into consideration when drawing up their plans.

In order to get the record data needed to do calculations of the relative speed, times and distances of athletes students have to use internet searching skills.

During the work of this unit students will need to combine logical, problem solving and arithmetic skills. All of these are needed in practical situations in their own lives.

Required Resource Materials:
A2 paper
A list of World and Olympic records
Activity:

In session 1 we ask the students to construct an athletics track. It has to have the following properties:

• be 400m on the inside
• allow for a 100m straight along one side
• have eight lanes
• have the lanes staggered for the 200m and the 400m
• an appropriate finishing line

Athletics tracks usually consist of two semi-circular ends joined by straight sections. Allow your students to choose any radius that they think fit for the ends. However, they should realise that it is hard to run fast around semi-circles that have small radii. On the other hand it is easier to run in a straight line than in a circle.

In session 2 the students need to know the area and perimeters of the various shapes on their tracks. This is so they can work out how much different items, such as turf to cover the infield, will cost. For this session students will need to find the cost of turf, bitumen (or some other track cover), drainage, and fencing cost. (Remember that there will need to be gates to let the athletes, officials, and ground keepers on to the track. Sometimes heavy equipment will also have to be used and that will need at least one larger gate.)

If possible you might invite a local constructor to give some idea of how a new track might be constructed and the costs involved.

They will need to decide in advance if they are going to cover all of the infield with turf or whether they will use some sections for pits for high jump, long jump and so on.

Having said that, try to keep things relatively simple.

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. (If you are giving the unit near to a Commonwealth Games, then substitute Games’ records for Olympic records.)

Find time to measure some of the class (or the school champions) over 100m. Alternatively find the 100m school record at each level and for both girls and boys.

In all of the calculations where events are compared it is assumed that the record holder is always performing to their record level.

### Teaching Sequence

#### Sessions 1 and 2

In these sessions, students will construct the plan of an athletics track on a piece of A2 sized paper and use their plan to measure some distances.

1. If possible take the class to a track before you start this session. If you are able to do this it might be useful for them to measure the straight sections and the radius of the semi-circles. At the very least discuss the important sections of a track. In particular let them talk about the various shaped sections; how the 100m straight fits in; and why the lanes have to be staggered for the 200m and the 400m sprints.
Tell me all that you know about the shape of an athletics track.
What shapes are the different parts?
Why do runners in some races start from different parts of the track?
What races need to do this?
What is done to make those races fair?
Where are the starting points for different races?
Where do the different races end?
What is used for the surface of the track?
2. Get the class to work in pairs to make a scale drawing on A2 paper of an athletics track. This may need to be discussed first to give them some clues as to how to go about the task.
How will you work out the length of the straight parts of the track? (There is no reason why every pair should have the same straight lengths.)
How will you work out the length of the curved parts of the track? (There is no reason why every pair should have the same curved lengths.)
How will you work out the staggered starts?
How will you scale your track?
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.)
3. 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.
4. If some students haven’t finished during the session time, allow them to take their tracks home to complete or find some other time to do this.
5. If possible have a reporting back time so that students can share their experiences. 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 field inside your track?
6. To save space, in one of the semi-circular ends there is a plan to put the shot put.
Would the world recorder holder put the shot onto the track from anywhere inside one of the semi-circular ends of the track?
7. It is hoped that the javelin could be thrown with the dead throw line being along the diameter of the other semi-circular end.
Whereabouts on your track would the World, Olympic and National records land?
8. Is there enough room on the interior of the track to hold the javelin and the hammer throw for an international event?

#### Session 3

In this session we perform some calculations relating to the dimensions of the track.

1. For this session students should find out how much turf costs to buy and lay; the cost of a material to cover the track; drainage; sprinklers; fencing, etc. They can do most of this by internet research.
2. Discuss the task and try to ensure that it is well understood.
3. Let them work in their pairs to do the calculations. They should present their calculations along with a balance sheet of the costs.
4. When they have finished discuss what they did and what costs they would expect in a real project that they have omitted.

#### Sessions 4 and 5

In this session, we look at the relative speeds of Olympic and World record runners on the track.

1. Discuss the relation between speed, distance and time. Give them some examples that concern travelling by car between two cities. Make sure that all three types of problem below are covered.
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?
2. Let each group choose three women’s events.
What was the average speed of the holders of each of these records?
3. Choose three events where the World, Olympic and New Zealand national records are different. Assume these runners were running against each other.
By how much would the World record runner be ahead of the other two runners at the finish?
4. Look at the men’s national records in the 100m, 400m, 400m relay, 1500m, and 10,000m events.
If the 100m runner could keep going at that pace for each of the other races, what would the records for each of these be?
If they competed in the same race, how far ahead of the other runners would the 100m runner be at the finish?
How much faster would the 100m runner be over 400m than the 400m relay team?
If the 10,000m runner kept going at that pace for each of the other races, how far outside the records would he be?
If they competed in the same race, how far behind each of the other runners would the 10,000m runner be at the finish?
5. Now use the school records or the times that you took of certain students over 100m.
Are these students going as fast, or faster on average, as 3,000m Steeplechase record holders?
Are these students going as fast, or faster on average, as 5,000m record holders?
Are these students going as fast, or faster on average, as 10,000m record holders?
Are these students going as fast, or faster on average, as Marathon record holders?
6. Discuss the results.
Was there anything that the students found surprising about these results?
7. 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.
8. Now you have had some experience with that let’s go back to your track. Suppose that a woman runner broke the existing National record for the 100m by 0.1sec. Estimate by how far she would beat the current record holder and show it on your track.
Now calculate this distance.
9. Repeat the last exercise with the men’s 800m World record and the women’s 1500m Olympic record.
10. Suppose that a woman runner beat the existing Olympic record runner in the 100m by 1m. Estimate by how many seconds she would beat the current record holder and show it on your track.
Now calculate this time.
11. Repeat the last exercise for the women’s National 800m record and the men’s World record 400m.
12. So far the men’s records are faster than the women’s. 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?
First make a mark on your track and then calculate the distance.
13. Repeat the last exercise for the National 800m record and the National 5,000m record.
14. Suppose that the woman World record holder could run the 200m at the same speed (not time) as she can the 100m.
Would she beat the men’s 200m World record holder or the men’s 200m Olympic record or the men’s National 200m record?
15. Allow the students to set there own problems.
16. Summarise the unit.

## How much room?

In this unit we explore the amount of room we have in our classrooms. We use this to decide if the "ideal" classroom size.

## Scaling Our School

In this unit we look at estimating the length and height of buildings at your school. We then compare these estimates with the actual heights or lengths. Students will then build scale models selecting appropriate scales using ratio to translate the "real" dimensions into the dimensions for the scale model.

## What Goes Around

This unit is made up of five stations which investigate the relationships between the area and perimeters of squares and rectangles. The unit could be taken as a series of sessions with the whole class or the students could circulate around the stations over the five days.

## How fast is fast?

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

## How Can You Measure This?

In this unit students, working in groups of 2 to 4, carry out and report on a series of investigations involving decisions about how to measure something. The four investigations suggested are:

What’s In a Newspaper?
Students calculate what fraction of a newspaper is devoted to news, sport, advertisements and other categories of information.

Are You a Square?
Students determine whether their height is equal to, greater or less than, the distance from end to end of their outstretched arms.

How Far Do You Walk?
Students work out approximately how far they walk in one year.

How Thick Is It?
Students decide how to measure a length that cannot be measured directly – for example the thickness of a wall of their classroom.

In each investigation the students follow the same sequence:

1. Make sure they understand the problem
2. Discuss and decide on three different strategies for tackling the problem
3. Complete a table to assess the merits of each strategy
4. Use the best strategy for conducting the investigation
5. Record their methods and results