The purpose of this unit is to engage the student in applying their knowledge and skills of measurement and algebra within the science learning area; to investigate a physical system.

Students develop their skills and knowledge on the mathematics learning progressions measurement sense and using symbols and expressions to think mathematically, in the context of time and motion in sports.

Students will apply their understanding measurement and algebraic skills, solving problems involving time and motion in the context of sporting trivia.

### Structure

This cross-curricular, context based unit has been built within a framework that has been developed with input from teachers across the curriculum to deliver the mathematics learning area, while meeting the demands of differentiated student-centred learning. The unit has been designed around a six session focus on an aspect of mathematics that is relevant to the integrating curriculum area concerned. For successful delivery of mathematics across the curriculum, the context should be meaningful for the students. With student interest engaged, the mathematical challenges often seem more approachable than when presented in isolation.

The first session is an introductory activity that is aimed to spark the imagination of students, to introduce the need for a particular idea or technique in mathematics that would enable them to explore deeper into that context. It is expected that rich discussion may be had around the context and around the nature of the mathematics involved.

The following five sessions are each based around a model of student-centred differentiated learning.

- There is a starting problem to allow students to settle into the session and to focus on the mathematics within the chosen context. These starting problems might take students around ten minutes to attempt and/or to solve, in groups, pairs or individually.
- It is then expected that the teacher will gather the students together to review the problem and to discuss ideas, issues and mathematical techniques that they noticed during the process. It may be helpful to summarise key outcomes of the discussion at this point.
- The remaining group of activities are designed for differentiating on the basis of individual learning needs. Some students may have managed the focus activity easily and be ready to attempt the
*reinforcing ideas*or even the*extending ideas*activity straight away. These could be attempted individually or in groups or pairs, depending on students’ readiness for the activity concerned. The students remaining with the teacher could begin to work through the*building ideas*activity together, peeling off to complete this activity and/or to attempt the*reinforcing ideas*activity when they feel they have ‘got it’. - It is expected that once all the students have peeled off into independent or group work of the appropriate selection of
*building*,*reinforcing*and*extending*activities, the teacher is freed up to check back with the ‘early peelers’ and to circulate as needed.

### Introductory session

(This activity is intended to motivate students towards the context/integrated learning area and to inform teachers of each student’s location on the learning progressions, multiplicative thinking and measurement sense):

An international tennis champion has an average serving speed of 200 kmph. The typical distance the ball travels diagonally from his racquet to just inside the opposite service line is approximately 20 m. **Estimate** how much time elapses from the moment of the serve to when the ball bounces (and thus the time the opponent has to prepare to return the service)?

In this activity, the teacher(s) will be able to locate their students on the measurement sense learning progression by observing their management of the quantities involved in the problem. This activity integrates mathematical skills and knowledge with the science learning area, the physical world. In this unit of learning activities, the SI units of measurement have been used to ensure validity of the physical relationships used in calculations. Students may be more comfortable measuring with derived units such as minutes or hours for time, but should be encouraged to convert these measurements into seconds the SI unit for time, ensuring clear and accurate mathematical communication.

Mathematical discussion that should follow this activity involve:

- What are the most appropriate units of time for the answer to this problem?
- Is the unit for speed kmph the most useful for this problem?
- How can we convert kmph to ms
^{-1}? - Approximate and average distances and speeds are used in this activity. Would it be appropriate to work with much more accurate information given the variability of the players performance and where he will aim each serve?

### Session 2

Focusing on problem solving involving the relationship between physical quantities; speed, distance and time.

**Activity**

A cricket pitch is just over 20 m long. The fastest of the fast bowlers can manage to bowl the ball at a speed of over 40 ms^{-1}. If the reaction time of the batsman is 0.2 s, does he have time to react to such a fast bowl, determining how he should swing his bat?

Discussion arising from activity:

- Would a good batsman have a higher or lower reaction time than the average person?
- If a batsman doesn’t have time to react after the ball has left the bowlers hand, how could he anticipate the stroke he should make?

**Building ideas**

In the game of cricket there are many speeds that need to be taken into account when batsmen try for runs. Find the following speeds in ms^{-1}.

- The bowler delivers a ball that takes 0.8 s to cover the 20 of wicket. What is the speed of the ball?
- The batsman swings the bat at a maximum speed of 50 kmph (but only for a very short distance).
- The ball leaves the bat and sails over the boundary, 60 m away, in a time of 2.4 s.
- When the fielder returns the ball (60 m), it covers the distance in 3.5 s.
- The batsmen take 3.0 s to sprint the 20 m length of the wicket.

**Reinforcing ideas**

In a game of cricket, a batsman can get the ball away at 28 ms^{-1}. The ball is caught at the boundary, 64 m away and returned by the fielder at 20 ms^{-1}.

- What is the total time for the return flight of the ball?
- The batsmen take 3.0 s to sprint the length of the wicket. Ignoring time taken to catch and set up a throw, and for the batsmen to turn around at the end of a run, how many runs should they aim to complete on this ball?
- Considering time taken to catch and set up a throw, and for the batsmen to turn around at the end of a run, how many runs should they aim to complete on this ball?
- What other factors need to be taken into account for the time it takes to run, or be run out?

**Extending ideas**

In a game of cricket, a batsman can get the ball away at 28 ms^{-1}. The fielder takes 1.5 s for the fielder to catch and throw the ball, with a launching speed of 20 ms^{-1}.

The batsmen take 3.0 s to sprint the length of the wicket and a further 1.0 s to turn and prepare to run again.

- Work out the horizontal distance the ball needs to be hit in order for the batsmen to attempt:
- A single run
- Two runs

- Is the score of 4 runs reasonable for a ball that rolls over the boundary (64 m away)?

### Session 3

Focusing on problem solving involving the relationship between physical quantities; speed, distance and time, with the need to convert into consistent units.

**Activity**

An Olympic sprinter runs the 100 m in just under 10 s. A cheetah can run at a speed of 120 kmph. By how many seconds would the cheetah beat the Olympic sprinter if they were to race 100m?

Discussion arising from activity:

- Would the sprinter have the same speed over the entire 100 m?
- Would you expect the sprinter to manage to run 200 m in just under 20 s, or 400 m in just under 40 s?
- Would you expect the cheetah to be able to run 400 m at 120 kmph?
- Would the cheetah have the same speed over the entire 100m?

**Building ideas**

Compare the following world record times* for different track events.

Distance | 100 m | 400 m | 4 x 100 m relay |

Time | 9.58 s | 43.03 s | 36.84 s |

- Find the average speed in ms
^{-1}of the 100 m record run. - Find the average speed in ms
^{-1}of the 400 m record run. - The 4 x 100 m relay involves four different runners and includes three baton changes. Discuss with reasons whether you would expect this to be run at a faster or slower speed than the 400 m race.

*The records are as at the time of writing this activity.

**Reinforcing ideas**

Compare the following world record times* for different track events. Times greater than 59.99 s are given in the format minutes:seconds or hours:minutes:seconds.

Distance | 100 m | 400 m | 1500 m | 3000 m | 10 000 m | 42 km |

Time | 9.58 s | 43.03 s | 3:26.00 | 7:20.67 | 26:17.53 | 2:02:57 |

- Find the average speed of the runner in each race.
- Which of these are faster than the urban speed limit of 50 kmph?
- Which of these are faster than the road works speed limit of 30 kmph?

*The records are as at the time of writing this activity.

**Extending ideas**

Compare the following world record times* for different track events. Times greater than 59.99 s are given in the format minutes:seconds or hours:minutes:seconds.

Distance | 100 m | 400 m | 1500 m | 3000 m | 10 000 m | 42 km |

Time | 9.58 s | 43.03 s | 3:26.00 | 7:20.67 | 26:17.53 | 2:02:57 |

- Find the average speed of the runner in each race.
- Graph these record breaking speeds against the distance of the race.
- Describe the relationship between world record pace and race distance.

*The records are as at the time of writing this activity.

### Session 4

Focusing on problem solving involving the relationship between physical quantities; speed, distance and time.

**Activity**

An Olympic sprinter takes just 37 steps to cover 100 m in 10 s. How does the the ‘cadence’ (step rate) of the sprinter compare with the wings of a hummingbird which have a flap rate of 80 times per second?

Discussion arising from activity:

- What is cadence?
- Use the ideas of cadence and stride length to suggest why a short and a tall athlete might have similar running paces.
- The giant hummingbird ‘only’ has a wing flap frequency of around 12 flaps per second. Suggest why this might be a lower frequency than the smaller hummingbirds.

**Building ideas**

A sprinter takes 120 steps to run 200m.

- There are two steps in a stride. How many strides did the sprinter take?
- What is the average length of each stride?
- The sprinter took 24 s to run the race. How many strides per second does the sprinter take?

**Reinforcing ideas**

A 200 m sprinter takes 22 s to complete the race, with an average number of 2.5 strides per second.

- Find the length of the sprinter’s stride.
- Use this calculation and/or practical investigation to determine whether a stride consists of one step or two.

**Extending ideas**

A 200 m sprinter with legs that are 1.0 m long takes 21.50 s to complete the race, with an average number of 2.4 strides per second.

- What is the length of each stride the sprinter takes?
- A competitor with the same cadence has legs which are 1.05 m long. Assuming leg length and stride are directly related, what time would you expect the competitor to run the 200 m race in?

### Session 5

Focusing on estimating using the relationship between physical quantities; speed, distance and time, with the need to convert into consistent units.

**Activity**

A famous cycle race covers 3540 km over a number of stages. The time for each stage is expected to be completed by a cyclist riding at an average speed of 42 kmph is just over 4 hours. **Estimate** how many stages are there in the race.

Discussion arising from activity:

- How accurate do you think your estimation is? Would it have been better to have used a calculator to solve this problem?
- How do the units of each quantity give clues as to how you might use that value in a calculation?

**Building ideas**

The 200 competitors in a cycle race that covers 3540 km are expected to take a total 500 000 pedal strokes. How far, on average, does each pedal stroke carry a competitor?

**Reinforcing ideas**

The 200 competitors in a cycle race that covers 3540 km are expected to require a total of 800 tyre changes.

- What is the average distance a single tyre will cover over the race?
- Explain the assumptions that you have made in your calculation.

**Extending ideas**

The 200 competitors in a cycle race that covers 3540 km are expected to require the equivalent energy of 50 000 hamburgers (at 2000 kJ of energy per burger).

- What is the average distance a hamburger will power a cyclist over the race? Give your answer with the appropriate units.
- Explain the assumptions that you have made in your calculation.

### Session 6

Focusing on finding and comparing travel distances, speeds and times using everyday measurements.

**Activity**

Which was the fastest method of transport in these endurance feats?

- Hang gliding 764 km in 11 hours
- Ballooning 37 000 km in 11 days
- Cycling 3540 km in 86 hours

Discussion arising from activity:

- Which of these pursuits relies the most on human power? Does this have any bearing on it’s relative speed?
- Which of these activities could be carried out continuously (without rests)? How do the times for these feats give clues about whether rests were taken?

**Building ideas**

In 2015, Kevin Carr completed his record breaking around the world run. He took 621 days to run 26 200 km.

- What was his daily running distance?
- Kevin averaged 12 hours of running per day. What was his average running speed in kmph?

**Reinforcing ideas**

In 2017, Mark Beaumont followed the 11, 320 km route described in the ballooning book ‘Around the World in Eighty Days’. The two differences were that Mark was on a bicycle and that he did in in 78 ½ days. How much faster was Mark on his bicycle than the fictitious balloonists?

**Extending ideas**

In 2016, Thomas Coville completed his solo sail around the world. He took 49 days, 3 hours sailing at an average speed of 41.5 kmph.

- What was his daily sailing distance?
- How far did he sail in his circumnavigation of the world?