Lesson 3

This lesson is designed to reinforce to pupils HOW mathematics can help them make the decision about whether the batsman in the photograph is IN or OUT.

In this lesson, pupils will use the factors from lesson two, namely the distance the bail has fallen and the distance the bat has travelled past the crease-line, to calculate the time taken by the bail to fall as well as the time taken for the bat to travel past the crease-line. In this way pupils will be able to decide whether the batsman in the photo was IN or OUT.

Pupils need to have learned the relationship between distance, speed and time and be able to use it to calculate time.

Pupils will need rulers and calculators for this lesson. They may also need stop watches.

You will also need to be able to show the video clips again, using a laptop computer and data projector.

A good way to begin the lesson would be by reviewing pupils’ answers to the previous night’s homework. If pupils did not complete lesson 2 homework then you will probably need to allow time for these questions to be done now. Pupils should be confident in their ability to use the speed-time-distance relationship to undertake simple calculations, as well as the use of scale factor in simple problems.

Now ask pupils to consider how the distances calculated in lesson 2, together with the speed of the batsman, can assist with solving the problem. What else do we need?

Ascertain whether pupils have managed to think beyond the calculation of the distances from the previous lesson and have seen the need for a calculation of the TIMES of both the bail to fall and the bat to travel past the crease-line. In order to calculate these times, we must have an idea of the SPEED of each object. (Advice for Teachers #3.1)

The aim is for pupils to gain a reliable estimate of the speed of a test batsman.

In pairs, ask pupils to come up with an estimate of the speed of a batsman, providing some justification. (For example, they could consider whether the batsman is likely to be running faster or slower than a champion athlete in the 100 m sprint event at the Olympics.)

They then decide how to calculate an estimate of the speed, either by obtaining primary data from a fully equipped batsman (Advice for Teachers#3.2), or by using a video-clip (Advice for Teachers #3.3). Instructions for showing video clips are supplied and are also relevant to lesson 1. It is important, during this discussion, that the pupils are aware that, as a first stage of making a decision, they are going to model the situation in the simplest way. (Advice for teachers#3.4),

Teaching and Learning Issue 3: Keeping a brainstorm manageable

This vignette is written in the voice of an observer of the first class where this sequence was trialled; at that time, we had not identified the issue of dealing with the many possible factors in two stages of modelling.

Bill, the teacher, opened up the discussion of how they might determine the speed of the batsman. What was implicit in his thinking, but not explicit in what he said to his pupils, was that initially it was going to be best to assume that the batsman was running at the same speed all the way down the pitch. Given this assumption, he wanted ideas on ways that the class could get a reasonable estimate of this speed. The pupils were enthusiastic participants, but immediately raised issues that were outside the model Bill was thinking of.

“We could measure someone out on the oval”

“Yes but they might be slowing down because they get tired”

“He might have made a final sprint when he realised that it was going to be close"

“Suppose he was straightening his arm just as he crossed, that would speed up the bat”

There were a range of other comments. Bill realised that he had a problem; he recognised that all of the comments were genuine and sensible and he recorded and praised each of them as they were made. However he only wanted to use ones like the first one in what was intended to be the next task and he did not want to just dismiss the others. He guided the discussion to the ones that he wanted to use, and I could see some of the students whose ideas seemed to be sidelined become less involved. It was only after the lesson that we thought of being explicit with the students about the issues of modelling and of beginning with a simple model, but recognising the assumptions and later revisiting these. In later trials this provided a way of keeping the initial calculations manageably simple, but not dismissing the students’ creative thinking.

Close Box

3.3 CALCULATION OF THE TIME TAKEN FOR THE BAT TO TRAVEL PAST THE CREASE-LINE

Pupils should use their findings for the speed of the batsman to calculate the time the bat has travelled beyond the crease-line, having first measured the distance the bat has travelled beyond the crease-line from the photograph.

Choosing one bail, pupils should estimate how long they think it took to fall to its current position shown in the photograph. This reveals the problems of trying to measure this empirically and that a formula will be needed. In a way that is appropriate to the class level, bring out the issue that the motion of the bail is different to the motion of the batsman and so requires a different formula. This area in one where it may be appropriate to make links to relevant science content. (Advice for teachers #3.5)

Introduce the formula for a body in free fall to calculate the time (t sec) taken for the bail to fall d metres,

Use the value of d determined from the photograph. Pupils need to assume that the bails fall straight down from the top of the stumps.

(The derivation of this formula is beyond pupils at this level, but might be of interest to advanced pupils.)

It may be appropriate to hold a brief discussion of accelerated motion, but be careful not to delve too deeply into this concept. However, there does need to be an introduction to the concept that the motion of the bail is different to the motion of the batsman. All objects fall at the same acceleration whereas all batsmen run at different speeds. Pupils can collect primary data for the speed of the batsman but it is very difficult to collect primary data on the falling bail. You could try this - pupils will soon realise that the time is too short to measure and that therefore they need to rely on the formula given to be able to determine the time the bail has been falling.

Pupils will need reassurance that the times which emerge from these calculations are SMALL numbers, of the order of one or two tenths of a second. This is to be expected. World records in events such as the 100 metre sprint are decided by tenths, hundredths or even thousandths of a second, and often in sport such short times mean the difference between winning and losing. You can now sympathise with the umpire in a county game of cricket who only has this very short amount of time to make this judgement by eye. The human eye can distinguish between one and two tenths of a second, but events which involve time differences of a few hundredths of a second cannot be judged by the human eye. Hence the need for technology!

Pupils should compare their calculations for the time taken by the bail to fall with the time taken by the bat to travel beyond the crease-line, and then review their response to the key question: Was the batsman IN or OUT?

Teachers should refer to the sample calculations provided as a guide to facilitating this work with pupils.

Write a clear argument for your decision using all the data and facts discovered and/or calculated in the lessons.

Write a list of any other assumptions which may have had an effect on your calculations.