SUPPORTING NOTES FOR LESSON 4

Advice for Teachers #4.1

There is important learning embedded here, which has links not only to other mathematical problems but also to everyday life, and may prompt pupils to ask questions of themselves in the future, such as what assumptions have I made?; am I justified in making those assumptions?; what is the likely impact of those assumptions on my conclusion? what level of doubt is reasonable? Your pupils need to have a sense of the significance of these questions so that the lesson will have purpose and meaning for them.

Advice for Teachers #4.2  

At this stage pupils will be aware that they have made estimates in order to calculate the values used in Lessons 2 and 3. Pupils need to enlarge the concept of estimate to include the concept of assumption. You may need to help pupils distinguish between the variables that were identified and assumptions made about those variables.

Pupils need to develop the awareness that in earlier lessons they made assumptions in order to make calculations, and to understand that making assumptions has implications for their initial IN or OUT decision.

They need to understand that, in general, assumptions play an important role in mathematical modelling because, often, making assumptions allows us to simplify a problem sufficiently so that we are able to study it.

Explain to pupils that they are not expected to have final answers these questions just yet, but after they have done some further exploration of two particular assumptions then they will better understand how variations in measurements can alter the IN or OUT decision.

Advice for Teachers #4.3

Pupils should offer suggestions which involve the need to try different values of running speed, distances and times to see if the batsman is IN or OUT. They might suggest that they could look at the decision in light of reasonable adjustments to the time that the bail was falling or the speed that the batsman was running.

At this point some pupils may perceive the need to vary one measurement while keeping another constant. (see Excel Demonstration) Note that varying speed and distance simultaneously is a more haphazard and unsystematic approach, which is likely to reduce pupils’ ability to draw conclusions. This is because students will not know which of speed or distance is responsible for any change to their original decision. You might choose to allow some pupils to take this approach, and then discuss the issue at a later point. Be aware that doing this involves risk-taking for both pupils and teachers as it may result in a less focused investigation for those pupils.

Advice for Teachers #4.4

The following explanation is included as background material for teachers. Try to avoid spelling out these understandings for pupils – let them emerge as result of them thinking about the ideas for themselves.

To determine if the batsman is IN or OUT, the time that the bat-tip has been over the crease-line must be compared with the time that the lower bail has been off the stumps. In trials it was found that the essential understanding of how to compare the times emerged quite slowly for many pupils.

·     If the bat time is greater than the bail time then the batsman can be judged IN.

·     If the bail time is greater than the bat time then the batsman can be judged OUT.

This leads to a somewhat counter-intuitive finding (and one which many pupils struggle to understand): namely, the slower the batsman has been running, the more likely he is to have been IN. This is because the bat-tip is over the crease-line for increasingly longer times. Once again, it is better if the pupils encounter this for themselves rather than the teacher announcing it before they have begun to play with data; it is also worth seeing if (at least some) pupils can develop an explanation themselves

Some classes may try, but not succeed in explaining the finding; some may, in either case the following three stage activity is likely to be useful, either to provide an explanation, or to affirm what has been worked out. Firstly, ask the pupils to consider three batsmen running at 3, 5, and 10 metres per second respectively and calculate how far each of them would have travelled in the 0.1649 seconds (calculated in lesson 2) since the bail had been removed (Answers 0.495, 0.825, &1.649 metres).  Secondly, get them to mark where the bat was when the photo was taken on a scale diagram of a pitch (See Standard cricket measures).  Finally ask them to measure back from this and, using the three figures just calculated, mark on this diagram where each batsman would have been 0.1649 seconds earlier when the bail was removed – only the slowest batsman is in.

Advice for Teachers #4.5

It is important pupils have time to grapple with this part of the Case because they are now being asked to re-introduce a degree of complexity into the model.

Pupils need to understand what it means to “make an assumption”. They also need to understand how to test a range of values of a particular variable against a constant value. (This is done by varying values that previously were assumed to be particular single values.)

Trials have shown that small group discussion is an effective methodology for assisting pupils to get to grips with these issues.

Advice for Teachers #4.6

It is very important that pupils’ ideas about the design of the investigation are valued. There is little point asking pupils to design their own investigation if their ideas are not incorporated into the whole. Of course, some pupils will probably need a greater degree of teacher input and guidance than others, but, at the very least, all ideas should be heard.

Using Spreadsheets

The use of spreadsheets enables the rapid production of a range of answers and/or graphs.

Depending on the facility and previous skill of students, the teacher may need to explicitly teach students how to explore the impact of one assumption using a spreadsheet. (see Excel Demonstration). In trials, teachers found it necessary to remind pupils of details such as the need to begin all formulas with the “=” sign, and that unit symbols could not be used in cells.

Pupils with a good knowledge of Microsoft Excel (or similar software) will be able to independently design spreadsheets relating the variables speed, distance and time (for the batsman) and distance and time (with g constant) for the fall of the bail.

During this investigation it should become apparent to pupils (if it has not already) that the number of decimal places used in determining the times can have an impact on whether an IN or OUT decision can be made. An interesting extension for some pupils might be to explore the effects of rounding distances and times to different numbers of decimal places.  Pupils should bear in mind the units of measure they are using (metres and seconds) and think about the level of accuracy that is meaningful

Variables and assumptions

A good list might look like this, but do not expect pupils to come up with all of these.

Excel Demonstration

Investigating the impact of changing the average speed of the batsman

The speed of the batsman is varied while the distance of the bat-tip past the crease-line is kept constant. By varying the assumed running speed of the batsman we may discover a speed above which the batsman is IN but below which the batsman is OUT. Reasonable lower and upper limits to choose for the speed of a batsman might be 4 m/s and 10.5 m/s, assuming 10.5 m/s is about the speed of an Olympic sprinter (100 m in 10 s). The range of values chosen by pupils will depend on the values they calculated in Lesson 3.

In Lesson 3 pupils will have calculated a value for the time that the bail has been off the stumps. (One reasonable (but not prescribed!) estimate of the time that the bail has been off the stumps is 0.1649 s (see Sample Calculations). In this investigation, this time will be compared with a range of times for which the bat tip has been past the crease line, calculated in a spreadsheet using the formula:

In Lesson 2 pupils calculated a value for the distance of the bat tip past the crease line. A reasonable result is 0.95 m (but, again, this is an example value).

Example 1

In the example below, there is a range of running speeds and the distance of the bat-tip past the crease-line is assumed to be 0.75 m. In each case, the batsman is OUT. This is because the time that the bail has been off (0.1649 seconds) is always greater than the time the bat has been past the crease line. 

The calculated values are in Table 1 and the corresponding spreadsheet formulas are shown in Table 2.

Table 1

Table 2

 Example 2

In the example below, there is a range of running speeds and the distance of the bat-tip past the crease-line is assumed to be 0.95 m. The batsman is OUT only for the speeds 5 m/s and 5.5 m/s. This is because the time that the bail has been off (0.1649 seconds) is less than the time the bat has been past the crease line I those cases. 

Table 1