LESSON FIVE: PUTTING IT ALL TOGETHER
| LESSON FIVE: PUTTING IT ALL TOGETHER | |
GOALS OF LESSON
This lesson is an important one within the structure of the Case as pupils have the opportunity to draw some important conclusions about the role of assumption, error and doubt in mathematical modelling.
FEATURES OF THE LESSON
Pupils work individually to explore the assumptions about the IN or OUT decision, using the investigation they designed in lesson 4 and refined for homework.
A whole class discussion is used to explore the notion of acceptable limits for doubt within the mathematical model – the idea of ‘reasonable doubt’.
SUMMARY
Spreadsheet exploration of assumptions |
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Putting it altogether – the concept of Reasonable Doubt (10-15 minutes). |
PREREQUISITES
Pupils should have prepared their investigation prior to the lesson.
PREPARATION
Computers are needed for this lesson.
Pupils will need a copy of another run-out photo for Homework. (Picture 2)
PRIOR LEARNING FOR LESSON FIVE
Students should be familiar with the following concepts:
Decimal numbers, such as 0.1 and 0.23, being able to rank them from least to greatest and their understanding of place value in this work;
The substitution of numbers for pronumerals into a mathematical formula in order to calculate a result. Pupils may have met this work in formal algebra, or in their study of measurement (formulae for area and volume);
The use of spreadsheets, especially in the way a mathematical formula is used to calculate a set of numbers in a column efficiently;
The difference between a CONSTANT, a quantity whose value is FIXED such as the length of a cricket pitch, and a VARIABLE, a quantity whose value can CHANGE, such as the speed of a batsman.
PLAN
5.1 EXPLORING AN ASSUMPTION
Each group of pupils conducts an investigation using a spreadsheet to vary one previously assumed value to see if it makes a major difference to the calculations and to their original decision as to whether the batsman is IN or OUT. (Advice for Teachers #5.1; Example 1; Example 2)
A challenge for capable pupils would be to use spreadsheets to further explore the relationship between distance fallen and final speed under constant acceleration due to gravity. Alternatively, some pupils may design an investigation that challenges the assumption that the bails fall straight down off the stumps.
An alternative approach for students finding difficulty in using the spreadsheets approach would be to allow students to do Example One BY HAND. In this example, the speed of the batsman is held constant at 10 m/s, and the distance the bat tip past the crease-line is varied from 0.1m to 0.9m in increments of 0.1m. The task is then to calculate the time taken for the bat to reach this distance in each case.
The students should work in groups and divide the task between them. They can then compare their results with the time taken for the bail to fall (0.165 seconds) and discuss which calculations lead to the batsman being IN, and which to the batsman being OUT.
5.2 BRINGING IT ALL TOGETHER : THE CONCEPT OF “REASONABLE DOUBT”
Bring the class back together for a closing discussion.
Compare some of the pupils’ new findings.
Pose the questions:
Did anyone discover speeds and distances for which the batsman is IN? What are they? Are these results surprising?
What is the effect of making assumptions on the IN or OUT decision?
If the questions above do not elicit good responses then more specific questions to ask the pupils are:
In what way does the time that the bail has been falling affect the IN or OUT decision?
How does your IN or OUT decision change if the assumed time is longer than the actual time?
How does your IN or OUT decision change if the assumed time is shorter than the actual time?
Make the point that you can only be sure of your IN or OUT decision within the error margins created by the assumptions made.
Remind pupils that in this lesson they have explored the notion of ‘reasonable doubt’, based on their investigation of assumptions.
Questions to lead the generalised discussion are:
How sure are you now about your original calculations?
What assumptions do you think create the most uncertainty?
What upper and lower limits on the time calculations would be acceptable so you would be sure of your decision within ‘reasonable doubt’? For example if you calculate a time of 0.17 seconds, do you think an error of 0.03 seconds either way is reasonable? In other words, is an interval of time between 0.14 sec and 0.20 sec a good measure of “reasonable” doubt? If so, is that sufficient to change a decision? Why or why not?
One way to summarise the lesson is to present a table which shows a spread of results. (Advice for Teachers #5.2)
Review your original decision : was the batsman IN or OUT? Clearly outline your argument.