LESSON FOUR: REASONABLE DOUBT
| LESSON FOUR: REASONABLE DOUBT | |
GOALS OF LESSON
At this stage of the Case Study pupils will have formed an opinion as whether the decision made by the original umpire was ‘correct’: but now they will be challenged to think more carefully about their calculations and results!
In this lesson pupils develop an understanding of the impact of assumptions and/or errors on the process of mathematical modelling.
FEATURES OF THE LESSON
Pupils are asked to explicitly think about and analyse the impact of the assumptions and/or errors they made in putting the model together. (Advice for Teachers #4.1)
Spreadsheets are used to facilitate the rapid production of values. (Using spreadsheets)
Through the methodology of discussion, investigation, and review, pupils closely examine the effect of one assumption on the IN or OUT decision. This mathematical analysis will be performed use the technique of keeping one measurement constant while varying another measurement. (Excel demonstration)
SUMMARY
Whole class discussion: What are assumptions? (5-10 minutes). |
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Demonstration: Using a spreadsheet to investigate the effect of one assumption (10-15 minutes). |
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Individual : Using a spreadsheet to investigate the effect of one assumption (15-25 minutes). |
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Whole class discussion: Sharing investigation plans (5-10 minutes). |
PREREQUISITES:
Pupils should have completed their homework from lesson 3.
| Pupils will need the formulas: time = | distance | and speed = | distance |
| speed | time |
Pupils will need to know how to use an Excel spreadsheet.
PREPARATION
You will need to ensure the availability of sufficient computers.
A data-show would greatly facilitate your demonstration of the Excel spreadsheet.
PRIOR LEARNING FOR LESSON FOUR
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
4.1 WHAT ARE “ASSUMPTIONS”?
4.1.1 Use a whole class discussion to explain the concept of assumption and the impact of making assumptions on the IN or OUT decision.
Ask pupils to share their responses to Lesson 3 homework question 1. Take a straw poll even in the event that not all pupils have completed the homework.
Who decided the batsman was IN? OUT? Who is still UNDECIDED?
How did you reach your decision?
Ask pupils to share their responses to Lesson 3 homework question 2. (Variables and assumptions) If your pupils have not completed this question, then pose it now.
4.1.2 Discuss the notion that a different decision may result from varying the assumption.
In your calculations you have made several assumptions. For example, one such assumption is that the batsman ran in a straight line. What are some other assumptions which may have an effect on your calculations?
Some questions to frame this discussion are:
You have completed calculations using a model of the problem which was simplified by making assumptions. To what extent do you think your results are in doubt? (Advice for Teachers #4.2)
Do you think the results could change depending on the assumptions made? For example, what if the average running speed of the batsman was different to the value you calculated last lesson?
How might your IN or OUT decision change if the assumed speed is slower than the actual speed?
How might your IN or OUT decision change if the assumed speed is faster than the actual speed?
How could you investigate this situation? (Advice for Teachers #4.3)
During this discussion, be prepared for the emergence of the counter–intuitive finding that the slower the batsman was running, the more likely it is that he was in when the bail was removed. (Advice for Teachers #4.4)
Teaching and Learning Issue 4: Allowing students to work through an issue
4.2 DEMONSTRATION: USING A SPREADSHEET TO INVESTIGATE THE EFFECT OF ONE ASSUMPTION
This part of the lesson should teach pupils how to use a spreadsheet to explore the effect of varying the speed of the batsman. (Excel demonstration) This will prepare pupils for their own spreadsheet investigations.
Following the demonstration of Example 1 ask pupils to explain what would happen if they start with a different assumed distance. (Answer: There may be results where the batsman would be 'in'.) Examples 2 and 3 could be used, or pupils could construct their own spreadsheet to investigate this.
4.3 GROUP WORK: DESIGNING A SPREADSHEET INVESTIGATION
In Lesson 5 each group of pupils will conduct at least one investigation of the assumptions relating to the key variables to see if these make a difference to the calculations and to their decision as to whether the batsman is IN or OUT.
Organise pupils into small groups.
Pupils now design a spreadsheet exploration of their own. (Advice for Teachers #4.5)
4.2 and 4.3 ALTERNATIVE OPTION
An alternative approach for students who are finding difficulties with the use of spreadsheets is to allow them to do some of the same calculations by hand. The concept of holding one variable constant whilst varying the other is still the approach to be used.
For example, assuming the bat has travelled 0.95 metres (or whatever value pupils determined in lesson 2) past the crease-line, ask the students to calculate the time it has taken the bail to fall, BY HAND, for each of the cases of batsman speed 5 m/s, 6m/s, 7m/s, 8m/s, 9m/s and 10 m/s.
¨ In which cases is the batsman OUT? Why?
¨ In which cases is the batsman IN? Why?
This approach can be used to replace any of the spreadsheet activities. Students can work in groups and share the workload, share their answers and complete the task together.
4.4 SHARING DESIGNS FOR THE INVESTIGATION OF AN ASSUMPTION
Hold a plenary discussion to ensure that pupils have grasped the technique using a spreadsheet to keep one measurement constant while varying another.
Ask pupils to share their ideas about how they will carry out their spreadsheet investigation. (Advice for Teachers #4.6)
On paper, prepare your spreadsheet investigation for Lesson 5.
Things to consider:
Which value will I keep constant?
Which value will I vary?
What spreadsheet formulas will I need?
What headings should I use on my spreadsheet?
How will I know whether the batsman is IN or OUT?