25 April 2009

Goodie for April 2009

Filed under: 8-12,algebra,education,math — admin @ 11:17 am

I loved this question. You can ask it in almost any class.
If Johnny has a 82% average for the 1st 9weeks and a 75% average for the 2nd 9weeks, what grade would he have to get on the final to receive 80% semester grade in this class.  The grades are weighted as follows: each 9 weeks average counts 40% and the final test counts 20%.
Source: MathForum@Drexel

How did I find it? I was working on a Build your Own Simulator Kit using the freeware GeoGebra. To build the simulator, the student must make the ball go from point A to point B as s goes from 0 to 1. How to explain that this is: P=A*(1-s)+B*s ?  You can see here. Then a colleague of mine said “This is weighted averages and can be used in algebra, probability and geometry and can be expanded to n weights. Too bad we don’t talk about that anymore. Those problems are really rich.” So I looked around to see what he was talking about and found this problem.  There is so much logic and usefulness in this problem – but not really any kind of special mathematics. Too bad we label it with such an awful name: weighted averages.  It is just a good question.

1. Check that the grading system makes sense: 2 × 9 week  + 1 × final test = 2 × 40% +1 × 20% = 100%. Right.

2. Let x=grade on final test.
Each 9 week grade counts 40% and final test 20% so we have:  82% × 40% + 79% ×40% + x ×20%
What does this expression equal? Well, we want an 80% total average at the end.
But we cannot write Expression=80% because it is obvious that we need another “%”.
(This is actually the hard part.)  We have not used “at the end“. What does “at the end” mean?  It means 100%.

So our equation is:    82% × 40% + 79% ×40% + x × 20% = 80% × 100%.

Cancel all percentages and solve to get  x=79%.

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