No class 10 in this blog – the first exam was last Thursday. I’ll discuss it in class next time, and discuss it here then.
Fresh start today – weighted averages (Chapter 6 in the text). After a quick review of the ordinary meaning of average (add up the values, divide by the count) I moved to weighted averages – those appear in the world much more often.
We did a fair amount of qualitative estimation – e.g. the average of 7 grades of 80 and 3 of 90 will be between 80 and 90, nearer to 80. I should go back and put more of this kind of thing into the book.
I followed the text pretty closely (for a change). The first serious work was grade point average computations. It was interesting to see how hard it was to raise a GPA when you’d already taken lots of courses – the qualitative thinking makes that clear.
I asked this question: “If full tuition is $14K and students on financial aid pay $7K what’s the average tuition?” Two students knew right away that we needed more information – the number and percent of each kind of student. Qualitatively, we could see that if no one had financial aid ( – it was just an empty promise on the university web page) then the average would be $14K, while if everyone had financial aid (thanks to a generous grant from the Gates foundation) the average would be $7K. Then I made up some numbers – say 13,000 students, with 9,000 on financial aid. That should and did lead to an average between $7K and $14K but nearer the former. Then we reworked the problem using only the percentage of students on financial aid to get the weights, showing that you needed only weights, not the absolute size.
At that point there was still – surprisingly – lots of time left in the class. I wondered if that was because
- everyone understood everything because it was easy
- no one really understood but didn’t want to say so
- the material was hard but everyone understood everything because my explanations were particularly clear
I hoped it wasn’t the last of these – because then the understanding would evaporate as soon as the students left the room. I’ll see what happens on the homework.
Nevertheless, I proceeded – I did the example in Section 6.4 of the text, in which all car/truck prices increase while the average price decreases. That paradox can happen because the weights have changed. (This is material I’d written up but never used in class.) I’m pretty sure some of the students got it – and pretty sure some didn’t. ‘Tis always so.
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