From Maura:
We started by going over the most challenging homework problem – the Chron. This problem is a great test of unit conversions. I can only get through it by using the units to set up the arithmetic. I’m not sure everyone saw the reasoning. It’s worth going through again until the units part of it clicks into place.
We then started chapter 3, percentages. I started with a familiar example of calculating the tip in a restaurant on a $45 bill. We went through several different approaches to getting the 20% tip, then moved to considering what the final bill would be. We went through that calculation, then I introduced the “1 + ” approach and explained why it works. At first students thought they should do the factoring each time, and they justly complained that it wasn’t any easier. Once they realized they could jump to the “1 + ” they started having fun with it. In fact, one student even announced that it was fun!
The real usefulness of the “1 + ” techniques comes in when successive percentage increases are calculated. We worked through an example using UMass Boston enrollments, with the assumption that enrollment in 2008 was 13,420 and that it increased by 12%; the next year, enrollment increased by 6%. I asked students to describe how enrollment increased over the two years. The students worked in groups and many of them first did subtraction. We talked about that as absolute change and put that in a context. Good. I pushed them to find another way to describe the enrollment growth and they puzzled over the total percentage increase. Many of them saw the 12 and the 6 and decided to add them, telling me that there was an 18% increase over the two years. I asked them to check it – and they realized that it wasn’t correct. After a bit of discussion, one student came up with the reason: the 12% growth is calculated on one number, but the 6% growth is calculated using a different number. Good! We then found the total percentage growth two different ways: making a ratio (absolute change divided by the original enrollment) and using multiplication: 1.12 times 1.06. We found a total growth of 18.7%.
The nice part about the “1 + ” idea is that you don’t even need to know the original amount (the original enrollment, for example). We did another example: in one year, you get a 1.5% salary increase. The next year, you get a 3.4% salary increase. What is the total salary increase over those two years? It doesn’t matter what your original salary is. We can just use multiplication to figure it out: (1.015)(1.034)=1.0495 or about 5% in total. That’s close to what we get when we add the percentages – but if it’s a salary increase that I’m getting, I’d rather have the extra little percentage that I get from doing it the correct way!
We didn’t get to talk about percentage decrease. That’s first on the list for next week.
From Ethan:
It’s Friday, blogging hurriedly about yesterday’s class (not the best way- right away is much better).
I thought about working the chron exercise, but didn’t. Having read Maura’s blog, maybe I should have. I’ll know better when I read the homework and see how the class did.
Started on percentages. I used a restaurant example too, with both tip and tax. We had an interesting discussion about how much a tip ought to be, and on the etymology of the word. Fun, but didn’t contribute much to learning quantitative reasoning.
I did introduce the “1+” trick as a one step way to compute a percentage increase. Their was initial reluctance. The payoff came when I asked how to compute the cost of a restaurant meal if your credit card statement (digression on how credit card companies charged merchant fees) told you that the food and the 20% tip together totalled $42.50. Just about everyone’s first guess was to subtract 20% of $42.50 from $42.50. When we checked that answer it was (of course) wrong. I think most people understood and appreciated the fact that you got the right answer by calculating $42.50/1.20.
I really should curb my willingness to digress!
I plan to spend Tuesday’s class going over the homework due Thursday – I will write the class encouraging them to start it over the weekend and come with questions.
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