From Maura:
At the beginning of class we talked through how I grade homework (being honest about subjective grading) and my expectations for homework. Part of the next assignment is to look at solution to the first homework and write about how you’ll do things differently. I’m curious to see what I’ll get.
We continued talking about unit conversions, with a big focus on writing out the steps using units, then filling in the numbers later. We went back to problem 1.2 – have you been alive for 1000 seconds, 1,000,000 seconds, etc. I wanted to work through it by converting 1000 seconds to minutes, one million seconds to days, and so on. We set up the calculation with the units and worked it out. I really like this problem because it is a nice (and fairly easy) example of unit conversions and because it points to the distinction between millions, billions and trillions – quantities that we hear a lot but rarely stop to think about. The jump from a million to a billion is a jump from a few days to just over 30 years; the jump from a billion to a trillion is even more staggering: from about 30 years to about 3 millenia! We talked a bit about examples of billions (state budgets) and trillions (federal budgets). I don’t think there are any trillionaires – although certainly there are billionaires.
From there we worked on currency conversion with the pretend example of buying a football jersey in London, converting from British pounds to US dollars. While we can use the web here, the reality is that when you travel you’ll need to do conversions on the fly. And it’s a nice way to think about setting up units! From there we went to the conversion of the cost of gas (or petrol): from British pounds per liter to dollars per gallon. The point of focusing on the units for the straightforward cases (where I know students – most of them – can set it up in their heads) is that when we get to trickier examples the units can help us. For this example, we started with
British pounds U.S. dollars gallons
—————- x —————- x ————
liters British pounds liters
then we put in the numbers. We do have different types of “fractions” here, and maybe that was part of the challenge. The first represents how we are measuring the cost of gas: in pounds per liter. The second and third fractions are our conversion fractions – that’s why we group dollars with pounds (both monetary units) and gallons with liters (both measures of volume). Thinking of it that way may also help.
I tried not to digress into the old English monetary system – back when 12 pence were in a shilling and 20 shillings in a pound and there were many other units that were used regularly.
The last exercise we did involved unit pricing. We compared the price per ounce for two boxes of cereal. Then we did it again with the assumption that we could use a coupon. There are lots of blogs (called “mom blogs”) that do this kind of math intensely and it’s fun to both read and learn from them. If I can, I’ll post some links later.
From Ethan:
I too talked about homework, and qualitative grading, and reading posted solutions to find models. I was generally quite pleased with the hw1 submissions – lots of interesting student backgrounds, and several people who actually read the book during those snow days and used what they learned (primarily about unit conversion) in their answers.
To motivate “write the units first” we spent the whole class on fuel economy. Starting with the well known miles/gallon, we figured out how to find the amount of gas used for a trip (miles * (gallons/mile)) and the cost of gas. Worked out the paradoxical fact that upgrading from 15 miles/gallon to 25 miles/gallon saved more than from 25 miles/gallon to 40 miles/gallon (for driving the same distance). Then converted euros/liter to dollars/gallon and discussed why gas was so much more expensive in France.
We did this work both by writing out the conversion equations (units first) and by asking Google for searches like
[1.25 euros per liter in dollars per gallon]
I too am curious to see how people do on the homework for Thursday – particularly how much they were able to take my advice and write more words.
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