Wednesday, October 22, 2014

How to Solve AMC 8 Problems: Year 2013, Problems 1 and 2

I have mentioned before that I don't expect my children, a third-grader and a fifth-grader, to be at all competitive in the AMC 8 exam this year, but nevertheless, the exam is actually quite accessible even to these younger students, since much of the math involved utilizes the four basic operations in ways that require only clever, logical thinking to suss out.

The exam is also a great, free opportunity for the kids to get some experience taking standardized tests, as well as to deal with specialized mathematical formatting (that dot is a multiplication sign!) and multiple choice trickery (is your answer one of the choices? It still may not be correct!).

And the AMC 8, in particular, is maliciously intent on asking its students to be very thoughtful and VERY observant as they work the problems, as you'll see in a minute.

Fortunately, all the AMC 8 past problems are available for study, and this is not only helpful when preparing for the exam, but it also makes an excellent part-time math curriculum, and the kids have been enjoying spending one day a week with me, going over a couple of problems and solutions each time. For each problem, I first have them work through the solution using a hands-on method, and then I show them the "short-cut" pencil-and-paper calculation that will also solve the problem.

Problem #1 from 2013, for instance, is obviously and easily solved using Hot Wheels. The kids actually set up 23 Hot Wheels in rows of 6, and then could easily see that they needed to add one more car to make the final row equal to the others. They counted by sixes to count the total cars, and I helped them connect this to the 6x table; you would use the 6x table to calculate how many total cars you would need, then use subtraction to calculate how many you would still need to buy.

Problem #2 from 2013 is only a little more complicated. Will glanced at the problem and immediately said, "Six!", AND was pleased to find that six is, indeed, one of the choices (see? Multiple choice trickery!), so I had her physically underline the parts of the problem that she'd used to calculate that answer, and then showed her that there were parts of the problem that she hadn't taken into account yet.

Syd drew a big fish--

--cut it in half to make two "half-pounds" of fish (it would be better to use half-pound weights as the manipulative here, but I don't have any at hand), and placed the sale price on each half:


They can both easily see, then, that a pound of fish at this price would, yes, equal six dollars, but that's the sale price, and the problem is asking for the regular price. 

Although we haven't done a lot with fractions and percentages yet, both kids know the easy ones, and so 50% wasn't a hard one for them to figure out. They simply doubled the sale price to find the regular price! 

To calculate this on paper, I pointed out to them that doubling is the same as multiplying by two, and it was a piece of cake from there. 

The kids LOVED working through these problems with me--most of this, I know, was the opportunity to play with toy cars and real money and draw fish, but nevertheless, some if it was definitely the fun of figuring out tricky math problems and discovering how to solve them. 

I love math so much precisely because of that feeling!

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