Wednesday, February 21, 2007

Developing Conceptual Understanding of PerCents using an SAT-type Problem

The following was inspired by a question from a recently released SAT. How do you think many students would approach this? Seems like an innocent quantitative or algebraic problem, but I also see some subtleties and important ideas to discuss.

No Calculator Allowed.
(a) In a school election, 480 students voted for one of 2 candidates. Candidate A received 24 more votes than candidate B. What percent of the number of votes cast did Candidate A receive?

Notes: This problem can be done mentally if one has strong number sense and a good feel for percents. You may want to encourage this but allow pencil and paper for those who want it. When reviewing it, certainly share several methods including an algebraic approach. Is this question appropriate for middle school? I think so!

(b) Now generalize: N votes cast for 2 candidates and Candidate A wins by k votes. Derive an algebraic expression for the percent of the votes that Candidate A received.


Totally_clueless said...

Hi Dave,

I certainly think both problems are appropriate for middle school, and the calculator should be strictly disallowed for problems that involve simple arithmetic operations.

I would be interested to see how your students react if you said 480 votes were cast for the 2 candidates and candidate A won by 25 votes.


Dave Marain said...

thanks tc--
one doesn't know for sure until we administer these to real students but I did discuss a similar question a few days ago with an SAT group I teach. Since I did not disallow the calculator, most students just naturally set up an equation and did the calculation with the calculator. A few did it numerically to arrive at 252 votes for candidate A, then used the calculator to convert 252/480 to a percent.
If I had said, find a mental math way to do this, perhaps some would have! They were all surprised at how easy the question was when I explained it using the 'split the difference' approach using 50% as a starting point. I was not shocked they weren't thinking like this however.
If I used 25 instead of 24 some would not have recognized the difference but some would!!

jonathan said...

I would have converted 12/480 to a percent... and that is a bit strange, I know, but I have good luck transmitting some of my number tricks to the kiddies.

Dave Marain said...

some students might have difficulty understanding your 'splitting' of 24 into 12 and 12; that's what makes this problem interesting IMO; i'm not sure i would refer to it as a 'trick' - it's a really great approach that can be transferred to many similar questions involving head-to-head compettions.
Again, the students i observed didn't look for any creative approaches. If I had required a mental math approach, they might have!

jonathan said...

I end up teaching lots of arithmetic shortcuts in my algebra class. The topics feed each other.

If I call them tricks, the kids like them more.

mathmom said...

Dave, did you ask the kids who did it numberically how they got the numbers? They may have use the same "splitting the difference" trick on the numbers that they could have used on the percents.

I think the numerical version is appropriate for middle school. The algebraic version is not (in my opinion) appropriate for pre-algebra, but would be good for those middle schoolers in Algebra I (or equivalent) -- this is only the "honors" kids in our district.