Our high schoolers continue to develop strong math reasoning through the use of the Interactive Mathematics Program (http://mathimp.org/). This program integrates arithmetic, algebra, geometry, pre-calculus, and statistics using problems that look a lot like brain teasers. The logic is that it does no good to memorize an algorithm and practice it forty or fifty times (as in traditional math programs) if the student cannot reason when to use it in real life. Here is a sample problem (from p. 276 of Book 1):
Three travelers met one night along the Overland Trail. The decided to have dinner together. Sam had seven cans of beans to contribute and Kara contributed five cans of beans. Jock didn't have any beans, but the three cooked up what they had. Each ate the same amount. After dinner, Jock offered the 84 cents in his pocket and said that the other two could divide it up in an appropriate way. The all agreed that in this way everyone would have contributed a fair share to the dinner. Jock thought that Kara's share of the money should be 35 cents, but Sam and Kara convinced him that this was wrong.
1. Explain why Jock might have thought that Kara's share was 35 cents.
2. Then explain what Kara's correct share should be.