Counterfeit Coins and Hiking in the Desert? Welcome to the MS Problem Solving Project

Math teacher Matt Notary, along with his department colleagues, has devised and implemented The Problem Solving Project for all students in the middle school. Designed to help students become more persistent in their solving abilities as well as shifting focus from getting the right answer to the process of doing mathematics, students have been gleaning the benefits of this innovative approach to teaching since it was introduced into the curriculum last year.  Assigned once each semester, all students in a particular grade level choose one of three problems to work on for two weeks.  Then they submitted their process answer packets to their teachers that included not only their work on the problem but a summary and reflection about their process.

To quote Notary, "Mathematicians often work hours, days, or even years on a single problem. The purpose of this project is to give students an opportunity to investigate a problem at length. The goal is not to solve the problem quickly; it is not even necessary to successfully solve the problem." He instructs students to "try to come up with a different strategy for working on the problem." Urging them not to use resources such as books or websites, but to work with their teachers on the process when they get stuck. Patience, resilience, persistence, and self advocacy are developed in students and they delve into the project, so is a new understanding of what constitutes mathematics.

Sample questions from the 8th grade project 2012-13

Counterfeit Coin

You have a pile of 24 coins. Twenty-three of the coins have exactly the same weight, but one, which is counterfeit, is heavier than the others (though it looks exactly the same). Your task is to determine which coin is heavier, and therefore counterfeit. You are given a balance scale, which will compare the weights of any two coins or sets of coins. What is the minimum number of weighings that you will need to do to find the counterfeit coin?  Extension: a) What if there are 64 coins and one is counterfeit? b) What if you don't know if the counterfeit coin is lighter or heavier? Does it change the solution?

Mathville Middle School

Mathville’s roads are designed as a grid of horizontal and vertical lines.  The horizontal streets are numbered with consecutive integers from 1 to 10, while the vertical avenues are labeled with consecutive letters of the alphabet from A to J.  A student, who lives at the corner of 1st street and Avenue A, attends Mathville Middle School, located at the corner of 5th street and Avenue E.  If the student always walks toward the school, but never diagonally, how many different routes are there from home to school?  Extension:  Try changing the location of the school and finding the number of different routes again.  Do you notice any pattern in your answers?

Hiking in the Desert

Three hikers head from home camp to an oasis.  They can carry only 10 days supply of food and water in each of their packs.  Since the oasis is more than 10 days away, they agree to try to get only hiker #1 to the oasis.  After walking together for a spell, hiker #3 refills the packs of hiker #1 and hiker #2 and returns home.  Hiker #1 and hiker #2 continue, and later hiker #2 refills the pack of hiker #1 and returns home.  Find the maximum distance from home to oasis such that hiker #1 can get there with hiker #2 and hiker #3 both safely home.  Extension:  If the oasis is 18 days from home and two hikers are expected to reach the oasis while the others return one at a time after refilling the packs of the others, find the least number of hikers needed.