Improving Problem-Solving Skills in Upper Elementary and Middle Grades

1. Improving Problem-Solving Skills in Upper Elementary and Middle Grades Dianna Spence Gregg Velatini Georgia Mathematics Conference 2010 Friday, October 15

2. “The Train Problem” Adapted from lesson plans used in the Elementary Mathematics Lab (EML), University of Michigan Authors: Ball, Deborah L.; Sleep, Laurie; Shaughnessy, Meghan (2010).

3. Introduction The GMC Train Company makes five different sized train cars. 1-passenger car 2-passenger car 3-passenger car 4-passenger car 5-passenger car These cars can be connected to form trains that hold different numbers of people.

4. Task #1 You can only use these 5 types of cars to build trains, and you can use at most one of each type of car in each train. What are the different numbers of people that the GMC train company can build trains to hold?

5. Task #2 A customer named Mr. Howe wants to order a special 5-car train that uses exactly one of each of the different sized cars. He wants to be able to break apart his 5 car train to form smaller trains, one to hold exactly each number of people from 1 to 15. In addition, he wants to be able to form these smaller trains using cars that are next to each other in the larger train.

6. Task #2, continued For example: If Mr. Howe purchased this train: He would be able to make a train but not a train Can the GMC Train Company fill Mr. Howe’s order? Explain how you know.

7. The Train Problem:Notes on Teaching • Students should explain why a given train does or does not meet conditions (justification/proof) • Strategies for organizing information • Conjectures (e.g., “red-white”) • Introduce students to idea of conjecture • Don’t “give away”. Make sure students convince themselves rather than letting them off the hook. • Have students work on this in several short sessions, rather than one long session. • For Task #2, recognize that if no such train is possible, this does NOT constitute “failure”