Formative Assessment 134

Assessment Problem: 

A school with 345 students had a sports day. The students chose between football, swimming, and bike riding. Twice as many students chose basketball as bicycling, and there are 30 fewer students who chose swimming than football. 120 students went swimming. How many students chose football? How many chose bicycling?

Common Responses: 

1) Example of “algebraic” solution:
M + S + C = 372
4C = S
M +18 = C
(C – 18) + 4C + C = 372
=>6C -18 = 372
=>6C = 390
=>C = 65
4(65) = S
=>260 = S
M + 18 = 65
=>M = 47
(This problem could be solved arithmetically with a guess and check method)
2) Example of “arithmetic” solution:
F = 120 + 30 = 150
B = 150 / 2 = 75
(Guess and check would also serve as a arithmetic solution strategy for this problem)
Example of “algebraic” solution:
120 + B + F = 345
2B = F
F – 30 = 120
120 + B + 2B = 345
=>3B = 225
=>B = 75
 

Mathematical Issues: 

When solving word problems, some students automatically apply algebraic methods that are unnecessarily cumbersome. Other students refuse to use algebraic methods (relying solely on arithmetic calculations) because they do not see the utility of using algebra. These problems help assess student’s ability to flexibly apply arithmetic and algebraic solution strategies.

Domain: 
Modeling & Word Problems
Theme: 
Algebraic Solutions
Common Core Standards: 
6.EE Expressions and Equations
Research References: 
Khng, K. H., & Lee, K. (2009). Inhibiting interference from prior knowledge: Arithmetic intrusions in algebra word problem solving. Learning and Individual Differences, 19, 161-268. MacGregor, M., & Stacey, K. (1996). Using Algebra to Solve Problems: Selecting,Symbolising, and Integrating Information. Paper presented at the 19th Annual Mathematics Education Research Group of Australasia Conference. MacGregor, M., & Stacey, K. (1998). Cognitive models underlying algebraic and non-algebraic solution to unequal partition problems Mathematics Education Research Journal 10(2), 46-60. Reed (1999). Word Problems Stacey, K., & MacGregor, M. (2000). Learning the algebraic method of solving problems.Journal of Mathematical Behavior, 18(2), 149-167. Van Dooren, W., Verschaffel, L., & Onghena, P. (2003). Pre-service Teacher's Preferred Strategies for Solving Arithmetic and Algebra