Mathematical Modeling and Word Problems
algebraic solutions, arithmetic solutions, strategies, flexibility
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Most word problems can be solved either through algebraic or arithmetic methods. Moreover, a given word problem is usually more efficiently solved with one or the other type of solution. To motivate appropriate use of algebraic and arithmetic solution strategies, students should be exposed to problems that are not easily solved with arithmetic solutions. Students should also be encouraged to develop flexibility in choosing whether an arithmetic or algebraic strategy is more appropriate for a given problem.
372 people are working in a large company. There are 4 times as many labourers as clerks, and 18 more clerks than managers. How many labourers, clerks, and managers are there in the company?
Example of “algebraic” solution:
Let L, C, and M represent the number of labourers, clerks, and managers, respectively.
Eqn 1: M + L + C = 372
Eqn 2: 4C = L
Eqn 3: M +18 = C
=> (C – 18) + 4C + C = 372
=> 6C -18 = 372
=> 6C = 390
=> C = 65
Substituting back: 4(65) = L => 260 = L
M + 18 = 65 => M = 47
Example of “Arithmetic” solution:
If there were 18 more managers, there would be the same number of clerks as managers and a total of 390 workers. In this group of people, there were four times as many labourers as clerks, so now there are two times as many labourers as clerks and managers combined. This means that two-thirds of the 390 people are laborers, or 260 people. Of the other 130, there are 65 each of clerks and managers, but this is 18 managers too many, so there are 47 managers.
A primary school with 345 pupils has a sports day. The pupils can choose between in-line skating, swimming and a bicycle ride. Twice as many pupils choose inline skating as bicycling, and there are 30 fewer pupils who chose swimming than in-line skating. 120 pupils want to go swimming. How many chose in-line skating and bicycling?
Example of “arithmetic” solution:
The number of students who chose in-line skating is 120 + 30 = 150. The number who chose bicycling is 150/2=75.
Example of “algebraic” solution:
Let I and B be the numbers of students who chose in-line skating and bicycling, respectively.
120 + B + I = 345
2B = I
I – 30 = 120
120 + B + 2B = 345 => 3B = 225 => B = 75
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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, or they are unable to create an equation that makes sense to them in the context of the problem. Students are not able to think flexibly about options for problem solving.
Students need practice selecting either an algebraic or an arithmetic strategy. Typically, when most students are in an algebra class, they are asked to solve all word problems algebraically to practice the algebra problem solving skills. However, this does not afford students the opportunities to learn how to flexibly apply either arithmetic strategies or algebraic strategies. Accordingly, teachers should encourage students to explore different solution strategies (both algebraic and non-algebraic) while learning algebra problem solving skills. Consider discussing in class which approach might be better and why.
If a problem can be solved most efficiently through an arithmetic solution strategy then it does not make sense to force students to use algebraic approaches. Algebra is a powerful problem-solving tool. However, in order to come to appreciate the value of algebra, students need to encounter problems that are not easily solvable using arithmetic calculations. If they are only asked to solve problems with algebra that could just as easily be solved with arithmetic calculations or by “guess and check,” students will likely not see the point in using algebra.
To help bridge the gap between arithmetic and algebraic solution strategies we should put concerted efforts into helping students make connections between arithmetic and algebra. Teachers should also recognize that formulating an equation is not an intuitive way for many students to represent a problem. As a suggestion, teachers should focus on helping students make connections between pictorial and symbolic representations of unknown values (Khng & Lee, 2009). We should also focus on developing students’ abilities to accurately represent word problems in equations (MacGregor & Stacey, 1996).
Encouraging students to write sentences describing algebraic equations will help them learn to model using algebraic equations. For example, students who can translate C=2B to "There are twice as many carrots as bananas" are close to translating "There are twice as many carrots as bananas" to C=2B.
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 teachers' preferred strategies for solving arithmetic and algebra word problems. Journal of Mathematics Teacher Education, 6, 27-52.
Walkington, C., Sherman, M., & Petrosino, A. (2012). Playing the game of story problems: Coordinating situation-based reasoning with algebraic representation. Journal of Mathematical Behavior, 31(2), 174-195.
- What have you learned about your students’ thinking regarding this topic?
- What have you learned that is an effective teaching strategy to help students understand this topic?