Problem Solving and the Common Core

Algebraic Representations of Word Problems

Not all problems need to be represented using algebraic symbols, expressions, or equations to be solved. Guess and check can be an efficient strategy that allows students to understand the quantitative, numerical, relationships within word problem solving.¹² “Guess and check is a powerful problem-solving strategy that can connect a conceptual understanding of word problems with a symbolic representation.”

Many of the one-step multiplication word problems discussed in the taxonomy and examples above can be completed without writing any calculations down on paper. In my classes, students are never discouraged from mentally computing their answers, nor will they be forced to show their work in the early parts of the unit. However, by introducing additional considerations and adjusting the constraints, one-step problems can be gradually adjusted to necessitate students’ eventual use of symbolic and variable representations of the problems. As students encounter and progress through different types of word problems, the use of variables can be encouraged by changing values frequently, and asking students to make predictions and generalize patterns.

Utilizing the Uniform Scenario, the problem can be extended further: Ms. Gonzalez thinks that there will be more athletes next year because there will be new uniforms. We know that a decent uniform costs $24 each, but there is also an initial set-up cost of $250 that will be charged to print all of the uniforms.

Questions: How much money will 200 uniforms cost? How much money will 250 uniforms cost? 375 uniforms? 450 uniforms? Is there a rule that can be used to describe the cost for any number, n, uniforms? How many uniforms can we afford with a $10,000 budget?

Introducing variables in conjunction with word problems can lead students to better understand their purpose and representations.¹³ In the context of the scenario, the beginning questions require multiplication and addition in order to determine the total costs. Once problems involve even larger numbers, the calculations are the same level of difficulty, but with multiple steps, the repetitive computations may allow some students to notice patterns that can lead to generalizations and formulations of rules, eventually leading to a linear model. Based upon prior knowledge expectations for this unit, some students will be able to construct a linear variable expression. Nevertheless, past classroom experiences suggest that if students were expected to respond to the prompt, “Write an equation representing the cost of n uniforms,” a majority of students will not be able to independently construct the equation. I anticipate needing to supplement this unit with direct instruction on specific topics as needed.

Extending the Scenario 

This curriculum unit was designed with the intention to grab students’ attention by using a context that directly impacts their experience and opportunities at Overfelt High School. Beginning with a lower than grade level entry threshold that is relatively basic in concept for students entering ninth grade, with the consideration of a wide range of student abilities, the connection with the context and problems could potentially draw students in. Also, the intentional use of friendlier whole numbers to begin the problems will allow students to practice the four step problem solving process. Multiplication and division one-step and two-step problems allow students to reach numerical solutions quickly. Emphasizing the fourth step in problem solving, students will continually revisit and look and consider their numerical answers in the context of the scenario.

Within the collection of problems, the questioning employed by the teacher can assist students with the conceptual development and understanding of purpose for variables, linear expressions, linear equations, and inequalities. By varying the questioning, the scenario can be easily extended to analyze and compare multiple linear situations within the uniform costs.

For example, by providing more information and using new questions for the same general scenario with uniform costs, but modifying the questioning, we can easily create multi-step word problems within the context that require addition, subtraction, multiplication and division. Furthermore, we can make open ended problems that can elicit strategies such as guess and check and variable representation of word problems as linear expressions (see Appendix C).

Example: Ms. Gonzalez and Mr. Delgado informed us that we do not need to buy only one type of uniform. We can choose two different types of uniforms if we want, but we still have only $10,000. Option A: The Name Brand uniform costs $32 each. Option B: Decent uniform costs $24 each. Option C: T-shirt with school logo costs $8 each.

Questions: If we buy 200 Name Brand uniforms, how many Decent uniforms can we buy? If choose to buy T-shirts and Decent uniforms, how many of each can we buy and still be within budget?

A benefit to using a real-life scenario is that real-life situations are often messy. The messiness of a problem can be introduced mathematically to the scenario as seen with the constraints on the budget and the lifespan of the uniforms. In addition to mathematical computations, the real-life nature of the participatory budgeting context must consider the personal and community values that influenced the initial project proposals.