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The “Box Method” For Teaching Ratio/Proportion Problems

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2009 Annual Conference & Exposition


Austin, Texas

Publication Date

June 14, 2009

Start Date

June 14, 2009

End Date

June 17, 2009



Conference Session

Bridging the Gap and Freshman Programs

Tagged Division


Page Count


Page Numbers

14.1266.1 - 14.1266.11



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Paper Authors

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James Sullivan Dallas Independent School District

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NOTE: The first page of text has been automatically extracted and included below in lieu of an abstract

The “Box Method” for Teaching Ratio/Proportion Problems


This paper details a systematic method for teaching high school students how to set up and solve ratio and/or proportion problems. Such problems frequently occur in a wide variety of engineering applications. The author, while teaching high school algebra courses, noticed a remarkable fact: Students were able to solve such problems correctly once the problems had been set up properly. In other words, their major difficulty was not in the arithmetic required to solve these types of problems, but simply in setting the problems up. After examining several textbooks, the problem became clearer: this important aspect of solving ratio/proportion problems has been neglected for many years.

The author theorized that this learning, as most other learning, takes place in very small “micro- steps”. Teachers are familiar with the solution to such problems and tend to gloss over the essential phase of setting up such problems. As indicated above, algebra textbooks also neglect this important aspect of solving such problems. Students need structure while they are learning this type of process, and this fact has been overlooked for too long in the pedagogy of such problems.

The author then developed a highly-structured, systematic means of setting up such problems. Students quickly began to set up the problems correctly. Once students had the problems set up correctly, completing the arithmetic details of the solution was easy for them. The result was an almost perfect success rate for students working on ratio/proportion problems. A few simple math errors remain, but student success rates have been dramatically improved using this method.

This paper details the “box method” and how it should be taught. Several examples are provided to illustrate the use of this method.


Ratio/proportion problems are a key area of mathematics often used in science, engineering, and business. Conversions of any type of linear direct variation, such as feet to inches, pounds to ounces, etc., are essentially ratio/proportion problems. Very often, similar triangles provide opportunities for using ratio/proportion analysis to determine the unknown lengths of the sides of these triangles. And maps, blueprints, and photographs often serve as the basis for ratio/proportion problems.

The arithmetic involved is usually minimal. In its simplest form, only one multiplication and one division are required. Yet students frequently have serious difficulties learning how to use such a fundamental tool. In fact, a 2007 report on standardized testing of mathematics found that when attempting to master standards for 8th grade coordinate geometry, “Students who are unsuccessful have the greatest difficulty with setting up and solving proportions from real-world

Sullivan, J. (2009, June), The “Box Method” For Teaching Ratio/Proportion Problems Paper presented at 2009 Annual Conference & Exposition, Austin, Texas. 10.18260/1-2--5803

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