Word problems are challenging. In middle school and high school, word problems for proportional relationships and linear functions are particularly challenging. This post shows an visualization approach to unpacking the unit rate component of such word problems.
Visualizing Unit Rate
Below is an excerpt from a handout used to introduce such word problems. It starts with the unit rate and unit quantity provided. This is more accessible for students and allows for a visual representation. This allows students to “see” the multiplication and make a connection to the word problem.
This is followed by situations in which the unit quantity is unknown. Students still draw a visual, but use the 3 dot symbol for etc. to indicate the unknown quantity.
The same problems are used but students now write the symbolic representation. First they write the expression with both factors provided, then with the unknown. The “?” emphasizes there is an unknown quantity before writing the variable.
The difference between the proportional relationships and slope is context and the ratios. The ratio for the former addresses the variables themselves. The ratio for the slope addresses the change in the variables. This arises from context. To flesh this out let’s use the pay as a function of hours, with $15 an hour for the hourly rate.
If we focus on the fact that to compute the total pay, we multiply by the hours worked by 15 we have a proportional relationship and the $15 an hour is a constant of proportionality.
If we focus on the fact that every increase of 1 hour results in an increase of $15 in our total pay, we have a linear function and the $15 an hour is a rate of change (or slope of the line). Because of the context, we have different constants (k vs m).
In my experience, many students struggle with translating word problems modeled by a proportional relationship or a linear function into a algebraic expression or equation. (Image below.)
Here is a link to a Jamboard that can be used to introduce the algebraic expression to model the unit rate. (See image below on how to copy and edit.)
Here is how you can use this to introduce modeling the word problem.
Start with the unit rate concept. In this case there is $45 “in” every hour. This is modeled in slide 1 (top 2 photos).
The next 2 photos show slide 2 in which the student duplicates the $45 image and fills 2 hours, with $45 “in” each. They complete the multiplication expression by multiplying by 2.
This is followed by the same steps for 3 hours (photo bottom left) and sequentially to 6 hours.
In the last slide there are no hours shown because the # of hours is unknown. This leads to using “X” to represent the unknown NUMBER of hours (I don’t let students get away with x=hours) and finally the algebraic expression (bottom photo).