## 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 Handout

Here is a link to the handout. It is in WORD format to allow you to enter your own problems.

## Multiplication Problems as Intro to Unit Rates and Slope (follow up)

I previously related elementary school word problems with math topics in secondary schools. The photo below shows a method to help elementary school students unpack the multiplication problem, to help middle school students identify the unit rate, and to help algebra students identify slope (you can focus on simple problems like this as an entry point to the linear function type problems).

In advance of this method, a review is conducted on the representation of multiplication using the groups of items model (below). By drawing a picture for the two parts of the problem that have a number, the students are guided to break the problem into parts and then to unpack the parts. The “5 boxes of candies” is represented by squares (or circles if you prefer) with no items inside. The “each box holds 6 candies” is represented by a single square with 6 items (dots) inside.

In turn, the drawing of the group of items leads to the multiplication statement, “6 candies x number of boxes.” I prompt students to include the items with the number as sometimes they will write this statement as “6 x number of candies”. I point out that 6 and candies go together. As seen in the previous blog post, the next step in this problem would be to replace “number of boxes” with the quantity given and then compute.

## Unit Rates

Real life applications in of themselves will not make a concept real for many if not most students with special needs. They likely need the concept broken down into more concrete form (CRA).

For this problem I would fudge the numbers and have 42 oz at \$6 (7 oz per dollar) and a 5 oz at \$1. I would have a photo of a “42” oz yogurt and cut it into 6 pieces and have 6 \$1 bills and match each yogurt piece with a dollar bill to help students visualize the unit rate.

I took this photo tonight at Big Y and threw together this idea on the fly.