## Brownie Pan Model - Multiplying Fractions

The Brownie Pan Model is an area model for multiplying two fractions. It is often used to introduce students to the idea of finding a part of a part in the context of sectioning a pan of brownies.

Strengths

• Helps to illustrate why the traditional algorithm works $\frac{a}{b}\times\frac{c}{d}=\frac{ac}{bd}$
• Highlights part-of-part meaning of multiplication of fractions

Weaknesses

• Sectioning can be challenging when fractions have large denominators or for students whose work is messy
• Can lead to confusion when students try to model fractions greater than 1 with a single brownie pan or square[3]For example, to show $3\frac{1}{2}\times\frac{3}{4}$ one would need $3\frac{1}{2}$ squares/pans each with $\frac{3}{4}$ shaded. Students who model this using 1 pan often lose track of what the whole is for each fraction.

Sources[4]Leave a note in the comments if you know of another source where this model appears.

• Connected Mathematic Project 3 - Grade 6 - Let's Be Rational: In problem 2.1, the text uses this model to introduce the concept of finding part of a part

How It Works

This model helps to visually represent finding a part of a part. It is often introduced with story problems like this:

Stephen eats $\frac{3}{4}$ of a brownie pan that has only $\frac{2}{5}$ left. What fraction of the whole brownie pan does Stephen eat?

Once students recognize that dividing the whole square into an array where the denominators are the dimensions, the process can be shortened into fewer steps.

In the shorter process below, students shade $\frac{2}{5}$ and $\frac{3}{4}$ of the whole by using horizontal and vertical lines. The overlapping region describes the product of the fractions.

In practice, this is done only in the final square (see: bottom square), but I showed an intermediate step for visual clarity.