To multiply fractions using a grid model, you should create a grid for each fraction you are multiplying. Then overlay one of those grid models onto the other and see where the shading overlaps. The overlapping shading represents the numerator of the product, while the denominator of the product is found by counting the number of rectangles in the combined grid model.
For Example
Chef Fractiona baked a cake for a party. At the end of the party, some of the remaining cake was split among the 6 guests. Each person is taking home \( \frac{1}{8} \) of the leftover cake.
How much of the leftover cake is taken home by the guests?
This question is asking "What is \( \frac{1}{8} \) of \( 6 \)?" You can use a grid model to help you solve this problem. Click the show me button to see how.
Step 1: Start by creating a grid model for \( \frac{1}{8} \). |
Create a grid with 8 sections, representing eighths. Shade 1 section. 
A large rectangle that is vertically divided into 4 equal columns and horizontally divided into 2 equal rows. There are 8 total sections. Of the 8 total sections, 1 is shaded blue. |
Step 2: Multiplication represents repeated addition. This means that \( \frac{1}{8} \times 6 \) is the same as \( \frac{1}{8} + \frac{1}{8} + \frac{1}{8} + \frac{1}{8} + \frac{1}{8} + \frac{1}{8} \). |
Shade until 6 sections of the grid are filled in. 
A large rectangle that is vertically divided into 4 equal columns and horizontally divided into 2 equal rows. There are 8 total sections. Of the 8 total sections, 6 are shaded blue. |
Step 3: Write the product. |
Count the number of shaded sections. This is the numerator of the product. The denominator is the total number of sections. So: \[ \frac{1}{8} \times 6 = \frac{6}{8} \] |
Step 4: Reduce the product if needed. |
The numbers 6 and 8 share a greatest common factor (GCF) of 2. Then: \[ \frac{6 \div 2}{8 \div 2} = \frac{3}{4} \]The party guests take home \( \frac{3}{4} \) of the remaining cake. |
Think you got it?
How well can you create and use grid models for fraction multiplication? Use the activity below to practice. Answer the question you find on each tab. Click the Answer button to check your response.
You may want to use graph paper to help you create your grid models. If you need graph paper, click below to download printable graph paper in Word or PDF format.
Multiply \( \frac{1}{2} \times \frac{1}{3} \) using a grid model.
Show your model and state the product.
The product is \( \frac{1}{6} \).
One possible grid model is shown.
A rectangle that is vertically divided into 3 equal columns and horizontally divided into 2 equal rows. There are 6 total sections. Of the 6 total sections, 1 is shaded green, 2 are shaded blue, and 1 is shaded yellow. The green shading represents the overlap.
If you need help arriving at this answer, click the Solution button.
Step 1: Create a grid model for \( \frac{1}{2} \). |
Create a grid with 2 sections. Shade one section. 
A rectangle that is horizontally divided into 2 equal rows. One is shaded blue. |
Step 2: Create a grid model for \( \frac{1}{3} \). |
Create a grid with 3 sections. Shade one section. 
A rectangle that is vertically divided into 3 equal columns. One is shaded yellow. |
Step 3: Overlap the grid models, and state the solution. |
One shaded section overlaps. This is the value of numerator. The grid is divided into 6 sections. This is the value of the denominator. The product \( \frac{1}{2} \times \frac{1}{3} = \frac{1}{6} \). 
A rectangle that is vertically divided into 3 equal columns and horizontally divided into 2 equal rows. There are 6 total sections. Of the 6 total sections, 1 is shaded green, 2 are shaded blue, and 1 is shaded yellow. The green shading represents the overlap. |
Step 4: Reduce if needed. |
The fraction \( \frac{1}{6} \) is already in simplest form. |
A recipe for a large pot of soup requires \( \frac{3}{4} \) tablespoons of salt. If the cook wants to make only \( \frac{1}{3} \) of a pot of soup, how many tablespoons of salt should the cook add?
Use a grid model to solve. Show your model and state the product.
\( \frac{1}{4} \) tablespoon
One possible grid model is shown.
A rectangle that is vertically divided into 3 equal columns and horizontally divided into 4 equal rows. There are 12 total sections. Of the 2 total sections, 3 are shaded green, 6 are shaded blue, and 1 is shaded yellow. The green shading represents the overlap.
If you need help arriving at this answer, click the Solution button.
Step 1: Create a grid model for \( \frac{3}{4} \). |
Create a grid with 4 sections. Shade three sections. 
A rectangle that is horizontally divided into 4 equal rows. Three of the rows are shaded blue. |
Step 2: Create a grid model for \( \frac{1}{3} \). |
Create a grid with 3 sections. Shade one section. 
A rectangle that is vertically divided into 3 equal columns. One column is shaded yellow. |
Step 3: Overlap the grid models and state the solution. |
Three shaded sections overlap. This is the value of numerator. The grid is divided into 12 sections. This is the value of the denominator. The product \( \frac{3}{4} \times \frac{1}{3} = \frac{3}{12} \). 
A rectangle that is vertically divided into 3 equal columns and horizontally divided into 4 equal rows. There are 12 total sections. Of the 2 total sections, 3 are shaded green, 6 are shaded blue, and 1 is shaded yellow. The green shading represents the overlap. |
Step 4: Reduce if needed. |
The numbers 3 and 12 share a greatest common factor (GCF) of 3. Then: \[ \frac{3 \div 3}{12 \div 3} = \frac{1}{4} \]The cook should use \( \frac{1}{4} \) tablespoon of salt. |
Rita makes the grid model shown below to model the multiplication \( \frac{1}{10} \times 4 \).
Explain why her model is incorrect.
A rectangle that is vertically divided into 5 equal columns and horizontally divided into 2 equal rows. There are 10 total sections. Three of the section are shaded blue.
The multiplication problem \( \frac{1}{10} \times 4 \) is the same as the addition problem \( \frac{1}{10} + \frac{1}{10} + \frac{1}{10} + \frac{1}{10} \). The grid model Rita created is divided into 10 equally sized smaller rectangles, but only 3 of them are shaded. To represent \( \frac{1}{10} \times 4 \), she should shade 4 of the smaller rectangles.
