Multiply Two Fractions With an Area Model

A step-by-step guide to using models to multiply two fractions

Here’s a step-by-step guide to using models to multiply two fractions:

Step 1: Identify the two fractions to be multiplied

Identify the two fractions that need to be multiplied. For example, if you need to multiply \(\frac{2}{3}\) and \(\frac{3}{4}\), these are the two fractions you will be using.

Step 2: Choose a model

Choose a model to represent the fractions. Some common models include number lines, area models, and rectangular grids. For example, you might choose an area model to represent the fractions. Rows and columns can be exchanged without changing the product, provided both factors use the same whole. The steps below use columns for the first factor and rows for the second.

To find the product of the multiplications, you can use the area model:

– The denominator in the first factor signifies the number of columns that you must draw.
– The numerator signifies the number of columns that you must shade.
– The denominator in the second factor signifies the number of rows that you must draw.
– The numerator signifies the number of rows that you must shade.

Step 3: Draw the model

Draw one rectangle to represent one whole. Keep this same rectangle for both factors. For this introductory overlap model, use fractions between 0 and 1; fractions greater than 1 require extending the area beyond a single whole.

Step 4: Partition the same whole

For \(\frac{2}{3}\times\frac{3}{4}\), divide the same whole into 3 equal columns and 4 equal rows. The resulting grid contains \(3\times4=12\) equal-area cells. Do not resize the whole for the second factor.

Step 5: Shade the appropriate parts

Shade 2 of the 3 columns in one color to show two-thirds of the whole. On that same grid, shade 3 of the 4 rows in a second color or with hatching. Both shadings must remain visible so you can identify their overlap.

Step 6: Count the overlap

Count only cells shaded by both patterns for the numerator. There are \(2\times3=6\) overlap cells. Count all 12 cells in the original whole for the denominator, including cells with just one shading or no shading. The product is \(\frac{6}{12}\), not the fraction of cells shaded at least once.

Step 7: Simplify the result

Simplify the result to its lowest terms, if necessary. For example, if the total number of shaded parts is \(6\) out of \(12\), you would simplify this to \(\frac{1}{2}\).

Using Models to Multiply Two Fractions, Example 1

Find the product by model.
\(\frac{4}{5}×\frac{1}{3}=\)_?
The model has \(5\) columns and has \(3\) rows.
\(4\) out of \(5\) columns are shaded, which shows a factor \(\frac{4}{5}\).
\(1\) out of \(3\) rows is shaded, which shows the factor \(\frac{1}{3}\).
The part where the shaded columns and rows overlap signifies the product. There are \(4\) overlapped sections.
So, \(\frac{4}{5}×\frac{1}{3}=\frac{4}{15}\)

Source and optional further study

The Common Core Grade 5 fraction standards include representing fraction products as rectangular areas. For optional book study, compare the Pre-Algebra for Beginners product page and its current contents, format and preview before buying. You can use this lesson independently of a purchase.

Frequently Asked Questions

What is using models to multiply two fractions?

It’s a visual method where you draw a rectangle (the unit square), shade rows for one fraction, shade columns for the other, then count overlapping cells. The overlap is the product’s numerator; the total cells are the denominator. The drawing matches the “multiply across” rule exactly.

How do you multiply two fractions with a model step by step?

Draw a rectangle. Split into horizontal rows using the first fraction’s denominator. Shade rows equal to the first numerator. Split the same rectangle into vertical columns using the second fraction’s denominator. Shade columns equal to the second numerator. Count overlap cells (numerator) and total cells (denominator). Simplify.

What’s the easiest way to multiply two fractions with models?

Start with unit fractions like \(\tfrac{1}{2}\times\tfrac{1}{3}\), one row shaded, one column shaded, one overlap cell out of 6. Once that pattern is clear, move to non-unit fractions where you shade multiple rows and columns. The setup stays the same; just the counts change.

When do I use models to multiply fractions?

Use models when you’re first learning fraction multiplication, when a problem requires a visual explanation, or when you want to verify a numeric answer. After the multiplication rule feels automatic, models become a backup tool for proving the rule works.

Common mistakes when modeling fraction multiplication?

Drawing rows and columns both in the same direction (you need one horizontal, one vertical). Miscounting the overlap. Counting only the double-shaded part for the denominator (no, the denominator is ALL cells). Drawing uneven splits that make counting messy. Take a moment to draw straight, even lines.

How does the model compare to the multiplication rule?

The rule is faster: multiply numerators, multiply denominators. The model is slower but shows WHY the rule works, the overlapping region is literally one fraction OF the other. \(\tfrac{1}{2}\) of \(\tfrac{1}{3}\) means “half of one-third,” which the model shows as half of a third-strip. Same answer, two ways of seeing it.

Can I multiply two fractions with models without a calculator?

Yes. The examples here can be drawn and counted by hand. A ruler can help make equal partitions. With large denominators, the grid can become cumbersome; use the model to understand the rule rather than drawing hundreds of tiny cells.

Real-world examples of multiplying two fractions?

If \(\tfrac{2}{3}\) of the students in a class are girls and \(\tfrac{1}{2}\) of those girls play soccer, then \(\tfrac{1}{2}\times\tfrac{2}{3}=\tfrac{2}{6}=\tfrac{1}{3}\) of the class is a girl who plays soccer. If a road covers \(\tfrac{3}{4}\) mile and you’ve walked \(\tfrac{2}{3}\) of it, you’ve walked \(\tfrac{2}{3}\times\tfrac{3}{4}=\tfrac{1}{2}\) mile.

Worksheet for modeling fraction multiplication?

For the examples here, draw a rectangle on paper and partition it into equal rows and columns. The linked lessons below offer additional explanations. A resource title alone does not verify a pre-drawn grid, answer key or particular book chapter.

How to teach kids to multiply two fractions with models?

Start by folding a paper square in half horizontally. That’s the first fraction. Then fold in thirds vertically. That’s the second. Unfold and count the small rectangles created by the folds. Shade the overlap. The folding makes the abstract idea physical. Once they can model it, introduce the multiply-across rule and show it agrees.

Related EffortlessMath Lessons

If a topic on this page feels rusty, these short lessons go deeper:

• How to multiply fractions using models
• How to multiply fractions and whole numbers
• How to use arrays to multiply fractions by whole numbers
• How to divide unit fractions by whole numbers
• How to divide whole numbers by unit fractions