Area model showing multiplication of fractions - 1/2 times 2/3

Fraction Visual Models Made Simple

What would your students say if you asked them to create fraction visual models?

In too many classes, I hear things like, “I prefer to just write the numbers,” or “I don’t get visual models.”

Many students learn algorithms for adding, multiplying, and simplifying fractions. And they can perform these operations without even understanding what they’re doing.

So when I hear that students “don’t like visual models,” it’s safe to assume they don’t actually understand how fractions work.

But the real question is ‘why?’ It’s not that fractions are all that complicated. If students start by learning to cut shapes into equal parts, it’s a small step to introduce fraction notation (numerator and denominator). And if they understand how fractions work, they see that operating with fractions is similar to operating with whole numbers.

If students are exposed to fraction visual models, it’s often from a diagram in their textbook. But to become fluent, students need to create their own visual models. But there’s no space for that in a textbook. And even if there were, how would we assess their work when there are so many ways to draw the same model?

As a result, many educators focus on procedural approaches. And some teachers attempt to introduce visual models without really being clear on how they work, let alone effective methods for teaching with visuals.

Studies highlight the disappointing results of this approach. For one, few children ever really master fractions. And many adults, even college students, struggle with basic fraction operations.

Try This Fraction Visual Model Assessment

Teachers often ask me for help with lessons on fractions. And whether it’s equivalent fractions, adding, subtracting, or mixed numbers, I always start with the same question: “Do the students know what a fraction is?”

When teaching a new group, I start by playing The Secret Whiteboard Game. It’s really just a quick formative assessment, but children enjoy it a lot more than a pre-test or “entrance ticket.”

Everybody stands, and I announce a question. They have five seconds to respond on their whiteboards. When I finish counting down, they hold their boards up. If they’re right, they remain standing. Otherwise, they continue to play from their seats.

I start with questions like, “What is ½ plus ½?” or “Draw a visual model of ⅗.” If students can’t answer these questions, we put the planned lesson on pause. Students need these fraction concepts before they can perform more complex operations with fractions.

I can easily get a sense of how well students understand fraction concepts by scanning the room after the first couple questions.

Teachers are often surprised to see their students have difficulty drawing simple fraction visual models. Especially at the middle and high school levels, we just assume that our students know the meaning behind fractions.

But this assumption can be dangerous. So if you haven’t yet, try The Secret Whiteboard game with your students the next chance you get.

Two Meanings of Fractions

If you discover that your students don’t understand fractions as well as you thought, don’t worry. Most students can learn fraction concepts quite easily if we approach them the right way.

And to start, we want our students to understand fractions two different ways: as parts of a whole, and as division.

There are more than two ways to understand fractions, but these two are fundamental. And the other meanings (parts of a set, ratio, rate, and length) can be derived from the first two.

Parts of a Whole

Students should first learn about fractions as parts of a whole. The denominator tells us how many pieces the whole is cut into. And the numerator tells us the number of pieces we have.

This concept and fractions notation are first introduced in third grade. But the foundations are built in first and second grade, by cutting shapes into equal parts and describing them as “halves,” “thirds,” and “quarters.”

Unfortunately, the first and second grade activities fall under the Geometry domain, which is not considered a ‘major cluster.’ So, many students make it to 3rd grade without understanding equal groups or the language of “halves,” “thirds,” and “fourths.”

When asking students to draw fraction visual models, notice how they divide the shape. This tells you whether they understand the meaning of the denominator. Though the numerator is on top, the concept of the denominator actually comes before the concept of the numerator.

Area model and number line showing three fourths (3/4)

Once students understand parts of a whole, they can extend this concept to other meanings of fractions. Students can think of 3/4 as three pieces out of 4, measurement (¾ of an inch), or parts of a set/ratios (three out of 4 students).

Fractions as Division

After learning parts of a whole, students are ready to learn about fractions as division. While parts of a whole is introduced in 3rd grade, fractions as division is usually introduced in 5th grade.

Start with the basics. If I have one whole, and divide it by two, the result is one half. 

One way to assess this understanding is to ask students what is 3 divided by 5. Many students struggle with this type of question. I often hear responses like, “you can’t divide three by five.” Other students will attempt to do long division.

Fraction visual model showing fraction as division, with 3 wholes split into 5 pieces

But a simple visual model can make clear why a fraction is equal to the numerator divided by the denominator. Draw three circles, and divide each circle five ways. Take ⅕ from each circle and combine them to make ⅗.

A story problem can also be a helpful complement here. “If five students share three cakes equally, how much does each student get?”

Understanding fractions as division provides a foundation for topics like rate, slope, and balancing equations.

Visual Models for Adding and Subtracting Fractions

Once students can create visual models for a single fraction, it’s a small step to create visual models of fraction addition.

Adding fractions with the same denominator is no different from adding whole numbers. To perform ¼ plus 2/4, simply add the numerators. Students can represent this by coloring in a circular or rectangular area model.

Fraction visual models showing addition of 1/4 plus 2/4, and subtraction of 3/5 minues 2/5

For subtraction, simply do the opposite. If I start with ⅗ and take away ⅖, simply cross out the ⅖ being taken away, leaving ⅕.

Students should also create models like ⅓ plus ⅔ so they recombine fractions to make a whole.

Adding fractions with unlike denominators is a good deal more complicated. Traditionally, students learned this skill before they learn to multiply and divide fractions. But in can make sense to teach multiplication of fractions first.

Multiplying fractions provides a foundation for equivalence and finding like denominators. I’ve noticed that students have more success adding unlike denominators when they already understand fraction multiplication.

Fraction Visual Models for Multiplication and Division

Understanding fractions as division is quite helpful when multiplying and dividing fractions.

Start with questions like “⅗ x 5.” Or “¾ ÷ 3.” Students can use reasoning to solve these problems before being exposed to the multiplication and division algorithms.

They should also be able to create visual models for these problems by combining what they know about fractions and about the meaning of multiplication. If a student draws 5 copies of ⅗, they’ll find that combining them makes three wholes.

If they draw ¾ and separate into 3 equal groups, they’ll easily see that each group has ⅓.

Multiplying a fraction by a fraction is a bit more complicated, but is easier if students think read the multiplication sign as ‘of.’ So ½ x ⅔ is read “½ of ⅔.”

This can be represented visually with an area model. Start with a rectangle, and cut vertically into two pieces, shading one. Next, cut horizontally into 3 pieces, shading two in a different color. The overlapping shading is our product, 2/6 or ⅓. 

Area model showing multiplication of fractions - 1/2 times 2/3

This demonstrates two important aspects of fraction multiplication. First, it shows why we multiply the denominators. When they cut the shape in both directions, they can see that the horizontal cuts multiply the vertical cuts. And the same is true for the numerators: if we take ⅔ of ⅖, we’ll have 4 squares shaded, 2 by 2.

fraction visual model showing a fraction divided by a fraction

To divide by a fraction, students will generally rely on the repeated subtraction meaning of division. So to divide ¾ by ⅛, start by drawing a model of ¾. Then, subtract ⅛ again and again. You’ll find you can subtract ⅛ six times, and thus, the quotient is 6.

This only works for simple fractions, but it can help students understand why the algorithm (keep, change, flip) works.

Equivalent Fraction Visual Models

A third key competency with fractions is equivalence. Students need to recognize that ⅔ is equivalent to 4/6, 8/12, and so on.

With fraction visual models, this is often pretty straightforward. If you start with a model of ⅔, simply cut each third into 2. Students can see we now have 6 total pieces, 4 of which are colored in.

It also works in the other direction. If they are trying to simplify a fraction like 8/12, they simply erase enough of the cuts until they are left with two pieces shaded out of three.

Fraction visual model showing equivalence of 8/12 and 3/4

Representing equivalent fractions is critical for adding fractions with unlike denominators. It’s also a foundation for several middle school standards, such as proportion, slope, and solving equations.

As with the other fraction models, equivalent fraction models work best with simple fractions. I wouldn’t want to bother creating a visual model to show that 56/64 is equivalent to ⅞. 

Once students build the concepts with simple fractions, show them how the algorithm connects to what is happening in the model. Then, they can simply rely on the algorithm for more complex problems.

Planning Your Fraction Visual Models Lesson

When planning a lesson with fraction visual models, what’s most important is that it’s hands-on. 

If the teacher is the only one creating the visual models, the students will get limited benefit. Consider using the workshop lesson model. This simple structure breaks a lesson into 3-parts: inspire, inquire, and reflect. You can learn more about the workshop model here.

A common mistake educators make is drawing the examples on the board, and having students copy them down. But this doesn’t tell us if students understand what they’re drawing! 

Instead, have them create visual models for new situations. And have them try it before you show them the algorithm. If I ask a student for a model of ⅓ x ½, I won’t accept a model of ⅙. They need to show how they got there. Using a visual model organizer can make the process easier. 

For more on best practices for teaching with visual models, check out this post.

Another way to incorporate visual models is with number sentence proofs. Students determine if an equation is true or false, and provide support for their answer. One way to show their thinking is with a visual model. Here is a complete lesson plan for number sentence proofs with fractions. It includes a rubric, key, and printable resources, so you can use this lesson in your class tomorrow.

If you’re interested in more stories and tips to support student-centered learning in your school or classroom, subscribe to our newsletter. When you register, we’ll send you a free Workshop Lesson Planning Template. This simple structure will help you plan engaging, inquiry-based lessons on any topic.

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About the Author

Jeff Lisciandrello is the founder of Room to Discover and an educational consultant specializing in student-centered learning practicesJeff Lisciandrello is the founder of Room to Discover and an education consultant specializing in student-centered learning. His 3-Bridges Design for Learning helps schools explore innovative practices within traditional settings. He enjoys helping educators embrace inquiry-based and personalized approaches to instruction. You can connect with him via Twitter @EdTechJeff

 

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