Few topics in math education produce the types of fear and frustration caused by fraction visual models. And its not just students – few teachers, parents, or school administrators learned about visual models in school. So when it comes to teaching them to our students, we are often at a loss.
But the reality is that fraction visual models don’t need to be so complicated. We shouldn’t think of them as something “extra” we need to teach. Instead, recognize that fraction visual models are a way to make conceptual understanding of fractions more accessible to our students.
Fraction Visual Models: Useful Even in High School
Case in point: several years ago, I tutored “Carly,” a high school freshman. Carly had been a strong math student from Kindergarten through 8th grade. But when she began algebra, everything changed.
She paid attention. She did her homework. But she got C’s and D’s on her tests. When her parents and teachers couldn’t figure out what was wrong, they asked for my help.
At our first session, Carly showed me her most recent test. I had her walk me through each question, and It quickly became obvious that fractions were the problem.
But why hadn’t the issue come up before? After all, fractions are a major topic from 3rd grade on up.
Carly had learned to add, subtract, multiply, and divide fractions. But she had never learned the concepts. She didn’t know that a numerator told us how many pieces we have. Or that a denominator told us the size of each piece. She had never learned that a fraction was also a quotient.
Tricks like keep, change, flip worked fine when she was doing arithmetic. But she was misapplying strategies, finding common denominators when she didn’t need them. She lacked the fluency to manipulate fractional expressions.
The good news is that, once we identified the problem, it was a simple fix. We spent most of our first few sessions on fraction visual models. First, she drew models of individual fractions, like 1/3 or 2/5. Next, we moved on to operations, like “2/7 plus 3/7,” or “1/3 of 6/10.”
She quickly developed her conceptual understanding of fractions. And mostly on her own, she began applying this understanding her algebra class. She got a B on her next test, and her grades steadily improved through the end of the year.
Why Are Fraction Visual Models Neglected in Schools?
Given the importance and effectiveness of fraction visual models, you think they’d be taught in every classroom.
But in too many classes, I hear things like, “I prefer to just write the numbers,” or “I don’t get visual models.” And not just from students.
Many students learn algorithms for adding, multiplying, and simplifying fractions. They perform these operations without even understanding what they’re doing.
So when I hear that students “don’t like visual models,” it’s usually because they don’t actually understand how fractions work.
If students are exposed to fraction visual models, it’s often from a diagram in their textbook. To become fluent they should be creating their own visual models. Of course, there’s no space for that in a textbook. And even if there were, how do we know if they have the ‘right answer,’ when there are so many ways to draw the same model?
As a result, many educators focus on procedural approaches. 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, continue struggle with basic fraction operations.
Assessing Fraction Understanding
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 pretest 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.
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 such assumptions can be dangerous. So if you haven’t yet, try The Secret Whiteboard game with your students the next chance you get. (For interactive, digital lesson plans that include a Secret Whiteboard warmup, and are designed for remote learning, visit our online store).
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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.
Start by making sure your students understand fractions in two different ways: as parts of a whole, or as quotients.
There are more than two ways to understand fractions, but these two are fundamental. And the other meanings (items in a set, ratios, rates, or lengths) 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 your students draw fraction visual models, pay careful attention to 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.
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. [Buy: Grade 5 Digital Lesson, Fractional Quotients]
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.
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. [Buy: Grade 6 Digital Lesson: Reasoning with Rates]
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 for adding fractions.
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.
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 learn this skill before they learn to multiply and divide fractions. But it actually make senses 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 it becomes a bit easier when students 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.
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.
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. [Buy: Fraction Division, Grade 6 Digital Lesson]
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.
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 handson.
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 3parts: 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. They can show their thinking either with an explanation, or 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.
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About the Author
Jeff Lisciandrello is the founder of Room to Discover and an education consultant specializing in studentcentered learning. His 3Bridges Design for Learning helps schools explore innovative practices within traditional settings. He enjoys helping educators embrace inquirybased and personalized approaches to instruction. You can connect with him via Twitter @EdTechJeff