Are you smarter than a 5th grader? When it comes to fraction operations, for many adults, the answer is ‘no.’
If you don’t teach math, you might be surprised at how much we expect of 5th graders. Here’s a sample grade 5 test question from the Louisiana Department of Education.
I don’t know about you, but this problem had me scratching my head. I’m pretty sure the answer is D. But I certainly couldn’t have solved this when I was 10.
So if your students struggle with fraction operations, you’re not alone. Many rely on tricks like keep, change, flip, but never develop conceptual understanding.
These students have a hard time applying their understanding to real world problems. And they forget what they learn much more quickly.
Ensuring students are fluent in fractions is essential for their overall success in math. Fraction understanding provides the basis for decimals, percents, and ratios. They also allow students to understand proportions, slope, and rational numbers.
When students don’t master fractions in elementary school, they struggle through middle school, Algebra, Statistics, and even Calculus (if they get there).
To help your students, first you’ll want to know why they’re struggling with fraction operations. Then, be sure to have the tools and strategies to address their misconceptions.
Why Do Students Struggle with Fraction Operations?
An over-reliance on “tricks” isn’t unique to fraction operations. Throughout K-12 education (and beyond), students often learn to calculate without learning to reason.
So why is the problem felt most acutely with fractions? Part of the reason is that fractions are harder to connect to their lived experiences.
Fractions also require that students build upon their prior knowledge with whole number operations. So when students lack those foundations, fractions are twice as hard (or ½ as easy).
The actual answer is different for each student. But here are the common pitfalls to look out for.
Fractions Aren’t Intuitive
One reason students struggle with fraction operations is that fractions are just less intuitive than whole numbers. Students constantly add and subtract whole numbers in their everyday lives, even without realizing it. On occasion, they even multiply and divide.
And while they know what it means to eat half a cookie, or measure ½ cup of flour, they don’t usually operate with fractions. Many think of a ‘half’ as just a half, not as a number with one in the numerator and two in the denominator.
Many Students Lack Foundational Concepts
Another reason students struggle is that they haven’t mastered foundational concepts. Namely, the meaning of fractions, and whole number operations.
If you ask your students to multiply 3 by 13, do they envision an array? An area model? Skip count by 10 three times, then skip count by 3 three times? Or do they just stack and calculate?
Students who can visualize whole number operations can build on their understanding of division to imagine a whole cut into four parts. They can extend that idea to add one fourth to another fourth. Or to multiply a fifth by three.
But since many students lack these conceptual foundations, they just see fractions as one number stacked atop another. The more tricks and procedures they learn, the more likely they are to forget them or use the wrong trick for the wrong problem.
But there’s an even bigger problem with algorithms sans concepts. Students don’t develop the ability to see the math that is all around them. They don’t recognize math as a language, or a study of space and quantity. They can’t see the beauty and the art within mathematics.
Instead, it becomes a collection of “math facts,” random rules, and secret symbols.
Fraction Operations Contradict Whole Number Rules
Even students who understand whole number concepts may struggle with fractions.
In early elementary, students focus on mastering the Base-10 system. They count by tens, combine tens and ones, and break up numbers by place value to operate. It’s all about making tens, breaking tens, and using zero as a placeholder.
But fractions aren’t built on tens. Fractions can split a whole into 3rds, 7ths, or 45ths. And as the denominator changes, the size of the “unit” changes. As does the units needed to make a group (one whole).
Fractions also contradict the ‘rules’ students learn for whole numbers. When you multiply 3 x 5, the result is larger than both. But when multiplying fractions, you never know. The product of ⅓ and ¼ is smaller than both. But multiplying ⅕ by 4 creates a product in-between the two factors.
When students have strong conceptual foundations, they can generalize what they know about whole numbers to understand fractions. But without this understanding, they might feel like they’re starting over, with a whole new set of rules!
Three Tips for Teaching Fraction Operations
The first step to helping your students with fraction operations is to learn why they struggle.
The next is to have specific, actionable steps to address those challenges.
First, make sure your students really understand the meaning of a fraction. Then, connect that to what they already know about whole number operations.
Both of these can be accomplished with the The Three Vehicles — inquiry-based lesson models that can support almost any math concept.
1. Review and Assess Fraction Foundations
Simply put, a denominator tells us how many pieces a whole is divided into. A numerator tells us how many of those pieces we have. Though the numerator is on top, it doesn’t mean much unless you know the denominator. That’s why I teach fractions from the bottom up.
At some point (usually around grade 5), students learn that a fraction can also be thought of as a quotient. The result of a numerator divided by denominator. So 3 ÷ 4 is the same as ¾. This can be shown visually as 3 wholes, each split into 4 equal parts, with the parts regrouped into a single fraction.
To assess their understanding, students must be able to do this without direction. If you have to tell them to cut or to shade, they don’t get the concept. They’re just drawing.
2. Connect Fraction Operations to Whole Number Operations
Once students understand fractions, they can operate with fractions the same way they operate with whole numbers. Assuming, of course, they understand the meaning of the operations.
Addition and Subtraction
When students understand the meaning of ⅓, they can begin counting up and counting down to add and subtract fractions with like denominators.
The next step is to make a whole, such as by counting up by 4ths or adding ⅓ to ⅔. Next, they can subtract to break a whole (1 – ⅙).
By counting up beyond a whole, they can begin to operate with mixed numbers (⅔ + ⅔ = 1 ⅓).
Adding and subtracting fractions with unlike denominators is more complicated. Students need to convert to equivalent fractions, which relies on skills developed by multiplying fractions. So students first learn those concepts before returning to addition and subtraction with unlike denominators.
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Once your students can add fractions, it’s a simple step to multiply a fraction by a whole number. Just think of ¼ x 3 as ¼ + ¼ + ¼.
Next, replace the multiplication sign with the word ‘of.’ So ¼ x 3 becomes “one fourth of three.” This connects fraction multiplication to whole number multiplication. 3 x 2 means three (groups) of two. So it makes sense for ⅓ x 4 to be one third of (a group of) 4.
It also extends the concept from multiplication of a fraction to multiplication by a fraction, which allows students to solve when both factors are fractions.
This Google Slides Activity uses visual representations to demonstrate the meaning of fraction multiplication: students slide fraction models to see what happens when we multiply.
In addition to supporting conceptual understanding, fraction multiplication models show students why we multiply numerators and denominators in the algorithm.
For more on connecting fraction multiplication to whole numbers, review the five meanings of multiplication.
Dividing fractions can also be anchored in what students know about whole numbers.
Start by dividing a fraction by its numerator, such as ⅗ divided by 3. This involves partitive division, in which the divisor determines the number of groups.
Next, divide a whole number by a fraction. Here we use quotative division, in which the divisor determines the size of each group. Dividing 2 by ⅓, involves splitting both wholes into thirds (divisor = group size), and counting the total groups (quotient = number of groups). This illustrates why we multiply by the denominator when dividing by a fraction.
Quotative division is also useful for dividing a fraction by a fraction, but only in some cases. To divide ⅔ by ⅓, simply create 2 groups of ⅓ each.
But what about ⅓ divided by ¼? It’s possible to imagine splitting a third into groups one fourth in size…but it’s not very intuitive. In this case, I return to partitive division and use what I call the “ghost copies” strategy.
If I were to divide 8 by 2 partitively, I am changing one group of 8 into two new groups, with four in each group. If, instead, I divide 8 by ½, I’m turning one group of 8 into one half of a group. To create one whole group, I have to make a ghost copy of my starting value (dividend).
The ghost copies strategy can be extended for quotients like ½ ÷ ⅓. Treat ½ as ⅓ of a complete group. Thus, we add two ghost copies of ½ to make one whole group, resulting in 1½.
The five meanings of multiplication is also useful for helping students with fraction division. Each meaning of multiplication, in reverse, applies to division.
Equivalent fractions may be the trickiest aspect of fraction operations. It’s hard to connect it to whole numbers, as there’s no other whole number equivalent: Eight is just eight, there’s no whole number equivalent to 8.
But a fraction like ⅓ can also be written as 2/6 or 10/30, and still have the same value. In fact, you could argue that equivalence is why we use fractions in the first place.
When students ask “why do we need both fractions and decimals,” a great answer is that fractions allow us to divide whole numbers into any size parts we want. With decimals, we are limited to factors of ten.
But I’m including equivalence here for two reasons. First, it’s critical to many later applications of fractions. Simplifying fractions and adding unlike denominators require converting equivalent fractions. As do decimal and percent conversions, working with proportions, and finding slope. The list goes on.
The second reason is that fraction multiplication can be used to teach equivalent fractions.
I teach fraction conversion as multiplying by one. To find an equivalent for ½, I can multiply by 3/3 (aka one), creating 3/6.
Students who can use area models to multiply fractions should connect the idea of multiplying by ⅓ to the idea of multiplying by 3/3. Both visually, and with the “multiply across” algorithm.
3. Use The Three Vehicles to Teach Fraction Operations
The Three Vehicles are inquiry-based lesson models that promote conceptual understanding. They can be used to teach almost any math concept, at any grade level.
The vehicles are built on multiple representations theory, the idea that any mathematical concept can be represented five different ways: physically, visually, symbolically, conceptually, and verbally.
We can define fluency as the ability to translate a mathematical idea among all five representations. This includes translating an expression into a visual model. Or explaining, in words, how a manipulative works.
The five representations define how mathematical ideas can be represented. And the vehicles teach students how to translate among them.
The first vehicle is the scale model. It combines physical and visual representations (also known as concrete and pictorial), as both serve a similar purpose. They represent the size of numbers and the meaning of operations.
Scale models are useful for building and assessing conceptual understanding. To use scale models as a teaching tool, first teach the principles of reading and creating scale models. Then, use the models as tools for helping students to learn through inquiry.
For example, if students understand fractions, and whole number multiplication, they can use their modeling skills to “discover” fraction multiplication, even without direct instruction.
It’s important for students to create their own models, rather than just interpreting models in the textbook or made by teachers. Reading models is helpful, but isn’t enough to achieve fluency.
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The next vehicle is the number sentence. Number sentences (equations and inequalities) build on the concepts developed with visual models, to support development of abstract understanding.
Number sentences allow students to manipulate expressions and work in multiple steps, making them much more useful than calculating with algorithms.
I use number sentence proofs to help students learn number sentences and connect them to the other representations.
The third vehicle, story problems, help with application skills and mathematical language.
I teach students to use the Polya Process for solving word problems. This approach emphasizes the importance of using multiple representations to solve word problems.
This vehicle comes last, because students can use scale models and number sentences as tools for solving word problems.
Teaching Fraction Operations in Your Classroom
I hope this article offered a useful overview of the main concepts needed for success with fraction operations.
You can find resources for teaching fractions with all three vehicles in our online store.
Or level up your inquiry-based math skills by enrolling in an online workshop. These are real-time sessions, conducted by a live facilitator. We offer separate sessions on each vehicle for elementary and for middle school teachers, so you can focus on the techniques and standards that are most important to your students.
Finally, if you’d like to incorporate this type of learning right away, download our Fractions Essentials Bundle. It has everything you need to get started, from interactive Google Slides activities, to lesson plans, answer keys, and more!
About the Author
Jeff 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