Students Stumped by Fraction Operations? Try this.
Few math topics cause more frustration for students and teachers than fraction operations. Part of the problem is that fractions don’t look or act like most numbers we deal with on a daily basis. To make matter worse, many adults don’t have a strong command of fractions. Ask ten adults to multiply 1/2 by 3/4 and you might wonder whether fractions were even covered when they were in school.
When I help schools develop their curriculum plans, I notice a trend. Teachers in middle and upper grades bemoan their students’ struggles with fraction operations. They wonder why elementary teachers never taught fractions. After all, fractions are supposed to be covered in grades 3 through 5.
The reality is that those teachers did teach fraction operations. The problem is that the students didn’t learn how to operate with fractions. At the very least, students didn’t retain what they’d learned beyond the unit test.
Importance of Fraction Operations
Though many of us would love to move on from fractions after grade 5, the reality is that fraction operations are an essential foundation for much of higher level math. Decimals and percents are actually types of fractions. In middle school, ratio, proportion, and slope all build upon fraction operations. In high school, Algebra, Statistics, and Calculus all involve rational numbers (numbers that can be expressed as fractions).
Whenever I begin working with a teacher on fractions, I have their students draw me pictures. “Show me what 1/2 looks like. Show me 3/4.” Time after time, teachers are shocked to see their students struggle to draw a simple fraction.
These are not just 3rd grade students, either. They include 6th graders working on fraction division, and Algebra students simplifying rational expressions. Some are surprised that students would have such gaps in understanding. But it’s actually pretty common when students learn procedures without concepts.
Teach Fraction Concepts, not Tricks
To truly understand fractions, students just need to understand numerators and denominators. A denominator tells us how many pieces a whole is divided into. A numerator tells us how many of those pieces we have. At some point (usually around grade 5), they learn that a fraction can also be thought of as numerator divided by denominator.
Once they understand fractions, they can operate with fractions the same way they operate with whole numbers. Adding 1/5 to 2/5 brings 3/5. Taking 1/8 away from 1/4 requires breaking the 4th into 8ths — much the same way taking 3 from 62 requires breaking up a ten. Multiplying 4/7 by 1/2 leaves you with 2/7 — half of what you started with.
Eventually, students can save time by using algorithms to operate with fractions. Sadly, many students learn algorithms without really understanding the meaning of fractions or of the operations. This approach has several drawbacks. For one, when a student can’t attach meaning to an algorithm, they won’t retain it for long. In addition, they get confused about which algorithm to use. “Do I need a common denominator to multiply?”
Perhaps a bigger issue is that algorithms cause students to feel disconnected from math. Rather than seeing math as a way to make sense of the world around us, it becomes a collection of random rules and secret symbols.
Teach Fraction Concepts with Visual Models and Number Sentences
I’ve had great success using visual models and number sentences to teach fraction concepts. Visual models can be used both as an instructional and an assessment tool. If a student can tell you that 3/5 times 1/3 is 1/5, ask them to draw it. If they can’t draw the picture, they don’t understand the concept.
There is more than one right way to use visual models, but there are some general rules. To be a visual model, it should be drawn to scale – otherwise it’s just a drawing. Also, the model should clearly show starting values and operations, not just a result. If someone else can turn your visual model into a number sentence, it’s probably good.
Number sentences are another great way to teach concepts, including fraction operations. A number sentence is a complete mathematical statement. They consist of two expressions on either side of a comparison symbol, such as an equal sign. Equations and inequalities are both examples of number sentences.
By proving or disproving number sentences, students deepen their understanding of number and operations. Manipulating expressions and equations can help them see how math standards are interconnected. Students should be taught the conventions of working with equations. Show them, for example, that each step goes beneath the step before it. Also teach them to only use one equal sign per row. Then, they can manipulate number sentences to discover new concepts and connections.
Diving into these hands-on activities is not always easy. That’s why I designed a course to cover every step of teaching math through hands-on activities. It can be taken either as a live, 2-Day Workshop or as a self-paced online course. The course includes video lessons, planning guides, and reproducible classroom resources.
Want to get started with hands-on math but not ready to take a whole course? I feel your pain! That’s why I’m offering our Hands-On Fractions QuickStart Guide at no cost. It includes everything you need to effectively teach fraction operations through inquiry. You’ll get lesson plans and reproducibles you can try out in class tomorrow.