Introducing Division with Fractions: 5 Teaching Tips for Success

“Yours is not to reason why, just invert and multiply!” Sound familiar? This is how many of us learned how to divide fractions, by using the “invert and multiply” algorithm. But did we really understand what we were doing? I know I didn’t. It wasn’t until I became an adult that the mystery of fraction division started to make sense. And it wasn’t inverting and multiplying that helped me. In this post, we’ll explore five teaching tips when introducing division with fractions that can help students make sense of this perplexing concept.

 

Teaching Tips that Help Students Understand

Division with fractions is often considered the most mechanical and least understood topic in elementary school math (Tirosh, 2000). Take 6 ÷ ½. Using the standard algorithm, all one has to do is flip the ½ to make it a 2, then multiply 6 x 2 and presto! The answer is 12. No questions asked, no interpreting the problem, no reasoning involved. 

When introducing fraction division to students, we’ve found that there are five things to keep in mind that help students develop understanding and focus on reasoning.

• Begin with Measurement Division
• Build on Whole Number Division
• Use Fraction Strips
• Offer Sentence Frames
• Pose Problems in Real World Contexts

 

Begin with Measurement Division and Build on Whole Number Division

We know that there are two types of division, partitive or sharing division and quotative or measurement division. Let’s say we have 6 cookies, and we want to divide the cookies among 2 friends. We know how many groups there are (2 friends) and we must figure out how many cookies each friend will get. This is called partitive or sharing division. With measurement division, we start with the 6 cookies, but this time we want to give each friend 2 cookies. The amount or number of cookies is already measured and what we must figure out is how many friends or groups we can give the 2 cookies to. 

Donna Curry from TERC (Curry, 2010) says that “With division of whole numbers, both types of division add meaning for students. With division of fractions, measurement division seems more effective.” We have also found this to be true, especially when dividing a whole number by a unit fraction. For the problem 6 ÷ 1/2 for example, we’ve seen that it’s easier for students to think about the question, “How many 1⁄2 ‘s are there in 6?” Or, to put a context to it, “How many half pint servings of ice cream are there in 6 pints of ice cream?”

When introducing fraction division, we recommend starting with whole number division problems such as 6 ÷ 2. For this problem, the question is, “How many groups of 2 are there in 6?” Then move to a division of fraction problem such as 6 ÷ ½. Building on what students already know about whole number division provides a bridge to dividing with fractions. 

 

Use Fraction Strips and Sentence Frames

After linking whole number division and division with fractions, have students make a fraction kit. Below is an example of a fraction kit using wholes, halves, fourths, eighths, and sixteenths. 

Directions:

Cut 12-by-18-inch construction paper lengthwise into 3-by-18-inch strips. Colors don’t matter, as long as each student has the same color strips for each strip. Notice how I don’t tell how many times to fold each time. I want to see if students can guess before we all decide on the correct number of folds. 

1. Take the dark blue colored strip and write “1 whole” on the strip.
2. Take the red colored strip, fold it ____ times to create 2 equal pieces. Cut. Write ½ on each piece.
3. Take the light blue colored strip, fold it ____ times to create 4 equal pieces. Cut. Write ¼ on each piece.
4. Take the purple-colored strip, fold it ____times to create 8 equal pieces. Cut. Write 1/8 on each piece.
5. Take the brown-colored strip, fold it ____times to create 16 equal pieces. Cut. Write 1/16 on each piece. 

Visual models can help give meaning to abstract ideas, and we’ve learned that fraction strips are very effective in helping students develop meaning when dividing fractions, and making connections between models (fraction strips and equations). The teaching sequence below helps students see what 2 ÷ 1/2 = 4 looks like using a physical model. 

1. Pose a problem that’s accessible to everyone such as 2 ÷ 1/2 =
2. Ask, “How many 1⁄2 ‘s are there in 2 wholes?”
3. Have partners work together to model the problem with their strips and figure the answer. 
4. Finally, have students share their answer using the following sentence frame: 

“There are ____ ____s in ____.”

1. Continue by posing a menu of problems like the ones below for students to work on:

1 ÷ 1/2          3 ÷ 1/2      1 ÷ 1/8       1 ÷ 1/4         2 ÷ 1/16    

Provide the following frames for students to use when asking about and answering the problems:

How many ____s are there in ____?

There are ____ _____s in _____.

These frames serve to not only support language production, but also to help students interpret the problem. In other words, what is the problem asking me to do? In the first problem, it’s to find out how many halves there are in one whole. Without the frames, students (and many adults) might think that 1 ÷ 1/2 means, “1 divided in half.” Sentence frames help students talk about their thinking and conceptualize problems.          

 

Watch a Division with Fractions Lesson

Watch Andrea Barraugh from Math Transformations introduce division with fractions to a group of students using fraction strips and sentence frames. Click here to watch the video. The password is “number.” 

In the lesson, you’ll see Andrea start with division with whole numbers, reviewing the two types of division (partitive and measurement). She then provides a bridge connecting whole number division to division with fractions using fraction strips and sentence frames.

 

Fraction Kits with Different Fractions

To expand students’ horizons, have them make another fraction kit using wholes, thirds, sixths, and twelfths.

This time, pose a word problem: 

“Ms. Chavez brought 2 pints of ice cream for a class party. Serving size for each student is 1/3 pint. How many students can she serve?” Use the fraction strips to model the problem and explain your thinking. 

While students find it accessible to model problems using fraction strips, it can be challenging when we ask them, “What equation would model this problem?” Knowing that the word problem above can be represented as 2 ÷ 1/3 = 6 can be tricky, as you’ll see in some of the student work samples below. Helping students make connections between models is important, whether they are operating on whole numbers or fractions.  

 

Pose Problems in Real World Contexts

Physical models such as fraction strips can help students visualize division with fraction situations. Word problems, like the one involving Ms. Chavez’ ice cream, can also make abstract ideas come to life. While fraction strips are handy physical models, students can also create their own drawings to represent division situations.

My colleague Kelly Miller posed some division with fraction problems to her fifth graders. She was teaching the following Common Core Math Standard:

Interpret division of a whole number by a unit fraction and compute such quotients. For example, create a story context for 4 ÷ (1/5), and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that 4 ÷ (1/5) = 20 because 20 × (1/5) = 4. 

Before I share some student work samples, try solving the following problems mentally and then draw models (pictures and equations) that match each story.

• The pizza slices served at Connor’s Pizza Palace are ¼ of a whole pizza. There are three pizzas ready to be served. 14 children come in for lunch. Is there enough pizza for every child? Show your math thinking.
• You have just bought 6 pints of Ben & Jerry’s ice cream for a party you are having. If you serve each of your guests 1/3 pint of ice cream, how many guests can you serve?

 

Analyzing Students’ Work

Let’s look at a few students’ work samples. The first one is a response to the pizza problem. Notice how Jose (below) does a great job of modeling the problem using a diagram, showing the three (square) pizzas and dividing up each pizza into 1/4 servings. He then clearly explains his reasoning about the picture model. 

What’s interesting is that his number model (equation) doesn’t match his diagram or his explanation. It looks as though he might have used the standard algorithm to arrive at 1/12. Using procedures that don’t make sense to students often get in the way of understanding. Fortunately, Jose can benefit from some specific feedback. Asking him questions such as, 

“The answers to your equation and your explanation and diagram don’t match. Why do you think so?” 

“Should the equation be 1/4 divided by 3 or  3 divided by 1/4?” 

“Does 1/12 as an answer make sense? Why or why not?”

These types of questions are ones that teachers can use in subsequent class discussions, benefitting everyone. 

For the second problem, Amanda (below) shows two correct models, an equation and a diagram that match the story. While her models are clear, they don’t necessarily speak for themselves. Encouraging students to explain their thinking with words can serve as a window into their thinking. Questions like “Why does your equation match the story?” or “What were the steps in your thinking when making your diagram?” can engage students in metacognition and provide the teacher with good assessment information. 

Both Jose and Amanda demonstrate strengths in their work. They use models that are accurate and that make sense to them, meeting at least part of the Common Core Standard that Kelly was teaching. Their work also shows areas for growth that their teacher can use to guide her instruction. 

Next steps might be to have students create their own story contexts for division with fraction problems. Kelly might also help students see, for example, that 6 ÷1/3 is equivalent to 6 x 3, highlighting the relationship between division and multiplication and laying the groundwork for exploring the invert and multiply algorithm. 

Laurie O. Cavey and Margaret T. Kinzel do a terrific job of explaining the invert and multiply procedure and provide some engaging activities for students (and adults) in their article, From Whole Numbers to Invert and Multiply (Vol. 20, No. 6 Teaching Children Mathematics, February 2014, NCTM). You may need to be a member of the National Council of Teachers of Mathematics (NCTM) to access the article. If you’re not a member, I encourage you to join! 

 

Yours IS to Reason Why

My math mentor (everyone needs one, right?) once said “Do only what makes sense to you.” She was talking about math, and I think it applies here with division with fractions. Starting with measurement division, building on what students know about whole number division, using fraction strips, providing sentence frames, and situating division of fraction problems in real world contexts all help students make sense of this difficult concept. We hope that you try out some of these tips. And thank you, Kelly, for sharing your students’ brilliance with us!