Developing Fraction Concepts through Equal Sharing Problems

Early fraction instruction can sometimes enable children to solve beginning fraction problems without providing a foundation for more complex fraction problems and concepts. Consider the following problems that students will solve in upper Elementary, Middle and High School:

Early fraction instruction should support students to develop an understanding that they can draw upon when they encounter problems such as the ones above. A common problematic way of teaching beginning fractions is teaching children that m/b means “m out of b”. Although m/b can sometimes mean “m out of b”, the problems above can’t be fully understood using a “m out of b” understanding of fractions.

m/b always means m divided by b. We want students to understand that m/b equals m divided by b (for all b ≠ 0) because this understanding applies to all problems that involve fractions or rational expressions. Children whose early fraction instruction is based on Equal Sharing Problems learn that m/b is the result of m divided by b. An Equal Sharing Problem is a word problem that involves dividing an amount (which can be represented with m) by a whole number of sharers (which can be represented as b, as long as b doesn’t equal 0) and obtaining a result that isn’t a whole number m/b.

Here are some examples of Equal Sharing Problems:

A.   2 friends want to share 3 cookies. How much cookie should each person get if they want to have the same amount and as many cookies as possible?

B. There are 4 blocks of clay for 6 art students to share. If they share the clay equally, how much clay should each person get?

C. 8 children were sharing 6 pancakes. If they share the pancakes equally, how much pancake will each child get?

We recommend that fraction instruction begins with Equal Sharing Problems. Since Equal Sharing Problems such as the ones above contain no fraction words or symbols, children can solve these problems before being taught fraction terms and representations. You can later connect fraction words and symbols to the fractional quantities that children have created in the process of solving the problem.

It’s best to pose Equal Sharing Problems without first demonstrating strategies for solving these problems. Virtually all children have shared snacks or toys with someone else and can use that experience to solve these problems. Provide children with paper, pencils, crayons or markers but do not provide fraction manipulatives. Children will create fractional quantities in the process of solving these problems which will help them to learn about these quantities. The methods children use for creating quantities gives teachers a window into a child’s fraction understanding.

Even though most State Math Standards don’t include fractions until second or third grade, it’s useful to start posing Equal Sharing Problems in kindergarten or first grade. Young students can provide their answers with a picture if they aren’t ready for fraction words or symbols. For example, here is how Alice solved the problem, 2 friends want to share 3 cookies. How much cookie should each person get if they want to have the same amount and as many cookies as possible?

She began her strategy with a pencil, but at the end of the process, she used a yellow and a brown crayon to draw different hairstyles to show which person got which portion of the cookies. Even though Alice didn’t use fraction words or symbols, she was beginning to develop an understanding that 3 shared by 2 is 3 halves or 3/2. (Other students in Alice’s class solved this problem by first giving each child a whole cookie and then dividing the final cookie by two.)

It’s useful to continue posing Equal Sharing Problems throughout elementary school and into Middle School. Seventh and eighth grade teachers have been surprised to find that even some of their most advanced students don’t immediately know that 8 people sharing 5 items equally will result in each person getting 5/8. These students don’t understand that 5 divided by 8 = ⅝ or that m divided by b equals m/b which is essential for understanding Middle and High School math concepts such as proportions, linear equations, and rational expressions.

Here is how Wesley, a fifth grader, solved the problem about 8 children sharing 6 pancakes.

Unlike the seventh and eighth graders mentioned above, Wesley understands and can explain why 6 divided by 8 equals 6/8. His understanding of fractions will provide a strong foundation for Middle and High School math.

Our Mentor Teacher Site under Resources has additional Equal Sharing Problems that you can pose to your students. This Equal Sharing Problems Reference Page will help you write problems for your students. In a future blog post, I will describe the developmental trajectory that children pass through when solving Equal Sharing Problems.

Additional information about Equal Sharing Problems is available in Extending Children’s Mathematics: Fractions and Decimals by Susan Empson and Linda Levi.

This blog post was supported in part by the U.S. Department of Education, through grant award number U423A180115 to Florida State University. The opinions expressed are those of the authors and do not represent views of the U.S. Department of Education.

Extending Children’s Mathematics: Fractions and Decimals by Susan Empson and Linda Levi contains much of this information and more about Equal Sharing Problems.