Early Strategies for Equal Sharing Problems
A previous blog described Equal Sharing Problems and the advantages of using these problems to develop students’ understanding of fractions. I suggest you read the previous blog before reading this one.
This blog describes the developmental progression of early strategies that students use to solve Equal Sharing Problems and what each strategy indicates about a student’s understanding of fractions. I have posed Equal Sharing problems to kindergarteners through seventh graders and have seen these early strategies at all grades. Although most fourth – seventh graders pass quickly through these strategies, it’s helpful for fourth – seventh grade teachers to understand these strategies. Students generate these strategies on their own without their teachers demonstrating the strategies.
Here is how fifth graders Jordan and Shirley solved the problem: Zayna and her friends are having a sleepover. The 4 girls are sharing 7 giant cookies. How much cookie will they each get if they are sharing them equally? Pause to consider how these strategies differ from each other and what they might indicate about these students’ understanding of fractions.
Shirley’s strategy
Jordan’s strategy
Shirley’s strategy indicates that she understands that a whole can be partitioned into parts. We don’t know if Jordan understands this concept from examining his strategy.
Consider this interaction I had with Brendan, a first grader who was solving this Equal Sharing Problem: Two brothers were sharing 3 cookies so that each person got the same amount of cookie and there were no leftovers. How much cookie would each brother get?
Brendan: [After hearing the problem read to him a few times and spending a minute in thought.] Caden would get one cookie and I would get one cookie and there would be one cookie left for my mom.
Ms. L: What if your mom said, “I don’t want any cookies today. I want you to share the cookies so that you each get the same amount and there are no leftovers.”
Brendan: Well, I would have one cookie and Caden could have two. I am a sweet brother and I like to help Caden because he is little.
Ms. L: That is super nice of you, but your mom said that you had to have the same amount. If you have one and Caden has two, is that the same amount?
Brendan: No, but if I get two cookies and Caden gets one, he might cry. I don’t want that to happen.
Understanding that a whole can be partitioned into parts is critical to understanding fractions. Equal Sharing problems are ideal for developing and assessing this understanding because these problems require students to partition whole items. When we provide fraction manipulatives that are already partitioned, we can’t assess if students understand that a whole can be partitioned into parts to create fractional quantities.
Consider how Adam, another fifth grader, solved the problem about 4 girls sharing 7 cookies. Pause to consider how Adam’s strategy differs from Shirley’s and Jordan’s strategies.
Like Shirley, Adam partitions whole cookies. Adam goes a step further in that he uses all of the cookies and each person gets the same amount. While Shirley created halves, Adam created both halves and fourths.
My interaction with Brendan, the first grader solving the problem where he and his brother were sharing 3 cookies equally, continued as follows:
Ms. L: What if your mom said, “You can’t have any cookies unless you can share them equally and there are no leftovers?” Is there anything you could do?
Brendan spent a couple of minutes trying to convince me that he was happy with one cookie, but I held firm that no one would get any cookies unless they could do what his mom asked. It would have been ideal to pull other students into the discussion, but I was working with Brendan after school and there were no other students in the room so I proceeded as follows:
Ms. L: When my kids were young, sometimes we would go somewhere, and they would have these really huge cookies. I would just buy one cookie even though I had two kids. Has that ever happened in your family?
Brendan: Yes. [long pause] Oh, wait a minute! I could karate chop the cookies.
Ms. L: What do you mean, karate chop the cookies? [Brendan showed me a chopping action with his hand.] Maybe you could draw that so that I could understand?
Brendan produced this drawing:
Brendan: I tried to make them the same. Cade gets the first piece.
Ms. L: How much would each person get?
Brendan: One slice for Cade, one for me, one slice for Cade, one for me, one slice for Cade and one slice for me. Three slices for Cade and three slices for me.
Brendan’s solution is correct even though he doesn’t use fraction words or symbols. Take a moment to consider how Brendan’s solution is different from Adam’s and Shirley’s.
The student work shared illustrates the different levels of beginning strategies for Equal Sharing Problems:
Pause for a minute to consider where each strategy in this blog falls along this continuum.
Notes about these strategies related to grade bands
I have seen all these strategies in kindergarteners through seventh grade. The strategies at the beginning of this progression are very common in kindergarten through third grade. It’s more common to see the strategies at the beginning of the progression in fourth through seventh grades when students haven’t solved Equal Sharing problem in earlier grades.
When working with kindergarteners and first graders, my goal for most students is that they eventually generate strategies like Brendan’s: They exhaust the amount to be shared, each person gets the same amount and they can show me their answer with a drawing. Sometimes I choose to introduce fraction words like half or fourth, but it isn’t necessary that K-1 students use or understand these terms. Fraction symbols ( ½ , ¼, ¾ and so on) are very complicated so I use them cautiously, if at all, with K-1 students. I typically stick to Equal Sharing Problems with 2 or 4 sharers when working with K-1 students.
When working with second and third graders, my initial goal is that students will generate strategies like Brendan’s where they exhaust the amount to be shared, each person gets the same amount and they show their answer with a drawing. Many third graders arrive at these solutions after only solving a few Equal Sharing problems if they have solved these problems in second grade. Once students generate these strategies, I introduce fraction words and support them to write their answers with fraction words as Adam did.
If a second grader generated Brendan’s strategy, I would ask, “How much of a whole cookie is one of those slices?” in hopes that they, or another student in the class could answer a half. I could then ask, “Brendan said each boy would get 3 slices, if each slice is a half, is there another way Brendan could say his answer?” In hopes that someone could say, “3 halves” (Note that there is nothing wrong with an answer like 3 halves, in fact, fractions greater than 1 are preferred over mixed numbers in higher math.) I wait until students are comfortable with fraction words (1 half, 3 fourths and so on) before introducing fraction symbols as “another way to write your answer.” With second and third graders, I typically stick with 2, 4, 8, or 3 sharers.
Fourth – seventh graders often have fraction misconceptions that get in their way of understanding the fraction, ratio and proportion concepts in their grade level standards. Equal Sharing problems are excellent tools to develop students’ understanding of fraction concepts at these grade levels. When students are supported to use their own strategies to solve these problems, and teachers facilitate discussions of students’ strategies after students solve these problems, fourth – seventh graders typically move quickly through this beginning progression of strategies. It’s important to expect and allow these beginning strategies even though they move quickly through this progression. Fourth – seventh graders have been introduced to fraction symbols and will often write their answer with fraction symbols. For the first few problems, I strongly encourage them to also write their answer with fraction words.
In the next blog, I will describe the progression of later strategies for solving Equal Sharing Problems. This progression is especially important for third – seventh grade teachers.
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.