Students often struggle with division problems that do not yield a whole number result. Whether dividing whole numbers in the lower grades or fractions in middle school, when students are asked to explain what the remainder represents, they become puzzled.
Context plays a vital role in making sense of the remainder. If a student is thinking about 5 ÷ 2, it can be helpful to have them imagine sharing five of something with a friend. However, it is important that the objects being shared could, in theory, be broken in half. Giving students objects that they are used to seeing split into pieces (like a sandwich, or cookie, etc.) instead of ones that are always shown as wholes (cars, or animals, etc.) gives them the advantage to start thinking about splitting one whole apart. Too often, students are told (or interpret that they are told) that “you can’t do 5 ÷ 2”. Using models that illustrate these contexts helps students make sense of what the “leftover” represents.
Although context can help students make sense of the remainder, it can also confuse them. This does not mean that we should ignore the context at the risk of confusion. Instead, we must embrace it and encourage students to persevere through their confusion.
Take, for example, the following problem— “Johnny has 218 pounds of hamburger meat. Each burger is 14 pounds. How many burgers can he make?” In this problem, many students will tell you that it is impossible to make 12 of a hamburger. They will figure out how many whole hamburgers Johnny can make (8) and “throw away” the leftover hamburger meat. However, In this case, we can probe students to explore deeper by asking them questions such as “Did Johnny use all the burger meat? How do you know that Johnny can’t make another whole burger?” These questions get students thinking about the remainder in a non-procedural way.
Once students recognize and explain why he can’t make another whole burger, they become more inclined to start thinking about what fraction of a burger he COULD make. When thinking about what is leftover, a major student misconception is the inclination to say that since there is 18 of a pound of burger meat left, then Johnny can make 8 18 hamburgers. Keeping the context at the forefront of this problem, allows students to think about what is left over in comparison to the serving size of the hamburger. Since 18 is 12 of the serving size of burger, the context helps students see that Johnny can actually make 8 12 hamburgers.
We can enable students to make sense of remainders by helping them differentiate between problems that require a whole number response and those that require a fractional response. Even when a problem requires a whole number response, it is important to discuss what the remainder represents instead of just casting it aside. We must continue to ask students to explain what the remainder means and how it relates to the problem. Even when we are not expecting students to work with fractional parts, helping them understand what the remainder signifies will empower them to have a deeper foundation for and understanding of future work.
Elizabeth Masalsky is a 6th, 7th, and 8th grade mathematics teacher at Battery Park City School in Manhattan. She has a post-baccalaureate in mathematics from Brandeis University and her master’s in secondary math education from Bard College. Elizabeth is a Math for America Master Teacher and continues her professional development through workshops with Math for America, Metamorphosis and Math in the City.