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Making Sense of Remainders

When learning about division in fourth grade, I remember our teacher marching us through an explanation of remainders. It went something like, “If the divisor doesn’t go into the dividend evenly, you’ll have something leftover called a remainder. Just write that number next to the quotient using a little ‘r’.

Math for me back then was a blur of explanations and procedures that didn’t make much sense and didn’t have any purpose or connection to real life, especially my own. I wonder whether things would have been different if my teacher had asked us, “What do you notice and wonder about this number that’s leftover?” Questions like this might have changed how I felt about math, which I hated at the time.

In our last post, we talked about measurement division, and how we can use children’s literature as a context for helping students understand what this type of division means. In this post, we’ll focus on helping students make sense of division with remainders. What is a remainder and what do you do with it? When is it appropriate to divide the remainder into fractional parts, and when does it make more sense to round up or down to the nearest whole number?

Context is Everything

Putting math problems in a context gives meaning to numbers & operations, shapes, and data. And this includes remainders as well. Take the following problems for instance. How are they alike? How are they different?

Share 21 cookies among four children.

Share 21 balloons among four children.

Share $21.00 among four children.

Do 21 ⌯4 on the calculator.

The problems above are essentially the same when you take them out of context (21 divided by 4), But when you embed 21/4 in a context, the meaning changes. What do you do with the leftover cookie? Do you split it into fourths and give one fourth to each child? What about the balloons? Since you can’t split up the extra balloons, is the answer just 5 balloons for each kid? Dividing money gives you a different meaning for figuring out what’s leftover. And then there’s decimal numbers to figure out when you solve the problem using a calculator.

In each case, students must use their math reasoning to make sense of the remainder and figure out what to do with it. And for each context, the answer is a bit different: 5 ¼ cookies; 5 balloons; $5.25; and 5 and twenty-five hundredths.

Engage Students in Productive Struggle

In their article Conceptualizing Division with Remainders (Lamberg & Wiest, Teaching Children Mathematics, March 2012), researchers tell the story of how a third grade teacher uses different contexts and concrete materials to help her students make sense of division with remainders. For one lesson, the teacher assembled four different bags of objects, one for each group of students:

Bag 1: 100-centimeter tape measure

Bag 2: 55 pencils

Bag 3: $56.88 in play money

Bag 4: 75 square tiles

As her third graders worked to divide the materials evenly, the teacher noticed that they struggled with what to do with remainders. One group that was trying to share 55 pencils among their four group members finally decided to just hide the leftover pencils because they didn’t want to have some in their group have more pencils than others.

The teacher followed up this lesson by facilitating classroom discussions, posing similar type problems, and having her students write and solve their own division with remainder problems until they eventually became more comfortable and skilled in solving them.

Giving students messy problems to solve is beneficial because in the process, students must make decisions. How do we evenly divide these materials? How many will each person get? What do we do with what’s left over? Posing these types of problems engages students in productive struggle, which can develop strong habits of mind such as perseverance and flexible thinking.

Offer a Variety of Division Contexts

Students benefit from solving a variety of division contexts in which they must make sense of the remainder. Lamberg and Wiest (2012) suggest posing problems that require the exact remainder to be retained (How many ounces of soda do each of 4 children get from 34 ounces?), problems in which the remainder should be discarded (How many 7-inch bracelets can be made from a 38-inch chain?), and problems in which the remainder requires the answer to be rounded up to the next whole number (How many buses that hold 60 students each are needed to transport 215 third graders on a field trip?).

Launching with Literature: 17 Kings & 42 Elephants

There are several children’s books that provide interesting contexts for making sense of reminders. One of my favorites is 17 Kings and 42 Elephants by Margaret Mahy (Dial Books, 1972), pictures by Patricia MacCarthy. You can show your students the book on video being read by Math Transformations consultant Jenn Carr here, or purchase the book for your classroom library if your budget allows.

I like this book because of its playful, rhythmic language and its colorful illustrations, and after reading it you can pose this math question: How many elephants will each king get if the kings share the elephants equally? The problem is a good one for reasoning about remainders, especially since you can’t divide up an elephant!

Launching with Literature: A Remainder of One

Another personal favorite is A Remainder of One by Eleanor Pinzes, illustrations by Bonnie MacKain (share with students using the link here). It’s a fun rhyming story about 25 ants who march around in different formations. When their queen demands that they march in even lines, one of the ants is always left out. Rows of 2, 4, 6, 8, and 12 ants always leave poor Joe the ant on the sidelines until he suggests that they organize in 5 rows of 5.

As a follow up to the book, my friend Juli Traci created an activity called Remainder Riddles. You start with any number 1-25 (25 is the number of ants in the story). Then you write 7 clues and see if your partner can guess what your secret number is. Can you figure the answer to this one?

Clue 1: When you divide my number by 1, the remainder is 0.

Clue 2: When you divide my number by 2, the remainder is 0.

Clue 3: When you divide my number by 3, the remainder is 0.

Clue 4: When you divide my number by 4, the remainder is 0.

Clue 5: When you divide my number by 5, the remainder is 4.

Clue 6: When you divide my number by 6, the remainder is 0.

Clue 7: When you divide my number by 7, the remainder is 3.

Kids have fun trying to solve these remainder riddles, and they learn more about remainders and divisibility rules by writing their own using a recording sheet like the one below.

The Game of Leftovers

Leftovers is an engaging game that helps students learn about division and remainders. See the directions for the game:

As children play, they create contexts for partitive or sharing division by rolling a die to determine how many ‘plates’ or paper squares they will use to divide a certain number of tiles.

Players start the game with 15 (or 20) tiles. Player 1 rolls the die, takes that number of plates, and divides the tiles so that there’s an equal number of tiles on each plate. If there are any tiles leftover, the player gets to keep them.

For example, If the number on the die is 4, Player 1 divides the 15 tiles putting 3 tiles on each of 4 plates, keeping the 3 tiles that are leftover. The remaining tiles go back into a cup for Player 2. Player 2 now has 12 tiles as a dividend when it’s their turn to roll the die. Players keep taking turns until all the tiles have been used, and they keep track of what happens by writing equations after each turn.

When students are finished with the game, teachers can facilitate a class discussion by asking questions that require critical thinking:

“Which number (of tiles) gave you the most leftovers? Why?”

“Which number (of tiles) gave you the least leftovers? Why?”

For a challenge, see this video of two fourth graders, Drew & Morgan as they explain the directions for Leftovers with 100.

Interpreting Remainders Requires Decision Making

Learning about remainders when dividing can be challenging for students, and it requires making decisions, especially when there’s a context. Do I just leave the remainder behind because it’s not needed? Do I split the remainder into pieces and turn it into fractions? Do I round up or down to the nearest whole number?

We can help our students make sense of remainders by embedding problems in a variety of contexts, posing interesting challenges launched from literature, having them play games, and giving them concrete materials to divvy and then make sense of the remainders.

What ideas do you have for helping students learn about remainders?

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