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January 15, 2015

# Division S.O.S.: What To Do About Remainders

My second year of teaching, my principal sent me to help score the state math exam. I spent three long days pouring over the fourth grade open response questions with a team of teachers. I’ll never forget one of the questions: “A squirrel collected 46 acorns for winter. It can hide 4 acorns in a single hole. How many holes will the squirrel need to dig to hide all of its acorns?” Student after student responded “11 remainder 2” to the question, and I had no choice but to mark it incorrect. After all, the squirrel will need twelve holes to hide all of those acorns, including the “remainder of two.” But I was crying inside as I looked at the effort many of the students had put forth, drawing diagrams with acorns tucked into holes, complex arrays, and elaborate explanations of their division strategies — just to be sunk by the remainder.

With that formative experience under my belt, I’ve always paid extra special attention to be sure my students truly understand division contexts and what to do if there’s a remainder. While division algorithms can be fairly rote, tackling remainders requires logical thinking and deeper understanding — a worthwhile use of time, in my opinion. Here are some of the math activities I’ve used with my students to help them figure out what to do about remainders.

## Introducing Remainders with Picture Books

It’s no secret that I love using picture books to teach math concepts. Even for upper elementary students, math picture books create a playful, low-stress context for exploring a concept without crowding the students’ thinking with lots of computation. Both A Remainder of One by Elinor Pinczes and Bean Thirteen by Matthew McElligott address the concept of remainders with the help of some charming insects.

Check out author McElligott’s website for lesson plans, extension ideas, and background about the book. I like to give my students dried lima beans to “play” with and ask them to act out other “unlucky bean” scenarios with a partner. They naturally discover that prime numbers always have an unlucky leftover bean.

Start a class discussion: "That pesky thirteenth bean is always left over! What other amounts have a leftover?"

## “Playing” With Remainder Strategies With a Sorting Game

In another lesson, I copy a variety of division word problems onto index cards like these sample questions. Students pull one card at a time from a bag and we have a whole-class discussion about what to do with the remainder for each scenario. I introduce our four main remainder strategies: “Just Drop It,” “Round Up,” “Sharing is Caring, and “Remainder Only.”

Once it seems like the concept is clicking, I have the students work on this sorting game in small groups.

To play this non-competitive “game,” students turn over one question card at a time and discuss as a team how they would handle the remainder. They place the question card into the correct pile under the remainder strategy headings. This is an easy activity to prep — especially if you invite students or parent volunteers to cut out the question cards — and has a big payoff in terms of mathematical talk.

## Applying Remainder Strategies With an Artsy Foldable

Once the students have experienced a wide range of division problems with varying remainder strategies, they should be ready to create their own problems to fall into each of the categories. This can be very challenging for some, and it helps to have a model or chart of the different types of questions on hand.

Sukie proudly shows off her completed "What Should I Do with the Remainder?" foldable.

I’ve found that students are often very comfortable writing one or two types of remainder problems, but get stuck on one or two. Writing their own problems is a great way to access how well the students really understand the various contexts for division.

Josie uses her planning template and a worksheet with sample questions to create her own foldable.

To make this foldable, students folded 12”x18” construction paper lengthwise and measured off four quarters. They cut flaps along the top fold of paper, and followed this foldable template to plan their writing.

The students' foldables are displayed to create a fun, interactive math bulletin board.

## Extending Remainders With the Leftovers With 100 Game

The Leftovers with 100 game is yet another slam-dunk from Marilyn Burns and the Math Solutions team. This game helps students think strategically about calculating remainders. The article provides a complete lesson plan for introducing the game with a guided exploration, and models a classroom conversation. I give my students several days to play the game with different partners to hone their strategic thinking — it also makes a great center activity for division rotations. For more about how I use games to teach math, check out my blog post about "Playing with Math."

My second year of teaching, my principal sent me to help score the state math exam. I spent three long days pouring over the fourth grade open response questions with a team of teachers. I’ll never forget one of the questions: “A squirrel collected 46 acorns for winter. It can hide 4 acorns in a single hole. How many holes will the squirrel need to dig to hide all of its acorns?” Student after student responded “11 remainder 2” to the question, and I had no choice but to mark it incorrect. After all, the squirrel will need twelve holes to hide all of those acorns, including the “remainder of two.” But I was crying inside as I looked at the effort many of the students had put forth, drawing diagrams with acorns tucked into holes, complex arrays, and elaborate explanations of their division strategies — just to be sunk by the remainder.

With that formative experience under my belt, I’ve always paid extra special attention to be sure my students truly understand division contexts and what to do if there’s a remainder. While division algorithms can be fairly rote, tackling remainders requires logical thinking and deeper understanding — a worthwhile use of time, in my opinion. Here are some of the math activities I’ve used with my students to help them figure out what to do about remainders.

## Introducing Remainders with Picture Books

It’s no secret that I love using picture books to teach math concepts. Even for upper elementary students, math picture books create a playful, low-stress context for exploring a concept without crowding the students’ thinking with lots of computation. Both A Remainder of One by Elinor Pinczes and Bean Thirteen by Matthew McElligott address the concept of remainders with the help of some charming insects.

Check out author McElligott’s website for lesson plans, extension ideas, and background about the book. I like to give my students dried lima beans to “play” with and ask them to act out other “unlucky bean” scenarios with a partner. They naturally discover that prime numbers always have an unlucky leftover bean.

Start a class discussion: "That pesky thirteenth bean is always left over! What other amounts have a leftover?"

## “Playing” With Remainder Strategies With a Sorting Game

In another lesson, I copy a variety of division word problems onto index cards like these sample questions. Students pull one card at a time from a bag and we have a whole-class discussion about what to do with the remainder for each scenario. I introduce our four main remainder strategies: “Just Drop It,” “Round Up,” “Sharing is Caring, and “Remainder Only.”

Once it seems like the concept is clicking, I have the students work on this sorting game in small groups.

To play this non-competitive “game,” students turn over one question card at a time and discuss as a team how they would handle the remainder. They place the question card into the correct pile under the remainder strategy headings. This is an easy activity to prep — especially if you invite students or parent volunteers to cut out the question cards — and has a big payoff in terms of mathematical talk.

## Applying Remainder Strategies With an Artsy Foldable

Once the students have experienced a wide range of division problems with varying remainder strategies, they should be ready to create their own problems to fall into each of the categories. This can be very challenging for some, and it helps to have a model or chart of the different types of questions on hand.

Sukie proudly shows off her completed "What Should I Do with the Remainder?" foldable.

I’ve found that students are often very comfortable writing one or two types of remainder problems, but get stuck on one or two. Writing their own problems is a great way to access how well the students really understand the various contexts for division.

Josie uses her planning template and a worksheet with sample questions to create her own foldable.

To make this foldable, students folded 12”x18” construction paper lengthwise and measured off four quarters. They cut flaps along the top fold of paper, and followed this foldable template to plan their writing.

The students' foldables are displayed to create a fun, interactive math bulletin board.

## Extending Remainders With the Leftovers With 100 Game

The Leftovers with 100 game is yet another slam-dunk from Marilyn Burns and the Math Solutions team. This game helps students think strategically about calculating remainders. The article provides a complete lesson plan for introducing the game with a guided exploration, and models a classroom conversation. I give my students several days to play the game with different partners to hone their strategic thinking — it also makes a great center activity for division rotations. For more about how I use games to teach math, check out my blog post about "Playing with Math."

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