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.
Download the complete Remainders Sorting Game as a PDF
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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This activity will teach students how to deal with remainders in real life by either rounding, splitting them evenly, or simply ignoring them.
Invitation to Learn
- 12 manipulatives
- Math journals
Teaching with the Brain in Mind, by Eric Jensen; ISBN 1-4166-0030-2
Background for Teachers
It often takes a leap of understanding for students to apply the procedural algorithm of division with remainders to real-world situations where remainders are encountered. A child who can easily calculate 40 divided by 6 = 6R4 will too often state 6R4 as the answer to the number of cars necessary to transport 40 children to a baseball game if 6 children can fit in each car. The activities in this section will first review the concept of division as proportional reasoning involving equal shares and then they will lead children to discover the three usual ways of dealing with remainders in real life: they are either used to round up to the next whole number, they are dropped and discarded, or they are split evenly among the participants.
Before beginning this lesson, students must be able to express remainders as fractions and decimals.
Invitation to Learn
Distribute a set of 12 counting objects to each child. (They may be cubes, blocks, chips, etc.) Tell the students that they each have a set of 12 objects. Then ask the students to divide their sets into four fair shares. Guide them to create four sets with three objects in each set. Discuss the term fair shares if it is not part of your usual vocabulary. It means every set has the same number of objects, the dividend is divided equally by the divisor. Then write the following equation on the board and ask the children to copy it into their math journals.
Ask what is different from the usual way of writing a division problem. They should notice that the number 1 is written above the divisor. What is significant about the number 1? Take several ideas from students. Lead them to discover that the 1 is implied in every division problem, because the quotient is how many items are in 1 fair share. Then have the children write the following two statements in their journals:
- 1 fair share contains 3 objects.
- 4 fair shares contain 12 objects.
Explore with the children the relationships between the numbers as they discover the proportions: 1/4 = 3/12; 1/3=4/12 and 1X12=3x4. Write all the true statements on the board and have the children list them in their journals.
Next, copy these three equations on the board:
Ask what is the question in the first equation. (The students are asked to form 4 fair shares from 12 objects.) What is the question in the second equation? (The students are asked to find how many fair shares of 3 each can be made with 12 objects.) What is the question in the third equation? (Students are asked to find how many objects must be used to make 4 fair shares containing 3 objects each. This case involves multiplication rather than division.) Have the children build each situation with their manipulatives, knowing that even though the problem looks the same each time, in the first instance the question is the number of fair shares in each set. In the second instance, the question is the number of sets, and in the third instance, the question is the total number of objects.
Introducing division as proportional reasoning prepares children for equivalency in fractions; proportionality in ratios, proportions and percents; and provides a more concrete understanding of division as the process of creating fair shares.
- Divide the class into three groups. Each script has enough parts for eight actors. Additional class members could be used to direct, create, and manage props, etc. If you have a really large class, you may wish to double one or more of the scripts and perform the same play(s) twice. The plays also work well in a readers theater format, shortening preparation time.
- Practice the plays: Round-up!, Sharing is Very Important! and You Just Drop It!
- Present each play to the whole class. During and after the presentations, the class completes the graphic organizer Playing with Remainders.
- As a class, discuss the different applications of remainders in the three plays using the graphic organizer to illustrate the different ways each play uses remainders in real life. You may wish to have the children trim the edges of the graphic organizer and glue it into their journals as a reference.
- As a whole class or in partners complete the worksheet Remainder Stories.
- Note: Another option for these plays is to use them as center activities, with each child participating in each play, using no audience but discussing each play separately as a whole class. This option may increase student engagement.
- After reading or acting out these plays, children could write their own stories or plays where the characters must interpret remainders correctly in real-life situations.
- Children new to the United States could be encouraged to set new plays in their homeland countries with names, food, and problem-solving situations common to their life experiences.
- Assign students to create two to five word problems at home using members of their families and either real or made up situations that require the correct use of remainders.
- Which use of a remainder is most common? Give students a few days to collect data at home about which scenario is most commondropping, rounding or sharing. They might be allowed situations on TV in addition to real-life occurrences. After collecting data, a bar graph could be constructed comparing the three types of remainders frequency.
- Formative assessment: Check for accuracy as students complete their graphic organizers, participate in the discussion following the presentation of the plays, and solve the word problem worksheet.
- Final assessment: In a word-problem test, students should be able to supply the correct answer and explain in words what they did with their remainders (dropped, shared, or rounded up).
Wiebe, A., (1989). Proportionality: A major concept in mathematicspart II: Remainderswhat are we to do with them? Aims newsletter, volume iii, No. 7, 6-7.
Dr. Wiebe explores the gap between abstract answers to division problems with remainders and real-life situations where students encounter remainders. Expressing remainders as fractions and decimals are explored and applied, and the choices of rounding up, dropping, and sharing remainders are introduced.
Martinez, J.G.R., (2000). Look smart. Early years, January 2000. Retrieved January 12, 2007 from http://www.findarticles.com.
Engaging children in math story problems is easier when the stories have real plots and good endings. By engaging students in the plot, they become interested in solving the math situations, rather than routinely solving a page of word problems. Additionally, the enthusiasm generated motivates students to write their own stories, developing new problems within the story context, and acting out the story line.