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A Remainder of One: A Math-Manipulative Lesson


  • Mathematics


  • K-2
  • 3-5

Brief Description

Use A Remainder of One (by Elinor Princzes, who also wrote the popular children's book One Hundred Hungry Ants) to develop understandings of even numbers, division, and remainders.


Students will

  • use math manipulatives to recreate "bug squadrons" formed in the book A Remainder of One.
  • recreate marching groups and attempt to create even rows as "Joe" did in A Remainder of One.
  • recognize when they have a remainder after grouping their manipulatives.
  • record all group answers/strategies in their math journals.
  • discuss what they learned; share strategies and results with others.


remainder, even, division

Materials Needed

  • A Remainder of One by Elinor Princzes
  • plastic bugs and/or other math manipulatives (for example, BINGO chips, pennies, macaroni)
  • math journals (or a sheet of paper on which students can record their observations as they work with the manipulatives)

The Lesson

Read aloud to students the book A Remainder of One by Elinor Princzes.

From Publishers Weekly
As they did in One Hundred Hungry Ants, Pinczes and illustrator Bonnie MacKain apply numerical division to a practical problem -- and explain it in an entertaining, visually emphatic way. Keeping to the insect theme, Pinczes introduces the "25th Army Corps," a regiment of 25 beetles on parade. Their blue bug queen "likes things tidy," and when the bugs march two by two, she notices that one bug brings up the rear. The "unfortunate Joe" has to stand aside rather than be a "remainder"; on the days that follow, Joe tries dividing the squadron into symmetrical rows of three, then four and, finally, five, when he is at last accommodated. Rather than endorse conformity, this rhyming tale focuses on Joe's search for a solution. And lest squadron-like precision trouble readers, each big-eyed "bug-soldier" has a unique patterned shell. MacKain even ensures that the same beetle characters -- one with a pointy nose, two wearing glasses, etc. -- appear in every spread, allowing readers to play spot-the-bug. Rendered in dusty blues and pasture-green with warm yellow, red, and pink accents, her linocut-style art vibrates with energy. Ages 4-8.
Copyright 1995 Reed Business Information, Inc.

As you read, pause to allow students to comment. Ask students for alternative solutions that they believe may be possible for "Joe's" problem.

When you have finished reading aloud, ask students to reiterate what occurred within the story as well as how "Joe" solved the problem with which he was faced.

Developmental Activities
Give each student 25 plastic bugs. (Macaroni or another manipulative can be used to represent the bugs.) Have students work individually as you reread A Remainder of One. As the story is reread, have students use the manipulatives to model the formation of the bug squadrons. Pause to be sure all students are participating and correctly modeling formations. Have students recreate each subsequent bug squadron from the story as you read aloud.

After completing the re-creation activity, collect all "bugs" from students.

Next, arrange students into groups of four. If any students remain, place them into the group with which you think the student will work best. Have students choose another math manipulative. (BINGO chips work well. So do pennies, goldfish pretzels, and many other things.) Students in the group decide on a number between 20 and 50 and select that many of the manipulative. For example, one group may choose to work with 36 BINGO chips; another might work with 44 pennies.)

Instruct students in each group to find ways to use their chosen manipulative to create marching groups just as "Joe" had to do in A Remainder of One. Students must record all answers/strategies in their math journals or on a special work sheet created for use with this lesson. If students have any "remainder Joe's" as they group their manipulatives, they must state so in their journal recordings.

As students are doing this problem-solving activity, walk around the room observing to make sure that all students are on task and participating. Spend time with each group. Ask for students' reasoning as they work with the manipulatives, and encourage new strategies.

Set aside time for each group to share up to five strategies they found when working with their manipulatives. (Each student in the group might share at least one.) Students should specify how many manipulatives they chose to work with, how they attempted to group them, and what the result was. Do members of other groups have alternative strategies that might have been tried?


Assess students based on

  • their ability to create even marching groups.
  • their ability to recognize a remainder.
  • their ability to use and share multiple strategies for grouping manipulatives.
  • their participation with other group members.
  • the completeness of their math journals.

Submitted By

Alicia Bittner, University of Pittsburgh at Johnstown (Johnstown, Pennsylvania)

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