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A Squadron of Bugs: Introducing Division with Remainders with a Book

  • Lesson
Number and Operations
Lisa Willman
Hillman, MN

This hands-on lesson uses the book, A Remainder of One, by Elinor J. Pinczes, to introduce division with remainder. Students will enjoy the story context as they explore different remainder situations and model division with arrays.

Before beginning the lesson, collect the cards with the numbers 1 - 5 from the deck of cards (ace will be used as 1). Also, make enough copies of A Squadron of Bugs Activity Sheet for each student.

pdficon A Squadron of Bugs Activity Sheet SquadronOfBugs(2) 

Read aloud the book, A Remainder of One. Read for the enjoyment, language, and overall concept of the book. You will read the book again later in the lesson. If you do not have access to the book, present the plot to the class in your own words: Soldier Joe is a bug who wants to march in a parade with the 24 other bugs in his squadron. But the rows in which the bugs form always seem to leave him as the "odd bug out." Joe and the rest of his squadron form two, three, and four rows before finally finding that five rows will allow Joe to march in an even arrangement of rows.

Tell students that each time Joe thinks of a new way to arrange the troop, they will try the arrangement. Give each student 25 counters. Give students 24 of the same color and one in a different color to represent soldier Joe. If you don't have enough counters, have students work in pairs or small groups. Allow time for students to explore with the counters before starting the activity.

Reread the book, stopping after the troop arranges themselves into two lines. If you do not have access to the book, tell students that the bugs in a parade first arrange themselves into two lines. Ask students to divide their 25 counters into two rows, with an equal number of counters in each row. Discuss what they discovered. Steer the discussion to the fact that there was an odd number of counters and that it couldn't be divided evenly. Remind students that odd numbers are those that have one of these digits in the ones place: 1, 3, 5, 7, and 9, so 25 is an odd number.

Many students will not know what to do with the extra counter. Some may want to add it to one of the lines and some may want to just remove entirely it from the equation. Explain that the remainder is important and needs to be accounted for in the equation. Write the equation 25 ÷ 2 = 12 R 1 on the board. Explain that in this case the remainder is Joe, the leftover of two even lines. Use different colors of chalk or markers to circle the dividend (25), divisor (2), quotient (12), and remainder (1) and to label what they represent. Use the same color for the remainder that is used for the counter that represents Joe. As you are circling each part of the division equation explain that the dividend is the number being divided, the divisor is the number that you divide by, the quotient is the answer after you divide one number by another, and the remainder is the amount left over after a division.  

Ask, "If you were trying to figure out  and did not have counters, how could knowing your multiplication facts help you?" Ask students to think of the facts that include 2. Ask, "Is there a number you can multiply by 2 to get 25?" [No.]

Continue reading the book. Each time the squadron rearranges themselves, have the students use their counters to explore what is happening in the book. The squadron arranges itself into three lines and then into four lines. Ask students to write the equations in their notebooks. Make sure to introduce various ways of writing a division problem, as shown below.

  • 25 ÷ 3 = 8 R 1
  • \frac{{25}}{3} = 8 R 1
  • ASquadronOfBugs IMAGE DivisionEquation 

If you were trying to figure out  and did not have counters, how could knowing your multiplication facts help you? Ask students to think of the facts that include 3. Ask, "Is there a number you can multiply by 3 to get 25?" [No.] "How close can you get?" [24.] 

When you have finished rereading the book, ask students to take 1 - 5 additional counters. Explain that this is their new squadron. From your deck of playing cards, randomly give each student one of these cards. Distribute the Squadron of Bugs activity sheet.

Have students share their division illustrations and equations. Reinforce the lesson concepts. Division is dividing equally among groups. If there is not enough to go into each group, then the answer has a remainder.


  • Moyer, Patricia Seray. A Remainder of One: Exploring Partitive Division. Teaching Children Mathematics (April 2000): 517-521.
  • Pinczes, Elinor J. A Remainder of One. New York: Houghton Mifflin Co., 1995.

Assessment Options  

  1. As students work on the pictorial representation of the new equation, observe if students are:
    • dividing groups equally in a systematic manner or just haphazardly moving counters.
    • correctly dividing counters by creating even groups, with remainders left out of the groups and by correctly writing division equations.
  2. Give each student an index card with a division expression written on it. For example, 27 ÷ 5. Ask students to draw a picture of what the expression represents and to evaluate the expression. Have counters available for students to use as they work on the problem. Also have students write out the solution 5 R 2.
  3. Give students a starting number (dividend) and have them come up with a division problem that will not have a remainder, and a division problem that will have a remainder.


  1. To have students practice and expand their skills with division with remainders, provide students with straight computation problems. Since the lesson focuses on dividends from 24 - 30, provide students with problems that involve 3 or 4 digit dividends and single digit divisors.
  2. Provide students with real-world applications that involve division with remainders. For example, choosing starting players for teams in a tournament will have remainders whenever the number of players is not a multiple of the number of starting players on a team. When a group makes sandwiches for a picnic and decides to put the same number of slices of cheese in each sandwich, there may be a remainder. Discuss what the remainder depends on: the total number of sandwiches (the dividend) and the total number of slices of cheese (the divisor). 


Questions for Students  

1. Why didn't 25 divide evenly into two equals groups?

[Twenty-five is an odd number and when you divide an odd number into two, there will be one left over. In this case, one bug did not have a partner.]

2. What is a remainder?

[A remainder is a number that is left over after the other numbers have been divided equally into groups.]

3. How many different remainders can a number have?

[The remainder always has to be less than the divisor. For example, if you were dividing a bug squadron into five rows, you could have 1, 2, 3, or 4 remainders. If you have five or more remainders, you have enough to add a new bug to each row.]

4. What is the divisor of 24 ÷ 3?


5. What is the dividend of 24 ÷ 3?


6. What is the quotient of 24 ÷ 3?


7. What is the remainder of 24 ÷ 3?


8. If you were trying to figure out 25 ÷ 5 and you didn't have counters, how could knowing your multiplication facts help you solve the problem?

[You could think, "What number times five will give me 25?" In other words, 5 × ___ = 25.]

Teacher Reflection  

  • How were concepts of division presented too quantitatively or too abstractly?
  • How did the lesson address the needs of tactile and visual learners in your classroom?
  • How could the lesson be differentiated for the advanced student in the classroom?
  • How was the integration of literature into math successful?
Number and Operations

The Quotient Cafe

3-5, 6-8
Use this applet to illustrate division and remainders by the division of food to aliens, dinosaurs, penguins and more.

Learning Objectives

Students will:

  • Explore division with remainders.
  • Write division equations.
  • Create a pictorial representation of a division equation.

NCTM Standards and Expectations

  • Understand various meanings of multiplication and division.
  • Understand the effects of multiplying and dividing whole numbers.
  • Identify and use relationships between operations, such as division as the inverse of multiplication, to solve problems.

Common Core State Standards – Mathematics

Grade 3, Algebraic Thinking

  • CCSS.Math.Content.3.OA.A.2
    Interpret whole-number quotients of whole numbers, e.g., interpret 56 ÷ 8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each. For example, describe a context in which a number of shares or a number of groups can be expressed as 56 ÷ 8.

Grade 3, Algebraic Thinking

  • CCSS.Math.Content.3.OA.A.3
    Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.

Grade 3, Algebraic Thinking

  • CCSS.Math.Content.3.OA.C.7
    Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 x 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers.

Grade 3, Algebraic Thinking

  • CCSS.Math.Content.3.OA.D.8
    Solve two-step word problems using the four operations. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.

Grade 4, Algebraic Thinking

  • CCSS.Math.Content.4.OA.A.2
    Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison.

Grade 4, Algebraic Thinking

  • CCSS.Math.Content.4.OA.A.3
    Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.

Grade 3, Algebraic Thinking

  • CCSS.Math.Content.3.OA.B.6
    Understand division as an unknown-factor problem. For example, find 32 ÷ 8 by finding the number that makes 32 when multiplied by 8.

Common Core State Standards – Practice

  • CCSS.Math.Practice.MP1
    Make sense of problems and persevere in solving them.