A Remainder of One

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SWBAT model division with remainders with an array.

Big Idea

Students gain experience with what a remainder is and model division situations with arrays.

Concept Development

50 minutes

Note: I left the following lesson for a guest teacher.  She reported that the lesson went VERY well, except that they ran out of time to complete the book activity. This is a fun lesson which truly allows students to move between concrete and representational division strategies.

Concrete math strategy: Each math concept/skill is first modeled with concrete materials (e.g. chips, unifix cubes, base ten blocks, beans and bean sticks, pattern blocks). Students are provided many opportunities to practice and demonstrate mastery using concrete materials.

Representational math strategy: The math concept/skill is next modeled at the representational (semi-concrete) level, which involves drawing pictures that represent the concrete objects previously used (e.g. tallies, dots, circles, stamps that imprint pictures for counting). Students are provided many opportunities to practice and demonstrate mastery by drawing solutions.

This is important in order for students to be successful with long division and making sense of remainders.  The Common Core State Standards require students to know more than just "how to get an answer" and instead support students' abilities to access concepts from a number of perspectives.  Students need to be able to see math as more than a set of mnemonics or discrete procedures. 


I start this lesson by asking students "What is division?"  Then I write on the board  48 ÷ 3. I ask students what that means? I list their responses on the board.

I then write on the board: Remainder of 1.

Give each student 25 one-centimeter cubes and a sheet of graph paper.  I then begin reading the story Remainder of One.

Cover art

This story tells of bugs that are getting in different equal sized lines.  There are 25 bugs and at various points in the book the bugs need to get in 2 equal lines, 3 equal lines, 4 equal lines etc.  The bugs that can not get into the lines because they would not be equal are the remaining bugs, or the remainder. 

As I read the story, I stop often so students can build the arrays in the book. Students create the rows of bugs and build the rows using centimeter cubes to represent the bugs.  Students take turns to share the array under the document camera. 

I guide a discussion about what the single centimeter cube represents (Lone Joe, the left over bug). I repeat this with each new line configuration in the story and have students model the array.

After the story, I explain to students that they will create books that show the arrays we modeled with the cubes while reading the story.  (See the interlocking book directions in the resource section for how to make a paper book.) Using the graph paper, students draw the arrays from the story, and then cut out the array.  For example, the first array in the story is 25 ÷ 2 = 12 r1. Students draw 2 rows with 12 columns (the bugs) with one left over to show 25 ÷ 2 = 12 r1.  Students label the array with the number sentence. See this example array from a student's division book.

I lead a discussion about the vocabulary term for each number in the number sentence. Students should record the terms on the last page in the book and write, “dividend ÷ divisor = quotient and remainder” on the current page.

For the rest of the lesson, students work at their own pace to model each new bug configuration in the story.  If students forget, they may use the book at their leisure. I  move around the classroom, helping and assisting students as needed.

Students continue with this procedure for the entire book.  Each time the bugs lined up in a new way in the book, students draw an array to represent the rows of bugs and any left over, or the remainder.

If students finish with time left, they decorate their book with crayons or colored pencils.