Division Bar Models

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SWBAT represent and solve division word problems by drawing a visual model.

Big Idea

There are two visual models that are very useful for division that will help us model every kind of division problem.

Think About It

7 minutes

Students work on the Think About It problem in pairs.  The models provided represent a groups unknown division problem and a group size unknown division problem.  The goal of this Think About It problem is to have students recognize and articulate the differences between the models. 

After 3 minutes of partner time, students have 1 minute of independent work time to tweak their written responses. I'll then ask students to share what is similar about the models, and what they've identified as being different about the models.  

Intro to New Material

12 minutes

To start the Intro to New Material section, I have students fill in a simple definition for division:  Division means sharing or splitting a total amount equally.

The focus of this lesson is on drawing the appropriate models.  Students will use these models throughout the year, with division problems.  In this lesson, I've chosen to use small numbers that result in expressions students can likely evaluate using mental math.  I don't want the arithmetic to stand in the way of students mastering the models.

I guide students to draw the correct model for the first problem in this section, which is a group size unknown problem.  Once we've read and annotated the problem, I ask students to flip back to the Think About It problem (I leave the current problem on the doc cam for students to see).  I ask students to vote with my fingers to let me know if the problem we're working on is more like the Goldfish problem (#1) or the apples problem (#2).  I then call on a student to share why our current problem is like problem #1.  I want students to identify that we don't know the size of the groups in this problem.

I'll then ask students what we should draw in our model, and I ask what each piece represents.  Once we have an accurate model, we create a division equation (using '?' for the quotient) and a corresponding multiplication equation (using the '?' for the unknown factor).  The final step is to answer the question in a full sentence.

We go through the same process for the second problem, which is a groups unknown problem.

The 3rd problem in this section asks us to write a word problem that could be represented by the provided model.  My students really enjoy writing story problems, and I encourage their creativity.  My two rules about the problems students write:  it has to make sense (please don’t split your little sister into 5 pieces) and it has to be respectful and school appropriate  (if anything in your brain says ‘hmmm, is this really okay to share with Ms. Seeger?, then the answer is probably no.’)  I model writing the story problem -  "I have $40 to spend on books.  If each book costs $4, how many books can I buy?"



Partner Practice

15 minutes

Students work on pairs on the Partner Practice problem set.  As students work, I circulate around the room and check in with each group.  I am looking for:

  • Are students drawing and labeling the bar model that represents the problem?
  • Are students writing the correct number sentence to solve the problem?
  • Are students answering in a complete sentence?
  • Are students including units in their answer?
  • Do students’ word problems align to the equation given?


I'm asking:

  • How did you know to draw the model like this?  What does this piece represent?
  • Is this a group size unknown or a groups unknown problem?  How do you know?
  • What are you looking for in this problem?  How is that represented in your model?  In your number sentence?
  • How does the model show division?
  • How do you know that the word problem you wrote can be represented and solved using the equation provided?


After 10 minutes of partner work time, I ask students to independently complete the Check for Understanding problem.  I ask for a student to present his/her model for the problem to the class.  I'll ask the student to explain each piece of the model.  

Before moving on to Independent Practice, I'll have 2-3 students share out one of the word problems they created with their partners. 

Independent Practice

15 minutes

Students work on the Independent Practice problem set.

Problems 7 and 8 give me good insight into how well students have internalized the differences between the two problem types.  I intentionally used the same numbers in these two problems so that students have to wrestle with the difference.

The final problem in this set asks students to write their own word problem, and then represent it with a bar model.  Students will sometimes ask me if they can use larger numbers for this problem.  I let them know that the can, so long as they can prove to me that the divisor they've picked goes evenly into the dividend, by showing either the long division or multiplication on the page.

Closing and Exit Ticket

8 minutes

After independent work time, I show students a model for Problem 6.  I intentionally put up a model that incorrectly represents this problem as a group size unknown problem.  I ask students to provide me feedback on my work.

Students then work on the Exit Ticket to close the lesson.

Responses to the first problem could look like the sample I've included. A sample of student work is also included.