Whole Numbers Divided by Fractions Using Models

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SWBAT to determine the quotient when dividing a whole number by a fraction where the quotient will be a whole number using a visual model.

Big Idea

We can use visual models to help us represent and solve problems involving dividing a whole number by a fraction.

Think About It

7 minutes

Students work in pairs on the Think About It problem.  After 2 - 3 minutes of partner time, I have pairs share out their strategies for solving this problem.  I'll ask for students to share until we exhaust all of the strategies that students used.  

My students learned how to divide whole numbers by unit fractions in 5th grade (5.NF.7.b), so this lesson is not entirely new for them.  Students are able to complete the Think About It problem, using a variety of strategies.  One of the purposes of this problem is to give students the chance to talk about their problem solving strategies with one another. 

In this lesson, students will use visual models to divide whole numbers by fractions (and all quotients will be whole numbers).   

Intro to New Material

15 minutes

In the Intro to New Material section, I show students how to draw three different visual models:  a number line, a circle model, and a rectangular model.  Throughout the lesson, students have access to these Steps for Drawing Models.  I project the steps on the document camera as students work in partners and independently later in the lesson.  

To start Problem A, I ask students what they know from the problem.  I also ask them what we need to find out.  Throughout the school year, students will need to be able to answer these questions when making sense of problems, and I want them in the habit of doing this early on.  For this model, we draw 6 circles to represent the 6 wholes in the problem.  We then split each of the circles into thirds, because we'll need to find out how many groups of 2/3 will fit into the 6 wholes.  I expect students to circle groups of two 1/3 pieces.  To help with organization, students will number the circled groups as they go.  In this problem, they will circle 9 groups.  

Problem B is very similar to Problem A.  For this model, I have students use rectangles to represent the wholes, rather than circles.  I ask questions of students, to have them help me construct my model.  For the written response, an exemplar answer can be: "The quotient is larger than the dividend because the divisor (3/4) is larger than the dividend (3).  This means that you can make at least the number of groups of the divisor from the dividend, with more left over."

For Problem C, I introduce a number line model.  I want students to take away the idea that any of the representations are correct and will help them get to the answer.  The models have different ways of representing the wholes, but the concept behind showing division with each is the same. 

circle and number line picture shows what the models can look like.  


Partner Practice

20 minutes

Students work on the Partner Practice problem set in pairs.  If it seems like students might need more practice with the models, this section can be presented as guided practice with more scaffolding from the teacher.  

As students work in pairs, I circulate around the room.  Because this lesson happens earlier in the school year, and students are still working on the routine of working together, I do a lot of listening during this lesson.  I want to be sure that students are making good choices about how to use their work time.  I'm also looking for:

  • Are students drawing a visual representation for each problem that accurately represents the number sentence?
  • Are students trying a variety of models in this problem set?
  • Is the student work neat and organized?
  • Are students writing a number sentence to solve the problem?
  • Are students finding the correct quotient? 


I'm asking students:

  • How did you know to draw the visual like this?
  • What does your quotient mean?  Why is it greater than 1?
  • Can you show this using the (other) visual?
  • How did you know what fraction to split the wholes into?


Independent Practice

15 minutes

Students work on the Independent Practice problem set.  

The problems in this section do not ask students to use a specific visual model.  As I circulate, if I notice a student is using all of one type of visual representation, I'll ask her to try a different model for the next problem.  I want my students to master each of the model types.  When they are solving problems independently, they can access whichever model they prefer.  It's also important, though, that they can interpret different visual representations.  I don't want them to come across a circle model, for example, and not be sure what to make of it.  

Problem 6 does not ask students to use any model.  In this lesson, I do expect that students will represent the problem with a visual model.  Eventually (later in this unit), they can move to using number sentences as the primary representation of the problem, but not yet!


Closing and Exit Ticket

8 minutes

After independent work time, I bring the class back together for a discussion about the work.  First, I ask students to turn to their partners and share out the real world scenarios they wrote for Problem 8.  I'll ask the class to raise a hand if their partner had a great scenario, and then will have 2-3 students share their problems out with the class.  Asking partners to volunteer their peers' strong work helps to build community.

I'll then ask students to think about how what they wrote for Problem 10 can help them determine the reasonableness of their quotients.  We did something similar in the previous unit, when working with whole number division.  I want students to have deep number sense skills that will help them check the reasonableness of their solutions. Having a sense of whether the quotient should be bigger or smaller than the dividend is an important concept to master.

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