Modeling Division

It’s impossible to overstate the importance of the third grade math curriculum. Students get their first real exposure to multiplication, division, and fractions—all critical math ideas. We must approach instruction for these important topics in a way that helps students develop a strong understanding and the ability to visualize the meaning behind the concepts. Without a strong foundation in these concepts, students will struggle in later years as more complicated skills are layered on. It’s nearly impossible to master division of fractions or decimals, for example, if you don’t really understand what division means. It’s a snowball effect that results in students falling farther and farther behind.

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Two instructional strategies for building understanding are adding context and incorporating manipulatives and drawings or pictures.

Add context

Literature is a very effective way to introduce abstract math concepts in context. The Doorbell Rang by Pat Hutchins is often used to introduce division in a relatable and engaging way—sharing cookies.

In the story, Sam and Victoria are ready to share the hot, delicious cookies their mother has just taken from the oven, and they have quickly calculated how many cookies each will get. When the doorbell rings, bringing two of their friends from next door, they realize the calculation has changed. Each time the doorbell rings, more friends arrive. The repetitive nature of the story allows children to see the division process unfold multiple times, reinforcing their understanding of the process.

My recommendation is always to read and enjoy the book first before using it for math activities. So you might read and discuss the book on one day and then revisit it another day for a math lesson. If you are teaching math using small group lessons, consider reading and discussing the book whole-class before meeting in small groups for the math lesson.

Use objects and pictures

So let’s talk about a couple of lessons we could teach using this book. While they can be done whole-group, moving them to your small group table enables you to have better control over the manipulatives and allows you to better assess student understanding through the conversations they engage in. Whether whole class or small group instruction, be sure to pair students up to encourage mathematical conversations.

modeling the story

To make the first lesson as concrete as possible, I suggest using paper plates and actual cookies to model the action from the book. You might use little mini cookies, like these Chips Ahoy cookies. We’re also going to throw a little problem solving into the lesson, which just means that I’m not going to tell the students what to do step-by-step.

Have the students turn and discuss with their partner what they recall from the story. After a minute or two, ask for a volunteer to summarize the story. Give each pair of students six small paper plates and twelve cookies. The Chips Ahoy containers hold about 14 cookies, so that’s just perfect! Remember, each pair needs six plates and twelve cookies, not each student.

Read the first page of the book. Notice that it doesn’t say how many cookies Ma made, so after reading the page, ask students if they remember how many cookies Ma made. If none of the students recall, remind them that she made 12 cookies. Probe the students’ understanding by asking how many children are sharing the cookies (2) and how many cookies they are sharing (12). Ask students to use their paper plates and cookies to show how many cookies each of the children would get. This is where the problem solving comes in, because my directions are rather vague. Once students have modeled the problem, ask them to describe what they did. Begin to introduce the language of division—How many groups? How many in each group? How many total?

modeling division

There’s no need to reread the book in its entirety, just skip to when the doorbell rings the first time and Tom and Hannah from next door arrive. Ask students how many children are sharing the cookies now and then have them use their plates and cookies to show the sharing. Discuss their findings—How many groups? How many in each group? How many total? How is this the same as the last problem? (12 cookies, equal groups) How is it different? (different number of children sharing, different number of cookies for each child)

modeling division

Repeat the process by reading the page in the book when two more children arrive and have students model what it would look like to share the cookies among six friends. Don’t skip the conversation following the hands-on work. When you get to twelve children, the students won’t have enough plates. Provide each pair of students with a white board and marker, and ask if they could draw a model representing sharing the twelve cookies among twelve children. Again, notice that I’m not giving specific instructions for how to draw the model. There are two reasons for this. First, if I am instructing them step-by-step how to draw the model, it doesn’t give me any information about their understanding. Second, it’s important for students to understand that there’s not one right way to draw a model. Give students enough time so that most can finish their models. Some students might have opted to draw detailed cookies, and they might not finish. That’s a good discussion point. As pairs of students show their models, generate discussions about the drawings. How are they organized? Does that make it easier to understand? What are similarities among the models? Do our cookies have to look like cookies (it’s a math drawing, not an art drawing…)?

Notice that in this lesson we didn’t write any equations—we just modeled the story using the manipulatives and transitioned to drawing a picture.

Connecting models to equations

With the next lesson, students will draw models to represent story situations involving sharing and then write equations to match.

Once again, pair students up to foster mathematical dialogue. Provide each pair with about 25 counters and a student whiteboard and marker. Present students with this story problem:

Six friends were sharing 18 mini candy bars. How many candy bars did each friend get?

Ask students to work together to model the problem using the counters. Watch how they do this, but don’t tell them what to do. And, as hard as it is, don’t jump in if you see them doing something wrong. Let them work through it. You will likely see various processes. Will they count out 18 counters from their pile to start? Some might do that and then “deal out” those 18 counters into six groups, much like you deal out cards. Others might use a trial and error approach using the whole pile of counters, for example, they might put two counters in each of six groups, count and determine it’s only 12, add one more to each group, and recount to find that the total is now 18. Still others might just know it’s 3, maybe because they know the relationship to 6 x 3 = 18. Watching the process is a big part of your formative assessment, because it speaks to each student’s level of understanding. After pairs complete their work, select pairs of students to share the process they used.

Next, have the students draw a model of their work on the whiteboard, and ask them to write an equation to match the drawing/problem. You might see a drawing such as the one below. Students might start by drawing all 18 of the mini candy bars. As they “deal out” the candy bars into the three groups, they cross them off the starting pile.

modeling division

But they also might start with the three circles and draw the candy bars in the circles while they count. When discussing different strategies, you might ask students about the efficiency of each strategy.

When you’re looking at the equation, don’t be surprised if some students write 6 x 3 = 18, because isn’t that what the model looks like? What a great opportunity to discuss the relationship between 6 x 3 = 18 and 18 ÷ 3 = 6.

Moving forward, you might differentiate by grouping students based on your observations of their level of understanding for this skill. Some students might need to keep using the counters alongside the drawing, while others might be ready to only use drawings. Students who need a challenge could be given the opportunity to write their own story and solve it. Remember, it’s quality, not quantity.

Here are some other stories you can use for your lesson. Don’t skip the conversation after each problem!

  • Sue, Rob, and Elissa bought a dozen cupcakes at the bake sale for $15 and shared them equally. How many cupcakes did they each get?
  • There were 16 markers in a box. Four friends shared the markers equally. How many markers did each friend get?
  • Ariel, Kevin, and Chris shared 15 marbles equally. How many marbles did each of the children get?
  • Kate got a package of 20 Pokémon cards for her birthday. She shared the cards equally with her 3 best friends. How many cards did each girl get?

Remember, adding context and using objects and pictures take the mystery out of division and build understanding!

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