Using Multiplication to Solve Division Stories

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Objective

Students will be able to explain their multiplicative thinking in the solving of a division problem.

Big Idea

Students typically see multiplication and division as two separate operations. This lesson is designed to help them understand their relationship.

Mini-Lesson

10 minutes

I begin today's lesson with several word problems, asking the students to verbally solve with their partners.  I have decided to not have them draw or write these solutions out as a way to "make" them use the facts they already know to solve for an unknown.

Sample problems may be:

Jennifer has 4 boxes of pencils. Each box has the same amount of pencils.  She has 20 pencils in all. How many pencils are in each box?

John has 15 gumballs.  He wants to put them into bags equally.  He has 5 bags.  How many canhe put into each bag?

Following several problems like these, I give a statement and pose a question.

Boys and Girls, you have been solving division problems. What are you doing in your minds to find your answers?  Are you using something you know?

Keep talking about this until you are able to have the class understand they are using what they know about multiplication to solve division. I would not go further than that in this conversation. There will be more time to delve into this concept as the unit progresses.

Active Engagement

25 minutes

I like to have prompts printed on labels when they are the main part of the lesson's activity, as it saves time from copying on the board.  After I pass out the labels, I ask the students to work on their own at first to show their solution to the question. I tell them all of the information is there and the math facts are complete already. Their task is to show/explain why it is true. I also have cubes available for them to model their thinking before writing. 

As students work, I watch to see if they are using equal groups naturally and discussing multiplication as they draw or write their thinking.  My goal is to just have the students work with factors in a division problem and practice communication skills around this concept of multiplication and division being related. They need to have the vocabulary in order for us to dig into their thinking and into deeper concepts. 

This student starts out telling me there are all sorts of ways to solve, but really wasn't giving me any detailed thinking.  With a few small prompts, she can focus her thinking and use some vocabulary to help explain her work.

When I speak with this student, I am really impressed with how he organizes his work, as we have done so much with arrays in the past.  When I ask him why he uses an array, he was at a loss for words.  Again, sometimes the routine students get into are so habitual, they can't explain why they are the best strategy. Common Core expects this metacognition to occur and I believe it is necessary for deep understanding.  Listen to the prompts I use with him. 

Wrap Up and Sharing

10 minutes

To wrap up this lesson, I suggest you pair students who used different strategies to share with each other. My students are working on agreeing and disagreeing with the math, not the person.  You may even have the students rotate to several journals.  Then pull everyone together and share a few "chosen" responses on the board. These would be the ones that most point toward the lesson of the day, which is "multiplication can be used to solve division".