Is It Multiplication or Division?

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Objective

Students will be able to show how they use multiplication and arrays to solve division problems in a real world context.

Big Idea

Many students see multiplication and division as separate operations and with no relationship. This lesson prepares them to have conversation about how the two are closely linked.

Mini-Lesson

15 minutes

I begin the lesson by asking my 18 students to think of a parade or a halftime show they have seen with marching bands. I ask them to share with their partner how the band members are usually organized.  Then I show this clip. It doesn't show array formation in every shot, but the students will respond to this as it is the college I attended and talk about all the time! I don't show the entire clip, either.  Usually I forward to the main show and go until the end!

Then we discuss why bands might organize into arrays for a parade or a  halftime show-at least to begin. 

At this point, I am not explaining to the students that we are dividing, just "organizing."

I have 18 students in my math class, so I asked them as a group to figure out how we might walk down the hall as a "band".  If you do this, make sure to have students share one at a time-the motivation can get loud!

My group finally settled on walking down the hall in a 3 x 8, or 8 x 3.  That is how we "marched" to lunch after our lesson:)

Active Engagement

25 minutes

The work for this is to begin with a number of band members and figure out different ways to arrange them for a parade. I have the prompt printed on labels, which you can find in the resources. Basically, the students need to work to divide the band members into as many possible configurations as possible, using equal groups of course. 

Again, the use of journals is so powerful. It offers my students an opportunity to grapple with several ideas and a chance to work on communicating their thinking.

My students are all unique learners.  I have included many clips in this narrative in order for you to see the various student levels and strategies. I am sure this is much like your class. 

This student uses his knowledge of the commutative property to work through his thinking.

As I work with this student, I am pleased to hear her questioning herself.  She tells me she doesn't think her attempt will work and why she thinks so. This is an example of Mathematical Practice 1 -Make sense of problems and persevere in solving them. I chose to stay with that and prompt her, rather than correct her first array, which equals 30, rather than 20.  

This student spends a lot of time describing his first array using the distributive property, which we just spent time on in our previous lesson path.  I was really excited to see him fix his mistake after we found it together.  My plan here is to move him a bit further into finding more ways to divide his band. 

 

I love that this student is working to find the "best" way for the band to travel down the road. His organization is around what he envisions being a real parade. Listen in and notice his use of the vocabulary.  He gets it!

In this clip, my student explains his organized approach to the problem and what he did when one idea doesn't work out. I encourage my students just to put a strike through an attempt that doesn't work as a record of thinking and as a learning tool.  We call these learning attempt, not mistakes, as they lead us to another idea. 

I have a few students that rely heavily on their knowledge of multiplication and this student does a good job using what he knows and thinking about why other factors won't work with a quick mental tests.

In this final clip, I find evidence that the students still think we are multiplying and not dividing. This is typical as many students conceptually understand multiplication better.  My next steps need to be to help them understand they are dividing even though they don't know it.  Also, this student is referring to a "quick array" because he finds arrays easy and quick to use.  They are not real quick arrays!

Wrap Up and Sharing

10 minutes

After the students work through this lesson, I ask them to bring their journals to the community center and sit in groups of 3.  They then share with their two partners their solutions.  If someone in that group also picked the same number of band members, they can compare.  Otherwise, the students are to compliment and comment using our talking moves.  

If there is time, I usually take a few good examples, or if the student is comfortable (which they usually are) I put up one with mistakes and work through it with the class. 

A home practice task might be to have the students do the same type of activity using a common number of band members that you choose.