Decomposing Arrays, Day 1

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Objective

Students will be able to deconstruct an array into two smaller arrays in order to find the total area.

Big Idea

Students need to practice deconstructing arrays with manipulatives in order to conceptualize the distributive property. This 3 day lesson takes the children through increasingly sophisticated activities to explore the "why" and "how" of this property.

Warm Up

10 minutes

In an effort to begin working with the distributive property, I want the students to have hands on experiences with decomposing large arrays and then composing the array again, using the equations (MP4). 

As a class, we built a large array with square inch tiles.  The array I chose to model the area of 12 x 9 would not be an easy one for my students to automatically multiply.  The question at the circle was, "How can we find the area of such a large array?"

As a whole group, the students discussed different ideas and the pros and cons of each.  Of course, one student said, "Let’s just count them." Immediately the response was, “That would take too long!”.  One person said, “Everyone can take 5 and we can go around in the circle and count by 5.”  That idea was not accepted because, “We might lose count”.  Finally someone said, "Let’s make smaller arrays and then add them up!"  So, that is what we did.  

Active Engagement

25 minutes

For pair practice, I gave the students 12 inch squares and asked them to create a configuration that would take up 12 square inches on the grid paper.  After they completed this task, I asked them to work and find several ways of decomposing the array and  finding the equations for each. They then worked to add the 2 or 3 products together.  My reasoning for doing this often with the students is that I want them to visualize the decomposing and composing of area so when we do get into the use of the distributive property for problem solving, they will hopefully have a reason to do it and they won’t be just doing it for the sake of following directions on a page. 

This video shows one of my students working through the "what" of decomposing.  She later moves on in the lesson to work through "why" we should decompose.  This may be the more difficult concept for your students to  grasp.  Just keep moving and working, it will come.

Sharing

10 minutes

As a wrap up, I ask pairs of students to show what they did to make their array easier for them to work with the numbers, and also how putting it back together worked.  I next have students extend their thinking in groups of 4, each partnership sharing with the other (MP3). 

I like this method of sharing out because it lends itself to more natural conversation than waiting for comments or questions from a whole group, where the raising of hands takes time and is somewhat random.

Closing

10 minutes

Following the double partners share, I ask students to write, in their journals, what they learned about solving the area of a large array. They could alternatively write about one idea they thought was important in their math work today.  Writing their thoughts is just another way of being succinct in communicating what they know, which is Mathematical Practice 6: Attend to Precision.

The journal entry I share here is by one of my students that has a harder time writing. However, the goal in writing in math is explaining your reasoning, and as you can see the math is explained very clearly.