Decomposing Given Large Areas, Day 3

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Students will be able to decompose large arrays in order to mentally compute the total area.

Big Idea

Students have been working with decomposing arrays in order to practice finding small areas and putting the totals together. In this third lesson, they will work on finding smaller areas that make the mental math more feasible.


During Day 1 the students worked on building out a large array with two or more colored tiles in order to find the total area. 

Today, I move the children to a more abstract model, drawings arrays on a graphic organizer. Next, they will decompose to find the area of a drawn rectangle.


5 minutes

The game I Have Who Has (downloaded from MathWire) is used as a warm-up today, as it was yesterday. The students try to beat yesterday’s time, while also preparing their minds to think multiplicatively. 

Today, instead of having students read their card as “I have 72, Who has 5 times 3”, I ask them to read it as “I have 72, Who has 5 by 3” because we are doing array work today.


10 minutes

I ask the students to get out their homework from the night before, which was to journal about how they would find the area of a large rectangle.  Students share with a partner and then I choose 2 or 3 pieces of work to display on the white board as examples of student thinking (MP1, MP3). 

When you do this, listen for evidence of the “why” breaking apart a large area is helpful.  This is the purpose of this string of lessons, in tandem with acquiring the knowledge, skills, and practice to do so.

Following the share, I transition to discussing how we often have to work with large areas and we don’t have actual tiles to help us.  I ask the students to copy into their notebooks the two rectangles I have written in mine, which are a 9 by 15 and a 7 by 12. 

For the first rectangle, I tell them they have to find the area with the following guidelines:

  1. They must not count each square
  2. They cannot use a calculator
  3. They must show their work

I send them off in pairs to work.

Active Engagement

20 minutes

The students have several different ways of approaching this task.  Some use rulers to measure and draw several smaller arrays, some work to find combinations of 5’s or 10’s, and some draw random lines and figure out the two array products to add together (MP4, MP5).

This is fabulous!  Although it may seem sloppy and unfocused with different outcomes for everyone, the comfortable struggle is very informative.  As you read the above paragraph, you can probably already imagine how I might organize my students into strategy groups for tomorrow’s lesson!

As students work, listen…I find the less I speak, the more I learn.  When students are done speaking, I insert a thought or comment and then move on, only to circle back to see what they have done with it. 

When you do this lesson, look for students really working to make “easy” combinations in their decomposing. Remember, in this task the algorithm has not yet been used, but seeing and using the concept is key.  Make sure to give them ample opportunities.

This student explains how he got to the "10th mark" to help him find his area.  He did not want to miss recess to do this work!!

 These girls worked with an organized division of the larger array, using "easy" numbers to work with. 

Wrap Up

15 minutes

I choose not to do a full class share at this point, as I know tomorrow I will be using student work as our opening.

Rather, I have students work on the second rectangle in their reflection journals as an assessment of our day’s work.  Here, I am looking to see if they use the suggestions and comments given during the work time and if they try to decompose into base 5 or base 10.