Area & The Distributive Property

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SWBAT calculate large areas using the distributive property of multiplication and addition.

Big Idea

Students develop spatial reasoning about large areas through composing and decomposing rectangles to determine area.


5 minutes

I begin this lesson with a review of folding skills used in our fraction unit to create fraction strips.  The students are given different sizes of paper, and I ask them to fold them equally into units to create fraction pieces.  I want to help the students make this connection between equal units of fractions and measuring an area. I choose different sizes of paper because I want the students to be able to compare eight units on a small piece of paper to eight units on a larger piece of paper. The size of the whole matters!  

Most of the students are able to fold their papers into approximately 16 units, and repeatedly folding after this amount can become difficult due to the increased thickness of the paper.  Some students fold only to eight units, and others work on problem solving to create even more units.  

Next, I have the students mark the lines of their folds with a pencil, crayon, or marker, to clearly show the grid they have created.  Many of the students begin marking these with fraction units, and others leave them blank.  

At this point, it does not make a difference if the students mark each section with a unit fraction such as 1/8  or if they discuss that a folded paper has 8 sections.  The folds are the model to support understanding of fraction units and area.This is where they can start to see the similarities and develop the connections between fraction units and area measurements.  


15 minutes

This section of the lesson requires two large pieces of construction paper. One piece is cut it into smaller rectangular pieces to demonstrate the distributive property.  

I first show the students the large unfolded piece of construction paper and ask, "What is the area of this piece of paper?" I know this is a vague question to ask the students, but I want them to apply their skills from the warm-up to problem solve how to figure the area of the piece of construction paper.  Their responses include fold the paper and count the area.  

I now tell the students this piece of paper represents a football or soccer field, and I can't fold a field. Finding the area is going to require another strategy.  Prior to the lesson, I cut a second identical piece of construction paper into smaller pieces. I was careful to always cut pieces with one side the same length so that students could create equal size units. I used construction paper that was 12" x 18", so the rectangles always had one side measuring 12".  I show these pieces of paper to the students and ask them, "How can I use these pieces to find the area?"   

The students quickly connect folding pieces into equal size squares to find the area of the large 12 x 18 piece of construction paper.  I then ask,  "Once they're folded what will I need to do?" Because the pieces are separate, the students realize they will need to add the rectangles together.  I model this with the students with the pieces I have cut.  For demonstration I had cut the one rectangle  6" x 12", and the remainder was cut into four strips of 3" x 12".  The 6" x 12" piece was folded so that there will four rows and two columns for a total of eight equal size units.  

Folding the longest side repeatedly in half, two times, resulted in four squares each three inches in height.  This was an easy size for the students to see during the demonstration.  These rectangles each resulted in four squares.  Since there were six strips of 3 x 12, the area of the demonstration field was 24 units.  

To explain to the students the distributive property we added 

Working Together

35 minutes

Students are given two pieces of construction paper measuring 9 x 12.  One is a full size piece, and the second has been cut into one rectangle of 4 x 12, one 2 x 12, and three 1 x 12.  My goal is for the students to use the distributive property to explain the area of the 9 x 12 piece of construction paper. 

These pieces are folded exactly as they were during the mini lesson to create three inch rectangles.  The total area of this piece of paper would be  units. 

Working in partners, students will fold the smaller pieces of paper into rectangles to create an array with the same unit.  They write the multiplication number sentences for the smaller arrays.  These multiplication facts are then written with addition to show the multiplication sentence for the distributive property.