Decomposing Rectangles to Find Area
Lesson 6 of 19
Objective: SWBAT use smaller rectangles to find the area of a larger rectangle.
During the first section of this unit, students will construct a house plan, find the area of the house plan, and calculate flooring costs. While finding the area is the focus of this unit, the first few lessons (where students explore the meaning of a polygon, construct house plans, and decompose rectangles into smaller rectangles to find the area) lay the foundation for finding the area of their home plans later on. This also provides students with a meaningful and purposeful context to find the area.
During the second section of this unit, students will investigate dog pen designs and will primarily focus on finding the perimeter, or amount of fencing needed for different dog pens. Students will also explore odd-shaped polygons by finding the area and perimeter of odd-shaped dog pens.
Prior to today's lesson, I visit a local home improvement store to gather five samples of flooring for students to chose from: Flooring Samples 1 and Flooring Samples 2. I try to pick a variety of flooring, including carpet, hardwood, and vinyl tiling. The vinyl tiling was my favorite find as I was able to purchase a square foot. Often, it's difficult for students to visualize exactly what a square foot looks like.
I begin the lesson by setting out the five different flooring options, labeled with the actual cost per square foot. Although we won't be using these flooring samples today, I want students to begin thinking about the flooring that they might install in their own home plans. I also want to provide students with a real purpose for finding area! Students immediately lit up and were full of questions, "What are those for?" "What are we going to do?"
I explain: Fourth graders, in about a week, you will be calculating the cost to install flooring in your home plan! You get to choose from the following floor samples. However, in order to find the total amount of flooring you'll need, you need to first find the area of each room in your home. Today, I want you to begin thinking about which type of flooring you like best for each room of your home. Take a few minutes to discuss your flooring ideas so far!
Over the next week, I leave the flooring samples out for students to view as they are lining up at the door. Each time students line up, I see them touching and interacting with the flooring samples and I hear great conversations! One student says, "I want to put wood floors in my kitchen and dining room. Then I want to put carpet in the rest of my house." Another student says, "I want to put laminate flooring in. It's cheaper than the wood floor and I like the look of it."
Goal & Lesson Introduction
I introduce today's goal: I can use smaller rectangles to find the area of larger rectangles. Students got out their math journals and wrote this goal at the top of a new page. I explain: Yesterday, you learned that there are many ways to find the area of a polygon. Referring to the anchor chart from yesterday, Finding Area Strategies, I reviewed the multiple ways to find area: multiplying length times width, counting by rows or columns, and decomposing. I continued: Today, we are going to explore decomposing further by breaking rooms down into smaller rectangles in order to find the area.
Prior to this lesson, I cut out the following rooms using large grid paper for each group of 2-3 students. I also cut out an extra set for teacher modeling. To build a logical learning progression that grows with complexity, I cut out smaller and larger rooms.
- Small Closet: 3ft x 4ft
- Large Closet: 4ft x 7ft
- Hallway: 3ft x 12ft
- Bedroom: 9ft 14ft
Examples of Decomposing
To connect today's lesson with yesterday's lesson, I show students the 3ft x 4ft closet from yesterday's lesson. Yesterday, a student offered one way of decomposing the 3 x4 closet, "We can cut the 3 x 4 in half so we have two rectangles that measure 3 x 2." Today, I want to encourage students to "think outside the box" by finding other ways to decompose a 3 x 4 closet besides cutting it straight in half.
I explain: Yesterday, you did a beautiful job decomposing this 3ft x 4ft closet into two smaller rectangles to find the area. I remember one student suggesting that we can just cut the room in half. Today, I want you to see if you can find other ways to decompose this closet into smaller rectangles! As you discover more and more ways to decompose this closet, I would like for you to make a list of ways in your math journals.
To get students started, I project my math journal and show students how to make a list of ways to decompose the closet: Examples of Decomposing.
One student suggests, "We can cut the top row off so we have a 1 x 4 and a 2 x 4." We drew a picture of this in our journals and labeled the dimensions of each rectangle. Then, students continued on by completing their own lists.
Looking back on this lesson, it would have been more helpful to provide students with grid paper for this activity!
After about ten minutes, we discuss all the ways to decompose a 3ft x 4ft closet. As students volunteer their ideas, I draw their representations on the board. During this time, students added other student's ideas to their own lists if they didn't have them already.
Decomposing Larger Rooms
To provide further practice, I pass out the cut outs of the first room (4ft x 7ft closet) to each set of 2-3 students and ask students to turn and talk: How can we use smaller rectangles to find the area of the larger rectangle? I hear one student count the sides and say, "We could cut this room into a 2 x 7 and a 2 x 7." Another student says, "Or we could cut it into a 1 x 7 and a 3 x 7."
After students discuss their ideas, I ask them to use a pair of scissors to cut the closet into two smaller rectangles.
Teacher Anchor Chart
I invite students to share how they decomposed the closet with the class. Just as before, students discovered many ways to decompose the 4 x 7 room.
Then, using one group's strategy, I model how to decompose the 4 x 7 room into two smaller rectangles: a 2 x 7 and a 2 x 7. I paste this group's idea onto the anchor chart, Anchor Chart: Decomposing Rooms. I then label the dimensions and model how to write an equation: 4 x 7 = (2 x 7) + (2 x 7). By encouraging students to represent the decomposing process using an equation, I knew students would also be expose to the distributive property. In addition, anytime students connect abstract equations with representations, they are also engaging in Math Practice 2: Reason abstractly and quantitatively. It is important for students to be able to visualize the meaning of equations.
During this time, students follow along by documenting how to decompose the 4 x 7 room in their math journals and how to write a corresponding equation. Some students use the modeled example while others represent other ways of decomposing the 4 x 7 room.
We move on to finding the area of larger rooms, the hallway (3ft x 12ft) and the bedroom (9ft x 14ft). Following the same process as above, students discuss ways to decompose the rooms within their groups, then they share out with the rest of the class, and, altogether, we record one way to decompose both rooms on the anchor chart and in student journals. Here's a Student Journal Example.
Here, students cut the grid paper room cut outs to model how to decompose the larger room into smaller rectangles: Students Decomposing.
As students explain their strategies, I will often draw their ideas on the board: Example 1: Student's Thinking Represented on the Board and Example 2: Student's Thinking Represented on the Board. I do this for many reasons. First, it's sometimes difficult for other students to hear and understand the student's thinking without a visual representation. Also, whenever I take the time to listen and draw a student's idea on the board, it is incredibly validating to that student, which results in a higher confidence and love for math!