Decomposing Large Rectangles

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SWBAT use smaller rectangles to find the area of a larger rectangle.

Big Idea

Students will explore how to use smaller rectangles and friendly numbers to find the area of larger rectangles.

Teacher Demonstration

20 minutes

Unit Explanation

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. 

Goal & Lesson Introduction

I start today's lesson by reviewing the learning goal that we began working toward yesterday: I can use smaller rectangles to find the area of a larger rectangle. I explain: You have all created beautiful home plans! Soon, you'll be tasked with finding the total amount of flooring needed for your home. In order to do this, you'll have to find the area of each room. Some of your rooms, such as a bathroom, are smaller, which makes it easier to calculate the area. However, some rooms are much larger, such as your living room. Turn and talk with a partner... How might you find the area of your largest rooms? One student sings: To find the area, count the squares... count the squares! Another student suggests: You can multiply the length times width. And another says: You can decompose!  

We then discuss which strategy for finding the area is would be most efficient when finding the area of a larger room, such as a 15 x 12 living room. Again, this unit takes place at the beginning of the year. Students will how to multiply multi-digit numbers later on in the year. Therefore, decomposing the larger rectangle into smaller rectangles in order to find the area is will be the most effective strategy for students to use at this point.

I continued: We already experimented with decomposing rooms yesterday, but today, it's going to get harder! 

Friendly Numbers

During yesterday's lesson, some students struggled with decomposing rectangles using friendly numbers. For example, when decomposing a 9 x 13 room, some students decomposed the room into (9 x 10) + (9 x 3). Others decomposed the room into a less helpful combination: (4 x 13) + (5 x 13).

I knew it would be important to discuss friendly numbers today. I first ask students: What are friendly numbers? Turn and Talk! After a few minutes, we come back together. One student says, "10 is a friendly number." I ask: Why is 10 a friendly number? "It's an easy number to multiply with." I then ask the class: What other numbers are friendly and easy to work with? One at a time, students suggest other friendly numbers, including: 2, 5, 15, 25, 50, 100. I make a list on the board. 

I explain: Today, I would like for you to really look for friendly numbers. Let me give you an example. I draw an 18 x 15 on the board and label the dimensions. What are some ways we could decompose this larger room? I let students think on it for a moment.

Before taking responses, I want to model a less helpful way of decomposing a larger rectangle. I ask and model at the same time: How about if I decompose the rectangle into an (18 x 11) + (18 x 4)? Did I make this problem easier to solve? Turn and Talk. After discussing, a student offers, "You need to use friendly numbers, like 10!"

I continue: Can anyone tell me how I can decompose this larger room using friendly numbers? A student suggests, "You can turn the 18 into a 10 and 8 and the 15 can become a 10 and 5. I then model this student's thinking on the board in the following fashion: 18 x 15 Teacher Example.

Turn and Talk: Why was this way easier? (Because we used friendly numbers!)

Student Practice: Part 1

30 minutes

Getting Started

With excitment, I share: Are you ready for a challenge today?! Students sat up with interest. I explain: I have a 15 ft x 9 ft dining room and I need help finding the area. I pass out the Array 9 x 15 and a 11x17 piece of construction paper (poster) to each group. I continue: Today, I'd like for you to work together with your group to decompose this room into smaller rectangles so that the area is easier to find! Once you decide the best way to decompose the room, you can cut the room into smaller rectangles and paste them on your poster. Don't forget to use friendly numbers and make sure you listen to everyone's ideas in your group before cutting the room into smaller rectangles! Ready! Set! Go! 

Monitoring Student Understanding

Once students begin working, I conference with every group. My goal is to support students by asking guiding questions (listed below). I also want to encourage students to construct viable arguments by using evidence to support their thinking (Math Practice 3). 

  1. What did you do first? 
  2. Can you explain why you _____? 
  3. What do you see?
  4. What did you just learn?
  5. Are you using friendly numbers? How do you know?   

During this time, I also encourage labeling, equations, words, and finding creative ways to communicate student thinking. 

After about 15 minutes, I announce: You have five minutes left to finish this challenge! This is the time to make sure you have labeled everything so that your math makes sense to others! Also I'd like for you to choose a team leader to share your ideas with the class in five minutes. Each group chooses a leader to share their thinking. Students were so eager to share that they could hardly listen to others, so I take the time to call attention to students' work by saying: Wow. Did anyone else think about it this way? That's amazing! I also made sure to praise students for writing equations: (10 x 9) + (5 x 9) = 135 square feet.

After everyone shares, we discuss which method made calculating the area of the original 15 x 9 easiest and is most helpful. Students agree that decomposing the 15 x 9 into (10 x 9) + (5 x 9) was one of the most efficient ways to find the area. 

Here are group work examples during this time: Example 1Example 2Example 3Example 4. I was proud to see some students providing a written explanation. Others showed their calculations and provided beautiful equations to represent their thinking. I also thought it was interesting how some students decomposed the rectangle into 2 smaller rectangles while others decomposed it into 3 or 4 smaller rectangles! 

Student Practice: Part 2

40 minutes

Increasing the Complexity

To provide students with increasing complexity during this learning experience, I decide to ask students to decompose an even larger rectangular room, measuring 25 feet x 22 feet! I tell students: I have another challenge for you to solve as a group, only this one is going to be even harder! 

I excitedly hand out an Array 25 x 22 to each group. Instead of passing out a new poster paper, I ask students to flip over their posters from the previous challenge and to use the backside. Students quickly jump in and began counting the dimensions! 

Following the same process as before, I conference with each group to monitor student learning. I also take the time to support positive responses and respectful listening by recognizing students and providing guidance on how to respond. 

Here's an example of a group Using Friendly Numbers (25 and 50) to find the solution successfully. They decompose the 25 x 22 into (25 x 4) + (25 x 4) + (25 x 4) + (25 x 4) + (25 x 4) + (25 x 2). They explain, "25 x 4 is 100. If there are 5 hundreds, that's 500 and then you have 50 left to add in because 25 x 2 is 50."

Again, each group chooses a team leader and we discuss each group's ideas. Here are group work examples: Example 1Example 2Example 3Example 4. I was impressed with the variety of strategies, student use of friendly numbers, and the lengthy equations. I knew this was a lesson I would want to teach again next year!