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 & Introduction
I begin by reviewing our learning goal: I can use smaller rectangles to find the area of a room. I explain: Today is the day you've all been waiting for! You get to start finding the area of each room in your home plan! Here's an example of a student home plan from a previous lesson:House with Walls Example 1. Students are really excited as they have a lot of pride and ownership in their home plans.
I continue: Remember, the whole reason we want to find the area of each room in our home plans is to figure the cost of installing flooring. I hold up a Flooring Sample and ask: If you went into Home Depot and wanted to get enough flooring for a room in your house, would you walk in and just say... I would like to buy this flooring for my bedroom. Students giggle.
I then ask, "Would the flooring department associate need more information? " One student raised her hand said with a smile, "They need to know how big your room is!" I continue: So they would need to know the total..... I pause to see if my students can fill in the blank. Sure enough, students respond back, "Area!"
I pass out one piece of grid paper to each student and explain: Today, we are going to find the area of one room in your house so that you can walk into a store like Home Depot and successfully get carpet! I ask students to get their home plans and to choose a medium-sized room to work with first. I want to provide students with guided practice finding the area of a room before letting them find the area of all rooms independently.
I show students how to reconstruct their chosen rooms on grid paper. If a student had a dining room that was 12 x 9 in her house , she cut out a 12 x 9 rectangle using grid paper: Example of Cutting out Room. To begin with, I want students to be able to work with and focus on one room away from the rest of the house plan outside of the confined space in-between walls. I also want students to be able to find the area of the room by cutting the room into smaller rectangles with a pair of scissors!
While students reconstruct their chosen rooms on grid paper, I walk about the room to check that students are accurately determining the dimensions. Some students struggle with counting the squares to find the length and width. I remind them to use their pencils to check off counted squares.
Once students are ready to go, I ask them to decompose their rooms into smaller rectangles with a pair of scissors: Example of Decomposing Room. I also ask students to keep friendly numbers in mind, such as 2, 5, 10, and 25 when determining how to decompose the room. For example, a student with a 9 x 18 room might choose to decompose it into (9x10) + (9x8).
Modeling a Student Journal Entry
I continue: Now that we have decomposed our rooms, we are going to take the time to discuss how to create a journal entry in your math journals for each room in your house. I ask students to get out their math journals.
Using the following Anchor Chart: Decomposing Rooms, I model the steps I want students to take as they find the area of each room in their home. As I demonstrate each step, students complete the steps in their own journals using their own rooms. The goal of this chart is to encourage students to engage in Math Practice 6: Attend to Precision. Often times, when students are given an assignment without clear expectations, the level of accuracy and attention to details decreases.
1. I explain: The first step is to write the name of the room next to the number one. I decomposed my bedroom, so I'm going to write "bedroom." I give students time to do the same in their math journals.
2. For the next step, we're going to show how we decomposed the room into smaller rectangles to find the total area of the room. I'm going to glue my grid-paper rectangles down on the paper. What else should I do? Students raise their hands and offer, "You should label the sides!" I then model how to label the sides 10 ft x 10 ft and 5 ft x 10 ft. I take the time to point out the importance of labeling feet. Again, students complete the same task using their own rooms. I walk the room to ensure students understood the directions and are attending to precision.
3. Now, we're ready for step number 3! For this step, I'd like for you to provide an equation that represents your decomposed room. Remember, equation is just a fancy name for a number sentence. I ask students to help me write an equation for my room. One student says, "You have a 10x10 plus a 10x5. I write: (10x10)+(10x5). Another student points out, "10 x 10 is 100 and 10 x 5 is 50. If you add 100 + 50, the total is 150." Again, to encourage precision, I ask: 150 what? What are we counting? A student answers, "150 square feet." Before moving on, I provide plenty of time for students to complete their own equations. As students finish, I ask them to check their work with others.
4. The next step is the easiest! I'd like for you to identify the total area of your room. I'm going to write, "Area: 150 feet squared."
5. At this point, I realize that the fifth step would be way too much for the students to complete for each room. I was wanting students to write a paragraph, explaining how they found the area for each room. Instead, I decide to take this time to write an explanation together on the anchor chart (instead of requiring students to complete a similar paragraph in their journals). I let students choose whether or not they wanted to include this step for today's room in their journal.
I begin by providing students with a standard sentence prompt: "I knew that..." Can anyone help me finish this sentence? A student offers, "I knew that I could use smaller rectangles to find the area of a larger rectangle."
I then write another sentence prompt: "So I..." I need help again! What should I write next? A student finishes the sentence: "So I decomposed the 15' x 10' into a 10' x 10' and 10' x 5'."
A student points out, "We should tell about the area of each of the smaller rectangles next." Per student suggestions, we then write, "The 10' x 10' is 100 ft squared. The 5' x 10' is 50 ft squared. Altogether, the square footage of my bedroom is 150 ft squared."
I always celebrate students for discovering mistakes and finding create ways to fix them! This will encourage students to take risks in the future and to know that mistakes are evidence that they are trying! Here's a picture of a student matching their grid paper up to their house plan: Verifying Measurements. You can see that she is two feet short.
I simply ask: How can we fix that?
She responds perfectly: I can just cut out another array that is is 2 ft by 12ft.
I excitedly share her problem solving idea with the class and celebrate her for finding an innovative way to move beyond a mistake.
Here, Example 1: Student Journal, a student decomposes her guest bathroom into (6x3) + (6x6) to get 54 feet squared.
Another student, Example 2: Student Journal, decomposes his kitchen into (5x5) + (5x5) to get 50 feet squared.
Although both of these students could have just multiplied 6 x 9 = 54 and 10 x 5 = 50, the process of decomposing will make more sense if students are able to practice using simpler rooms. This way, when students decompose larger rooms, such as 23 x 15, students can rely on this carefully developed understanding of decomposing.