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.
Finding the Area of a Closet
Today, we will start by finding the area of a small room (closet). Once students learn how to find the area of the closet, they will apply this understanding to find the area of larger rooms in their own home plans tomorrow.
For today's lesson, students will learn how to find the area of an array using several strategies. Teaching multiple strategies is important for many reasons.
First, I am teaching this unit at the beginning of the year, before teaching my multiplication unit. I want to make sure all students have a range of strategies to use, even if they don't know their multiplication facts.
Ultimately, I want to lead students to decompose (break apart) a rectangle into smaller rectangles to find the area. This is important as many of the students' rooms in their house plans are larger rectangles, such as 17 ft x 9 ft or 24 ft x 18 ft.
Also, by teaching a variety of methods, students will be engaged in Math Practice 2: Reason abstractly and quantitatively. Students will have a better chance at conceptualizing the abstract concept of area if they know how to arrive at the solution using more ways than one.
To prepare, I cut out several 3x4 Closets using large grid paper. I make sure there's enough for each group of three to have a copy and I cut six extras for modeling strategies later on.
I always try to find ways to transform lessons into hands-on learning experiences as often as possible. This helps with developing a deeper conceptualization of concepts. I love when students walk into the classroom and ask, "What are those for?" They simply can't wait to get their hands on new materials!
I pass out a 3x4 Closet to each group of three students and explain: This is a closet from my house plan. What if I want to replace the flooring in my closet? What information do I need to know before I replace the flooring? Students offered the following suggestions: the measurements, length, width, amount of space/area.
I then ask students to turn and talk: I think there's only one way to find the area of my closet. Do you agree or disagree? After a few minutes, we discuss student thinking as a class. Some students point out how they would count up the squares. Others say that they would count the rows of three. As a class, we decide that there's more than one way to find the area of a room.
Lesson Introduction & Goal
I begin by writing the goal on an anchor chart: I can find the area using multiple strategies. Students follow along by writing the goal at the top of a new page in their math journals as well. Here's what the completed anchor chart will look like at the end of this lesson: Finding Area Strategies.
I ask students: Why do I need to know more than one way to solve the problem? Students discuss the importance of checking answers for correctness. Also, one student points out that a single strategy doesn't always work with all problems.
Recording Strategies on Anchor Chart
Turning to the anchor chart, Finding Area Strategies.pdf, I explain: Today, we are going to work together to find the area of my closet using as many strategies as possible. One student says, "That's easy! It's 12!" I then explain: I love that you're already thinking! However, our focus today is going to be finding more and more strategies. Turn and talk with your group. What is one strategy you could use to find the area of my closet? Students grab their 3x4 grid papers and immediately begin discussing how to find the area.
A couple minutes later, we discuss this question as a class. A student volunteers to share her strategy. I paste one of the 3x4 grid papers on left side of the anchor chart and record the student's name and strategy on the right side. I try to to use student names in anchor charts as often as possible as it promotes student ownership. Also, later on, the student will GLOW as other students refer to the strategy as "Sara's strategy." This validates the student's mathematical thinking!
Constructing Viable Arguments
By providing an opportunity for students to share and provide evidence of their thinking, students are also engaging in Math Practice 3: Construct viable arguments and critique the reasoning of others. During this time, I encourage students to listen carefully to other students' arguments and to respectfully disagree by kindly saying, "I respectfully disagree because..."
During this time students also keep track of strategies in their journals. Here are a few examples:
The first student suggests, "We can just find the length and width and multiply. So 4 x 3 = 12." I ask: Twelve what? The student clarifies, "12 squares." I ask: Can we also call them 12 square units?
After each student shares a strategy, I ask students to turn & talk: Can you please reteach _____'s strategy to a partner? After a few minutes, I ask students to also begin discussing other strategies to find the area!
Then, another student shares his/her strategy. We follow this process (student shares, record strategy, all students turn and talk) until all of the following strategies have been shared:
Students B: Count by 3s (3...6...9....12)
Student C: Count by 4s (4...8...12)
Student D: Count by 2s (2...4...6...8...10...12)
Student E: Count by 1s (1...2...3...4...5...6...7...8...9...10...11...12)
Decomposing the Closet
At this point, one student offers, "We could count by 6s!" Knowing that I only had a little room left on the anchor chart, I ask students to think of strategies besides counting. Students were a bit stumped so I held up a hint... a pair of scissors!
One student says, "Oh! I know!" This sparks a sense of a excitement. Soon others raise their hands! I call on a student and he shares, "You should cut it in half!" Following the student's directions, I cut the 3x4 into two 3x2 arrays and ask: Like this? The student says, "YES!" I then question: But how does that help me? He continues, "If you cut it in half, you can find the area of the smaller rectangles: 2 x 3 = 6 and 6 + 6 = 12."
This was the perfect opportunity to teach the meaning of decomposing. As a class, we develop a simple definition for decomposing, Decomposing: breaking down into smaller parts.
When I teach vocabulary, I try to use TPR (Total Physical Response). As a class, we will develop a simple definition for a vocabulary word as well as hand movements. TPR activates multiple parts of the brain and promotes a stronger memory connection. Often, students are able to recall the meaning of vocabulary words by recalling the hand movements.
Today, we discuss and come up with the following definition and hand movements: Decomposing! (start with hand as a fist, like an apple about to be decomposed) “breaking down into smaller parts” (wiggle fingers)
Next, we practice the new vocabulary word several times. To review the meaning of area, throughout the unit I say, Turn and Talk: How do you find the area of a polygon? Students will use the hand movements to recall the definition!
To bring closure to this lesson, we discuss the strategies on the anchor chart as a class.
First, I ask: Which strategy do you like to use and why? One student points out that he likes the first strategy, multiplying 3 x 4 because it is the quickest way to find the area. Others like counting because it helps them keep track of the squares. Some students like decomposing because it helps make the problem easier to solve.
Next, I ask: Which strategy is the least useful strategy to you and why? Most students agree that counting squares, especially one by one takes the most time and is the least useful.
Tomorrow, we will build upon this lesson as students will begin decomposing each room in their home plans to find the area.