Finding the Area of Odd-Shaped Dog Pens

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Objective

SWBAT find the area of odd-shaped dog pens.

Big Idea

Students will decompose odd-shaped dog pens into smaller rectangles to find the area.

Opening Activity

30 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. 

Reviewing Perimeter

Prior to changing our focus to area, I want students to be given the opportunity to collaborate and discuss answers from the previous activity on perimeter.

I ask students: Why is it important to share your work with others? Students respond, "So you can look at each others answers." "So you can check your work." I remind students how to provide respectful feedback using the prompt, I respectfully disagree because...

Today, you are going to be working with a group of students to compare answers and provide feedback. If you have different answers, I want you to figure out which answer is correct. Then I want you to discover where mistakes could have been made. 

I group students together, taking sets of partners from yesterday's lesson and pairing them up with another set of partners. Then, I ask all students to get a red pen for making revisions.

Mathematical Discourse

In this video, the students are successfully Checking Answers. All students end up agreeing on the first problem and student engagement is high. 

Here, I watch a group get Different Answers. Immediately, I saw some kids double checking their own answers. Other kids were uninvolved. Then, one student explained her calculations. It is clear that they need a more successful way of communicating ideas. I asked the students to use a model so that all students could understand their thinking. Getting a Model. Student engagement increased, but still wasn't perfect! In this video, Solving Again, some students are willing to go back and solve while other are taking  a "sit back approach." To increase student engagement, I ask all students to recalculate: Finding Mistakes. This is certainly easier for some than for others! One boy realizes he forgot two measurements of 5 feet. One girl decides that she must have multiplied 10 x 7 instead of adding. This is a powerful experience that encourages student reflection and mindfulness.

Here's an example of Student Revisions during this time.

Teacher Demonstration

10 minutes

Goal & Lesson Introduction

I started with our two-day goal: I can find the perimeter and area of odd-shaped dog pens. I explain: Yesterday, you focused on finding the perimeter of odd-shaped dog pens and today, you'll focus on finding the area! 

Vocabulary

We began by reviewing both the Horizontal Vocabulary Poster and the Vertical Vocabulary Poster from yesterday. I ask students to turn and talk: What is the difference between a horizontal line and vertical line? 

We also reviewed the meaning of variables (missing number) by drawing Teacher Model A and Teacher Model B on the board with two missing dimensions.

I ask students: Can anyone show me how to solve for y? Can anyone show me how to solve for x? 

Finding the Area by Decomposing

At this point, students were ready to move on to finding area. Directing student attention to Teacher Model A, drawn on the board, I ask students: When we have an odd-shaped dog pen like this, what do you think we should do? Should we just multiply the length times the width? Students respond: "No, you need to decompose!"

Okay, so we need to find the  area of smaller rectangles? Does anyone see the smaller rectangles in this odd-shaped dog pen? Many hands went up! After providing some wait time, I hand the marker to a student to demonstrate. He successfully draws two lines, creating two smaller rectangles. 

Then I ask students to turn and talk: Now, how would you find the area of this dog pen? After a couple minutes of discussion, one student says, "A one by two is 2." Two what? "Two feet squared." Another student says, "Then you times 4 by 3... no... 4. Four times four is 16 feet squared." Now what do we do? Almost every student responds, "Add them up!" Others shout out, "It's 18! It's 18." How do you know? "16 + 2 = 18... ft squared."

How can we know for sure? Can we draw squares on the inside? I model how to divide the smaller rectangles into rows and columns: Teacher Model on White Board. I want students to see that they really can trust this method! Also, this helps solidify their thinking. 

We continue this same process with a harder model, Teacher Model B.

Student Practice

30 minutes

Preparation

For today's lesson, students will continue using the Odd Shaped Dog Pens from yesterday.

  • I purposefully created Dog Pen A and Dog Pen B with grid lines.
  • Dog Pen C and Dog Pen D a bit more challenging with no grid lines. However, I still had rectangles inside the figures. 
  • Finally, Dog Pen E and Dog Pen F were the most challenging: no grid lines and no rectangles! 

By starting off with a simpler task, students were able to develop the skills necessary to complete the more challenging tasks. 

Choosing Partners

Students will also continue working with the same partners as yesterday. (I already assigned partners to students, taking into consideration ability levels, communication skills, and behavior.) 

Continued Practice

I drew the example Chart on the board and asked students to do the same in their journals. I explain: Today, I'd like for you to work with the same odd-shaped dog pens as yesterday, only today, I'd like to know how much space Jedi and Josie (my dogs) will have in each dog pen. Please start with Option A and move on to Option B when you are ready... then C, D, E and F. 

Monitoring Student Understanding

Once students begin working, I conference with as many students as possible. 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 did you just learn?
  4. How did you find the missing dimensions?
  5. Will this strategy always work?

Student Conferences

As students began working with Options A and B, I notice some students counting the squares whereas others were decomposing and multiplying side lengths. In this video, you'll see a student calculating the area using side lengths while her partner shows concern about the next odd-shaped figure without squares: Calculating Area

In this video, a student decomposes one of the more complex shapes to calculate the area: Area of Option F

Student Work

Here's an example of student work during this time: Student Journal Example. Most students were able to successfully find the area of all odd-shaped dog pens!

Closing

15 minutes

When students finish, they meet with other students from other groups to check and revise their work.

During this time, I provide further instruction and support for students. Here's an example of how a student progresses over the course of this time. 

At first, this student is struggling with finding the area of a rectangle without squares. He also doesn't understand why we multiplied instead of adding (like with perimeter). I show him how to add squares to help solidify his understanding. I don't want him to multiply the length times width without understanding why we multiply: Learning Progression 1.

Next, this student shows how he split the larger rectangle in the middle into smaller squares. However, he uses incorrect dimensions. Instead of correcting him, I ask guiding questions to help him arrive at the correct answer: Learning Progression 2.

In this last video, you'll see that I highlighted one side without saying anything more. He realizes the correct side length. As he counts the squares, he sees that counting sometimes isn't as accurate as multiplying the length times width: Learning Progression 3

I am so proud of his learning progression. He did great!