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.
Wood or Vinyl Fencing?
I began by presenting a real-life problem with fencing... choosing the type of fencing! I explain: There are many types of fencing, but I am most interested in two types of fencing: a wood panel fence or a vinyl (plastic) panel fence. I draw a picture of a panel and explain: A panel is a section of fencing.
To help students help me decide on the type of fencing, I show students a Lowes video clip, Fence Installation Tips: Choosing and Planning Your Fence. Before starting the video, I ask students to make a list of pros and cons in their journals. We make a t-chart with Pros and Cons as headings. Then we split the t-chart in half. On the top half of the paper, students took notes on the pros and cons vinyl fencing. On the bottom half, students took notes on the pros and cons of wood fencing: Pros & Cons of Fencing Panels. Here's a student's journal: Student Example.
After the video, I ask students turn and talk: Which type of fencing should I get? Vinyl Panels or wood panels? We then discuss their thinking as a whole group. I encourage students to develop evidence-based explanations, such as I think you should buy wood panels. They'll cost less money than the vinyl panels or If you buy the vinyl panels, you'll save money on paint and stain.
The goal of this opening activity is to provide students with the opportunity to experience what it is like to make decisions and install fencing in the real world.
Goal & Lesson Introduction
I began by introducing the lesson's goal: I can determine the perimeter of a dog pen when the area is specified. I explain: Today, I would like your help building a dog pen that provides my dogs, Jedi and Josie, with a certain amount of area (or space). I wrote the following problem on the board: Mrs. Nelson wants a rectangular dog pen that measures _____ square feet. How many vinyl panels does she need? I ask students to also write the same problem in their journals.
I continue: Let's write the letter "n" here. (I wrote the n on the blank line.) Let's say that the letter "n" represents the amount of square feet (or area) on the inside of the dog pen. It is also called a variable because the letter "n" can vary or change. One student asks if it has to be an "n" and if the letter can be any letter. I explain that variables can be any letter. In no time, most students replace the "n" with the first letter of their names!
I divide students into groups of two or three, purposefully placing students and taking into consideration ability levels, communication skills, and behavior. I try to place a variety of levels and strengths into one group as students can learn so much from one another.
Area & Number of Panels Chart
I create a T-chart below the problem: Problem & Chart and explain: Let's place Area (or n) in the first column. Then, let's put the number of vinyl panels needed in the second column. While writing 36 square feet in the area column of the t-chart, I tell students: I wonder how many vinyl panels I'll need if I want an area of 36 square feet.
I hand out popsicle sticks to represent panels. However, to engage students in Math Practice 5: Use appropriate tools strategically, I also allow students to try choosing other math manipulatives to model their thinking.
I purposefully don't tell students the length of each panel, hoping a student will speak up and ask for more information to solve the problem. About three minutes into problem solving, one student... and then another... and another said, "We need to know how long a panel is!" I was so proud! I pass out information from a local hardware store: Vinyl Panel Fencing.
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).
As students are problem solving, I notice many of them struggling with one popsicle stick representing more than one unit. It was great to seem them overcome this obstacle and realize that one unit can stand for 6 feet.
At first, this group, 6 x 6 using a Geoboard, struggled with finding a way to represent a 6' x 6' dog pen using a 4 x 4 geoboard. It was great to see how they chose to have one unit (rubber band stretched between two pegs) represent 6 feet.
In this video, students explain how they created a 6 x 6 Pen on a Grid. They have a beautiful representation, but are still developing an explanation. There's some confusion on the difference between area and perimeter.
One most students have a solution, we gather back together as a class and discuss student thinking. Students come up with a variety of models for a 36 foot-squared dog pen using a variety of manipulatives. Here are examples of students using unifix cubes and popsicle sticks: 6 x 6 using Unifix Cubes and 6 x 6 using Popsicle Sticks.
Increasing the Complexity of the Task
At this point, I challenge students to create dog pens with the following areas and add them to the chart.
Most tools decide to use the tools they were most successful with from the last activity: grid paper and/or popsicle sticks.
It didn't take long for me to realize the challenges of the larger numbers. Several students became confused pretty quickly! I decide to change my plans and bring students back together so we can solve these problems as a class.
I invite all students to sit on the front carpet with their individual white boards. First, I draw the 6 x 6 on the board and ask students to do the same on their white boards. I also create the 6 x 6 model using four popsicle sticks on the floor: 6 x 6 Popsicle Sticks.
72 Square Foot Dog Pen
Then, I ask: What would you do to the 36 to get to 72? Students realized that we need to "double the 36." I drew a double box: 6 x 12 and ask student volunteers to come up to the board and label the box.
I continue: How do you know the area of this dog pen is 72 feet? A student comes up to the board and explains, "A 72 is just two boxes of 36."
How many panels do we need? Another student comes up to the board and marks each side, "One... two... three... four... five... six panels."
Does everyone agree? Does anyone disagree?
108 Square Foot Dog Pen
Students start seeing the pattern and quickly go to work on their white boards, trying to find the number of panels needed for a dog pen that has 108 square feet.
After giving students time to solve, we come back together and a student volunteers to draw this on the board: 6 x 18. Another student manipulates the popsicle sticks on the floor: 6 x 18 Popsicle Sticks.
144 Square Foot Dog Pen
We move onto 144 square feet. Again, students excitedly quickly got to work, drawing groups of 36 square foot boxes. I gave students plenty of time to show their thinking on their own boards, White Board Problem Solving, prior drawing a model on the front board: 12 x 12.
Then, a student said, "Wait, I have another way" and drew a 6 x 24 on the board! This was a great opportunity to ask students:
What is the area of the 12 x 12 model? (144 square feet)
What is the area of the 6 x 24 model? (144 square feet)
Does each dog pen require the same number of panels? (No, one needs 8 panels. The other needs 10 panels).
Why does the 6 x 24 take more fencing than the 12 x 12? Several students came up to the board to try explaining this. One student says, "If the space inside is more spread out and longer, you'll need more fencing. If the area is pushed together, you'll need less fencing."
216 Square Foot Dog Pen
With only a few minutes left, I challenge students find how many panels I would need to fence in a 216 square foot space. We actually didn't an adequate amount of time to solve or discuss this problem fully. A couple boys who were quite confused before this activity came up to the board and show their thinking while the rest of us got ready to go home. At the end of the day, I look up at the board and proudly admire their work, Students Showing 216 Feet Squared, as they were looking for and making use of structure... since 2(6 x 6) = 72, then 2(6 x 6) + 2(6 x 6) + 2(6 x 6) = 216, or as they had written 72 + 72 + 72 = 216.
Also, after finding the number of panels for each scenario, we completed the chart on the board Completed Chart.