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.
I begin my explaining: A couple days ago, I showed you a video of my dogs, Jedi and Josie in our backyard. I asked you to help design a dog pen using vinyl fence panels to help keep my dogs safe. Today, we are going to talk about a new kind of fencing: wire fencing.
I show students a picture of Wire Fencing that I can buy at a local hardware store. Then, I ask students to discuss: Why would you want to use wire fencing instead of panels? (You have more flexibility with side lengths. For example, with the vinyl fence panels, all side lengths had be multiples of 6 because the fence panels were 6 feet long.)
Goal & Lesson Introduction
I begin by stating the goal for today's lesson: I can determine the largest dog pen that can be made when given a specific amount of fencing. I explain: Well, last night, I went to the store and I bought 16 feet of wire fencing. One excitedly student asked, "Did you really?" (I had to admit that it was a made up story!) I know that I can place this fencing in many different arrangements, but I need help finding the biggest arrangement possible. I restate and write the question on the board: What it the largest rectangular dog pen I can make using 16 feet of fencing?
Next, I invite students to get out their white boards and to find all the possible dog pens that can be made using 16 feet of fencing. Students eagerly began!
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).
I love watching this student draw his fencing out first: Starting Off. You can tell that this helps him visualize the length of fencing he had to work with.
Some students work through confusion between area and perimeter: Oh… Perimeter!. I enjoy asking guiding questions, hoping they will discover their mistake without me directly pointing it out!
I am so proud when I see a student using one line to represent more than one unit: One Line = 2 feet. This is a concept that this student struggled with yesterday (one 6ft panel can equal one popsicle stick), but it seems to make more sense today.
Finally, it is great hearing how students found all of the arrangements. For example, these students explain how they just used the guess and check method: Finding Pens with 16 feet of Fencing.
Recording Dog Pens
At this point, I want to bring students back together to discuss their findings and to look for patterns! Prior to this lesson, I cut out the follow arrays using large grid paper (one inch by inch) so that I can readily paste them on the Teacher Model.
At first, I ask students: What is the smallest dog pen that I could make for my dogs?
One student says, "A one by seven." I paste this on the poster and purposefully arrange this on the poster in way that students could easily see a pattern later on (1... 2... 3...4..). Then, I label the dimensions.
I continue in the same manner... What's the next biggest? How about the next? Each time, I call on a new student and he/she successfully gives me the next biggest pen. I can't help but think: Yes! They're really getting this!
When finished, I ask: What is the answer to my fencing problem? What it the largest rectangular dog pen I can make using 16 feet of fencing? Students respond, "The 4 x 4 is the biggest dog pen."
Same Perimeter, But Different Area
I want to draw attention to the fact that rectangles can have the same perimeter, but different areas. I ask: So all of these arrangements have the same perimeter, but different areas? Are you sure? How can we know for sure?
I pull out a 16 inch string that I had cut ahead of time. If this string is 16 inches long and each square on my grid paper is a 1 inch square, how can we use this string to check our work? After discussing, I model how the string fits perfectly around each arrangement. After checking each arrangement, the students respond with astonishment. Here, students recreate this demonstration: String as Perimeter.
Looking for Patterns
I want to engage students in Math Practice 7: Look for and make use of structure, so I ask students: Do you see any patterns?
I love listening to student responses as they build with complexity. At first, they saw that one dimension increased by one while the other dimension decreased by one: Noticing Patterns 1. Then, another student noticed that the wider arrangements have a larger area: Noticing Patterns 2.
Finally, I listed the arrangements on the board as below:
After several minutes of students talking and whole class discussions, students notice: "the numbers we're multiplying add up to 8." To represent student thinking, I drew this picture, Complex Pattern 1, and ask: So you're saying that one side + the other side always equals 8? Why does it equal 8? A student points out, "Because two sides is half the perimeter and 8 is half of 16."
Applying the Pattern
Often times, students discover patterns that are only applicable to one scenario. It's important to teach students to see if patterns "always work." I decide to provide students with two other situations to test. I explain: I wonder if this always works! What if you're finding all the arrangements when you want the perimeter to be 10ft? What would you do? Students came up with the following: Complex Pattern 2.
To see if we can "trust this pattern," we apply the same conjecture to a perimeter of 12, Complex Pattern 3.
Drawing a Conclusion
Do you think we can draw a conclusion about finding rectangular dog pens with a specific perimeter? One student says, "It will always equal 8 if it's 16." I respond: And 8 is what of 16? Students respond, "Half!"
Through class discussion and some teacher direction, students came up with the following conclusion: When we add one length and one length (On the board, I wrote L + W) it equals... 1/2 of Perimeter (Complex Pattern 4).
To provide students with further practice with this concept, I explain: I've been thinking. I wonder if my dogs need a bigger pen. What if I had 24 feet of fencing? Can you help me find the biggest arrangement?
I hand out an 11 x 17 "poster" paper to groups of 2-3 students, along with larger grid paper (similar to my teacher model). Again, students were highly engaged, ready to find the biggest pen possible!
I also provide students with a piece of string longer than 24 inches so that students would have to cut the string in order to check that each rectangular dog pen had a perimeter of 24 inches.
I watch for students using a "guess and check strategy" and students who were looking for patterns. I am happy to see this group using patterns: Using Pattern to Solve.
Here, Making Precise, students explain how they are making their work more precise by carefully labeling and numbering. I love watching them take such pride in their work.
At the end of this activity, we post student work, Example 1: Student Work, Example 2: Student Work, Example 3: Student Work, on the board. I celebrate students who used patterns, students who were precise, and students who checked the perimeter using the string.