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 & Lesson Introduction
I started with our two-day goal: I can find the perimeter and area of odd-shaped dog pens. I explain: Today, you will focus on finding the perimeter of odd-shaped dog pens and tomorrow, you'll focus on finding the area!
I continue: Do dog pens always have to be rectangular? Why might I build a dog pen that isn't a perfect rectangle? Turn and talk! We then discuss their thinking as a class. Some students suggest that I might have to work around a tree, garden, swimming pool, or other obstacles in the backyard.
I invite students to move closer to the board with their white boards, markers, math journals, and a pencil. I want to make sure I have the attention of all students! Prior to the lesson, I drew the Goal & Chart as well as the Teacher Model A dog pen (Modeling on White Board). Without my asking, students began writing the goal at the top of a new page in their math journals. I celebrate these students: I just love how _____ is writing the goal at the top of his paper! Soon, all students are completing this task.
I also ask students to recreate the chart in their journals as we will be using it later on.
Horizontal & Vertical Lines
We discuss the difference between horizontal and vertical lines. I first show the Horizontal Vocabulary Poster and we act out a horizontal hammock together (we acted out a horizontal hammock with a straight arm rocking back and forth). Then, I show the Vertical Vocabulary Poster. Again, we acted a vertical vine out together (we acted out an up and down vine with one arm). Turn and talk: What's the difference between horizontal and vertical lines? Students act out each vocabulary word as they explain the difference. I am filled with a feeling of pride that my students listened so well!
Looking for Patterns
I explain: Today, we are going to be looking at this odd-shaped dog pen together. I point to Teacher Model A, drawn on the board. I purposefully made the horizontal lines blue and vertical lines green to match our vocabulary posters. I also used very simple numbers as I want students to focus on the patterns and process of finding unknown numbers instead of being challenged with the calculation of larger numbers.
I ask: Does anyone see a pattern? One student suggests: It goes 1...2....3...4....skips 5... then 6. Another student points out: 1 + 2 = 4, 1+ 3 = 4, 2 + 4 = 6. And another student says: All the horizontal lines are blue. And all the vertical lines are green.
I wrote "Patterns" up on the board and began listing some of their thoughts: Patterns.
To push their thinking further, I ask: Does anyone see a relationship between the green lines? Turn and talk! After a few minutes, we come back together and finally, one student points out the pattern I've been hoping they would notice: The two smaller horizontal lines equal the bottom line... 1 + 3 = 4.
Then another student excitedly says: And the two smaller vertical lines equal the bigger vertical line.... 2 + 4 = 6.
It was at this point that I asked students to turn and talk: Can you explain the patterns that (student's name) and (student's name) pointed out? I do this for several reasons. 1. I want to make sure students understand. 2. I want students to practice communicating mathematical reasoning. 3. I want any struggling students to hear the pattern a second time.
Finding Unknown Sides
I then erase the side measurements and create Teacher Model B. I purposefully make this model one step harder. I want to build a clear and understandable learning progression. I label the unknown sides with variables (letters that represent unknown numbers). I then review the Meaning of Variable on the board.
Next, I ask students to use their whiteboards to show me how they found the missing sides. While we discuss student solutions as a class, I write an Algebraic Equation to show students another way of showing their thinking. Students quickly caught on and were excited for the next "challenge!"
To provide students with guided practice, I change the dimensions of the sides and monitor students as they "solve for x and y." Sometimes I use different variables and ask for student input: Who would like to help me come up with the next variable? Of course, students offered the first letter of their names!
Teacher Model C
Teacher Model D
Next, I explain: For each of these odd-shaped dog pens, we have had to add to get lengths of the missing sides. Sometimes you'll need to subtract as well. I create Teacher Model D. As students calculate the missing dimensions, I conference with and support students during this time. Again, we discuss student solutions solutions as a class and move on to the last couple of challenges!
Teacher Models E & F
Following the same process as above, students solve and discuss Teacher Model E and Teacher Model F. Here, you'll see how I model the student's thinking on the white board: Teaching Model E on White Board and Teaching Model F on White Board.
At this point, students are ready and excited to apply the patterns they discovered while finding unknown side lengths!
Prior the lesson, I made copies of Odd Shaped Dog Pens and cut them into half-sheets. I place these in order on the white board tray.
By starting off with a simpler task, students were able to develop the skills necessary to complete the more challenging tasks.
I assign partners to students, taking into consideration ability levels, communication skills, and behavior.
I explain: I've come up with several odd-shaped dog pen arrangements for Jedi and Jozie, but I want to know which one will take the least amount of fencing. Pointing to the last model on the board, I continue: If I want to build a fence around this odd-shaped dog pen, would I find the area or the perimeter? Students agree... in order to find the amount of fencing, we will find the perimeter because the perimeter is the distance around the outside.
I model how to fill in Chart using the most recent teacher model on the board. Equation: 24 + 8 + 16 (Vertical Sides) + 16 + 4 + 12 (Horizontal Sides)
Students jump right in. They solve the first odd-shaped dog pen, fill in their charts, and return to the board to get the next odd-shaped dog pen! Here are examples of student charts: Student Journal and Student Journal 2 during this time.
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).
At the end, students discover which odd-shaped figure will take the least amount of fencing: Least Amount of Fencing.