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.
Students at the 4th grade level need to be able to apply their understanding of area and perimeter to solve real world problems. Instead of pulling a problem out of textbook (which is often unrelated to student lives), I want students to truly experience the necessity of math in real life by solving a problem closer to their hearts.
To increase student engagement, I show students a picture of my dogs, Jedi and Jozie. Students are immediately excited and several move up closer to the screen!
I tell students that I need help solving a fencing problem in my backyard, but I wanted to first show them a video (Backyard Video of Dogs) to help them understand the needs of Jedi and Jozie. Students were sitting on the edge of their chairs at this point! Throughout the video, they respond with Awww...., side comments, and giggles.
Then, I explain that sometimes my dogs wander off to the other side of the yard where I can't see them. This worries me because we have eagles and hawks in our area. I want an enclosed dog pen that keeps Jedi and Jozie where I can see them!
Goal & Introduction
I begin by revealing today's goal: I can explain the difference between area and perimeter. I continue: For the past couple of weeks, we have learned all about area. Today, we will turn our focus to include perimeter as well. By the end of today's lesson, I want you to understand and be able to explain how area and perimeter are different measurements.
Vocabulary & Song
To review the concept of perimeter, I ask students to turn and talk: What is perimeter? This is also my opportunity to listen for any misunderstandings or to identify students who don't know. After a couple minutes, students share ideas, such as "It's on the outside."
I share and explain the Perimeter Vocabulary Poster. 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: Perimeter: The measurement (acting out a measuring tape) of the outside (making rectangle with hand and pointing to the distance around the hand).
Next, we practice the new vocabulary word several times. To review the meaning of perimeter, throughout the unit I say, turn and talk: What's the difference between area and perimeter? Students will use the hand movements to recall the definition!
We also take the time to review our Area Vocabulary Poster and I teach students the next part or our Area & Perimeter Song. I have found that singing math songs provides one more layer of instruction that helps solidify a concept.
Then, I give each group of 2-3 students a bag of popsicle sticks. I know that smaller groups of students allow all students more access to math experiences.
Instead of asking students to find the area of a polygon with a perimeter of 12 units, I want students to begin discovering and creating mathematical conjectures about perimeter in a real world setting. So, I begin by saying: Today, I'm going be giving you fence panels to construct different size dog pens for Jedi and Jozie. First, I'd like each of you to take out one fence panel (a popsicle stick) and make a rectangular dog pen with a perimeter of one unit (fence panel). Students are slightly confused at first and finally decide, "That's impossible!"
I then ask students to create a dog pen with a perimeter of two units.... and then three. Students were ready this time, "We can't do that!" Finally, I ask them to make a dog pen with a perimeter of four units. Students catch on quickly, constructing a square with four sticks. One student asks, "Does a square count?" This is the perfect opportunity to remind students that a square is a special kind of rectangle.
We begin to document the dog pens on our class Anchor Chart. Students also create the same chart in their journals: Example 1: Student Chart, Example 2: Student Chart, Example 3: Student Chart. I want students to look for patterns between each dog pen and engage in Math Practice 7: Look for and make use of structure! Later on, I'll encourage students to look for patterns by asking them what they notice.
Next, students experiment with constructing a dog pen with a perimeter of five units (impossible) and then six units of fencing. One group made a 1 x 2 dog pen. We discuss other possible arrangements. Another student explains that we can make a 2 x 1 dog pen. Then, students agreed that 2 x 1 "is the same" as a 1 x 2.
It was at this point that I asked, What are you noticing about area and perimeter? One student says, "As the perimeter gets bigger, the area gets bigger." Another student points out, "You have to have an even number to make a rectangle." I start another chart called Conjectures and I add both of these students' conjectures. I explain: Conjectures are conclusions that we can draw based on the evidence we know so far. They can be proven correct or incorrect! Be watching for evidence that either proves or disproves any of our conjectures! We will add to and modify this chart throughout the week.
Moving on to constructing a dog pen with a perimeter of 8 units, some students come up with a 1 x 3 while others create a 2 x 2. I am excited to see this as I want students to begin to see that rectangles can have the same perimeter, but different areas.
Next, students move onto creating a dog pen using 9 units of fencing (impossible).... and then 10 units of fencing... After students have time to experiment with each, we discuss their findings as a class. One student shares how he made a 1 x 4 dog pen using 10 units of fencing while another student explains how she made a 2 x 3 dog pen using 10 units of fencing.
At this point, I ask students to continue using their popsicle sticks to create rectangular dog pens for Jedi and Jozie independently with their partners. This time, I challenge students to construct the largest dog pen with a perimeter of 12 panels of fencing (12 popsicle sticks).
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).
During this student conference, Confusion Between Area & Perimeter, it is clear to me that students are still developing a clear understanding of area and perimeter. This is why it is so important to teach multiple lessons and to teach these concepts with depth!
In this student conference, Comparing Fenced Areas, the student explains how two pens can have different areas, but the same perimeter. I love how she notices the more narrow the pen, the smaller the area.
To gain further information on my student's understanding of the lesson concepts, I ask them to respond to the following prompt in their journals: Area and perimeter are very different. I encourage students to explain each and to provide examples. Here's an example of a Student Explanation.