Rectangles with the Same Area and Different Perimeters

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Objective

SWBAT recognize that rectangles with the same area can have different perimeters.

Big Idea

Rectangles that have the same area can have different perimeters.

Think About It

7 minutes

Students work on the Think About It problem in partners.  After 3 minutes of work time, I bring the class back together to share out opinions.  I ask students to vote with their thumbs, showing whether or not they think it is possible to have rectangles with the same area and different perimeters.  

First, I'll ask someone who thinks this is impossible to explain his/her thinking.  I'll then ask someone with a thumb up to explain how this is possible.  I'll show work on the document camera, and call on other students to add to the shared thinking.      

Intro to New Material

15 minutes

To start Example 1 in the Intro to New Material section, I have students turn-and-talk with their partners to list out all of the factor pairs of 20.  After 2 minutes of work time, I have students show me on their fingers how many factor pairs they came up with.  My expectation is that students organize their work in T-charts, as this is how we've organized factor pairs in previous units.  

Next, we draw the rectangles on the grid.  We discuss that there isn't a need to draw the 2 different orientations of the rectangles - a 2 x 10 rectangle will have the same area as a 10 x 2 rectangle.  I ask students to label each side of the rectangles they draw.

As we work through the same process for Example 2, I ask students which dimensions are most reasonable for a garden.  Although a 1 x 18 garden is certainly possible, it isn't likely the most practical layout.  

I then have students determine the perimeters for each of the rectangles, and cold call on students to share which dimensions yield the largest perimeter and which dimensions yield the smallest perimeter. I ask the class why they think the 1 x 18 rectangle has the largest perimeter.  I want students to come to the generalization that the dimensions that are furthest apart are going to create a long, skinny rectangle.  The long side will result in a larger perimeter.  Conversely, the dimensions that are closest together with yield rectangles that have smaller perimeters.  

 

Partner Practice

15 minutes

Students work in pairs on the Partner Practice problem set.  As students work, I circulate around the room and check in with each pair.  I am looking for:

  • Are students determining all the possible dimensions given the area of a rectangle?
  • Are students organizing their work in T-charts?
  • Are students correctly calculating the perimeter of each rectangle?
  • Are students correctly identifying and explaining which dimensions yield the shortest and longest perimeter?
  • Are students using the correct units?

I am asking:

  • How did you determine all of the possible dimensions?
  • How did you know which dimensions would yield the shortest or longest perimeter?
  • What units should you use? Why?

 

After 10 minutes of partner work time, students complete the Check for Understanding problem.  This problem does not have the support of a graph for the rectangles, so I am able to see if students are grasping the idea that a larger difference between the dimensions results in a larger perimeter, and vice versa.  I ask a student to share his/her response to the CFU question, and then allow other students to add to the thinking, to create a strong response. 

 

Independent Practice

15 minutes

Students work on the Independent Practice problem set.  As students work, I check in with each student.  I supply graph paper as a support during Independent Practice, if there are students who need to practice a bit more with this scaffold.

I engage in conversation with students about Problem 7.  I ask students for the logical dimensions of a farm.  There are multiple possible answers here.  For example, both   a 4 x 10 and a 5 x 8 farm are reasonable.  The key point here that I want students to draw out is that a 1 x 40 farm is not very practical.  

 

Closing and Exit Ticket

8 minutes

After 15 minutes of work time, I bring the class back together for a conversation about Problem 8.  Some students may have difficulty with the figures in this problem, because they aren't rectangles.  We talk about how the figures can be encased by rectangular fences.  Once everyone is clear on the shape of the fencing, I have students turn and talk with their partners about which one would be less expensive to fence in.

Students independently complete the Exit Ticket to close the lesson. An exit ticket sample shows what student work can look like.