Rectangles with the Same Perimeters and Different Areas

Students work in pairs on the Think About It problem.  There will be students who attempt to use a factor chart, as was used in the previous lesson, to try to figure out dimensions that would lead to an area of 100, rather than a perimeter of 100.

When we're back together as a whole class, I start by underlining 100 feet of fencing.  I ask students if this would be used for perimeter or area, and demand that justification be given.  I want students to talk about the units and identify right away that we're talking about perimeter.  Some students will also talk about the context of the problem and how fencing typically relates to perimeter. 

While we do not talk through all of the possible combinations of addends for this problem, I do have students discuss what a 1' x 49' field would look like, and whether or not it'd be practical.  Students figure out the areas of the fields that are 1' x 49', 2' x 48', and 3' x 47'.  I then have students make a prediction about what the largest field would look like.

To start the Intro to New Material section, I ask students what they know about rectangles.  If I get responses about rectangles that are far from the response I'm hoping for, I'll use 2 colored markers to highlight the top/bottom and left/right sides of a rectangle on the board.  I want students to identify that opposite sides of rectangles are congruent.  Using this information, I'll reference the Think About It problem by asking 'if the perimeter of the field is 100 ft, what would the sum of 2 adjacent sides be?'

I then shift to talk about the first example in this section.  Students identify that for a rectangle with a perimeter of 14, the adjacent sides will have a sum of 7.  We then construct an organized list of all of the addend pairs that have a sum of 7.  Students draw the rectangles on the grid. I ask students what they notice about the dimensions that lead to the rectangle with the largest area.  

I guide students through the second example.

Students work in pairs to complete the Partner Practice problem set.  As they work, I circulate around the room.  I am looking for:

• Are students determining all the possible dimensions given the perimeter of a rectangle?
• Are students correctly calculating the area of each rectangle?
• Are students correctly identifying and explaining which dimensions yield the greatest and smallest area?
• 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 greatest and smallest area?
• What units should you use? Why?

After partner practice, I ask students to share out their strategies for solving problem 2.  Some students will look for dimensions that add to 14.  Other students may have found the perimeter, given the dimensions.  Another part of the discussion involves checking all answer choices, as in this problem both c and d are correct.

Students work on the Independent Practice.  A student work sample is included. As students work, I am looking for organized lists of addends.  There will be students who are able to figure out the largest and/or smallest areas without the lists, and I allow this when the problem doesn't ask for all possible rectangles. 

For the final part of problem 1, some students will claim the farmer can fit 25 pigs in his pen if they are not reading carefully and thinking through the problem.  If I see 25 as an answer, I tell the students they are squishing in all the piggies and suggest they read the problem again.

After independent work time, we discuss problem 9 as a class.  The first part of the problem provides me an opportunity to reinforce the importance of organized work.  I pull a popscicle stick and display the student work for the first part of the problem on the document camera.  I praise the list of addends or give constructive feedback, as needed.

Part B of this problem requires students to recognize that the 6' x 6' room will have the largest area, calls for them to compute the area of the room, and gives them an opportunity to practice multiplication with decimals.

Students complete the Exit Ticket to end the class.  An exit ticket sample is included.