*1. Take the seven rectangles and lay them out in front of you. Look at their perimeters. Do not do any measuring; just look. What are your first hunches? Which rectangle do you think has the smallest perimeter? The largest perimeter? Move the rectangles around until you have ordered them from the one with the smallest perimeter to the one with the largest perimeter. Record your order.*

Here’s my hunch, from smallest to largest perimeter:

C

E

D

B

F

A & G (tied)

*2. Now look at the rectangles and consider their areas. What are your first hunches? Which rectangle has the smallest area? The largest area? Again, without doing any measuring, order the rectangles from the one with the smallest area to the one with the largest area. Record your order. *

Here’s my hunch, from the smallest to the largest area:

C

D

B

F

E

A

G

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*3. Now, by comparing directly or using any available materials (color tiles are always useful), order the rectangles by perimeter. How did your estimated order compare with the actual order? What strategy did you use to compare perimeters? *

I used the color tiles as my nonstandard unit of measurement, and recorded the length and width of each rectangle. Here’s what I came up with:

A: length = 5, width = 3

B: length = 6, width = 2

C: length = 8, width = 1

D: length = 2, width = 5

E: length = 3, width = 4

F: length = 7, width = 2

G: length = 4, width = 4

To calculate the perimeter, I used the equation 2 * (length+width):

A = 2 * (5 + 3) = 16

B = 2 * (6 + 2) = 16

C = 2 * (8 + 1) = 18

D = 2 * (2 + 5) = 14

E = 2 * (3 + 4) = 14

F = 2 * (7 + 2) = 18

G = 2 * (4 + 4) = 16

So the actual order of the rectangles, from the smallest to largest perimeter would be:

D & E

A, B, & G

C & F

Okay, so my prediction was way off! The main one being that I thought C had the smallest perimeter, and it actually was tied with F for having one of the largest! I guess I found the “slenderness” of the rectangle really deceiving!

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*4. By comparing directly or using any available materials (again…color tiles), order the rectangles by area. How did your estimated order compare with the actual order? What strategy did you use to compare areas? *

To calculate the areas of the rectangle, I started with some previous knowledge I had already obtained from calculating the perimeters (each rectangle’s length and width, using the color tiles):

A: length = 5, width = 3

B: length = 6, width = 2

C: length = 8, width = 1

D: length = 2, width = 5

E: length = 3, width = 4

F: length = 7, width = 2

G: length = 4, width = 4

To calculate the area, I used the equation length times width:

A = 5 * 3 = 15

B = 6 * 2 = 12

C = 8 * 1 = 8

D = 2 * 5 = 10

E = 3 * 4 = 12

F = 7 * 2 = 14

G = 4 * 4 = 16

So the actual order of the rectangles, from the smallest to largest area would be:

C

D

B & E

F

A

G

I did much better at predicting the area, rather than the perimeter. In fact, I only made two mistakes: not realizing that E’s area is smaller than F’s, and that E’s is also equivalent to B’s area. Again, the slenderness of F compared to the chunckiness of E made me believe that E had a larger area.

*5. What ideas about perimeter, about area, or about measuring did these activities help you to see? What questions arose as you did this work? What have you figured out? What are you still wondering about? Share your responses to all the questions in 1-5 on your blog. Spend some time discussing your answers with your blog partner. *

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This activity helped me see that certain first impressions of a shape can be misleading to what it’s perimeter or area truly is. For example, I kept thinking that the slender rectangles would have less of an area compared to the other chunkier rectangles. But I guess it’s understanding that it just depends on the how long (the length of) the slender rectangle is, compared to how wide (the width of) the chunky rectangle is.

Another thing I noticed is that rectangle C had the one of the largest perimeters, but also the smallest area out of the set. I think this would be important to discuss with my students if I were doing this activity in the classroom, because it brings up the question: How do we measure what’s larger? One might argue that rectangle C is the largest rectangle because it has one of the largest perimeters, but another might argue that it’s the smallest because it has the smallest area. In the end, I think it all depends on content. For instance, think if these rectangles as swimming pools, and I had to swim around the wall of them. I would say that C is one of the largest swimming pools because its perimeter is one of the largest, and would take me the most time to swim around. But on the other hand, what if I had to fill each swimming pool with as much water as it can hold? Then I would say C is the smallest swimming pool because it would take me the least amount of time to fill it up, since it holds the least amount of water, having the smallest area out of all of the swimming pools. Hopefully this discussion would lead my class to think about the different perspectives of measurement, and the content and why they’re measuring something. It could also lead students to become more precise with their solutions and explanations, stating “C has the largest perimeter in the set”, rather than “C is the largest rectangle”.

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