We are a few days into a mini unit on area and perimeter, and we have been doing a number of "hands on" investigations. The CCSS states that fourth graders need to:
"Apply the area and perimeter formulas for rectangles in real world and mathematical problems."
It would be very easy to look at this and think, "Man--the area and perimeter formulas are pretty simple . . . I'll teach them those, do a few problems about carpet and grassy fields and we should be good to go!" The simple fact is, we must remember that true understanding of mathematical concepts involves careful study, a variety of investigations, and applying new learning in a variety of ways. Over the last week we have built shapes with one inch times, used cheese crackers to find rectangles with certain perimeters, and used grid paper to explore different perimeter and area questions.
Today I decided to ask students to apply some of the learning we have done to moderately open ended task. Here is the task I gave them:
Students could use the open space at the bottom of the investigation sheet or sheets of grid paper to work on finding the 3 rectangles that would fit the rules. Students busied themselves guessing and checking, sketching and figuring, and then asking someone to "prove" whether or not the rectangles they found did, indeed, fit the rules. I walked around asking questions and having students prove to me with numbers and/or drawings that their three rectangles worked.
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| Some students crossed off efforts that weren't successful. Others wrote comments like "too small" or "perimeter too large". |
Things went along pretty smoothly for most, but we did have some work to improve along the way! Two students were really struggling with the guess and check method and were having a hard time visualizing the different types of rectangles that can be made with any given perimeter. I asked, "What do you do when you get stuck? You've already tried a picture--now what?" and after a short pause . . .
Victory! We want students to be able to draw from a wide range of strategies to help them get "unstuck" and that's just what happened. Within a few minutes, the tiles magically arranged themselves into a great solution!
Another student was really struggling keeping track of perimeter as he counted. He had learned a good strategy . . .
but he kept losing count and was getting a different answer each time. I asked him how he thought he could better keep track so he didn't keep losing his place. He really didn't have any ideas, so I gave him a little think time. Eventually I asked, "What do I do when I am solving a problem that is a little tricky or has lots to remember?" He lit up and said . . .
"You write stuff down!"
. . . and so he did!
Success! This simple strategy helped him see that he did, indeed, have the right answer! He just couldn't keep track of it in his head.
So here's where it gets fun! As a few students began to finish, I called a time out and brought the group together to explain the next steps. (You thought we were finished, right?) We talked about the measurement unit we just finished, and I explained to them the next step would be for them to BUILD their rectangles with 1/2 inch wide strips of paper. We reviewed the definition of a rectangle and I asked the students to experiment a little with using the strips to build exact models of the rectangles they had just found.
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| They measured . . . |
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| and cut . . . |
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| and arranged in rectangles! |
NO! I am a HUGE believer in taking pride in one's work, so guess what we did? We glued those rectangles down to make geometric art and we displayed them proudly in the hall for all to see! Check out some of our finished work!
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I love to find ways to combine problem solving, art, hands on learning, questioning, and teamwork, and today was a blast! One of my easily frustrated students said as he hung up his work, "This project was the perfect amount of challenge for me!" I couldn't agree more!
I can't help but remind you that it is Teacher Appreciation Week! Today is the first day of the big sale over at Teachers Pay Teachers, and we'd love for you to check out all the resources we have waiting for you! Sunday's blog post right here on ATUE has one featured product from each one of us, and we know you'll love them. We hope you have a peaceful week at school--and know that we APPRECIATE each and every one of you as you take time out of your busy day to see what we have going on!
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