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 ``` From: Student 1, age 9, 4th grade To: GeoPOW Staff Subject: GeoPOW: Congruent Rectangles ``` Answer: ```The perimeter of the larger rectangle is 114 units. ``` Explanation: ```First I calculated the area of a small regtangle. This is one seventh of the area of the larger rectangle, so it is equal to 756/7 = 108 units2. I call the long side of the larger rectangle the base, and its short side the height. The long side of the small rectangle is one third of the base. Looking at the top side of the larger rectangle, I noticed that the short side of the small rectangle is a quarter of the base. So we can say: (base/3)x(base/4) = 108 From this I worked out basexbase = 3x4x108 basexbase = 3x4x2x54 basexbase = 3x4x2x2x27 basexbase = 3x4x4x3x9 basexbase = 9x4x4x9 So I found that base = 36. Because basexheight = 756, I could calculate height = 756/36 = 21 units The perimeter of the larger rectangle is 2x(36+21) = 114 units. ```
 ``` From: Student 1 ``` ```[The student looked at answer 1] ```
 ``` From: Student 1 ``` ```Actual written comment: First I calculated the area of a small regtangle. This is one seventh of the area of the larger rectangle, so it is equal to 756/7 = 108 units2. I call the long side of the larger rectangle the base, and its short side the height. The long side of the small rectangle is one third of the base. Looking at the top side of the larger rectangle, I noticed that the short side of the small rectangle is a quarter of the base. So we can say: (base/3)x(base/4) = 108 From this I worked out basexbase = 3x4x108 basexbase = 3x4x2x54 basexbase = 3x4x2x2x27 basexbase = 3x4x4x3x9 basexbase = 9x4x4x9 So I found that base = 36. Because basexheight = 756, I could calculate height = 756/36 = 21 units The perimeter of the larger rectangle is 2x(36+21) = 114 units. ```
 Problem Solving Notes interpretation Solves the problem completely. strategy Very efficient and effective method. accuracy All his arithmetic is correct and is presented correctly. Communication completeness Talks about every step that he took to find the answer. clarity He could explain it a little more clearly - slightly confusing. reflection Didn't do any reflecting - his comment is a repeat of his answer.
 ``` From: [the mentor] To: Student 1 Date: Sep 8 2002 6:20PM Subject: Re: GeoPOW: Congruent Rectangles - posted 09/02/02 ``` ```>The perimeter of the larger rectangle is 114 units. Hi [Student 1]. You've done a pretty good job with this problem - your math is right on. But you could work on your explanation a bit. I've made some suggestions below that might help you make it a bit stronger. We emphasize the explanation in this project, with a goal for you of writing an explanation that would help another student learn how to solve this problem and problems like it. >First I calculated the area of a small regtangle. This is one seventh >of the area of the larger rectangle, so it is equal to 756/7 = 108 >units2. Say more here - how do you know they all have the same area? It's worth reminding the reader what "congruent" tells us about the rectangles. >I call the long side of the larger rectangle the base, and its short >side the height. The long side of the small rectangle is one third of >the base. >Looking at the top side of the larger rectangle, I noticed that the >short side of the small rectangle is a quarter of the base. I like how you point out "looking at the top side of the larger rectangle". You might have done the same thing for the previous part, where you say that "the long side of the small rectangle is one third of the base." How do you know that? What should the reader be looking at so that they say, "Oh, yes, of course it is"? >So we can say: >(base/3)x(base/4) = 108 >From this I worked out >basexbase = 3x4x108 >basexbase = 3x4x2x54 >basexbase = 3x4x2x2x27 >basexbase = 3x4x4x3x9 >basexbase = 9x4x4x9 > >So I found that >base = 36. I think that the section above would be easier to read if you spaced things out a bit, as in (base/3) x (base/4) = 108 And instead of "From this I worked out", explain how you worked it out. What did you do? I like the way you found the different factors and moved them around to find the answer. But I think that instead of going from base x base = 9 x 4 x 4 x 9 right to base = 36, it might be clearer to write base x base = 36 x 36. Then it's really easy to see how you got what you did. >Because basexheight = 756, I could calculate >height = 756/36 = 21 units > >The perimeter of the larger rectangle is >2x(36+21) = 114 units. You might also say a little bit more about what perimeter is and how you find the perimeter of a quadrilateral. One last part that we are asking students to think about this year is to be more "reflective" when you are "finished". There is some information about this on the web page I pointed you to, but basically parts of that might include checking your answer somehow, talking about whether or not your answer seems "reasonable", or maybe even finding another way to solve it that will confirm your answer. This part is hard, but we are hoping that we can help students get better at it. Thanks for your submission, and I am looking forward to your revision. -[the mentor] Scoring Summary: Problem Solving Interpretation: Practitioner Problem Solving Strategy: Practitioner Problem Solving Accuracy: Practitioner Communication Completeness: Practitioner Communication Clarity: Apprentice Communication Reflection: Novice (Read more about these scores at http://mathforum.org/pow/scoring.html.) -- [the mentor], for the Geometry Problem of the Week http://mathforum.org/geopow/ ```

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Annie, annie@mathforum.org
December 2002