A Math Forum Project

Learning to Mentor
Replying Example 2

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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