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Topic: Find all the ...
Replies: 5   Last Post: Jun 18, 1997 7:27 PM

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Chih-Han sah

Posts: 75
Registered: 12/3/04
Re: Find all the ...
Posted: Jun 17, 1997 3:29 PM
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The entire problem depends on:

What does one intend to do as a *follow up* to this

For the sake of argument, let us simplfy the question:

Consider only squares.

First the 2 x 2, then the 3 x 3 board.

Here, one does not tell the student what the answer is.

After that, consider 4 x 4 and 5 x 5.

In each case, the student has to answer the question:

How do you know that you have a square?

After that, raise the question, suppose we now have an n x n board?
What do we do? [The general answer is quite difficult. At the K-12
level, one should not expect the students to obtain just a good upper
estimate and understand how to reformulate the problem in terms of
counting lattice points (with duplications) in certain regions of the

Depending on the class, this may be too difficult. However,
in the case of 7 x 7, the students may or may not have
noted that there two congruent squares (each of area 25) which can
be placed in different ways other than translating the vertices
along the edges. This leads to an *over count*.

Han Sah,

From Tue Jun 17 14:33:49 1997
Comments: Authenticated sender is <>
Subject: Find all the ...
Priority: normal
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Regarding activities of the type "Find all the ..." is it better to
tell the students ahead of time how many ... exist or to ask them
to decide when all have been found?


Find all the noncongruent quadrilaterals that can be formed
on a 3 x 3 geoboard. (From NCTM Curriculum and Evaluation Standards
for School Mathematics, Addenda Series, Grades K-6, Geometry and
Spatial Sense, Chapter 7 Sixth Grade, p.49)

Phil Green

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