What does one intend to do as a *follow up* to this question?
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 plane.]
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, firstname.lastname@example.org
From email@example.com Tue Jun 17 14:33:49 1997 Comments: Authenticated sender is <firstname.lastname@example.org> To: email@example.com Subject: Find all the ... Reply-To: firstname.lastname@example.org Priority: normal X-Mailer: Pegasus Mail for Windows (v2.23)
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)