From: Jeanne (for Teacher2Teacher Service)
Date: May 16, 2004 at 22:38:21
Subject: Re: Manipulatives for square roots

Hi Paul,

I thought I'd share a different approach than what Suzanne has shared.
I've used geoboards and the Pythagorean theorem to help students see 
sqrt(18) = 3sqrt(2) and sqrt(20) = 2sqrt(5).
 _
|_|  Using the Pythagorean Thm, students can see that the length of the
diagonal of a 1 by 1 square is sqrt(2).

 _ _ _
|_|_|_|  Using the Pythagorean Thm, students can also see that the
|_|_|_|  length of the diagonal of a 3 by 3 square is sqrt(18).  BUT
|_|_|_|  draw this diagonal on the adjacent picture, and hopefully, you can
also see that this diagonal is exactly the same length as the diagonals of
three 1 by 1 squares or 3sqrt(2).
 _
|_|  Using the Pythagorean Thm, students can also see that the length of
|_|  the diagonal of this 2 by 1 rectangle is sqrt(5).

 _ _
|_|_|  Using the Pythagorean Thm, students can see that the length of the
|_|_|  diagonal of this 4 by 2 rectangle is sqrt(20).  BUT draw this
|_|_|  diagonal on the adjacent picture, and hopefully, you can also see
|_|_|  that this diagonal also has the same length as the diagonals of two
2 by 1 rectangles or 2sqrt(5).

I can't use the geoboard to show that the sqrt(24) = 2sqrt(6) because I
can't express either 6 or 24 as the sum of two perfect squares. (20 = 16+4;
5=4+1; 18=9+9)

Hope this helps.


 -Jeanne, for the T2T service


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