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Identifying perfect squares

I’ve been doing the distance formula in Geometry this week.

I started it off with an accelerated version of my 7th grade pythagorean lesson. Basically, a quick discussion of this applet, followed by the second worksheet from this post, and finally the application to to the coordinate plane.

It was a great lesson. There were a months worth of Aha!s, and the kids were happy to actually kind of get why it worked, rather than just plugging numbers into a formula.

Their big hangup was getting the square roots. More often than not, the problems they’ll see on standardized tests involve pythagorean triples. Sometimes not, but when they do end up with a perfect square, they’ll need to be able to make a quick guess at it.

The big improv moment was teaching them how to make an educated guess:

1 - 100 - 1
2 - 400 - 4
3 - 900 - 9
4 - 1600 - 6
5 - 2500 - 5
6 - 3600 - 6
7 - 4900 - 9
8 - 6400 - 4
9 - 8100 - 1

The second column gives the range to guess for the 10’s digit, while finding the last digit of the number in the third column gives a hint as to what to guess for the ones digit. As an example – 5184 is between 4900 & 6400, so the 10’s digit will be a seven, so the two possible guesses (if it is indeed a perfect square) are either 72 or 78. Since it’s on the low end of the range, you try out 72, and if that doesn’t work, you know you have an imperfect square. The biggest danger was the tedium of having to do multiple guesses to find a square root, after learning how to make this table, a lot of them were getting very good first guesses, and were quickly able to tell if it wasn’t a perfect square.