Friday, December 01, 2006

Next Lesson: Properties of Square Roots

I'm catching up a bit - this was Thursday's lesson...

The homework due for this lesson was a review/exploration of the multiplication and division properties of square roots. During the Do Now, students practiced this. I gave them problems that couldn't be simplified also (i.e. root 3 + 2root 5) because students often confuse the properties and apply them to addition and subtraction as well. They seem to need to hear about it quite a few times before it sinks in.

After this, I did some direct instruction. I started with some historical background on radicals which I just read about recently. The checkmark part of the radical sign is thought to be a manuscript form of the letter "r", which of course stands for "radix", Latin for root. It was first used in the early 1500s. What I didn't know was that the extended top bar is the last remaining holdover of the old notation of grouping, called the vinculum. I decided to explain this to students so they would understand why "you do what's in the radical first", and it actually did seem to click for some of them.

Following this, I showed them how to reduce square roots by finding the prime factorization and pulling out pairs of like factors. Or, as one student recalled from his 8th grade teacher, it's like two convicts trying to escape from jail, where one of them gets out but the other gets shot. I tried to go for a more mathematical explanation... We also looked at rationalizing the denominator, which was new to them. This is a holdover from the days before calculators, when dividing by a radical was quite a chore. But, the concept of rationalizing is very useful in all those fun limit problems in calculus, so I thought we should still go ahead and learn it.

After this, there was pair work practice, with lots of different types of problems combined together, the hardest being simplifying fractions like (5 + root3)/(4 - root6).

I'll post this lesson on ILoveMath.

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