Yeah, it’s sort of a grab bag—a miscellaneous compilation of word problems, review from yesterday, and so on. You can just get them started working on the assignment after you’re done going over yesterday’s homework.

Toward the end, however, they are going to start running into trouble. This is when you introduce rationalizing the denominator. You may want to bring the whole class together to see who can figure out how to rationalize 13+113+1 size 12{ { {1} over { sqrt {3} +1} } } {}. It’s a great opportunity to review our rules of multiplying binomials: ( a + b ) 2 = a 2 + 2 a b + b 2 (a+b ) 2 = a 2 +2ab+ b 2 which is why multiplying by 3+ 1 3 size 12{ sqrt {3} } {}+1 doesn’t work; ( a - b ) 2 = a 2 - b 2 (a-b ) 2 = a 2 - b 2 which is why multiplying by 3- 1 3 size 12{ sqrt {3} } {}-1 does.

But please be very careful here, because this particular topic has a very subtle danger. A lot of teachers communicate the idea that denominators should always be rationalized, “just because”—because I said so, or because somehow 2222 size 12{ { { sqrt {2} } over {2} } } {}is “simpler” than 1212 size 12{ { {1} over { sqrt {2} } } } {}. This is one of the best ways to convince students that math just doesn’t make sense.

What I’m trying to do with this exercise is demonstrate a real practical benefit of rationalizing the denominator, which is that it helps you add and subtract fractions. It’s difficult or impossible to come up with a common denominator without doing this first!

And of course, we have the “you never understand a function until you’ve graphed it” question. Talk a bit about the graph after they get it. They should be able to see that the domain and range are both ≥ 0 ≥0 and why this must be so. They should see that for large values, it grows very slowly (like a log), but without the drastic behavior that the log shows in the 0 ≤ x ≤ 1 0≤x≤1 range.

Homework:

“Homework: A Bunch of Other Stuff About Radicals”

Content actions

Add module to:

My Favorites Login Required (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own Login Required (?)

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks

Footer

This work is licensed by Kenny M. Felder under a Creative Commons Attribution License (CC-BY 2.0), and is an Open Educational Resource.

Last edited by Kenny M. Felder on Jan 10, 2009 1:29 pm -0600.