Our very first skill in Algebra 2 this year was this:
IN1: I can simplify radical expressions.
Even though it isn't particularly related to the rest of the first unit (it's called "Introductory Algebra Skills", but it's really functions and transformations, focusing on linear and absolute value functions), there are a number of reasons I wanted to start with this skill:
- In the change over to the new (now one year old) Oklahoma Academic Standards, this content migrated from Algebra 2 to Algebra 1. Meaning my students (who mostly took Algebra 1 two years ago) missed out. So, starting with this is one way of bridging the gaps between the two subjects and the two sets of standards.
- To a certain extent, this topic stands on its own, so I saw it as a way to get into our course quickly and let my kids start to get the feel of the rapid pace required in Algebra 2.
- The skill is not particularly difficult, assuming students have a strong grasp already of concepts like radicals, exponents and prime factors. This way, I get to quickly evaluate their understanding of these concepts, and hopefully rectify anywhere they've gone astray.
- Within the next few weeks, we'll be graphing functions such as y = x² - 8. I would like for them to give me the x-intercepts as ±2√2.
I know that writing out all the factors individually isn't the fastest method, but I'm a strong believer in the idea that we shouldn't give students the fastest method first. In this case, there are a lot of shortcuts, but I'm consciously not teaching them explicitly. I'm certainly dropping a lot of hints that they exist. But if a student really understands the principles here, they'll be able to find those shortcuts themselves. If a student isn't able to find those shortcuts, that tells me they still need to develop their understanding of radicals until they can.
There are two key reasons why I think this method is solid, despite its slowness. Firstly, it always works for this type of problem, no matter how many factors the number has, or how many variables we throw under the radical. And secondly, it exposes what's happening underneath all those strange mathematical symbols. Radicals remove repeated multiplications (or "exponents" as we like to call them). I want my students to see that repeated multiplication in their mind any time they see an exponent, even if they haven't written it out.
I feel like this second page could be improved a bit. I state the "rules" for adding and subtracting radicals, but there's not much in the way of justifying why these rules exist. Our class discussion did tease this out a bit more. But I think in future I want to put more emphasis about why addition and subtraction are possible to simplify in some situations but not others.
These are my thoughts right at this moment. We can simplify when we add and subtract between things we know the relative size of. For instance, we know 3x and 2x are respectively 3 and 2 times the size of x, regardless of whatever x happens to be. So we can add them and get 5x. But we can't add 3x and 2y, because we don't know how x and y relate to each other. (If we happen to actually know how they relate to each other, it turns out there is a way to add them.)
Same thing with √2 and √5, or ³√2. We don't know how they relate. We could always express them as decimals, but that's the thing we're trying to avoid so we can maintain an exact value (and that's trivial with a calculator anyway, and ∴ boring). 3√5 and 2√5 are related, on the other hand, so we can add them and get 5√5.
On the inside of each of these foldables is a blank grid. These are spaces for students to write practice questions in. For this sort of thing, I usually write a few questions on the board for them to fill in here. I like to make them up on the spot, because it lets me gauge how the group is going and adjust the examples I think they need to see. (I'll admit, this does sometimes backfire as I'm standing at the board, and my mind goes blank.) I don't typically give them enough questions to fill the whole page, but I encourage them to copy in any other practice questions they want to keep as examples, off of worksheets or wherever else. I like to remind them that their INB is their only reference they have access to on quizzes, so this is good motivation for some of them.