Sunday, December 16, 2012

Algebra 2 & relationships

We've been working on our algebra 2 curriculum over the past year, both to align it with the Common Core and make it seems less like a random collection of function things. Last year we had the idea to make it very data-based; the idea was to have students gather a ton of data, and then base our study of functions on those data sets.

There are a few problems with this approach. For one, our district decided that prep periods are a luxury that cannot be afforded, so we don't have the time to prepare such a radical departure from our current curriculum.

Second, and more problematically, I think this would put way too much emphasis on the applied part of mathematics, and would make it difficult to meaningfully teach any aspects that aren't as practical. If we're motivating the study of functions based on answering questions about real data, I'm not sure how I'm going to help students understand removable discontinuities. Or motivate them to solve difficult equations algebraically when a graphical method yields a solution quickly. And so on.

Mathematics is not only about answering practical questions about the real world and data. It is certainly useful for that, but it's also much more. Thus a solely data-based approach to teaching algebra 2 would be a mistake, in my opinion.

So we need a more flexible approach if we're going to be true to all of mathematics. Thinking about the essence of our algebra 2 course, and really algebra in general, what we're dealing with is relationships. Linear, polynomial, rational, radical, exponential, logarithmic, and trigonometric, to be specific. We learn about these relationships by looking them in different lights - algebraically, graphically, and numerically.

Currently, a major problem with algebra 2 is that, to students, it feels like a ridiculously long laundry list of things to learn. I get the feeling that they don't understand that the equations, graphs, and tables that we work with represent the same relationships. I try to teach the connections, but they're mostly lost on students. And it's not surprising - the focus of the course isn't on understanding the relationships, it's on the set of techniques and facts about them. Without understanding the relationships themselves.

So, my idea: refocus the course on the relationships. One way to do that might be to start numerically. For example, we might start with the following:
(1, -3), (10, 1019), (-1, -4.5), (?, 59), (2, -1), (3, 3), (0, -4), (0.5, ?) }
I think this would really put the focus on the relationship itself. We're dealing with two quantities, and there is some way to predict one from the other. However, it is difficult to figure that out simply with numbers, so translating the problem into different forms - equations and graphs - takes on more meaning. 

Our new semester starts in about a month; I'll let you know how it goes. Any wisdom you have to offer would be appreciated.


  1. Thoughtful post. I haven't touched Algebra 2 since I was in high school years ago. However, I get what you're saying about relationships. I agree that it's important for students to understand how different concept are related, especially if they represent the same thing in different ways! It seems like there's still room for real-world data in that approach. In that case you're looking for the relationship between real-world data and the mathematics that might seem purely abstract to the students at first. With your time constraints, you don't need to do it all the time, but I definitely think it would still be a valuable experience where you can fit it in. Good luck!

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  2. One way to show relationships is using these "Link" sheets that I found out about at a conference. The teachers at the conference were using lingo like "Rule of Four" which I had never heard of, but it makes sense: on a single sheet of paper, for a single function, show four representations of that function: (equation, graph, table, and word problem/questions). The Algebra II teacher at our school and I took the link sheets back with us and I've used them a bunch in Precalculus and even some in Physics. I've got a whole folders of Algebra II link sheets from this conference that I've never used because I only teach Precalculus, and I can send them to you if you'd like!

    1. Thanks for the comment - I really like the idea of those sheets! Our new semester starts on Tuesday, and we're looking to make more than a couple of changes to algebra 2. I would love to see those sheets - could you send me a link or email them to me? (Gmail: kevin.krenz)

    2. Just sent you the documents. Oh, and you should add "Desmos" to your list of awesome, free, cross-platform (browser based) technology. It's almost like having a fast, free TI Calculator on your computer, but better because things like window re-sizing and finding intersection points is way easier than on a TI.