If you want students to master something–call it R–and R is a means to S, then work on S; R will slip unobtrusively in under the radar….” -Bob and Ellen Kaplan, Out of the Labyrinth, p. 87

I need an overarching theme, question, or mission for Algebra 2 that transcends and motivates the required skills.

The content goals of Algebra 2 are to invert, transform, solve, and apply graphical, tabular, and analytic representations of linear, quadratic, exponential, sinusoidal, and rational equations. That’s R.

What’s S?

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[…] I could make Algebra 2 into a modeling course […]

To me, it is the application of those items that is the s. Like the way a rocket or a basketball lands in the hoop. I have a document of all these different types of data that shows up into those forms. I can send it to you if you want.

Hello Telannia,

Could you send me that document. I am an Algebra 2 teacher and I would love to have it.

Thanks for the offer, Telannia – I have a large collection of these, too (though not in one document – that’s cool!). So what would be the single question (or two) you would pose at the beginning of the year that students would be answering as they did all those applications? How would you phrase that overarching goal so it was intriguing enough to inspire 10 months of effort?

[…] I count myself fortunate to have never taught Algebra II, the current configuration of which in California probably violates some of the conventions and treaties the US has signed w/r/t human rights abuses. […]

It’s got to be something along the lines of “How do we describe a world that’s almost never linear?”

That’s not a student friendly question, but that’s what Algebra 2 meant to me when I taught it.

This seems like a good answer.

If you’re looking for a hook, maybe something along the lines of (example 1: linear, example 2: linear, example 3: non-linear). Then maybe you can get some WTF reactions out of the students and capitalize on that to get a more intuitive sense of what “non-linear” means.

I have stopped just reading the awesome things everyone is doing and start contributing my efforts in all of this. I have been writing about the process of creating Essential Understandings for Alg 2, and the next step is coming up with methods of the learners demonstrating their understanding. You can find the ongoing efforts at http://blog.mrwaddell.net.

I am at a standstill at the moment waiting for the district to come out with their blueprint and standards for next school year. They are changing it bcs of the Common Core rollout. As soon as that document is in my hands, I will be updating things again on the Alg 2 front.

Glenn: Looks great! My own skills list for Algebra 2 has been going through the same flip-flop: organize by function family or by skill? I’ve ended up thinking of it as a matrix: function family x skill ….

I was at a workshop where the guy said “Functions are at the heart of Intermediate Algebra” (the college equivalent of Algebra II). I liked that. It’s not framed as a question, though. I’d love to be able to frame it as a question. I liked Andrew’s question: “How do we describe a world that’s almost never linear?”

I wonder if there are other questions that get at this.

The only time that I taught Algebra II was as a student teacher, but as a math and physics teacher I would venture to say that it gives you the tools to describe and analyze motion.

One “S” is definitely the description of motion, as was described above… physics classes treat math as the “language” of what they’re doing. A math class that used the physics as the motivation might be an interesting different lens on the same issues.

I’m currently struggling through the same issue in my attempts to rework how I approach my my Alg2 classes. Any progress that is made on this question will help me immensely. Here are some quick thoughts that I’ve had.

* Who uses math? (1) Scientists: (Optics — Rational functions; Astronomy and Climate — Sinusoidal functions) (2) Finance and Business: (Interest, Supply/Demand — Exponential functions) (3) Everyone (Statistics, if you’re in NY). [http://weusemath.org/]

* I like the “describe the world that’s almost linear idea” but it would be better to lead students into the realization that the world isn’t linear. I’m not sure how to fix that. What about using a car as a model, and start the year off with a series of graphing stories challenges. So, we start with a car at a constant speed. Then we go to a car that speeds up, and we graph the odometer against time. Then we graph the motion of the wheels. How do we find the language to talk about these things?

* A series of counter-intuitive problems that are only accessible with the math that we teach ’em. This list is hard for me to pull together. Are some infinities bigger than others? (functions, inverse functions) Which door should I open? (If you have to teach Probability) But I can’t think of any cool, counter-intuitive problems that you need (say) rational expressions to answer.

mbp

I had the pleasure of attending the Greer conference at Philips Exeter two times. One of the times, I attended a session that was taught by a fellow from Minnesota (I can’t recall his name right now) who had a terrific set of problems. His school had developed a series of problems with the theme of “Goat on a Rope” and these problems were of increasing complexity during the course of their curriculum. The goat was tethered to a pole, the goat was tethered to a corner on a barn. He was tethered to a post on a circular pen, etc.

I have wrestled with this idea for a while but I LOVE the idea of starting within the framework of a single course to develop a string of problems on a theme.

Me too. Someday maybe our final exams can read: “Given a goat and a rope, organize them in different ways such that measurements of features of the system have (1) a quadratic relationship, (2) an exponential relationship, ….”

Such great discussion! It’s got me thinking about relationships . What was going to be a quick comment ballooned into a post!

Definitely bookmarking this one. Was just discussing with another fellow math teacher about how much of a potpourri A2 must seem to most kids, and how after teaching two sections this past year (first time through for me as a teacher) that I haven’t really found the hook that binds the subject together.

Some of algebra 2 should be spent mastering solving word problems of various types, which are generally introduced and attempted in algebra 1 with varying degrees of success. There doesn’t need to be a whole lot of time spent. When I took algebra 2, the first few weeks were spent learning the finer points of certain types of word problems. I realize that talking about distance, rate and time problems, mixture problems and work problems is viewed as anachronistic and needless, but I tend to disagree. Vehemently disagree.

As much as I love Algebra 1, I hate Algebra 2. Other than helping students score better on the ACT it is difficult to justify a lot of the Algebra 2 content. I feel so much like we fall into this trap of teaching like we are training little math professors. Yuck.

Thus, the importance of this post. Glenn’s list of topics for A2 is probably pretty typical but as it stands it is a list of skills that is begging for some problems to solve. What problems can we solve if we had those skills? Are they worth solving? Will our students see them as worth solving? If we use the Linear Funtions section of Glenn’s list, then a problem like “Meadows and Malls” from IMP 3 is perfect. It is essentially a linear programming problem in 6 dimensions and includes everything in that list plus some work with 3 dimensional space. For me, this type of problem makes connections and at least presents a plausible use for the topics.

As a tangent here, looking at Glenn’s list makes we wonder why we pretend technology doesn’t exist. What does the classroom look like if topics like 3.2, 3.5, 3.6, 3.7, and almost everything in the quadratics section were handled by the technology so the focus of the class is not skills and procedures, but going after some complex and interesting problems? I probably won’t find much agreement here – I mean students have to be able to factor by hand right? Now what those complex and interesting problems look like, I’m not sure…yet.

@Glenn I am in no way criticizing your list – just using it as a point of reference. Thanks for having it there!

Sorry that I’m coming to this a little late to the game, but I share the frustration of many posters here about teaching Algebra 2. I love the topics taught in Algebra 2, but there is no coherence to them. And the only reason that I love the topics is because SOMEHOW I was able to perceive the “forward flow” – when I learned them, I was dimly aware that there would be deeper results about the way the world actually was coming down the pike. And that I would need to really understand these disconnected topics in order to perceive that coherence later. So, I was patient, and put up with the shotgun nature of the course I was taking as a high school sophomore, and then did it again (more deeply, but still) as a junior in precalc, and _then_ to calculus. To calculus!

Now that I’m teaching, I know in my heart that “to calculus” is not S.

I think S is the connections that exist between the topics that are taught in A2 and the deeper understandings about the world that are possible as a result. But the standard A2 curriculum never allows for those real connections to develop, and that’s a tragedy. I think technology is going to finally allow some of that to start to happen (e.g., Wolfram|Alpha, GeoGebra), but until the top-down framework of “what must be taught” by the Common Core folks is obliterated, those connections are never going to be given the proper time and space to be developed for our students.

(As an aside, I am constantly amazed at the value of blogs….both from the standpoint of what the original poster puts out there (thanks, Dan) and the comments from the thoughtful responders!)

[…] all my talk about Algebra 2, it has worked out that this year I’m teaching precalculus instead. So I’m trying a […]

I’ve been thinking I should reply to this one for a long time. I certainly wonder if someone out there has come up with anything further after the initial flurry of conversation!

There’s a nice book by Richard Sisley called Data, Models, Predictions that pretty well summarizes what I see as the overarching theme of this course: you get some data, you realize your existing tools aren’t going to make a good model for it, so you learn a new family of functions and use it to model the existing data or predict future data.

This course incorporates a fair bit of trigonometry and statistics along with a lighter dose of polynomials and the usual amount of exponents and logs. It’s also got a big unit on the binomial theorem that ties it to binomial probabilities and the resulting approximations to the normal curve. It then uses the normal curve as another family of functions that you can use to model things — not one you usually see in algebra 2, but you can use the same ideas (stretching, shifting, and so on) to work with it, and use it to make some very important predictions if you’re at all interested in surveys!

The integration of the probability and statistics content into the rest of the algebra 2 curriculum is one of the great features of this approach. More to the point of the original post here, I think the S here is “Understand data, build models for it, and predict future data”.

@Joshua. That sounds like a great book. A quick Internet search yielded no results. Any help on where to check out a copy?

I will email the author and find out what the current status of these books is. Maybe he’ll come post here, or I’ll post his email address so you can get a personal copy electronically.

The main difficulty with using material from his books directly is that this book presupposes that you know a lot about transformational geometry and coordinate geometry, since that’s an even deeper unifying theme for the whole four-year sequence through calculus. Students using this book will have had a whole course of “baby linear algebra” where they learn enough about matrices and transformations to do a lot of geometry and to develop an understanding of lines and parabolas.

Joshua, I tried to find that book online and failed. Do you have a link to it?

Dick Sisley, keckcalc@earthlink.net , is the author, and you can probably get an electronic copy from him at a low price (or free). You can also distribute electronic copies to students for a reasonable price.

[…] year, Dan Goldner asked a question that struck me even when I had nothing to do with math teaching: what’s the theme of Algebra II? The comments on this post are excellent, and Dan followed it up an excellent first stab: […]