Wednesday, July 30, 2014

My Work-in-Progress Algebra 2 Sequence (for Next Year)

Last year, I felt pretty good about my Algebra 2 sequence. Some of the students struggled with the formal assessments in the class, but I don't think the fact that they struggled was tied to the way the material was sequenced. (Re-quizzes definitely helped them, but I also saw them learning things like managing their responsibility as the year went on and the material ramped up in complexity.) This year, as usual, I want to do better.

I think that my ideal Algebra 2 sequence, assuming that the students come in only knowing how to solve basic one-variable equations, would look like this:

Unit 1. Review solving equations, clearing fractions, and manipulating formulas. The reason why I would start with this topic is that it allows me to see immediately who is struggling with topics from the years prior, and who isn't. Introducing fractions at this point of the year also gives me an opportunity to keep spiraling back to it in every other topic. A nice problem to use at the very start of the year is the classic pool border problem or any visual pattern that extends linearly (to review the idea of inductive thinking and what a variable means for generalizing patterns).

Unit 2. Linear Functions. After reviewing the meaning of a solution in Unit 1, I feel that it's super important for kids to see that when there are multiple variables, you can now intuit an infinite number of solutions to an equation! Using exploration to plot some of those solutions allows us to see a linear pattern emerge. Unit 2 is all about understanding the connection between predictability of elements and algebraic forms. In Algebra 2, I cover both slope-intercept and point-slope form, the latter I start with letting the kids figure out that collinearity has everything to do with slope, and then from there they can simplify the slope formula m = (y2-y1)/(x2-x1) into the point-slope form. This year, I think I am also going to throw some unit analysis in there to help explain why slope m has to be change of y over change of x, and not vice versa (in order for mx + b to work out to have the same units as y.) Along with linear functions last year, we did a bungee-jumping regression project including a significant lab write-up, and I plan to repeat that next year.

Unit 3. Systems of Equations Setup and Solving via Graphing Calculator. Following basic review of things from Algebra 1, the most important baseline skill for a student's algebra success is the ability to go from language to symbols. I always take some time (even in Algebra 2) to go through how to write basic equations of the form PART + PART = TOTAL  or  PART*PART = TOTAL, and then I give them a chance to put that to use by writing various systems equations and solving by graphing. A trick I learned from a former rock-star colleague is that you have to always teach and thoroughly practice the graphing skills first, if you want to have a fighting chance of the kids using the graphing calculator later on. If you make the choice of teaching algebraic approaches first, most kids who are afraid of thinking flexibly will always resort to the algebra, even in the cases when the calculator is clearly more efficient and less error-prone. Similarly, kids will be reluctant to check their answers using technology, unless the mechanics of doing so is already second-nature. In this unit, I teach them how to graph, zoom, trace, find intersection, and look at the table to help them with figuring out the appropriate zoom. (I don't like the Zoom Fit feature of TIs, since they're a bit buggy.) For you GeoGebra-lovers, don't worry, the kids will use graphing software later on. 

Unit 4. Systems of Equations Algebra. Now that the kids already know how to set up word problems and to solve by graph, we are ready to delve into the various methods of manually solving a system. This year, I will start with the puzzle explorations for systems to help the kids really get what it means to substitute. After they learn both elimination and substitution methods, they will then practice setting up and solving systems involving fractions (spiraling back to fractions is always a good idea) and word problems, and to use their graphical solution from the calculator to check. I wrote about this before, but I always require on tests that kids solve each complicated problem twice, using two different methods, to reinforce their understanding of graphical and algebraic connections.

Unit 5. Inequalities in the Coordinate Plane. If time allows, I want to spend a short amount of time on inequalities this year. (I did so last year as well, but it was sort of scattered.) Following systems is a good time, because I can then use linear programming problems to drive home the usefulness of the constraints and the graphing.

Unit 6. Quadratic Functions. The way I teach quadratics is by building it up from linear patterns, and I drive home the connection between dimensionality and degrees via this type of side-by-side comparison. The recurrent problem sometimes is that kids don't really understand dimensions from Geometry. (If you're a Geometry teacher, please give some TLC to this very important idea!) We do go into various forms of quadratics and I teach them both how to factor and how to sing and apply the quadratic formula. We do some completing the square, but not enough to master it in Algebra 2, only to see that you can get things from standard form into vertex form. It's important for them to recognize that the quadratic formula can be broken down into various useful parts (discriminant and axis of symmetry) before we move on, so that they could sketch graphs based on any given function equation. Last year, I really drilled the kids to be able to sketch linear-quadratic systems, which, although they probably will not remember the specifics of the procedure, definitely helped to reinforce the idea of connections between graphs and algebraic forms. I didn't do a quadratic-specific project last year, but this year I plan to do a bridge modeling project using all three forms of the quadratic function, as I have done previously in other classes. Dan Meyer's pennies and circles task is also nice to use during this unit to review the idea of regression in the context of quadratics. Sometime early in the quadratics unit, I feel that it is very important to explore the idea of constant second differences between the sequence elements. This sets the stage for other types of patterns to come and helps to reinforce the difference between linear and quadratic patterns.

Unit 7. Transformations. Following quadratics is a good time to talk about general function transformations. The same rock-star colleague had advised me that kids think this topic is too abstract. They will not retain it if you start by teaching g(x) = a*g(x - c) + d, but they will retain it if they can think of a concrete (ie. quadratic) pattern that they already are familiar with. I do these with explorations on the computer, and I have a pretty scaffolded plan if you want to grab it to take a look.

Unit 8. Exponential Functions. I think after quadratics as a big unit, the most natural next major topic is exponential patterns, if your students are following the trajectory of discussing various sequences. Kids can see geometric sequences everywhere, and it is so useful in their lives to understand compounded growth, that I think exponential sequences should be introduced as early as possible to contrast with linear patterns. Here, sustainability issues should really be discussed, both in terms of inflation of costs (of living and education and debts) and our unsustainable human growth / depletion of resources. I teach exponent rules inductively, and they go along with this unit but are assessed separately. I had the idea last year of asking the students to do a sustainability PSA (public service announcement) project, and I will really try that this year with more careful planning / pacing. I have not decided if I will teach logs this year yet, only because not all the teachers in our department can agree where in our curricula (Algebra 2 vs. Precalc) that should be taught.

Unit 8. Inverses of Functions. I didn't do a full unit on this last year, but I think a full unit on inverses, domain, and range should logically follow the introduction of basic function types, because it helps to introduce all the interesting forms that the kids may wish to use in their function pictures project. The wrap-up of this unit should be a functions picture project (via GeoGebra or Desmos), in which they show at least two types of "sideways" functions as part of their included functions. For me, the functions pictures project in Algebra 2 needs to be accompanied by explanations of the transformations, to reinforce the connections between algebraic form and graph.

Unit 9. Polynomials. I taught polynomials as the very last topic this past year, and absolutely loved it! I loved the particular placing of this unit at the end of the year because it allowed us to spiral back to factorization and quadratic formulas, while covering deeper ideas like u-substitution and complex roots. (I didn't do complex roots during the quadratics unit, since there were already so many skills there.) The kids also learned to apply the Rational Root Theorem, of course, and reinforced their understanding of the root. We did a quick maximization problem and some backwards problems working from remainders and factors to finding missing coefficients, and I was so happy to see how well the kids did! If we have time, we'll do a stocks project along with this unit. If so, I'll have to dust that one off from the archives...

Ok, that is a lot! I've just pretty much laid out my entire Algebra 2 curriculum for next year. Whew! Let me know what you think and where you think the missing corners are! xoxo.

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