I’ve decided to reflect on the interleaved algebra 2 course that I’ve designed and taught over the last couple of years. My hope is that by taking a more outward-facing approach to the curriculum, it will help me make it better. This is the 1st post in the series.
Unit 1 (Problems 1-24)
- Function notation & analysis
- Domain & range graphically
- Function arithmetic
- Trigonometric ratios & circles
- Factoring the greatest common factor
Ah, the opening unit. The point where summer fades away. Students often find graphs to be their mathematical friends, so I lean on them a lot, especially in the first few units. I pair that with some analytical stuff by getting into function notation, function arithmetic, and factoring with GCF. Over 90% of the kids come to me from geometry (the others are taking it concurrently), so I also like jumping right into trig. The ratios are fresh in their minds and they get excited when they see them in the first few problems (e.g. 2, 8, 18). The problems reintroduce sine, cosine, and tangent and we begin thinking about the ratios in the coordinate plane with circles. It’s not explicitly stated in the concepts, but I also go hard with using interval notation when representing domain and range; it sets up so much of our future work on intervals. It’s a small thing, but I like how I frame factoring as “rewriting as multiplication” (problem 7).
Two things that I’m left thinking about: I don’t focus on domain and range in any other representation other than graphically. For this unit, I think this is ok, but as of now, I don’t devote any problems to it later on, either. This is an issue. Also, sadly, the function arithmetic that we do sort of dies at the end of this unit, as I haven’t yet included problems that loop back to it in future units.
In the end, this unit is full of pretty straightforward concepts and the students perform well. This is by design. With such a big contrast in how they learn math as compared to prior years, I find it helpful to build success in for them at the start. It’s a needed confidence booster for the long year ahead.
Unit 2 (problems 25-50)
- Intervals of increasing & decreasing
- Extreme values
- Visual sequences & sequence notation
- Factoring the difference of perfect squares
- Trigonometric ratios & circles
- **Systems of linear equations (2)
By studying where a function is increasing and decreasing and its extreme values, we continue the work on graph analysis that we started in unit 1. Again, interval notation comes in handy here. This problem set also introduces sequences. The intricacies of sequence notation can be tricky for students, but the few problems that I begin with (37, 38) have combated that nicely through the years. We also continue our factoring journey as we study the difference of two perfect squares. Now that I think about it, I wish I had included some difference of squares expressions that had a GCF. And I still don’t know how I feel about obscure expressions like (2x-1)^2 + (3x+8)^2 and asking students to factor them, but hey, that’s why I march to the beat of the Regents exam.
This is an important unit in the development of trigonometry because the kids learn about a unit circle along with defining sine, cosine, and tangent in terms of x and y. We also explore the signs of the three ratios depending on where the terminal side of the angle lies when it’s drawn in standard position. (Annoying: so many kids at this point start using the mnemonic ASTC because of the DeltaMath problems I assign.) Speaking of the terminal side of an angle, lost in all this trig work is the explicit learning of how to sketch an angle in standard position. I assign them problems from DeltaMath, which are really nice, but there’s nothing in the problem set that targets this skill. This results in me creating an opening problem to discuss it with the students at some point during the unit. This is haphazard and sloppy.
**Throughout the course, we come across topics and skills that are included in the problem set, but are not tested on that unit’s exam. I’ve found that there’s no need to test on EVERYTHING that we study during a single unit. Certain concepts, and certain problems, are stepping stones to broader ideas. I simply use them to navigate us to much deeper, more important parts of the curriculum. In this case, wrestling with systems of two linear equations will, over time, eventually lead us to solve systems of three linear equations (along with non-linear systems) in later units.
Unit 3 (problems 50-77)
- End behavior
- Function translations
- Recursive sequences
- Difference & sum of cubes
- **Trigonometric ratios & circles
- **Experimental Design
- **Systems of linear equations (2)
In this unit, we continue our work on graph analysis, sequences, and factoring. Naturally, I am able to spiral in our previous graphing work (domain/range, intervals of increasing/decreasing, extreme values) in almost all of the end behavior problems (e.g. 57, 65). Factoring the difference or sum of two cubes always seems so formulaic and boring to me, but I know of no better way of approaching it. That said, this year I changed problem #59 to include an example of using the formulas, and it really helped. The students also get their first taste of a reciprocal trig function when they learn about cosecant. I define it plainly, which I’m ok with for now. I fold the ratio into several problems in the set to help build familiarity with it, especially in relation to the other ratios. When doing this, we continue to discuss terminal sides of angles and sketching triangles in the coordinate plane. Submerged in this set is rationalization. I touch on it with one problem (61), follow it up with DeltaMath, and bring it in again with problem 74. This basic level stuff is good enough to be able to help us with future problems. We also mix in our first look at transformations of functions when we do problem 53. I use a Desmos activity for this that I like, but I know could be better — especially when it comes to horizontal translations.
The sequence notation that they learned during the last unit is put to good use when I contract a throat virus (which renders me unable to speak) on the day they must investigate recursive sequences in problems 67-69. It’s a fun, memorable lesson. I drop two more linear systems problems in the set for good measure (60, 73). On a related note, I really like problem 72 because it combines function transformations with systems of equations. The equation f(x)=g(x) gets some attention, which is a nice.
I’ve sprinkled in a couple more problems on bias during this unit, which seemed like second nature to the kids. I feel the problems are useful, but they seem to linger with no stronger connection anywhere in sight. This will soon change when we start getting deeper in statistics in upcoming units, but maybe I can adjust for this next year?