I’ve been working on a big project over summer: a set of student notes with fully worked examples for an entire year of Algebra 2.
You might be wondering why I did this. I’m not even teaching High School currently, as I’m taking a couple of years to attend grad school. While I’m doing some teaching, it’s been as a TA for Calculus 1, certainly not for Algebra 2. Even so, I’ve still spent a lot of time thinking about Algebra 2 curriculum over the past year, and had a lot of conversations about it with Sarah, as she is has been teaching Algebra 2 since last August. It turns out, even after taking some time away, my passion is still for high school math.
These are the ideas I’ve had for the function of these notes:
They’re aiming to meet the Oklahoma Academic Standards, though in places they step back to strengthen the conceptual foundation, and in others they go beyond the standards. Eventually I’ll produce an alignment document to explain all the links.
This is not a complete curriculum, but I see it functioning as a “skeleton” on which a complete curriculum could be built.
The notes focus almost purely on the mathematics, not on “real world” applications (with a few exceptions in statistics topics). This is not because these are unimportant; on the contrary, they are vital. But I believe these are better addressed using methods other that pre-prepared notes.
The intent is that the notes would be hole-punched and kept in a binder. This means if a teacher doesn’t like how I’ve done something, they can change it. Remove the parts you don’t like. If you don’t think there are enough examples, add more.
Teachers can incorporate the notes and examples as they wish into their lessons. While textbooks give they answers to examples away from the start, with fill-in-the-blanks, the teacher can choose at what point in the class discussion they make the correct answers known.
You might remember that I was working on a book of Algebra 2 practice questions. That’s still ongoing, but it’s been overtaken somewhat by these notes. But that’s okay, because I see these as two aspects of the same long-term project. Having the notes planned out should make planning questions a lot easier.
If all of this sounds good, here’s the great news: I’m going release the notes as I work on them. And while they’re still just a first draft at this point, that first draft is entirely done, and hopefully in a usable form for the upcoming school year.
Over the weekend, I needed a break from working on grad school assignments. At the same time, Sarah needed a card sort on the vertex form of quadratic functions. Being the nice husband I am, I thought I’d help. Being the nerd that I am, I did it in LaTeX.
This took a bit longer that I’d expected to create. I’ve a little experience creating graphs, so they weren’t too bad, but tables in LaTeX can be a little fiddly to get right. I also wouldn’t claim that took much thought went into choosing which functions to use. But, I’m pretty happy with the result.
You can download the files here. Included is the shuffled set of cards (pictured above), the cards sorted into the correct order, and a zip file of the original .tex files in case you want to modify them at all.
Edit (Oct 23): There were a couple of mistakes in the original version, it has been fixed now.
She did get it laminated today, and it looks like this:
What I love about this is the way it demonstrates the differences between functions, expressions and equations, but also shows how there’s a connection between them too. This poster goes well with the set Sarah created about solutions, roots, zeros and x-intercepts; that’s how she’s got them in her room, after all!
Now, the poster doesn’t capture all the particulars of these algebraic tools; it’s just a simple poster showing one example. There were some comments on twitter about the fact this poster doesn’t completely define what a function is, particularly absent a discussion of sets. Also, there was concern that the equation example implies all equations are homogeneous. Both criticisms have an element of truth, but also miss the point of a poster that, by its nature, only has one example. I think posters serve the best as either reminders of topics already discussed or starting points to launch a deeper discussion or investigation. Please, never put a poster on a wall and assume that means you’ve taught a topic. For an example of how I’ve discussed functions before, see these notes that I used in Algebra 2 last year.
This one may be a little tricky to put together, so if you want some guidance, keep reading:
Choose your three colors. The effect works best (and helps communicate the idea) if the middle color appears to be a blend of the other two. If you don’t want to think too hard, just use blue-green-yellow like me, but I’d love to see other color combinations too.
Print the first page (function) on the first color (blue, in my case), and the second page (equation) on the last color (yellow).
Print either page on the middle color (green). For this page, you only need the expression rectangle, which appears on both pages. Cut off the rest of the paper.
Stack your three pages with the expression rectangle of each page overlapping, with the middle color on top. Make sure you line up the rectangles as perfectly as you can.
Continuing the theme of blogging I didn’t get around to last year, here are some diamond problems for INBs.
If you haven’t heard of these before, the solution to these puzzles consist of four numbers. The top number is the product of the left and right numbers, and the bottom number is the sum of the left and right numbers. The puzzle consists of the diamond with two numbers filled in; the task is to determine the other two numbers.
If the puzzle gives the left and right numbers, the solution is pretty simple. Even so, I provided some of these on the first page so students could understand the problem before moving on to the actual challenge: finding the left and right numbers when given the top and bottom.
You may have noticed that the solutions to these puzzles are related to quadratics. For instance, the first example matches the factorization x2 + 9x + 14 = (x + 2)(x + 7). The magic of these puzzles is that they start teaching kids this vital skill for Algebra without it being obvious that’s what happening.
If you’re teaching middle school or even upper elementary school, I really recommend giving your students these puzzles. You can change the difficulty as you need, introducing negatives when students are ready, and decimals or fractions when you really want to up the challenge. The puzzle is a logical thinking problem which is not too difficult to understand, and ties directly in a skill needed in Algebra. As it happens, I used these in Algebra 2, because that’s what I was teaching, but I wish I’d known to use these puzzles when I used to teach 7th grade. Your students’ future Algebra teachers will love you for it.
There’s also a last-five-minutes activity I used a few times related to this. Once students know how the puzzles work, have a student invent their own puzzle and write it on the board for the rest of the class to solve. I had some (very competitive) students who were very determined to give their classmates a problem they couldn’t solve, and others who were equally determined to solve these problems.
These are the notes I wrote for adding and subtracting rational expressions last year in Algebra 2.
On the first page, I wanted to make the connection between rational expressions and fractions explicit, so I started with a reminder of how to add and subtract fractions. The two examples were chosen deliberately; the first only required changing the denominator on one fraction, while the other required changing both denominators. Students need to deal with both types of problems with rational expressions.
The next part introduced two simple problems, with only one denominator being changed between them. They are both problems that require simplifying, as I wanted to emphasize the need to do this from the start. (Students had already seen how to simplify rational expressions and stating excluded values.)
The next page was about finding the lowest common multiple. I had students use a strategy which emphasizes the definition of the LCM, having them multiply by factors so that the two expressions are the same.
And finally, a couple of examples putting all of this together.