Tag Archives: Algebra 2

Motivating 1/x

We’re deep in a functions unit in FST (year two of a decelerated Algebra 2 course), and I love it. I love the concept of a relationship that takes inputs and produces outputs. I love visualizing functions with graphs. I love that functions feel natural and intuitive. I’m trying hard to share this enthusiasm with my students. They’re doing well with it so far, and it’s interesting to see how they think about functions.

Last class, I wanted to introduce f(x) = 1/x. I love this function. I love the discussions about division by zero and division by really large numbers and how the graph represents those ideas visually. My colleague shared a fun investigation with me, and I am so glad that I tried it out. At first I was hesitant because I know that I don’t explain directions well, but I focused on being very explicit and modelling each step. The kids investigated the breaking point of spaghetti. I wish I had some photos, but the students placed a dry spaghetti noodle over the edge of the table and hung a paper cup on the end of the noodle and added pennies one by one until the noodle snapped. The fun factor was definitely there- the kids enjoyed predicting when it would snap and liked watching the pennies crash to the floor.

Besides being fun, the activity modelled the function effectively. The kids recorded their data (length of spaghetti vs number of pennies), and I used Desmos to display some class data.

hyperbolaPenniesSpaghetti

Voilà, a hyperbola. The investigation gave the kids a good understanding of how the function behaves and why the graph looks the way it does. In retrospect, I should have done more of a “Noticing and Wondering” activity with the graph, but instead I just asked some questions like “What happened as the length of the spaghetti got longer?” and “What happened if the length of the spaghetti was really small?” which probably did too much of the thinking for them, but oh well.

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Filed under FST / Algebra 2, fun, graphing, group work

Teaching Polynomial Long Division

I confess: I think polynomial long division is kind of a waste of time. It’s a tedious process that doesn’t really involve much mathematical understanding. And when you use synthetic division, there’s even less understanding involved. So I say just skip it.

Unfortunately for me, my school’s current FST (2nd half of Alg 2) curriculum includes polynomial long division. The reason is so that we can factor and solve equations like y = 9x^3 – 31x -10 …but I’m not entirely convinced that that’s very useful either. Math class needs to move behind problems that wolfram alpha can solve for us in 3 seconds.

Anyway.

So back to teaching polynomial long division. It actually went well. I really emphasized CCSS Standard for Mathematical Practice #3: Make sense of problems and persevere in solving them. I told my FST kids that there are problems in math (and in life) that are long and challenging and that require stamina and perseverance. For example, these long division problems will test your mathematical stamina, but stick with it and don’t give up.

So many of them took that as a challenge. They wanted to prove that they could stick with the problem all the way through. It was lovely. So maybe there is something to be said about polynomial long division after all.

It was also great when I told them to use zero placeholders for “missing” terms (like 0x^2 in my example above) because right after I said that I forgot to use a placeholder in my example, so then it became completely clear why placeholders are useful when terms weren’t lining up. Yay for making mistakes.

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Filed under FST / Algebra 2

Factor Craze

I didn’t have the greatest teaching day today, so I thought I’d try to remind myself that I CAN do this teaching thing by describing an activity from a few days ago that was successful.

Factor Craze, which I think I found via Fawn Nguyen, is one of NCTM’s monthly “Problems to Ponder”, and it asks:
Which numbers have exactly three factors?
Which numbers have exactly four factors?
Which numbers have exactly five factors?

This problem was a great introduction to factoring with my FST (2nd half of an extended Algebra 2) kids. They saw factoring last year, but this year I wanted them to really understand how they were coming up with the equivalent expression instead of following a list of steps from the teacher. So I used Factor Craze to spark some conversations about factors.

I have my students seated in groups, but I had them think on their own for a minute before working with their group. I actually started with the question Which numbers have exactly two factors?, which may seem rather elementary for high school juniors and seniors, but as I suspected, many had very little knowledge or experience with the concept of prime numbers.

Most groups started by writing down examples of numbers that had the required number of factors, but I prompted them with, “What’s a way to describe ALL numbers that have exactly ___ factors?”. All groups eventually came up with prime for exactly two.

When they got to exactly three, most groups found out that 4 and 9 worked. I asked them if there was anything special about numbers 4 and 9. “Oh, oh! They’re perfect squares! Perfect squares have exactly three factors!”

So I respond with, “Do all perfect squares? What about 16 and 25?”

“16 doesn’t work. Oh. But 25 does!”

So I say, “Nice. So some perfect squares but not all perfect squares. What type of perfect squares work?”

And so on. Most groups figured out that squares of prime numbers have exactly three factors. Only one group in each class was able to delve into exactly four factors before we ran out of time.

I really liked how this problem posed a challenge for every student. For some, just remembering what prime numbers are like was a challenge. For others, it was recognizing a theme for exactly four factors. Either way, all students were developing an understanding of factors.

We later moved on to greatest common factors and factoring expressions, and I think laying the ground work with Factor Craze made a difference.

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Filed under FST / Algebra 2, group work, productive struggle

A Favorite Math Lesson Resource

I want to share one of my favorite resources for rich math lessons: the Mathematics Assessment Project, which is a collaboration between the University of Nottingham and UC Berkeley. I love the structure of their Formative Assessment Lessons. According to the website, these lessons are designed “to reveal and develop students’ conceptions, and misconceptions, of significant mathematical ideas and how these connect to their other knowledge”, as well as “to assess and develop students’ capacity to apply their mathematics flexibly to non-routine unstructured problems, both from the real world and within pure mathematics”.

When I taught middle school summer school, I used a couple of these lessons with great success. I love how they include matching and sorting activities that require the students to come up with their own mathematical justifications. When I tried these out in summer school, I had small classes and had the students do it individually, but this year I am definitely going to follow their suggested structure of using small groups because I want to see more mathematical discourse among the kids. I also want to do a better job of creating the final product and displaying the work on posters. In my summer school class, once the kids finished sorting they would just toss the cards in the trash (or worse, leave them scattered on tables or the floor for me to clean up). So this year I will definitely use the small group format and require the posters, which will hopefully get more students to explain their thoughts and reach a deeper understanding.

Conveniently, I was able to find a MAP lesson for each of the first four units in my Functions, Statistics, and Trigonometry (FST) class. (Students at my school have the option of the Geom–>Advanced Algebra–>Pre-Calc route, or the Geom–>Algebra 2–>FST route. So the curriculum for this class is comparable to the second half of an Algebra 2 class, I think. First year teacher here, so I can’t comment too much on “typical” math curriculums.)

Unit 1 – Our curriculum calls this first unit “Symbolic Manipulation” which is a decent description for it. It’s basically a review of Algebra topics they’ve previously encountered, such as the distributive property, combining like terms, and factoring. The MAP lesson I plan to use is Interpreting Algebraic Expressions. I like how this lesson helps students distinguish between expressions like (5n)^2 and 5n^2, and it also uses area diagrams that will come up again when I’m teaching factoring. I plan to try this lesson right away the first week of school, and it might be a fairly simple for this class content-wise, but that’s okay because it will also serve the purpose of introducing the students to this type of group work and what it’s like to share their thinking, which will probably be a very new experience for them.

Unit 2 – This one is all-things quadratics. The MAP lesson I want to use is Forming Quadratics. This is a great matching activity, and I like how it incorporates all of the different forms for quadratic functions and helps the students identify key features of the graphs.

Unit 3 – Polynomials. Not my favorite unit (I find long division, rational root theorem, etc to be somewhat tedious), but I can’t let that show. I think I will definitely emphasize SMP #6 – Make sense of problems and persevere in solving them during this unit. The MAP lesson here is Representing Polynomials, which is another great matching activity which connects what students already know about finding zeros to 3rd-degree polynomials and their graphs. The extension activity is also intriguing.

Unit 4 – Functions and Function Transformations. There are two awesome MAP lessons that could be incorporated here. The first one, Interpreting Distance-Time Graphs, was the one I used with my summer school class last summer (8th graders who would be taking Algebra 2 in the fall… so a very different group than the one I’m talking about now). Perhaps this one would be too simple for my FST kids, so the second option, Functions and Everyday Situations, might be a better fit. I like the open-endedness of Distance-Time, but I like how Everyday Situations has students translating between the algebra and the graph.

These lessons will be great formative assessments, and  I’m glad I can incorporate one into each unit. Hopefully you got a chance to check out the MAP website and lessons if you aren’t already using this great resource. If you’ve used these in your classroom before, what do you like about them? What about them is difficult to implement? Thanks in advance!

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Filed under formative assessment, FST / Algebra 2, group work