# Category Archives: FST / Algebra 2

## Fun with Expected Value

I just taught expected value in FST and really enjoyed it. The two main tasks I used were: The Carnival Candy Game and Dan Meyer’s Money Duck.

The Carnival Candy Game

You’re at a carnival and you get to pick one piece of candy from a bag. The color candy you draw determines how much money you win. I used starbursts, and I set it up like so:

The students didn’t win money; rather they won that many starbursts. (I had a different bag of starbursts for prize winnings because I made sure that the candy drawn was replaced each time to keep the probabilities the same for everyone.)

This was enjoyable because naturally all the kids wanted to pick the purple one. Not surprisingly, most picked pink, yellow, or red, but I have 45 FST students (two classes), and the 44th student did pick the purple one.

Then I asked them to calculate the expected value for their prize winnings when playing this game.

Then I said, suppose it costs \$5 to play this game. What does that mean for the player? What does it mean for the carnival game host?

Money Duck

Love the Money Duck. The students were very engaged by the idea of the money duck. I basically followed Dan Anderson’s lesson plan for this one. Like Dan’s students, and as I commented on his post, my students also wanted to determine the possible/impossible distributions based on what they saw in the video instead of in theory. I slightly fixed that in my second class by stopping the video after the first \$1 money duck, explaining that the video was made up, and stressing that we were interested in what is possible, not necessarily what the company actually does.

Like Dan, I had my students come up with company names, probabilities, and price. They then had to compute expected value and their profit. I also compiled the data in a spreadsheet, but didn’t really do anything with it, unfortunately. If I did it again I would like to have the students do some more sharing and comparing between groups.

And then things got even better. Today was the grand opening of a new Cabela’s nearby my school, so several of my male seniors told me how they all skipped class this morning (well, some of them probably had open campus 1st period… I hope) to get in line at the new store because the first 500 customers received a gift card up to \$500. One of them said, “But Ms. Cummins, they didn’t tell us how many were for \$500″. It turned out that they all got \$10 gift cards except for one who got a \$25. It was perfect. I told them I was going to write a test question about that.

Filed under FST / Algebra 2, fun, probability

## A cold day, followed by a beautiful display of student initiative

Yesterday we had a cold day! It’s like a snow day, except it’s really cold out. With wind chill, temperatures around here were -35 F. The timing was good because somehow I ended up being really sick yesterday. So I didn’t particularly enjoy myself on my day off (in fact, I felt terrible), but thankfully I could nap by the fire, drink tea, and spend the day recuperating.

Anyway, I wanted to post about a proud moment from my FST class today. These kids are used to a lot of hand-holding and spoon-feeding, and many of them rarely do independent work (unless I really hound them). Most days, I’ll hear this from at least one FST student: “I’ll be honest, Ms. C, I’m not gonna do this.”

These kids are mostly seniors who’ve been placed in “lower track” math classes their whole life, so changing their mindset isn’t easy. But they did elect to take 4 years of math in high school, plus they’re all good kids, so I know it’s worth it to keep trying.

Today, I told them I would walk them through one example of each type of problem (unit circle stuff), but that was it. No more.

A few kids said, “Aw, can’t you keep going.”

“Nope. I said that was all I was going to do as a class.”

Here is where one kid said, “We can keep doing them as a class, I’ll just go up there.” And he did.

The awesome thing was this kid didn’t know how to solve the problems. But he was willing to go up there and try to figure it out. It probably helped that he’s in the drama club and is an anchor on the school announcements.

So he starts to play the role of the teacher. “Ok, so let’s do problem 2: 495 degrees. We need to find an equivalent rotation between 0 and 360 degrees. How do we do that?”

Miraculously, the rest of the kids played along.

“It’s 45 degrees.” “No, it’s 135 degrees.” “How’d you get that?”

The 135 degree kid explains his thinking, the kid at the board follows along, agrees, and writes down 135.

I quickly snap out of my state of shock and try to remember good techniques for facilitating student discussions.

So I ask, “S, could you please repeat how you got 135?”

So he does.

“Thank you. Can someone summarize or rephrase what S just said?”

Someone does.

And, oh man, it was beautiful. Students were participating without any prodding from me. I managed to remember to ask good questions (Who can rephrase that? Who did it differently?) and to occasionally ask for a collective pause to let something sink in for everyone before moving on. Most importantly, I remembered not to interrupt too much.

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Filed under classroom management, collaboration, culture, FST / 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.

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.

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.

Filed under FST / Algebra 2

## Making Corrections is Valuable

I love, love, love having the kids make corrections on quizlets (formative assessments) and tests (summative assessments). It requires them to actually look at my feedback and to maybe even learn from their mistakes.

For corrections on the last test in FST, I also had the kids write down one thing they’re proud of or one thing they thought they really learned. I got some great responses back, and I hope it helped remind them of what they did well rather than just focusing on mistakes, so I’m glad I had them do that.

Sample responses:

“I did really well on my factoring. I was worried about it and it went better than I thought. Happy about it.”

“I did well on the quadratic formula.”

“I liked Part 1 because I didn’t get any points off, and I did good with my negatives.”

“Last year I feel like I didn’t get a single quadratic problem correct. I feel like I understand them a lot better this year.”

“Factoring went well and I really have cemented the material in my brain.”

“I learned how to graph equations and find the x-intercepts.”

“I think I really mastered the factoring aspect of this unit.”

“I showed my work.”

“I slowed down this test!”

“Overall, I did OK.”

“I did well at taking my time and going through my work.”

“Test was easy but J. did better than me, so I’m salty. I’m over it. Otherwise, test went pretty well. What really helped was coming in and reviewing with you. Thanks, Ms. Cummins.”

Sometimes these kids drive me crazy, but sometimes they can be thoughtful and serious and make me proud.

Filed under formative assessment, FST / Algebra 2, grading

## 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.

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!