Posted in Enhanced Math III, Instructional Strategies, Precalculus


The most amazing thing happened in 5th period today.

Setting the scene: We’re in Enhanced Math 3 Honors, which (to make a long story short) is an accelerated honors class that includes all of Honors Precalculus and most of Math 3. We started working on inverse trig functions before break, and this is the first day back from winter break, so we’re bound to be rusty.

The task: Inverse or not? from Underground Mathematics

We started as a Stand and Talk. Per Sara VanDerWerf (my fave!), students walked 17 steps away from their desks to find a partner from another table, no writing, just talking for a full 2 minutes. I handed them the prompt: One of the graphs shows 𝑦 = arctan(tan π‘₯). Another shows 𝑦 = tan(arctan π‘₯). Which graphs are these, and why?

Inverse or not 1

Side note – or TANGENT (haha, see what I did there?!?) – this is a good example of a task that I like for a refresh of prior knowledge. I hate reviewing the same material from last year or last semester all over again in the same way. The kids are bored, I’m bored… it’s like we’re assuming they never learned it in the first place. And yet, it seems wrong just jumping into new material that builds on what they’ve learned without acknowledging that they may have forgotten something. What did we learn before break? We learned about inverse trig functions, specifically their domains and ranges. I could just re-teach the same thing in the same way (ugh) or just keep teaching like it was yesterday and not two weeks ago… or I can exercise the same concepts in a different way, like with this task.

Anyway… back to 5th period. It was awesome.

After students talked in pairs, I had them share out some of the ideas they had. Teams were unsure whether 𝑦 = arctan(tan π‘₯) matched graph A or graph B. One student said that it must be graph B because the domain of 𝑦 = arctan(π‘₯) is (-Ο€/2, Ο€/2). Another student said that it was actually the range of 𝑦 = arctan(π‘₯) that is (-Ο€/2, Ο€/2), which is why they had excluded graph C. One student said that the domain of 𝑦 = tan(π‘₯) is (-Ο€/2, Ο€/2), which is why it must match graph B, while another student argued that 𝑦 = tan(π‘₯) has a domain of all real numbers except odd multiples of Ο€/2 and that (-Ο€/2, Ο€/2) is only the restriction to make the tangent function one-to-one. One student very eloquently stated, “the domain for this function 𝑦 has to be the same as the domain for tan(π‘₯).” That convinced the class that graph A was the only possible logical conclusion.

Whew… already airing misconceptions, using rich vocabulary, strategic thinking… and we haven’t even gotten to the best part!

We moved on to the function 𝑦 = tan(arctan π‘₯). They had been working with the same partner for a while, so I sent them back to their teams of 4 to check in. Amazingly enough, exactly half of the teams thought this equation matched with graph B and half thought it matched with graph C. I gave them a minute to bolster their arguments with the “believers” aka their teammates who agreed with them. Then they had to find a “non-believer” (someone who thought differently) and convince them to join their side. A little trick I picked up from Andrew Stadel – students had to face their body toward opposite walls (one labeled B, one labeled C) so they (and I) could visibly see who agreed with which graph.

Amazing conversations. MP 3 all over the place. Lots of moving bodies, students visibly changing sides. I paused them after a few minutes and asked, “If you were convinced to change sides, what was the argument that convinced you?” One student and his partner explained that the domain of 𝑦 = arctan(π‘₯) is all real numbers, then outputs a value between -Ο€/2 and Ο€/2, which 𝑦 = tan(π‘₯) then takes in and outputs all real numbers. The only graph with both domain and range unrestricted is graph C.

I saw a lot of students nodding and moving their bodies… except one. D just looked at everyone and said, “but I just don’t understand how the range of arctan(π‘₯) doesn’t matter!”

There was a pause… then every single student in the class started explaining and arguing and yelling to try to convince D. There’s absolutely no way he could have understood anything they were saying in the cacophony of voices. To any random passerby, I’m sure it was chaotic and loud, but to me it was the most incredible symphony. Students found their voices and felt confident to use them without any kind of prompting from me.

OK… so maybe it doesn’t sound as amazing as it felt during 5th period. TBH it wasn’t even as good in 6th period when I did tried to do the exact same thing. But I overheard one student say as she walked out the door, “I’m going to convince you that I’m right during English next period” and another stayed after to ask me questions (which I did not answer) and walk through his thinking (to which I shrugged) until he felt confident that 𝑦 = tan(arctan π‘₯) “matches with graph C, darn it! my brain hurts…”

And isn’t that what we want for our students? To be so engrossed in a problem that they can’t let it go? To feel like their brain has been exercised to the point of exhaustion? To feel confident that they have reached the correct conclusion without verifying the answer with an “expert?” I don’t feel like I’m doing this every day, or even once a week, but it feels amazing for that one small moment when everything clicks.

I love teaching.

Posted in Algebra 2, Enhanced Math III, Math III, Precalculus

My favorite lesson: Introducing polynomial functions

We’re studying polynomial functions in my Enhanced Math 3 class right now, and I love it. There’s just something about the predictable nature of their graphs… but really, I think it’s because I am in love with this card sort.

I originally created this card sort as part of a conference presentation tracing the progression of standards on linear functions from 6th grade ratios to derivatives in Calculus; we stopped on polynomial functions along the way, focusing on the relationships between the linear factors of a polynomial and key features in its graph. I was serving as Teacher on Special Assignment at the time, so I had no classroom and no students to test out this lesson, but the teachers that attended our sessions really enjoyed the task.

This is now the third year that I’ve used it with students, and I think I made it even better by creating “task cards” for each round of the card sort, like the one below.

task 1

I had done this card sort successfully with teachers several times, but the night before I first used this with students, I worried that students wouldn’t look as closely at the equations and graphs to notice the patterns in the behaviors near the zeros (my ultimate goal for task #1). So I added an additional incorrect graph and equation pair for them to find. I thought this was brilliant *pats self on the back* … my students, of course, found the wrong graph in about 2 seconds (it’s card G7, shown below).

graph example

“We just plugged in numbers. The y-intercept in the graph is positive, but it’s supposed to be negative.” (oops… so much for looking closely)

The question I really needed to ask was “Now that you’ve found the incorrect graph, what’s wrong with it? How can you fix it?” Some groups needed more nudging than others, but ultimately every group figured out a pattern and adjusted the equation accordingly:

“The (x+2) is squared so it should look like a parabola around -2, and (x-4) is cubed so it should look like a cubic around 4. We would change the exponents so that (x+2) is cubed and (x-4) is squared and then it would match the graph.”

This year, with the addition of the task cards, I formalized this as a checkpoint question at the end of the first task. I borrowed this strategy from a complex instruction task shared by Rick Barlow. When a team tells me that they are ready for the next task, I ask one person at random(ish) in the team the question. If they answer correctly, the team gets the next task; if they answer incorrectly, I reiterate that every member of the team must be able to answer the question, they need to get together to ensure that is true, and I’ll return again when they’re ready.

task 2

zeros example

In task #2, students have to match the zeros to the graph and figure out what the word “multiplicity” means. This task (and the remaining tasks) employs my favorite card sort hack: the blank card. If there are 10 cards to match, I leave one card blank so that students have to apply what they’ve learned to create their own card for the remaining match. This pushes students to be less passive during the card sort.

Tasks #3 and #4 are similar to task #2 in that they have sets of 10 cards to match, with one blank card in each set. These cards focus on the degree and end behaviors of the graphs, and the intervals on which the function is increasing/decreasing and positive/negative.

task 3

task 4

Exit slip: Students had to create an equation to match a graph projected on the board and ask at least one question, either something that confused them or a “what-if” question they want to explore further. Every student was able to identify the correct zeros and multiplicity, and almost all identified that the leading coefficient should be negative given the end behavior of the graph. Most of the questions they had pertained to the leading coefficient and y-intercept, which we had not really addressed in this task, but I knew we would confirm in the next class.

What I love most about this task is that students are able to discover all of these patterns in polynomial functions. I did absolutely no direct instruction; I monitored student progress and nudged them along by asking questions or focusing their attention on particular cards.

This was only the second day of this unit. On the first (short) day, we completed this task from Illustrative Mathematics. In the days following this card sort, we did a Connecting Representations task that I created (more about the Connecting Representations routine here and here) and some practice with graphing and writing equations of polynomial functions, before going into any of the algebraic techniques traditionally associated with polynomials: factoring, long/synthetic division, possible rational roots, irrational and non-real roots, etc.

Resources:Β  Card sort (PDF, word doc) and task cards (PDF, word doc)

I love sharing ideas… if you try out this task with your students, please let me know how it goes in the comments below or on twitter! @kristiedonavan

Shoutout to Rick Barlow – the complex instruction tasks you share on twitter give me great ideas for how to improve my own class structures.

Shoutout to Sara Van Der Werf – did anyone catch it? Look at task #1 again… I steal all my best ideas from her. The reason I created this task in the first place is because I need students to see it before I show them. πŸ™‚

Shoutout to Martha Barrett – thank you for your unwavering support and for working with me to build our Linear Functions Roadmap presentation.

Posted in Calculus, Instructional Strategies, Precalculus

Resurrecting the blog

Haven’t blogged in almost two years… how the time flies! I keep thinking of things I want to write about, but there’s never enough time in the day.

My original intent with my blog was to help me become more reflective in my own practice. Although that is still a goal for me, between grading papers, creating a new course, and trying to actually have a life with my own family, I’m not sure that reflecting daily in blog form is in the cards for me this year. So instead, I’m going to try to focus on the results of trying something new, strategies and resources that have been particularly effective, and slight tweaks that we teachers can make in our classrooms that pay big dividends for students.

This year I’m teaching AP Calculus BC for the first time and Enhanced Math 3 for the second time. I’m also reviving our Math Lab course, which I originated at our school about 7 years ago; after I moved to the District Office for a few years, the course fizzled out, but I’m determined to make it a success this year! In addition, our school has adopted an 8-period alternating block schedule with anchor day after a 6-period traditional schedule for nearly 40 years. Life is never dull!

One thing I’m trying to be very intentional about this year is instructional strategies that get students out of their seatsΒ during our now 85-minute classes; I’m encouraging all of my colleagues to do the same!

Strategy:Β Stand and TalkΒ 

Thanks to Sara VanDerWerfΒ (@saravdwerf) for this great strategy. I have students stand up and put their hands up, find a partner from a different team and high five; any student with their hand still raised is still looking for a partner. Then I give them a discussion prompt to work on together. Sara recommends giving each pair a problem/prompt written on a half-sheet of cardstock so it will hold up for multiple classes.

In Calculus, we sketched a graph of a function that meets all of the criteria described by limits. I laminated this half-sheet so that students could write on them with dry erase markers and I could still reuse them in the next class.

stand and talk 1stand and talk 2Β  stand and talk 3Β Β stand and talk 4

Students drew their graphs, then paired up with another pair to check each other’s work. Even though most pairs turned into “kneel and talk” or “lean over a desk and talk” so they could write on the cards, students got out of their seats, worked with other students outside of their teams, battled misconceptions (they hadn’t experienced a function with more than one horizontal asymptote), and practiced describing function behavior with limits.

In Enhanced Math 3 (Integrated Math 3 + Honors Precalculus accelerated class) we’re working on composition of functions and inverses. Since I only have one section, I wasn’t concerned about reusing the prompt, so I just printed on regular paper and collected this as their exit ticket for the day.Β Stand and talk inversesI left this pretty open to see what they would do with it, but I had several student pairs who did this very quickly (maybe too quickly???), so I reminded them of our learning target: “Mathematicians understand and apply inverse functions in multiple representations” emphasis on multiple representations. One team said, “at first we said yes, then when we sketched the graphs, we realized we needed to restrict the domain of the quadratic function to make it match the reflection of the square root function.”

When we analyzed the results of the exit ticket together the next day, we noticed that the range of the quadratic function was the domain of the square root function; when we found the range of the square root function, we realized that was the appropriately restricted domain of the quadratic function!

So far just two stand and talks have convinced me this is a must do, ESPECIALLY for block classes! πŸ™‚