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 𝑥). Another shows 𝑦 = tan(arctan 𝑥). 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 𝑥) matched graph A or graph B. One student said that it must be graph B because the domain of 𝑦 = arctan(𝑥) is (-π/2, π/2). Another student said that it was actually the range of 𝑦 = arctan(𝑥) that is (-π/2, π/2), which is why they had excluded graph C. One student said that the domain of 𝑦 = tan(𝑥) is (-π/2, π/2), which is why it must match graph B, while another student argued that 𝑦 = tan(𝑥) has a domain of all real numbers except odd multiples of π/2 and that (-π/2, π/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(𝑥).” 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 𝑥). 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(𝑥) is all real numbers, then outputs a value between -π/2 and π/2, which 𝑦 = tan(𝑥) 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(𝑥) 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 𝑥) “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 Instructional Strategies

Same and Different

… or the easiest routine you’re probably already using in your classroom!

I love using the Same and Different routine with a variety of students, for a variety of reasons. It’s easy, it’s versatile, and it’s powerful learning and discussion. I think teachers do this naturally, and we can find even more success when we do it more intentionally.

First, the basics: what do I mean by Same and Different? I generally do this with the same basic structure every time:

Same and diff 1

  • Post a simple prompt: two equations/graphs/tables/diagrams etc. next to each other with the questions, “What is the same?” and “What is different?”
  • Give students 2-3 minutes of individual time to write down as many similarities and differences as possible.
  • Team brainstorm: individuals take turns sharing with their teams and together they make a collective list.
  • I vary the whole class share out: sometimes I’ll have them post their list on poster board and do a gallery walk; more often we’ll make a whole class list taking one same from every team, then one different (repeat as necessary)

I try to leave tasks open enough so I can see what students come up with, but usually I have a particular concept that I want to highlight. Here are some of my instructional goals when I use the Same and Different routine.

Intervention to correct a key misunderstanding

This is fresh on my brain because this was the purpose for a Same and Different routine I just did on Monday with my Math Lab classes (support class for struggling math students). In reviewing a recent assessment with one of our amazing Math 1 teachers, we noticed that students were using interval notation to describe the domains of both discrete and continuous functions. We designed this Same and Different task as an intervention:

Same and diff 2


We’ve been working on key features of functions and they identified both graphs as having the same intercepts, same rate of change, same max and min; all teams said that the first graph is discrete and the second is continuous. I was so proud of them using great vocabulary! We also had a little preview of their next unit: one student asked if the first graph is still linear because there’s not a line going through the points, and another student said, “I think yes because they both have a constant rate of change.” Wooooooo!

Another note: I love Same and Different because everyone can access a task. One student (who is enrolled in a below Math 1 level course) said “the first graph has just dots, but the second has a straight line connecting.” Her teammate (in Math 1, normally does not feel confident to talk in math class) said, “Oh that means the first graph is discrete and the second is continuous.” 😍


Some teams said both graphs have the same domain and range, which is exactly the conversation I wanted to have. One student said (I’m paraphrasing), “The second graph is continuous, so we have to include all of the inputs in between the whole numbers, like -3.5 and 0.2. But the first graph is just the separate points, nothing in between, so we have to write the domain and range as separate points.” Light bulbs going off everywhere, and on my follow-up exit slip, almost every student correctly identified the domain and range of one continuous and one discrete graph.

Highlight key differences to prevent common mistakes

In addition to intervening to correct mistakes, I love Same and Different for preventing mistakes by highlighting key differences between very similar-looking equations. This Same and Different task was a favorite that I did last year with my Precalculus students:

This generated a lot of discussion between students. To set the stage: we had spent a day exploring polar graphs in general, then a day exploring limaçons specifically. This was their opener on day 3. (Their ideas, I just scribed on the board; stars indicated ideas they weren’t confident with yet that they wanted to verify with Desmos)

img_1535 img_1536

These equations look similar but their graphs are very different, and it’s easy for students to confuse them; this task helped students to walk through those similarities and differences so they would be able to recognize and produce graphs of limaçons in the future.

Again, this task is accessible; any student can at least identify that the equations have the same numbers, same structure, the numbers are switched, and one is sine, one is cosine. These are all very important characteristics to notice, and I can guide them to see how they connect to the key features of the graph.

The task below didn’t generate as much conversation, but allowed students to see two very similar-looking conditional probability statements are very different:

Here’s another one I loved doing with my Math 2 classes:

This one was really great. Whenever we’re learning about parabolas, I want to make sure that students don’t assume that every graph they create will be a parabola. That “little 2” that every student can notice makes a big difference! We talked about key features and transformations and realized that maybe these graphs are more alike than they are different, which brings me to my next topic…

Highlight key connections between concepts/procedures

In that last task, students realized that all of the transformations we had done in Math 1 with linear and exponential graphs, still applied to quadratic (and later, absolute value) graphs! By making this connection, I did not “teach” nearly as much of graphing quadratics; we thought in terms of transformations and key features of the parent function.

How powerful is this? I gave the following task to my students, let them notice the similarities and differences, and suddenly I saved 2+ days of “teaching” operations with complex numbers that I can now use to focus on cool patterns and deeper problems.

Not all Same and Different problems have to look like this, nor do they have to revolve around whole class discussion. Here’s a homework task that I used in my Enhanced Math 2 class:

Guess what? By making the connection between these four nearly identical equations, solving equations with absolute values and square roots became just one more tool in their toolbox, rather than an entirely different idea to learn.

Activate prior knowledge

As seen in several tasks above, we can create greater coherence between topics by making connections to prior knowledge. Here’s another task I used to activate prior knowledge at the beginning of our sequences and series unit in Precalculus:


Students have so much knowledge already; we just have to figure out what it is and how we use it to build new ideas.


What’s so great about this strategy?

I love Same and Different for so many reasons. Comparing and contrasting is a key cross-curricular skill and an important life skill. More importantly IMO, the prompt is easily accessible by all students. Anyone can look at two graphs and say one is just dots while the other has a line through, or the numbers are switched, or there’s a little 2 on that one. They don’t have to have the right vocabulary or some profound observation; that’s where we as teachers come in, to build that vocabulary or guide that important connection.

Also – they’re so easy to make up! What is a mistake you want to correct? An idea you want to highlight? Try making a Same and Different task and share it on Twitter: #samediffmath. Tag me too (@kristiedonavan); I’d love to see what you come up with!


Resources: Here are all of the Same and Different tasks that I’ve created so far (PDF, PowerPoint). Please feel free to use in your classroom – let know how it went!

Shoutouts to Brian Bushart (@bstockus) for the site Same or Different? and to Bridget Dunbar (@BridgetDunbar) for working to create more secondary Same and Different tasks.

Posted in Assessment, Calculus, Instructional Strategies, Intervention

Exit Slips and Intervention

I’ve been trying to figure out the “best” way to write this blog post… finally I decided to just write it in whatever imperfect way it comes out because I think it’s one of the most powerful things that we can do as teachers and I want to spread the gospel far and wide!


But it’s not just the exit slip, it’s what you do with it. You’re getting some information about your students’ understanding, and then following up in a very strategic and targeted manner.

Here’s what I did recently with an exit slip on limits and continuity.

Exit slip 1

Intervention #1: Pulled small groups

I gave this exit slip at the end of a class period on Tuesday. On Tuesday afternoon, I quickly sorted exit slips into “yes” and “no” piles based on their answers to #1(a)-(c). Then I sorted the “no” pile into subcategories based on the mistake that was made. I ended up with these categories:

  • On #1(a) – students who answered 2, which is the value of f(-2) not the limit as x approaches -2;
  • On #1(b) – students who answered 3, which is the limit from the right not the limit from the left;
  • On #1(c) – students who answered 2, which is the limit from the left and the value of f(3);
  • On #1(c) – students who answered 1, which is the limit from the right.

I made new student teams based on like mistakes. On Wednesday, I projected a PowerPoint slide with their new teams; as they worked on some practice problems together, I visited each group to address the misconceptions in their exit slips and answer any remaining questions. (In the past I have pulled small groups to work with me at a side table, but this year I’ve decided to regroup teams temporarily and visit each team.)

Intervention #2: Targeted whole class intervention

When I looked at students’ responses to #1(d), I noticed that all students had said that the function was not continuous at x=1, but most did not reference the definition of continuity in their explanations. This told me that I needed some whole group instruction around the definition of continuity, specifically in how to justify their reasoning. So after I addressed each team’s individual needs on #1(a)-(c), I asked students to discuss the prompt below in their teams.


We had a discussion of the three conditions required for a function to be continuous at a point, and students were able to specifically identify the reasons for which each function was not continuous at x=0.

Intervention #3: Peer teaching

Finally, on problem #2, the mistakes were all algebraic/computational and had nothing to do with limits: students either forgot how to factor a difference of two cubes or made a mistake in dividing. In every team, there was at least one student who had correctly factored, simplified, and evaluated the limit, so those students supported their teammates in finding their mistakes and making corrections.

So how much time did this take?

Well… it has taken me longer to write this blog post! 🙂  It took me about 5 minutes to sort the exit slips into piles for intervention #1, maybe another 10 minutes to read the responses for #1(d) and create intervention #2, and about another 5 minutes to look at the work for #2 and make sure that each team had at least one student who had completed the problem correctly.

It took me about 25 minutes of class time to check in with every small group (and would probably have been less if I remembered my timer… oops!). And this is happening as the rest of the class is working on another task, so I’m not losing any time; I’m actually gaining time because I’m not reteaching to the whole class.

Is it worth it?

Absolutely! On their quiz last week, every student correctly identified a two-sided limit, a one-sided limit, and a limit that did not exist; nearly every student was able to correctly show that a function was continuous at a point using all three parts of the definition.

In all honesty, I think this is one of the simplest changes a teacher can make to improve student learning. And although this example is from my Calculus class, I have used this strategy in every class, at every level, with every student population. IT WORKS!

Shoutout to Cassandra Erkens of Solution Tree, who has been a consultant in my district for many years now, and who has helped me to refine my assessment and intervention strategies every year!

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 seats during our now 85-minute classes; I’m encouraging all of my colleagues to do the same!

Strategy: Stand and Talk 

Thanks to Sara VanDerWerf (@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 2  stand and talk 3  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. 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! 🙂