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 Uncategorized

Small wins

I’m a big believer in celebrating small wins. Even if something doesn’t go well, I try to find something positive to celebrate.

Here are a few small wins I’ve been celebrating in my classes recently:

In Math 2, we were having a partner quiz on right triangle trigonometry. D and her partner had set up a problem and solved for a side length, then said, “Wait… that can’t be right… we got that the leg is 6.9 but the hypotenuse is 6 and that’s the longest side… let’s try that again.” Similarly, L solved for a missing angle and told his partner, “This angle is supposed to be more than 45 degrees, I can tell because the opposite leg is longer than the adjacent leg, but I got that the angle is 37 degrees. I think I used the wrong trig function.” Can we pause for a moment and appreciate the sense-making going on here? And the perseverance and confidence in identifying their mistakes? Oh, and that D is a senior repeating Math 2 who has been telling me all semester that she sucks at math, and that L got so frustrated during a quiz earlier this year that he crumpled it up and threw it on the floor? The growth these two have shown this year is remarkable.

M is a student in my Math Foundations class who likes to doodle more than do math. I hadn’t seen a lot of quality work from her in the past few weeks, so I pulled her up to the board to solve a system using substitution, just to get a feel for where she was. She started working on the problem, looking at me for approval after each step; I’m just over here, poker face on, sipping my coffee, giving her absolutely nothing back. She did the whole problem, start to finish, perfectly with no help. When I said, “M, that was perfect!” she smiled, and said, “really?!?!?” “Yeah, really! You’ve got this!” She then went on to explain to her teammates how to do the problem, then did all of the remaining problems on her assignment. Then she created another problem for her teammate to solve. And – the MOST AMAZING PART OF ALL – M has not been off task once in class since (that was 2 weeks ago!!!).

K was eager to share that she earned a perfect score on her most recent Math 3 quiz, and L earned her first score of 4 ever! These young women are in my Math Lab class and have been working their tails off to be successful in Math 3. It’s amazing to see their hard work paying off.

Former student J wrote me a Christmas card that said “Calculus is hard! Thank you for teaching me to value making mistakes and learning from them.” I am so incredibly thankful for that because I feel like I’m not doing as well emphasizing that this year, and it keeps me striving to improve.

Surprise win – I gave the system of equations pictured above to my Math Foundations class (thank you IM) – it looks pretty intimidating at first. Many of my students tried starting at the top or didn’t try at all. I encouraged them to pause and look at the whole problem, to find the part that they think looks the easiest, and one student said, “the bottom equation looks easy, I can just divide by 2.” Once everyone saw that, the problem didn’t seem so complicated anymore. Every single student was able to find values for all 4 variables – even if it took the whole class period, they stuck with it! Something about this problem just clicked for them, and now they are solving much more algebraically complicated systems using substitution with ease.

I am so incredibly proud of these wins from my talented math students. Please join me in celebrating their hard work!

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:  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. 🙂

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

Posted in Assessment

The case against “The Case for Not Allowing Test Retakes”

We interrupt our regularly scheduled programming to bring you this important message…

Last night on Twitter I came across this tweet: 

Go ahead and read the article… I’ll wait 🙂

The article really struck a chord with me, and not in a good way. Actually it sent me on a 10-tweet rant, and when I woke up this morning, I still felt compelled to write about it.

Anyone who has worked with me knows my passion for Standards-Based Grading. I started this in my first year of teaching (before I even knew that SBG was a thing), and I am continually striving to improve our grading system so that it meets the needs of both students and teachers. We have modified every year to address different needs/issues that arise. It’s never been perfect. It’s still not. But the pros so outweigh the cons that I would never go back. [As a side note, here’s a pervasive problem in education – if something doesn’t work perfectly the first time, we throw it out instead of working to solve the problems we have. But I digress…]

The author bases this article on his “experience with policies around allowing test retakes, dropping late work penalties, and prohibiting zeros.” I am not clear on what these experiences are – what this teacher has tried, what issues arose, what he did to try to fix them – but I’ll build my arguments here on my own experience.

The traditional policies—giving each assessment only once, penalizing late work, and giving zeros in some situations—help most students maximize their learning and improve their time management skills, preparing them for success in college and career.

I find this statement problematic for many reasons. First, the idea that these policies help “most” students. How many is most? 99%? 51%? There’s a big difference. Yes, we have a lot of students who have figured out how to play the game of school, but what about the other 1-49% of students who are not helped by these policies? I see grading reform as looking for ways to move closer to “all.” I started using SBG to help students help themselves. I wanted all students to be able to look at an assessment and know where they excelled and where they needed to improve. I found that students did not understand how to “maximize their learning” with “traditional” grading policies, so I looked for ways to make the learning process more coherent for all students. Am I reaching every single student? Not yet… but now I’m better able to strategically intervene with students based on specific areas that they haven’t yet grasped.

But this author doesn’t focus on content learning as much as improving “time management skills, preparing them for success in college and career.” So let’s talk more about these interpersonal skills that are taught through “traditional” grading practices.

Deadlines motivate force me to finish something. I am a procrastinator/perfectionist: I am so sure that the longer I keep working on a project, the better I can make it, so I tend to wait until the last minute to submit. (This is why it takes me so long to blog. EDIT: this is why I’m still working on this blog post 2 days later…) But if I’m past a deadline, do I keep going? If I don’t get “credit” for my late work, what’s the point? Oh well, maybe next time. Except what have I learned? I’ve learned that if I don’t submit something on time, I never have to learn that content; in fact, my teacher won’t even let me turn in that project to show what I’ve learned. I have NOT learned how to “improve [my] time management skills” so that this won’t happen again. I am NOT prepared for college because I still don’t know how to “self-assess and then self-advocate to get the help that [I] need.” I need a teacher to help me learn how to do this.

The author purports that these new policies are designed for students who are intrinsically motivated, and that many students need extrinsic motivation. I’m not sure what evidence there is to support the first point, but I would argue that traditional grading practices decrease motivation. On a traditional 100 point scale for grading, a zero is detrimental. Let’s say there are four major assignments in my class; if I earn a zero on the first assignment, even if I earn 100% on the next three, the maximum score I can earn is 75%. If the best grade I can earn is a C, even if I try my very hardest, why bother? Using SBG, on the other hand, I’ve seen an increase in motivation and buy-in from students because they know that I am continually assessing their progress. They know that if they don’t get it now, they can keep working at it and keep learning. Students learn at different rates and we must honor that in both our words and our actions, which is why the following quote hurts my heart the most:

In math classes, where concepts constantly build on one another, traditional policies hold students to schedules that keep them learning with the class. This makes kids practice and check their practice on a timeline determined by the teacher that will have them ready to test before the test.

As much as teachers plan lessons and try to keep a certain pace, it is student learning that truly dictates the timeline of the class. Please don’t misunderstand me: I don’t stop moving forward until every single student gets a concept – that’s not good for anyone. But it’s also not good to stick to a timeline for the sake of a timeline, at the expense of student learning. I find that SBG allows me to continue with the course at a reasonable pace, while allowing for students to continue to improve their understanding at their own pace.

But isn’t math sequential? The author points this out in his paragraph on mental health:

My math class builds sequentially: Mastery in early units helps students be successful in the following units. In my experience, traditional policies motivate students to maximize their learning in the first unit, which helps them on every later unit.

The great thing about math being sequential is that we’re constantly practicing the skills that we’ve learned! For example, in my Algebra class, we would learn the distributive property early in the course. Students would make mistakes (distributing a negative!) that would lead to lower scores on assessments. However, we would use it all the time in every unit that came after, and guess what… they got pretty good at the distributive property! Of course, this happens in every class, but with SBG, their grade now reflects their current understanding, instead of their quiz score from September.

Since the author brought it up, let’s talk about mental health for a second. I’m going to ignore the HUGE leap from soft deadlines to Snapchat and instead focus on this statement in particular: “Under retake policies, parents at my school have expressed concerns about how overwhelmed their children become due to being caught in a vicious cycle of retakes.” Although my entire school hasn’t enacted these policies (yet), I have seen the opposite happen in our math classes. Instead of stressing out over an assessment, students feel more relaxed because they know that one bomb won’t tank their grade, that we’ll keep working at it until they get it. I also employ highlight grading to refocus assessments on learning rather than grades. I got several sweet notes this year from students who specifically named these practices as helping them “relax and focus on learning rather than stressing about grades.”

The author’s third point is on teacher effectiveness. I’m not going to lie; the first year of reforming grading policies is tough stuff. But I find that to be true with everything! The first time you try something new, it feels clunky and time-consuming because you’re having to make conscious decisions about everything. Think about your first year teaching: those lesson plans probably took a long time. Now they are second nature, so it doesn’t take you nearly as long. Same with implementing SBG. Once you have a system in place, it becomes second nature. The author points out, “every minute writing and grading retakes or grading long-overdue work is a minute that I’m not planning effective and creative instruction,” but I have come to find that I can’t plan a lesson without knowing where my students are. It is the assessment that feeds the instruction, and with SBG, I know exactly where my students are and SO DO THEY. Ultimately they are better equipped to make decisions as to how to improve, and they start doing the heavy lifting, not me. Presto changeo, teacher effectiveness.

Finally, and most importantly IMHO, I’d like to address specifically the fundamental mindset shift that must occur before making any changes in grading policies, the mindset that seems to undergird this article and many arguments like it. Many teachers think that the goal of changing grading practices is giving students better grades, which is reflected in the author’s focus on options (“test retakes, dropping late work penalties, and prohibiting zeros”) rather than interests (increased student learning). Although I’m thrilled students are earning better grades, I’m more concerned with aligning student grades with their current understanding.

Shoutout to @mrdardy for sharing this article and getting me all fired up! 🙂

Posted in Assessment, Calculus, First day

AP Calculus BC: reflection and ideas for the future

I loved teaching AP Calculus BC last year! I was a little intimidated at first… I hadn’t taught or even used Calculus in 15 ish years. It was a great experience and I can’t wait to do it again! But alas… not enough sections this upcoming year. I do want to reflect on what I did and what I would like to do in the future so I don’t forget.

My favorite lesson was actually the first day of school. I showed three linear graphs and asked students to write down everything they knew about them.

First day graphs

After talking about key features – intercepts, end behavior, domain and range – and linear vs. constant functions and a line that’s not even a function, I surprised them: these aren’t lines!

First day graphs 2

Students worked in teams to match each graph with an equation and come up with a convincing reason why. Then teams presented their solutions via gallery walk.    

First day student 2   img_1357

We touched on domain and range, even and odd functions, and other key features of functions; we also began exploring the ideas of limits and local linearity. Students really latched onto the idea of “zooming in” on the graph, which ultimately paid huge dividends throughout the course. I found myself saying weekly, if not daily, “remember the first day of school when we zoomed in on those three graphs?”

Another win last year was practicing derivatives and (some) antiderivatives at the same time. This is one of my favorite teaching strategies in general; if you can do something one way, you should be able to think “backwards.” I only did basics, like the table below, but it really got students thinking both directions.

I like to give a little peek into what’s coming, and since we already knew that the derivative of a constant is zero, it wasn’t too challenging to sneak in conversation around this “+C” business early.

Here’s another one, this time after trig derivatives and chain rule:

In the future I would like to do this even more intentionally and often.

Around the same time, we were sketching derivatives given a graph, so I threw in some antiderivatives in the same way: if this is the derivative graph, what could the original graph look like? Here they could really see the “+C” as teams came up with different vertical translations of the same graph.

As much as I loved exploring and wondering about these different ideas, the biggest struggle I had was the order in which to introduce concepts. Everything is so connected, and I didn’t want to muddle their understanding by doing too much at once. I ended up sticking pretty close to our textbook; especially in my first year, especially since my PLC teammate would also be following the textbook, especially since the instructor of my APSI used the same textbook at his school and provided me with a bunch of already aligned supplemental resources (!!!), it felt important to stay consistent.

Now that I have a little more “big picture” vision, I would do things a little differently.

First, standards-based grading! I started this in second semester last year, when we began studying series. It was a natural break to begin something new, but the content turned out to be perfect for the kind of self-reflection and improvement that I needed from students. (I’ve been implementing various versions of SBG every year I’ve been teaching; this might be my favorite version so far, but still far from perfect.)

Learning Objective 4.1A – Determine whether a series converges or diverges.

This learning objective encompasses A LOT of different tests for convergence, and what I found is that our textbook gave very bare answers to homework problems, so students thought they were doing great when they were actually missing big chunks of explanation that would communicate sound mathematical reasoning (and let’s be honest, would cost them points on the AP exam). By changing my grading strategy, I was able to better communicate my expectations and give feedback for improving.

After our first quiz on the topic, I gave students their highlighted quizzes back (no points marked, my typical M.O.) along with a Google doc for reflection. This included a rubric for them to score their own progress on this learning objective.

Rubric series

I showed an example of a 4 to the whole class, emphasizing three parts to communicating whether a series converges: conditions, evidence, conclusion. Many students were providing the evidence, but not first checking that the conditions for using the convergence test have been met or providing an appropriate conclusion; we looked at two examples of 3s that illustrated these omissions. Then students made corrections to each problem on their quiz and gave an overall score for their understanding with evidence to support their score. Ultimately they were very accurate and I only had a couple of conversations with students whose evaluation of their learning was different from mine.

The best part – every student earned a 4 on their next assessment of the same learning target. After a whirlwind first semester, I finally felt like I was doing something right!

I always give cumulative quizzes/exams, so this quiz also included derivatives from first semester, along with a rubric for self-evaluation similar to the one above. I got some very honest, “I thought I was a 4, but this evidence says that I’m currently at a 2” evaluations. They realized for themselves that they still had some work to do on the “easy” stuff! I really hope to start this from the very beginning of the year when/if I get to teach this course again… I think the potential for continuous learning here is huge.

The other thing that I would really want to do is address the sequence of content; I still don’t have a good feel for this yet. In my other courses, I try to space out related content so that students have time to practice content before trying to build on it. For example, in EM3, we teach unit circle and trig functions (all no calculator) in unit 2, practice it throughout homework in units 3 and 4, then unit 5 brings in inverse trig functions after they have built up fluency. I don’t know yet what this looks like in Calculus, but I would like to attempt it. Thoughts so far:

  • Do some work with series earlier in the course. Chapter 10 in our textbook is all about series and it is HEFTY. It could easily be two units with all of the convergence tests taught earlier and practiced before needing them to test endpoints for intervals of convergence later.
  • Teach derivatives and integrals simultaneously??? Maybe start with graphical and tabular interpretations of both, then move to analytic approaches? Can this be done??? I have more questions than answers.
  • I felt like I needed more time for teaching and practicing integration techniques, so find a way to do this earlier. My students sucked at u-substitution, honestly because I think I sucked at teaching it. Maybe by teaching chain rule and “backwards chain rule” together, students will become more fluent. Maybe by separating it from other integration techniques and letting them get good at one technique at a time, they can feel more equipped when I ask them to compare and contrast strategies. Maybe Coaching Questions would be a good routine to use practice using integration techniques.
  • Polar area – I feel like I can sneak this in earlier with other area integration problems. Still not sure.
  • The construct of teaching all AB content first, then moving to BC content, seems forced. Is there a reason for doing this, other than my textbook said so? Yes, the AB content is foundational for BC content, but can/should it be better integrated?

And there we have it folks – a lot of partially formed ideas about AP Calculus BC! I really do hope to teach this class again. Shoutout to my department chair – if you’re reading this, in case I haven’t said this enough, I would LOOOOOOOOVE to teach AP Calc again in the future. 😉 😉 😉

Posted in Uncategorized

Summer time = idea time

Blogging is hard. I really admire people who can carve out the time to do it regularly! I haven’t found a groove yet.

But one of my goals for the next year is to give myself more grace. There is so much that I want to do that I don’t always get done and I feel bad for it. Instead, I’m going to try to read more often, play with my kiddos, watch movies with my husband, lay in the hammock with a glass of wine, and try not to let all the things I “should” be or “could” be doing to do creep into my head.

Because there’s a lot of things! Summer is when I start to dream about the new school year: setting up new routines to make my classroom run more smoothly, inventing far too elaborate projects, trying yet another new way to implement homework that supports students and doesn’t make me crazy, etc. etc. Next year, I’ll be teaching 4 different courses, so I have even more ideas than usual swirling around in my head.

In my classes, I try to encourage students to have a “rough draft mindset.” Math thinking can be messy as we’re playing with a problem and trying different strategies. When we find our pathway through a problem, we edit and revise, seek feedback from peers, and rewrite our solutions to make our thinking clear and coherent to the reader. Now it’s my turn to employ this same process for planning for next year! I thought I would start writing about all of these ideas I have swirling around in my head, and turn some of these half-baked ideas into a whole series of blog posts. I’ll seek feedback from peers, and maybe… just maybe… they’ll eventually turn into real live lesson plans! (GASP!) Here are some of the ideas that I hope to expand on in a series of posts:

This year will be the third year of existence for Enhanced Math 3 (think Math 3 + Honors Precalculus all in one school year) and my third year teaching it. I’m really proud of all of the work I’ve put into this course and I think it’s a really great course! My goal for next year is to move to standards-based grading. I have convinced my colleague that this is an interesting idea, but he’s not ready to move forward with implementation quite yet. We’ve landed on a compromise that I like a lot! We’ll have students track their progress and use evidence to argue for the grade they think they should have.

I love teaching Math 2. I taught it two years ago when I first came back to teaching full time and I was disappointed to not teach it again last year. But I’ve used up all of my ideas for this course already! 🙂 I think this year will be more about refining and improving already existing structures, like standards-based quizzes, Ready Set Go homework, and spacing units for mastery over time.

AP scores just came out and I’m really proud of my AP Calculus BC students! I have so many ideas for what worked well last year and how I want to improve… I want to expand on the SBG structure I tried second semester, reorganize units, and rely less on the textbook and more on bigger conceptual ideas, like my first day of school lesson that paid dividends the entire year! Too bad I’m not teaching it again 😦 but I want to record all of my ideas so I don’t forget for when/if I have the opportunity to teach this course again!

I’ve already mentioned standards-based grading a lot. I have been implementing and revising various versions of SBG for 12+ years, since before I knew it was called standards-based grading. It’s a passion of mine and although I realize that it’s not the cure-all for problems in education, I have seen such tremendous transformation in math instruction at our school after implementing these principles. I could write 100 blog posts about our journey and how much I love what we do at my school.

The class I may be most excited and nervous about is our brand new Math Foundations course, which will be offered concurrently with Math 1. We have eliminated our Math 1AB course, which was originally designed to be a two-year Math 1 course (Math 1AB + Math 1CD); we eliminated Math 1CD a few years ago, so students who take Math 1AB then take Math 1 the year after. Starting this year, every ninth grader will enter high school taking Math 1 (or above), and those who have been identified as needing additional support will also take Math Foundations. This will be more structured than our Math Lab class (which I’m also teaching) which will be geared towards Math 2 and Math 3 students this year.

Of course, how could we start a new school year without talking about the first week of school? I plan to continue the good stuff I’ve done (thank you Sara van der Werf) and sprinkle in a few new ideas (thank you Howie Hua). But I also don’t want to overwhelm myself and my students with too much.

With that, my summer planning series is born! Even just thinking about writing about planning has helped me clarify my thinking. I look forward to writing my rough draft about my ideas, seeking feedback from peers and making revisions, and making the final product in my classroom next year the very best it can be.

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! 🙂

Posted in Math II, Quadratics

My undying love for Desmos

For the past two years, I’ve been out of the classroom working as the high school math specialist for my district, so I’ve been totally jealous of everyone using Desmos classroom activities with students. I’ve presented trainings on how to use Desmos classroom activities and Activity Builder; I use the Desmos app on my phone every day; I bug my teachers to use Desmos at every opportunity; I am convinced it’s the most powerful math learning tool there is; I just haven’t gotten to use it with my own students … Until now. And it was every bit as glorious as I thought it would be.

First up: Marbleslides Parabolas. I have never seen students so intent on a problem in math class before. I mean, I know the students find me FASCINATING and everything, but seriously, the amount of learning going on in the room was amazing. I started taking pictures and they didn’t even notice! I set students up in pairs on chrome books and let them go. I was a little afraid that students would start changing numbers randomly without paying attention to the cause and effect, but from every pair I heard some version of “change that number so we can move it over to the left.”

Only a couple of sticking points: I know it shouldn’t, but it still surprises me that students weren’t willing to just hit the “Launch” button and watch what happened; they were trying to get each graph perfect before launching. I need to work on growth mindset with this group. This population struggles with the need to look “smart” in front of their classmates and aren’t willing to put themselves out there if it means being wrong. Additionally some groups got hung up on the domain; they wanted to “move the parabola to the right more” when they really meant that they wanted to change the domain. It took a lot of questioning to coach them to see how the domain restrictions were affecting the graph.

Next day, we followed up with another Desmos activity, Quadratic Transformations, that I borrowed generously from Mary Bourassa‘s Quadratic Transformations part 1 and part 2. Basically I loved her activities and wanted to condense them down to one day to solidify what we had learned from Marbleslides the day before. Students were engaged and thinking deeply, and they were able to apply what they learned to graphing and writing equations in vertex form the following day.

As a teacher, I never sit still while students are working (actually u have trouble sitting still no matter what… ADHD anyone?). I love hearing students’ conversations, seeing what they’re doing, and challenging their thinking (read: bothering them). So I walked around the classroom as they worked and didn’t really take advantage of the teacher dashboard. I think that’s going to take me some time to get used to. Maybe if I accessed the teacher dashboard on my iPad, I could still walk around the room and feel comfortable doing so. Does anyone have any tips for using the teacher dashboard effectively?

I’m looking forward to my next Desmos activities; I have two planned for next month: Building Polynomials #1 and Roots of Quadratic Functions: Looking for Special Cases.

I love my husband. I love my two daughters. I love my friends and my colleagues and my students. And I love Desmos.