The power of the feedback loop

Note: I have some phenomenal colleagues in my school. One of them gave a powerful presentation about some changes she made in her classroom, and I asked her to write a guest post on it! The kicker: she’s not a math teacher. She teaches French. But pedagogy can transcend the subject matter at hand, and this is one of those cases. So enjoy!


When I adopted a no-homework model for my classes several years ago, my role as a teacher shifted drastically. I was no longer strictly giving instruction, but rather facilitating the movement from one activity to the next and offering on-the-spot feedback and answering questions that my students might have. The goal was to remove myself from the equation as much as possible and put the students at the center of their learning. With all of the emphasis placed on class time, it became incumbent on the student to focus completely and participate thoroughly in each activity. It also became incumbent on me to come up with a system that would allow me to objectively and accurately calculate the quality of student note-taking and participation during class.

The rubric I currently use in my French classes was designed to allow for effective and efficient use of class time, which, in turn, facilitates maximum learning. It looks like this:

  • Is punctual
  • Is ready to work at the start of class
  • Takes active notes, keeps an organized notebook
  • When speaking to the teacher, uses French only
  • Engages in activities in French
  • Engages in activities for the duration of the time indicated

Each of the six components is worth 1 point per class day, for a potential total of 36 points per cycle. I designed a page that has this rubric at the top and a box for each day of the cycle underneath, and I keep a copy of it on my clipboard at all times:



Whenever a student makes an infraction, I point it out to him or her and I write it down immediately in the box corresponding to the day of the cycle. On day 1 of cycle 3, for example, I noted that three boys were not prepared to work at the beginning of class. I also collect the students’ notebooks daily and write down any issues regarding the quality and organization of their written work in these boxes as well. You can see an example of that on day 2 of cycle 3, when two boys passed in notebooks that had missing or incomplete notes. At the end of the cycle, I calculate the points lost and keep a running tally of total points in my gradebook.


In my work this year with several colleagues regarding the importance of feedback, it became apparent to me that it would be useful for my students to have the opportunity to see and discuss the breakdown of the information from these pages. So I organized a table that allows for the student to see when and how many points were lost for each component. I also included on the page the overall GPA, as well as a list of commendations, areas for improvement, and suggested challenges. I then scheduled 10-minute individual conferences during breaks and community time to discuss the results. Below is an example of one of these reports :


Student : Jean-Paul de la Montagne

Total Notes & Participation points

  • mid-semester 1 : 139/156
  • semester 1 : 97/108
Total infractions Distribution
Is punctual 1 Cycle 2
Is ready to work at the start of class 2 Cycles 3, 6
Takes active notes, keeps an organized notebook 13 Cycles 2 – 7, 9
Speaks French only (with the teacher) 0
Concentrates on activities / Engages fully in activities / Participates for the expected duration 12 (chatting, following instructions) 3-10


  Mid-semester 1 Semester Average
GPA 89.74 94.7 92.2

 Commendations :

  • accurate accent
  • ability to properly formulate full, complex sentences
  • frequently volunteers answers/comments during large group work
  • notable increase in use of French with peers

 Areas for improvement :

  • consistency in the quality of note-taking
  • drop the habit of chatting

Suggested challenges :

  • read Daniel Pennac’s L’œil du loup
  • watch movies, listen to songs in French

 This intensive participation grading model allowed me to remove subjectivity and emotion from my participation grades. It also eliminated the potential for students or their parents to debate the grade. The final step of conferencing with each of my students was the piece I’ve been missing all these years. These conferences yielded almost 100% reduction in the behaviours that hinder productivity and learning, not to mention costing students points.

My ultimate take-away from this experience is that providing students direct feedback on the quality of their notes and class participation resulted in the kind of behaviour modifications that have made for an even more effective learning environment. In a no-homework class where every minute counts, this is key. I am so excited about what this experience has taught me, and am looking forward to refining it in the future.


An Prelude to Unit Circle Trigonometry

This year in standard precalculus, one of my collaborators shared an approach to dipping our toes into unit circle trigonometry that I loved. She learned it from at her old school. I love how math teaching ideas spread, percolate, adapt!

Here’s the gold: don’t start with a unit circle. We’re going to prime students with thinking about x and y coordinates for other shapes. In our case, we used:

We placed a bug at (1,0) and had the bug walk counterclockwise around the figure at “unit speed” (one unit per second). And we asked questions about the position of the bug at various times, and had students create x(t) and y(t) graphs. The choices for the shapes that my colleague chose to use was inspired. Why?

They start out easy. The square is easy to come up with x(t) and y(t) graphs. The diamond (okay, geometry peeps, I know it’s a square!) is also pretty easy, except students have to recognize that it takes \sqrt{2} seconds to go from one vertex to the next (so they have to pick a good time scale on their x(t) and y(t) graphs). It also… dum dum dum… harks back ye olde 45-45-90 triangle! Great when you’re about to start unit circle trig, no? And of course the last one is trickiest, because it requires the use of some deeper thinking and involves the pythagorean theorem and/or knowledge of 30-60-90 triangle!

You can imagine the great discussions that could arise, right?

Here are the x(t) and y(t) graphs:




 Some great topics of conversation:

  1. Why do some graphs have “horizontal” segments and others don’t?
  2. What is similar about the x(t) graph and the y(t) graph for each figure? What is different?
  3. What do all six graphs have in common?
  4. Explain the slopes of the graphs for the triangle.
  5. Highlight the part of the original shapes which corresponds to when y(t) is negative. And the parts of the y(t) graphs where y(t) is negative.

What’s nice is that the term “periodic” came up naturally (so we could define what a period is). The idea of domain and range came up naturally. And, whoa, neat, some of the graphs were “the same shape but one was just a shift left/right of the other” (*cough cough sine cough cosine*).

I also love that this approach brings up parametrics for free! And the backwards question — giving x(t) and y(t) graphs and coming up with the path of the bug — is golden.

IMHO the introduction of unit circle trigonometry through this approach was marvelous. I am going to share with you the document we made for this (.docx ; PDF on scribd). However, I freely admit that I think this document didn’t lead to the smoothest classes. It felt like a series of exercises instead of a series of puzzles. Looking back, there was too much structure involved. I would have liked a bit more experimentation and play, and a little less formality, at the start. I have a few thoughts about this — especially around my attempt to have students make predictions, but I know I could have done a better job. (I see Desmos as being a possible tool if I were to modify this. I also wonder if it could be “gamified” in some way.) My only thought right now is to have a set of 5 shapes to start with (not the square, diamond, triangle) and 14 possible graphs. And students need to find an x(t) graph and a y(t) graph for each of the five shapes (and 4 graphs are left over).

Being critical, overall I would give this approach an A- for “idea” and B for “execution.” As I noted, I could have structured things to be more smooth

Why A- for “idea” and not an A? It has a contrived framing. A bug is walking around a path. That framing doesn’t quite make me “excited” to study it. I’m not hooked as a student. Is there a related question or framing that could get me hooked? Any thoughts?

If you do end up using this idea, please share any changes you made in the comments if you remember… I’d love to hear how the general idea morphs when used other classes!


Multiple Representations for Trigonometric Equations

I have to say that we’re doing some pretty neat stuff for trig this year in precalculus. I’m working with two other teachers and totally writing everything we’re doing from scratch. I had about 3 days to teach solve some basic trigonometric equations. They are basic. Like 2\sin(x)+5=4.7. But we’ve put a lot of thought into what we’re teaching, how we’re teaching it, and why we’re teaching it — and more complicated trig equations just didn’t make the cut. [1]

Besides not-a-lot-of-time, the other bugaboo I was contending was how to deal with inverse trig. Long story short, I’ve found a way to teach inverse trig which makes me fairly happy in my advanced precalculus class. But because of our time constraints, I decided that we could get my standard precalculus kids solving trig equations without understanding the theory behind the restricted domain of inverse trig functions. :) Why? They learned years ago in geometry that if they have a triangle like the one below


they could get an angle, like angle A, by writing: \sin(A)=\frac{3}{5}. And then using the inverse of sine, they could get A=\sin^{-1}(\frac{3}{5})\approx 36.87^oThey know about the inverse trig functions already. So I wanted to exploit that fact.  And if organically a question about what the calculator was doing when spitting out an answer, and why it only gave one answer, I promised myself I would address it. (This year, no question like that arose.)


A quick last note, before I shared how I approached these few days in class, I decided to totally eliminate the use of the term “reference angle.” Kids would discover the relationships among the solutions of trig equations on their own. No need for new terminology here. Just logic.

Day 1: Three important “do nows”


This led to a great discussion. Every group decided the “top left equation” was going to be the easiest. And every group decided that the log and tangent equations were going to be the hardest. When I pressed them on why, they said it’s because they forgot logarithms from last year, and that tangent was just kinda tricky. They could “undo” a square root or a square, but they didn’t really know how to “undo” a logarithm or tangent function.

Next I threw up this slide. I just wanted to remind kids that sometimes there are more than one solution to equations — even simple equations they know. I also wanted them to see that they knew something about the tangent equation. They knew it had infinitely many solutions — even though they might (right now) know what those solutions are!


Finally, I wanted to do a serious review of special angles and their relationship with the unit circle. So I had kids spend 5 minutes solving these basic trig equations.


Obviously I put the unit circle on there as a prompt to get them thinking. And YES, that last trig equation, with the 3/7ths, was done on purpose. I asked kids after they got stuck on it if there were some of these they would not want to appear on a pop quiz. They all recognized that the 3/7th one was bad because it wasn’t one of the coordinates associated with the special angles.

This laid the groundwork for the packet.

[docx editable version: 2017-04-24 Basic Trigonometric Equations]

Kids had good conversations and were able to solve equations like \sin(x)=0.3 and \cos(x)=-0.8 using the unit circle/protractor, a detailed graph of the sine and cosine waves, and using their calculators to get fairly precise answers.

Their nightly work was simply to finish the sine and cosine questions in Part 1 (questions #1-4).

Day 2: Expanding Understanding

I started with an awesome “do now.”


I thought this was going to be a quick 4-5 minute discussion. But kids took 3-4 minutes just to really talk in their groups. And I had them share their thinking. It led to kids talking about “efficiency” and “conceptual understanding” themselves! They all pretty much though the unit circle was the best way to solve it — even with the annoyance of the protractor — because they liked the conceptual understanding it provided. They thought the calculator did the work quickly, and was more accurate, but it annoyingly only gave one of the solutions (so you had to use logic and the unit circle to figure out the second solution), and you could easily forget the meaning of what you were doing. I was so proud of what they were saying. Super awesome metacognition! All in all, this was probably 7-8 minutes.

Then I let them loose on the tangent questions in the packet (Part I #5 and 6). They initially had to solve \tan(x)=1.1 using a protractor. Every single group remembered tangent represented slope. Most groups reasoned that if \tan(x)=1, they would get 45^o and 225^o as their solutions. And since this slope was slightly greater than 1, the angles would be slightly different, just a few degrees higher. It was lovely. (And exactly what I hoped would happen, which is why I chose to use 1.1 in the equation.) But one group literally drew a line with a slope of 1.1 and measured the angles associated with that. I wasn’t surprised that a group did that, but I expected a few more to do so. (I had this group share their thinking with the rest of the class, at the end of the period.)

Then kids spent the rest of the class working on select questions in Part II (8, 9, 11) and Part III (13, 14, 15).

For nightly work, kids finished any of those problems (#8, 9, 11, 13, 14, 15) that they didn’t finish up in class.

Day 3: Polishing Things Off

I started with a question that I wanted to reinforce after the previous class:




We did a bit of review of some unrelated Algebra II ideas to help set them up for our next unit on polynomials. And then…

… to work! I had kids discuss problem 13 in their groups first (since I could see that being a place where a kid, at home, might get trapped… and I wanted them to use each other to get unstuck). And then they compared their answers to the other nightly work questions — and used a solution sheet I gave them to see if they were correct. Then I set them loose on using Desmos to do Part IV. The rest of the period was spent working on finishing up the problems that weren’t assigned in the packet (the ones they skipped).

Pretty much all groups were working together amazingly, and when I went around to check in on different groups, everyone was getting all the questions correct. The biggest problem was actually finding a good window in Desmos! If that’s the biggest problem, I’m golden.


What I loved:

Okay, so I’m going to toot my own horn here. Although the packet may “look” simple, I have to say the only way to see why it’s so awesome is to actually do it. The choice of having kids solve \sin(x)=0.3 and then immediately solve \sin(x)=-0.3 was on purpose, to generate good conversations with kids about reference angles without using that term. The choice of \tan(x)=1.1 was done specifically to exploit their understanding of \tan(x)=1. And the fact that they’re constantly looking at the same question through three different lenses (unit circle, wave, calculator) is deliciously sweet. And then — at the very end — they get to see the solution a fourth way, by using Desmos to graph these equations to find a solution? SO COOL. Because the very last thing we had done in this class was learning transformations of sine and cosine graphs! [2]

This packet, and associated “do nows” and conversations, did what I was hoping for. It highlighted multiple representations. It had kids thinking deeply about the meaning of sine, cosine, and tangent. It had kids develop a way to understand multiple solutions to trig equations by simply using logic and what they know. It had kids recognize that the more they understand trigonometry, the more ways they have to solve a trig problem. And no kid got derailed because they didn’t understand inverses deeply.


[1] I could argue a case for these type of equations, as well as a case against them. But considering our goals and what we’ve already done with trig, I think we’re making the right decision. Why? Because our goal isn’t solving algebraic equations writ large, and I could see solving something like 2\sin^2(2x-180)=5 being useful for that. But for getting a deeper understanding of the trigonometric functions? I see less value. (Not no value, mind you, but less…)

[2] We did this in a deliciously marvelous way. I hope to blog about it!

A Beautiful Mistake

In Precalculus, we’re working on solving basic trigonometric equations. A student was working on this problem:


And he made an error on his calculator and accidentally typed \tan^{-1}(-0.1). He got an output of -5.711^o. I think he realized his error when comparing his answer to his partner, who typed in the right expression into his calculator: \sin^{-1}(-0.1)\approx -5.739^o.

And his curiosity was piqued. Was it a coincidence that the two results were the same?

Of course my curiosity was piqued too. How could it not be? And his question led me to trying to figure this out on the fly. Why were the two results so close? A difference of about 0.028^o. I tried to wrap my head around that… Even in the context of these 5^o results, that is so miniscule!

So in this short post I’m going to share what I did at this moment. In total, this took about 3 minutes.

  1. I acknowledged it was so bizarre that the two results were so close, and that the question of why that might be was an awesome question. I said to the student: let me share how I’m going to think about this with you, and maybe we can figure this out.
  2. I throw desmos on the screen. The rest of the kids are working in their groups on something else, so I’m just working with this one kid and his partner. I switch desmos to degree mode, get a good window, and type in the following:
  3. I zoom in around y=-0.1.

    and then I make the sine curve disappear, so we only saw the tangent curve. And then I made the tangent curve disappear, so we only saw the sine curve. I said: “if these curves weren’t different colors, would you be able to tell them apart?” (Leading question. Obvious answer prevails. No.)

  4. So I said: it’s weird that around here, for small angles, the sine graph and tangent graph look the same. But that’s not true for most angles. So I’m wondering what it is about sine and tangent which make them both similar for small angles.
  5. And then it strikes me. So I share my insight: “What is the meaning of tangent again, graphically?” And we review that tangent is slope, which is steepness, which is rise over run, which is y over x, which is sine over cosine.
  6. So I write on the board: \tan(x)=\frac{\sin(x)}{\cos(x)} And I say: let’s look at what happens for input angles close to 0^o. And here he has the insight that for these angles, the denominator is really close to 1. So we’re left with \tan(x)\approx\sin(x). [1]
  7. I was elated at this. At the question, and positively giggly that I was able to figure it out using graphing and simple logic. And I remember saying that “This was the most interesting math thing I’ve thought about this whole week! Thank you!”

Why did I want to write a blogpost about this? Not because it was a good learning experience for the kid who asked it. I literally did all the thinking and shared my insights as I had them with him. (So it shows him he has a teacher who values his questions and enjoys problem solving, but it didn’t really push forward his content knowledge much.)

The reason I wanted to write it is because I immediately saw that this could be an amazing learning opportunity for students next year if I design it carefully. I could see spending a good 20 minutes of class on this question. I give groups giant whiteboards. I give them a prompt (which I will draft below). I have some hint envelopes at the ready. And I encourage the use of desmos (which would encourage some graphing work!).

Last year I had a student who accidentally typed something incorrectly in his calculator. He typed \tan^{-1}(-0.1) instead of \sin^{-1}(-0.1). He realized he had an error only after doing a super careful comparison of his answer with his partner. Their answers differed by a minuscule amount, a mere 0.03 degrees. Imagine that angle! How small that difference in angle is! This student was left wondering if this was just a strange coincidence or not. It turns out that it is not a strange coincidence, and there is a reason that the two outputs were super similar. Your task is to figure out why! Use Desmos! Talk to each other! Go to the whiteboards! Exploit what you know about sine and tangent! Figure out what the devil is going on!

What I love about this question is that its concrete, but also brings up so much conceptual knowledge. Kids have to understand what inputs and outputs of inverse trig functions are. Kids have to know what sine and tangent represent on a unit circle. Kids might even look at graphs! But I could see different groups getting at an explanation in two different ways… Some using a unit circle. Some using desmos like me. And maybe some using some method I haven’t thought of!

I also thought what a fun question this could be if translated for a calculus class. A consequence of the fact that the graphs look the same for small angles is that their derivatives will also look the same for small angles. And also the taylor series approximations for sine and tangent will be similar-ish — for the lowest order term, in any case!

[1] Admittedly some handwaving here. That’s why we have calculus!

Graham’s Number

TL;DR: If you have an extra 45-60 minute class and want to expose your 9th/10th/11th/12th graders to a mindblowingly huge number and show them a bit about modern mathematics, this might be an option!

In one of my precalculus classes, a few kids wanted to learn about infinity after I mentioned that there were different kinds of infinity. So, like a fool, I promised them that I would try to build a 30 minute or so lesson about infinity into our curriculum.

As I started to try to draft it — the initial idea was to get some pretty concrete thinkers to really understand Cantor’s diagonalization argument — I decided to build up to the idea of infinity by first talking about super crazy large numbers. And that’s where my plan got totally derailed. Stupid brain. At the end of two hours, I had a lesson on a crazy large number, and nothing on infinity. You know, when that “warm up” question takes the whole class? That’s like what happened here… But obvi I was stoked to actually try it out in the classroom.

In this post, I’m going to show you what the lesson was, and how I went through it, with some advice for you in case you want to try it. I could see this working for any level of kid in high school. Now to be clear, to do this right, you probably need more than 30 minutes. In total, I took 35 minutes one day, and 20 minutes the next day. Was it worth it? Since one of my goals as a math teacher is to try to build in gaspable moments and have kids expand their understand of what math is (outside of a traditional high school curriculum): yes. Yes, yes, yes. Kids were engaged, there were a few mouths slightly agape at times. Now is it one of my favorite things I’ve created and am I going to use it every year because I can’t imagine not doing it? Nah.

We started with a prompt I stole from @calcdave ages ago when doing limits in calculus.



Kids started writing lots of 9s. Some started using multiplication. Others exponentiation. Quite a few of them, strangely, used scientific notation. But I suppose that made sense because that’s when they’d seen large numbers, like Avagadros number! I told them they could use any mathematical operations they wanted. After a few minutes, I also kinda mentioned that they know a pretty powerful math operation from the start of the school year (when we did combinatorics). So a few kids threw in some factorial symbols. Then I had kids share strategies.

Then I returned to the idea of factorials and asked kids to remind me what 5! was. Then I wrote 5!!. And we talked about what that meant (120!). And then 5!!! etc. FYI: this idea of repeating an operation is important as we move on, so I wouldn’t skip it! They’ll see it again in when they watch the video (see below). While doing this, I had kids enter 5! on their calculator. And then try to enter 120!. Their calculators give an error.


Yup, that number is super big.

Then I introduced the goal for the lesson: to understand a super huge number. Not just any super huge number, but a particular one that is crazy big — but actually was used in a real mathematical proof. And to understand what was being proved.

Lights go off, and we watch the following video on Graham’s number. Actually, wait, before starting I mention that I don’t totally follow everything in the video, and it’s okay if they don’t also… The real goal is to understand the enormity of Graham’s number!

I do not show the beginning part of the video (the first 15) because that’s the point of the lesson that happens after the video. While watching this, kids start feeling like “okay, it’s pretty big” and by the end, they’re like “WHOOOOOOAH!”

Now time for the lesson… My aim? To have kids understand what problem Ronald Graham was trying to understand when he came up with his huge number. What’s awesome is that this is a problem my precalculus kids could really grok. But I think geometry kids onwards could get the ideas! (On the way, we learned a bit about graph theory, higher dimensional cubes, and even got to remember a bit about combinations! But that combinations part is optional!)

I handed out colored pencils (each student needed two different colors… ideally blue and red, but it doesn’t really matter). And I set them loose on this question below.pic3

It’s pretty easy to get, so we share a few different answers publicly when kids have had time to try it out. The pressure point for this problem is actually reading that statement and figure out what they’re being asked to do. When working in groups, they almost always get it through talking with each other!

One caveat… While doing this, kids might be confused whether the following diagram “works” or if the blue triangle I noted counts as a real triangle or not:


It doesn’t count as a real triangle since the three vertices of the triangle aren’t three of the original four points given. During class I actually made it a point to find a kid who had this diagram and use that diagram to have a whole class conversation about what counts as a “red triangle” or “blue triangle.”. Making sure kids understand what they’re doing with this question will make the next question go more smoothy!

Now… what we are about to do is super fun. I have kids work on the extension question. They understand the task (because of the previous one). They go to work. I mention it is slightly more challenging.


As they work, kids will raise their hand and ask, with trepidation, if they “got it.” I first look to make sure they connected all the points with lines. (If they didn’t, I explain that every pair of points needs to be connected with a colored line.) Then I look carefully for a red or blue triangle. Sometimes I get visibly super excited as I look, saying “I think you may have gotten it! I think you may… oh… sad!” and then I dash their hopes by pointing out the red or blue triangle I found. (So here’s the kicker: it’s impossible to draw all the line segments without creating a red or blue triangle… so I know in advance that kids are not going to get it… but they don’t know this.) After I find one (or sometimes two!) red or blue triangles, I say “maybe you want to start over, or maybe you want to start modifying your diagram to get rid of the red/blue triangle!” Then they continue working and I go to other students.

(It’s actually nice when students try to modify their drawings, because they see that each time they try to fix one thing, another problem pops up. They being to *see* that something is amiss!)

This takes 7-8 minutes. And you really have to let it play out. You have to ham it up. You have to pretend that there is a solution, and kids are inching towards it. You have to run from kid to kid, when they think they have a solution. It felt in both classes like a mini-contest.

Then, after I see things start to lag, I stop ’em. And then I say: “this is how you can win money from your parents. Because doing this task is impossible [cue groans… let ’em subside…] So you can bet ’em a dollar and say that they can have up to 10 minutes.!That it takes great ingenuity to be successful! What they don’t know is… you’re going to get that dollar! Now we aren’t going to prove that they will always fail, but it has been proven. When you have six or more dots, and you’re coloring all lines between them with one of two colors, you are FORCED to get a red or blue triangle.” [1]

Now we go up a dimension and change things slightly. Again, this is a tough thing to read and understand so I have kids read the new problem aloud. And then say we are going to parse individual parts of it to help us understand it.


And then… class was over. I think at this point we had spent 35 minutes all together. So that night I asked kids to draw all the line segments in the cube, and then answer the following few questions:


These questions help kids understand what the new problem is saying. In essence, we’re looking to see if we can color the lines connecting the eight points of a cube so that we don’t get any “red Xs” or “blue Xs” for “any four points in a plane.” Just like we were avoiding forming “red triangles” and “blue triangles” before when drawing our lines, we’re now trying to avoid forming “red Xs” and “blue Xs”:


So the next day, we go over these questions, and I ask how this new question we’re working on is similar to and different from the old question we were working with. (We also talk about how we can use combinatorics to decide the number of line segments we’d be paining! Like for the cube, it was _8C_2 and for the six points it was _6C_2 etc. But this was just a neat connection.) And then I said that unlike the previous day where they were asked to do the drawings, I was going to not subject them to the complicated torture of painting all these 28 lines! (I made a quick geogebra applet to show all these lines!) Instead I was going to show them some examples:


It’s funny, but it took kids a long while to find the “red X” in the left hand image. Almost each class had students first point out four points that didn’t form a red X, but was close. But more important was the right hand figure. No matter how hard you look, you will not find a red X or blue X. Conclusion: we can paint these line segments to avoid creating a red X or blue X. Similar to before, when we had four points, we could paint the line segments to avoid having a red triangle or blue triangle!

So now we’re ready to understand the problem Graham was working on. So I introduce the idea of higher dimensional cubes — created by “dragging and connection.” I don’t take forever with this, but kids generally accept it, with a bit of heeing and hawing. More than not believing that it’s possible, kids seem more enthralled about the process of creating higher dimensional cubes by dragging!


And then… like that… we can tie it all together with a little reading:


And… that’s the end! At this point, kids have been exposed to an incomprehensibly large number. And kids have learned a bit more about the context in which this number arose. Now some kid might want to know why we care about higher dimensional cubes with connecting lines painted red/blue. Legit. I did give a bit of a brush off answer, talking about how we all have cell phones, and they are all connected, so if we drew it, we’d have a complex network. And analyzing complex networks is a whole branch of math (graph theory). But that’s pretty much all I had!

In case it’s helpful: the document/handout I used: 2017-04-04 Super Large Numbers (Long Block).

[1] I like framing this in terms of tricking their parents. We’ve been doing that a bunch this year. And although I understand some teachers’ hesitation about lying to their students about math, I think if you frame things well, don’t do it all the time, it can be fine. I don’t think any student felt like I was playing a joke on them or that they couldn’t trust me as their math teacher because of it.

An Introduction to Simplifying Trigonometric Expressions (and perhaps a preview of Trig Identities)

Note: This is a guest post written by one of my friends/colleagues Brendan Kinnell.

For a recent job-interview demo lesson, I was tasked with introducing simplifying trigonometric expressions and/or trig identities…my choice! And, geesh, that seemed like a LOT to tackle in less than an hour with students I’ve never met. Sooo, with some serious help from my colleague and an awesome activity from Shireen Dadmehr, I was able to cobble together a fairly solid introduction to simplifying trigonometric expressions with nod towards trig identities.

First, we opened with a nifty warm-up. I had four different problems on half sheets of paper to give out (one easyish algebra problem, one easyish trig equation, one tough polynomial equation, and one tough trig equation). Each student received one problem, and I didn’t announce they were different. I tried to be sure that no two students near each other received the same problem. I told them they had 2 minutes to solve these. “Do the best you can on the Warm-up — this is just to see what you remember. There are a few problems on the back in case you finish early.” (I didn’t really care about the problems on the back, but just in case some students breezed through them, I wanted them to have more to do.) [Intro to Trig Identities Warm-up file]


A few kids with the easier ones solved them pretty quickly, but most other students with the tougher ones were writing frantically. After two minutes (I may have given three minutes in the end), I stopped everyone abruptly and revealed how not all the problems were the same. I made a *BIG* deal about how unfair it was that some people got really “much easier” problems to solve. Specifically, we focused on the polynomial equations. I had one student share a solution to the “easier” equation, and then I walked them through how you can solve the “more difficult” polynomial equation by recognizing a binomial expansion. In the end, the two equations are the same. The same? Yes. The same! “So, if you had to choose which problem you would solve, which one would you pick?” Hopefully, they all agree that the non-expanded form is preferable. But I didn’t want to kick the expanded form to the curb — perhaps some students like the expanded version, and that’s okay. But in the end, they are algebraically identical.

With a bubbling energy in the room, I didn’t even bother to review the other trig-based warm-up questions. We went right to the next part of the lesson. Direct Instruction! Seriously.

I gave them all a little handout with same basic reciprocal trig identities and the basic pythagorean identity.


We quickly filled these out together and then I “worked” through a simplification problem with them [Simplifying Trig Expressions file].


I had prepared about 10 different sheets of paper that had a new expression that was just a slightly altered form of the previous expression. I taped the first one to the board and then I wrote an equals sign next to it. I grabbed the next sheet and taped it alongside it. And then I wrote another equals sign. I had a third sheet ready to go with a bit more simplification. I taped it to the board, and wrote another equals sign. I briefly explained each step, but more often emphasized how at any point, I could use the expression on ANY of the cards to replace the original. It was my choice. Choice!

And finally…

Every now and then I would untape on of the later sheet and hold it up next to the original expression. “See? These are equal. They don’t look anything alike, but they are fundamentally identical.” Or move any two sheets next to each other. Doesn’t matter. All equal.

Eventually, the original expression gets you to something really simple. I think here you can really play up the fact that it’s surprising that this weird trig expression is essentially just sec x (I think).

So in the end, the kids had this sort of “train” of equivalent expressions, each implementing a slightly different sort of simplification technique. And then, the engaging part.

Finally, I used this brilliant matching activity ( where each group of students (groups of 3 worked really well) gets a bunch of these cards cut out and all jumbled up. (Although I think they are already jumbled if you want kids to cut them out for you!) Ask them to find two cards that “match”. A match being any two cards that are equal by an obvious simplification technique. Students might start slow, but matches will emerge. Give them time.

And what is really cool about this activity is that students will start to recognize that not only are there pairs, but there are trios of matches…and sets of four(!). Students were super pumped to find more than just pairs. “We found a triple!” “We got a four!” They were finding three and four cards that formed a “train” of simplification like the giant one that was currently taped to the board.

In the end I ran out of time to finish the activity, so I didn’t get to see it through. But I believe that Shireen has designed this so that there are four different “trains” of various lengths (up to six or eight cards in length for some of them, I believe. I suggest that you give students time to justify the order of each part of these “trains.” And in the end, I would hope that they could appreciate that every step along the way reveals a new way to denote any other expression in that “train”, and each of these new expressions is available for them to choose.

A Strategy For Groupwork Intervention

The Problem

I care a lot about how groups in my class work. I know some teachers switch groups regularly, but I actually keep my groups together for many weeks (over the course of the school year, kids have about 6 different groups total). The point is: I want kids to learn how to work collaboratively, and for group dynamics to evolve and grow over time — hopefully for the positive.

Although it is important to me, I have only started becoming intentional about developing groupwork. I did a couple of activities at the start of the year to generate conversations about what groupwork is and how it can be effective [1], I have kids have conversations about groupwork when they switch groups, I have kids write reflections about their group and their place in their group, and sometimes I have groups create “group goals” which they write on fancy colored cardstock and keep in their group folders (and when I remember — which admittedly isn’t often — I have them revisit, reread, and add/revise them). All of this is scattered, but it does show kids that I highly value how they work together and I’m watching them work.

But what I realized is that although kids have conversations about groups and groupwork when they have a new group, and they reflect at the end, there was no safe way for kids to “course correct” if their group wasn’t working for them. Most of the time, groups work well, but I’ve had a few times when people felt they weren’t valued or listened to, or the group never formed a cohesive whole and were more like three or four kids working independently. And if you’re in this situation, how do you “course correct”? This whole reflection was triggered by an individual meeting I had with a student a few months ago where they were in a situation where the group wasn’t really working, and he didn’t know what to do. At that point, I talked to him about his agency in the scenario, and ways to be brave and speak up in a thoughtful and vulnerable way (and if he wanted, I could facilitate). But kids don’t know how to do this. It takes a lot of courage to tell friends and/or classmates that you aren’t learning as well as you could be partly because of their actions.

I wanted to create a safe way for kids — after spending a good amount of time with their group — to have The All-Important How Is This Going? Conversation.

Part I of My Solution

Here’s what I did… The next time I switched groups, I explained to my classes what I was grappling with, and that we were going to try something out. As a whole class, what kids valued in a group, and I took notes. Here’s an example of one class’ responses:


I wrote everything they said down. I asked if anyone had any questions about anything written, or if they disagreed, and we made refinements. Then I left the conversation. It was good to have this conversation because kids had just joined a new group.

That weekend, I created a personalized “form” addressing every single thing on the list:

It took a while, but not as long as I thought. (And since many of the things that kids came up with overlapped from class to class, I reused a lot of these questions… And I think if I do this again, I will be able to use many questions from this bank again.)

Part II of My Solution: Four Weeks Later

So here’s how I used this to have The All-Important How Is This Going? Conversation…. After four weeks, I had kids individually fill out this checklist at home. Notice some of the questions deal with the kid themselves…


… and some of the question deals with the group…


After kids fill out the checklist, they are asked to note a few things:


Part III of My Solution: Four Weeks and One Day Later

Students used these questions to have a facilitated conversation. The facilitation wasn’t awesome — it was too structured, so I have to rethink that. But in essence, each person in each group had a chance to talk about one or two things they thought was going well for the group — and then the rest of the group got to respond (did they see things the same way? did they disagree slightly? completely?). And then the important moment when kids got to talk about what they felt wasn’t going as well. It was here that I loved being a fly on the wall. Kids were honest, and responding positively to hearing what others were thinking. Just for those minutes when this was happening, I knew that this was a worthwhile endeavor. (I think I also asked kids to talk about some specific changes they could commit to to address these things moving forward.) There was, in fact, a lot of agreement among group members about what wasn’t working! Finally, I had each kid publicly make a pledge, based on what they learned about themselves while filling out the checklist, of something they were going to consciously work on.

I decided since we did this four weeks in, I would keep the groups together for another three or four weeks to allow for changes.

Part IV of My Solution: Hasn’t Happened Yet

I am asked to write narrative comments on my kids twice a year. I usually include a specific paragraph for each kid about their group. But I am going to try something different. I am going to have kids draft (either individually or collectively, I’m not sure) their own comment on their groupwork — and about any changes or evolution that might have happened. And I’m going to include their own comment on their groupwork in my narrative comments.


[1] Activities from Jessica Bruer, her slides, my notes from her presentation; Sometimes I do this.