The Friendship of Calculus: A Girl’s Journey Into the Unwavering Depths of The Third Dimension

One of my multivariable calculus students did her final project based around a book we read and discussed in class. It is called The Calculus of Friendship by Steven Strogatz. In it, the author writes each chapter about his own life and relationship with his former calculus teacher through the lens of some mathematical puzzle or concept.

My student wanted to do something similar, exploring her her multiple identities with her mathematical experience through the lens of multivariable calculus concepts. With her permission, I am putting up her three chapters here. It was a powerful experience listening to it as she read it aloud during her public presentation. I entreat you to read it. And although it may seem strange, there are many parts of it that are worth standing up and reading aloud. If you do that, you can inhabit my student’s voice for a while and really hear what she’s trying to say.


The Friendship of Calculus: A Girl’s Journey Into the Unwavering Depths of The Third Dimension

by Brittany Boyce


Chapter One: The Fourth Dimension

The fourth dimension as described in the dictionary is “a postulated spatial dimension additional to those determining length, area, and volume.” The key word in that definition is postulated. The fourth dimension is not something we can see, hear or touch, it comes from our imagination. In the times of early human life, the Mystics saw the fourth dimension as a place where spirits resided, since they did not inhabit our 3-dimensional world and were therefore not limited to our earthly confines. Albert Einstein, in his theory of special relativity, called the fourth-dimension time, but also concluded that time and space were inseparable. But what truly is the fourth dimension? In life, we try to make meaning of the world, what it will bring, what it will mean, how it will help us grow or not, and how it will change. Although we have a certain plan on what we want our world to look like, it is not something tangible that we can hold on to or grasp. The 4th dimension is something we can only imagine. We use the 3rd dimension, what we know and live through to help us envision the 4th. We assign colors and densities to certain points in space, and that helps us paint a picture that we can live with, but we are never truly satisfied.

In 1884 Edward A. Abbott, published a book about the problem of seeing dimensions that are not our own. In “Flatland: A Romance of Many Dimensions,” Abbott describes the life of a square living in a 2-dimensional world, which means he lives with triangles, rectangles, circles, and other two dimensional creatures, but all he sees are other lines because everything is flat. When the square finally has the chance to visit the third dimension with the help of a trusty sphere, a new world opens up to the square. Yes he is a shape like his 3rd dimensional counterparts, but he never took the chance to step out of his world and never sought to understand other worlds because he was never encouraged. At first, the square did not have the ability to comprehend the 3rd dimension, because for his whole life he only knew two dimensions. When the sphere takes the square out of the 2nd dimension, the square is finally able to see that there is a lot more to the world than just flat shapes like himself. The square was able to learn that other shapes have depth, color, height, etc. and because he was so amazed he turned to the sphere and asked what was beyond this dimension. The sphere, like the square, was appalled, unable to comprehend a world that wasn’t his own.

In this way, the sphere is like each and everyone of us. We are unable to comprehend other worlds, simply because we haven’t lived in other worlds. Our levels of privilege and different experiences explicitly prohibit us from knowing what each other’s lives are like. But does that mean we shouldn’t try? Does that mean we should just sit down and not try to understand anything simply because it is different from our experience? The answer to that question my friends, is a simple no.

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It was spring of junior year 2016 and I was sitting with my dean at the time, Mr. Brownstone, in his office going over my course registration. Now, this was it. This was the end, the last course registration I would ever have to do, the icing on the cake that would make me and my resume look appealing to the college of my dreams.

We were going over each class to make sure they looked okay, english looked good, history looked good, languages looked good, and art looked good. The only problem was if I was going to decide to jump into the deep end that was Multivariable Calculus. At first when I heard that Multivariable Calculus was an option I avoided that conversation like the plague with all my previous math teachers.

“You’re taking Multi right?” Mr. Brownstone said.

“Can I take Math Apps instead? They’re both different types of advanced math right?” I replied with a slight chuckle. He looked and me and laughed and replied with a hard “No.” There was no way he wasn’t letting me take Multivariable Calculus, and there was no way he wouldn’t make me step up to the challenge. As a kid who was already succeeding, I did not see the point in taking something extremely hard, but I went along with it anyway.

See that’s the thing about Mr. Brownstone and many other faculty members at Packer. They look out for you by pushing you to your limits and although in the moment you hate them, it’s always worth it in the end. Multivariable Calculus had already had its reputation of being a class, that would really “challenge you,” to put it nicely. Mr. Shah also already had a reputation of being one of the hardest teachers in Packer, so just thinking about this class was making my stress levels rise.

As a junior going into what would be the second half of the hardest year of my life, I didn’t think I was ready for this level of mathematics. I had always prided myself on being good at math and I enjoyed the subject as a whole but all the new variables, operators, and symbols in calculus had opened the door to a whole new side of math that scared me to be honest. Not that an integral sign is physically scary in anyway, but I was scared of the fact that I might not be able to do it. I was scared of needing help because growing up I was taught to be independent. Help was a foreign concept to me because I’ve always been told that based on my skin color no one was willing to help me and so I always had to fight for myself.They had always taught me to be independent because independence was power, and power was success.

Multivariable Calculus had always been a puzzling topic to me. What is it? I still couldn’t tell you. I was already confused by the addition of the alphabet, Greek and English, into the mathematical world, so when I heard that there could be multiple variables added into equations that I would soon be required to solve, I was even more worried. I remember thinking to myself that Mr. Shah would be too hard of a teacher for me and that the material would be too confusing. There was a part of me that thought that I would lose my status of “intelligent” and that I would let down all the people who told me I could be successful regardless of my background. In taking this class, I felt a certain pressure to do well as a poor, young, black, gay woman because not many others like me had this opportunity to study at such a high level in high school. Going to a place like The Packer Collegiate Institute, where I was one of few, always reminded me of my duty to the marginalized communities.

This type of math, meaning calculus, had always felt like a very distant topic to me. I never could picture myself being a “mathematician” because even though I was passionate about math and I had always been good at it, when I looked in the mirror, I never saw a mathematician.

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So it was September of 2016 and my first day of Multivariable Calculus with Mr. Shah had finally arrived. I had no idea what to expect and I was scared out of my mind. It was my second day of classes as a senior in highschool. The pressure was on. I had a chance to prove that I could be as great as everyone thought I could be. So here was my shot, my ticket to the big time academia.

Overall, looking at my new math teacher, Mr. Shah, he didn’t look so intimidating. However, his reputation still preceded him. See that’s the thing about Packer teachers, there are some that you can’t mess with. Some that are so passionate about what they study that they try to imbue you with that same passion in the form of school work. They expect so much of you, and give you so much work to better you, that you can’t help hate and respect them for it.

Every Packer upper schooler knows who I am talking about. Firstly in the sciences, there is Dr. Lurain, an exceptional chemistry teacher who often appears and often is very serious, but will light up and burst out in laughter in appreciation of a good chemistry joke. Next, in the languages there is Mr. Flannery, an inspiring Latin teacher who pushes his students to the breaking point every week with his famous tests. You will always catch one of his students learning lines, memorizing vocab, or reading some famous classical story. Mr. Flannery is no joke, but he has a devout dedication to each and every one of his students. The list goes on and on, but Mr. Shah was one of those teachers. Students told me how they were required to write essays on their tests or be so thorough in their answers to get full credit. But, he didn’t have the demeanor of a mean and strict teacher, he was very passionate about math and he didn’t look like he planned to intentionally make my life a living hell.

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In the first few days of Multivariable Calculus with Mr. Shah, I remember thinking “okay, come on hit me! I can handle it.” I was expecting some complex problem that I couldn’t handle or some other problem that required, some “higher math” that required prior knowledge I didn’t get a chance to grow up with. Instead, Mr. Shah nurtured us, all of us.  He taught us not to be scared of Multivariable Calculus. He taught us that we were prepared for 3-dimensional calculus, and the third dimension was just a step up from the second dimension. He made us aware that we already fought hard enough through Calc I & II with Mr. Rumsey, which was a battle of its own, to be sitting here together taking the same class. He never said it was going to be easy, but he made us feel like we were prepared from the bottom up. But, this comfort and reassurance is not something everyone in the world has the privilege to have.

At times, going up a dimension can seem scary. Most often, in our world things can be complicated enough, which causes us to forget that there are things that are higher than ourselves and more important than ourselves. If you’re like me, you use the fact that two-dimensional calculus was already hard enough, so why study 3-dimensions? Why go beyond what you already know? What’s the point?

The point wasn’t to solve the problem right every time or to be able to understand the most complex things first. It was to be willing to take that step into the unknown in the first place. I had an amazing opportunity to try to understand a world that didn’t necessarily welcome me with open arms. I wasn’t lucky because I had the intellectual ability to take Multivariable Calculus. I was lucky because I was one of few students who had an instructor that made me feel like I could understand the higher maths. Not many kids my age have the ability to study the higher maths, or to even believe that they could study the higher maths, especially students of color, women, and LGBTQ+ students. Today’s education system lacks mentors that have the ability to push kids in the right direction and to make them believe in themselves regardless of their social status. What is unique to my experience is that as a woman of color, low socioeconomic status, and who is proud to say that she is a part of the LGBTQ+ community, I had people around to support me. There was never one time I felt that my peers or teacher didn’t think I was worthy enough to be there taking that class because of my gender, race, sexuality, or socio economic status.

However, although my reality was brighter and more positive than other students who share my identity and do not have the same support system I do, I cannot just be grateful and move on with my life. I must think about those who have to fight harder, speak louder, and do better than I do to hold their place in the classroom and the community of the higher maths. I must bring attention to their fight even though I only know my own.

Chapter Two: Line Integrals

A line integral is essentially integration of a function along a curve. But, that means nothing to most of you. On each curve there are an infinite number of points that trace the path of the curve, determining what it will look like, how it will behave, and how it can be analyzed. Not each point is worth more than another in value or in status, but each plays an integral role in defining the curve. Let’s just say, all points are created equal. But what does that curve really mean? What can it do for us and what can we do for it? Sure it can be pretty to look at or cool to trace, but it all means nothing if we can’t make something out of it or give meaning to it.

That’s where our friend the line integral comes in. To many, it looks like a weird “s.” To my readers, three of these majestic creatures in a row means that I am switching directions or switching to a different moment in time. But to a mathematician, the line integral gives meaning to the curve. It takes the path traced by the infinite amount of points and cuts it into infinitesimally small pieces and adds it all together into the culmination of a single amount, quantity, and meaning. The line integral represents the culmination of everything we’ve been through and the addition of all those infinite moments into one big picture called life. But, while you may have all the pieces and the trajectory, solving the line integral and finding the meaning behind the trajectory, will not always be easy.

Often times, in school we as children are set on a given path or a chosen trajectory, let’s call it f(x). We are given a curve C, and we are told to follow it. We get the grades, play the sports, and be the children our parents want us to be. But what does it all mean when we have hit all the points, traced the path, and completed it? What is it supposed to mean? How are we supposed to evaluate our lives when we haven’t even begun to make any choices for ourselves? And how are we supposed to deal the the fact that we may never make meaning of our chosen path even though we might have all the tools?

The creators of calculus dared to confront this problem through math, because of course, it was the only option. To them, doing the work, solving the integral and making meaning of such a path, was more important than perhaps what the integral meant numerically. Frankly, to be the most cliche, it’s about the journey, not the destination. Not all integrals are meant to be solved in the most complex way or with calculus; sometimes it only takes simplest geometric proof or the simplest meaning of life that can propel you in the right direction, or help you move forward in the problem.

Do you ever wonder how long it takes to change your life? What measure of time is enough to be life altering? Is it four years like high school? One year? A 2-semester calculus class? A semester long, history course? Can your life change in a month? A week? A single day? We’re always in a hurry to grow up, to go places, and get ahead. But when you’re young, one hour or even 50 minutes can change everything.

Through integration, a curve becomes a series of tiny straight lines, working together towards one common quantity. Through integration, life becomes a series of tiny moments working together towards the culmination of you and what your life means. However, sometimes it may be hard to make meaning of a certain time in your life. Sometimes that moment may be unsolvable and that can be frustrating. But, the important thing to remember is that each infinitesimally small piece or small moment works to affect the meaning of your life. Each small experience adds something to your journey.

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I had made it through the first semester of Multivariable Calculus feeling like I could actually pursue mathematics in college. But, I wasn’t completely sure what helped me get here. There was some small moment along my path, where it just clicked. There was something about the elegance of Multivariable Calculus that caused me to light up during every class. Surely there were days that I was tired and defeated, and felt that I could not take anymore of Mr. Shah’s high expectations; But, something about the math itself always brought me back to that stillness I felt. The stillness that was almost calming at the sight of an elegant proof or after spending time doing hard rough algebra, fighting and wrestling with exponents, variables, and symbols to finally get an answer. I didn’t know it then, but that stillness was my ability to feel passionate about math. I had a willingness to understand the concepts behind the algebra I was doing, and had come to appreciate the conceptual approach rather than the hard hitting, laborious algebra I was used to my whole life.

For the more complex conceptual solutions, sometimes I felt cheated, when the very complex parts of the problems were reduced by simple geometric approaches. I saw the immense power of calculus, and I didn’t want it to be reduced or lessened by geometry. There was something about putting my head down and jack hammering through the hard work that always pleased me, but I soon learned that it wasn’t cheating, nor did it lessen the power of calculus in any way.

One simple solution to a complex integral we often faced in class was the integral of cos^2(x) from zero to kπ, k being a multiple of ½. Now for all you mathematicians out there, you know that this integral is no joke. There is no simple u-substitution or power rule you can use to solve this, it must be solved with integration by parts, which is a method that requires some of that “jack hammering” I loved so much.

The proof of ∫ cos^2(x)dx using integration by parts, goes as follows:


Using this integral, the area under the curve on the interval 0 to π/2, makes the integral equal to π/4. While this solution did take intuition and elegance, such as turning the ∫ sin^2(x)dx into ∫ 1 – cos^2(x)dx. Then, adding the ∫ cos^2(x) to both sides to make the proof simple algebra. We learned to tackle complex integrals like this using integration by parts in Calculus I & II,  but 3-dimensional calculus builds on 2-dimensional calculus, so complex integrals always popped up in daily problems. I admired the hard work that calculus required, and the instincts that one could gain from solving such problems, but let’s be real, no one is going to remember the solution to a random integral because one random integral is not that important to all of math. So the question we’re faced with is do we fill our minds with random memorizations of quantities representing areas under curves or do we find another way to remember?

One day, Mr. Shah gave me and my fellow peers a new tool to add to our mathematician’s tool belt. He gaves us geometry. He took us back to our roots and showed us that sometimes simplicity is the ultimate sophistication. So we tackled the same solution.

What is ∫ cos^2(x)dx?


We were essentially tasked with finding the area shaded above. Sometimes when you’re in the middle of solving a problem, and this integral pops up, you can’t result to algebra every time. Sometimes the matter is too urgent and the problem can’t wait for you to do all this algebra. So Mr. Shah showed us one single shape that would change the way we would approach any integral for the rest of our math careers.


Now look closely. The area under the curve is equivalent to exactly ½ the area of the blue rectangle. Now the graph tells us that the length of the rectangle is 1 and the width of the rectangle is π/2. That makes the area of the rectangle  π/2 • 1 =  π/2, making half the area of the rectangle π/4. BOOM. One complex integral simplified with the power of geometry. This proof amazed me. I was astounded by the elegance of such a simple solution. I mean a seventh grader could do this.

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Calculus was never meant to be unreachable. Renowned mathematician Edward Frenkel once said, “mathematics directs the flow of the universe, lurks behind its shapes and curves, and holds the reins of everything from tiny atoms to the biggest stars.” The beauty that math holds has become a privilege unreachable to those who are marginalized everyday for their skin color, race, and sexuality. Everyday students of color and women are told that they cannot or should not see this beauty, the beauty of math is held from them until they climb and fight to the point where they are so bruised, broken and beaten that they give up. There are increasingly low percentages of black and Latinos in high-paying, high-status jobs in finance, science and technology. Since  perceived intelligence in the higher math communities are increasingly influenced by racial prejudices it is getting harder and harder for students of color to believe that they can be something more than the stereotypes. Fundamentally, this is a question about power in society.

Being a student of color who had to claw tooth and nail and go to highly selective programs to even be in a place like Packer, I have experienced that loss of a love for education. Being a black girl who was able to show her intelligence at such a young age, I was set on the path to success. Do your school work, get a good job, be successful. But at the time, I didn’t really know what it meant to be successful. I still don’t know what it means. Most of the time, success is dependent on whether or not I beat the system. I was never told to do what makes me happy. I was told to do what makes me money. I never had the privilege of growing up studying what interested me, or what I was passionate about, and I never knew that having the chance to delve into European history or a new language was a privilege. I was too busy preparing for survival. I was busy getting a head start on the material I needed for the future, so the pressure and the rigor of a predominantly white and male setting wouldn’t defeat me.

There are kids out there who don’t get to enjoy and love knowledge because they are not taught that knowledge is beautiful, they are taught that knowledge is power, and that power is the key to success. Academics never become leisure activity because survival is more important than leisure. They are set on their own path, and asked to make meaning of that path without loving the path in the first place. At the end of their trajectory, they are left at a crossroads, choose another path that they truly love with the possibility of failure or never love a path at all.


Chapter Three: Path Independence

Path Independence shows that the value of a line integral of a conservative vector field along a piecewise smooth path is independent of the path; that is the value of the integral depends on the endpoints and not the actual path C. Now wait a second, am I hearing that vector calculus thinks that it’s about the destination not the journey? Frankly, I don’t blame the creators of this theorem. Most of our world thinks life is path independent. People think that they can see past their privileges and just go on with their lives and that every accomplishment they achieve is independent of a third party. But is our world truly conservative? No pun intended. Do we live in a world where, as one of my favorite bloggers puts it, “instead of recognizing our unfair privileges, we just build walls around us and project out way of life as normal. Any story you tell about how you got where you are that doesn’t include land theft, profiting off of forced, unpaid labor, illegal occupation, murder, assault, theft, psychological and physical warfare, exploitations, and a culture of complicity is, you know, a lie.”

If it is then what’s the point of me fighting so hard to hold onto my passion for mathematics? What’s the point if my journey, which might be ten times harder than someone else’s is recognized in the same or even a lesser fashion than someone who got to the same endpoint. Isn’t there supposed to be beauty in the struggle? Value in someone’s journey? What’s the point in finding the meaning of your path if it is weighed the same as everyone else’s path who started and finished at the same places you did? How are we supposed to try to learn and value the experience of others if we just value where we’ve ended up? Does this mean that the situation you are born in, something that you can’t control, has some type of influence on the overall meaning of your path? It shouldn’t.

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The problem of the conservative vector field that is our world always had a place in our Multivariable Calculus classroom although we sometimes didn’t know it. Every Day 4, when we had class for 90 min, we would hold a book club. Mr. Shah would assign us a different piece of literature to read regarding math, whether it was Flatland by Edward A. Abbott, or The Calculus of Friendship by Steven Strogatz, or Love and Math by Edward Frenkel. As a senior, already up to my eyeballs in work, I disliked him for giving me this reading on top of all the math problems he had already assigned me. I never knew it then, but what Mr. Shah was doing was important work. He was showing us what is was like not to be path independent. He made us value the stories of the mathematicians before us, so that we could know how hard it could be for the person sitting right next to us to be successful in the mathematical community. He made sure to make us feel the responsibility we had to the ethics of the math community. We discussed the politics of math, the religion of math, and the inequities of math every week.

He showed us that while learning the material itself was important, the story behind the material is just as important. In life and in math, there are multiple approaches to solving problems. Often times in math class, Mr. Shah highlighted when two students had different approaches to the same answer and would even have them write it on the board for the whole class to experience. Each approach would have something different. Maybe a trick, a new tool, or even a slight adjustment. When I thought about the way a problem was solved, I never really saw the value in the different approaches, all that mattered to me was that the same answer was achieved.

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In my last quarter of my Packer career and a Multivariable Calculus student, we returned to pure learning. We watched a series of lectures, which was considered our preparation for college math and the whole host of difficulties that would come with it. As the time was winding down, and I began to think about what my final project would look like, I admired Mr. Shah for making us do something that we were interested in and that was meaningful to us. I had never gotten the chance to do so, while still preparing for “survival.” Once again, I hated all the stress and work it brought me. But, I was very grateful that I had gotten a chance to make meaning out of my experience. While the culmination of my mathematical trajectory or “path” at Packer was not completely numerical or quantitative, the experience of finding meaning through math has been life changing.

Math is beautiful, and I only hope that seeing this beauty no longer becomes a privilege in this world, but a necessity. Everyone deserves to believe that they can be passionate about something and not be deemed a failure. No kid should have to carry the weight of their struggle alone. We must not be path independent, we must be aware of the stories that are around us.

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.