This post is going to share the talk. If you scroll to the bottom, you’ll get access to the slides and the handout.
Abstract: Don’t like the way the textbook approaches a concept but are intimidated by creating your own content? Bowman and Sam both write their own content from scratch. We’ll share the simple lesson-design tricks we use to write investigations that lead to vibrant discussions and a-ha moments. You will leave ready and excited to write your own content!
Hack #1: Old Problem, New Problem
The Important takeaway: This is the simplest of all the hacks. You might already do this naturally, and textbooks sometimes have questions that switch what students are traditionally given and what they are asked to find. If you’re hankering to see if students have gotten what they’re doing conceptually, mix things up. Just look at a problem and see if you can’t refurbish it by maybe giving them some information and “the answer” and asking them for some other piece of information that they traditionally are given. When you do this, kids will think harder, talk a heck of a lot more with each other (because the problem is more abstract), and you’ll often have many different responses that lead to great whole class conversations.
My favorite slides (one content, one funny):
Relevant blogposts:
Hack #2: Thinking Before Mathing
The Important takeaway: Too often, mathematical notation and premature abstractness get in the way of student thinking instead of being the tool for efficiency and communication that it is for those of us that already understand the concept. Let students play around with ideas in their heads, with their own framing, and own vocabulary, before you develop abstract structures. Let them do it their own, inefficient way before you show a better, more efficient, “correct” mathematical way – the right way won’t stick unless they’ve created something in their brain to stick it to!
My favorite slides (one content, one funny):
Relevant blogposts:
Hack #3: Make Math Magical Again
The Important takeaway: This hack takes some time, but it is worth it. You are trying to build up a moment of surprise and curiosity for kids – something that will make them want to learn more. (It’s like watching a magic trick. You’re in awe, but you desperately know how the trick was performed because magic isn’t real.) You have to think about something you find interesting and really dig deep to figure out for yourself why it is interesting. That takes some thinking! But once you find the answer, I’ve found it often points directly to a way to get kids to appreciate that thing. Often times, I’ve found that having kids explore uninteresting things is powerful because it gives context for the interesting outcome (e.g. appreciating that the complex solutions to polynomials when plotted aren’t that interesting, but solutions to x^{n}=1 are interesting). Also, like in magic, misdirection can also work. Have kids think they are working on one thing, but actually have them accidentally stumble upon another thing can be powerful (e.g. algebraically finding properties of very different looking trig equations like x-intercepts and vertical asymptotes, but as students work, they find out the very different looking equations actually produce the same graph).
My favorite slides (one content, one funny):
Relevant Blogposts:
Hack #4: Toss ‘Em An Anchor
The Important takeaway: Math instruction doesn’t always need to go from skill to practice to application. Instead, application to some interesting context, whether that be abstract or “real world” can actually drive student learning, and help them learn the more mundane skills and contexts. Great anchors are both natural to the mathematical context, and sticky – tangible, novel, memorable, easy to refer back to.
My favorite slides (one content, one funny):
Relevant Blogposts:
Photos of Me and Bowman Presenting:
A photo of Bowman, me, and my colleague who came to support me!
Some Tweets about the Presentation:
Click to view slideshow.
Resources:
Slides (with one taken out…):
That being said, I really wish I could participate in this initiative that Raj Shah (no relation!) shared with me a while ago. But because of life stuff I might not be able to. But one of the biggest things I want to do is bring joy into the math classroom as a core value, and this does that. And I love the idea of a collective joyful math moment for students and teachers all around the world! I’ve done a bit of exploration with this initiative — exploding dots — and I think it’s fabulous and full of wonderment. What it takes? At minimum, 15 minutes of classtime! I highly recommend you reading the guest post I asked Raj to write (below), and joining in this worldwide effort to celebrate the interestingness of mathematics!
Always,
***
Last year, my school’s awesome director of communications contacted the math department to let us know that the one issue of the magazine she publishes four times a year was going to focus on math. And she wasn’t kidding! The cover of the magazine had most of my multivariable calculus kids on it (thinking deeply at the math-art show I helped put on last year)!
One of my favorite things is that the feature article with an alliterative title, Making Math Meaningful, was simply the transcript of a roundtable discussion we had. A bunch of math teachers got in a room around a big table, and we were led by our director of communications who had done her research and come with some questions. There was a digital recorder in the center of the table. And through talking with carefully crafted prompts, we got to think deeply and collectively about our own practice. I can’t even tell you how interesting it was to listen to my colleagues during that facilitated conversation, and how proud I was to be in a school with such like-minded folks that I have the opportunity to learn from. (If you’re a department chair or academic dean, consider doing this!)
I wish I could just post a PDF of the article for you to read, but alas, the whole magazine is online but can’t be downloaded. Here are two quotations to whet your appetite:
…
So if you want to read about a department that is doing strong work moving towards inquiry-based learning, and read the words of real teachers having a real conversation playing off of each other, I highly recommend you:
That is all!
Marbleslides Challenges
I love Sean Sweeney. He’s everything good in the world, packaged in humanoid form! He’s so welcoming and kind to everyone… he wants everyone to feel part of things. At the Desmos Fellowship, he was the person I felt most safe saying “I have no idea what the hell I’m doing” and he would hunker down and help. I think many others felt the same. Okay, enough of the love fest. I am going to share his my favorite which I desperately want to use in my classroom. First, a little note. There is a difference between reading something on a blog and experiencing it. More and more, I’m recognizing that. I think if I read about this, I’d think “cool story, bro” and be like “okay, I could do this, but is it really worth it?” But experiencing it like we did during his short presentation, it’s like “I MUST DO!”
Sean has made a number of Desmos marbleslide challenges (if you don’t know about this, google it). Here’s a gif from his blog. The idea is that the marbles drop and you have to create stuff on Desmos to make the marbles hit the stars.
He shared one with us, and everyone in the giant room got obsessed with drawing functions that would let us “win.” For our challenge, people used ellipses, used lines, used piecewise functions, use quartics. It was inspired to see all the different approaches, and all the play that resulted.
What was lovely about Sean’s facilitation is that he paused us after a while (note: a teacher trick is to say “I’m going to pause your screens in 5… 4… 3… 2… 1…”). You knew from the cacophony of groans that we were in a good place. Then he shared out different approaches. The diversity of “answers” for the challenge was fascinating.
He made this a regular thing in his classes. I love his poster which shows the diversity of responses:
So how can I use this? I’m not sure yet. I need a way to keep it light and fun, but also with all that my kids have on their plates and their lack of time, I don’t know if they would take the time to do it without some incentive. After teaching kids how to restrict the domain of a function/relation, and reminding them of all they have at their disposal that they’ve learned about (trig, circles, lines, parabolas, step functions, etc.), maybe I need to have a 10 to 15-minute in-class challenge (with kids working in pairs, so they are comfortable). And then do it again two weeks later, in class (but not in pairs). And then… announce that we are going to have regular marbleslides challenges. And the winner(s) will get the bonus question on the next assessment without having to do it. Or maybe buy some cheap plastic trophies which get displayed proudly in class? I want kids to work on the marbleslide challenges outside of class because part of this for me is that I want kids who might be slower at processing or coming up with ideas to have the time to execute their vision. I don’t want this to be a timed thing. Though maybe each time I introduce a new challenge, I give everyone 5 minutes in class to work on it.
What I have to make sure to do is share publicly the diversity of answers, like Sean did with his posters.
I also had an idea about how to score it. Something like 1 point for each star. But maybe if we’re learning about conics, or tangent, or something else, I’d give a bonus point for using those functions. And maybe an additional possible bonus point or two for any additional creativity (teacher’s choice)?
Sean’s posts are here and here.
SQUIGLES
David Butler also presented a my favorite on squigles. The poster and his blogpost are here.
I am not one for acronyms, really. They often are forced. But what I like is that these are used to teach student math helpers how to work with other students. From David’s post:
SQWIGLES is an acronym that we use to help our staff (and ourselves) when teaching in the MLC Drop-In Centre. It is a list of eight actions we can do to help make sure our interaction has a better outcome and make it more likely students will learn to be more independent.
It was originally Nicholas’ idea to have something like this. He wanted something to help the staff choose what to do in the moment, and also to help them reflect on their actions and choose ways to improve. We noticed that our staff (and ourselves) needed something focused on actions rather than philosophies, because then it could be used on the fly to choose what to do. Telling staff they need to be “encouraging” or “socratic” is not all that helpful when they don’t know how to put it into action. Yet this is what many documents giving advice to tutors do. So we decided to focus on the actions instead.
The reason I wanted to blog about this is because I think it might be helpful to share with the student tutors at my school. We have a peer tutoring program called TEACH (probably an acronym, since I always see it written in upper case… but for what, who knows!). And I haven’t inquired if and how students get trained. But I’d love to do a short 10 minute presentation on this, and maybe do a few scenarios where kids can practice tutoring while other kids watch (fishbowl?) and take notes on which of SQUIGLES happened. (Not all need to happen! Just look for them.)
I think I should also have this on my desk, since I work with students one-on-one a lot and having that reminder can’t hurt!
Porfolios
I went to Cal Armstrong’s session on documenting student learning. Over the years, I keep on getting inspired to have kids make portfolios that they turn in to show evidence of different traits. And this came up again in that session. James Cleveland has done it. Tina Cardone has done it. I want to do it. But aaaah! The time to make it into a reality! Argh! But I really would love to make explicit some values — maybe not standards of mathematical practice (… or maybe throw of a few of them in there…), but things like perseverance or active listening or seeing a problem in a different way or acting with courage or helping someone understand something by asking good questions or recognizing your own a misunderstanding or changing for the positive as a group member in someway. And have kids document these moments or interactions. And then at the end of a quarter, turn them in. (But have a check in halfway through the quarter!) It would mean that they are looking for these things, looking to do these things. And recognizing that I value these things. Maybe they have a choice of things they can include — not all of them? Maybe they can take videos or photographs or write paragraphs or draw a comic — it can open-ended how they demonstrated this quality or action.
There is something that I think happens in my school. Kids form facebook groups (or maybe on some other kind of social media) for their classes, and I suspect lots of backchannel communication about the class happens on this group. I suspect a lot of it is positive and uplifting and helpful. I would love to encourage kids to submit that sort of stuff in their portfolio also, if it demonstrates whatever qualities were asked for!
I don’t know if I’m going to do this this year. But maaaaaybe?
Preview, not Review: Student Intervention
Kat Glass gave a my favorite on intervention with students who were failing. Part of it was a powerful and important note about language and using code-words instead of saying what you mean. We don’t have many kids that fail classes in my school. But one thing that did strike home was that sometimes when working with kids who are struggling, we put all our emphasis on remediation and it’s like we’re always playing catch up. But sometimes we need to remember that with a struggling student, one tack that we can’t overlook is previewing upcoming material. It can help kids be more engaged and confident in class, and it sets a good tone moving forward.
I do this sometimes, but I need to remember to do this more frequently. Although I do lots of discovery based work, I don’t think that previewing some of it with a kid, and working through some of the discovery with them one-on-one, and then them seeing some of it happen again in class is a bad thing. I’ll just have to remind them that they need to be careful about not letting other kids have the same insights they had — and their role is to help without telling.
Don’t play with your food, damnit! Play with your math!
I love the idea of having kids engaging in recreational math. I don’t have much time to encourage that in my curriculum — or at least the only way I’ve found for that to happen is with my explore math project [posts 1, 2, 3; website]. Some kids get some extra math problems to work on at math club (usually problems from math competitions or brilliant.org), and kids do math problems on our math team. But that isn’t the spirit of what I want to bring to my school. I want to get kids just fooling around with math for fun! Tinkering! Thinkering! Building! Collaborating! So that’s why I fell in love with Joey Kelly (@joeykelly89)’s my favorite presentation. Where he shared with us Play With Your Math.
He and a friend created it. Right now it has 15 sheets of paper that can be printed out, each with a challenge. The name, inspired. Design wise, fantastic. But the problems are captivating, easy to dive into, and many have this open-endedness that can lead to obsession. When I was at the Desmos Fellowship a couple weeks ago, they had these for us to work on as a way to get to know each other. Each table had a different one and we were encouraged to play, and meet others who were playing, and then move to a different table and meet and play when we felt like it. The one I spent all my time on, trying to come up with a strategy? One that I know will get my kids in competitive mode? Poster 5:
I liked getting to know people and I liked these problems! At TMC we were given poster 14 and I became obsessed. And eventually, I solved it (and a second more complicated one). But it took A LONG TIME and I DIDN’T CARE. I refused to go play boardgames at gamenite until I had climbed this mountain!
I need to brainstorm if and how I am going to use these in my school. Some initial ideas:
1. Leave copies of these in the library for kids to use. Or put many copies of all of them on a bulletin board for kids to take, so when they’re board and standing there, they just grab one and start thinking.
2. Use these when I need to fill a long block (we have double periods one out of every five times we meet our kids) and I don’t have a good idea.
3. Plan an Upper School math night, where we gather at a space in the school, do math, order pizza. Like PCMI’s “pizza and math” (was that what it was called? we can do better!). These can be the amuse bouche or the main event!
Math Art!
Speaking of recreational math, at TMC17 there was so much math art. I just wanted to share some of it!
Captivating! I hope at some point to learn how to make crochet coral. It feels like once I get in the rhythm, it could be so soothing. Actually, I wonder if it would be fun to have a MAKER MATH club where we make math stuff together. And create our own math art gallery. Things like the things shown here, but also like these, and origami (demaine and lang), and a menger sponge made of business cards, and design and 3d print these optical illusions, and carefully color in pictures from Patterns of the Universe, and create our own mathart coloring pages. If you are reading this and have ideas of things that we could make, let me know in the comments! You probably can tell this is something I’m actually totally *feeling* (FYI, for me, the definitive math art page is @mathhombre’s page here.)
How To Adult: Let’s Buy A House
So @rawrdimus gave a my favorite on how to adult. He was teaching calculus and wanted to keep his seniors engaged. So he came up with this project that had kids pick a few houses and figure out what they’d need to buy it. He was the banker (a hilarious banker) and gave them two different mortgage options (a 15 year and a 30 year, with different interest rates) and they had to figure out their monthly payments.
I know come the spring, the kids in my calculus class will have their attention wane. So I think something like this could work (this investigation on wealth inequality worked a few years ago)! But right now it’s a little bit like trying to put a square peg into a round hole. I need it to have some more calculus before I do something like this though. Maybe we’ll spend some time talking about e or we’ll do something with summing (in)finite geometric series, and maybe seeing that as a riemann sum? I think it’s totally doable — I just need to think a bit more! But if you want to get a sense of why I’m trying to make this happen, just watch Jonathan’s presentation and you’ll totally get it. (Here’s his blogpost.)
Hey, You Guys! Words Matter
A while ago, I realized when I said “you guys” it was super gendered. So I just sort of said to myself I’ll say “y’all.” When I wrote emails to my classes, I pretty much say “Hi all!” And then… and then… someone brought up the “you guys” issue at a faculty meeting at our school, and in my head I was like “I don’t do that!” But for some reason instead of that reminder doing good, and reinforcing what I was doing, I found it impossible to not say “you guys.” Like when someone points out you say “um” a lot, or say “like” a lot. You just, um, like, end up, like saying it, um, more.
Glenn Waddell spoke about “you guys” at TMC, and it resonated with a lot of people.
So I think I have a plan. Thanks to a huge discussion on twitter (sorry, don’t remember who to cite), here are my options:
The + others was cute… someone recalled they would say: “Humans… and others…” which made me laugh! I think I my lean towards nerds and my loyal subjects because I like whimsy. And as another teacher I love says about her classroom: “It’s a benevolent dictatorship.” @mathillustrated said it’s fun to mix them up. We’ll see what I’ll do!
A thought: I should post this in my classroom so I can refer to it! And tell students what I am trying to do. And have them catch me if I say “you guys” (which of course will make me say it more!). And have an ongoing tally of how many times I say it. And when they reach a certain amount, I’ll bring them some treat. I like the message it sends: I care about words because I care about you. For some of you, these words don’t matter. But I’m doing this for the others of you for whom these words do matter. Also: help me get better because I need to be, and I’m happy to be called out when I mess up.
Other ideas that came up:
@EmilySliman has renamed ‘homework’ as ‘home learning’ [I called it ‘home enjoyment’ because of another colleague, but they have since left me! So I am free to rename it as I please!]
@gwaddellnvhs has renamed ‘student’ (passive) to ‘learner’ (active) [“Learners learn, and students study. I don’t care how much you study. I care how much you learn.” paraphrased from here]
@chieffoulis has renamed ‘tests’ as ‘celebrations of knowledge’ (someone else uses ‘celebrations of learning)
Now do I think things like this will make a difference? Probably not. Calling something “home enjoyment” won’t make kids enjoy it. But it’s stupid and goofy and that’s worth something. And I don’t doubt that making an effort to change language might make a difference to some students. And it can prompt discussion where I get to talk about my values and philosophy around teaching. (“Why do you call tests ‘celebrations of knowledge,’ your majesty?”) I try to live and act those values, but sometimes talking about them can help too.
Kids Say The Darndest Things: Another Classroom Culture Thing
I was having dinner at Maggianos with a TMC 1st timer, @pythagitup. Over dinner, he was telling me about a quote board he did where he put funny things kids said up on display. The beaming of his eyes as he recounted his classes and their quote boards made me know he had done something special. I begged him to write a blogpost about it, which he kindly did here. Here are his top 12 quotes:
I just got sad as I was writing this part of the post, because I remembered that I don’t have my own classroom. I usually am in two or three different classrooms and share the space with other teachers. So doing things like this are trickier. Sigh. It did remind me of one year in calculus. Years ago. 2012-2013. Back then, I was actually a funny-ish teacher. Like pretty goofy. And that particular calculus class was gads of fun. Good and strong personalities. I don’t know why but in recent years, I have lost that spontaneousness and goofiness that I used to have. I’m much more even keeled. I don’t know what happened. Does that just naturally happen when you grow older? I am up at the board a lot less now-a-days, so maybe that’s it… less class-teacher-class-teacher interaction? Whatever it is, I’ve changed. But back then, we had a goofy class. And all year, a student was secretly taking notes on funny things I said, or funny things kids in the class said. And she gave it to me at the end of the year. It was one of the most meaningful things a kid has done. You want to read some of it? Thought so. Wait, you said no? TOO BAD MY POST DEAL.
The post about it is here.
Promoting Kindness & Gratitude
I want to do this explicitly in my classroom. I tried a post-it wall of kindness/gratitude once, but that didn’t *really* take off in the way I wanted it to. I probably should have blogged about that to share a failed venture, and why it failed (namely: I saw it as a tack on unimportant thing, so I didn’t build time in class for kids to do it, and also kids have difficulty sharing kindness/gratitude so helping them see different things as kindness/gratitude would have helped too). [I see “nominations” as a way to do this too, and also related to the material! post 1, post 2]
But I saw something super nice. @calcdave was wearing a clothespin clipped to the collar of his shirt. I couldn’t read it but I asked about it. He then gave me a huge bear hug… which I thoroughly enjoyed because @calcdave is awesome and who doesn’t want a hug from him… and then looked at the pin. On the front, it said something like “hugs!” and on the back it said:
And then the person take the pin off and puts it on the other person (I think that’s important… they pin it on!). This then continues… from person to person to person. I love that @mrschz got it from me, and has now bought clothespins, painted them, and written on them. She’s all in!
I am not comfortable hugging my kids. I’m not that teacher very often (until they come back from college and visit). But I could see this going in different directions.
(a) Making 10 pins, each with one side blank, and the other side saying things like “high 5! #youmatter” or “two good things! #youmatter” or “fistbump! #youmatter” [and the person who inquires gets a high 5, 2 good things said about them, or a fistbump], and then the clothespin travels. I like the blank side because the clothespin then begs the question… and having different responses
(b) Making a bunch of pins and giving them all out to one class and explaining the purpose. I would have to do this with a class that is totally into stuff like this. I can imagine certain classes having a majority of kids who groan and then throw the clothespin away. So I’d have to choose wisely and come up with a good framing/rollout.
This idea originated with Pam Wilson, who is a true gem.
When this idea made its way on twitter, @stoodle pointed out that @_b_p has done something related in his classroom. And I remember reading this, being like OH MY GOD I NEED TO DO THIS and then promptly forgetting about it. The TOKEN OF APPRECIATION. I mean the name itself gets me giddy!
But I like this idea for a few reasons. First: it is done only once a week. It doesn’t take away from classtime. I can do it during my long blocks (once every seven class days). Kids have all week to think about who they are going to give it to. Kids also get to alter it, so at the end of the year, it is a recollection of good.
I know people are going to hate me for saying this, but this upcoming year, I have small classes. I’m at an independent school, so my classes tend to be small. But I think I remember my tentative rosters being even smaller than usual. I like to have larger classes because I like the chaos and interaction and cross pollination of ideas (though not the grading nor comment writing). But I wonder with small classes this year, will this work? I need to think more about this.
Crouching versus Sitting
This wasn’t at TMC but I saw it on twitter and wanted to affirm its truth for me.
I am fairly good about this. I have kids sit in groups of 3 but the tables can fit 4, so I tend to just hunker down with groups when talking with them. In most classes, I almost always drag a chair with me from one table to another which doesn’t have one. I agree there is a huge difference between crouching and sitting. There is value in crouching… it sends the message “I’m here to sort of briefly check on you and see what you’re doing but I’m likely going to move on… things are on you… so persevere.” I tend to sit when (a) I need to ask the group a set of questions to see their understanding, (b) a group seems to be getting stuck beyond productive frustration, (c) when a group is having a heated or interesting conversation and I want to listen in [I tell kids to ignore me and just continue, which I know they can’t really do but they do a pretty good job] or (d) when my feet are tired and I just feel like plopping down somewhere. Ha! Just kidding!
First, you can see her session description, which then framed how I approached the problem:
And then this is what she gave me (but it was hand drawn):
From the session description, I knew I had to find the ratio of the side lengths, so I could find exact trig values for angles other than 30, 60, 90, 45.
Rachel also gave me a “hint page” which she told me to look at when I was stuck (and to time how long it took me before I opened it). Let’s just say I’m extremely stubborn, and so as long as I think I have the capability to solve something and I am not completely stuck, I knew I wasn’t going to open it. Turns out my stubbornness paid off, and I ended up solving it.
In this post, I wanted to write a little bit about my experience with the problem. Because now when I look at that triangle, I have an duh, there’s an obvious approach to use here and everything I know points at that obvious approach. And the answer feels really obvious too. It is funny that I’m almost embarrassed to post this because there are going to be people who see it right away, and I worry (irrationally) (math pun) that they are going to judge me for not seeing it as quickly as they did. Even though I know being good at math has nothing to do with speed. And that it was important to go through the steps I did!
It took me over an hour to solve this problem. I had to do a lot of play and make a lot of random leaps before I stumbled across the “obvious approach.” And I needed to do that in order for me to mine it for lots of things. It was true problem solving. And I know I really deeply understand this because at first the problem looked flummoxing and interesting, and now it looks obvious and somewhat trite. That’s my metric of how I know I deeply understand something. There are still certain things that I teach that I don’t deeply understand: like how the cross product of two 3D vectors yields a third vector perpendicular to the original two. I have done the math, but it’s non-obvious to me why the crazy way we compute cross products give us something perpendicular.(When I only understand something by doing brute algebra, I rarely feel like I get it.)
I’m going to try to outline the messiness that was my thought process in this triangle problem, to show/archive the messiness that is problem solving.
That appraoch didn’t work. Nothing popped out. I saw 54s and 18s and 144s pop out. But those weren’t angles that helped me. But I did then realize something nice… 36 is a tenth of 360! So I was going to use a circle somehow in this solution. Obviously!
At first I drew all ten vertices for a 10-gon. I started connecting them in different ways. I thought I could exploit the chord-chord theorem in geometry, but that wasn’t good. I tried in that second diagram to extract part of the circle diagram and investigate it more. And the third was just more of the same. At one point, I was like and was thinking I could somehow think of this as a problem on the complex plane, where each vertex was and then look at the real parts for the x-coordinate and the imaginary parts for the y-coordinate. Clearly my mind was whirring, and I was going anywhere and everywhere. I actually thought maybe this complex plane thing seems ugly but it will be so elegant. But then I realized I didn’t know where to go if I labeled each of the points on the complex plane. Done and done and done. At this point I put the problem away. Nothing was working.
Nope. Didn’t help. But for some reason, this diagram and looking at the 72 reminded me of something I hadn’t thought of before. This is the leap that helped me get to the answer. And I can’t quite explain why this diagram sparked this leap. Which sucks because this is that moment that led to the rest of the problem for me! But I immediately remembered something about 72s and pentagons. And it hit me.
I remembered the 72 degree angle appeared in a star. And this star was related to a pentagon. And that the pentagon had something about the golden ratio tied up in it. So I knew that maybe the golden ratio was involved in the answer. And where does the golden ratio appear? When there are similar triangles and proportions. I had my new approach and my inroad that I thought would work. Two triangles next to each other failed. Circles failed. But star/pentagon might work!
Initially, I was thinking I could do something with the law of sines. Because if you think about it, this is the ASS case — where you have that 36 degrees (circled), the side I labeled 1 (circled), and the other side I labeled y (circled). But you note that last side could be in two different places, which is why there are two ys circled. I still think there is something fun that I could do with this. But as I was doing this, I realized I was making things more complicated.
I knew that the golden ratio came out of a proportion. So I abandoned the law of sines for the proportion. I simply set up a proportion with the two similar triangles. I first found “?” by doing . So ? was . This was exciting. I knew the golden ratio came out of solving a quadratic. Yeeeeee! At this point, my excitement was growing because I was fairly confident I was almost at the solution.
Then I labeled the part of the leg that wasn’t ? as (since the whole leg length was 1). Finally I looked at the third triangle in the diagram that wasn’t similar to the original triangle. It was isosceles and has legs of and so I set them equal and solved and not-quite-the-golden-ratio came out! (There was a mistake I made where I set and got . But I then found it and rewrote the equation . This was the most depressing part of it. Because I couldn’t find my error because I was so tired. I went through my work multiple times and nothing. But taking some time away and then looking with fresh eyes, it was like: doh!)
And so that was the end. I found if the original triangle had leg lengths of 1, the base was going to have a length of .
I was so proud. I was on cloud nine. I was telling everyone! SO COOL!!!
It probably took me in total 90 minutes or so from start to finish. So many false starts at the beginning, and one depressing transcription error that I couldn’t find.
The point of this post isn’t to teach someone the solution to the problem. I could have written something much easier. (See we can draw this auxiliary line to create similar triangles. We use proportions since we have similar triangles. Then exploit the new isosceles triangle by setting the leg lengths equal to each other.) But that’s whitewashing all that went into the problem. It’s like a math paper or a science paper. It is a distillation of so freaking much. It was to capture what it’s like to not know something, and how my brain worked in trying to get to figure something out. To show what’s behind a solution.
Back to geometry. A few weeks ago, I met with one of the teachers at my school who is going to be teaching advanced geometry. I shared all my materials with her electronically, but I met to talk through things in more detail. But this meeting reminded me of something I’ve felt acutely for a few years: a curriculum is more than a set of papers.
As I wrote each piece of the geometry curriculum (or as I worked with my colleague as he took the lead), I had so much whirring around in my mind. I knew the intentionality of the questions and their ordering. I knew where kids would stumble. I knew where I asked questions that had no answers — on purpose — to get kids to think. I knew that I included a particular question in order to prompt a class discussion. I knew there were placed I needed kids to call me over to have a discussion with each group individually. I knew I had included questions which were designed for me to verbally ask follow up questions. And of course I knew which things were hastily designed and didn’t work out so well when teaching.
But as I was attempting to go through my materials with her, it struck me pretty hard how hidden and implicit all those things were in that collection of papers that she had.
A real curriculum needs so much more, if someone else is going to successfully use it instead of me. When creating materials for other people in my department, who are teaching the same material, I started writing comments/notes in Word when I had a teacher move that I had in mind when crafting the problems:
It’s also a good reminder for me in the future. These notes help me and my colleagues remember what I was thinking of when writing my stuff. When I started doing this, I realized how a curriculum is a set of problems/activities with the intentionality behind the problems and teacher moves spelled out.
In the past few years, I’ve had the fleeting and recurring thought: hey, I should organize all my geometry, precalculus, and calculus files neatly, and put them online in a systematic order for anyone to access. Maybe all of it will be useful to someone, maybe bits and pieces. I still sometimes think that. But what keeps me back from doing it is that gnawing feeling in the back of my mind: things need to be spelled out so someone else understands the flow and intention of each thing. And how to use it in the classroom. Where to stop. How to start. If there were any important “do nows” that weren’t captured in the sheets. Or knowing that someone was written as extra practice or to reinforce an idea that a class in a particular year wasn’t getting.
Over the past two years, it’s become harder and harder for me to open my feedly app and read blogposts. (I find most of my blogposts through twitter now.) It’s just been hard to find the time, and I get overloaded. And I haven’t had time to blog much either. And that sucks. But one thing I love about blog posts — that you can’t get on twitter/facebook/ed research — is that they often illuminate hidden ideas and bring to life something inert. Like when I read a blow-by-blow about an activity/problem set/ worksheet. Something that shows me the thinking that went into creating it, or better yet, how things unfolded in a classroom. What teacher moves happened? What were students thinking? [1]
If I wrote materials… and had a blogpost about how each day unfolded with those materials… that would be a curriculum at its best in my eyes. Because life is breathed into it. It becomes three dimensional. It involves people. The teacher. The students. And it makes explicit what is happening and why. [2]
Note: Funnily enough, Sadie posted a great piece on the idea of “curriculum” the day after I started writing this one! It is definitely worth a read.
[1] I like writing these kinds of posts — though they take a long time. Here’s a recent one: https://samjshah.com/2017/04/28/multiple-representations-for-trigonometric-equations/
[2] Obviously I won’t ever have the time to do this. But it’s nice to fantasize about. An extensive 180 curricular blog. Writing this post also reminds me that I need to get back to regularly reading blogposts.
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.
∫ ∫ ∫
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.
∫ ∫ ∫
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.
∫ ∫ ∫
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.
∫ ∫ ∫
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.
∫ ∫ ∫
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.
∫ ∫ ∫
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.
∫ ∫ ∫
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.
***
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:
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 :
SEMESTER 1 REPORT CHART
Student : Jean-Paul de la Montagne
Total Notes & Participation points
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 :
Areas for improvement :
Suggested challenges :
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.
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 and 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 and graphs. The diamond (okay, geometry peeps, I know it’s a square!) is also pretty easy, except students have to recognize that it takes seconds to go from one vertex to the next (so they have to pick a good time scale on their and 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 and graphs:
Square:
Diamond:
Triangle:
Some great topics of conversation:
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 and 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 graph and a 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!
UPDATE:
The amazing Bowman Dickson created four different desmos applets (which I said I wanted in the post above). I am including them here (with his permission) so whenever I teach this again, I have them at my disposal. Note to self: the equilateral triangle applet has a slightly different position than my equilateral triangle in my packet, so I need to either remember to change my document or change the desmos applet!
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 , by writing: . And then using the inverse of sine, they could get . They 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 and 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 using a protractor. Every single group remembered tangent represented slope. Most groups reasoned that if , they would get and 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 and then immediately solve was on purpose, to generate good conversations with kids about reference angles without using that term. The choice of was done specifically to exploit their understanding of . 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 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!
And he made an error on his calculator and accidentally typed . He got an output of . I think he realized his error when comparing his answer to his partner, who typed in the right expression into his calculator: .
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 . I tried to wrap my head around that… Even in the context of these 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.
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.)
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 instead of . 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!
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 was. Then I wrote . And we talked about what that meant (). And then 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 on their calculator. And then try to enter . 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.
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 and for the six points it was 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.
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 (http://mathteachermambo.blogspot.com/2014/10/trig-identity-match-up-activity.html) 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.