Intro to Math: Circles and Squares

School is back in session! One of the students I tutor apparently has no homework tonight, otherwise I don’t know when I would have had time to put this into writing – things got busy in a hurry.

I’m doing the whole #teach180 thing so if you’re desperate to see more frequent updates from the classroom you can follow me on Instagram, @multiplefactorsi. I’ll be posting every day, held accountable by a reminder on my phone. I hate tiny little red numbers on that screen, so you know I’ll be following the rules.

Anyway! The first week of math class I always do “intro to math.” We started with some talk about what topics in math we were going to do over the year, and then we spent some time on an arts integration project inspired by these tasks for small children. This would have been awesome if we had more time, or maybe they would have gotten old – who can say? A couple students finished them beautifully at home, and others have kept their unfinished projects in case of momentary boredom. I’ll attach that whole assignment.



PDF of the assignment: day-1-arts

The second day was all about seeing things from multiple perspectives and integrating multiple viewpoints into collaborative work. We started with a number talk, something I loved when I first saw it but just never used. I took the one from YouCubed’s “Week of Inspirational Math,” week 2 day 1. I had been very nervous that our conversation would be nothing like the thought-provoking and joyful example video, but this went amazingly. Here’s our board:


Inspired by Sarah at Math = Love, who was in turn inspired by other bloggers, we then launched Broken Circles. That was great! I love hearing students talk about math, but I also loved the no talking rule. It’s also an inspired touch that one circle completes itself. In one group, the person with the “A” pieces sat there, self-satisfied while the rest of the group struggled, and it was kind of glorious. It’s almost like the point of this task was to show that working together and paying attention is crucial.

Between that and our next collaborative task, we went over our group norms. I translated them into Spanish for an extra touch, and perhaps that will inspire me to do more group work in Spanish class? Time will tell.


The end of Wednesday we started the Pentomino task – I blogged about Pentominoes last time I did it, SO long ago! This year I didn’t leave as much time and consequently they didn’t come up with quite as many combinations, but I do still think it was a useful exercise in visualization and pattern recognition. I’m kind of in love with my independent reflection for that task, attached. Why indeed can you not build a 6×6 square?

Independent Reflection PDF: pentomino-ir

Things are shaping up, but I already know I won’t have nearly enough time to blog as I’d like : 0


Summer PD Part I

Summer has officially started, and here in DC its arrival is never subtle. I can’t walk three blocks without being drenched in sweat, my water bottle is never full enough, and honestly I am loving every minute of it. Last summer I felt like I didn’t have enough to do and got sad about it, but this summer I’m trying to take advantage of my unstructured time. In addition to doing significantly more yoga and binge-watching Orange is the New Black, I’m working on accomplishing these things:

1. Design a statistics class for seniors, based [loosely?] around the AP curriculum.
2. Learn statistics – I somehow never took a course
3. Re-design my classes for 9th, 10th, and 11th graders, because now that I’m teaching all 4 grades there is no way to put them on different tracks, so I’ll have to differentiate like crazy. Plus most of the fun visual geometry stuff was in math 1 which no longer exists, so I need to find places to put that in 2, 3, or 4, probably 4 which means I’ll have to significantly enrich it. Like uranium. 4. Write curriculum for Spanish that assumes that they were all with me last year, because most of them were, but that will accommodate any new students.
5. Design a service learning course that continues Hope’s original vision but that I can feel ownership over.
6. Seriously plan the **** out of everything because I am scheduled with zero planning time next school year. ZERO.
7. Create an arts integration plan based on the Kennedy Center workshops I attended last year.
8. General professional development

I’m mainly here to blog about the last one, but this made me realize I’ll have plenty to share throughout the summer. It’s also really sinking in that I will be at a serious loss of unstructured time in the near future.

Anyway, PD. In the past, we’ve all read the same book over the summer and then discussed it on the retreat. This year we couldn’t decide on a single book, so everyone was tasked with just going out and finding professional development opportunities and then coming back and sharing.

So far, I’ve read one book in the name of PD: Building a Better Teacher by Elizabeth Green. I linked to her “Why Do Americans Stink At Math?” article last summer, and some of that same content was in the book. A lot of the book actually centers around math education, and excitingly for me, around one teacher, Deborah Ball. This is so exciting for me because Deborah Ball is the dean of the School of Education at the University of Michigan [GO BLUE], so I have taken her class and feel well-versed in her philosophy of teaching. I can also hope that her greatness had even the slightest impact on my own teaching. The book focuses on her work, in conjunction with another Michigan teacher, to help new teachers create investigative, collaborative classroom environments like her own.

For a great example, watch this video of some elementary students having a spirited yet respectful debate about even and odd numbers.

The book also focuses on Doug Lemov, whose book Teach Like a Champion provides extremely practical classroom strategies, revealing the secrets to managing an effective learning environment. He takes what seems like a magically smooth classroom and boils it down to concrete actions that any teacher can learn to take. While I was doing my student teaching, my mom mailed me this book and I remember reading it on the elliptical after school, dreaming of my future orderly classroom full of excited learners. More on that later.

The two visions of a productive classroom – Ball’s lively class discussions and Lemov’s orderly SLANTing – seem somewhat at odds, but what they have in common, and what the entire point of the book is, is that teaching is a skill that can be taught, not just an ability that some are born with. The immediate question then becomes: if I went through Michigan’s superior teacher training program and read Teach Like a Champion cover to cover, why am I not the best teacher in the world?

I surround myself with very positive, supportive people who would answer that with “but I’m sure you ARE a great teacher,” and that’s nice, but in reality there is no way that I internalized every piece of effective pedagogy. There is a huge difference between being inspired by something you hear in class and writing it down, and knowing how to actually apply it in my own class.

My major takeaway from this book is that no matter how amazing teacher training is – and the book implies as I’ve always assumed that mine was extra good – these skills need to be reinforced. What I really want is someone in my classroom frequently, not in a frightening evaluative way, but in a helpful “are you doing what you learned was best?” questioning and guiding way. I’m sure that after 4 years in the classroom I’m doing a lot of things better, but I’ve also strayed from some original intentions and picked up some bad habits.

What I’ll Share at the retreat: a) They should all watch the video of the 3rd grade number debate, because it will challenge some assumptions about the level of discourse we can expect from our students, and it will give us a jumping-off point for explicitly teaching them to communicate and hear each others ideas. b) We need to revisit doing classroom observations and assessing ourselves. That’s going to be incredibly hard to schedule because we’ll be down a couple staff next year, but I think it’s important. I’m tired of people assuming that I’m doing amazingly in my classroom just because the kids aren’t unhappy, because I want to be better than that.

Exams then travel!

Last week was exam week. These are fairly traditional around here – some of the students even say it doesn’t quite feel like our school. I think spending a week taking exams is an important skill to have, so that’s totally OK.

But also, I like to include a super exciting group portion on the test. One of my classes was dealing with Bell numbers – you should glance at the Wikipedia page for a sense of the deep math behind those if that’s your thing. The formula for the nth number looks like this:


Cool, huh?

The nth Bell number counts the number of ways to partition a set of n things into groups. For example, the Bell number for 3 is 5 because you can split these 3 things 5 different ways:


Bell numbers can also count the number of rhyme schemes of a poem with n lines. So naturally I asked them to write 5 different 3-lined poems with different rhyme schemes. Such as:

I hope you know addition
Because it’s a good addition
To your math repertoire

One of the other classes had to use Newton’s Law of Cooling and logarithms to solve a murder mystery. An invisible dead body, separated from them by caution tape, lay in the corner of the room. They had to test its poison levels and model them according to exponential decay. Newton’s law requires the use of delicate thermometers and other equipment.


I think the amount of fun they were having approached the amount of fun I had making those. End behavior humor : )

Another class’ exam revolved around, among other things, the tetrahemihexahedron. Here’s a picture of my cat investigating one:


Unlike Olive’s, which was mainly the sniff test, the students’ investigation involved cutting out the net and folding it (harder than it seems!), and then re-creating that out of origami, and toothpicks and marshmallows.

I’m always impressed with how focused and collaborative the students are with these group portions. Group work probably won’t work as an all-the-time thing, but it can be super effective.

Oh! But the main reason I’m taking to the blog is that I’m leaving tomorrow morning. I’ll be co-leading a 2-ish-week senior experience travel component. It’s not math teaching, but it’s still teaching, so I’ll definitely post about it when I’m back home! See you in 2 weeks!

Playing with Blocks … But with an Algorithm


In my Math 2 class we’re working on geometry topics, specifically lines and angles. To investigate the relationship between the number of sides in a shape and the sum of the interior angles, first we looked at paper triangles. We ripped the corners off to verify that added together, the angles create a straight line, aka 180 degrees. Then I asked them to figure out the measure of each individual angle in an equilateral triangle. Excitingly, the class came up with 3 different methods of doing this, that was a nice moment.

To investigate farther, I gave them each a large handful of pattern blocks. Before giving them the assignment, I let them just play around with the blocks. This was a great way to deal with students finishing the previous assignment at different times – some just got more time contentedly playing with blocks.


Most of the angles of these shapes are 60 degrees or 120 degrees, just not the white quadrilateral. I thought this was kind of boring, but I didn’t hear any of them expressing that. It seemed like it was helpfully repetitive for them, giving them a process.

Over the summer I read this professional development book about executive function, which warned that the types of students I have in my classes, students with ADHD and LD, may have a difficult time in math unless they have an algorithm. It specifically called out investigative learning and explorations like this as being a bad idea for students with executive function difficulties. This upset me because I had spent most of the summer extremely excited to implement some of these investigative approaches into my classes, convinced that it would be perfect for my learners. Having a process – in this case, comparing the 60-degree angled triangle to the angle in question – was definitely important for them, but I think having the ability to move pieces around was just as important. Having them accept as fact that the big angle in the blue pattern block was 120 degrees is not nearly as convincing as holding it up to a green triangle and seeing the straight line.

After this investigation, I absolutely understand what the book was getting at. My students do prefer to have a process and a set of steps. Unfortunately for them, not everything in the world has a set process. I don’t mind scaffolding the things that do and giving them as many algorithms as are out there, but even (especially?) students with executive function challenges need a space to explore. I do slightly regret not assigning this as a group task, because students could definitely have benefited from working together more.

My tentative plan for Monday is to write the sequence 180 360 540 720 on the board, and ask them what these represent. I want them to tell me next that the angles increase by 180 each time, and that the relationship is (n-2)*180. Some of the students aren’t quite done with the investigation yet, and I’d hate to spoil it for them. I think I need someone to come collect data on me in this class, to see if I’m spending more time with some students than others. I feel like at the end of class there’s always a surprise student who I never noticed was lost.

Giant Flags

We’ve been working on transformations in the coordinate plane in two of my classes, and in each that work is leading us in different directions. The best part about writing my own curriculum is that I can orchestrate these mathematical story arcs with themes and threads and a plot progression.

In Math 2, we finally finished our work with probability, elections, and fairness. I say finally in spite of really enjoying the work we did. It was just a long long long unit. The first task I assigned was a flag enlargement project. Each student was given a very small flag, a scale factor, and a huge sheet of graph paper with a 1-inch grid. For some cross-curricular action, I asked them to look up the origins of their nation’s flag. It turns out that most of the time red is on a flag, it is symbolic of bloodshed. It also turns out that all of my red markers are completely dead by now.

flag proj 1

Most of the class was able to complete this task using linear measurement and multiplication alone. My student with the Seychelles was struggling until I handed her a protractor – so much easier!flag proj 2

This experience with flags leads perfectly into similar figures, which leads perfectly into proofs. See what I’m saying about a story arc?

Determining if two triangles are similar isn’t the most exciting thing, yet I’m very excited about it. Remember all of those rules, SSS, AAA, and SAS? These say that even though being similar means that every single angle is the same and every single side length is proportional, with triangles you can figure this out by testing just three things instead of everything. Because of their work with the flags, they had the exact prior knowledge to understand this concept. Most of them had not measured angles (nobody except for the Seychelles) and yet their angles were the same – that’s SSS! The case of the Seychelles verifies AAA. By the time we got to SAS I was looking pretty credible, plus they had fun with some SASsy puns, which I will consider an emotional connection with the mathematics.

I gave them a fairly boring worksheet of similar triangle practice where they had to determine if two triangles are similar. Examining angles and ratios (I used the language “is the scale factor the same for every pair of corresponding sides?” to connect to dilations) and using AAA, SSS, and SAS to justify their conclusions was secretly their very first exercise in proof.

And let’s talk about proof for a minute. The way that mathematical proof is taught in school is ridiculous and everybody knows it. Lucky for me, I have the leeway to not only create a plot progression in my class, but also completely abandon traditional proof pedagogy. I’m skipping these guys up to more college-level proof. It’s not more difficult but it is more logical, more interesting, and more like the way that mathematicians actually operate. Next week we’re going to start on proof by mathematical induction, a highly algebraic proof type. Our first exercise: prove that adding any 3 consecutive numbers results in a sum divisible by 3. I love this because you can actually test it out and convince yourself that it should be true. And then the proof. I’ll let everyone know how the rest of this goes.

Intro to Math Week

The first week of regular classes was an “Intro to Math” for all 4 groups.

Day 1: Syllabus (boring) & growth mindset (inspiring!)

Day 2: Establish group norms & introduction to Pentominoes

Day 3: Pentominoes Task

For their first collaborative challenge, each group had to come up with as many different ways to form a 5×5 square using Pentominoes.


Between the 4 classes, the students came up with 193 distinct ways to form the squares! I’m very impressed with their teamwork and problem-solving skills so far.

To wrap up their work in groups & introduce an element of individual accountability, I gave them an independent reflection that is a little bit more abstract. We didn’t finish that in class last week so we’ll go back to it on Monday.

As classes get into full swing I’m very quickly realizing that:

1. I do NOT have enough time to cover everything! At our school, every Tuesday is a field studies day, plus students all either do service learning or an internship on Fridays. Classes are already only 50 minutes long, but on Thursdays they’re 30 minutes because of the school meeting & P.E. (I’m teaching yoga!). The schedule is awesome and the students get lots of valuable experiences they wouldn’t ordinarily be able to have, but it’s a huge challenge for an integrated curriculum.
2. Teaching 4 totally new classes is going to be really difficult. Most of the material I have taught before, but in a different order, or a different way, and certainly to different groups of students. I alternate between being REALLY EXCITED about trying new stuff and REALLY OVERWHELMED at how much work I have to do on a constant basis.

I hope everyone else has started off the school year well!