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


Rose Curves, Tattoos, and Weekly Journals

Last year and the year before, I stared every Thursday’s class with a journal entry. The students wrote to me in response to the questions “how was your week?,” and “what did we do?,” as well as some third thing. This third thing was almost always a writing prompt about the math that we had done that week, or that we were about to do. Then I wrote back to them on little post-it notes. It created this nice sense of ownership of their learning, offered them a chance to reflect on and write about their week of mathematics, and provided a space to give me feedback about the class. I would always dread reading these on weeks I had deemed bad, but nobody ever wrote anything terribly negative.

Towards the end of last year I stopped doing this. I felt that it was taking too much time out of the day Thursday, which at our school is only scheduled to last 30 minutes (impossibly short!). I also felt that writing only one day out of the week wasn’t organic since the most prompt-worthy topics might have been covered on Monday. I told myself I would have them stop and write every week whenever it seemed like a good time to put math into sentences, but since the schedule wasn’t forcing me to do it, I hadn’t been since.

NEW SOLUTION: weekly journals will still be due each week, but there is no specific class time carved out to do them. They are expected by class on Monday, and students can do them after school, in the morning, over the weekend, and whenever they get done early in class. I will see if it works and let you know. I already got a “good luck with that” from one fellow teacher.

I polled the math 4 students about journals (yes or no about bringing them back) on an assignment from Monday involving plenty of purposeful writing. I will likely do a whole separate post about that because my polar unit got really excellent this year, but basically they had to discuss the visual and numeric patterns of the number of petals on rose curves with fractional coefficients. This is the best visual that I have found for that. I summarized the results on excel:

tats roses

In addition to the more mathematical questions, I asked them about the journal (the aye’s have it! good thing I was going to re-institute it anyway!) and which rose curve I should get as a tattoo. Yet another thing to stay tuned for – new math tattoo!

My major conclusion right now is that I should do more polls. When I was in high school we would do this thing we called “carpool car poll” where anyone could just declare a poll and we’d all have to say our opinions on a topic. I want a catchy phrase like that and more chances to give opinions.

Intro to Math Week Year 3: Arts Integration Edition

Just finished my last academic day of the first academic week of the school year. As it has been for the last couple of years, this week is “intro to math week,” and this year I focused specifically on arts integration, keeping the mathematical standards of practice front and center.

On day 1 we spent some time on the syllabus, focusing on class procedures and previewing the upcoming units. Thanks to keeping this blog I have plenty of pictures of previous classes doing the work they’ll be doing, so it’s not just a line about functions or data or whatnot on a sheet of paper. Then I had them each take a Math Attitude Survey. Somewhat unsurprisingly, most of the new ones had some pretty negative associations with math. Kids also weirdly LOVE sharing bad experiences in math classrooms, so it’s a nice first day icebreaker. I took everyone’s three words and put them into a word cloud:

math word cloud 2015

I’ve always been too scared to re-administer the survey at the end of the year, but I think I should do it, or at least have them give me three words again.

The next day was all about arts integration. I had them re-examine the mathematical practices and compare them to the National Core Arts Standards, since I had personally noticed so many great links. Some of them were a little unsure at first, or had trouble understanding what individual standards meant, but they didn’t shy away from asking each other or me for clarification, and the resulting conversations were pretty cool.

art and math

Then we watched this TED talk by Daina Taimina, or at least most of it – it’s super long [for my kids’ attention spans & non-university math levels] and the important parts are at the beginning and end. I love this talk so much though. It inspires me to see her construct her own understanding, especially given her initial troubles with the subject, I enjoy seeing stereotypes smashed, and is the perfect example of arts integration. The art form makes the mathematics more concrete and comprehensible, and the mathematics provides structure and context for some interesting-looking pieces.

Typically I have some major emphasis on group norms, but I didn’t want to insert something else random into this week, so instead today we just had an extended journal reflection and then dove right into our first topics. I’ll teach them the group norms as we start doing our first group tasks. I realized earlier that I’ve never actually blogged the group norms, so I’m working on a separate post about that.

So far things are going well BUT hey it’s only the first week.

“Learn What You Missed Week” and MI stations

Greetings! I have recently returned from this year’s senior trip. As you can see from my wrist accessories, we went to the Rock and Roll Hall of Fame in Cleveland and the Franklin Institute in Philadelphia.


We can worry about how cool my tattoo is later. Anyway, before the trip happened we were back in class for 2 days after exams. I will be honest and say that I was not extremely in favor of this decision – I did not feel like I could come up with 2 days worth of engaging lessons on “going over the test.” Ick. Instead of trying to fix every mistake they made on the test and magically make them understand what they hadn’t just one week prior, I created an initially simple-seeming activity.

For each of my classes I identified the three most-missed objectives. In case you are curious, for math 4 they were modular arithmetic, fractals, and summation notation (sigma). For math 3 they were complex number operations, data, and exponential growth and decay. Finally, for math 2 they were factoring, similar figures, and probability. Then for each of the 3 objectives I found or created (usually created) a task for each of the 9 intelligences in Gardner’s theory. Those are: interpersonal, intrapersonal, existential, naturalistic, logical/mathematical, visual/spatial, verbal/linguistic, musical, and kinesthetic.

So that’s 3 classes x 3 objectives x 9 intelligences = 81 tasks. Luckily the art objective for exponential growth also worked for fractals so actually just 80 but still! Luckily again, these went really well, and it was all worth it.


This was the kinesthetic data task, where they had to throw the paper glob a bunch of times and record the distance, then find the mean, median, etc. I like the looks of that full page of mathematical text – that is not the work of an un-engaged math learner. This also shows the bin of algebra tiles, which I’ve never seriously used but actually really like. I experimented with them for the kinesthetic factoring task, and I think they have potential.


I printed the tasks on colored paper and cut them out so that they could just select one and then grab it and do it. Can you slightly see that the big one has music notes on it? I think that particular music task was one of the ones that was a bit of a stretch actually – they had to look at the notes and graph the melody on a complex plane, which probably doesn’t have musical meaning. The best ever musical task was the musical data one. I drew inspiration from this & had them record data about their favorite songs’ danceability, valence, and speechiness.


There’s a lot going on in this picture. The picture of ghosts was for a task inspired by this which is very neat. The reason why it is ghosts is probably because of the manipulatives I created to practice multiplying complex numbers. Those are in the blue cup in the front of the picture. They are pretty much algebra tiles, only instead of “x” blocks there are positive and negative ghosts to represent i. The positive ghosts are smiling and the negative ones are frowning. It was cute, initially confusing, and potentially very effective.

I’m done teaching until the fall but I think I will still have plenty of things to write about. We shall see.

Group Exams and Assessment in General

Exams are over! As a teacher, I secretly love exams – I love grading in brightly-colored pens, noticing trends in student performance, and somewhat shamefully, I enjoy a few days of just sitting. It is a really interesting balance, though, of stressing the importance of demonstrating knowledge on a culminating assessment without heightening any students’ anxieties. All last week and now this week I’ve been on the verge of sending mixed messages, applauding students for doing well but reassuring ones who didn’t knock it out of the park that their grade won’t drop down catastrophically.

The rest of the year I give short, low-pressure assessments as soon as students feel they’ve mastered a particular objective. I think that making these as small a deal as possible helps get more accurate data – students aren’t nervous about their performance, they’re just demonstrating their knowledge.

I tried to make the exam pretty non-threatening as well, even though it inherently feels high-pressure. A big part of that is the group exam. This is usually a multi-part task with multiple entry points to maximize student participation. This year for math 4 I had them build three iterations of the Koch snowflake out of craft sticks. I was going to have them build 4 but we would NOT have had time, or probably space. Then they had to find the area of each iteration based on the patterns of exponentially-increasing triangles. It turned out very nice!

snowflake1 snowflake2

My favorite part about this was that they walked in for the exam to see all the tables pushed to the walls and just knew something was up. I’ll attach the text of the group exam as well: Math 4 Group Final 2015

I should note that I do a group final as well as a standard individual final because of the way exams are scheduled at my school – I have each class for 1.5 hours before lunch, then another 1.5 hours after lunch. I didn’t think that it would benefit anyone to write a 3-hour math test. These tasks could work in a non-assessment situation, too, though.

For math 2, we’ve been doing all this great work with polyhedra so I had them continue for the group exam. As a sidenote I’ve been very inspired by this myself and have been hard at work crocheting pentagons for a dodecahedron – I hope to post that at some point. This exam was centered around the tetrahemihexahedron, which I think is amazing. I had them cut out nets and fold them with basically zero guidance – they did an amazing job intuiting how these should be folded. Then I asked them to create toothpick-and-marshmallow structures with things in common with this concave shape, such as the cuboctahedron below with the same vertex figure, 2 squares and 2 triangles:


That’s actually one I made, theirs was kinda lopsided, but they tried SO HARD. Most of the time my philosophy is that trying hard is NOT the same as doing good work, but sometimes I let lopsided polyhedra and the like slide.

Here’s the full text of the task: MATH 2 GROUP EXAM final 2015

So I’m almost done in the classroom, but I still have enough odds and ends to be posting well into the summer. I’m actually about to head back to school now to see the seniors present their projects, which is always a really nice culmination of some interesting independent student work. Plus I made cookies!

Since November

Haven’t blogged since November? Oops. In my defense, a great deal of the time between now and then was just winter break, exam review, and then exams. I’m also basically teaching the same things as last year – in Math 4 they just finished up the Math Words Task and are moving on to the Fractals Task, and in Math 3 we just finished up our discussion on Zero and asymptotes. In terms of new things, here are four exciting moments in math:

IM 3 Infinite Series Argument
infinite sum

Before just telling them the formula for an infinite sum, I wanted the kids to really understand what infinite series really mean. We all went outside and chose a little tree. I walked halfway to the tree, then half that distance, then half that distance. I regret not having them walk halfway themselves, but I was cold and wanted to go back inside quickly. Next year I will be a better teacher. I asked, “will I ever get to the tree?” Consensus? Basically.

Before our whole-class discussion I had given them the pie graph to color in and asked them to make a prediction, basically the exact same situation as the tree only with 1/3 in there instead of all 1/2’s. Most of them said “it looks like it’s going to be 3/4 because you keep adding smaller and smaller pieces.” At least two of them simply couldn’t accept that infinite pieces added to a finite area. This led to an excellent debate and demonstrated a clear need for the a/ 1-r formula.

IM 2 Probability and Combinations

If I’m honest, I am not a big fan of probability. It doesn’t always make sense to me, there are lots and lots of formulas, and it never really rings true to me. In an attempt to make my probability unit feel a little less contrived, I took a picture of all of my gym socks. They just felt like a probability word problem to me. I asked the kids to, among other things, tell me the probability that I reach into my sock drawer and get a matching pair, and it’s super unlikely. I also received plenty of unsolicited advice about keeping sock pairs together.

When we got to the combinations part, it was even easier to keep things real-world. I taped some stations around the tables asking about potential boxes of cupcakes or rice bowls at ShopHouse. Have you been there, BTW? It’s Vietnamese/Thai Chipotle and it’s delicious! There are 2,304 different combinations to choose from and 192 are vegan!

combination stations

It’s disappointingly difficult to read these but the right one is the ShopHouse menu, the middle one is about getting a dozen assorted cupcakes at The Sweet Lobby (they won cupcake wars!), and the left one asks them for the probability that they can unlock my phone in the 5 tries you get. The probability is 1/2000 so I didn’t feel too nervous leaving my phone out.

This definitely felt real-world. One of the kids almost refused to do the cupcake problem because a dozen seemed too expensive. Two of them saw the ShopHouse menu and were like “OK, I want brown rice, chicken, red curry…” before I asked them to read what the problem was asking. Students in other classes did the problems just because they were curious, which I obviously liked.

To get students to do practice problems like they sometimes just need to do, never underestimate the excitement that some tape and colored paper provides. The math of counting menu items from an actual menu is not different from the math of a worksheet that says “at Todd’s sandwich shop, there are 5 types of bread, 3 types of cheese, and 8 types of meat…” BUT the investment level certainly is.

On the test, I actually wrote a really similar problem to my un-exciting example just there, only the sandwich shop was called Jenny’s Sandwich Jubilee and you could potentially choose a banana bread cottage cheese pastrami sandwich! RIP Mitch Hedberg.

Speaking of Jubilees …
The word “jubilee” is really having a moment in my classroom. Did you know that the word “Jubilee” is a Biblical-times word for a period of debt-forgiveness? Thanks NPR! Evidently they had a jubilee in Iceland for mortgages, and we listened to the story to compare mortgages there with the mortgage formula we learned for US mortgages.

Then we did a search-and-rescue that spelled out JUBILEE. Meanwhile in math 2 I not only named the fictional sandwich shop Jenny’s Sandwich Jubilee, I also asked them to compare potential re-arrangements of JUBILEE, HOLIDAY, and REVELRY.

This was a lot. I should just blog more.

A quick love letter to Desmos

Desmos is so lovely. Unlike GeoGebra, which is great, too, it can graph polar things!

desmos polar

It is super user-friendly: to get theta you can click the button for it, or you can just type “theta” and it knows what you mean. It is very easy to edit and delete functions. Making a slider is as easy as typing a non-variable letter and just clicking on it. Idea that I’m having too late: type in r = 2 + bsin(theta) and drag the slider to show how cardioids and limaçons and the loopy ones are related!

I had the students do an investigation about polar graphs in pairs. Last year I gave them some work ahead of time about what polar coordinates were, but I decided to postpone that this time. I thought it might be better to let them discover more on their own.

I’m attaching my PDFs, but mainly I just gave them a good amount of each type of equations and asked them to sort and then to generalize.

I cut these functions out and put them in an envelope: polar task chart and functions
These are the instructions: polar task