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

Factoring Amazing Race

Sometimes I have a strong temptation to use my blog solely for successes and never mention anything that isn’t great within my classroom. But I recently had a realization so simple that I don’t know how my teaching could possibly have been effective before it! Realization: students won’t work together effectively until they are together on a team.

kids amazing racing

We had this excellent all-school camping trip last week, but as a consequence class did not take place for an entire week. I was concerned that they would have forgotten everything, so for my Math 2 class I created an Amazing Race game to review and practice binomial factoring. All along I had been encouraging them to work together and ask each other questions, but everyone was either lost and totally un-focused, or focused on doing their own work, not motivated to help others.

amazing race cards

I had them in 3 groups for the Amazing Race. I put three students who were on the verge of full understanding with an extremely competent but extremely quiet student. My thinking was that the quiet student could shine and the on-the-verge students would quietly listen in order to gain full understanding. This group was fairly effective although never fully cohesive – I mainly saw them interacting one-one-one rather than as a full group. I put four on-the-verge but potentially less-motivated students together so that they would all have to participate fully in order to understand. This went well – some of the best conversations came out of this group. Then I put a brilliant and patient student with a very silly but very smart student and two students on the struggle bus. I think that at least one of the strugglers is doing really well after that, but the other one is still struggling. I’m feeling sad about this but I’ll catch up with her on Make Up Work Day.

Launch – Explore – Discuss

Happy Saturday, blogosphere! I’m currently located at school, about to do my second round of ACT/SAT prep of the day, and furthermore drink my third cup of coffee of the day. In between, my colleagues and I were tabling at a conference for twice-exceptional students. Tabling typically involves me excitedly explaining our school to anyone willing to listen, and ramping up imaginary rivalries with our fellow small schools. I did try to tone that down this time, though, because I suddenly want to do some classroom observations (aka spying) at other small schools.

The congregation of the church we rent our space to is slowly arriving, and that is the only thing weirder than being at school by myself. There are also two cats in the office building instead of just one like usual,

None of this is what I had intended to discuss! I have been implementing and blogging about collaborative learning for years, explaining that we had learned all of these techniques and ideas in college and I had wanted to try it. All this time I had been ignoring what I have recently discovered is an important element: the discussion. What you’re supposed to do is launch the task by explaining the directions and process, have the kids explore the material on their own, and then have a discussion as a full class to bring together their ideas. Since I was so focused on having the kids prove to me and themselves that they knew the material as an individual, I’ve been skipping the last step.

Such a poor choice! Wednesday we spend most of class on a discussion of the Rectangle Pattern Task I discovered last year. I love this task so very much. It’s the physical embodiment of “look for and make use of structure.” As I had learned to do in college, I structured the discussion around which groups had discovered what in a nicely-phrased way, making sure that everyone would have a chance to speak. Here are my notes:


The groups shared the patterns they had recognized. They predicted step 0 as just 2 blue squares, then determined that each time you add 4. To find the pattern, I wrote that out like this:


This made it easier for them to see that the nth iteration would have 2 + 4n blue squares. Love it!

To discover the green and red patterns, which are quadratic, we went more visually. That’s the squares you can make out in my notebook. The students described that the pattern of green squares is that on each side, you add another row, and that row will include 2 more than the previous. I drew this out as they described it:

structure A

Then I asked them to find the largest square in the first one. They said 1. I asked for the largest square in the second. They said 4 at first, then clarified that that was 2 by 2. For the last one, someone pointed out that there are many different 2 by 2 squares in there, which is true. I asked them if they could re-arrange the boxes to create a larger square than that (should I have waited to pose this?). One poetically described that process as “you take the wings off of the bird, and you put them on the bottom.” I illustrated it slightly differently, and then gave everyone paper to cut in order to create squares. This was the end result here:

structure B

The pattern for one side is then n squared plus n!

The next day the room was entirely silent and productive as brilliant math learners completed their independent reflections and then determined the patterns on yet another set of box designs. Next week we’ll discuss those, then I want to give them an assessment on this [which I have yet to write], and then we’ll work on factoring in a hopefully visual / non-terrible way.

Group Norms

Working collaboratively is an incredibly important skill, and increasingly so. So in addition to teaching math, I’ve been working on strategies for teaching communication and collaboration skills.

I got a jump on this during bonding week this year. Every year we divide the school into small groups of maybe 5 or 6 and task them with creating boats. We have this rotation – year 1 these boats are strictly cardboard and duct tape, year 2 strictly trash bags and duct tape, and year 3 cardboard, trash bags, and duct tape. It’s year 3 but this picture is from last year:

boats for blog

Chaos, right? But super fun chaos, and we get more buy-in than you’d expect. Even still, we had noticed this tendency for boat group communication to be subpar. The groups would usually defer to the ideas of whoever was loudest, kids sometimes got mad that their idea didn’t feel listened to, and without a teacher consistently in the room (which would ruin it in other ways), it was totally OK to not participate for kids who didn’t feel like it.

So this year we had them do limited-resource challenges in small groups, and then reflect on these group dynamics upon entering their boat groups. We orchestrated the groups so that nobody was in a group with someone they just worked with, so they could talk openly about their experience. I wrote up a set of reflection questions:

Communication Discussion Questions:
1. What was the most challenging part of communicating with your previous group?
2. Did you feel that your ideas were listened to?
3. Did you always feel comfortable sharing your ideas?
4. Did you feel that you listened to other ideas? Why/ why not?
5. What qualities in someone else make them easy to communicate/collaborate with?
6. What qualities in someone else make them difficult to communicate/collaborate with?
7. Do you find it helpful when someone steps up to be the leader, or does that make you more reluctant to share?
8. What can you do as a group member to make sure nobody is dominating the discussion?
9. What can you do as a group member to make sure quieter people are joining in?
10. What can you do as a group member to make the decision-making process easier?

So now in my math 2 class we’ve started our first group task, the rectangle pattern task I discovered last year. Before breaking into groups we went over the group norms, based on concepts we talked about in my college classes and the book Designing Groupwork by Elizabeth Cohen. Group norms:

For a productive group…
1) Stay in your group
2) Ask the other group members first
3) Everyone is accountable for what the group is doing
4) You are responsible to ask for and offer help

I like the last one best although the first one is a close second.

Best question ever: “wait – are we doing this to practice working in groups or to learn the math?” Answer – both!!! I think I exclaimed that with three exclamation points in class as well.

Tomorrow we finish the task in math 2 and launch the vector fields task in math 4.

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!

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

Rectangle Pattern Task

I am so into this lesson by Cynthia Lanius! She wasn’t on my radar before but I’m definitely taking a look at her website. I like that the first word of the title is “fun!” Fun, along with Mutual Respect and Safe Place, is one of the three pillars of HGS, and so it is near and dear to my heart.

Even though these rectangle patterns are definitely fun and look deceptively simple, they contain some fairly complex patterns.

lanius rectangle

Can you sense the quadratic equations?!

I changed the task just slightly. I thought it would help them to synthesize their thinking on the patterns if I initially asked them to describe them in words before writing an equation. In fact, for most of the students, I taught a teacher-led lesson on writing expressions before I had them do that. Some did not need any help at all, so I just furnished them with every single problem in this article (which is also great!).

My favorite question on the task as written was the third one, “Will the design use 42 blue squares in any stage? Will it use 102 red squares? Will it ever use 830 squares in all? If so, state the stage number for each answer.” I also loved that it wanted them to predict step 0, that definitely helped them grasp some of the pattern.

I asked them to predict stage 8 by observing the patterns, and to my great surprise every single group actually drew out what step 8 would look like – even though they had all CLEARLY gotten that the blue ones just added 4 every time, and the other colors were two pyramids whose new steps were just 2 boxes longer than the previous ones. Some of them even counted every single box instead of relying on these patterns – disappointing! I bet if I had them predict step 20 they wouldn’t have done that, 8 is too small a number I guess. NEXT TIME!

I overall loved this problem because it is so approachable and has patterns of various levels of complication. It’s got something for everyone! This is encouraging me to try use more visual methods for algebraic topics, especially boring things like factoring that use important skills and practices but that are so boring they don’t deserve a fancy different synonym for boring. Stay tuned for that (if it works).