Visual Patterns

Have you seen this excellent website? It was like nondenominational math teacher Christmas when I happened upon it!


I definitely blogged last year about the Rectangle Pattern Task, which I’m in love with. I love the mix of linear and quadratic patterns, and the low floor high ceiling aspects. So great! This website has over 200 similar things, aka I do not have to create my own. I emailed Fawn Nguyen who created the site just to say THANKS, and she emailed me right back with the answer key. I did not think I would need it, because I’m super conceited when it comes to my own high school math knowledge, but a couple of those threw me for a loop. I was actually having a ton of trouble with the third one!

This year I decided to use that Rectangle Pattern Task as a springboard into differentiating between linear and quadratic growth. We took some notes which you can kind of see in the background of the next picture, behind some of the visual patterns:


We’re going to do linear modeling as the very next topic, plus we do a ton with quadratic factoring and quadratic modeling later on, so it was really important for them to know what the graphs look like, what the growth is like, and what the equations are like. I’m glad we’re starting with this because we’ll keep referring back to this.

Today I had them in pairs sorting the patterns into a pile for linear, a pile for quadratic, and a pile for neither (the “neither” was a fractal, which is exponential growth). Then they wrote the equations for the linear ones. I was going to start having them write the equations for the quadratic ones, but it was too many objectives for one day. We’ll get to that later when it actually makes sense. Although honestly the linear ones have a clear procedure that always works, and the quadratic ones do not, at least not one which is obvious to me. I’ll keep puzzling with it.


I’m going to see how things go on the assessment, but I felt like I was seeing some breakthroughs. And if they aren’t ready for the assessment, I can give them like 200 practice problems. YAY.


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.

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).