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.

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).

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