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High Ceiling math problems

If you are like me, you are always looking for problems that cause our learners to think beyond just the process of doing mathematics.  I call these problems high ceiling problems because they engage the learner in thinking.  Recently, I have been drawn to Dr. Jo Boaler's website:, where she has been posting all levels of mathematics each week.

I like this video:

Here is one of the problems below, how do you see this pattern growing? What would the 5th case look like? What would the 100th case look like?  Let's chat about this and see what creative ideas we can come up with.  AND if you have a place to post your pictures on the web for us to share your thinking - then please post your visual so that the community can try to "see" how you are thinking.

Here is a link to this lesson:

Growing squares math problem


Stephanie Lindberg's picture

I like that video. We have been working a lot with growth mindset in our classes and this video reminds me of that. This activity looks so fun! Thanks for sharing. 

I had a hard time explaining the first set of squares. But I did come up with an equation. I noticed that the top row always grew by one more block. And every case added another row and column. 

The last one had me confused, too! 

BrookeIstas's picture

I like the idea of noticing the top row growing by one block....did anyone else see it a different way?

Jose Vidot's picture

I am working as an adjunct professor and I just started as a Curriculum Developer in a  Community College where we are on a quarter system. More often than not, the scope of curriculum is rather broad. Many of the professors feel that this constraint on time to cover expected curriculum prohibits this type of activity. I beg to differ.

This activity could be done at home with a paired collaboration. Using Google  or ZOOM conferencing I can assign this activity to my students. The prompts that are in the activity could be put in writing and then we could have a discussion in class the day following the activity. I see great benefit in helping students think quantitatively beyond the confines of a simple text book. We are in essence growing math skills; preparing students for this modern society.

donnawparrish's picture

Thanks for posting that problem, Brooke, and please pardon this late response to it. We "Boaler-ites" do see such problems as ripe opportunities to grow our brains. Though I had read Mathematical Mindsets, that was one problem that I just had not gotten to. I shared it with my "high-ish" level math class without thinking much about it, confessing I had not worked on it and that I was wondering if they could help me direct my thinking. The top student in the class pronounced the problem  "BS" and would not even think about it...interesting. Other students scratched their head about it until someone asked for graph paper and colored pencils.  Others thought that was a good idea and followed suit.  As they colored they shouted out about the patterns they saw - well, they were excited but not really shouting.  The "BS" student growled for a while but eventually joined the color-fest.  There are a LOT of patterns (and thus a lot of equations) in that problem and describing them in understandable terms is a worthwhile activity. (Which math practice is that? Aren't we doing several of the math practices?)  After students and I worked on the problem mostly out of class, we did find an equation for the number of squares in the "nth" diagram, but that was almost an anti-climax. The fun was in the exploration. The "BS" student came up to me after we had the equation and said, "Oh, a quadratic - that was pretty cool."

I have always thought of algebra as the generalization of arithmetic, but some people I admire (like Donna Curry) define algebra as the math of patterns. Perhaps both trains of thought are correct and maybe both trains are the same. Woo-o-woo-o!