Can never go wrong with border problem # talk popularized by @joboaler different reps lead to vastly different looking expressions #MTBoS pic.twitter.com/jDlSsUPUvL— Martin Joyce (@martinsean) December 23, 2016

— WithMathICan (@WithMathICan) December 26, 2016I also saw a video of Cathy Humphreys demonstrating this task.

It all starts with the image below (I found it by googling "Math border problems," make sure you don't forget the plural form and math or you'll get a bunch of articles on immigration policy). This is a picture of a square with a side length of 10. The border has been shaded orange. How many orange squares do you see without counting one by one?

And then the magic happens...

In the particular class pictured below, students got answers of 37, 36, or no answer at all. I put 40 up their because I think some thought that but were too afraid to share it.

As you can see, there are 5 distinct methods that students in this cass used. The most common I believe is the one in the bottom left where the multiply 10 by 4 and subtract 4 because the corners were overlapping. I made sure students understood that it was subtracted because each corner was counted twice. This leads to 4x-4.

One student worked around that by saying there's a spiral of 9 square lengths around the border. It's hard to draw, but you can see it in the upper left. Therefore they got 9*4, which leads to 4(x-1).

Another said there were 10 squares on the top and 10 on the bottom. Therefore the left and right sides had 8 each, so 10*2+8*2. We didn't write that one as an algebraic expression.

Karin said she saw the square in each corner for a total of 4 corner squares, and 2 rows of 8 and 2 columns of 8.

Finally, Sebastian saw it similar to the visual pattern from yesterday, as a square with a total of 10^2 or 100 squares subtracted by the inside white square which is 8^2 or 64. 100-64 gets you 36. This leads to x^2-(x-2)^2.

I think I'm going to revisit this warm-up again, but slightly different as a Contemplate then Calculate instructional routine that I saw David Wees present at a Global Math department webinar. I can't find it right now but I'll post a link once I ask him.

Nice problem. which long predates Jo Boaler. The 3-D version is "the painted cube".

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