# The Border Problem

I have a new favourite problem!!  It looks like this:

Without counting one by one, determine how many shaded squares are in this 10×10 grid.

This is not a new problem.  I think it’s appeared in many books and possibly even textbooks throughout the years.  My first exposure to it was actually in Jo Boaler and Cathy Humphreys’ book “Connecting Mathematical Ideas”.  I loved watching the video of middle school students explaining the way they saw the problem and then eventually generalizing the methods for an nxn square, but I always wondered what it would look like in a high school class.  Luckily one of my good friends allowed me to come in and try this problem with her class.  It was awesome!  Like exactly what I was hoping it would be!  Maybe even a bit better!  Here’s how it went:

1.  We put the graphic above up for a moment and asked kids to give us a thumbs up when they knew the number of shaded squares.  I asked someone to explain how they were thinking and then wrote that method on the board.  “So is this how you thought about it?” I asked.  They agreed.  “Did anyone think about it differently?”  It kept going until we had a list like this:

4.  We had already touched on the idea of equivalence with the 6×6 list (the fact that all the expressions should equal the same number), so I brought it up again with the nxn list.  “How do we know that these expressions all are equal?” No volunteers, but I was out of time.  I left them with that challenge for homework – to pick two of those expressions and show to me how you know they are equal.  My purpose was to hopefully get kids to start thinking about equivalent expressions and also like terms, the big idea being that even though expressions look different they actually mean the same thing, and we can manipulate them to make them look differently.

Because I wasn’t able to be there the next day, I’m not sure what kids did to show that the expressions were equal.  The classroom teacher told me that the next day they pulled the algebra tiles out to start modelling these expressions with them and proving equality.  I think that is a great way to prove this, and also a great lead in to combining like terms, distributive property, etc.

## 6 thoughts on “The Border Problem”

1. sec2pi

Awesome! I love the way you problematized the topic and worked from the general case towards abstraction. I also think it is neat that there are so many big ideas included in the problem.

2. Greg Heid

Awesome – I love asking students “Will this work for any size…any shape…any sequence…?”.
I’m putting this in my unit “Solving Linear Equations” – CCSS standard 8.EE.C.7 “Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms”.
This will be a great start to something that students should have some prior knowledge to.
Thanks!

3. Cleargrace

I like the tie in to the formula and equivalent expressions. Students do not have a lot of confidence in expressions that are equivalent. It sounds like they were really learning!

4. Sandra Corbacioglu (@sandrasoupmaker)

Sometimes the problems which seem the most straightforward generate the most discussion and are not straightforward at all. I also saw this problem this summer in the Boaler course and enjoyed it. Did you walk around this summer asking people this question? That’s also a fun thing to do outside of the classroom, with adults.