This classic problem is a great way to make connections to the language of mathematics, specifically around the meaning and uses of variables.
The pdf above links to a chapter from A Collection of Math Lessons from Grades 6 through 8 by Marilyn Burns and Cathy Humphreys. The chapter has extensive teacher notes with lots of student voices and methods. It details one way teachers can use the Border Problem to help students observe and identify patterns and then make predictions and generalizations.
As a frame, Marilyn Burns begins the lesson by asking her students to think about what the following statement might mean:
Algebra is the generalization of arithmetic.
When we generalize, we look for what is true in general, not just in a specific case. How can we find rules from the computation we do repeatedly? This lesson based on a visual pattern can be used to bridge the gap between students’ current understanding of mathematics and the algebraic reasoning they need to develop.
The two url links above will take you to two videos which show the Border Problem in action in the classroom. It is helpful to watch the videos and take note of the questions the teachers ask to help students build understanding of algebra as a generalization of arithmetic.
Look below for resources related to the Border Problem:
A summary of the Border Problem lesson with steps for teachers to follow.
This short presentation can be used with students to present the Border Problem, review ways of seeing and move to written descriptions and equations.
The Border Problem – In Out Table Worksheet gives students an opportunity to find a pattern in the number of squares in grids 3 through 12.
This matching activity is for a follow-up for a later class after exploring the Border Problem with students. In this activity, student groups match visual representations of ways of seeing with written descriptions as a preparation for writing equations for way of seeing.
For math teachers with access to a computer lab, you might use this Desmos digital version of the Border problem. Students will first construct expressions with numbers to determine the number of tiles that border a pool. Then they’ll use those numerical expressions to help them write an expression with variables. Then they’ll put the algebraic expression to the test, and see if it helps them find the tiles for lots of pools very quickly.
Jo Boaler, along with a few colleagues, used the Border Problem as an example of visual thinking that positively affects our students’ learning in the paper, SEEING AS UNDERSTANDING: The Importance of Visual Mathematics for our Brain and Learning.
Also check out The Language of Mathematics: The Meaning and Use of Variable by Glenda Lappan. It is not about the Border Problem specifically, but she writes about students developing an understanding for how to use variables.
