“Naturally Occurring” Algebra

My grade nine extended class (similar to GCSE top set) have an algebra unit at the beginning of the year. I am assigning all the skills work for homework: simplifying, expanding brackets, the mechanics of solving equations, and so on. In class I am trying to give experiences that answer the unit question, “What did algebra ever do for anyone?” I’m trying to show how algebra’s power to solve problems is naturally arising from good problems.

Background: We have spent two lessons talking about types of sequences and formulas (such as 4k +12) that can lead to sequences.

Today I asked them to watch as I demonstrated something on the board in silence. I did the same thing you see in this video. In one minute, I demonstrated in silence how to find the sum 1 + 2 + 3 + … + 9 + 10.

Then I asked them to describe to their partner what they saw. That took another one minute.

Next I asked them a series of questions. First, could they adapt my method to find the sum of the first 100 positive integers?

Then, I put this list up on the board. What I like about this list (also from nrich) is that there is no time wasted on easy repetition. Each item is a little harder and provides a lot to talk about.

After about 20 minutes, I asked two students to come up to the board and put their solutions to the first two problems. As the lesson ended, some students were starting to attack the last part. What a beautiful and naturally occurring use of algebra!

Do you have any classroom problems that show how powerful algebra can be?