nrich – In Pursuit of Great Mathematics Teaching

Animal Algebra: General Terms of Sequences

Use linking cubes and make these three animals.

algebra animals

Ask students to discuss:

How do they see these animals growing?
What does the next animal look like? How many cubes will be needed for its {head/body/front leg/hind leg}?

Using these answers, bring students to see the general term for the number of cubes used. I’ve used this picture to help:

animal-expanded

Now do a similar exercise using growing L-shapes, growing T-shapes, growing Z-shapes, growing animals of their creation.

I have used this lesson countless times and it continues to be a favourite. I like that general terms are introduced without the method of making a table of values. I want to avoid reducing geometric sequences to a meaningless sequence of numbers.

Excellent similar ideas are found at visualpatterns.org and this nrich problem about cable bundles. (The nrich problem has sample student work, making it good for teacher workshops, as well.)

Does anyone know where this idea is from? It’s not original to me and the second image in this post is a screenshot from a long lost book. I have also seen that Colin Foster had an idea like this in his (amazingly free) book, Instant Maths Ideas for Key Stage 3 Teachers: Number and Algebra.

a mathematics lesson that worked

Update:

Thanks to Colin, I think the original idea is from Paul Andrews’ book, Linking Cubes and the Learning of Mathematics. It’s available for sale from the ATM and I highly recommend it. (I have just bought a new copy.)

“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?