While we were working on measuring circles; circumference, area of circles and volume of cylinders, I noticed that manipulating the formulas was a mystery for some students. For example, the formula for the area of a circle is:
most students could find the area when given the radius but if I gave them the area and asked them to find the radius they were stumped. I could just show them how but I realized that the issue lay in algebra so that is where we have returned.
Algebra is often a huge stumbling block for students. I’ve been doing a lot of reading and research that I may post about later but for now I’ll focus on what we are doing in class.
First we revisited that the equal sign means that both sides of the equation are balanced or the same. Surprisingly, some people don’t really realize that. To reinforce this idea we used Cuisenaire Rods as models.
The students made a “train” with the yellow and black rod.
They had to find other trains that were equal in length to the yellow and black together and then write down equations to express equality.
This photo shows our progression through simplifying the equation. The letters represent the colours of the rods. The rods can also be given numerical values. The distributive property came out during the discussions! I was so excited I actually jumped up and down and squealed a little bit!
At this point, I knew I had to be explicit about the commutative property of addition and multiplication and the distributive property of multiplication over addition (which you can see in the photo above).
I struggled with how to present it and luckily Kyle Pearce wrote a post that helped me out (and I attended three online webinars presented by Kyle). These animations by Kyle really helped me to present the distributive property first with numbers (modeled by Cuisenaire Rods) and then with variables. We did this on the SMART board with Cuisenaire Rods but, alas, I did not take photos! I know some of the students did though so I may update this post with their photos.
Next the students practised a few times on their own (we NEED a Cuisenaire Rod App for the iPads!) like this (sorry I can’t embed it):
Then, still following Kyle’s lead we did this:
I did a few more examples using Alge Tiles and then the students proceeded to practice making models and writing down the simplification of the expressions. Some still needed a fair bit of scaffolding, but they are getting there. I also had some students who did not like models and just wanted to simplify using the numbers and letters only. I think it’s important to use the models though so they aren’t simply memorizing procedures.
This went fairly well, but there are still a few students struggling. We will move forward into solving for the variable by performing the same operation on each side of the equation. Then we will circle back to formulas for circles and apply what we’ve learned about algebra to measurement concepts. Most posts to come to show you our path.
Don’t forget, we’ve also learned the origin of the root word circ in English! It all ties together in room 133!