Wednesday, November 6, 2013

Solving Systems with Manipulatives

In the Grade 10 applied Math course in Ontario students are required to solve systems of two equations in two unknowns. The overall expectation in the curriculum guide reads " By the end of the course students will solve systems of two linear equations, and solve related problems that arise from realistic situations."

First and foremost let me address 'by the end of the course":

1) OK so by the end of the course- not at the end of the unit on solving systems - with that in mind I will emphasize that I cycle the curriculum by doing activities as I have written about here.

2) By slowly building the idea of solving a system throughout the course students gain control of their learning: solving systems by trial and error; using manipulatives to solve systems like Ax+By=C; solving y=mx+b systems with a table, a graph, a graphing calculator, and eventually with the equations; solving systems like Ax+By=C with equations.

3) Because it is spaced out (like me) the students build confidence and adapt the growth mindset that we want our students to have.

So how about the manipulatives????

Setting the Scene
Mr. "O" walks into a candy store and buys 3 JubJubs and 4 Smarties for 26 cents, you go into the same store and buy 2 JubJubs and 7 Smarties for 24 cents. How much does it cost for one JubJub and one Smartie at this candy store?

Bring in the Manipulatives
Students are given manipulatives to represent JubJubs and Smarties and lots of Pennies. Then they assign pennies to JubJubs and Smarties until it "works".

It looks like this.

Letting them Struggle / Explore /Play
We do a few so that all students can experience some success and I choreograph the learning.

Here are some other photos of other questions. I know you can figure out the questions!

Eventually we have done a few and this is what the board looks like.

The students tried #4 for a long time until someone screamed "I have tried everything! This one doesn't work!"

Ah Ha!
Then I gave them this one.
Mr. "O" walks into a candy store and buys 6 JubJubs and 2 Smarties for 22 cents, you go into the same store and buy 3 JubJubs and 1 Smartie for 11 cents. How much does it cost for one JubJub and one Smartie at this candy store?

It was great to listen to them share their answers. Here is what the board looked like on that one.

Create your Own

Next I asked students to create their own example. Here are a few samples.


1) Students loved this and found it easy once they got the hang of it.

2) It took a while for some students to realize that the prices could not be different for themselves and Mr. "O".

3) I loved that all students could do this activity. I can't imagine starting with the algebra and doing elimination now that I have tried this with manipulatives. We will get to the algebraic solution  "by the end of the course".

4) They enjoyed creating their own examples.

If you try this or have tried something like this I would love to here about it.

Stay the course!


  1. I really like the way this lesson supports concrete understanding of equivalence, substituting for the variable, and multiplication (groups of). Algebra making sense!

  2. I can't wait to try this! Thanks for taking the time to document it so well.