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