# Smoothing the path from simpler to more complex systems of equations

We played with Venn Diagram puzzles, and now my students had a better sense of what it means to be a solution to a system. So, onwards! But I noticed that my 8th Grade curriculum jumped pretty quickly from questions like this

…to questions like this.

To bridge that gap, I paired an estimation question and a worked example activity.

The purpose of the estimation question was just to make sure everyone was ready for the example. I wanted to remind kids that each of these equations would produce a line, and I wanted to remind them that their existing techniques wouldn’t work to provide a precise solution.

I was hoping that there would be a nice example activity as part of the Algebra by Example collection. Sadly, nah, it jumps into too much, too quickly for my group. So I was forced to make my own (doc here):

I find it really necessary to add a “Ready for more?”-type question to these, as some kids really need time to study the example carefully. A good reminder that it all works together — a good extension for fast-finishing students is an accommodation for my slower-finishing students.

I ran the example activity using the routine that I described here.

What makes this bloggable, for me, is that it shows how various things can work together. A worked example, an estimation question, connecting graphical and algebraic representations — there are people who advocate for any one of these, but they only really work for me when carefully aligned.

• The graph/equation piece is an important bit of non-procedural knowledge.
• The estimation bit makes sure students understand what a type of problem is asking.
• The example helps students focus on a useful procedure, and the prompts to explanation make sure they understand it.
• The “ready for more?” challenges students while giving me more time to help those who need it.

Today was a day when every piece mattered for me.