Plan With Me...Infinite and No Solutions

So I'm planning this lesson for the upcoming week and I have 3 slides that I feel like are a good start. I'm just about to tweet them out and ask what I can add to them when I decide to check the MTBoS search engine first. I land on this great blog post about using Desmos to check answers after combining like terms by graphing.

And instantly my lesson just got better. I get to use Desmos for the first time with my freshman and we are just beginning!

Here's a general outline and my thought process.

I will ask students to share out some answers and I will write them on the board. I will have one chrome book per group of three students and ask them to each take turns typing in an expression from the board (I'm thinking 6 so each student types in 2 and purposely include wrong ones).  But what if I don't get very many answers?

I will ask them what they notice as they type in each equation.

We will discuss the connection between the expression and the line.


I will ask students to prove me right or wrong. I'm thinking I'll have to explain that they either need to solve for x or plug in random values and see what happens. Some kind of work will happen which will lead us to graphing it in Desmos and seeing if it is the same line or not.


Some kind of work happening, either with Algebra or Desmos leading up to the fact that they graph two parallel lines which have no intersecting solution and how they simplify to the same slope with different y-intercepts.

Is that it? Now we just practice?

What are some good questions I can ask? What needs to go in their notes?


  1. 2 things first- what level class is this, since I know you teach every math class. And I don't know what you meant in the first slide since the = disappeared.

  2. This is an Algebra I lesson. The equal sign didn't disappear; there is no equal sign. They are giving me equivalent expressions to the one I provided.

  3. "I'm thinking I'll have to explain that they either need to solve for x or plug in random values and see what happens." Is that necessarily the case? I'd say have students solve for x, then dial things back a few steps. In your example problem, a student might simplify the right side of the equation to -5-9x. I would write -5-9x = -5-9x on the board (or whatever) and ask students, "Why can I stop here and claim every number will work?" I can see this question about the algebraic representation tip-toeing towards the visual representation (the lines look the same). I'd ask a students a similar question about the no solution case.

    1. Thanks Tom, that's a great question to pose.

  4. Do you use INB's? If yes, I would follow up with a good foldable!

    1. And good luck, let us know how it goes, I love Desmos for equivalent expressions!

    2. I do use INBs but the pages I made are not a foldable. Now I may have to rethink that! :)