# Equations, expressions and inequalities, oh my!

I was in a grade 9 class today trying out a lesson we co-planned about equations.

The discussion with the teachers when planning the lesson was around whether or not students understand what an equation is, and if they understand what it means to solve an equation.

This is what we did:

• Students, working in pairs, a set of cards revised algebra sorting cards. Students were given the instruction to organize the cards into groupings that made sense to them. I reinforced that their groupings had to make sense, in that, every card in the group should be there for an obvious reason- they should all have something in common.
• We then asked the students to visit 3 other teams and try to figure out how they grouped their cards. This was an interesting task and involved a lot of arguing.
• We then did a stay and stray. Each pair of students choose one person to stay with their groupings and then the other partner visited the other teams to check their guesses for how their grouped their cards and to ask for any clarification.

The groupings that students came up with included: grouping by type of numbers in the expressions (these have fractions, these don’t), by whether or not there were variables, by the number of terms on either side of the equation, by formulas, just numbers, just one variable, etc. What was impressive was the level of engagement on this task and the conversations that they were having. They were talking about equations, exponents, fractions, binomials, distributive property, etc. They were really thinking about how they wanted to group them and were often struggling with where to put the last few cards they had.

Students all returned to their seats and then I shared with them how I had grouped the cards and asked them to try to figure out my thinking.

This is what I showed them: algebra card groupings. (note: there are a few equations in the first column that are actually identities and I thought of putting them in a separate group but decided not to have this conversation yet. I think a nice follow up to this conversation would be to give them cards with just equations and talk about which ones are sometimes true, always true and never true) They had a tough time with this. I gave them some time to talk with their partner about it and then we did some sharing. A few of them took some guesses but we often would find one of the cards out of place based on their guesses. For example, a student said that “group 3 has more than one thing on both sides of the equation” but then realized that there was one like this in group 1 too. Eventually we got to group 1 being “things we can solve”, group 2 being “things that don’t have equal signs” and group 3 being “things that aren’t true”. I helped them with the terminology equations, expressions and inequalities.

We spent a bit of time talking about the card that said “x + 1 = x + 2” and why that wasn’t an equation. One student explained that “since the x’s both have to be the same number it is impossible – you can’t add 1 to a number and then 2 to a number and get the same answer”.

We moved the conversation to the first column of “things that can be solved” and talked about what does it mean to solve. Students actually seemed to be very comfortable with the idea that a solution to an equation is an x value that makes the statement true.

The last task was to choose one of the following: a) make up an equation that has a solution of x =2 b) make up an equation that has a solution of x = -3 or c) make up an equation that has a solution of x = 1/2.

Most students chose to do a), a few did b) and no one tackled c) – but it is early in the unit and the teacher will likely have them return to this and try to do c) later.

I was impressed with the complexity of some of the equations they came up with. Here are some pics:

equations with solution x = 2

equations with solution x = -3

Considering that the students haven’t actually worked with solving equations yet this year they seemed to have a pretty good understanding of how to create equations with a specific solution.