This week we learned a great way to distinguish between the different options of how many solutions exist when working with two equations in a system of equations. I was working with a group of students who were struggling to really see what they were trying to do when identifying how many solutions are possible. They just didn’t understand what it meant for a system of equations to have one solution vs. no solutions vs. infinite solutions. So, we started adding the strategy of hand gestures, and it really seemed to help some of them understand this topic differently.

One major misconception I saw was that students kept thinking that if the slope was different, then there must be no solution. I think they were confusing the term “solution” with “answer”. I had to get them to understand that when there is one solution then yes, the lines are not equal- they are different from each other. But, there *is *a solution to the question where do these line intersect? There is one point that these two lines share, and *that *is what we’re trying to figure out.

#### A Super Simple Tip for Teaching How Many Solutions in a System of Equations

I started using hand gestures each time we were presented with a new system of equations to talk about what they would look like, and then talk about how many coordinate pairs would be the same for both lines. It was a simple strategy, but it could quickly and clearly show just what was happening:

**No solutions**

**One solution**

**Infinite solutions**

#### Getting Students Involved

And then I remembered, learning by doing is so much more powerful than learning by seeing. I asked students to use their hands to illustrate the relationship in the systems of equations.Every system of equations we saw, they had to use their hands to illustrate how these two lines were going to relate to each other.

As students used their hands to represent the two lines it was a way for them to explain their thinking. That was really powerful and got everyone explaining each system- so there was lots of mathematical “talk” without just using words. They had to show what the lines would look like and then say how many solutions they believed there would be.

Don’t you just love quick (and free!) strategies that help students really get it?!

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