# To Accumulate a Rate — Integrate!

Teaching High School Mathematics

Today was much more enjoyable in Algebra than I had anticipated.

I had a 30 minute period today because of an early release Friday and a Martin Luther King assembly. In my 11th grade Algebra class we recently finished solving systems of equations by: Graphing, Substitution, and Elimination. The next topics are either solving equations by graphing each side or looking at systems that have no solution or infinitely many solutions. We couldn’t possibly squeeze those in… 🙂 So… Here’s what I did today:

On the board at the start of class I had written (4,2) and I told students to write a linear equation that included this point as a solution. I also asked them to try to write a different equation than everyone else.

This was harder than I anticipated. I got three students to share and they all shared a different variation of the same equation. I had to explain how to go about it more than I anticipated, but once I did, students created a list of different equations. Here’s a list of the equations my class came up with:

x-2=y                   y+2=x                   x-y=2                x+y=6                2x-3y=2                   4x-4y=8

2y=x                    3y-2=x                 -y+x=2             3x-2y=8            -2x+5y=2                 -x-2y=-8

I asked my students what these equations had in common and followed that up with what they would notice if we graphed them all.

I wasn’t planning to actually graph them, but decided to after there was still some uncertainty… here is the graph:

We all agreed that the lines all shared the point (4,2).

It was fun to see some lights clicking as they realized how this connected to solutions of systems of equations.  The point the graphs share is the solution because that point will make both equations true.

Every student then picked a point in secret and wrote it down on their paper. I told them that if they wanted to choose a challenging point they could use large numbers, decimals, or fractions. Then each student wrote two linear equations so that their secret point was a solution to them both. After trading equations with a neighbor, they tried to find their neighbor’s point.

Here is the spectrum of what the students did…

Some struggled to even come up with two equations.
Some came up with 2 equivalent equations without knowing it.
Many came up with 2 different equations.

Some solved their partner’s equation quickly.
Some struggled to solve their partner’s equation.
Some didn’t bother to find a partner.

I tried to go around and help those who needed it and also told those who finished to try to create a harder problem for their partner.

One student didn’t have a partner and needed a little help to get his equations finished, but when he did had an equation that would be a little harder to solve. None of the coefficients of the same variable were the same or opposites.

We were in the last few minutes of class and on this awkward short day students were starting to get antsy. I grabbed this one student’s paper with his permission and told the entire class that this student had a good one and I challenged them to all solve his system of equations.

At this point, there was 3 min left in class and most of my students were trying to solve his system the entire time without putting away their stuff. 3 finished with the correct answer and I sent the rest off to finish it on their own.

I felt great about the lesson especially since I came up with it right before class and even switched it up on the fly.

For today with time limitations, and because I hadn’t thought about it much beforehand, if I noticed students writing two equivalent equations I told them so and had them write another. I was thinking though that this could be a good transition into “infinitely many solutions.” Students would not be able to figure out their partner’s point.  I could ask everyone to try to solve this “tricky” system and everyone would either get stuck or find solutions that were not what the person had chosen. This would lead perfectly into a great discussion about when this might happen.

Well… That was a good day!

1. Janessa Slattery
June 8, 2017

Could you have a desmos question where they type their equation for the graph, do some trial and error to make sure it works and then you show the class graph with all their answers without you having to go in and type them all yourself?

Does that make sense? I know teachers can show results and can show all graph answers on one graph… That is what I am talking about.

• kaleb40
June 9, 2017

Great idea! I came up with this on the fly so I didn’t have a Desmos activity set up, but in the future that would be a perfect fit for this lesson.

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This entry was posted on January 15, 2016 by in Algebra and tagged , , , , .