As a teacher only halfway through my second year, I sill have A LOT to learn. Teaching takes years to fine tune; whether it be lessons or classroom management, my abilities improve little by little. For example, I still struggle with classroom management, but I am a whole lot better at keeping a class on track and motivated than I was a year ago.

That is what I love about teaching; I will never reach a point where I can’t improve on something.

Today, algebra was working through some practice problems on solving systems by graphing. *(I teach algebra 4 periods in a row). *As the morning went on I noticed across the board in every class students were struggling with a particular part of the same problem.

**Solve by graphing**

**x + y = 0**

**3x + y = -4**

They had no issues graphing the first equation; but for the second, what I saw was they would solve for y; **y = -3x – 4, **the issue was the graph bottomed out at -6, so what they did was from -4, went up three, right one.

Every. Student. made this mistake.

We have gone over a few questions like this (which is teacher talk for **not my issue, students need to learn the stuff). **

I am uneasy about that reasoning. I’m sure students take the bait on problems like this and make mistakes, but the fact that this was happening over and over and over again is interesting to me. Is there some other way to approach graphing lines that would have students thinking a different way?

This particular issue is an easy fix, but, gets me thinking about students memorizing a set of steps to solve a problem, then running into big issues when a problem doesn’t follow those steps. I want to reach a point in my teaching career where, to quote David Cox, I can make an impact on students to flip their thinking from:

**“If I know the rules, then I can do the math”**

to

**“If I do the math, I can know the rules”**

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Dan – I also wonder about the automaticity of turning this into slope-intercept form. How do you feel about this consistent instinct of your students? In my Geometry class I have been showering love on the point-slope form (I also teach Calculus so I have skin in this game!), and I have been talking about how to recognize intercepts right away in standard form. I am impressed that your students were comfy with the first line you wrote, I often see problems with interpreting the origin as a y-intercept.

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One thing that came to mind quickly is for students to think about what each equation should look like graphed before they ever put pencil to paper. (Linear; Negative slope so decreasing; negative y intercept) If they think about this FIRST, they should recognize when the graph doesn’t match what they think should happen. This also helps them develop a sense of intuition where math is concerned.

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