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Tuesday, November 18, 2008

Flashes of Brilliance

I'm sure I'm not the first one to think of this, but this lesson was pleasingly effective. In past years graphing inequalities in two variables was something that seemed easy for the kids in class, but they could never remember what was up when we reviewed it later.

This year, to start off, I had the kids pick the coordinates of three different points (x, y) on the plane, test them in the given inequality, and mark the point with a closed circle if it came out to be true, and an open circle if it came out to be false. After a few minutes I had to say, "If you only got true points, find a point where it comes out to be false." We ended up with a screen that looked like this:

We discussed the significance of the boundary between the blue and white spots, and also tried a few non-integral points. Then I had them try a different one on their own. They developed a very serviceable method for indicating the "true region". One class wanted to write the words "True" and "False" instead of shading, which isn't bad, but I warned them that eventually we'd need to know where two of them overlapped, so they shaded.

This seems like a stunningly obvious approach now that I've tried it, but I thought I'd share since it took me 4 years to think of it. Does anyone do anything similar, or have a better way?


  1. I've used those little circle stickers - red for "false" and green for "true", then we put them on a piece of poster paper.

    It works well for systems of inequalities too!

  2. I love this idea! We are starting inequalities on Monday. I think I will try it!

    Mrs. H

  3. This is a great idea! I'll share this with the teacher of the student who responded to "what are you learning?" with "I don't know we're shading in graphs."

  4. I'm learning that I need to keep going back and back to showing that, for instance, a line is comprised of all the ordered pairs (x, y) that make an equation true. And a shaded region is the same idea for an inequality.

    If others try it let me know how it goes, and if you have any suggestions for improving or applying it elsewhere.

    Jackie, I didn't introduce systems of inequalities this way, but we spent a good 5-10 minutes with the graph of x >=3 and y < -2 indicating closed/open circles for points in/not in the solution - it felt like the classes were really with it.

  5. "The graph of the line is the set of points that make the equation true." is something I find myself saying repeatedly at repeated times throughout the year. It's right up there with "whatever you do to one side, you have to do to the other"

  6. Don't forget "if you are multiplying two things together and getting a product of zero, one of the things must be a zero!" I should find a way to record that so I can just press a button and play it.

  7. Kate,

    I think if you start this with the lines themselves, it comes easier.

    I graph a few solutions to Ax + By = C, notice they line up, draw the line, find more solutions, show that THEY fall on the line, try a few non-solutions, show they fall OFF the line, then pick points on the line that we have not yet named, show that they are solutions, pick points off the line, show they are not solutions.

    Beating this one near to death pays dividends. The line is a picture of all the solutions. Everything on the line is a solution. Everything off the line is not a solution. Repeat, repeat.

    I do the same as you with inequalities, but with X's and O's or colored chalk. It feels like an extension of the line exercise, and goes very quickly, with good understanding.


  8. Excellent point Jonathan. I should do more of that with graphing lines. I run into the problem teaching graphing lines that they already spent quite a bit of time on it in 8th grade and already think they know everything. I need to find more ways to surprise them and make it fresh. For example, I like to show them how to graph the intercepts of an equation in standard form instead of converting to slope-intercept. But really I should be making the point that if they can just find any two points, they have the line. The intercepts just happen to usually be really easy.


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