Monday, March 7, 2011

Relief At an Easy Problem

Many of the students in my Beginning Algebra course are nervous about their abilities. And I push them to really think about what's happening. So when I asked them to solve a matching problem in the book that felt easy to them, they loved it! Relief washed though the room.

They're just learning to graph linear equations. I've begun to realize how hard it is for many people to see a point as being the equivalent of two pieces of information (the x-coordinate and the y-coordinate). One visual object turns into two numbers. So of course they struggle with graphing lines. The matching problem from the book gave 4 equations, all y = 5x+b, with 4 different numbers for b, along with their graphs. They liked doing that.

Today I'm going to start by asking them to graph 2x+3y = 6. That will take a while. Then I'm going to hand out the sheet I just made with all 8 possible equations like this (± 3y, ±6, and swapping the 2 and  3) and all 8 graphs, so they can do another matching problem. That will go quickly take over an hour, but I think they'll really enjoy it, and will begin to think about how the same numbers can make different lines. (Embarrassing how wrong my estimates of difficulty level can be. That sheet was definitely not easy. But they worked hard on it, and I do think it worked out well.)

[How do I show a Google doc file here?]


  1. Wow, that sounds like a foreign language I once knew ... your method sounds like a very effective approach. Everyone likes to taste success!

  2. Actually, I worry that I don't offer them enough tastes of success. Something I need to keep working on...

  3. "I've begun to realize how hard it is for many people to see a point as being the equivalent of two pieces of information (the x-coordinate and the y-coordinate)."

    I had the good fortune of having Robert Davis as a professor at Rutgers. One of the "warm up games" he invented for kids as part of the Madison project was very good at introducing kids to the x,y coordinates. It involved creating a grid of dots and asking the kids to pick two numbers and if the numbers were on the grid that team got an X on that spot then the other team would go. Then once they had three in a row he would cross a line through the dots and say team X|O wins. Part of the beauty of this lesson was that the only rule is no one can ever tell anyone the rules. The kids have to figure out how it works on their own. I wrote a blog post about it awhile ago here: here

    I should really make a video as it is easier to explain things that way. Also you can extend if from picking to numbers (where the kids usually pick positive) to negative numbers, and even operations where for example they have to pick for numbers and you pick an operation to perform to get the X and Y.

  4. Yes... I often forget how completely and utterly rare it is for the students to be able to just do something and get it right. We tend to keep them at the edge of that frustration level... when a little more time spent practicing what they know, successfully, would go a long way at many levels.
    I think this contributes to a Mathew Effect in math -- them's what has, gets... if you're already behind, you keep getting further behind. You are so accustomed to not quite mastering it that you stop expecting it.

  5. To embed Google docs, you need to publish them as a webpage (top right under the "Share"), then choose "HTML to embed in a page" option. Copy the HTML code to the clipboard.

    In Blogger, choose the "Edit HTML" tab and paste the code where you want it embedded... It won't be especially "pretty", but you can tidy up the HTML code if you feel brave enough!

    If you change the doc, it will also change dynamically in any blog posts too, so be careful!



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