Thursday, January 07, 2010

Introducing Linear Inequalities

To show that a line is a representation of an infinite number of points, I like to give my algebra 1 classes an equation, like y = 2x + 3, and then give each student a couple of different ordered pairs - some that are solutions and some that aren't.  I have them each work out their points, and then go to the board to plot an open or closed circle, depending.  Once all the students sit back down, we look for patterns and see that all the closed circles fell on a straight line.  Discuss, and voila.

This extends nicely to linear inequalities (and systems of equations and inequalities).  On Tuesday, my algebra 2 students were reviewing linear inequalities so I did this activity with them.  I really like it, because it is engaging, and it helps build a mental picture that they can rely on later on when they are struggling through graphing problems on their own.  My students often get stuck on the "pick a test point" part of the process; but now, I ask them if they would have plotted a closed or open circle based on their result, and to think about what the picture on the board looked like.  This usually helps them see which side of the boundary line to shade, and to be able to explain why.

Here is what the board looks like after students plotted their points:





Then, we looked for patterns.  Usually, a student will come to the board and draw some sort of line after getting frustrated with trying to explain it in words.  Then I reveal the shading:



And there are usually some audible "ahhs" and such.  Another great benefit of this is that the string of open circles on the boundary helps students see what the dotted line is all about, and why changing the inequality to include an equals sign would create a solid line - a string of closed circles.

Here is my lesson that goes with this.  And the keynote.

7 comments:

Mrs. H said...
This comment has been removed by the author.
Mrs. H said...

Sorry, Dan, my other comment was messed up so I removed it, but, great idea! I hope to try this soon.

Kate Nowak said...

I think you and me were separated at teacher birth. I like how you explained more of the flow of your lesson though.

Dan Greene said...

That's funny. I looked at your post and I actually remember reading it when you wrote it. I wonder why I didn't comment on it at the time. Or maybe that's actually where I got the idea and just forgot? Hmm... am I old enough to start forgetting things like that?

Ben Blum-Smith said...

Awesome Dan & Kate. Makes totally clear that the graph is a locus of points fulfilling an algebraic relationship.

(Btw. did the word "locus" ever trouble you due to similarity with "locust"?)

Kate Nowak said...

Hee. My first lesson of the Locus unit has a slide that says "A locus is a collection of points. A locust is a bug." (with a picture)

EastCassian said...

Thanks very much, was looking for a lesson to introduce the concept of a linear inequality, that's brilliant! :)