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I've been trying something new with my level 3 geometry. We were doing triangle inequalities and I was adapting a lesson I read about in

We then discussed what the relationship between the sides needed to be in order for a triangle to be possible (i.e. the sum of the two shorter sides must be greater than the longest side). After a couple of simple problems determining whether a triangle would be possible given three lengths, I gave them two lengths. I asked them to determine what the third side could be. They began by calling out a few possibilities. After they had manged to determine the possible whole numbers it could be, I asked if that was all the possibilities. Thay said yes. I asked if it could be a decimal ("Are you saying that we cannot have a side that is 45.5?") One student said we couldn't use decimals because there would be too many possibilities that way. I replied that if we used decimals, there would indeed be a lot of possibilities - an infinite number, in fact.

I asked if there was a way we could show all the possibilities, it being obvious to them that we could not in fact list individual numbers. One of my students (lets call her Erin) said we could graph them. Alleluia!!!! Erin is also in my level 3 algebra II class, and often purports to be lost. I had not thought of graphing as a way to show all the possibilities - I had been looking for the students to say "write an inequality". But graphing was even better - we had recently done graphing compound inequalities in algebra II and here was an honest to God

Anyway, the point is, I was excited about how the lesson went. The students really got into figuring the spaghetti triangles out, and even got into how we could show

*The Mathematics Teacher*, using uncooked spaghetti as the side lengths of the triangle. You have the kids break the spaghetti into pieces, first to try and make a triangle, and next break it into pieces that will*not*form a triangle. The author said that about half of his kids were able to determine the breaks that would make a triangle impossible in the time he had allotted for it. All of my kids managed to figure it out in less than three minutes, which I wasn't expecting. (Although maybe some of them were looking around and seeing how their classmates did it. )We then discussed what the relationship between the sides needed to be in order for a triangle to be possible (i.e. the sum of the two shorter sides must be greater than the longest side). After a couple of simple problems determining whether a triangle would be possible given three lengths, I gave them two lengths. I asked them to determine what the third side could be. They began by calling out a few possibilities. After they had manged to determine the possible whole numbers it could be, I asked if that was all the possibilities. Thay said yes. I asked if it could be a decimal ("Are you saying that we cannot have a side that is 45.5?") One student said we couldn't use decimals because there would be too many possibilities that way. I replied that if we used decimals, there would indeed be a lot of possibilities - an infinite number, in fact.

I asked if there was a way we could show all the possibilities, it being obvious to them that we could not in fact list individual numbers. One of my students (lets call her Erin) said we could graph them. Alleluia!!!! Erin is also in my level 3 algebra II class, and often purports to be lost. I had not thought of graphing as a way to show all the possibilities - I had been looking for the students to say "write an inequality". But graphing was even better - we had recently done graphing compound inequalities in algebra II and here was an honest to God

*application*of them. Not only that, but a student thought of the way to apply it. I had the students describe for me exactly how I should do it - it took a little while, but we eventually had everything right - 2 open circles and a line connecting them. The discussion (I think) led to a better understanding of compound inequalities for all. We then wrote the inequalities from the graph, and it was good practice for*moving between different mathematical representations.*Anyway, the point is, I was excited about how the lesson went. The students really got into figuring the spaghetti triangles out, and even got into how we could show

*all*the possible side lengths in the last problem. It was*much, much better than a lecture. And it was reinforcing earlier concepts that they had learned.*
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