## Friday, 17 May 2013

### Similar Triangles

Today we started the unit on trigonometry.  We begin with similar triangles so I came up with this investigation - what makes two figures similar?  Students worked in groups of three, armed with rulers and protractor in order to measure the side lengths and angles of three sets of similar figures.  As a group they had to write down the properties of similar figures on their whiteboards.

They quickly determined that the angles all had to be equal, but didn't see the relationship between the sides.  I found that I had to ask a lot of questions to get them thinking about those side lengths.
me: "Which sides should you compare?"
student: "The two longest."
me: "Can you tell which sides to compare without knowing all the lengths?"
student: "No.  Yes...  Angles!"
me: "Okay, so now compare them."

The shapes I created were interesting in that several sides on the larger shape appeared to be 1 cm longer than the corresponding sides on the smaller shape.  They were sure they had the right answer until I pointed out the side where that didn't work.  One group even calculated the differences between corresponding sides:

They could see that increasing/decreasing by a fixed amount was not the right strategy.  It was very interesting to see who came up with the idea of trying to find a number to multiply each of the smaller sides by to get the larger sides.  They erased and calculated the ratio of corresponding side lengths and voilà!  Scale factor!

That was followed up with the second side of the sheet to solidify what they had figured out.

This activity was simple yet pushed some of them to think when they thought they already knew the answer.  They needed to be able to explain their reasoning to each other and to me.

Whether they remember any of it on Tuesday remains to be seen!