Teacher education, whether traditional or TFA, places great emphasis on essential importance of planning. I am not overstating when I say that many ed school instructors think that planning is teaching. Ed school instructors–and teachers as a group–are planners, the type who get nervous if they don’t know what they are teaching in two weeks, let alone three days out.
Me, I often don’t know what I’m teaching the next morning.
Like today, for instance. I’m in the middle of special right triangles. I introduced the isosceles right triangle before Christmas. Penetration was weak; the ones who were able to work the problems correctly still weren’t quite sure about it.
I always teach the pattern–x, x, xroot2 (to save me typing symbols) and show the students how easy it is just to use the pattern to assign values. Know one, know all. I derive the relationship with the class, I explain that it’s a ratio, and to me it’s just obvious from that point. But it hadn’t been obvious in my class two years ago, and it clearly was taking a barely tenuous hold for most of my students. So I mulled it over the holidays, and suddenly wondered why I didn’t just teach it as a proportion. After all, they need to review cross-multiplication. And then it occurred to me that special right triangles are, in fact, similar triangles, even though the curriculum separates them out and rarely makes the connection. So why not introduce proportion now? I checked around and the other geometry teachers don’t teach it this way, but it was worth a shot. Oh, by the way–this part was planning. I didn’t write any of it down, but it was all in my head.
After reviewing ratios, proportions, and cross-multiplications for a day (my top students didn’t review and did far more complicated problems), I re-introduced special right isosceles as a ratio and had them solve problems using the ratio. Much better penetration, although it’s still early. But even my weakest students were able to set up problems and cross-multiply, even if the last step of solving for x still left them a bit confused.
But through all this planning, in the back of my mind, for three weeks, 30-60-90 triangles were a black box. I’d have done a lot of lecturing by that point, and I needed to break the routine. 30-60-90 triangles are more complicated than isosceles rights, too. So things needed shaking up. But how?
The idea came to me at 6:30 am. Fortunately, it was a late start day, so after the staff meeting and the coffee run, I still had 10 minutes to think through how to write the instructions on the front board. And two of my students came in early, so they generously copied the instructions on the back two boards while I rounded up rulers and protractors and scratch paper.
- Your group will construct (create) 4 equilateral triangles, which are triangles with _____ equal sides and angle measures of _____. (I then asked the students to fill in the blanks). You will need four triangles: 2″, 3″, 4″, and 5″. Each person creates one triangle.
- Draw the base side with your assigned length. (In a different color, a line example)
- Use the protractor to construct a 60 degree angle. (In a different color, the 60 degree angle mark with an x.
- Use the ruler to draw the second side of the same length from one end of your base line through (or towards) the 60 degree mark. (In a different color, two sides of the triangle are now complete)
- Connect the remaining two sides with a third. Confirm (make sure) that the third side is also of the same length.
This worked really well. The kids had trouble with the length of the second side in some cases (connecting it with the angle mark rather than using the assigned length), and as always, some of them refused to start until I came around and issued a reassuring personal invitation. But it was a great review of equilateral triangles, let them do a little bit of construction (which I don’t emphasize at all) and reinforced the important idea of SAS congruency and the rigidity of triangles. Many students really “got” that there was only one possible length for the third side, and made that connection to earlier lessons.
What, you may ask, does this have to do with 30-60-90 triangles? Well, almost all equilateral triangle questions end up using 30-60-90 triangles, so it’s helpful to let them see this relationship. The last thing I did was ask the students to visualize what would happen if they folded their triangles in half. What would they have? Everyone saw that it would be a right triangle, created in the middle (we will review the “isosceles altitudes bisect” theorem tomorrow, briefly). They realized that one angle wouldn’t change, and one would be cut in half. “Oh!” said one student, pointing to the board where I’d written the two special rights. “30-60-90!” and the class all made agreement sounds.
And tomorrow, they will have new instructions.
- Get out your triangles from yesterday. (Or hang, draw, and quarter the person who was supposed to keep them safe and then ask me politely for extras.)
- Create a table for your four triangles, with length columns for Short Leg, Medium Leg, and Hypotenuse (Long leg? Hmm.) and one column where you will find the ratio of the medium leg to the short leg.
- Using a ruler, find the side values and add them to the table.
- Using a calculator, find the ratio of the medium leg to the short leg by actually dividing the medium leg by the short leg. (I will review the fact that all fraction statements are division problems, something I do frequently).
- When you’ve finished, put your values on the board. (where I have tables for all the different triangles).
This will give me a chance to discuss measurement error. The students will all see that they have the same values, which they’d expect, and that everyone has very similar ratios.
And THEN, after that, I will go through the Pythagorean Theorem for this kind of triangle. They will see that the short side and the hypotenuse have a ratio of 1:2, and we can use that to generalize (which we’ve already done for special rights).
I never would have come up with the “bridge” of constructing equilateral triangles and turning the ratio into a discovery class if I hadn’t let it percolate for three weeks. You can’t force these things.