I’m guessing that most of you reading this are familiar with the awkward acronym for the Math Twitter Blog-o–Sphere – one of the joys of tapping into this community is that they are remarkably generous about sharing ideas and resources. Today in our Geometry classes (I teach only one of the five sections we have at our school) we used an activity written by Kate Nowak (@k8nowak on twitter). It is an activity based in GeoGebra and allows the student to explore the ratio between lengths of legs in a right triangle. You can find the document we used here . I modified (very slightly) the document that Kate originally posted here. Next time I use it I will tweak it a bit. I have only twelve students in my class and chose not to explicitly team them up. They talked with their neighbors as they are usually encouraged to. However, the directions either need to be tweaked so that team references are excluded or I need to clearly team them up. I am also debating question 7. A number of students did not make the explicit leap from using the ratio they found on page 1 and using it here. I don’t necessarily want to give away too much but I may add a little prompt that they should consider the work that they have already completed. We set up a google spreadsheet and in the next couple of days I will refer to this repeatedly to show that different students working on different triangles were arriving at the same ratio. We make a big explicit deal about scale factors between similar figures. I do not think we spent enough time pointing out that scale factors within figures will also match up for similar figures. I will definitely make this more of a point of emphasis next time through my text.
I cannot thank Kate enough for sharing this activity. My students worked well and I am convinced that they will have a more solid grasp of trig ratios moving forward. As I plan out the rest of the unit I am also going to be borrowing from Sam Shah’s latest post about trig. You can find that over here.
Man – the benefits my students are reaping from people that they will never meet – such as Kate Nowak, Jennifer Silverman, John Golden, Jed Butler, Sam Shah, Pamela Wilson, Meg Craig, and so many more – is just remarkable.
So in Geometry today we began to study the ‘special’ right triangles and I had an idea last night that I wanted to try. I handed each of my students two pieces of paper, a ruler, and a protractor. On the first page I asked them to draw an isosceles right triangle on each side and asked them to have the legs of their triangles be different lengths. I polled the students and had them tell me one of their leg lengths. I then asked them to find the length of the hypotenuse and tell me what number they get when dividing the hypotenuse by the leg length. I, of course, got a variety of answers all of which hovered around 1.4. Some students used the Pythagorean theorem and gave me decimal approximations. Some used the Pythagorean theorem and gave me radical answers. Some measured the hypotenuse with their rulers. I asked them why these answers seemed so close to each other – I specifically avoided the word similar here. Luckily, one of my students told me that all the isosceles right triangles were similar to each other. I pushed back a bit and asked what that had to do with ratios within one triangle. We usually discuss similarity ratios between triangles. The explanations from the students were not as concise as I hoped but we all seemed comfortable that rations within a triangle will be the same when looking at two triangles (or in this case 12) that are similar to each other. Since a few students used radicals we had the exact ratio in front of us and a quick solution using algebra confirmed that the ratio was the square root of 2. Success!
Next up I asked them to draw two equilateral triangles and construct an altitude. Now I asked for the ratio between the altitude and a side length. These answers all hovered around 0.87. We were running out of time now so I did a little more telling than I wanted to but we saw the ratio for the three sides of this new right triangle were 1 : square root of 3 : 2
I have to say I was pleased with their persistence, with their measuring/equation solving, and with the idea that we could see these ratios without simply giving them formulas to try and remember. I may be an incurable optimist, but it feels to me that these ratios will be easier to remember at this point. Now I need to have the discipline to avoid using the words for the trig ratios for at least a few days. I am going to steal ideas from Kate Nowak (here is her trig blog post) and Jennifer Wilson (you can find her trig wisdom here) as I attempt to shepherd my Geometry students through the tangle of right triangle trig. I feel that we had a good start today!
Our school has a two-week spring break at a silly, early time in the year. We have been back for a week now and I feel like my students and I are all getting back in the groove again. I know that the dreaded senior slump will continue to pick up momentum but at least I am still seeing some energy and engagement from most of my seniors.
I have a few posts bubbling in my brain and I suspect it’ll be a busy blogging week. Tonight I want to briefly touch on my AP Calculus BC class. We are just settling in to our last major required topic of the year, the Taylor / Maclaurin polynomials. I wrote a little GeoGebra demo (you can find it here) and I started off by showing them (without revealing the mechanics behind the scenes) a polynomial approximation of increasing degree for the trig function y = cos x. We played a little noticing and wondering and saw that at certain stages the polynomial did not change. It did not take long to deduce that this happened at the odd powers of the Taylor polynomial. This led to one student remembering something about the symmetry of cosine, another student mentioning that this was a y-axis symmetry and, finally, a third student mentioning that this is even symmetry. So the lack of development due to the odd powers of the Taylor made a little sense. We then switched to y = sin x (as in the link above) and, unsurprisingly, saw that the even powers seemed to do little or nothing here. We did a little more noticing and wondering watching the Taylor expand on GeoGebra. I should note that all of this was centered at x = 0 (or, in the Taylor notation, we had a = 0) GeoGebra’s sliders allowed us to begin shifting that value and some interesting (and ugly/scary) things started happening to the Taylor equation. My kiddos quickly saw that the equation seemed to be undergoing a simple horizontal transformation – at least in the x terms. The coefficients were changing in some mysterious ways. Finally, we looked at the Taylor series for y = e^x. One of my students asked a great question at this point. He asked – Why are there all those factorials in the bottoms? I skipped this question around the room a bit to see if anyone wanted to make a guess. They quickly observed that exponents in the numerator were clearly attached to the factorials int he denominator but – understandably – they had no solid guesses. Without giving away all the mechanics (we have plenty of time for that) I asked what the derivative of x^7/7! is. I was told it would be 7x^6/7! Correct for sure, but unsatisfying. I must have made my unsatisfied face because one of my students offered a much cleaner version of that answer as x^6/6! Again, I did not go into the mechanics at this point, but there did seem to be some sense that this was an interesting thing to note. I was pleased by the power of the graphics of the GeoGebra applet. I know that I could do something similar in Desmos but I don’t know the commands there as well as I do in GeoGebra. I will start class off tomorrow with the power series we derived for e ^ x and I’ll ask for derivatives and integrals of that. Should be fun to see them realize in this format why the derivative of e^x is itself.
Fun to be back and excited to unfold Taylor’s series’ with my students. This was one of the genuinely awe inspiring topics when I studied Calculus. I remember being amazed by this idea and it’s mechanics. I hope I can share that wonder.