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.
A quick post here – I want to share something delightful that a few Geometry students did this morning. We had our last test of the winter term today and here is one of the last questions:
Prove that the points A (x, y), B ( x + 1, y + 3), C (x + 4, y + 5), and D (x + 3, y + 2) are the vertices of the parallelogram ABCD. Prove this is true by one of the following two methods:
- By showing that one pair of opposite sides are congruent and parallel.
- By showing that both pairs of opposite sides are parallel to each other.
So, I was hoping that the majority of my students would take the quick and easy option of calculating slopes rather than messing with distances. I also hoped that the coordinates having variables in them would make them slow down, be careful, and remember a touch of algebra. I grade page by page and I have graded three of the papers with this problem on it. One student said ‘We can let x and y be 0 so the coordinates are (0,0), (1,3), (4,5), and (3,2)’ I love this thinking. She avoided the worry of dealing with the variables here. It’s a little slippery to determine just how clearly she was thinking here. She might have just been dodging a bullet. One student said ‘I will first transform this parallelogram by the vector <-x, -y> and then we will have the coordinates A’ (0,0), B’ (1,3), C’ (4,5), and D’ (3,2)’ Now, it is ABUNDANTLY clear that he knew exactly what he was doing. I’m so delighted by this that I felt I should share.
This and my great AP Stats classes today made for a pretty terrific day!
When I last dropped by my own blog here to write I was cooking up an idea for my AP Statistics class. I wanted to write a good activity to explore data using my FitBit. I was lucky enough to win a FitBit Flex in a raffle and I’ve been fascinated by it for the past month. I had four volunteers who also shared their FitBit data with me and I put my data (identified) with the data of my four brave volunteers (not identified) and developed what feels like a pretty good activity. Yesterday I displayed my data in an EXCEL sheet before deciding I was better off in a Google sheet. We looked at my data together and with Desmos open we transferred some data two columns at a time and looked at lines of best fit. We tossed around some questions that seemed interesting, we questioned some of the data presented (especially the first data line on the sheet identified as Doherty Data), we made guesses about what relationships were hiding. We discussed the impact of height and weight and just generally had a pretty good time noticing and wondering together. Last night I combined all the data into a Google sheet (which can be found here) and I condensed some of the questions that came up yesterday and wrote them up on a Google form (which can be found here) and today I just set my kiddos loose. We have access to a computer lab so everyone had their own space to work. I dealt cards at random to spread the sheets around, I wandered and answered questions about moving columns on google sheets and how to make Desmos (like this graph) work its regression magic. I had discussions about resting heart rate, about whether calories burned or active calories were more interesting to look at. We remembered the dangers with extrapolation when discussing the y-intercept of these regression equations. We tried to figure out which mystery person might be taller or which one might be heavier. There is real joy in listening in on these conversations, but my biggest highlight today is that I got to show off the spirit of my classroom to a visitor today. He remarked on what a treat it was to witness ‘sense-making’ in action. I want to revisit my questions and make them better next time around, but I am pretty pleased so far. Tomorrow, we’ll start class with about ten to fifteen minutes to wrap up this activity and then we’ll share our discoveries with each other.
Right now my AP Calculus BC class is taking their final test of the term. I hope I am as happy grading those as I am thinking about my AP Statistics team right now.
Our school hosted a ‘Maintain, Don’t Gain’ campaign through the Thanksgiving and Christmas holidays. Those of us who volunteered to be weighed in before and after were candidates for a raffle if we met the goal of no weight gain. I managed to lose 2.2 pounds and got lucky in the raffle by winning a FitBit Flex. I hooked it up on Jan 28 and I am thinking it will help me in AP Stats next week. My cherubs have a test this Thursday and then five more school days before our two-week break. We hilariously call it spring break even though it feels nothing like spring in these parts. Anyways, I am thinking of downloading all my data into an EXCEL sheet and challenging my AP Statistics scholars to dig into this data. As an added bonus, I know a number of them wear a FitBit as well, so we might be able to get a nice data set out of all of this. What I am wrestling with are the following questions/concerns:
- I do not know how sensitive FitBit is in its calorie counter. I have lost some weight in the past month (yay me!) and I do not know if that would interfere with looking for a connection between steps taken and calories burned.
- I am not sure how consistent FitBit is with correlating steps and distance. Are there any FitBit pros out there who can let me know about their experience with this? You can comment here or tweet me @mrdardy
- I want to ask some structured, guiding questions but I do not want to lock them in to my ideas of what might be interesting. I just do not know how focused they will be or how sophisticated they are as statisticians at this point.
- Debating whether this is better as an individual project idea or a small group one. I am inclined to think that groups are better here. Any thoughts or advice about this?
So, I am shamelessly asking for help and wisdom here. I thank you in advance for any smart comments/tweets/emails/etc
A pretty interesting conversation unfolded in Geometry this morning. We are getting ready to explore similarity, so I gave the kiddos a quick assignment on solving proportional equations with one variable. This was meant to be pure review. When we started talking about these problems I, of course, heard talk about cross-multiplying, cross products, and even heard one student exclaim something about the old keep-change-flip idea. I decided to stand firm and talk about why we were able to do what we do with these proportional equations. We started simply with the equation like One of my students was taking a vocal lead in discussing cross products and I asked her what equation to write next. She told me to write . I agreed that this was correct and most of my students recognized what she was doing. I then asked them to pause and wrote the following equation I asked everyone what they thought the value of x had to be in this situation. They all seemed to agree pretty quickly that x must be 6. So I got them to agree that an equation with one fraction on each side AND the same denominator demanded that the numerators would be equal. They all seemed to think I was making too big a deal out of this. I then asked them if I could do the following to . I asked if I could multiply the right hand side by while multiplying the left hand side by. One student protested that I need to do the same thing to each side of the equation. I, of course, agreed with her but I asked her to look more closely at what I was doing. She agreed that I was doing the same thing even though it looked different. Most of my students still seemed to think that I was making too big a deal out of this. Next came the payoff. I picked the following problem from the homework: I pointed out that our cross product idea was not really a comfortable fit here. My KCF student quickly suggested that we clear the fractions out of the problem by multiplying by 20. I agreed that this would certainly work but asked if I could try something different. So I wrote the following equation: I was immediately met with resistance. I begged for patience and made them a promise. I told them that I would carefully explain why I was doing what I was doing and that if they unanimously decided that they did not like this approach, then I would cease and desist. I pointed out that we were, again, doing the same thing to each side even though it looked different. I made an argument that multiplying by smaller numbers decreased my chances of arithmetic mistakes and I pointed out that this technique made the common denominator for the problem obvious. The equation became . I saw some signs of visible relief as they saw that this was now a pretty easy equation to process. Combining like terms gave us and a conclusion that . I then solved the problem the more standard way by multiplying everything by 20 to begin with. I felt like it was a bit of a triumph when they voted that this new technique did not need to be banned from our vocabulary. I know that this is not revolutionary, but I certainly think that I made some strides here. My students are well-trained in mechanics and they know what works. I want to have serious conversations about the ideas behind why these techniques work.
Back at it again tomorrow!
PS – Thanks to David Wees and Zach Coverstone for valuable assistance in learning some LaTex for this post. I hope it looks right when I hit publish