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
I have a math related post brewing in my head, but I want to take a moment to relate a story from this past crazy week. In NE PA we have had cold and snow for days and the last time our high school started at 8 AM was a week ago on Thursday. We had a scheduled day off on Monday , three two-hour delays and a late start due to a faculty meeting. My children go to our lower school about three miles away and they take a bus isn’t he morning. On these delay days I am not entirely confident of timing with my kids on the bus and my students lining up for morning class. So, I tend to take my children to school a bit early and get to class. With other young children on campus we share trips down the avenue. Friday morning as I was waiting in the side stairwell with my two kids, waiting for a neighbor to pick them up, my phone rang. It was another neighbor offering to take my boy in with his daughter since they each have band practice. The idea that multiple people are offering to take my kids to school is pretty heartwarming.
Last night my son’s middle school basketball team was invited to have a scrimmage during half time of the varsity game and then they joined the big boys for a pizza party. A great community night. Tonight in the dining hall my five year old daughter was playing tag with a sophomore in the high school running around the place.
there are certainly some drawbacks to living Ina dorm and being on duty pretty much all the time. The past few days remind me of the benefits of this situation.