On Friday in Geometry we were continuing our conversation about triangle centers and I asked my students to find the point where medians coincide in a scalene triangle. There is a good amount of algebraic detail in these problems but my students were doing a nice job pushing through this problem. After finding the centroid, I asked them to form a new triangle from the three midpoints we needed when considering medians. We found the perimeter of the original triangle and I asked also for the perimeter of the triangle formed by the midpoints. One of my students theorized that the new triangle would have one-fourth the perimeter of the original triangle. I asked the other students to quiet for a moment to hear this guess. Before asking GeoGebra to check his answer he quickly corrected himself and said he was thinking about area, not perimeter. A beautiful realization on his part that this triangle formed by midpoints would divide the original triangle into four equal areas. Just as we were congratulating him for this guess another students asked about equilateral triangles. He wondered aloud whether the midpoint triangle in an equilateral triangle would form four equilateral triangles. I realized he was asking whether the triangles formed in the scalene we were looking at were also congruent, not just equal in area. A quick question from me confirmed my guess so we drew our attention again to the GeoGebra sketch we had up. He was able to identify where the congruent angles were that allowed us to prove congruence for the triangles.
This conversation was a wonderful way to end our day on Friday. I am delighted that my students are comfortable enough to make these guesses out loud and even more delighted that they are making such good guesses right now. I pointed out how helpful it is to play with GeoGebra to check these guesses and I hope (I hope hope hope!) that some of my students are making a habit of this.
Last week in my Geometry class we had a fantastic conversation about a homework problem. Here is the problem in question –
I wish that I could take credit for having written this, but I am certain that I ‘borrowed’ it from somewhere. Likely from the fantastic resources shared with me by Carmel Schettino (@SchettinoPBL)
So, this is the kind of problem that I expect only a minority of my students to navigate successfully on their own, but I am convinced that almost all of them will benefit from thinking about a problem like this one, from a little active struggle along the way. I KNEW that this would be asked in class if anyone took the time to do the HW I assigned, so I was pleased that it came up. I started by telling my students that I LOVE this problem and asked them if they could guess why. One student said ‘Because it’s so hard’. I laughed that off and said, yes it is hard but I love it because it ties together a bunch of important ideas. Off we went on solving this. I started by asking a couple of questions that probably seemed a bit irrelevant at first. I asked why they knew that the y-intercept was (0, 3) and that the x-intercept was (4, 0). Before they could answer I made sure to mention that they knew this without looking at the graph. We eventually arrived at the realization that we know whether a point is on the line or not by looking at the equation itself. If a point makes the equation true, then that point is on the line. If not, then not. This is the kind of thing that I think my students know but being reminded regularly sure does help reinforce it. I hope! So, I thought I had set the hook here for the rest of the problem. We talked about what we know about squares and we talked about how to identify points on the square without knowing their real coordinates. We got a little lazy, and I was okay with that,by calling the bottom right corner (x, 0) and the top left corner (0, y). This gave us no choice but to call the top right corner of the box (x, y). At this point I paused and asked them to remind me what needs to be true about points on a line. Then I asked them to remind me of what we know about a square, therefore what we know about x and y for that mystery point (x, y). It wasn’t easy to get everyone to agree with our conclusions, but I think we got there. We agreed that the x and the y had to equal each other. We agreed that the y coordinate had a definition based on x. We agreed that this was an equation we could solve even though it was not a bunch of fun to solve it. After all of this work it felt like the problem should be done, students were pretty sad to realize it wasn’t. We still had a conclusion to make about the triangles created. One of my students was pretty insistent that they needed to be congruent because their angles had to match up. This was not the time to launch into a conversation about similarity and I decided it was not the time to talk about the restrictions of AAA conclusions between triangles. We have talked about equilateral triangles of different sizes and we are (mostly) okay with that, but I felt that that conversation would be a diversion here. Instead, we kept at the calculating and we looked at side lengths. Once we agreed that they were not congruent, I pointed to the slope of the line and talked about the fact that his instinct was foiled by the fact that x and y lengths were not changing at the same rate. The whole conversation took quite some time, might have been 15 minutes by the time the whole thing was done, but I felt that we had done some important heavy lifting.
If you recognize the above problem as your own, feel free to claim it and let me know. Know in advance that I am very grateful for such a rich problem to tie together ideas of distances, slopes, line equations, properties of squares, and triangle congruencies all into one tidy package!
I’ve been at this high school math gig for a good long while now but I periodically have to remind myself of a couple of important facts. The most important one is that not everybody’s mind works like mine. Just because I like a certain way of thinking, or dislike a certain way of learning, I should not assume all my students will agree. In fact, I can be pretty certain that all of my students will not agree, there’s too many individuals for that to work.
When I studied Geometry I did not like physical drawings and constructions. In part because I am a bit inept when it comes to controlling something like a compass, but also because getting my hands engaged does not seem to fire too many of my neurons. So, when I wrote my Geometry book a couple of years ago I did not include much in the way of hands-on manipulations. The past couple of years of working through the text with our students has pointed out the weakness of this approach. So, I put my head together with one of my talented colleagues to try and make an activity that would trigger some neurons for those students who come to life when they get their hands busy. I had been using a pretty cool activity I ran across from Jennifer Silverman but I made pretty flimsy paper copies to work with on a pipe building activity where kids had to manipulate bent angle joints with different pipe lengths. It’s a great activity but using simple paper copies dragged the activity down. We invested in some packs of AngLegs this year and my colleague wrote a pretty cool activity modeled off of our pipe building activity. You can find his document here.
I was impressed as each of the seat groups in my class played with the AngLegs making some discoveries about combinations that worked and those that would not. We discussed, without naming it yet, the triangle inequality theorem to explain why some combos did not work. But the real fun, and the clever heart of my colleague’s activity, was when I asked one student from each group to come to the front of the room. When they left their group the remaining group members were given the following task – I slightly modified the original document on the fly – I asked them to make and measure a triangle. Find six measures, the three side lengths and the three angles. They then put the triangle away where it could not be seen. I sent the volunteers back and their teammate gave them three pieces of information. I left it to each group to decide what information to share. Once given three clues the volunteer student needed to manipulate the AngLegs to copy the triangle described. What ensued was a terrific conversation about what information is necessary to guarantee that I have to make the same triangle. We used this as a launching pad to discuss congruence theorems for triangles. I have some great links in the text to some wonderful GeoGebra activities up on the GeoGebraTube site but I know that many of my students do not do these explorations. I also know that some just need to get their hands dirty, so to speak. Some kids were able to recreate the triangle but admitted that it was a bit of luck. Some stumbled upon the ambiguous case of the Law of Sines without being told that this is what happened. Some realized that they had no choice but to create the correct triangle.
I was really pleased by the level of engagement and I am now thinking about ways to use the AngLeg sets again soon when we start talking about side and angle bisectors. I want to have tables create and draw their own triangles before we stumble into discoveries about concurrence of these bisectors. This will feel, I hope, a little more authentic than me just giving them a prescribed triangle which may feel a bit like I am just luring them into some pre-prepared trap. I think that this activity we ran benefited my students and we have referred to it on a number of occasions already. The grouping of three or four students together at a time helps and allowing them to get their hands busy has helped. Looking forward to loosening up a bit more and letting my students be more tactile in their approach to Geometry. I’ll still show them the GeoGebra and introduce them to Euclid the Game but I need to remind myself that they are not a bunch of mini Dardys in the room.
Today was a pretty blah day until my last period class. My first three classes all had assessments so I had no fun conversations and I watched work pile up. As I came in to my last class of the day – my Geometry class – one of my Geometry teammates was waiting in my room to share that his students had been making some great strides in GeoGebra. He told me that a number of his students were really beginning to dig into what GeoGebra could do for them, especially now that we are talking about transformations. I used Geogebra extensively when writing my text and I borrowed from resources around the web for activities. One of them was an activity called A-Maze-Ing Vectors which had been created by the amazing Jennifer Silverman (@jensilvermath) and we used that activity the past two years. My teammate who had been waiting to share his good news had asked me this past summer about modifying this activity. We had had trouble completing the activity in one day and it did not take up enough for two solid days. He also had an idea about combining vector transformations on objects more complex than points. He created a pretty wonderful adaptation of the activity (you can find it here) and my students worked through it yesterday. I opened class today by projecting the last page on my AppleTV where we had to navigate a triangle through a maze and I invited a student to come up and draw on the TV (with a dry erase marker, don’t worry!) and I cannot tell you how great the conversation was in class. I sat down – a commitment of mine based on my #TalkLessAM session at TMC16 – and just watched the fireworks unfold. Kids were challenging each other, going up to the TV to draw their ideas, debating distances, talking about slope, worrying about vertices colliding with walls and discussing the option of rotating the triangle as it moved. I was SO thrilled with the engagement and the level of conversation. I credit this to a number of factors. The original activity was terrific and my colleague’s rewriting of it is creative and concise. Kids like drawing on a TV – it feels naughty or something. I sat down and got out of the way. Kids had worked this through the day before in their table groups and were invested in both supporting their teammates and making sure that their memory and their perspective was clearly heard. They were supportive of each other and slightly defensive if someone else had a different approach. After a pretty uneventful day at the end of the week it would have been easy to just limp tot he end of the day, but these kids brought each other to the finish line for the week sprinting. I am optimistic that we can pick up with a similar level of energy on Monday.
A quick post here reflecting on a great solution presented by one of my AP Calculus BC kiddos. They had their first assessment on Friday. At this point, we are doing a quick and deep review of last year’s work. In our school, BC students have already completed AP Calculus AB and we spend this year digging deep and moving into the BC only topics. So, one of the questions I posed was in two parts:
- For what value(s) of x is x^10< x^6
- For what value(s) of x is x^7 < x^3
This, by the way, was a non-calculator assessment. I will be writing soon about my wavering on this issue. One of my students presented the following work:
- x^10 < x^6 becomes x^4 < 1 and this is true whenever |x| < 1 (other than x = 0) so the intervals are (-1, 0) and (0, 1)
- x^7 < x^3 becomes x^7 – x^3 < 0 which becomes x^3 (x^4 – 1) < 0 since x^3 < 0 for all x < 0 we need x^4 – 1 to be positive. This is true when 1 < |x|. So, the overlap here is x < – 1. If x^4 – 1 is negative while x^3 is positive, then 0 < x < 1.
What knocked me out here was that he divided in one case (when it was safe with even powers) while he subtracted in the other case with odd powers. Now, I have not had the opportunity to ask him about this yet, but I have to imagine that this was not just luck. I think he had some instinct, and I want to gauge how conscious this instinct is, that there is a problem with dividing by x^3 which can, of course be negative. I get to labor on labor day, we have classes here on this holiday, so I will quiz him a bit about this. I’ll report back.
Thanks to Sam Shah for catching a mistake in my earlier version of this post!