Tag Archives: Geometry


This post is a few days late due to, well, you know, life getting in the way. When I last checked in with you here I was preparing to put into action a plan to lag my homework with Geometry and have a series of HW assignments incorporating more review of past skills. My kiddos took their first quiz of 2016 yesterday and I plan on grading them tonight so I will have some data to back up (or refute) my reflections at that point.

What I have noticed so far:

  • Review assignments – at least the ones I have written – make my students pretty frustrated. I certainly do not want frustration to be the go to emotion for my students when I ask them to work, but I am willing to have that as a  stepping stone if I can help usher my students into a place where they are more comfortable with problem sets that do not depend on a small set of skills and ideas. I realize that I am combating years of habits and expectations.
  • We have more time to practice some of the new(ish) skills that I am hoping that they develop. We spent days in a row visiting some linear equation writing skills and some  ideas about linear combinations.
  • When we did  finally get to HW concentrating more on single sections of the text I was not receiving quite as many questions as I had been expecting. This I am taking as a positive sign. I will be more convinced that it is a positive sign if I see some stability on their quiz. We spoke briefly about the quiz today and I suspect that I will see some hesitant work. If the mistakes are more algebraic in nature I will feel better about the development of their Geometry skills and ideas.
  • We took a day in our lab to play with GeoGebra and I realize that I have to do something more consistent next year to encourage/require my students to engage with GeoGebra more frequently. What should have been a productive activity drawing some connections about the ‘centers’ of a triangle that we are examining, too many of my students were either distracted playing with zooming in and out on various images or they were flummoxed by some of the commands that we ‘learned’ in the fall. One of my new colleagues and I are brainstorming ways to make check-ins with GeoGebra a regular and meaningful part of our life in Geometry.


Part of the way that I am organizing out of school work right now is by asking my students to read based on class discussions while they do not practice those specific skills for a few days. I am not at all convinced that they are reading as I request, but I also do not think that they were doing to reading under our old structure either, so that is a wash. I will write again tomorrow after I grade the quizzes and I will check in to see if the data backs up my observations in any meaningful way.


MTBoS New Year’s Resolution

Every year we all experience this, right? We have big goals for the new year, we make promises and many of them fall by the wayside. I am going to be modest about my new year’s aspirations (at least in public!) and I am making a vow to myself to try something new for my Geometry kiddos this new calendar year. Awhile ago I got into a terrific twitter exchange with Henri Picciotto (@hpicciotto) and some other folks ( I wish I could remember everyone involved, I am pretty sure that Julie Reulbach (@jreuhlbach) was part of this) about HW. I mentioned that I will sometimes include problems in HW that touch on ideas we have not talked about yet in class. This brought up a conversation about leading vs lagging homework ideas and Henri is particularly articulate about these ideas over at his blog space (http://www.mathedpage.org) We tweeted about the idea of letting ideas percolate either before instruction or after instruction. I spoke about my feeling that it is important to have students struggle with an idea a bit to (maybe) help them appreciate a new idea/definition/formula when it does arrive. This, I think, is sort of like Dan Meyer’s series of if ______ is the headache, then ______ is the aspirin essays. One of the results of this terrific, spontaneous twitter chat was that I walked away with a commitment to instituting lagging HW assignments in my class. I wrote out a careful pacing calendar with the optimistic idea that I have a solid sense of how long these conversations will take and a hope that mother nature will not interfere by dumping snow on us at some point. Obviously, this calendar is not set in stone. What IS set in stone is my commitment to the pacing of the HW assignments. What I did was write out a pacing calendar that delays by three or more days any HW that directly relies on reading/instruction of a new section of my Geometry text. I do still feel a bit of a commitment to daily practice out of class – I know, this is another conversation completely – so I wrote a series of review HW assignments that reach back to old ideas/skills but I tried to do so in a way that is thoughtful and leads to preparedness for the new ideas we are discussing. I also wrote a series of entrance slips that I will start class with the day after we first discuss a section together. These will be collected and marked, but not graded. I am hoping that these will provide both me and my students a way of monitoring their developing understanding of new ideas. After three or more days they will final have a HW focused on a certain section of our text. Meanwhile, in class we will still be following a pace similar to what we followed last year in our first run through my Geometry text so I have some sense that this is a reasonable pace for our students. However, instead of going home and immediately practicing some new set of skills, they will be looking back either weeks (at the beginning of the chapter) or days (after a few days in) to ideas that have, hopefully, been percolating a bit in their brains. They will have had time to think about these ideas, they will have had an entrance slip check in on their facility with the ideas at hand, and they will have had further class time combinations of lecture and discussion to play with these ideas and build them up together. A few days afterward they will go home with focused practice and their assessments will also be lagging a bit to line up with the HW practice. I anticipate that there will be some concern expressed by my Geometry team and by my students because, you know, change is a bit of a challenge. This is especially true once you have established a rhythm and pace that you are comfortable with together. One of my three colleagues has expressed a desire to try this as well but the other two have not commented yet and I have no expectations that anyone else needs to follow me on this path. As department chair I kind of want to test the waters on this so that I can report back on the inevitable speed bumps as well as the successes that we encounter. If this works the way I think that it will, it will radically alter how I think about pacing and how my students think about HW. My biggest wish for this endeavor is that this practice will enhance retention and help my students think more about connections in the ideas we work with in class. I feel more confident about taking this leap in our Geometry course first for a couple of reasons. I feel more intimately familiar with the contours of this course since we are using a text that I wrote a couple of summers ago. I also feel that the Geometry course lies a bit outside of the vertical tower that much of our math curriculum builds. If we slow down a bit and there are not topics near the end of the course that we reach, it feels that there are fewer consequences in terms of what future courses and teachers will expect of these students. Also, these students are younger than those in my other two courses (a Discrete Math elective and AP Calculus BC) and I am optimistic that these younger students might be more flexible with the idea of changing their habits.

If you have not seen my Geometry book yet and want to take a look, you can download it from my Dropbox at this link. If you want to look at my Chapter Six pacing calendar, entrance slips, and HW assignments, you can find them all in this Dropbox folder

I am hoping that January will be a productive month for this blog space as I reflect and report on how this experiment unfolds.

Brief Notes on a Good First Week Back

As I wrote about the other day, I tried something brand new with my Geometry kiddos this week. I had found online somewhere recently a three day packet exploring reasoning and proof (you can find it here) and I had my students in small groups. I had three groups of three and one group of four (I know, I am very lucky (spoiled?) to have such small classes) and they all grappled with one of the open problems in the set and gave brief presentations on Tuesday. Yesterday we conducted a fishbowl discussion. I had never done this before as a teacher or as a student, so I felt a little anxious about it. Since I had not taken the time to ‘train’ my class, I left this as a pretty open exercise. There are two pages of definitions to grapple with in the handout linked above. I had seven students in the fishbowl for the first round and I joined the other six in the fishbowl in the second round. Every student drew a card at random as they came in to decide which group they were in. I instructed the outside group to just quietly observe rather than to take notes on the inside participants.  Both rounds went pretty well – in my opinion – but what was best about the day was the talk at the end looking back on the exercise. I have tried to make this first week back for Geometry rather open – ended. I wanted to try and make some important points about learning, and about classroom culture, about proof and logic.

I wrote already about the frustration one girl expressed during the spaghetti exercise where she wanted A right answer to the exercise. I took that opportunity to talk about different approaches, to try and emphasize our desire for efficiency when we can find it, but, more importantly, my desire to hear their voice and thoughts not just an echo of my voice and thoughts. They get too much of that from adults already. Yesterday as we reflected on the exercise two girls shared really interesting observations. One said that when she was inside the fishbowl (I was outside at the time) she felt really anxious about saying something out loud that might be wrong. She said she was more relaxed when she was outside but she felt she understood definitions better when she was inside. This is HUGE. This kind of self-awareness is so important. I asked her to think about that and think how she can use that realization moving forward in our class. I hope that she decides that she understands better when she is more actively engaged in the conversation around her. The other girl remarked that she knew that she understood better when she talks and I seized on that and challenged her to make talking in class a real commitment.

It’s been fun to be back – our school’s last full day of classes before this week was November 12. I appreciated the rest (other than grading finals – a post for another time) and I am glad to finally be prepared in advance for all three of my courses, but I sure did miss the interaction of the classroom and I have been thrilled with how my Geometry students (my youngest class) have come back ready to go. I have asked them to deal with different situations than they normally do and they played along beautifully. I am so pleased and I hope that we have made some important points about our time together. I also hope that I can hold myself to the most important lesson I learned this week. Unfortunately, it is one I have ‘learned‘ numerous times – my students are better off when they speak more and I speak less. I need to make this my mantra – especially if I want to effectively integrate some other changes in my classroom in the upcoming new year. That’s right, I do not want to let myself wait until August, 2016. I want some serious changes as of January, 2016.


Getting Back to Business

So, our school works on a trimester system with Thanksgiving Break (a full week) marking the end of the fall term. We also have fall term finals, so my last full day of classes was November 12. I set myself some lofty goals for the break and met about 80% of those goals. My number one goal, by far, was to do what I could to plan out our next fourteen days for all three of my preps. We have fourteen days of class until the long winter break begins.

I found out late in the summer that I was teaching a new course (around August 10) and I also have two brand new colleagues in my  department. I have not been able to spend as much time mentoring them as I had planned to. The combination of this disappointment, along with perpetually being only a few days ahead of my Discrete class made the fall term a pretty stressful one. I have three preps, five sections, and my chair responsibilities. Luckily, I have a pretty light student load this year.

So, I have my calendar mapped out for Geometry and AP Calculus BC and I have about ten of the fourteen days of Discrete taken care of. Overall I am pretty pleased. Add in the naps and the time with my wife and kiddos and it has been a good break with just enough productivity thrown in.

I am starting off my Geometry kiddos with a three day workshop on Reasoning and Proof. I found this somewhere on the inter webs recently but I cannot recall where. You can find the link here and if you recognize it, please let me know. I am pretty excited about this. I think that it will be a lively way to restart my classes and I am optimistic that the students may make some inroads into understanding the logical structure of proofs. We had a great activity with making peanut butter and jelly instructions for each other earlier in the year. I think that this serves as a nice follow up and I am happy that there is such time between them. My optimistic hope is that the students will make that connection on their own without me pointing it out. This unit has a similar idea with sentence strips outlining the process of making spaghetti. I do know that when I do the PBJ activity again in the future I will scaffold it a little more carefully in advance so that more of the students will have a solid idea how to approach that. If you want to read about our PBJ adventures you can look at this post or this one.

I am also already committed to a project for my winter break. Right before Thanksgiving I engaged in a lengthy and lively twitter discussion with Henry Picciotto (@hpicciotto), Elizabeth (@cheesemonkeysf), Peg Cagle (@pegcagle), Julie Reulbach (@jreulbach), Mattie B (@stoodle), and Chris Baldus (@Chrisbaldus04) We were discussing HW strategies. When to preview ideas, when to lag and let ideas catch up, how to possibly blend those strategies. It was an amazing conversation with people from all around and at least two of whom I am certain that I have never met. One of those great examples of why engaging with twitter has improved my practice. So, I am too weary to rewrite my HW sets that I wrote last year when we rolled out the Geometry text I wrote. But, I realize that the time before January will allow me to write a few more sets that I can use as buffers near the beginning of the year while I let ideas settle in and percolate for my students. The assignments that they would have been working on the night they were introduced to an idea will now come three or four days later. In the interim we will concentrate on in class discussion and practice and I will write some homework sets that concentrate more on helping to cement definitions and some new mechanical skills along the way – along with reminding them of highlights from 2015. I am excited to do this and I would not have had the motivation to do so without the urging of those virtual colleagues who took the time and care to share with me their ideas and experiences. I am a little anxious because change = bad for too many of my students, but I am convinced that the time off will allow me to think deeply about how to be as intentional and clear as possible with my students. The other fear I had and came to grips with is this – I am one of four Geometry teachers at our school. I am also the chair of the department and the author of the text. My ego keeps creeping in and wanting everyone to follow my lead because of both of my roles here. I came to peace (thanks Julie and Elizabeth!) with the idea that I do not have to have everyone on the same page AND with the idea that I can be a better leader in this process next year if I go through it myself this year. I will still share out my old (and new!) pacing guides and homework assignments. I will simply make it as clear as possible that not everyone needs to agree with this HW strategy and with the timing of assessments that this will entail. If the students are not doing homework concentrating on, say, section 6.4 until three days after we introduce that section in class, they cannot be held responsible for that material on an assessment until they have had time to practice. Consequently, assessments will lag behind where we are in class as well. I need to rethink my ideas about what review days mean and look like, but this kind of rethinking is one of the things that makes this job such a joy.


Catching Up with the Past Week

So there are a couple of activities this past week that I want to write about. However, I have been swamped with meetings so I have fallen a bit behind.

In AP Stats we have finished our required curriculum as of 8 days ago. I am a big baseball fan and my favorite team is the New York Mets. They are having a pretty wonderful start to their year (or at least were until the last few days) so last Friday I posed the following question to my kiddos: Given that the most optimistic projection I saw for the Mets’ season had them pegged as an 87 win team, what is the likelihood of their current record (which, if I remember correctly) is 10 – 5? I liked this for a few reasons. First, it concerns baseball and likely would have a positive outlook for my Metropolitans. Second, it was not so focused on the most recent material at hand. My Stats students tend to know recent material well but struggle remembering other procedures that have not been practiced as recently. Third, it generated some nice thinking out loud about what approach to take. Being more of an algebra stream guy myself I immediately placed this in the context of a probability problem and was prepared to go down a Pascal’s triangle/binomial theorem path. Most of my Stats students don’t tend in this direction so their conversation focused instead on comparing proportions – the 87 – 75 projection with the 10 – 5 proportion. They suggested running a two proportion z test and looking at the corresponding p-value. This opened up the avenue for me to sneak in my approach and make a connection pretty visible to them. Turns out that we felt that we had enough evidence to reject the null hypothesis of the Mets being an 87 win team in favor of believing that they will exceed that win total. Their recent 5 – 5 run of games might adjust that but I do not want to know this – so I will not re-run the test right now! After we checked our trusty TI to find the p-value of this test I reminded them of the probability approach and we set up the appropriate term of the binomial expansion. Guess what happened? This calculation matched the p-value of the two proportion z test!!! This is one of those ideas that we discussed but somehow seeing these results side-by-side seemed eye opening to my kiddos. A triumph on a number of levels!

In my morning Geometry class we dipped our toes into an exploration of radians yesterday using the ProRadian Protractor designed by the fantastic Jennifer Silverman (@jensilvermath) and using an activity that she designed. I wrote a follow up HW assignment that my kiddos worked on last night. I also linked to a fabulous web site that allowed my students to explore radian measure and I shared these notes with my colleagues. There is also a lovely GeoGebra applet (also designed by jennifer Silverman) that is linked from the worksheet. I was totally excited to explore this activity with my students and I had a really nice chat with one of my teammates.

I handed out the radian protractors as well as our regular old angle protractors and we had a nice conversation about similarities between the two protractors. We had a lovely discussion about this but, looking back on yesterday , I think I allowed too many clues to seep into the conversation too quickly. Jennifer’s activity is a terrific one and I got in the way by loading too many conversations in at the beginning of the class. By having students come to my screen and try to identify where one radian measure would lie on the circle AND by having the protractors side-by-side I reduced the mystery element that I think should have been part of the classroom activity. I think I took away the opportunity to discover what was happening here. I did have one student give a GREAT explanation of why the quadrilateral radian measure was twice the triangles radian measure. She invoked a proportional idea and referenced our (n – 2)*180 formula. I had a number of students quickly see that the ratios we had been working with before (\frac{x}{360}=\frac{arc}{2\pi r}=\frac{sector}{\pi r^{2}}) could be easily extended to add one more simple fraction of \frac{x}{360}=\frac{\theta }{2\pi }. That definitely felt like a triumph. So, the lesson I learned here – and I hope I remember it for next year – is to be a little more minimalist in front loading this conversation. I think that we can touch on all of these resources and really let the discovery sink in, but I feel I nudged them a little too much this time around. So the plan for next year is to hand out the radian protractor and work through the worksheet. Then hand out the angle protractor and talk about comparing them. Then, the next day after some time to think, show the web app and have them identify where one radian is. Let this unfold a little more slowly.

A Quick Geometry Snippet

So, we were reviewing this morning in Geometry getting ready to finish up our circle unit. I was reminding the kiddos of the angle and arc relationships we have been discussing. I had been writing things like x=\frac{1}{2}\left ( a+b \right ) where x is an angle formed by the intersection of two chords and a and b are the measures of the intercepted arcs. However, today as I drew my diagram for it one of my students suggested I mark is as seen below. Interior Angles

I think I am delighted by this and will write it this way from now on. It feels to me that it is a more natural way to think about this relationship instead of having a coefficient of 2 or a coefficient of 1/2 that might not seem at all intuitive. I then drew the following Exterior Angles

I’d love to hear from some other teachers about whether this seems at all like an improvement over the more standard way of writing these equations.

That’s all for now, just needed to get that off of my chest!

Highlights of a Stressful Week

So, there have been many scattered thoughts on my mind in the past week but there are also three things that happened that are just completely awesome.

  1. My pal John Golden (@mathhombre on twitter) steered a number of his teacher training students over to a post on my blog and twelve of them chimed in with comments. Totally cool! One of them decided to follow my blog and I took the time to respond to each of them. LOVE the idea of new teachers in training dipping their toes into this rich world of teachers blogging and sharing. I am also flattered that John thought my virtual home here was worth a visit.
  2. I woke up Wednesday morning with a message on twitter from a teacher in Louisiana who asked if he could use my Geometry book at his school next year. I am so excited by the idea that this work might be used at another school.
  3. In my Geometry class this week we are talking about angle and arc relationships. One of my students stayed after class one day this week and she had this to say. “You know, I was thinking, when will this be important? I mean, when will I need to find an arc length like that? Then I realized that the work we are doing to find that length is what is important. Pretty cool.” Wow.

Some Fun Geometry Action

On the heels of learning some right triangle trig I am really trying to develop more proportional logic with my students. Just this week we had a really productive conversation about the following problem.

Screen Shot 2015-04-09 at 6.45.44 PM

Being a bit of a bull in a china shop sometimes, I proposed that we should find the height of each triangle, find each chord length and find the height of the trapezoid by finding the difference of the heights. Not elegant, I know. I was trying to make sure that we remembered some right triangle trig. that we remembered our area formula for a trapezoids, and that we try to develop some patience in solving multi-step problems. That was my plan, but as with many school plans, it did not quite unfold that way. One of my students who is a bright and quick problem-solver pointed out that simply finding each triangle area would be enough. I understand that his solution is pretty much the same as mine, but it certainly sounds more efficient. But as soon as I acknowledged that his idea was more efficient than mine another student trumped each of us. She pointed out that I had already asked them to consider the ratio of areas between the two triangles. So, if we know one area, we can automatically know the other area. If we know both areas, we find their difference as suggested by the first student who chimed in. I was so happy that she took my clue from within the problem and that she was clever enough to really save time and energy this way. I made sure to compliment her in class and I bragged about her work to two of my colleagues yesterday. Oddly, this morning when we were reviewing before today’s quiz I reminded her – and the rest of the class – of her clever idea. She had no memory of this conversation. Sigh…

I’m trying to process this and figure out what it might mean for my classroom practice. I understand that I should be more excited by my students’ ideas than they often are. I understand that I will remember context of conversations more easily than they will because I am not dealing with the cognitive load of trying to learn/understand the conversation. I am simply coming at it from such a different place. What I don’t understand is how a student can be so in command of an idea but then not remember the creative process that made her arrive at this clever conclusion. I discussed this in the faculty lounge right after Geometry today and one of her other teachers intimated that this might simply be modesty on her part. I am not sure how much faith I put in that reading.

So, while I am a bit frustrated and confused, I am choosing to focus instead on the positive energy of yesterday’s conversation, on the clever ideas that my students brought to the table, and on the fact that my students did a really nice job on their quiz today.

Borrowing from the MTBoS

I’m guessing that most of you reading this are familiar with the awkward acronym for the Math Twitter Blog-oSphere – 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.

Show but Don’t Tell

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!