Our Geometry Support Site

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So, I’ve talked about Khan Academy before. They make instructional videos for students to watch and learn from.


I’ve talked about #flipclass before. In this model, the teacher makes videos for the students to watch and learn from.


I like the idea of students having instructional videos to watch and learn from. But, in both Khan Academy (and other related sites) as well as flipped classes, the students are recipients of the videos. We decided that we didn’t want them to simply be recipients. We decided that we wanted them to be producers of the resources, too. So, we decided to just let the students create the math help videos.

Here’s what they did: Today I unveil the @PennGeometry Beta for your review and feedback. Currently, there are videos created from one practice test about triangle similarity. We would like to expand and continually improve.


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All of the videos are designed, produced, and completed by our students. Some of my students were sensitive to the boring math video and tried to add their personality to the video. We would love some comments, “likes”, and constructive feedback. Please check it out. Eventually, we want to add our voices to the overall mathematics community to support those in need.

Check it out and let us know what you think. We look forward to hearing from you.

The Power of Network: The Wedding Cake Problem

I have another wonderful story about the power of the wider math community to support its own. Earlier in March I presented at MACUL 2014 in Grand Rapids, MI. During that presentation, I led the group through an experience with the Wedding Cake Problem, which ended up being a wonderfully energetic interaction.

Sitting in that meeting was a gentleman named Jeff who teaches at a school in Michigan. He wrote me an email some time later that included a message and the following photos:

“I’m planning on using the cake problem this week as review in my Trigonometry class, as well as later in the year with my Geometry students. Well, here are a few improvements, well, really just pictures. See attached.
Pans are 5″, 7″ and 9″. I think I’m just going to give my class the actual pans without the pictures.”








Now, these photos change the dynamic a bit, don’t they? Let’s do pros and cons… What do you like better about giving the problem with these pictures? What do you prefer about the original problem?

I would love some feedback (especially if you have tried either one with your class).

The Power of Network: Triangle Similarity

I want to share a story that shows the power of an effective PLN.

In a previous post, “proof and consequences: circular reasoning“, I begged for help solving a problem with students struggling to see their own logical crisis that was leading to predictable and consistent problems.

Several people reached out to me with suggestions. Thanks for that. I would like to highlight one specific suggestion that I tried to today and it worked just exactly as the designer predicted.

The suggestion was made by @nerdypoo.

From the comment:

“(i drew an example of this on the train home from work where i drew two isosceles triangles — the first had congruent legs 2 & 2 and the second had congruent legs 3 & 3, so scale factor of 1.5, but the first triangle was an isosceles right triangle and the second had an angle of a bit more than 90. i can send a jpeg if you want!)”

I loved this idea. And yes… I did. I did want that .jpg.

So, here’s a portion of what she sent back.

Dutch Triangle Idea Original

Today, I tried it in class. I began by putting up this image…

Dutch Triangle Idea Starter

… and asking the students to vote on whether the triangles were similar, not similar, or we don’t know. Overwhelmingly most students voted that they were similar. The thoughts they articulated were mainly that they could find the legnth of the missing side (which they claimed to be 3 cm long) and then could use SSS to show a consistent scale factor.

Then I showed them this image and asked them to vote again.
Dutch Triangle Idea

A lot of votes changed. Many changed from “similar” to “not similar”. A few others changed from “similar” to “don’t know”. An additional piece of information revealed an assumption. The assumption was that finding a consistent scale factor in two pairs implied the third. Perhaps an assumption that the angles were congruent.

It was essential that I made sure the students knew that I wasn’t changing the situation from the first question to the second. I was simply revealing information that was hidden. Those angles were never congruent. They simply didn’t know that, but most them assumed they were. But every person who voted that the first two sets of triangles were similar were making an assumption, an assumption that they didn’t recognize before. An assumption that shouldn’t be made because sometimes it’s incorrect.

A forum for real PD and authentic learning

It’s time we talked about Twitter. Actually, I’ll let my professional colleague Shauna Hedgepeth do it for me…


That person is not a salesman. She’s a teacher. Like me and like you. Question: When was the last time you heard a teacher so excited about a professional development?

Twitter is something that most district-provided, one-size-fits-all, canned professional development packages aren’t: individualized and timely. Through Twitter, an educator has the ability to reach out to people… a lot of people. A lot of people who teach what you teach. A lot of people who see what you see. That’s a lot of brainstorming…

… oh and It’s free.

I understand that Twitter has a reputation of being a sounding board for famous people’s monotonous daily blah-blah, but, well I’ll quote educator Rushton Hurley:

“If you’re Twitter feed is full of people telling you they just ate a sandwich. That’s YOUR fault. Follow people who say interesting things!”

Perfect. Twitter is what you make of it. I’m increasingly finding my professional development needs, ideas, and improvements are coming from ideas that I’ve shared on Twitter or stolen from people who have willing offered them on Twitter.

We are in an unprecedented time of connectivity and teachers don’t have to feel isolated anymore. It’s no longer a time when educators have to feel burden to come up with unique solutions to the problems that every educator faces. There’s a conversation going on right now and, well, I’ll let George Couros close this post (oh, by the way, I read this quote on Twitter.).

“Isolation is a choice educators make. If you’re isolated, you are CHOOSING not to connect.”

Reflecting on #macul14

So many great things went on these past few days in Grand Rapids, MI. When I consider that the last time I was along the Lake Michigan coast, I was in Ludington to see Dan Meyer, I’d say West Michigan has treated me and my students pretty well.

For those of you who aren’t familiar MACUL is Michigan Association of Computer Users and Learning. They are an organization dedicated to supporting Michigan teachers in the pursuit of making education better through the effective use of instructional technology. This was their annual conference.

The best thing about this particular experience is that there was a little of everything from academic talks like Erica Hamilton’s (@ericarhamilton) talk about Teacher Integrated Knowledge, to incredibly practical, I-could-totally-do-this-tomorrow talks like Bree Davey’s (@studiobree) talk on student blogging. There was the inspirational talks of Rushton Hurley (@rushtonh) to the intensely technical and energetic Leslie Fisher (@lesliefisher) teaching use the finer points of how to use iStuff to take pictures that don’t suck. It’s a lot to take in. Here are some summaries of my favorite sessions:

Erica Hamilton – Teacher Integrated Knowledge – Erica (soon-to-be Dr. Hamilton) did a fantastic job of detailing the how teaching becomes more complex in the 1:1 format. She spoke about the different types of knowledge that a teacher naturally has to draw from in the process of doing his/her job. (Content knowledge, curriculum knowledge, pedagogical knowledge, environmental knowledge, etc.) Switching to a 1:1 format add a few others that aren’t there (at least not nearly with the intensity) otherwise, which can make reinventing teaching to maximize 1:1 seem really, really intimidating because if you are, for example, going to ask your students to go out and make short videos, you might need to teach them what to consider to make a video effective (Lighting, stabilizing the camera, speaking clearly, editing tools, etc.). That isn’t part of most curricula. 1:1 puts teachers face-to-face with having to acquire those kinds of knowledge. My favorite idea that Erica kept coming back to: “It all comes back to what do you want students to learn? What do you need to teach to do it? What tools are available?” Excellent, excellent talk.

Ben Rimes and friends – #MichEd PLN – #MichEd is a PLN dedicated to connecting educators in Michigan with the goal of spreading ideas. This is a dedicated group of educators with a podcast and a weekly twitter chat. They are dedicated to the idea that we all need to grow and all we should have to do is ask the experts around us for help. We all have something to ask and to offer. It seems this panel discussion was best summed up by George Couros the next morning when he said “Isolation is a choice teachers make. If you’re isolated, you’re choosing not to connect.” Speaking of connecting, it was fantastic to get to meet these folks face to face for the first time.

Tara Becker-Utess – Flip Class Model – This was an important talk for me to go to because I was always a little bit uncomfortable with the flipped class model. Tara (whom I am proud to know personally these past 10 years) didn’t quite draw into the realm of the full believers (for reasons I can explain more if your curious), but she did a very nice job of explaining the philosophy behind flipping. I was relieved to find that I was able to identify with a ton of the spirit. Tara made some fantastic points, especially the absolute need for teachers who flip to plan very, very well for their time in class. “If you were used to using 20-30 minutes to lecture, you just got 20-30 minutes to plan rich activities for your students. That is usually a shock to people who are flipping for the first time.” (Those are probably more paraphrased thoughts than actual quotes, to be fair.)

I also had the privilege of presenting a one-hour session. If you are curious what it was about (or if you attended and want to revisit) I invite you to check out the “MACUL 2014 Presentation” link at the top right of my blog to get the details. Thank you for your kindness, warmth and enthusiasm during my session. It definitely did not go unnoticed.

Proof and Consequences: Circular Reasoning

I’m frustrating my students in ways that I don’t want to. I’m not sure exactly what to do about it. In geometry there’s proof. With proof comes a certain logical structure. Once you know this structure, it is terribly difficult to unknow.

Currently we are dealing with similarity, which involves using SSS, SAS, and AA postulates to prove whether or not two triangles are similar.

Suppose I gave this image to a student and asked them to find whether or not triangle FES and triangle GHS were similar.


Let’s suppose the student divides 54 by 24, and also 58.5 by 26. Both times the student gets 2.25 as a solution. The student assumes this is a scale factor and applies it to GH, finding that FE = 45. The student then divides 45 by 20 and gets 2.25 for a third time. That’s three pairs of proportional side lengths and BAM! Similarity proven by SSS.


Me, the teacher, is there is tell the student that he or she isn’t quite right. (You see the mistake, right?)

The student assumed similarity before it was proven. Then proceeded to use the assumed scale factor to find the missing side length, which ensured that the third quotient was going to be the same as the first two. This is circular reasoning. They are similar because FE = 45. FE = 45 because they are similar. I have seen this play out countless times.

I have addressed it with little success. I can’t seem to make sense to the students why that argument is weak. It sounds like “geometry teacher says we can’t, so do what geometry teacher says.”  (especially when the very next question asks for FE, which is 45… because the triangles ARE similar…). I can’t stand using my authority as a teacher to enforce a math idea that the students are perfectly capable of actually learning.

I’m trying to decide how picky to be with this. I have a hard time allowing that circular reasoning argument to be called correct, although it is clear that the student has learned a lot about similarity, proportionality, and the structure of a proof.

But the more I push the point, the more frustrated I get and the students don’t seem to be getting any significant gains. I just continue to enforce that “math teachers are just picky like that.”

I am hoping for some help on this one. I’ve tried a lot of things, but that thing you like that works really well for you… I haven’t tried that one. Toss it my way. I want to see how well it works.

Conversation starter: Is failure an option?

Let’s talk about students failing classes, specifically in high school.

Let’s suppose a teacher spent the last ten years teaching high school math. Let’s suppose further that the same teacher hadn’t had a single student fail his or her class for that entire span. This teacher is going to have that data met with a fair amount of suspicion, whether it is fair or not.

Let’s suppose a different teacher spent the same ten years in a comparable district teaching high school math. Let’s suppose further that for that time span 3 out of every 4 students who started that teacher’s math class left with a failing grade at the end of the year. This teacher is going to have this data met with outrage (and in all likelihood would never have made it 10 years like that.)

So, ten years without a failing student is suspicious, it’s potentially evidence of a rubber stamp course. 10 years at a 75% failure rate is outrageous. It is potentially evidence of a course that is unnecessarily difficult for a high school math course.

So… what’s an acceptable number of failures?

Actually, let me ask this question a different way…

How many students should be failing? It seems a little strange to suspect that anyone should fail a high school course, but is there an amount of failures that demonstrate a class is healthy and functioning properly? Is that number zero? Is it 5%? 10%? 20%?

I’ll tell you what motivated this post: I am aware that some schools impose mandatory maximums of failure for their teachers. It might be 12% or 7% or 2%. In these districts, each teacher in the district needs to make sure that at least 88% or 93% or 98% of students earn passing grades for their class each semester.

The implication is that if more students than the accepted maximum fail to earn passing grades, it is a reflection on the inadequacy of the course, the instructor or the support structures. But, I’m not sure if that’s true. And besides that, how does a district or community decide the acceptable percentage of failures?

There is another side of this argument that says that a school should be prepared to fail 100% of their students if the students don’t meet the schools requirements. This is the only way to motivate students to reach for the standard of proficiency that the community has agreed upon. There would certainly never be an instance where a teacher had to fail 100% of his or her class, but if the students didn’t meet the requirements, the teacher would have the support of the school and the community to every student, even if that meant 100% of them.

We should probably figure this out because failure numbers are starting to work their way into the mainstream, as demonstrated by this Op-Ed from the New York Times which asserts that perhaps Algebra should be reconsidered as mandatory for high school graduation because nationwide, math provides a stumbling block and the subsequent failures are leading to increased dropout rates. (This seems like a highly contentious point in itself, but it doesn’t mean that it isn’t driving decision-making in some communities.)

Here are the issues in play here:

What portion of the responsibility of a single high school student successfully earning a high school credit is the school’s and what portion is the student’s?

What are the costs of high standards? If we want to increase rigor, there is almost certainly a trade off in that there will be an increased number of students who are unable or unwilling to go through the more rigorous process to earn the credit.

What are the implications of community with class after class of students who know that the teachers are pressured to pass a certain percentage of their students? Is this effect overstated?

Has this ever been studied? I’m not sure if there’s ever been a comprehensive, research-based statement made on the topic of student failures and what the optimal percentage are. And if that’s the case, then should we be making decisions based on “what seems too high” or “what seems too low”?

I am looking for some conversation on this topic. Let me know what you think. Links to posts or articles by people that you trust are appreciated, too.