Friday, December 20, 2013

SBG Final Exam (Redux)

Last year I created a new way to administer a final exam. I got sick of writing a cumulative final, let alone trying to grade it. I refuse to use multiple-guess assessments, so I was left with an insurmountable pile of grading during finals week. 

I was very happy with how my SBG Final Exam turned out, so I'll be bringing it back this year, with only a couple of tweaks on last year's inaugural run. 

The basics:

  • Each class covers 25 standards within a semester
  • Students pick which standards they'd like to demonstrate proficiency on during the 90 minute final exam period
    • Sign up is done through a Google form
  • Students must pick a minimum of 10 standards, but they are welcome to pick more
    • Blank assessments beyond the minimum 10 will be thrown away
  • Each assessment will be brief - approximately 5-6 questions per standard. 
  • ALL grades will be recorded, for better or for worse
    • Yes, a student's grade can be lowered, but that shouldn't happen if only standards below proficient are chosen
  • Students are encouraged to pick the standards which offer the best opportunity to show growth
    • A student shouldn't spend too much time on standards that are 'beyond hope'
My overall thought process through all this is that if a student has demonstrated proficiency over the course of a semester between formative and summative assessments, why should they need to take yet another assessment on those topics? This setup allows students to focus their study on only those standards that have yet to be mastered. 

The downside? The prep work. I need to write 25 new assessments for each of my 3 subjects. Fortunately, this is only a one-time issue. It does get a little tedious to print out and collate each student's final exam, but I believe that it's worth the effort to ensure that I'm collecting accurate data on what my students have learned. After all, isn't that what we're supposed to be doing? 

Saturday, October 5, 2013


I've been a modeler for going on three years now and I've never really reached the expectation of what a modeling classroom looks like. My first year came close, but my second year was fraught with overreach on my part, as I tried to implement my new geometry curriculum while also rolling out SBG across all of my preps. This year, I'd actually consciously decided to scale back my efforts on the modeling front simply because I couldn't take the never ending stream of disappointment. I wouldn't go back to lecturing and be "sage on the stage," but I wasn't going to force all-modeling-all-the-time down students' throats. 

And this went on for all of September - I might have half-heartedly tried whiteboarding a worksheet here and there, but I wasn't forcing in depth discussions on anyone. My concerns lay with class size in physics (32) and one of my geometry classes (33), but also with maturity with another geometry class. My third geometry class is only 17 kids, and I started to think that maybe I could actually connect with a class that small, even if they weren't really 'ready' for such a paradigm shift. So for the first worksheet of the second unit (linear equations), I figured I'd give modeling another shot. 

My hope was that since linear equations is all a review of 9th grade algebra, kids would have a chance to acclimate to collaboration in a lower-stress environment. Wednesday was the first day of whiteboarding and I went through the "rules" of discussion with them, stressing respect above all else. By the end of the hour, we'd gotten through 2 of the 12 problems, but the class actually showed promise. 

On Thursday, I put instructions on the front board to put the whiteboards back up and we'd pick up where we left off. Some students voiced concerns about how long we'd be doing this, and I'd reply "as long as it takes." Thursday actually progressed fairly well for the first half of class, but then the natives got restless and started focusing more on going through the motions so that they could be done than on ensuring everyone understood how to answer the questions. 

So on Friday, I repeated Thursday's instructions, but said that the quiz would be at 11:38 (15 mins before the end of class). They only had a couple of problems left to cover, so they actually got through them with time to spare. When it looked like the discussion was mostly over, one student came to me privately and asked for help on a question that the class had just gone over - he'd been to shy/scared to ask for help. I politely reminded him that I wasn't going to answer stuff like that and that he needed to ask his peers for help. Dejectedly, he went back to his desk and sank into his chair. 

It was clear that at least one student wasn't getting anything from the discussion, and since we had 15 minutes before the quiz, I told the group what I was going to do. I said that since everyone had claimed to have consensus on the entire worksheet, that I would score the quizzes, but give everyone the lowest score for each of the two standards. I reminded them that they are a class, and that the point of this exercise was to work together so that everyone could master the content. And if they'd truly done that, it shouldn't be an issue. As expected, they were upset. "That's not fair! You can't do that!" My personal favorite was "you can't punish everyone because a few kids choose to not even try!" 

So for 15 panicked filled minutes, they collectively sought out anyone who was struggling and helped tutor them on whatever skill they were lacking. I mean, they really tutored everyone. It was probably the first time throughout the three days that they worked as a team toward a common goal. The 'smart' kids were polite and respectful and seemed genuinely interested in helping the students who were struggling. 

As I started passing out the quiz, a few students were still worried about the threat I'd made, so I offered a compromise: I'll still give everyone the lowest score, but I also said that they didn't need to take the quiz independently. That seemed to placate everyone, but amazingly enough, very few students took me up on the offer. When they were done, I asked why they didn't work together and students mostly said that they didn't need to - the quiz seemed easy and they all felt fairly confident about how they performed. 

I realized through all this that the thing that I love most about teaching is the chance to try new things. I don't hate lecturing, I hate only lecturing. And I might have grown frustrated with modeling over the last two years because I was only modeling. I really enjoy surprising students with different ideas, even if the ideas blow up in my face. This might make some wonder why all teachers don't do stuff like this, but remember what I said up top: I don't do these things in my classes over 30 because I can't take the stress. I honestly feel restricted by class size most of the time and am relegated to simpler lesson plans like lecturing and worksheets. If I could change ONE thing in education, it would be a firm cap of 20-24 on all classes. 

Monday, September 9, 2013


This will be my 6th year teaching and my 3rd year as a 'Modeler.' I have never experienced any professional development / curriculum framework like Modeling. Why else would I take it upon myself to create a geometry curriculum based around the modeling philosophy? 

But I find myself at a crossroads; I've yet to have a huge amount of success with Modeling Instruction, both data-based success as well as student opinion-based success. Classroom management has always been a weak spot for me, and I'm starting to think that Modeling is actually making my issues worse. 

It's only the 4th day of school, and it's already spectacularly clear to me that my students are well behaved so long as I'm fulfilling the traditional sage-on-the-stage role (which I abhor, btw). Show a quick video clip? They listen attentively. Try to facilitate a discussion about the clip? Good luck. Detail a process using a document camera projecting onto a large screen? Students silently take notes. Stop the 'lecture' to have students explore a concept on their own (or in small groups)? Fugetaboutit.

In talking with other teacher in 'better' school districts and I've always just assumed that 'better' students don't give their teachers as much grief as my students give me. But seeing how cooperative everyone is when I dim the lights and lecture with a doc cam has made me think twice about that. Obviously  my students *can* behave and take class seriously, but they often choose not to. 

The simple solution is put Modeling on the back burner for the sake of my own sanity, but I honestly don't think that's going to have a significant impact on the amount on the % of students who fail my class (the only metric that my district cares about when determining my worth). Besides, deep down I know in my heart that Modeling is a better way to get to deeper understanding of content. Just because students are quiet and (seemingly) attentive doesn't mean they're learning anything. 

It all boils down to the paradigm shift all teachers who embrace Modeling must deal with, I'm simply still struggling to find an effective method to facilitate that shift. Students aren't paying attention to anything non-lecture because they're been taught that lecture is all that matters. The assumption they make is that if I'm not delivering content, then the content must not be that important. So I guess my goal for this year is to drive home the notion that just because I'm not the focus of the class, doesn't mean major breakthroughs aren't happening. 

Saturday, June 22, 2013

Reflections on the SBG Final Exam


  • Huge time saver on the grading side of things
    • Better than a cumulative SBG final exam, still worse than a MC final exam, but I'm OK with this
  • Students seemed to appreciate the opportunity to focus their studying on only content areas that they were weakest in
  • Since every student was working on their own stuff, the opportunity to cheat was gone
    • That doesn't mean they didn't try however
  • A lot of students asked "why don't other teachers do this?"
  • Huge time suck on the preparation side of things
    • Creation of 34 different assessments is hopefully a 1 time chore
  • Organization was a little cumbersome, tracking all that paper
  • 15 standards was too much for most students
    • I still think it's an acceptable goal, but most students simply have no appreciation for how much time should be spent on a problem when one truly understands the material. The ACT math section is going to be a harsh dose of reality. 
Improvements for next year:
  • Minimum of 5 assessments (see below)
    • Possibly a requirement that students with a grade below B be working for the entirety of the 90 minute exam period
  • Better tracking of what standards are chosen
    • I emphasized that students should pick their lowest scores to ensure that their grade doesn't drop, but I did have some fail to heed that warning. Not sure what they expected to happen.
  • Some form of prioritizing standards that haven't been assessed in a long time
    • Related: low prioritizing "easy standards" that aren't incredibly relevant to the course overall

I would say the most surprising aspect of the whole experiment was how many students' first reaction was to try and cheat the system. Honestly - here I was working myself to the bone to given students a chance to demonstrate what they had learned on their terms, and their immediate response was "how can I twist this to my advantage?" IT WAS SETUP FOR YOUR ADVANTAGE YOU LAZY SACK OF CRAP! Sorry, had to get that off my chest. 

For example, the most common question I got leading up to the final exam was "what happens if I leave everything blank?" Naively, the first time I heard this I thought "well, nothing happens." After all, one of the major points of the SBG philosophy is that grades should not behavioral rewards or punishments, but measures of student proficiency. Why should I care if a student opts out of a chance to demonstrate proficiency? After all, at least half of my students were on track to earn a D or worse leading up to the final exam. If a student is somehow satisfied with those results, who am I to stand in their way? However, I quickly saw that such a philosophy would lead to mass chaos if allowed to spread on a wide scale, so I haphazardly tried to toss in a minimum of 5 assessments out of the 34 total covered throughout the semester. It at least kept students awake and working for the 90 minute exam period. 

Overall, most students either maintained their proficiency level (expected), or demonstrated a slight gain (hoped for). I don't think anyone did more than move a single gradation (a B to a B+ for example). I did allow students to lower their scores, not as a punishment, but because the grade should reflect student proficiency. If a student demonstrated 'C' level proficiency back in March, but 'D' level proficiency in June, the original C shouldn't be kept for old time's sake. Interestingly enough, this was probably the sole complaint I heard about the entire setup, but I didn't understand why. Every other teacher in the school simply weights the final exam at 20% of the semester grade and let's the numbers fall where they do. Are students just oblivious to how poorly most of them perform on cumulative final exams?

Verdict: I will totally be repeating this format in the future.

Tuesday, June 4, 2013


Are bad ideas that get results still bad ideas? 

I feel almost dirty right now. I had an idea last night. We were going to be working on our last worksheet of the year today and I've been struggling all year to get kids to even bother putting pencil to paper. We've tried everything. Modeling style discussions disappeared a while ago because we couldn't have meaningful whole class discussions when only 2 kids did any work. We tried picking students at random to work through problems in front of the class, but again, when no one does any work, that quickly becomes a nightmare. I tried checking worksheets for a completion-based grade and that worked for a very short while, but soon we were back to square one. Quizzes have been open notes all year, with the idea being that if students are completing the work, it'll be right in front of them on the assessment. Still, maybe 10% of students are actually completing the worksheets. 

So I decided to give students an incentive to get the work done. I know that's not a new idea, but I hate anything that makes the reason to do get work done something other than learning. My idea was to give students a raffle ticket for every question on the worksheet that was 100% complete. This means detailed steps, the correct answer, and proper units. The raffle tickets will be put into a bucket and one will be drawn for a prize. The prize will be some token I have laying around, but I refuse to make it academic (and made that clear to the students). SBG makes extra credit a non-issue anyway. I told students that they were welcome to work together, but doing so would inflate the number of tickets out there and decrease an individual's odds of winning. 

Also, my rules dictated that I would NOT help in any way. All I would do is pass out tickets based on the number of complete problems I saw on a worksheet when it was put in front of me. I would not say which question was wrong or why. It was fun to see which students figured out that doing problems one at a time was the guaranteed way to keep track of which problems were correct. 

Sadly, a decent chunk of students (maybe 1/3) still did nothing. But that means that 2/3 of the classes were actively getting their work done! And they have no idea what they MIGHT get! And they know it's NOT academic! 

I feel really weird about how well it worked. I guess the real evidence will come from assessment data to see if getting them to try will yield to proficiency, or if they really did just copy answers to get a raffle ticket. 

Thursday, May 30, 2013

Choose your own final exam!

At the end of the first semester, I gave the traditional final exam, but with the SBG flair. It was cumbersome to say the least. To fit 25-30 standards onto a single test expanded the test to about 8 pages. In the end, it was a good test in that students were able to finish it and it adequately covered the entire semester. But after 3 days of proctoring finals, I had 2 days to grade 150 exams that were each 8 pages long. Never again.

So leading up to the end of this semester I had an idea: if the core of Standards-Based Grading (SBG) is to pinpoint specific content that students have mastered, why bother making them take a final exam full of content that they've already demonstrated proficiency on? Why not let students focus their efforts on only the standards that they've struggled with?

Here's what I did: I wrote individual assessments for every standard (30 for geometry, 24 for physics and 30 for astronomy). Then I wrote a Google Form to allow students to tell me which standards they'd like to take on the final. Once I sifted through the data, I was able to give each student an exam tailored to their specific needs. The major stipulations were that students had to pick at least 10, and they had to start by picking the standards which hadn't been mastered yet.

Here's the form I'm having students fill out (please don't submit anything - it'll only confuse me):

I'm really excited to see how this all pans out. Gotta be willing to try something new, right? 

Friday, April 19, 2013

Day 132: Intersecting Secants

Unit 9 is the first unit of my new experiment that I hadn't given any thought to last summer when I wrote this all out. It was one of those "I've done a lot, I can finish the rest as we go next year" thoughts that doom all teachers in the summer. As a result, much of the instruction in this unit has been direct because creating student-centered discovery lessons apparently takes a LOT of time and energy. That, coupled with a lot going on in my personal life has left me just trying to get by. 

I'm at least still making every effort to demonstrate WHY the ideas we're discussing are true. No joke, the prepared materials that I was given when I started teaching were nothing more than: "This is the theorem, this is how it's used, now you try." No attention was given to the why & how questions, which is what led me to create this curriculum. 

For secants, the main idea we're discussing is the relationship between the exterior angle and it's intercepted arcs. Similar to chords, but when the angle is outside the circle, we're looking for the difference between the two arcs, not the sum. 

After a brief exploration demo, students were given the class period to work on U9 WS3.

Thursday, April 18, 2013

Day 131: Chords

For whatever reason, I never interpreted chords as a subset of secants. Rather, I thought of chords as a separate classification of segments, akin to tangents. What's interesting (to me anyway, I get that I'm not 'normal') is that it seems as though every math textbook ever written handles the chapter on circles different than every other text. There is SO MUCH you can discover with circles, so it ends up forcing a value judgement as to what you what students to know (and how to present it).

I went with:

  • Intersecting chords form an angle within the interior of a circle. The measure of that angle is half the sum of the two intercepted arcs.
  • If a chord is bisected at a right angle, the bisector is a diameter of the circle (I always thought that it was neat that you could find the center of a circle with 2 chords)
  • Intersecting chords break each other into 4 pieces (2 pieces each). The product of the 2 pieces of one chord is equal to the product of the 2 pieces of the other.
  • Equal chords intercept equal arcs and are equidistant to the center of the circle. Probably one of the more confusing and not obviously relevant idea. 
On the upside, I haven't had any students pronounce it with the 'ch' from 'church' sound. Progress?

Reflection 75% of the way through

I'm certainly noticing a subconscious shift in my teaching style in this unit, which is most likely the result of a lot of influencing factors. Essentially, I'm seeing that I'm implementing less and less of my vision, which was to incorporate the modeling method and student discovery based learning into geometry. I've slowly morphed into a 'traditional' teacher with instruction that looks more like a lecture than anything else. I tell myself that it could be worse - the curriculum materials that i was given when I started teaching here 5 years ago were very basic fill-in-the-blank notes, low level homework quizzes, and multiple choice tests. I can say with confidence that what I'm doing now is light years beyond where I was. 

Some possible reasons why this is happening:

  • After 8 months, I grew tired of beating my head against the wall, trying to force a paradigm shift on unwilling participants. Path of least resistance and all. (FWIW, I hate feeling like this, but I can't deny reality)
  • The content in this unit (circles) is a mile wide and an inch deep. And what stinks is that it doesn't even offer itself to adaptation. As in, it would be hard to go narrow and deep because of how interconnected everything is. Plus, the ACT loves circle questions, so I feel obligated to show 10th graders everything they might see next year on the most important test of their lives (I hate saying that, too). 
  • These students (either 10th graders in general, or possibly just 10th graders in my school) are simply not mature enough to handle a modeling approach. MAYBE, if the whole school shifted gears and the philosophy was being reinforced by every teacher a student saw, but when you're the lone wolf, students feel justified in resisting change. "If I can just get through this class, I can get back to a teacher who'll just tell me the answers and I can memorize how to get them." 
Looking forward to next year and I am absolutely going to continue with this curriculum that I have spent all year developing. I might try the modeling approach with whiteboards & group discussion with the hopes that the class of 2015 was an anomaly (something that's actually talked about openly in the school). 

I will certainly NOT go back to the teacher led, "sage on the stage" method. I hate it. In terms of grades, I'm not seeing any significant difference between that method and my method, so there's no real reason to switch back, other than getting tired of listening to students complain. And let's be honest, if I allowed student complaints to effect me, I wouldn't have lasted this long in this job. 

Wednesday, April 17, 2013

Day 130: What to do with homework

Given how few students are using class time to complete required assignments, I am going to remove that luxury from the schedule. Instead, I'll make sure that they've seen a few examples, and simply require that the work be completed outside of the class to be checked for a grade (based on completion, not correctness) the following day. 

Literally 95% of students are doing absolutely nothing when it comes to homework, so I'm simply wasting class time giving them the opportunity to socialize. I'm not naive enough to think this is magically going make everything better and have a positive influence on achievement, but at least I won't be wasting time. 

What I hate about this situation is what it indicates for my success in implementing the modeling approach in a math class. The original intent was to have students discuss their work collaboratively and come to consensus about the content in true 'modeling' style. We never had much success with discussions because so few students completed the work, so there was no jumping off point from which to start a discussion. So I gave class time to complete the work, in hopes that would spur an eventual discussion. But, even with a full hour of in-class time to work and ask for help, nearly whole classes are still doing absolutely nothing (I mean literally showing a blank worksheet on Day 3). 

I have tried skipping the in-class time to work and simply pushing students directly into the discussion, but then i struggled with maintaining an environment conducive to a discussion with a class of 30+ 10th graders (and those were my 'honors' students). 

I'm still convinced that modeling instruction is the best approach, but I'm finding it difficult to be the only teacher in a building of 55 that takes such a radically different track. Modeling requires buy-in from students, and if they're given what they want (traditional instruction) 5 of the 6 hours in a day, they're never going to see a need to shift their views of what schools can be. 

Tuesday, April 16, 2013

Final Exams in SBG

I love Standards-Based Grading. I mean, I really love it. I would marry SBG if human-grading philosophy marriage was legal (let's get on it D.C.). But here's the thing: it's VERY difficult (and completely counterproductive) to implement a single, high stakes, summative assessment (aka: final exam).

This year, I implemented SBG across all three of my classes (Astronomy, Geometry, and Physics) which means that at the end of first semester, I gave final exams that tried very hard to assess an entire semester of knowledge on a single test. All three final exams were 8 pages long and covered an average of 25 standards each. Writing the tests was hard enough. Students are given "super sized" class periods to take finals, so time allowed didn't cause many headaches.

But then came grading. 

Our finals ran Tuesday, Wednesday, and half-day Thursday with grades expected to be posted by Friday afternoon. Which meant that from about midday Wednesday (when the first couple of classes were done), I had 48 hours to grade 8 pages of non-multiple choice questions (MC is also counterproductive to SBG) for ~150 students. It was a rough couple of days. 

Looking forward to the 2nd Semester final exams, I see the need to do something different. While I do offer reassessments to all of my students, very few take me up on the offer. The core philosophy around SBG is encouraging students to show growth (hence, the reassessments). If a final exam is supposed to be a final assessment of student mastery, why not combine the two? 

Here's my thought: Forced Custom Reassessments. If I already have reassessments ready (which is true for Geometry, not so much for the other two classes), why not simply give each student reassessments on the 10-15 standards they've demonstrated the lowest level of proficiency? 

Flaws in my plan that I've already thought of:

  • Setup is going to be killer.
    • I need to custom organize a final exam for each of ~150 students. 
  • A student who demonstrated proficiency 4 months ago (by the time of the final) might not still be proficient. 
    • I know, but I'm willing to make that sacrifice. 
  • Grading could still be a pain. 
    • Sure, but I doubt it will be any worse than it was. Gotta try something. 
  • Overall student grades CANNOT go down.
    • If you only test on things you suck at, how much lower can your grade go?
  • Students will be able to focus their studying.
    • Kinda the whole point of SBG.
I would very much value some input on this idea if you could find some time to leave a comment. 

Day 129: Skills Review & Practice

This time I picked Area of Quadrilaterals for the skills review, and each section raised the class average 0.5 on my 4 point scale. I made a point to emphasize how pleased I was with the results, even though the overall grades themselves were still very low. For example, my 5th hour went from 1.1 to 1.6. 1.6 is still a failing grade (passing line = 2), but it's progress, and that's what I'm looking for. If students can make another gain of similar size before June, they'll pass the class. 

The whole argument for SBG is to encourage student effort in the hopes that they'll begin to make the connection between effort and success. I've been so bogged down with creating this whole curriculum, as well as implementing SBG in my other classes (in addition to my regularly scheduled teaching), that I've lost sight of the real focus behind what I set out to do. I need to make more of an effort to highlight student growth and hard work. It's not even that I've instead been publicizing high achievement, it's that I don't do either. I make grades available, both in class and online, but I never talk about them and ensure that students understand where we're at as a class and how far we've come. 

After the skills review quiz, I gave the students the class period to work on the worksheet handed out yesterday. More thoughts on that tomorrow. 

Monday, April 15, 2013

Day 128: Formalization of Conclusions

We went through the major ideas to come out of the computer lab investigation from Friday. Having the pre-made applet on hand was very helpful - students who hadn't completed the work could still participate in the discussion just by looking up at the screen. 


  1. Radii are congruent (not what I was expecting, but important nonetheless)
  2. Distance from point of tangency to point of intersection for two tangents is equal
  3. Line connecting center of circle to point of intersection is an angle bisector
  4. The angle formed between radius and tangent is a right angle
  5. The angle formed by intersecting chords is equal to half the sum of the intercepted arcs
Students then began work on U9 WS2. 

Friday, April 12, 2013

Day 127: Investigation with GeoGebra

I created a set of instructions for a student led investigation of the conclusions for tangents to circles and the angles and arcs formed by chords. Most students were able to follow the instructions for the construction, but struggle with the conclusions. At least with the tangent half of the exercise, some students did make some relevant conclusions, but the chord angle/arc equation eluded everyone. 

Additionally, I made applets in GeoGebra for students to play around with and get around making the construction themselves (and also to help guide them toward the conclusions). 

Tangents to Circles
Arcs and Angles formed by Chords

Thursday, April 11, 2013

Day 126: Discussion and Assessment

As usual, even after 1.5 class periods to work, most students weren't finished with their work. I did still check the worksheet for completion at the start of class (to confirm my suspicion), then went through as many problems on the worksheet as we could to still allow 15 mins at the end of class for a quiz. 

I don't normally give quizzes on the day we go over something, because it's not a measure of retention. Possibly just a confirmation bias, but results on the quiz were much higher than usual. My main reason for doing this was that I have tomorrow planned in the computer lab, and I hate trying to proctor a quiz in a foreign environment. 

Wednesday, April 10, 2013

Day 125: Skills Review Quiz & Practice

I'd almost forgotten about giving Skills Review quizzes, but I found some over spring break while I was cleaning, and start of the 4th Marking Period seems like a great time to get back in the habit. So, I picked the lowest standard from the oldest unit of the semester which dealt with angles formed by parallel lines intersected by a transversal. Surprisingly, most students didn't complain too much and better yet - the results actually raised the class average on the standard from 1.9 (2 is passing) to a 2.4 (about a C-). 

After a few examples, students spent the rest of the hour working on U9 WS1. 

Tuesday, April 9, 2013

Day 124: Special Segments & Angles in Circles

Simple lecture discussing chords, secants, and tangents. Then a simple discovery (led by myself) regarding central angles and inscribed angles). I generally hate lecturing, but the students are so conditioned to it, their behavior is MUCH better when I lecture. I don't think they're necessarily learning more, but they're giving me less grief, and that's really more important, isn't it? (I'm totally kidding - just exhausted from staying up to watch my Wolverines lose the NCAA title game last night). 

I used the document camera, because that tends to go over better than the tablet & projector combo does (no idea why - they students just don't like that). I used to rely on the tablet so I could make PDFs of the notes, but now I can use an app called CamScanner on my phone to snap pics of the notes and upload them to the cloud as PDFs. Cool beans. 

Also getting better at making interacting applets in GeoGebra to show basic ideas (quadrilateral inscribed in a circle was today's attempt). All of this stuff is available at my class website:

Monday, April 8, 2013

Day 123: Unit 9 - Circles

Whoo - a new unit! Circles might be one of my favorite topics (how many times have I already said that this year?) There's just SO MUCH you can you do with circles and I guess I'm as mystified by the perfection of the shape as the ancients were. 

In any event, to ease students back into school after spring break, today was a very simple intro discovery lesson. 

  • Draw point A
  • Have students agree on a common distance (pick a number between 1 and 10), and draw a new point X cm away from point A
  • Repeat Y times (pick another number between 10 and 20). 
    • (Amazingly, this is the step that killed everyone. You should have seen the misinterpretations)
  1. What is the longest possible segment you can make connection 2 of the points you drew?
  2. What is the relationship between your answer to #1 and the distance we used to start?
  3. Find the perimeter of the shape by measuring the point-to-point distance at every gap and adding them up (you should have heard the groans)
  4. What is the relationship between the perimeter and the longest segment?
And then I gushed about how awesome Pi is. They didn't care. 

Thursday, March 28, 2013

Day 122: Unit 8 Assessment

I remember a day when I didn't dread giving assessments, because I had no reason to fear an altercation arising that could possible lead to a suspension. One of my classes has so many students that have completely given up (not just on geometry, but on school in general) that the thought of corralling them during a test terrifies me. 

It's constant giggling and whispering, but the administration simply says "give them a zero" for violating test procedures. Regardless of the fact that zeroes don't exist in SBG and that's a ridiculous way to treat grades, these particular students could care less about a zero. A few of these kids don't even turn the tests in! (I don't worry too much about test security because I don't think I've ever given the same test twice in the 5 years I've been teaching). 

So my only alternative is referral to the SRC (Student Responsibility Center) which involves a call home and a reinstatement meeting with me. If students are quiet and have given up, I have no recourse to ensure that I am not being measured by their scores. 

Oh well, tomorrow is spring break!

Wednesday, March 27, 2013

Day 121: Review

Today was basically a mop-up day as we prepare for the Unit 8 Assessment tomorrow and spring break on Friday. Students continued to work on the worksheet from yesterday, but I did spend some time ensuring that they at least saw proper setups for a few of the problems. I also handed out a brief summary of the unit so students could have a reference to help them as they create their note sheets. 

Tuesday, March 26, 2013

Day 120: Examples & Practice

I have adjusted my approach somewhat since September in that now I will do more examples for the class before we begin a worksheet. My rationale had always been that if I ensure that the class is properly suited, that I shouldn't need to show them how to do a problem. The constant hand-holding in high school is something that really bugs me. Additionally, I've learned through trial & error that the students will never be satisfied. Don't do any examples and they'll be upset. Do some examples, and they'll simply postpone getting upset until they encounter a problem that doesn't look like the ones you showed them how to do. Show them those and they'll just get upset on the quiz that looks different still. 

Even more frustrating through all this is how many students use this experience as fuel for their negative opinions of me (that's not a guess, I've spoken with some students and given evaluations across my classes). It's about more than what they *want* school to look like, it's sad to see that they think this is what school *should* look like. "Show me how to do X, I'll do X in front of you, tomorrow I'll do X on a quiz, next week I'll do X again on a test, and in a few months I'll do X on the final." 

But what about Y & Z?

Anyway, after another example of AoRP, this time given the apothem and solving for the side length, the students worked on U8 WS2. 

Monday, March 25, 2013

Day 119: Area of a Regular Polygon

Call me old fashioned, but I think the derivation for the area of a regular polygon formula is one of the most amazing aspects of high school level Euclidean geometry. It encompasses so much knowledge - it marks the the point when everything we've been learning finally comes together.

Which is exactly the reason why the students HATE this unit. 

Worst of all, most of the content that students need to get through this unit (as well as most of 2nd semester) came from 1st semester, and too many students have the "out of sight, out of mind" philosophy when it comes to the semester breakdown. It all gets back to the fixed vs. growth mindset ideas, but SBG throws a nice wrinkle into the mix. I finally settled on just saying that students will be assumed to be proficient on ALL standards covered in 1st semester. So if you need to know how to use Pythagorean Theorem to solve for the missing side of a kite, then it's on you. I will still give a proficient score of 2 (the passing line) to students who can demonstrate the proper setup (showing that the diagonals of a kite for a right angle) even if they can't solve it. 

In any event, I work through the discovery for the area of a regular polygon (how about AoRP from now on?) on my tablet as a derivation that the students are expected to follow. By the end, they're exasperated and are hoping that there's a shortcut that I just didn't show them at first (they know that is something I would totally do). Unfortunately, the only "shortcut" (after the central angle, right triangle trig, interior triangles and multiplication) is to catch the perimeter popping out. The students are generally upset that they'll be expected to do all that work for just ONE problem. Babies. 

One pleasant surprise was how many students still have their laminated trig tables that I gave them back in 1st semester AND they have a basic memory of how to use them. Awesome!

Friday, March 22, 2013

Day 118: Catch up

I lost 2nd hour due to testing yesterday, so 5th hour went to the computer lab for independent review/practice, and I thought I'd do something 'fun' for 6th hour to intro the next unit to them. Big mistake. 

Since Unit 8 will be about polygons in general, I figured we'd give constructions with a compass another shot. I had step-by-step flash animations from mathopenref on the screen and I switched between that and a document camera showing my own construction as I went and yet it devolved into total chaos. Students immediately become distracted and talk with their neighbors which forces them to miss a key step, so they'll interrupt and ask to go back so they can see the last step, which doesn't make any sense without the explanation. And of course if I don't kowtow to their needs, they'll simply talk and act out more causing more of the class to go off track. Fun times. 

In a perfect world, I would teach the ENTIRE class with a compass. It's such an incredible tool and it's been around for thousands of years. And I'm forced to resort to the SAFE-T compass with no points because the students can't be trusted with sharp objects. I'm also down about 10 compasses since September between students breaking them or stealing them. 

Sorry, I need to complain less here, but this lesson really got under my skin. 

Thursday, March 21, 2013

Wednesday, March 20, 2013

Day 116: Exterior Angles

I did a demo of the sum of the exterior angles in a polygon, recapped the exploration from yesterday, and set students to work on the first worksheet of Unit 8. 

If you've never seen the demo, it's both simple and powerful. Draw a convex polygon on a piece of paper. Create one exterior angle at each vertex around the perimeter (be sure to work in the same direction around the exterior). Number and cut out each angle, making sure to note where the angle was on the cutout. The pieces easily reassemble into a central circle, showing the sum total to be 360. 

This could have been done as a student activity and with more shapes to emphasize the independence of the relationship, but for time considerations I simply showed other examples in Geogebra and set the students to work. 

I should note that we also pointed out how to determine the measure of a single interior or exterior angle IF the shape is regular. For some reason, this equation gave students fits for days. 

Tuesday, March 19, 2013

Day 115: Unit 8 - Polygons

To kick off the new unit, I randomly assigned students to groups and had them draw a shape that had either 5, 6, 7, or 8 sides. Once drawn, they labeled the vertices, measured the interior angles and added them up to find the sum. 


  • Using a protractor is a still HUGE hurdle. It really shouldn't be, but results were off by over 100 degrees in some cases. 
  • Allowing any shape seems like it makes the task easier, but students had no idea how to measure a reflex angle, which most likely contributed to our error problem. 
  • In the future, I might direct students to round measurements to the nearest 5 degrees in the hopes that it will help.
The pentagon had decent results all day, but as the number of sides increased, the errors are compounded, and the results get further and further off. I used Geogebra to try and fill in the gaps and show the ideal case. Some students either noticed the pattern or reasoned through what the relationship between # of angles and total has to be which was surprising. 

In the end they all seemed to think it was easy, which is usually a sign that something went terribly wrong. 

Monday, March 18, 2013

Day 114: Unit 7 Assessment

I might have set a new record for "Students who turn in a blank test with their name on it" today. In light of that, it was actually fairly amazing to see how "well behaved" they were. I use quotes because they weren't actually well behaved, but considering that they did literally no work for 55 minutes, I'm surprised they weren't worse. 

Overall the results of the test were very poor. It was also very difficult to grade as I tried to implement a 'justification' standard into SBG. 

If a student graphs the 4 vertices of a quadrilateral and says it's a parallelogram because the opposite angles are congruent, how do your grade that? They are demonstrating some understanding of properties of parallelograms, but there is no way they can actually tell that their assertion is valid. What about students who say the sides are parallel, which can be shown on the graph, but they don't actually determine slopes of opposite sides to demonstrate the validity of their claim? 

This test did take a while to grade, but as has been the case with SBG, I'm very happy with the detailed information about what students know vs. what they don't that comes out of it. 

Friday, March 15, 2013

Day 113: Review

We spent the hour in the computer lab giving students a chance to prepare for the U7 Assessment on Monday. 

Students had the option of working with the Carnegie Learning Tutor (each student was moved to the current unit), watching/working with Khan Academy, or making a note sheet to be used on the test. 

Thursday, March 14, 2013

Day 112: Area & Perimeter

Today was partially lost to more (optional for our school) standardized testing, so the Unit 7 Assessment had to be moved to Monday. I generally hate Monday assessments, but in order to get Unit 8 fit in before spring break, sacrifices must be made. 

The only topic left in Unit 7 is area & perimeter of quadrilaterals  so I used Geogebra to show how we can find shortcuts to solve for the area of various shapes. I don't bother with formulas for perimeter - as long as students understand what perimeter is (and how to use the Pythagorean Theorem to solve for diagonal distances), I assume they'll be able to figure it out. 

In general, our theme for finding area is based around the idea of making a rectangle either from the quadrilateral directly (parallelogram), or making the rectangle around the shape and noticing that it's area is double the shape's area. 

In my head, understanding how to determine area is more useful than memorizing a bunch of area formulas, but students are unfamiliar with a general approach and generally feel more comfortable with the formulas. Except, most students cannot reliably use a formula as a result of very weak algebra skills. Or, they lack the ability to differentiate horizontal/vertical distances on a graph from diagonal lengths. Or they have no idea how to identify a base and a height.

Wednesday, March 13, 2013

Day 111: Formative Assessment

Students spent a large portion of class on the second quiz in Unit 7. This quiz was slightly longer than normal, so more time was allotted. It doesn't appear to have increased results any. 

In general, the correlation between student engagement and content is based primarily on ease of understanding the material. What I mean is that students in large part actually worked diligently on the first worksheet of the unit on parallelograms, but refused to work on the second regarding trapezoids and kites. If anything, we actually spent more time "learning" about trapezoids and kites when the computer lab time is factored in, so all I think of is that these shapes are less familiar to students, so their fixed mindset becomes an obstacle. If something seems easy (most likely because it's familiar), then students will given it a shot. If something is new and appears difficult, students take the "if I don't try, then I didn't really fail" approach. 

I'm guessing if I looked back across ALL of my standards since September, I'd find proficiency across the 'easier' ones (measuring segments & angles) and a marked lack of progress on the 'hard' ones (right triangle trig for example). 

Dealing with fixed vs. growth mindset is one of the biggest frustrations I have. There is a lot of pressure to make the content relevant and engaging, but I cannot do that if students refuse to move beyond the first (and easiest) step. To me, right triangle trig is amazing for its (apparent) simplicity and its almost endless applicability. It's just a bunch of proportions after all. But as soon as you give most students a glance at the endgame, they quickly state "oh well I'll never do that" as justification for why they don't need to learn it. 

This is a pervasive problem outside of just geometry, but it becomes incredibly difficult to teach 2nd semester geometry to students who couldn't be bothered to learn the content from 1st semester. How do you teach surface area and volume of polyhedra to students who never learned how to: name polygons, determine the area of a regular polygon, use trig to solve for missing sides of a right triangle, determine a central angle from dividing 360 into equal parts?

Tuesday, March 12, 2013

Day 110: Discussion

It's getting to the point that a class period set aside for going over a worksheet ends up just being more time spent working on the worksheet. So, instead of simple cycle of the EDGE method (Explain Demo Guide Enable) taking 3 days at most, a simple lesson takes about 5 days. 

Here's reality right now:

  • Day 1: Investigation. Some student centered activity in the vein of "organized play" in which students can discover the upcoming relationships without knowing what's coming.
  • Day 2: Formalization. Class works together to take notes and structure the discoveries just encountered. Examples given. 
  • Day 3: Practice. Students work on practice problems to become better acquainted with the material.
  • Day 4: Review / more practice.
  • Day 5: Quiz. 
Here's what I would prefer:

  • Day 1: Investigation. Some student centered activity in the vein of "organized play" in which students can discover the upcoming relationships without knowing what's coming.
  • Day 2: Formalization. Class works together to take notes and structure the discoveries just encountered. Examples given. 
  • HW assigned on Day 2, due on Day 3
  • Day 3: Quiz
The issue is that students won't do HW because they're either incapable or simply refuse to work independently. If they come across a problem with a solution that is not immediately apparent, they will do nothing until I guide them to an answer via a step by step procedure. 

If they won't work at home, I have to give them time in class to work. But there are 35 of them in class, so if I can't get to each and every one of them in class, they still won't work. Which means I need to give more time on the 2nd day to make sure everyone gets something done. This means there is often little or no time to actually go over the practice, which in turn creates a mentality of "the teacher never showed me how to do every problem and never gave me the answer to every problem, therefore I cannot be held accountable for demonstrating understanding on the quiz."

I honestly have a non-insignificant number of students who posses this mentality and will simply refuse to take quizzes in protest. 

Monday, March 11, 2013

Day 109: Practice and justification

Wow, a complete & uninterrupted week of school lays before us. That shouldn't feel weird, but the 3rd Marking Period is always like this. 

Students were given the hour to work on U7 WS2 dealing with trapezoids & kites. We did address the idea of isosceles trapezoids at the start of class, and outlined everything we know about quadrilaterals to this point. 

As part of the SBG implementation, I created a general standard for reasoning because I was having a hard time separating a demonstration of knowledge about an idea (like opposite angles congruent in parallelograms) from being able to explain WHY that is true. So at least for this unit, S7.6: I can appropriately justify my reasoning will be a part of almost every question. 

Friday, March 8, 2013

Day 108: Formalization of Conclusions

We reviewed the conclusions sought after from the computer lab investigation and developed a set of theorems that seem to hold true for trapezoids and kites. To avoid an information overload situation, we left any mention of isosceles trapezoids until next week. 

So far, we know:

  • Trapezoids in general have very little to offer us (One pair of parallel sides? Consecutive supplementary angles? That's it?)
  • Kites are neat, but incredibly confusion to describe in words (Two pairs of sides are congruent, but not that pair. No, not that pair either. Just take a guess). 

Students were then given the remainder of the class period to begin working on U7 WS2.

Thursday, March 7, 2013

Day 107: Killing time

It's ACT/MME test week, so the schedule is a mess. I used yesterday/today to give students more time to finish the trapezoid & kite investigation in Geometer's Sketchpad. 

Sketchpad is an amazing piece of software, but it's not terribly intuitive or user-friendly, especially dealing with high school students that aren't especially tech-savvy. As hard as I try to write directions that are clear and explicit, many students become frustrated and give up before reaching final conclusions. 

The focus of the activity was to help encourage the need to empirical evidence to justify conclusions. Rather than use a ruler and protractor to make measurements, we can use the computer to do the hard work for us, but we still need something to back up a statement like "the diagonals of a kite are perpendicular." 

Moving forward, I've begun to use Geogebra in place of GSP. Very similar functionality, but 1) Geogebra is free, and therefore can easily be installed on the school's computers, 2) Geogebra has a much more intuitive interface (IMHO). I've also begin thinking about interactive java applets I can make and upload to my class website for student exploration while at home. I know you can do that with GSP, but I lack the most recent version, so I was having difficulties getting it done. 


I going to start the arduous process of posting everything I've made for this curriculum online. They'll be uploaded to Google Docs, so feel free to download and do what you wish with the materials. The only things I will NOT upload are assessments, but if you're a teacher and would like to see them, send me an email (bfuller181 [at] gmail [dot] com and I'd be happy to oblige. Oh, and same goes for any requests for the original Word 2010 documents - I know that sometimes the conversion to Google Docs causes problems. 

Wednesday, March 6, 2013

Day 106: Continuing Investigations

The ACT/MME is wreaking havoc with our class schedule this week, so I planned the trapezoid & kite investigation to take two days. The fact that students don't have access to GSP at home also factored in to my decision. 

I actually was out sick today, so a sub covered my 2nd hour class. I hope to be back tomorrow and oversee the 5th and 6th hour classes as they finish up. 

Friday we'll reconvene as a class and formalize the results of the investigation, and start practicing. Which reminds me, I have a worksheet to write. 

Monday, March 4, 2013

Day 105: Investigation of Trapezoids & Kites

First stop was a quiz on last week's worksheet, and then off to the computer lab so the students could explore properties of trapezoids and kites. 

My hope was that if students now have a firm understanding of *how* we test ideas about shapes (use slope for parallel/perpendicular, distance formula for length, use midpoint for bisecting, etc), then then can use computer software to readily create shapes that would be problematic to explore with a pencil on graph paper. 

I chose to use Geometer's Sketchpad because that's what I've been using since college geometry, but after spending the last few days exploring Geogebra, I'll probably switch over to that in the future. 

I wrote up VERY explicit instructions, detailing what students should do, step by step, to achieve a proper shape. "Use the segment tool to draw a line segment, select a new point and create a parallel line by clicking here," etc. Of course, my best efforts were thwarted by students who absolutely refuse to read directions and immediately claim they did, but couldn't understand them. Students would call me over to ask why their quadrilateral didn't look right, and I could quickly see that they skipped step #4. "No I didn't!" they would emphatically deny. "Well actually yes, you never placed the two new points as instructed." I would reply. Then they'd get sheepish and laugh and admit the must have missed that part. 

That, in a nutshell, sums up most of my difficulties teaching 10th graders (the probably isn't generally as bad with the 11th and 12th graders in my physics and astronomy classes). A complete and total refusal to even bother paying attention to given instructions, no attempt and determining what went wrong on their own, and complete denial of any wrong doing when confronted. I understand this is simply a matter of maturity, but this strikes me as behavior one would encounter with 6th graders who might be 11 or 12 years old. I have some experience working with the same age level I do now in other school districts and I never saw maturity levels like I do now, which leads me to believe the issue is somehow correlated to community and socio-economic status. It provides a launching point for a very interesting discussion of fixed vs. growth mindsets, but that's a discussion for another venue. 

Friday, March 1, 2013

Day 104: Discussion

Again tried to make whiteboarding happen. I'm torn between wanting to give students a chance to confirm answers to the worksheets (I generally refuse to do that directly myself) and wanting to make the best use of class time. 

In 2nd hour, most of the class worked earnestly to get their assigned problems on whiteboards, but they chose for presentation style (over whiteboard parade style) and then couldn't keep quiet long enough to get through more than 4 problems (out of 20 total). 

I didn't give 5th hour the option, because I know they can't handle presentation style, so I had them stick with whiteboard parade. While the task eventually got accomplished, you can see the bulk of the work being done by the few (even though the worksheets are now graded). 

6th hour also got whiteboard parade, and for similar reasons.

Results will show on Monday's quiz.

Thursday, February 28, 2013

Day 103: Recap & Practice

The first part of the class was part review (snow day yesterday), and part organizing our notes. On Tuesday, we made a Venn Diagram of the parallelograms, but didn't list the conclusions directly (just categorized the shapes). So I spent ~15 minutes making a list of everything that was true for each shape. 

Obviously, we started with parallelograms in general and wrote the definition and the 4 conclusions we found. Then we branched off to rectangles, and from there we actually moved down to squares, since we never formally discovered the major properties of rhombi. I liked the way it shaped up because we had a list of things that were true for squares, and some of them weren't true for rectangles. So logically, properties such as diagonals being perpendicular and bisecting their angles must have come from rhombi. 

I insisted to the students that any justification for their work MUST come from the conclusions that were now cleanly listed on the board. 

Students were then given the remainder of the hour to work on U7 WS1 regarding Parallelograms & Rhombi. 

Picture of the notes