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.

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. 

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. 

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. 

Upsides:

• 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. 

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. 

Conclusions

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. 

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

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. 

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. 

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: www.mrfuller.net.

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.