Friday, December 21, 2012

Day 74: Holiday "Party"

Students complained that yesterday's test was too long, so I allowed them ~15 minutes today to finish it up. Surprisingly  many students said they were done and didn't take advantage of the chance to look over the test. 

I made a point to NOT tell the students I'd be doing this because I wanted them thinking it was over so they wouldn't go home and study more. I don't think that was an issue. 

Approximately 25% of my students were absent (excused or not, I don't really care), so they missed out on the opportunity to finish up. 

And I don't throw any type of organized party, it's just a chill day to watch a non-religious holiday special and relax. 

I did do something new this year, I wrote a note to each class (not to each student, I don't have that kind of time), printed them with a holiday print and gave one to each student along with a candy cane. The theme of the note was positive, trying to keep everyone motivated as we approach finals. They genuinely seemed to appreciate the gesture and I'm happy I did it, as much fun as printing, folding, and taping candy canes to 150 notes was...

Thursday, December 20, 2012

Day 73: Unit 4 Assessment

Includes the 6 standards from Unit 6 in addition to a rehash of S2.5 on Ratio & Proportion. 

Every year I give tests right before the holiday break (mostly because the students are going to fight any sort of productivity anyway) and every year I head into 2.5 weeks off with 150 tests to grade wondering why I would do such a thing to myself. Oh well. 

Wednesday, December 19, 2012

Day 72: Online Practice

Back to the computer lab for more time using the Carnegie Learning software. In my head, this is a good resource for students to practice in a different environment. Some work well in the online setup, while some do not. For that reason, I do not require any level of progress with the online practice, just as I do not collect and grade homework. I try to stress to the students that it's practice that has exactly as much value as effort you put into it. 

In addition, being the day before the Unit 4 Assessment, I figured students could spend the hour writing their note sheet if they so chose. 

Of course, those were my *ideas* for what would happen. Reality is never that simple. 

Tuesday, December 18, 2012

Day 71: Practice

Two items on the agenda: the third Skills Review Quiz of the year and discussion of U5 WS1: Ratio & Proportion. 

Technically, we're still in Unit 4, but I didn't want to assess early and try and start a new unit before the holidays, so I'm just shoe-horning the first segment of U5 in there before break. Ratio & proportion *should* be major review, especially since we ALREADY reviewed it back in U2 as we discussed angles, but who knows. 

I'm really liking the idea of more and more quizzes (both skills review and 'regular') because it (hopefully) takes some of the stigma/anxiety away from assessment. If they happen 3 times a week, they can't be a big deal, right? Of course, that's my opinion, the students expressed very different (read: mostly negative) views. 

Monday, December 17, 2012

Day 70: Ratio & Proportion

I backed myself into somewhat of a corner, scheduling wise. We finished Unit 4 last week, but I didn't want to rush the test and start Unit 5 before the holiday break, so I decided to stretch out Unit 4 a bit. I already added the review of simplifying radicals last week, and I decided to spend a couple of days this week quasi-starting Unit 5 by reviewing ratio & proportion. 

Technically, we already did some ratio/proportion stuff back in Unit 2 when we learned about angles as being fractional pieces of circles, but a) the students weren't very good at it then and 2) Unit 5 is all about the trigonometric ratios and similar triangles, so I figured the review would do us good. Mind you, ratio & proportion is a topic from *before* algebra, so there really isn't any reason why students shouldn't have this down by now. Alas...

So I created U5 WS1 and simply presented it as another worksheet in this Unit 4 (nobody seemed to notice). I already had a standard from Unit 2 dealing with this (S2.5), so I can easily reassessment an old standard on the U4 Assessment this week. 

Friday, December 14, 2012

Day 69: Online Practice

It was meant to be a simple day in the computer lab to keep students practicing "old" material, but very little ended up being accomplished (warning: the following will border on a rant about access to technology in public schools).

My school has 5 computer labs with antiquated desktop computers that are rapidly falling apart.  The "Math & Science Computer Lab" only has 32 machines in it, for class sizes that (in my case) are all over 32. Ok, fine, I generally don't average 100% attendance, so it can work out. In 2nd hour yesterday, a handful (maybe 3 or 4) of the machines weren't working. Either they can't log in to the school network, or the machine is stuck in a constant boot loop, or it simply can't access the website we use for the online tutor. It ended up not being a big deal as I had more than a handful of absences (it was Friday after all). 

By 5th hour, the number of non-working machines had risen to 10 (without my knowledge of course). Not only is this my most challenging class, but even after a number of students have been removed from the class, I still had more than 22 show up. So I decided to send a half dozen to the school media center where there are handful of loose machines that students can use. 

In 6th hour, I attempted to do the same thing, but the media center was booked. As was the computer lab in the media center. And the computer lab across the hall. I was left with 33 students and 22 working computers to incorporate a required part of the curriculum. 

As a result, not much got done. Only a handful of students chose to ignore the distractions and work diligently with the online tutor. Most chatted with friends, listened to music, or played games. I know, I technically have the power to insist that they not participate in those actions, but 1) what do you say when by default, 10 kids are not going to be able to participate in the lesson? and 2) I've learned that the more discipline issues I handle 'appropriately,' the more reinstatement meetings I have to sit in on, and the more plan hours I lose. That doesn't mean I simply avoid dealing with behavior, but I tend to reserve judgement for only the most serious offenses. 

Next week is the last week before the 2 week holiday break. We'll wrap up/review Unit 4, go over ratio & proportion to prep for Unit 5 (Trig ratios), and take an assessment. 

Thursday, December 13, 2012

Day 68: Practice

I'm building up the habit of weekly skills review quizzes. The math department at my school had always encouraged the idea, but I never saw much use/need under a traditional grading scale. As students forgot the content over time, skills review seemed more like a punishment, because most students would "lose" points and lower their grade. 

With SBG, Skills Review quizzes are simply mandated reassessments, which don't generally do any 'harm' to a student's grade. In a lot of cases, students perform just as well as they did in the past, but some students show improvement, and that's important to have evidence of. And since a student's grade is no longer broken into ridiculous marking period grades (just in my class, don't tell anyone - I don't think I'm supposed to do that), but instead the semester grade is simply a running average of all the standards we cover, skills review quizzes can have an immediate and drastic impact raising a students grade (also thanks in large part to the "decaying average" calculation that ActiveGrade lets me use). 

Additionally, with the traditional nonsensical sequencing of the material, I couldn't really blame students for forgetting the content, as it had no logical structure. Material covered in September didn't truly affect material learned in December, so why not forget it? With my sequencing, truly mastering linear equations is absolutely crucial to everything else we do, and will only become more important in the second semester as we transition into more complex problems with proofs. 

I'm honestly toying with the idea of making the final exam consist of only one question with MANY parts, demonstrating the connectivity of everything we've studied. The only thing to be on the look out for in that case is providing students who can't answer part a. a way to answer part b. (and so on). 

After the skills review quiz, we went over the worksheet on simplifying radicals and called it a day. 

Wednesday, December 12, 2012

Day 67: Simplifying Radicals

Started the day by taking a quiz on the 30/60/90 triangles, the results of which demonstrated an incredible inability to transfer knowledge to new situations. The worksheet contains a variety of 1 triangle and 2 triangle problems (where the hypotenuse might become the long leg of a separate triangle). Students can do the simpler variety, but will stop completely at the more complex until I cover up one triangle and ask them to keep doing what they'd been doing. So on the quiz, I created *gasp* three connected triangles and gave them one side length, asking them to determine a side on the far side of the diagram. 

I can't even guess how much kids left it blank or wrote a big question mark over the picture as if to say "You never taught us this so I obviously can't be held responsible for it." And these were students who did just fine on the simpler problems on the other half of the quiz. Ugh.

After the quiz we reviewed how to simplify radical expressions. Most students remember a factor tree type method from 9th grade, which is actually unfortunate, because that method is terribly unreliable (at least, the pieces of it that the students remember is). The biggest surprise continues to be the trouble students have with the square root symbol itself. They understand that add/subtract/multiply/divide are operations, and once the operation is complete, you can stop writing the symbol, but they don't make that connection to square roots. A LOT of kids will say that sqrt(144) = sqrt(12). I'm trying to combat that by reminding them that sqrt is a button on a calculator. Once you've pressed the button, you can't tell people (by writing the symbol) to keep pressing it. 

Tuesday, December 11, 2012

Day 66: Discussion

The biggest hurdle I've faced in trying to adapt a modeling approach with geometry is the students. It's possible that 10th graders are inherently unable to handle such a collaborative approach, but I personally think it's a more local phenomenon. The issue that is truly making things difficult is the size of my classes. I'm beginning to think that 35 10th graders might not ever be able to pull off student led class discussions. 

And I would be ok with that if all it meant was that I had to do more of the leading. But inevitably, the cycle we've fallen into is that when it's time to go over a worksheet, we might only get through 2 problems because the students are simply unable to focus in a constructive fashion long enough to accomplish anything significant. Some students then take the view that if I don't personally go over the answer to every problem in detail, I don't have the right to move the class forward and assess them on their knowledge. This sort of learned helplessness is obviously a learned behavior, but it's an incredible impediment to try and undo in 10th grade. 

In any event, we went over as much as we could from U4 WS3 on the 30/60/90 triangles. There's honestly not that much to go through, if you have the 2 models down regarding the connections between short leg/long leg and short leg/hypotenuse, all that's left is identifying the legs in a triangle, substituting into the proper model and solving. I am at least making headway in convincing the students that the geometry aspect of the work is very simple and basic, but it's actually their struggles with algebra that are holding them back. 

Monday, December 10, 2012

Day 65: Practice/Recap

We first finalized the discussion of properties in a 30/60/90 triangle, and then students were given time to work on U4 WS3. I tend to use "extra" days like today to get all of my classes back on the same schedule, since they always all take different amounts of time to get through the discovery phase. 

Friday, December 7, 2012

Day 64: 30/60/90 Right Triangles

Agenda Item #1: Quiz on Pythagorean Theorem & Distance formula. Results were shockingly good. 

Agenda Item #2: Discovery of the relationships in a 30/60/90 triangle. 

Of all my "teacher led student discovery ideas" for this class, this one was the trickiest. I wanted student to be able to create a triangle on graph paper (as per the norm), so they can easily set the side lengths. But x and x*sqrt(3) don't lend to integers very well. Closest I could come was 11 & 19, which leaves an angle of ~59 degrees. Close enough.

Then I wanted students to be able to enlarge the triangle to test any ideas/gather more data. Oof. Finally settled on "short leg + 1" & "long leg + 2." Again, close enough. 

Then I had students work the Pythagorean Theorem to solve for the length of the hypotenuse. I had been stressing that we shouldn't evaluate the radical, but I had to break the rule to make potential relationships a little easier to spot. So the hypotenuse for the first triangle is ~24, the next is ~26, then ~28 and so on. 

Most students saw that "short leg * 2 = hypotenuse." Not bad. Some came up with the idea that "long leg + 3 = hypotenuse," which was true based on the data we had, so I quickly used GSP to show a MUCH larger triangle, worked the numbers, and showed that it didn't work. I love that they're trying ideas though. 

I didn't really expect the whole "sqrt(3)" thing to pop out, but I wasn't sure how to point it out. By the 3rd iteration, I went with the idea that addition, "short leg + some number" doesn't work, because the number changes (2, then 3, then 4, etc). Which is ok, because when the number we're adding doesn't work, that's just multiplication. So there must be some number that we multiply by the short leg to get the long leg. But what?. 19/11 ~ 1.73, and test a few others. They should remember that sqrt(2) ~ 1.41, so that's not it. And sqrt(4) = 2, so it has to be something between 2 and 4. Why not 3? And that works. 


After we've hammered out the ideas, I use either GSP or a protractor to measure the angles, because we have to figure out what was so special about this triangle in the first place. That was the easy part. 

Thursday, December 6, 2012

Day 63: Practice

After recapping the Pythagorean Theorem & Distance formula, I worked through a few examples, and allowed the students to work on U4 WS3. 

Unrelated to geometry aside, a few weeks ago, some coworkers and I were chatting during lunch and either he or I brought up the idea of somehow incorporating QR codes (those funny 2D barcodes) into our classes somehow. I already have a class website, so I figured they could be permalinked to specific parts of the website (calendar, where notes might be stored, etc.), but I wasn't sure how useful that would be and it sounded like a hassle. 

Then, last week I was the DMAPT (Detroit Metro Area Physics Teachers) meeting and a teacher shared that she puts the answers to worksheet questions in QR codes that are placed in the empty space of a problem. I didn't realize you could link a random paragraph of text to a QR code, so I thought this was genius.

So now I'm using the QR Code Generator (there are tons of sites like this, this is just the one I use) to attach hints to worksheets. For example, in U4 WS3 on 30/60/90 right triangles, I wrote 4 short lines that summed up the notes. 2 lines to help identify the short & long legs and 2 lines laying out the model that's used to solve for the missing pieces. 

Some of the students knew what they were and how to use them, but I showed the classes quickly anyway. It does kinda suck that not every kid has a device that can read them (or they have a WiFi only device like an iPod and there's no WiFi in school), but it's something. They seemed appreciative, and it only takes about 5 minutes to create a code, so I hope to keep doing it. 

Wednesday, December 5, 2012

Day 62: Distance Formula

The Distance Formula is one of the biggest reasons I created this new curriculum/sequence of content. In nearly every current Geometry book I've seen, the Distance Formula is lumped in with a bunch of random gibberish in Chapter 1. It doesn't make any sense to a student, it's just a thing they're expected to memorize and know how to use. It's antithetical to modeling for sure, but it's just bad teaching. "Don't question why it works, just trust that it does." Really? And we're not even going to cover Pythagorean Theorem until Ch. 9 sometime in March? REALLY?

How about this? Instead of throwing random formulas at students to see what sticks, let's actually ensure the students understand how the formula came to be. Better yet, let's make the students derive the formula themselves!

So, I built off the activity that we used to discover the Pythagorean Theorem. Every student has a right triangle drawn on graph paper, and I asked them to draw a set of x-y axes around the triangle. Then, they need to identify the coordinates of the vertices (labeled A, B, & C). 

We had a brief tangent to discuss the difference in labeling sides vs. labeling angles (lower case vs. upper case respectively), and then I asked how to determine the length of sides a & b? Most students can glean that they can simply count the grid spaces since the sides are horizontal & vertical. At this point, they can (hopefully) recognize that strategy doesn't work on c since it's on an angle. but we know Pythagorean Theorem, so they can use that. But what if we don't have a triangle? What if *gasp* we don't even have a graph?

So I lead them to the idea that the length of side a could be written as (x2-x1) and similarly, side b could be written as (y2-y1). Then, we can substitute into P.T. and while it may look atrocious, the end result is a model that can determine the distance between ANY TWO POINTS. 

How is this not a better way?

Tuesday, December 4, 2012

Day 61: Examples

We had our first Skills Review quiz of the year, covering slope & linear equations. Even though it had been mentioned as early as last week, it's written on the board, posted on the website, and was mentioned explicitly in class yesterday, I still had a few students express their shock as such a discovery. After the quiz, one student responded with "can I be excused from this? I've forgotten all of this material." Thanks for proving my point. 

I spent some time addressing the continuing confusion between segments & angles. One of the most frustrating things I see in Geometry is: "Determine the measure of angle DGO." with a student response that reads "6 inches." Answers like that was actually one of the main reasons I wrote this new curriculum and spent so much time (all of Unit 2, about 4 weeks) going over segments & angles. And yet, the confusion persists. In some cases, the issue is quite literally that students do not know/remember the meaning of the words "measure" and "calculate" (vs. "determine"). 

We wrapped with an example working through the Pythagorean Theorem and I handed out U4 WS2, but we haven't covered the distance formula yet. 

Uploaded files to Google Docs

I've uploaded the files (overview and worksheets) to Units 1, 2, & 3 into Google Docs. They are public, but un-editable. For obvious reasons, I did not upload any assessments I've created (if you'd like to see those, please send me an email privately). 

Unit 1 - Linear Equations

Unit 2 - Segments & Angles
Unit 3 - Squares
The Google Doc converter might err when uploading .docx, If something is unusable, let me know and I'll try to upload a fixed version. 

Monday, December 3, 2012

Day 60: Quiz & Discovery of Pythagorean Thm.

I had told my students we'd go over WS1 today and then take a quiz, but we spent 3 days on that friggin' worksheet, so I opted for just the quiz instead. I know, I'm a terrible person. Besides, with SBG, the quiz doesn't have major implications. Instead, it only serves as a benchmark indicator for where they're at today and the later assessments will have more weight to overwrite the quiz results. 

As for the Pythagorean Thm. discovery, I'm torn with how best to implement it. I've done it as a teacher-led demo, but students tend to tune out when they hear "you don't have to write this down, just follow along." I tried it with my 2nd hour class today as a student-led investigation, but most ended up with triangles too big for the graph paper I gave out, so they couldn't create the squares off each side. I don't like the idea of putting on too many restrictions in the directions, because then it feels specific to the one triangle in front of a student, not a general model for all right triangles. I might try giving out a full sheet of graph paper (I generally cut the sheets in half to save paper) and see how that goes. 

As it stands, I'm confident we'll finish Unit 4 before the holiday break but we only have 9 days of school after break before finals - I'm not sure if that will be enough time for all of Unit 5, but we'll see. Overall, I'm happy with the pace of the class - I had anticipated the class being 10 or 11 Units, and we're almost halfway through.