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...

## Friday, December 21, 2012

## 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.

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.

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.

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.

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.

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.

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.

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.

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.

Viola.

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.

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.

Viola.

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.

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?

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.

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 1 - Linear Equations

Unit 2 - Segments & Angles

## 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.

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.

## Friday, November 30, 2012

### Day 59: Discussion

Wanted to spend the hour going over U4 WS1, but so few students have bought in to the idea of struggle and work outside of the classroom, that we had to spend the hour simply giving class time for students to do the work.

Again, the prevailing mindset is that if something looks new/different, that must mean that they (the students) are not capable of completing the problem. They absolutely refuse to even attempt to forge connections from notes & previous examples in order to tackle a new challenge. And of course, they then become very upset with me that I won't simply show them how to do it.

Couple of things I noticed from student questions:

Again, the prevailing mindset is that if something looks new/different, that must mean that they (the students) are not capable of completing the problem. They absolutely refuse to even attempt to forge connections from notes & previous examples in order to tackle a new challenge. And of course, they then become very upset with me that I won't simply show them how to do it.

Couple of things I noticed from student questions:

- Some students are confusing area with distance. When given a picture of a triangle drawn on a grid and asked to determine the length of a side, they'll count half-boxes at the edges. Don't think I've ever seen that one before (but I've never relied so heavily on examples from Geometer's Sketchpad either). I might need to pay better attention to vocab in the future ('spacings' instead of 'boxes').
- The idea of a model is so foreign to most students that they're unsure of how to transfer it to new situations. We created the model for diagonals & side length of squares in Unit 3 for the express purpose of applying said model in this unit with isosceles right triangles. Students have the model written down in their notes and remember using it, but are unsure of how to apply it to a situation that isn't 100% exactly the same (since these are triangles, not squares).
- I really hate how much calculators are foisted upon students these days. They have very little of what I call "number sense" because the magic little doohickey always solves the problem for them. Case in point: the square root of 2. It's almost as though students see the radical sign as some sort of variable that MUST be dealt with. I've been trying to hammer home the idea of precise vs. approximate (as a result of rounding) and why it's totally OK (and even preferred) to leave answers in radical form. My non-honors class literally lost their stuff when I tried to explain the difference.

I'm guessing Unit 4 will take up most of my time leading up the holiday break, but I don't want to start a new unit before 2 weeks off. We only have 9 school days after the break before the end of the semester, so I'll need to work out if/how much of unit 5 we'll get to before the final exam.

## Thursday, November 29, 2012

### Day 58: Online Practice

Spent the day in the computer lab with the students working with the Carnegie Learning software. Did this mostly because my 5th hour was a day behind the other classes, so I gave them the class period to catch up while the others worked online.

I created a custom sequence of the Geometry units in Carnegie to better match the sequence I had written for my class. The only problem is that because I don't "force" the students to complete Carnegie by given deadlines, they don't work very hard. And now, 3 months into the school year, most students are still in the first unit of Carnegie, when the class material is aligned to the fifth unit of Carnegie. When asked, I told one student that Carnegie is just practice. If you choose not to practice, you're choosing to not be at your best.

We're going to start having skills review quizzes next week, so maybe the poor pacing of the online practice will actually come in handy.

I created a custom sequence of the Geometry units in Carnegie to better match the sequence I had written for my class. The only problem is that because I don't "force" the students to complete Carnegie by given deadlines, they don't work very hard. And now, 3 months into the school year, most students are still in the first unit of Carnegie, when the class material is aligned to the fifth unit of Carnegie. When asked, I told one student that Carnegie is just practice. If you choose not to practice, you're choosing to not be at your best.

We're going to start having skills review quizzes next week, so maybe the poor pacing of the online practice will actually come in handy.

## Wednesday, November 28, 2012

### Day 57: Area & Perimeter

I love it when a plan comes together.

We started class with a recap of the challenge from yesterday. I recorded the legs and estimated areas from a selection of students, and then solicited ideas for how to relate the three quantities. Because of the estimation with partial box counting, very few triangles worked perfectly, but they worked.

Of course, most students were stumped after trying addition, seeing that it didn't work, and not knowing what to do next. A couple remembered the correct equation before we even got to this point, but a select few actually stumbled across the connection and didn't make the A=1/2 b*h connection until we were done.

It's fun getting to see that light bulb go off; even more so when it's something "simple" that they saw in middle school and didn't recognize. It's as if most students don't even realize that there IS a back story to where the content comes from. They're so used to being spoon fed a bunch of stuff that they're expected to memorize and regurgitate, that they never stopped to think about what's "behind the curtain" so to speak.

And that's kinda my whole point in doing this, so I'm calling today a win.

## Tuesday, November 27, 2012

### Day 56: Unit 4 (Right Triangles)

Ok, it's time to actually start covering material that might be new to students. I started by stressing the purpose of my curriculum because many of the students have noticed that I'm doing everything differently than the other geometry teachers in school. We're doing things differently so that everything is built from stuff we've learned. Unit 3 was all about squares, so we're going to start Unit 4 from there and go forward.

Major point to stress in the "review" is the diagonal to side length ratio. Looking over the U3 Assessment, many students still count diagonal spaces on a graph in the exact same fashion as horizontal & vertical spacings.

So with a diagonal drawn in a square, we very obviously have two triangles. Coming from a square, we know the sides are equal and there are 90 degree angles in the "corners." Intro the vocab that 2 equal sides --> isosceles, so we can call this triangle an "Isosceles Right Triangle" (IRT).

Also coming from the unit on squares, we know that the diagonal acts as an angle bisector. Half of 90 is 45 degrees, so each acute angle in the IRT must be 45 deg. 90+45+45 = 180 which conforms with expectations since the four 90 deg angles in the square summed to 360, and we cut the square in half.

I'm trying to keep everything very explicit as we build the unit from the ground up. I make a point to stress that we only know the angles of an IRT sum to 180 because we only knew the 4 angles of a square sum to 360. We can branch out later and verify that the idea holds true for all triangles.

I then tasked the class with a challenge. Draw a right triangle that is NOT isosceles on graph paper, keeping the sides to integral lengths. Determine the area by counting boxes (remembering that we defined area in U3 with squares) and try to find a connection between the lengths of the legs of the right triangle and the area.

I really thought kids were going to just start with the assumption that A = 1/2 b*h and claim they were done, but they actually didn't see the connection, presumably because I never used the terms 'base' and 'height.'

We'll recap the challenge at the start of class tomorrow and start work on U4 WS1: Area & Perimeter of Right Triangles.

Major point to stress in the "review" is the diagonal to side length ratio. Looking over the U3 Assessment, many students still count diagonal spaces on a graph in the exact same fashion as horizontal & vertical spacings.

So with a diagonal drawn in a square, we very obviously have two triangles. Coming from a square, we know the sides are equal and there are 90 degree angles in the "corners." Intro the vocab that 2 equal sides --> isosceles, so we can call this triangle an "Isosceles Right Triangle" (IRT).

Also coming from the unit on squares, we know that the diagonal acts as an angle bisector. Half of 90 is 45 degrees, so each acute angle in the IRT must be 45 deg. 90+45+45 = 180 which conforms with expectations since the four 90 deg angles in the square summed to 360, and we cut the square in half.

I'm trying to keep everything very explicit as we build the unit from the ground up. I make a point to stress that we only know the angles of an IRT sum to 180 because we only knew the 4 angles of a square sum to 360. We can branch out later and verify that the idea holds true for all triangles.

I then tasked the class with a challenge. Draw a right triangle that is NOT isosceles on graph paper, keeping the sides to integral lengths. Determine the area by counting boxes (remembering that we defined area in U3 with squares) and try to find a connection between the lengths of the legs of the right triangle and the area.

I really thought kids were going to just start with the assumption that A = 1/2 b*h and claim they were done, but they actually didn't see the connection, presumably because I never used the terms 'base' and 'height.'

We'll recap the challenge at the start of class tomorrow and start work on U4 WS1: Area & Perimeter of Right Triangles.

## Monday, November 26, 2012

### Day 55: Standardized Testing

WAAAAY back on Day 3, I avoided posting a long rant about the standardized testing that took up an entire class period. I'm going to try and avoid the full rant, but rather offer an explanation of what's going on.

I'm required to give a Quarterly Common Assessment to the core classes I teach (physics and geometry). This means I give the

For physics, I use the Force Concept Inventory which is far better than any assessment I could write. But for geometry, I volunteered to create a diagnostic / common assessment that all the math teachers could use. Right now it's a 45 question test with 15 questions covering pre-algebra, 15 covering algebra, and 15 covering geometry. The original idea was to give a diagnostic test to students who enter our school at random times throughout the year so that we could place them in the appropriate math class. I co-opted the idea for the common assessment for "simplicity."

To recap: I gave a 30 question test (no geometry) in the first week of school. Today I gave those

There's a debate here about the merits of "growth" when measured like this, and I think my opinion shines through with the emphasis I added. 5 class days lost to (IMHO) invalid data.

I'm required to give a Quarterly Common Assessment to the core classes I teach (physics and geometry). This means I give the

**at least 5 times throughout the year to my classes so that the school can track growth.**__exact same test__For physics, I use the Force Concept Inventory which is far better than any assessment I could write. But for geometry, I volunteered to create a diagnostic / common assessment that all the math teachers could use. Right now it's a 45 question test with 15 questions covering pre-algebra, 15 covering algebra, and 15 covering geometry. The original idea was to give a diagnostic test to students who enter our school at random times throughout the year so that we could place them in the appropriate math class. I co-opted the idea for the common assessment for "simplicity."

To recap: I gave a 30 question test (no geometry) in the first week of school. Today I gave those

**with an additional 15 tacked on. I will give that same 45 question test again at the start of the 2nd semester (don't want to waste time before finals), again at the start of the 3rd marking period (early April I think) and again before school lets out in June.**__exact same 30 questions____.__*The exact same test*There's a debate here about the merits of "growth" when measured like this, and I think my opinion shines through with the emphasis I added. 5 class days lost to (IMHO) invalid data.

## Tuesday, November 20, 2012

### Day 54: Assessment

Unit 3 Assessment.

My prediction: students will struggle with the length of diagonals in squares, creating images with symmetry, and solving for properties of squares when NOT initially given the side length.

My goal for the Thanksgiving break is to create the U3 Reassessments and the structure of Unit 4 so I can be ready for the rest of the year leading up to the X-mas break.

My prediction: students will struggle with the length of diagonals in squares, creating images with symmetry, and solving for properties of squares when NOT initially given the side length.

My goal for the Thanksgiving break is to create the U3 Reassessments and the structure of Unit 4 so I can be ready for the rest of the year leading up to the X-mas break.

## Monday, November 19, 2012

### Day 53: Review

Task #1: finish going over U3 WS4 from last week.

Task #2: Give students time to create a note sheet for use on the Unit 3 Assessment tomorrow.

With Task #1, I tried to make the deeper connections that I had hoped for when we started this unit. Through some guided discovery, the class was able to see that lines of symmetry have to be angle bisectors on both ends, perpendicular bisectors on both ends, or one of each on each end. That's something at least.

With Task #2, I stress that even though students are allowed note sheets, less than half of students actually use them on the tests. If there's a greater indicator of what students are willing to do to be successful, I haven't found it yet. Of course, some students who don't use note sheets are happy with grades in the C range, which is another issue altogether.

I'm ok with note sheets as I'm confident that I can write a test that requires a deep understanding of the content to be successful. Were I giving a simple true/false (or even multiple guess), then yeah, note sheets might be too much of an advantage. But that's why students hate my tests (and in some cases, by extension, me); because they can't be aced just by memorizing a bunch of facts and properties. Heck, they generally can't receive good grades by even memorizing a procedure (like, always add two numbers and divide by 2 to get midpoints). Of course, this is exactly my goal, so I'm ok with it.

Task #2: Give students time to create a note sheet for use on the Unit 3 Assessment tomorrow.

With Task #1, I tried to make the deeper connections that I had hoped for when we started this unit. Through some guided discovery, the class was able to see that lines of symmetry have to be angle bisectors on both ends, perpendicular bisectors on both ends, or one of each on each end. That's something at least.

With Task #2, I stress that even though students are allowed note sheets, less than half of students actually use them on the tests. If there's a greater indicator of what students are willing to do to be successful, I haven't found it yet. Of course, some students who don't use note sheets are happy with grades in the C range, which is another issue altogether.

I'm ok with note sheets as I'm confident that I can write a test that requires a deep understanding of the content to be successful. Were I giving a simple true/false (or even multiple guess), then yeah, note sheets might be too much of an advantage. But that's why students hate my tests (and in some cases, by extension, me); because they can't be aced just by memorizing a bunch of facts and properties. Heck, they generally can't receive good grades by even memorizing a procedure (like, always add two numbers and divide by 2 to get midpoints). Of course, this is exactly my goal, so I'm ok with it.

## Friday, November 16, 2012

### Day 52: Origami

Easy day. Work through some origami creations while identifying types of symmetry in things that aren't everyday common shapes like squares.

If you'd like the link to the website I used (downloadable instructions and animations) as well as any worksheets I've created, check out www.mrfuller.net and search the past calendar under Geometry.

If you'd like the link to the website I used (downloadable instructions and animations) as well as any worksheets I've created, check out www.mrfuller.net and search the past calendar under Geometry.

## Thursday, November 15, 2012

### Day 51: Discussion

Very unproductive day. Goal was to work through a representative sample of the worksheet problems so that we would see all the connections there were to see. Unfortunately, so many students had completed so little of the assignment, that we weren't ready to go over it. By the end of the class, we'd only managed to get through 2 or 3 problems.

I'm still very happy with my progress under a new curriculum. I'm very excited to see how the new sequence unfolds based on how structured the content has been thus far. However, I'm more than a little concerned that as the course content builds, students that haven't been working to their full potential are going to find themselves simply unable to participate in class (if you can't identify the slope of a line and you can't solve for the midpoint of a segment, how much luck are you going to have discovering general properties of parallelograms?).

I think the biggest lesson I'm learning this year is that the method of delivery and the sequence of the content are not the deciding factors in whether or not a student is successful. Do whatever you want - "flip" the classroom, lecture, model, whatever. If the students have no appreciation for the importance of education and are not willing to struggle to reach a desired goal, no fancy new approach will work.

Sorry, I try to avoid being negative, but it's the start of the 2nd marking period and I've been reflective this week.

I'm still very happy with my progress under a new curriculum. I'm very excited to see how the new sequence unfolds based on how structured the content has been thus far. However, I'm more than a little concerned that as the course content builds, students that haven't been working to their full potential are going to find themselves simply unable to participate in class (if you can't identify the slope of a line and you can't solve for the midpoint of a segment, how much luck are you going to have discovering general properties of parallelograms?).

I think the biggest lesson I'm learning this year is that the method of delivery and the sequence of the content are not the deciding factors in whether or not a student is successful. Do whatever you want - "flip" the classroom, lecture, model, whatever. If the students have no appreciation for the importance of education and are not willing to struggle to reach a desired goal, no fancy new approach will work.

Sorry, I try to avoid being negative, but it's the start of the 2nd marking period and I've been reflective this week.

## Wednesday, November 14, 2012

### Day 50: Symmetry in Practice

Very loosely structured day. Spent the first half just trying to firm up our ideas about symmetry from the intro day yesterday. Settled on a basic definition that symmetry results from any action that creates a result indistinguishable from the original image. I tried to word it like this to avoid pigeon holing us into thinking that symmetry only dealt with folds and lines, because rotations are just as important.

After the recap, I had students work on Unit 3 Worksheet #4. Some students work, but many did not. Some would work through the first 4 problems, definitely the easiest problems they'll ever see. Then they'd quit, either thinking that the content is so easy that they don't need to bother doing any more, or being so completely stymied by the transfer of skills to connect with midpoint and angle bisectors that they lose all hope.

I make a point of talking about each of these pitfalls and stressing the need to work through struggle and try to practice every style of problem to make sure that you understand all aspects of the content, but my pleas often fall on deaf ears.

After the recap, I had students work on Unit 3 Worksheet #4. Some students work, but many did not. Some would work through the first 4 problems, definitely the easiest problems they'll ever see. Then they'd quit, either thinking that the content is so easy that they don't need to bother doing any more, or being so completely stymied by the transfer of skills to connect with midpoint and angle bisectors that they lose all hope.

I make a point of talking about each of these pitfalls and stressing the need to work through struggle and try to practice every style of problem to make sure that you understand all aspects of the content, but my pleas often fall on deaf ears.

## Tuesday, November 13, 2012

### Day 49: Symmetry

I actually spent a fair amount of time prepping myself for this unit because I had high hopes of delving very deeply into symmetry. I had never really thought about the underpinnings of symmetry and how you can discover properties that can be proven geometrically. I found some lessons online that got into perpendicular and angle bisectors and read them through enough times that I felt confident. I planned out the week to build to cementing these ideas and prepping for the Unit 3 test before Thanksgiving. And then, I went through the intro day.

Nothing bad happened during the intro day, in fact, I would probably label it a success. I probed for students preconceptions of symmetry, jotted down the class ideas and worked through some basic examples of line and rotational symmetry. But at the foundation, I realized that what I had hoped to accomplish was way beyond what most of my students were capable of getting through, at least on the order of a few days.

So, I decided to re-plan my week to focus on making connections between symmetry and everything we'd talked about in Unit 3 to date, bisectors and midpoint to name a few ideas. Let's hope that works.

Nothing bad happened during the intro day, in fact, I would probably label it a success. I probed for students preconceptions of symmetry, jotted down the class ideas and worked through some basic examples of line and rotational symmetry. But at the foundation, I realized that what I had hoped to accomplish was way beyond what most of my students were capable of getting through, at least on the order of a few days.

So, I decided to re-plan my week to focus on making connections between symmetry and everything we'd talked about in Unit 3 to date, bisectors and midpoint to name a few ideas. Let's hope that works.

## Monday, November 12, 2012

### Day 48: Recap of Unit 3 to date

Between the students who never take notes (and somehow still find ways to get upset with me when I won't answer straightforward questions such as "what's an angle bisector?") and the students who miss class (and never both to check the website or ask a friend for the material that they missed), I used today to walk through the entirety of Unit 3 to date, writing out 5 pages of review notes on the tablet (so the file could be uploaded to the class website to never be heard from again).

Unit 3: Squares (so far)

Unit 3: Squares (so far)

- Definition
- Properties (Area, Perimeter)
- Diagonals (relation to side length, perpendicular bisectors, angle bisectors)
- Midpoint formula
- Using bisectors outside of squares

And now we're ready for symmetry.

## Friday, November 9, 2012

### Day 47: Online Practice

Back to the computer lab for more practice online with the Carnegie Learning Software.

I stress to the students that most are stuck in Unit 1 online (linear equations) and that I set up the software to run alongside the class which is in the equivalent of Unit 5 online. Of course, they want to be skipped ahead to catch up, but I refuse pointing out that they're behind because they don't use class time efficiently and because they're not practicing at home, which is one of the major reasons we have the online tutoring software.

Surprisingly, my regular geometry class was actually the best behaved and most on-task in the lab. I might readjust how I spend time with that class and get them in the lab more often.

I stress to the students that most are stuck in Unit 1 online (linear equations) and that I set up the software to run alongside the class which is in the equivalent of Unit 5 online. Of course, they want to be skipped ahead to catch up, but I refuse pointing out that they're behind because they don't use class time efficiently and because they're not practicing at home, which is one of the major reasons we have the online tutoring software.

Surprisingly, my regular geometry class was actually the best behaved and most on-task in the lab. I might readjust how I spend time with that class and get them in the lab more often.

## Thursday, November 8, 2012

### Day 46: Discussion of Angle Bisectors

We spent the day going over U3 WS3 on Angle Bisectors. I have about 15 minutes of class for students to work, and then I led Q & A for no more than the setups of some of the problems. We did not do anything like a whiteboard session due to time constraints.

I was very surprised at how little transfer students are willing to even attempt. They can recite the definition of an angle bisector, but cannot use that information to set up an problem involving algebraic expressions for 2 angles and solve for the unknown. They claim "I don't get it," but as I walk through the questioning techniques with them ("what is an angle bisector,?" "what does that mean for the angles,?" "how would that be set up in an equation?") it's clear that they do "get it," at least at a very basic level, they're just either completely incapable of, or simply refuse to, attempt to apply things they know to new situations.

I was very surprised at how little transfer students are willing to even attempt. They can recite the definition of an angle bisector, but cannot use that information to set up an problem involving algebraic expressions for 2 angles and solve for the unknown. They claim "I don't get it," but as I walk through the questioning techniques with them ("what is an angle bisector,?" "what does that mean for the angles,?" "how would that be set up in an equation?") it's clear that they do "get it," at least at a very basic level, they're just either completely incapable of, or simply refuse to, attempt to apply things they know to new situations.

## Wednesday, November 7, 2012

### Day 45: Geometric Constructions

Every year I subject myself to a lesson along these lines, every year I get the same result, and every year I keep coming back for more. Didn't Einstein have something to say about that?

Anyway, I finally decided to bust out the compasses, but only for my honors classes. I really think the compass is an amazing piece of technology and truly underscores the idea of congruence without equivalence, but those ideas are incredibly abstract and very difficult for my students to grasp, so the lesson almost never lands the way I hope it would.

I started by showing the different varieties of compass and emphasizing that they all do the same thing; create a set of points that are all equidistant from a common point. Using just a compass and a straightedge, the Greeks were able to discover most of classical geometry in the absence of a decimal numbering system. Pretty awesome, right? Well, not to 15 year olds anyway.

I demo the basic construction of copying a line segment using a document camera. At this point, I haven't given out the compasses, because all heck will break loose when I do (I've learned at least that much in my years trying to do this). I have a pre-printed packet of 4 basic constructions that I hand out along with the compasses, and then we walk through creating a parallel line through a point while showing a flash animation on the screen.

No joke, it actually worked a little this year. No idea why, but the results were more along the lines of what I hope for. What was lacking was the appreciation aspect of it. Kids don't really care about why a compass is useful or what we can do with it. Nobody wanted to explore the rest of the constructions at the end of the period or see what else they could do.

I hope I can devote another class period to at least the angle bisector construction, but it's hard to justify spending so much time on a skill that isn't really going to enhance their understanding of the content.

Anyway, I finally decided to bust out the compasses, but only for my honors classes. I really think the compass is an amazing piece of technology and truly underscores the idea of congruence without equivalence, but those ideas are incredibly abstract and very difficult for my students to grasp, so the lesson almost never lands the way I hope it would.

I started by showing the different varieties of compass and emphasizing that they all do the same thing; create a set of points that are all equidistant from a common point. Using just a compass and a straightedge, the Greeks were able to discover most of classical geometry in the absence of a decimal numbering system. Pretty awesome, right? Well, not to 15 year olds anyway.

I demo the basic construction of copying a line segment using a document camera. At this point, I haven't given out the compasses, because all heck will break loose when I do (I've learned at least that much in my years trying to do this). I have a pre-printed packet of 4 basic constructions that I hand out along with the compasses, and then we walk through creating a parallel line through a point while showing a flash animation on the screen.

No joke, it actually worked a little this year. No idea why, but the results were more along the lines of what I hope for. What was lacking was the appreciation aspect of it. Kids don't really care about why a compass is useful or what we can do with it. Nobody wanted to explore the rest of the constructions at the end of the period or see what else they could do.

I hope I can devote another class period to at least the angle bisector construction, but it's hard to justify spending so much time on a skill that isn't really going to enhance their understanding of the content.

## Monday, November 5, 2012

### Day 44: Angle Bisectors

I spent a lot of time thinking about this lesson. I really wanted to stay true to my goal of making the content sequential and discoverable, but I wasn't sure how to arrive at the idea of an angle bisector without so much else in the course. I finally settled on a plan that I was happy with, but it was more convoluted than the students could handle and I'm not sure it will end up being worth the hassle.

Here's what I did:

- Have students draw square ABCD that is at least 5 x 5 on graph paper.
- Draw diagonal BD
- Place four points anywhere along BD and label them F,G,H, and I (in order)
- Measure the distance from each point to the opposite corners A & C (so there are 8 lengths total)
- Make conclusion

The goal here was to kind of back in to the notion of angle bisectors, but all this part of the activity does is conclude that points along the diagonal of a square are equidistant from the corners. How can you show that the angles are in fact equal? That's where I got stuck.

So I very briefly addressed the idea of triangle congruence using SSS. If the sides are all the same, then the triangles have to be the same, and if the triangles are the same, then the angles have to be the same. But in actuality, that only deals with corresponding angles - I'm pretty sure you'd need to dig deeper, into the Triangle Sum Theorem to prove that each angle was 45 deg and bisected from the corner of the square.

No school tomorrow (Election Day), so I'll try to think of some way to tie it all together so we can close out Unit 3 with angle bisector construction (pray for me) and symmetry.

## Friday, November 2, 2012

### Day 43: Discuss U3 WS2

Goal was to go over the worksheet assigned yesterday. Surprisingly, most students attempted at least the easiest problems (finding the midpoint from the graph and when given two endpoints).

I've settled into somewhat of a groove in which I handle worksheet discussions different in the honors classes vs. the regular class. In honors, I have each group whiteboard ~3 problems and utilize the "whiteboard parade" method in which the kids have a chance to walk around the room and check the other boards and ask questions. There is almost too many kids and too many simplistic problems to justify the "one WB presentation at a time" method.

For the regular class, I basically can't relinquish control of the class like that. I need to maintain order and that means that we actually avoid group work of that nature. Instead, I passed my tablet PC around the room and had randomly picked students solve problems on the tablet while it projected to the screen. They like using the tablet, but there's still the issue of what students choose to do while waiting for someone to write their solution down.

I've settled into somewhat of a groove in which I handle worksheet discussions different in the honors classes vs. the regular class. In honors, I have each group whiteboard ~3 problems and utilize the "whiteboard parade" method in which the kids have a chance to walk around the room and check the other boards and ask questions. There is almost too many kids and too many simplistic problems to justify the "one WB presentation at a time" method.

For the regular class, I basically can't relinquish control of the class like that. I need to maintain order and that means that we actually avoid group work of that nature. Instead, I passed my tablet PC around the room and had randomly picked students solve problems on the tablet while it projected to the screen. They like using the tablet, but there's still the issue of what students choose to do while waiting for someone to write their solution down.

## Thursday, November 1, 2012

### Day 42: Midpoint (cont.)

Agenda item #1: U3 Quiz 1 on the basic properties of squares.

Agenda item #2: Recap the Midpoint discovery w/GSP from yesterday.

Agenda item #3: Start work on U3 WS2 - Midpoint

As mentioned yesterday, it was incredibly surprising at how little math sense students have. They can visually point out the middle of a segment, and they appear to understand the concept of average, but are generally unable to combine the two ideas. My theory is that they have only experienced math as a set of seemingly unrelated procedures one follows to arrived at a previously known conclusion.

I stressed the idea of the midpoint formula being a model that we've built that *always* works. It's not a procedure in the sense that "you should always take two numbers and add them, then divide by 2," because that procedure falls apart if you're solving for the other endpoint when given the midpoint. Instead, if you use the model as derived and substitute in the knowns and solve for the unknown, you can't go wrong. Judging by their quiz results, they bought into that methodology, but the algebra skills are still fairly weak.

Agenda item #2: Recap the Midpoint discovery w/GSP from yesterday.

Agenda item #3: Start work on U3 WS2 - Midpoint

As mentioned yesterday, it was incredibly surprising at how little math sense students have. They can visually point out the middle of a segment, and they appear to understand the concept of average, but are generally unable to combine the two ideas. My theory is that they have only experienced math as a set of seemingly unrelated procedures one follows to arrived at a previously known conclusion.

I stressed the idea of the midpoint formula being a model that we've built that *always* works. It's not a procedure in the sense that "you should always take two numbers and add them, then divide by 2," because that procedure falls apart if you're solving for the other endpoint when given the midpoint. Instead, if you use the model as derived and substitute in the knowns and solve for the unknown, you can't go wrong. Judging by their quiz results, they bought into that methodology, but the algebra skills are still fairly weak.

## Wednesday, October 31, 2012

### Day 41: Midpoint Discovery

The original plan was to have students continue on with the same graph of their square that we've been working with for a week now and determine the relationship between the coordinates of the endpoints of the diagonals and the coordinates of the midpoint. Students being what they are (teenagers), so many of them have lost their graphs or made such a mess of them that to continue working with the same data would be a fruitless venture.

Instead, I took the classes to the computer lab to do the same activity using Geometer's Sketchpad. As much as I love GSP, I don't love trying to teach students how to use it. It's an incredibly powerful program, which means there are a TON of little things you need to know how to deal with if you want to get consistent results. Sadly, even with the most direct instructions, my students see the program as an obstacle and will often quit at the first sign of struggle.

In the end, most students got the construction created successfully, but struggled with the discovery aspect of the activity. Most of them really fight the idea of being creative in math, they are so accustomed to being spoon fed a procedure which must be learned and memorized to be regurgitated later, that the freedom to create something new is a foreign concept in this specific environment.

Some students did arrive at the idea of "middle," but couldn't make the mathematical leap to the idea of 'average' (further reinforcing my suspicion that even students who are generally "successful" in math have little idea what they're actually doing).

I will adapt the instructions to include more scaffolding, specifically the suggestion that the x coordinates should be considered separate from the y coordinates.

Instead, I took the classes to the computer lab to do the same activity using Geometer's Sketchpad. As much as I love GSP, I don't love trying to teach students how to use it. It's an incredibly powerful program, which means there are a TON of little things you need to know how to deal with if you want to get consistent results. Sadly, even with the most direct instructions, my students see the program as an obstacle and will often quit at the first sign of struggle.

In the end, most students got the construction created successfully, but struggled with the discovery aspect of the activity. Most of them really fight the idea of being creative in math, they are so accustomed to being spoon fed a procedure which must be learned and memorized to be regurgitated later, that the freedom to create something new is a foreign concept in this specific environment.

Some students did arrive at the idea of "middle," but couldn't make the mathematical leap to the idea of 'average' (further reinforcing my suspicion that even students who are generally "successful" in math have little idea what they're actually doing).

I will adapt the instructions to include more scaffolding, specifically the suggestion that the x coordinates should be considered separate from the y coordinates.

## Tuesday, October 30, 2012

### Day 40: Practice and Discussion

Students worked on U3 WS1: Properties of Squares. This was something of a catch-up day so that all the classes could get back on the same schedule.

This was a worksheet that I created over the summer when I was still in the planning stages of this whole experiment and unfortunately, I did not look back over it between the development and the deployment stages of the unit. What that left me with was a worksheet that was essentially a rehash of the discovery activities we'd been working on for the past 4 days. Oh well.

This was a worksheet that I created over the summer when I was still in the planning stages of this whole experiment and unfortunately, I did not look back over it between the development and the deployment stages of the unit. What that left me with was a worksheet that was essentially a rehash of the discovery activities we'd been working on for the past 4 days. Oh well.

## Monday, October 29, 2012

### Day 39: Practice with Squares

Today was spent recapping the conclusions we made last week and giving students time to practice with U3 WS1 - Basic Properties of Squares.

The only "new" content discussed today was the relationship between the length of a square's diagonal and its side length.

Student minds always surprise me - when I asked the class to try determine the length of a diagonal by counting boxes (like they'd done with the sides), I'd expected them to be stymied. "But Mr. Fuller, those boxes (corner to corner) aren't the same as the horizontal/vertical ones" I imagined them saying. Nope - they'd declare an answer - 5 (for example) - which always happened to be the same as the side length for the particular square we were looking at. And then it hit me: Because it's a square, the diagonals have a slope of 1 (or -1) and they were simply counting the spaces on the grid that the diagonal passed through, just like we did for the sides. Huh.

So let's run with it: Projecting at image of our graph onto the screen (using Geometer's Sketchpad, so it looks precise), I can stand at the screen with a meter stick and have a volunteer who's close by read the length of a side. "57 cm" they'd say. Ok, now I'll pivot the meter stick at the vertex until it aligns with the diagonal. VERY clear that the 57 cm of the meter stick that's exposed doesn't reach the far corner of the square. Conclusion: Diagonal lengths are longer than horizontal/vertical lengths. But what's the relationship?

At this point, my original idea (back in July when I was still optimistic) was to have students "discover" the scale factor of sqrt(2), but I was running short on time and we needed to get going. Instead I worked at the computer and took suggestions (using GSP) for what mathematical operation might connect side length to diagonal length. This is why GSP is awesome - I can perform a calculation and leave the result on the screen while I alter the square to see if the result changes. I couldn't ask for a better visual for testing a mathematical hypothesis.

Anyway, we try adding the side length to the diagonal and see if the sum is a constant. Nope. Subtraction? Nada. Multiplication? Still nothing. Division? Jackpot. Out pops 1.41 as this unshakable constant.

Here's where I finally broke down and became "lecturer" for a moment. I asked: "If that number had hypothetically come out as 3.14, what would you think?" Students immediately jumped on Pi. Awesome. So I explained that there are a handful of very famous numbers in math that you'll start to recognize as you spend more time with math. So I lead them to the idea of sqrt(2) and thought we were done.

Nope. The number of students that have NO IDEA how to work with exponents and radicals in a 10th grade geometry (honors or regular - it made no difference) class was astounding. Worse yet - there was no shame on the part of the students for not remembering the concept. Instead, they got frustrated with me that I wouldn't simply tell them the answer, insisting instead that they should be able to figure it out or ask a neighbor for help.

Still a solid day.

The only "new" content discussed today was the relationship between the length of a square's diagonal and its side length.

Student minds always surprise me - when I asked the class to try determine the length of a diagonal by counting boxes (like they'd done with the sides), I'd expected them to be stymied. "But Mr. Fuller, those boxes (corner to corner) aren't the same as the horizontal/vertical ones" I imagined them saying. Nope - they'd declare an answer - 5 (for example) - which always happened to be the same as the side length for the particular square we were looking at. And then it hit me: Because it's a square, the diagonals have a slope of 1 (or -1) and they were simply counting the spaces on the grid that the diagonal passed through, just like we did for the sides. Huh.

So let's run with it: Projecting at image of our graph onto the screen (using Geometer's Sketchpad, so it looks precise), I can stand at the screen with a meter stick and have a volunteer who's close by read the length of a side. "57 cm" they'd say. Ok, now I'll pivot the meter stick at the vertex until it aligns with the diagonal. VERY clear that the 57 cm of the meter stick that's exposed doesn't reach the far corner of the square. Conclusion: Diagonal lengths are longer than horizontal/vertical lengths. But what's the relationship?

At this point, my original idea (back in July when I was still optimistic) was to have students "discover" the scale factor of sqrt(2), but I was running short on time and we needed to get going. Instead I worked at the computer and took suggestions (using GSP) for what mathematical operation might connect side length to diagonal length. This is why GSP is awesome - I can perform a calculation and leave the result on the screen while I alter the square to see if the result changes. I couldn't ask for a better visual for testing a mathematical hypothesis.

Anyway, we try adding the side length to the diagonal and see if the sum is a constant. Nope. Subtraction? Nada. Multiplication? Still nothing. Division? Jackpot. Out pops 1.41 as this unshakable constant.

Here's where I finally broke down and became "lecturer" for a moment. I asked: "If that number had hypothetically come out as 3.14, what would you think?" Students immediately jumped on Pi. Awesome. So I explained that there are a handful of very famous numbers in math that you'll start to recognize as you spend more time with math. So I lead them to the idea of sqrt(2) and thought we were done.

Nope. The number of students that have NO IDEA how to work with exponents and radicals in a 10th grade geometry (honors or regular - it made no difference) class was astounding. Worse yet - there was no shame on the part of the students for not remembering the concept. Instead, they got frustrated with me that I wouldn't simply tell them the answer, insisting instead that they should be able to figure it out or ask a neighbor for help.

Still a solid day.

## Friday, October 26, 2012

### Day 38: Discovering Diagonals

The goal seemed so simple: task the class with creating three hypotheses related to the diagonals of a square. Problem #1: Students ignored that bit about diagonals and just regurgitated previously known facts like the sides are congruent. Problem #2: Students don't make any distinction between an obvious observation and a hypothesis that will require investigation.

The idea was that with each student having their own unique square, any hypothesis they created could immediately be checked, albeit informally, with other squares for validation. The three conclusions I was looking for were: pairs of diagonals are always the same length; the point of intersection cuts the diagonals into equal pieces (haven't defined the word midpoint yet); and that the diagonals are perpendicular.

The first two are fairly easy for students to "discover" if they're willing to put forth the effort to measure the segments. The last one is often thought up, but students have no idea how to prove it. I tried to lead students to the connection between "90 degrees" and "perpendicular" and hope they make the connection back to Unit 1 and see slope as a method (these squares are on graphs for a reason).

For each class I made the brief point about why we spent these 3 days the way we did - I could have simply told everyone those formulas and conclusions on Day 1, but in the long run they wouldn't have a deep understanding of what they were actually doing. Some kids bought my explanation, but others (generally the regular Geo class) would just complain "this is stupid" and "just tell us the answers already."

I don't know how to reach kids that have such a combative attitude toward learning. All the literature I've read claims that if students can take ownership of their education, they'll change their attitude, but I have NEVER seen that work in practice. I still think this is a better way to teach, but I'm stymied about how to handle students who actively fight against the class methods.

The idea was that with each student having their own unique square, any hypothesis they created could immediately be checked, albeit informally, with other squares for validation. The three conclusions I was looking for were: pairs of diagonals are always the same length; the point of intersection cuts the diagonals into equal pieces (haven't defined the word midpoint yet); and that the diagonals are perpendicular.

The first two are fairly easy for students to "discover" if they're willing to put forth the effort to measure the segments. The last one is often thought up, but students have no idea how to prove it. I tried to lead students to the connection between "90 degrees" and "perpendicular" and hope they make the connection back to Unit 1 and see slope as a method (these squares are on graphs for a reason).

For each class I made the brief point about why we spent these 3 days the way we did - I could have simply told everyone those formulas and conclusions on Day 1, but in the long run they wouldn't have a deep understanding of what they were actually doing. Some kids bought my explanation, but others (generally the regular Geo class) would just complain "this is stupid" and "just tell us the answers already."

I don't know how to reach kids that have such a combative attitude toward learning. All the literature I've read claims that if students can take ownership of their education, they'll change their attitude, but I have NEVER seen that work in practice. I still think this is a better way to teach, but I'm stymied about how to handle students who actively fight against the class methods.

## Thursday, October 25, 2012

### Day 37: Graphing Data

Just because this is a math class, it doesn't mean that students shouldn't 1) be increasing their graphing skills and 2) learning how to analyze data to look for evidence that support a hypothesis.

So we built on yesterday's activity and collected the distance around the outside and the boxes inside every student's square. While displayed on the screen, I tasked the class with graphing "boxes inside vs. distance outside." There was a brief review of the importance of scale and which axis is which, but most students did just fine. There were some gripes that they had to plot *gasp* 25 data points, but I stressed that we needed a lot to smooth out the relationship in case anyone made mistakes with their measurements (which happened more than it should have).

In the end, I used a document camera to show the results of their hard work. The only major conclusion I was hoping they'd get to is that the relationship is NOT linear. They should have learned about parabolas and what general equation would create what we're seeing, but very few did.

I then used Excel to create my own graph of the same data. I did this so that I could easily show different potential trendlines to see which looked the best. I also displayed the correlation coefficient, but only described it as "the higher this number, the better the fit." Students quickly saw that the polynomial trendline actually fits better than the linear (it helps to have a couple of students make BIG squares to make this obvious). Then I had Excel provide the quadratic that describes the curve, and explained that our 'y' variable was really "Area" and 'x' was "Perimeter." They obviously knew these words, but seemed to be OK with my intended misdirection.

The end result is A=0.06(P^2) (the constant will vary based on the data, the three classes hovered around 0.06).

I continued on with a deeper analysis that I stressed was just FYI - students would not be responsible for doing this on their own. Essentially, I approximated 0.06 to 0.0625 so that it could be written as 1/16. Including the constant with the P^2, I rewrote the equation as (1/4 P)^2 which combines with the idea that P = 4s to create A=s^2.

Looking back, the honors classes were able to follow and at least some appreciate the derivation. This was a complete cluster in my regular class and threw everything off track for a full day. They simply don't have the attention span or the appreciation for something that's not required, as those students often struggle with connecting enhancement with greater success.

So we built on yesterday's activity and collected the distance around the outside and the boxes inside every student's square. While displayed on the screen, I tasked the class with graphing "boxes inside vs. distance outside." There was a brief review of the importance of scale and which axis is which, but most students did just fine. There were some gripes that they had to plot *gasp* 25 data points, but I stressed that we needed a lot to smooth out the relationship in case anyone made mistakes with their measurements (which happened more than it should have).

In the end, I used a document camera to show the results of their hard work. The only major conclusion I was hoping they'd get to is that the relationship is NOT linear. They should have learned about parabolas and what general equation would create what we're seeing, but very few did.

I then used Excel to create my own graph of the same data. I did this so that I could easily show different potential trendlines to see which looked the best. I also displayed the correlation coefficient, but only described it as "the higher this number, the better the fit." Students quickly saw that the polynomial trendline actually fits better than the linear (it helps to have a couple of students make BIG squares to make this obvious). Then I had Excel provide the quadratic that describes the curve, and explained that our 'y' variable was really "Area" and 'x' was "Perimeter." They obviously knew these words, but seemed to be OK with my intended misdirection.

The end result is A=0.06(P^2) (the constant will vary based on the data, the three classes hovered around 0.06).

I continued on with a deeper analysis that I stressed was just FYI - students would not be responsible for doing this on their own. Essentially, I approximated 0.06 to 0.0625 so that it could be written as 1/16. Including the constant with the P^2, I rewrote the equation as (1/4 P)^2 which combines with the idea that P = 4s to create A=s^2.

Looking back, the honors classes were able to follow and at least some appreciate the derivation. This was a complete cluster in my regular class and threw everything off track for a full day. They simply don't have the attention span or the appreciation for something that's not required, as those students often struggle with connecting enhancement with greater success.

## Wednesday, October 24, 2012

### Day 36: Intro to Squares

Ok, time to actually start implementing the modeling part of the class. My main idea for this unit was to take everything we've already covered in measuring distances and understanding angles and start out with the simplest possible shape, a square, to help students get comfortable with the idea of discovering the content on their own.

We started making sure that everyone knew the definition of a square. I know it seems simple, but I learned a long time ago to never assume students know anything they "should" know. Students had a lot of good (and scattered ideas), so to ensure consensus, I simply googled "definition of a square." Google is becoming so powerful that it didn't simply give me a link to the answer, it actually just answered the question.

Then I took suggestions for the different ways we could measure the lengths of sides. I wanted students to understand that there are multiple representations of the same idea. Students suggested the typical meters, cm, feet, inches, etc. Some suggested simply counting boxes on the graph which was nice to see. Nobody suggested using some form of coordinate subtraction , so I led the class to that idea. At this point, I kept things horizontal/vertical, so we're not getting into the actual distance formula just yet, just the ideas behind the Ruler Postulate (I do NOT use that phrase in class).

From here we transitioned into other things we could quantify from our pictures. I asked all students to count the total distance "walked" around the outside edge of their square. Some knew this was perimeter, but I explicitly avoided the word. Then we counted the number of boxes inside the square. Again, did not use the word area.

Looking back, I will need to make sure that all students are measuring side length and perimeter with the same units here. I'll probably go with boxes just to avoid the mess and confusion that comes with the rulers.

We started making sure that everyone knew the definition of a square. I know it seems simple, but I learned a long time ago to never assume students know anything they "should" know. Students had a lot of good (and scattered ideas), so to ensure consensus, I simply googled "definition of a square." Google is becoming so powerful that it didn't simply give me a link to the answer, it actually just answered the question.

Then I took suggestions for the different ways we could measure the lengths of sides. I wanted students to understand that there are multiple representations of the same idea. Students suggested the typical meters, cm, feet, inches, etc. Some suggested simply counting boxes on the graph which was nice to see. Nobody suggested using some form of coordinate subtraction , so I led the class to that idea. At this point, I kept things horizontal/vertical, so we're not getting into the actual distance formula just yet, just the ideas behind the Ruler Postulate (I do NOT use that phrase in class).

From here we transitioned into other things we could quantify from our pictures. I asked all students to count the total distance "walked" around the outside edge of their square. Some knew this was perimeter, but I explicitly avoided the word. Then we counted the number of boxes inside the square. Again, did not use the word area.

Looking back, I will need to make sure that all students are measuring side length and perimeter with the same units here. I'll probably go with boxes just to avoid the mess and confusion that comes with the rulers.

## Tuesday, October 23, 2012

### Day 35: Reflection

Unit 2 Assessments were handed back with feedback and an overall unit 2 grade (NOT a grade specifically for the assessment - something the kids are still struggling to accept).

Students had the option to review their test and ask around for clarification, or continue to work on the puzzles from yesterday. A surprising number of students not only wanted more puzzles, but harder ones as well. Cool beans.

Students had the option to review their test and ask around for clarification, or continue to work on the puzzles from yesterday. A surprising number of students not only wanted more puzzles, but harder ones as well. Cool beans.

## Monday, October 22, 2012

### Day 34: Logical Thinking

Both because I hadn't finished grading the Unit 2 Assessment yet, and because I hadn't really had time to get comfortable with how I wanted to start Unit 3 (Squares), I had students work on logic puzzles for the class period. My thinking was that instead of forcing formal proofs down students' throats, I'd rather just see them be able to thinking sequentially and justify their reasoning. Even with the angles in Unit 2, a LOT of students would label angles as "complimentary" without any reasoning for their choice.

So I made 100+ copies of some free puzzles I found online and showed them the basics. The honors classes took to them like a fish to water, while the 5th hour class actually ended with a couple of referrals because of how many students simply refuse to do anything that requires independent thought.

Overall, I'm happy I thought up a plan B that fit within my goals for the class.

So I made 100+ copies of some free puzzles I found online and showed them the basics. The honors classes took to them like a fish to water, while the 5th hour class actually ended with a couple of referrals because of how many students simply refuse to do anything that requires independent thought.

Overall, I'm happy I thought up a plan B that fit within my goals for the class.

## Friday, October 19, 2012

### Day 33: Assessment

Unit 2 Assessment. Not much to report, other than my continual amazement at student behavior during a test. Both in complete inability to sit quietly, and in the expectation that test day is the day to finally ask for help.

Because I'm a genius, I'm giving tests in 4 of my 5 classes today, so I'll have 130 exams to grade over a weekend in which I have absolutely no free time.

Because I'm a genius, I'm giving tests in 4 of my 5 classes today, so I'll have 130 exams to grade over a weekend in which I have absolutely no free time.

## Thursday, October 18, 2012

### Day 32: Review & Practice

I made a point to get U2 Quiz 4 graded and back to students ASAP in preparation for the Unit 2 Assessment. SBG has really made me refocus how I spent my energy in terms of what I grade, and what I do when I'm up against a deadline like a formative assessment. In the past, if I was pressed for time, I would just grade quizzes after the fact. But why bother? If the quiz is meant to give targeted feedback to help guide studying, it's not worth anything if students take the test before seeing their results.

We spent the day in the computer lab with students having the option to either practice with the online Carnegie software, look over their quiz results, or write their note sheet for the Unit 2 Assessment.

We spent the day in the computer lab with students having the option to either practice with the online Carnegie software, look over their quiz results, or write their note sheet for the Unit 2 Assessment.

## Wednesday, October 17, 2012

### Day 31: Assessment & Sketchpad

Hectic day. We took the 4th and final quiz (Angle Pairs) of the unit before the formative assessment. Afterwards, we rushed to the computer lab to work through an intro activity getting students familiar with Geometer's Sketchpad as we'll start using GSP as a discovery tool starting next week.

So far, the quiz results are mildly disconcerting. Generally, students can classify angle pairs, but have no idea what to do with that information. They're easily flummoxed by algebraic expressions in place of discrete angle measurements. For example, saying "Angles 1 & 2 are a linear pair. Angle 1 = 50 deg and Angle 2 = ?" is solvable, but the exact same problem with Angle 1 = 2x+10 leaves students stumped.

This confirms prior observations that even students who have been successful with math (specifically algebra) in the past have no real understanding of what they are doing and why they are doing it. I see the same phenomenon in physics with students who can graph anything in math class, but can't wrap their heads around the idea that 'y' is really just a label for the dependent variable which can be anything (same for 'x').

We spend so much time and effort drilling rote procedures into math students, that we (and more importantly, they) have lost all perspective as to our purpose.

So far, the quiz results are mildly disconcerting. Generally, students can classify angle pairs, but have no idea what to do with that information. They're easily flummoxed by algebraic expressions in place of discrete angle measurements. For example, saying "Angles 1 & 2 are a linear pair. Angle 1 = 50 deg and Angle 2 = ?" is solvable, but the exact same problem with Angle 1 = 2x+10 leaves students stumped.

This confirms prior observations that even students who have been successful with math (specifically algebra) in the past have no real understanding of what they are doing and why they are doing it. I see the same phenomenon in physics with students who can graph anything in math class, but can't wrap their heads around the idea that 'y' is really just a label for the dependent variable which can be anything (same for 'x').

We spend so much time and effort drilling rote procedures into math students, that we (and more importantly, they) have lost all perspective as to our purpose.

## Tuesday, October 16, 2012

### Day 30: Discussion

Whiteboard discussion of U2 WS4. With so many students (35), I've found that the best practice is giving each group 3-4 problems to whiteboard so that the entire worksheet is covered, with (hopefully) a little overlap. Rather than go over each board independently, I let students spend ~10 minutes walking around the room checking boards, taking notes and asking questions. Then we'll spend 10-15 mins as a group answering any lingering questions.

This works pretty well in my honors classes because most of the class will arrive having completed (or at least attempted) the homework. In my regular class, so many kids never attempt the practice, that we'd spend all hour just trying to whiteboard the problems and even then it would really only be 4-5 kids doing the work. Then the discussion is pointless, because the students who don't participate simply copy the work off the boards thinking that will be sufficient (which is obviously isn't).

At it's best, I love this idea. It's dynamic - it gets the kids working together, moving around, and holding them responsible for seeking help. With an efficient class, we're easily doing 3-4 distinct things in a single class period which helps the day seem shorter.

At it's worst, this strategy only benefits the kids who take it seriously. Of course, that's true with most methodologies but sometimes I fear what students are telling parents who don't understand the modeling philosophy, especially when there aren't any pre-printed notes or a useful textbook to supplant the classroom environment.

This works pretty well in my honors classes because most of the class will arrive having completed (or at least attempted) the homework. In my regular class, so many kids never attempt the practice, that we'd spend all hour just trying to whiteboard the problems and even then it would really only be 4-5 kids doing the work. Then the discussion is pointless, because the students who don't participate simply copy the work off the boards thinking that will be sufficient (which is obviously isn't).

At it's best, I love this idea. It's dynamic - it gets the kids working together, moving around, and holding them responsible for seeking help. With an efficient class, we're easily doing 3-4 distinct things in a single class period which helps the day seem shorter.

At it's worst, this strategy only benefits the kids who take it seriously. Of course, that's true with most methodologies but sometimes I fear what students are telling parents who don't understand the modeling philosophy, especially when there aren't any pre-printed notes or a useful textbook to supplant the classroom environment.

## Monday, October 15, 2012

### Day 29: Catchup

This was essentially a day for students to work on U2 WS4 in groups. It allowed my 5th & 6th Hours to catch up with my 2nd hour so that we'd all be on pace for Friday's Unit 2 Assessment.

The main problem with allowing students time in class to work is how many of them will waste the time thinking that they "get it." This means they're not prepared to be productive in a class discussion the following day and they don't have any notes to use as a reference on open note quizzes. By the time feedback from the quiz arrives to indicate that they do not in fact "get it," they're short on time to fill in the gaps before a formative assessment.

The related issue there is how many students simply refuse to think through problems on their own, and demand step-by-step instruction on every problem before they'll even take an assessment seriously.

The main problem with allowing students time in class to work is how many of them will waste the time thinking that they "get it." This means they're not prepared to be productive in a class discussion the following day and they don't have any notes to use as a reference on open note quizzes. By the time feedback from the quiz arrives to indicate that they do not in fact "get it," they're short on time to fill in the gaps before a formative assessment.

The related issue there is how many students simply refuse to think through problems on their own, and demand step-by-step instruction on every problem before they'll even take an assessment seriously.

## Friday, October 12, 2012

### Day 28: Online Practice

Spent today in the computer lab working on the Carnegie Learning software. Students often get frustrated with how many problems the program makes them complete which I partially understand, but they refuse to acknowledge that pressing "hint" and guessing random answers tells the computer to assign more problems until mastery is achieved. Like most other work, students want me to walk them through every problem step by step until they reach a solution without any of their own effort.

I do not require this work to be completed on any specific time table, but it is one of the components I mention to parents about why a student's grade might be what it is and I also might require completion before reassessments are administered. It only requires an internet connection, so students are free to work through the problems at home.

We'll go over U2 WS4 on angle pairs on Monday and spend the rest of the week reviewing and preparing for the Unit 2 Assessment on Friday.

I do not require this work to be completed on any specific time table, but it is one of the components I mention to parents about why a student's grade might be what it is and I also might require completion before reassessments are administered. It only requires an internet connection, so students are free to work through the problems at home.

We'll go over U2 WS4 on angle pairs on Monday and spend the rest of the week reviewing and preparing for the Unit 2 Assessment on Friday.

## Thursday, October 11, 2012

### Day 27: Pairs of Angles (cont.)

Continuation of yesterday after a brief quiz on segment & angle addition. Assigned U2 WS4 on angle pairs which will be due Monday.

## Wednesday, October 10, 2012

### Day 26: Pairs of Angles

2nd Hour is starting to move a little ahead of my other 2 classes, so I might focus on what they are doing to avoid getting confused. And because they don't make me rethink life choices I've made.

So we went through different ways to pair up angles today. Sadly, it was more lecture oriented than I would typically strive for, but even when I lecture, it's not really a lecture. I just couldn't think of how to get students to "discover" the terms

I used Geometer's Sketchpad (my favoritist thing in the whole world) to walk through examples and stop to document key terms. It makes it easy to show *why* the angle addition postulate (words that I never actually utter) arises from adjacent angles. I just speak in terms of "How could we write an equation to relate these three angles?" That way they're pressed to think instead of recall a certain postulate or theorem.

We'll continue tomorrow with linear pairs and vertical angles, then practice using Carnegie on Friday. I expect to be done with Unit 2 early next week and have a formative assessment by next Friday.

So we went through different ways to pair up angles today. Sadly, it was more lecture oriented than I would typically strive for, but even when I lecture, it's not really a lecture. I just couldn't think of how to get students to "discover" the terms

*adjacent*,*complimentary*,*supplementary*, etc.I used Geometer's Sketchpad (my favoritist thing in the whole world) to walk through examples and stop to document key terms. It makes it easy to show *why* the angle addition postulate (words that I never actually utter) arises from adjacent angles. I just speak in terms of "How could we write an equation to relate these three angles?" That way they're pressed to think instead of recall a certain postulate or theorem.

We'll continue tomorrow with linear pairs and vertical angles, then practice using Carnegie on Friday. I expect to be done with Unit 2 early next week and have a formative assessment by next Friday.

## Tuesday, October 9, 2012

### Day 25: Assessment and Discussion

First portion of the class was the 2nd quiz for Unit 2 on proper use of a ruler & protractor. It never ceases to amaze me how many students think during an assessment is the proper time to state "I don't know how to do this" and ask for direct instruction.

After the quiz we went over WS3 on classifying and naming angles. I made a specific point to emphasize the difference between those two words as well as the importance of evidence when declaring an angle to be "right." I really want students to understand the theme of the class as being the need justify reasoning without relying on an "I think..." mentality.

The rest of the week will be spent on the last part of the unit which deals with identifying angle pairs and learning how to solve for their measure. This is another one of those brilliant ideas I had in terms of sequencing, but I never really got around to thinking about *how* to put it into action. Here's hoping for the best!

After the quiz we went over WS3 on classifying and naming angles. I made a specific point to emphasize the difference between those two words as well as the importance of evidence when declaring an angle to be "right." I really want students to understand the theme of the class as being the need justify reasoning without relying on an "I think..." mentality.

The rest of the week will be spent on the last part of the unit which deals with identifying angle pairs and learning how to solve for their measure. This is another one of those brilliant ideas I had in terms of sequencing, but I never really got around to thinking about *how* to put it into action. Here's hoping for the best!

## Monday, October 8, 2012

### Day 24: Classifying Angles

More review than I had expected. I had originally come up with an idea to provide a bunch of angles of various sizes on laminated cards and have students sort them into groups. As many groups as they wanted and using whatever classification scheme they could come up with. We would then share approaches from different groups and lead the discussion to a consensus of ideas revolving around acute, obtuse, right, and straight. I was even prepared to allow the class to call those groupings whatever they wanted (big, small, perfect, line, whatever).

But it turned out that angle classification is one of the few things that has been retained from prior math classes. So we quickly reviewed what the classifications are and moved directly into the practice with U2 WS3.

Most students could work through the classification just fine, but working with angles and segments that have been broken into pieces was a challenge. Every year it is and every year it surprises me. Often, students will think that it's so easy they don't need to write an equation to solve for the unknowns, but when they're given more challenging problems they don't have *any* type of tested approach and will usually quit before they even try.

I don't think students struggle setting up the equations once they see a few examples, but they do struggle with identifying their own limits and seeing that what might be easy today is about to get a heckuva lot more complicated and that our goal is to set in place skills & procedures that can be applied to ANY situation.

But it turned out that angle classification is one of the few things that has been retained from prior math classes. So we quickly reviewed what the classifications are and moved directly into the practice with U2 WS3.

Most students could work through the classification just fine, but working with angles and segments that have been broken into pieces was a challenge. Every year it is and every year it surprises me. Often, students will think that it's so easy they don't need to write an equation to solve for the unknowns, but when they're given more challenging problems they don't have *any* type of tested approach and will usually quit before they even try.

I don't think students struggle setting up the equations once they see a few examples, but they do struggle with identifying their own limits and seeing that what might be easy today is about to get a heckuva lot more complicated and that our goal is to set in place skills & procedures that can be applied to ANY situation.

## Friday, October 5, 2012

### Day 23: Measurement practice

Not much to report - the class was spent allowing students to work on U2 WS2 which dealt with using rulers and protractors correctly.

I know a million different ways to show/explain this skill, but I have to find a consistent way to ensure that students are internalizing what we're doing. Generally, students think this is so basic that it's a waste of class time, but the truth is always revealed come quiz time. That might be an investigation to itself - how to work with students who consistently think they know material that they very obviously do not?

I was a little optimistic that students seemed to connect our work with "angles as portions of a circle" to why protractors are half circles. A little disheartening to see so many use the ruler part of a protractor to measure an angle though.

I know a million different ways to show/explain this skill, but I have to find a consistent way to ensure that students are internalizing what we're doing. Generally, students think this is so basic that it's a waste of class time, but the truth is always revealed come quiz time. That might be an investigation to itself - how to work with students who consistently think they know material that they very obviously do not?

I was a little optimistic that students seemed to connect our work with "angles as portions of a circle" to why protractors are half circles. A little disheartening to see so many use the ruler part of a protractor to measure an angle though.

## Thursday, October 4, 2012

### Day 22: Measurement & Assessment

My students' philosophy in regards to quizzes and tests really needs an adjustment. I know their 9th grade math teacher gave "homework quizzes" probably 2-3x per week in lieu of grading actual homework, so it's really surprising that they are so unprepared for regular assessments. I allotted 15-20 mins for an 8 question quiz which looked *very* similar to the worksheet we've been dealing with all week on angles & proportions, and a lot of the students wanted extra time.

Two issues at stake here:

1) If you need more than 20 mins to answer 8 questions, you don't know the material. Extra time is not going to change all that.

But this attitude is quite prevalent and something I've been thinking a lot about recently. My theory is that students are so trained to work recipes as procedures for doing problems in math & science that they honestly have no idea *what* they're doing. So they look at assessments as random - maybe their recipe will yield something fruitful, maybe it won't. If you offer a retake, they won't study, they'll just hope for "better" questions the second time around.

In short, more time = maybe it will come to me.

2) "If I fail this quiz taken on the 4th day of new material, I'm going to fail the class."

Both of these problems are related to mindset. Again, students are trained to think about assessments as worth points and if they fail, the just lost a bunch of points, never to be heard from again. That's why I love SBG so much - I could care less about points. Show me what you know. Do you think I expectation on Day 4 of a new unit? That's ridiculous. What's important is that we get it worked out before the end of the unit.

I tried to explain to my classes my thoughts on the matter, but as per usual, they don't listen to much I say (hence modeling - have them experience what it's like to be ignored by people you're trying to help). I could see some heads nodding in agreement, but there is a vocal minority that hates the new system. They don't care about learning, progress, or growth. They just want a grade.

But what is a grade if it doesn't reflect what you can (and can't) do?

Two issues at stake here:

1) If you need more than 20 mins to answer 8 questions, you don't know the material. Extra time is not going to change all that.

But this attitude is quite prevalent and something I've been thinking a lot about recently. My theory is that students are so trained to work recipes as procedures for doing problems in math & science that they honestly have no idea *what* they're doing. So they look at assessments as random - maybe their recipe will yield something fruitful, maybe it won't. If you offer a retake, they won't study, they'll just hope for "better" questions the second time around.

In short, more time = maybe it will come to me.

2) "If I fail this quiz taken on the 4th day of new material, I'm going to fail the class."

Both of these problems are related to mindset. Again, students are trained to think about assessments as worth points and if they fail, the just lost a bunch of points, never to be heard from again. That's why I love SBG so much - I could care less about points. Show me what you know. Do you think I expectation on Day 4 of a new unit? That's ridiculous. What's important is that we get it worked out before the end of the unit.

I tried to explain to my classes my thoughts on the matter, but as per usual, they don't listen to much I say (hence modeling - have them experience what it's like to be ignored by people you're trying to help). I could see some heads nodding in agreement, but there is a vocal minority that hates the new system. They don't care about learning, progress, or growth. They just want a grade.

But what is a grade if it doesn't reflect what you can (and can't) do?

Subscribe to:
Posts (Atom)