The rectangle portion of the investigation took longer than anticipated, but we covered a LOT of (review) material, and the kids were generally following along, so I'm ok with it.

Today we formalized the conclusions about rectangles, noting that not much of what was true for squares holds up for rectangles. I tried to stress that squares are a very specific shape and we're building "up" to more general ideas, eventually leaving to parallelograms and then quadrilaterals, but I'm not sure how much of the "big picture" talk they're actually absorbing.

Tomorrow we'll show how one simple tangent ratio can lead us to know a TON of angles in the parallel line & transversal set up, all thanks to simple ideas like linear pairs, vertical angles, and the triangle sum theorem.

## Thursday, January 31, 2013

## Wednesday, January 30, 2013

### Day 86: Investigating Rectangles

Today was a multi-layered activity that ties a lot of 1st semester content together to create new situations to be analyzed.

Start by graphing a line. Create a line perpendicular through point D. Create a second perpendicular line through point C. Connect C & D (essentially, make a rectangle).

Now:

Start by graphing a line. Create a line perpendicular through point D. Create a second perpendicular line through point C. Connect C & D (essentially, make a rectangle).

Now:

- Are the sides the same length? Prove with the distance formula.
- Is the diagonal equal to sqrt(2) * side length? Prove with the distance formula.
- Are the diagonals angle bisectors? Prove using a tangent ratio.
- Are the diagonals perpendicular bisectors? Prove using slope and midpoint.

From this activity, we'll build into determining all of the angles and look for commonalities, leading toward the idea of parallel lines and the angle pairs that are formed.

## Friday, January 18, 2013

### Day 84: Jeopardy

I made a Jeopardy style review game with the help of Jeopardy Labs and played with 2nd and 6th hour. I felt 5th Hour wouldn't get much out of the game, so I gave them an additional period in the computer lab.

The game is public and can be found here: 1st Semester Jeopardy Review

The game is public and can be found here: 1st Semester Jeopardy Review

## Thursday, January 17, 2013

### Day 83: Computer Lab & Review

I gave students the option of continuing the use the Carnegie Learning software online, but I also encouraged them to check out Khan Academy and search for video tutorials dealing with the standards that they're struggling with (they all got a updated progress report of ALL standards earlier in the week). Some kids actually watched a decent number of videos and claimed they helped, some worked on their final exam note sheets, but a shocking number of students did absolutely nothing.

I'm honestly tempted to create either an online form or a paper log that students can sign admitting they they are in fact wasting class time, and I can give that to either parents or administration to avoid culpability.

I'm honestly tempted to create either an online form or a paper log that students can sign admitting they they are in fact wasting class time, and I can give that to either parents or administration to avoid culpability.

## Wednesday, January 16, 2013

### Day 82: Assessment & Review

Last quiz before the final on interior & exterior angles. Overwhelmingly, kids basically get the idea that the three angles of a triangle sum to 180, but the exterior angle thing is still eluding them. Suddenly, any three angles add up to 180, regardless of where they're located. Kind of like how every triangle is a 30/60/90 triangle, regardless of given information (like the question dictating that the triangle is in fact, isosceles).

After the quiz the students were given the schedule for the remaining days and instructed to begin creating a note sheet to be used on the final exam.

After the quiz the students were given the schedule for the remaining days and instructed to begin creating a note sheet to be used on the final exam.

## Tuesday, January 15, 2013

### Day 81: Discussion

Today might have been the first day I let my honors class operate on a different schedule than my regular class. Actually, all three hours ran a little differently, but for different reasons.

2nd hour: Went over the worksheet via quasi-traditional "Whiteboard Parade." Groups whiteboarded a couple of problems each, then everyone had the chance to walk around and go over the boards at their own pace (no formal presentations).

5th hour: I 'rewarded' the class with an extra day to work because I was so happy they were actually working for a change. I also pointed out what I was doing to them (that their efforts were being recognized and I was slowing the pace of the class down for them as a result). I'd never been against slowing down, but if the class isn't going to do anything, then there's no reason to go slow.

6th hour: I tried to do the WB parade like in first hour, but it doesn't work as well with this group. They might get the problems whiteboarded, but everyone just sits in their seats and talks with their friends instead of checking their work and correcting themselves. I tried reminding them of the next day's quiz, but they either think they get it (they don't), or they just don't care.

I'll figure something out eventually.

## Monday, January 14, 2013

### Day 80: Practice

Students were to spend the hour working on U5 WS3 dealing with the interior and exterior angles of a triangle.

I've discovered it's very hard to write questions to assess proficiency on the Exterior Angle Theorem. In most cases, students can work through the problem using the interior angles and a linear pair. The one true problem involves representing the exterior angle with an algebraic expression (such as {x-25}deg), doing the same for one of the opposite interior angles, and then giving a measurement for the second opposite interior angle.

Students often struggle with the idea of an algebraic expression representing the angle. They'll very often pull out the x and use it alone as a representation of the angle (instead of the whole {x-25}deg). Additionally, if they do manage to solve for x, it's hard to get them to understand why they're not done, and that they need to substitute x back into the expression and then solve for the angle measure.

Those hurdles have more to do with student proficiency in algebra than in geometry, so I try very hard to separate the two aspects while grading, and SBG helps a lot with that. I promise every student that if they can set up the problem correctly, but never managed to do any algebra, they'll pass the class (with a D). That idea got a lot of them attempting more problems than before, so I hope it's momentum that will continue into the second semester.

I've discovered it's very hard to write questions to assess proficiency on the Exterior Angle Theorem. In most cases, students can work through the problem using the interior angles and a linear pair. The one true problem involves representing the exterior angle with an algebraic expression (such as {x-25}deg), doing the same for one of the opposite interior angles, and then giving a measurement for the second opposite interior angle.

Students often struggle with the idea of an algebraic expression representing the angle. They'll very often pull out the x and use it alone as a representation of the angle (instead of the whole {x-25}deg). Additionally, if they do manage to solve for x, it's hard to get them to understand why they're not done, and that they need to substitute x back into the expression and then solve for the angle measure.

Those hurdles have more to do with student proficiency in algebra than in geometry, so I try very hard to separate the two aspects while grading, and SBG helps a lot with that. I promise every student that if they can set up the problem correctly, but never managed to do any algebra, they'll pass the class (with a D). That idea got a lot of them attempting more problems than before, so I hope it's momentum that will continue into the second semester.

## Friday, January 11, 2013

### Day 79: Assessment and Investigation

So the sequence of the week went as follows:

- Day 1: Investigation
- Day 2: Formalization / discussion of results
- Day 3: Examples & practice
- Day 4: Review of practice
- Day 5: Quiz / new investigation

I feel that most educators would agree that this is a solid sequence of instruction. And yet, it's totally not working. I actually felt pretty good about the students' level of understanding on Day 4, but the quizzes were a complete abomination. Students were taking 20 and 30 minutes to complete a 4 question quiz, asking for direct help while taking it, and still turning in mostly blank papers.

They'll be graded this weekend to complete the feedback loop, but most students will deposit their work directly into the circular file and go about their business.

Sorry for being negative, but with final exams approaching and the worst grades I've ever seen in geometry, it's becoming a burden to handle.

## Thursday, January 10, 2013

### Day 78: Discussion

I still have no idea how to handle going over a worksheet in class. I've tried at least a half dozen different incarnations, and nothing seems to really get the job done. At the crux of the issue is that regardless of how much class time is devoted to practice, students will simply not do the work. So it becomes impossible to go over a 15 question worksheet, because only the first 3 questions were answered with any regularity.

Today I used the document camera and had students volunteer only their worksheet - I didn't make the student approach the camera and discuss their work, I simply presented it and we talked about it as a group. The approach actually seemed to work for 2 of my classes, but I would later discover that while most students were attentive, they still only completed the questions that we went over directly. Nobody is applying the discussion to their own knowledge and working toward understanding.

At every turn, students claim "I tried, but I didn't get it, so I stopped." Those that ask for help never get answers directly, they simply get Socratic questions that will lead them toward understanding. And the students that have the patience to hang with me will eventually "get it," but even those students aren't internalizing the process to be able to repeat the questioning on their own. Without someone holding their hand and leading them step-by-step, they're completely bumfuddled and refuse to even attempt a solution.

In talking with other teachers in my school, this seems to be a wider problem with the Class of 2015 as a whole (not necessarily specific to me or my methods), but that doesn't make it any easier to deal with. We're at the halfway point, and maybe half of my students have given up completely, which means it's going to be a long second semester.

Today I used the document camera and had students volunteer only their worksheet - I didn't make the student approach the camera and discuss their work, I simply presented it and we talked about it as a group. The approach actually seemed to work for 2 of my classes, but I would later discover that while most students were attentive, they still only completed the questions that we went over directly. Nobody is applying the discussion to their own knowledge and working toward understanding.

At every turn, students claim "I tried, but I didn't get it, so I stopped." Those that ask for help never get answers directly, they simply get Socratic questions that will lead them toward understanding. And the students that have the patience to hang with me will eventually "get it," but even those students aren't internalizing the process to be able to repeat the questioning on their own. Without someone holding their hand and leading them step-by-step, they're completely bumfuddled and refuse to even attempt a solution.

In talking with other teachers in my school, this seems to be a wider problem with the Class of 2015 as a whole (not necessarily specific to me or my methods), but that doesn't make it any easier to deal with. We're at the halfway point, and maybe half of my students have given up completely, which means it's going to be a long second semester.

## Wednesday, January 9, 2013

### Day 77: Examples & Practice

Trying VERY hard to stress the meaning of sine, cosine, and tangent this year. To that end, I'm insisting on using language like "the sine ratio for a 25 degree angle is equal to 0.4226" as we read "sin 25 =0.4226."

I still haven't mentioned that these ratios can be found using a scientific calculator (and I don't have any immediate plans to do that), so I'm requiring students use the trig tables I've provided. Some students know how to use the calculator, and that's fine, but my theory is that a greater understanding is achieved through the table.

Also stressing the use of a mnemonic to help remember sine = opposite / hypotenuse and the like. I know everyone and their brother uses "SOH CAH TOA," but I've found that while most students can repeat that phrase, MANY of them will misspell it, which immediately defeats the purpose. Last year, a student suggested "Some Old Hare Came A Hoppin Through Our Apartment" which I immediately fell in love with. Much less room for error, except most of my students don't know what a "hare" is, so I changed it to "hipster." I also made a large poster of the phrase and hung it visibly where all can see.

What's funny is that the poster has been up all year. Some kids have asked about it, and I simply said we'll explain it when the time comes. But a lot of kids expressed a fair amount of surprise that it even existed. These are the same students that haven't noticed the 4' x 8' whiteboard devoted to a weekly calendar for the class, can't seem to remember where to find extra copies of worksheets, and don't understand how to use the Turn-In Bin properly. *sigh*.

Anyway, today the students worked on Unit 5 Worksheet #2 practicing sine/cosine/tangent problems after seeing some examples worked through.

I still haven't mentioned that these ratios can be found using a scientific calculator (and I don't have any immediate plans to do that), so I'm requiring students use the trig tables I've provided. Some students know how to use the calculator, and that's fine, but my theory is that a greater understanding is achieved through the table.

Also stressing the use of a mnemonic to help remember sine = opposite / hypotenuse and the like. I know everyone and their brother uses "SOH CAH TOA," but I've found that while most students can repeat that phrase, MANY of them will misspell it, which immediately defeats the purpose. Last year, a student suggested "Some Old Hare Came A Hoppin Through Our Apartment" which I immediately fell in love with. Much less room for error, except most of my students don't know what a "hare" is, so I changed it to "hipster." I also made a large poster of the phrase and hung it visibly where all can see.

What's funny is that the poster has been up all year. Some kids have asked about it, and I simply said we'll explain it when the time comes. But a lot of kids expressed a fair amount of surprise that it even existed. These are the same students that haven't noticed the 4' x 8' whiteboard devoted to a weekly calendar for the class, can't seem to remember where to find extra copies of worksheets, and don't understand how to use the Turn-In Bin properly. *sigh*.

Anyway, today the students worked on Unit 5 Worksheet #2 practicing sine/cosine/tangent problems after seeing some examples worked through.

## Tuesday, January 8, 2013

### Day 76: The Trig Table

I hate calculators. Ok, that's a bit extreme. After all, a calculator is just a tool - you can't hate a tool (or at least, you shouldn't, because it's an inanimate object incapable of malice), but you can hate how some people use the tool.

I don't know what it's really like nationally, but I get the feeling that students are handed calculators as early as 4th grade and they never look back. In 6th grade they get a fancy scientific calculator, and in 9th grade they get an even fancier graphing calculator. My district has always been on the lookout for grants and giveaways so we can get more graphing calculators and put them in everyone's hands as soon as possible. But why? How much quadratic graphing are they doing in 9th grade algebra? Even if they do cover it, shouldn't we be reinforcing the underlying theory by manually graphing the functions?

My students have very poor number sense. Take away their calculators and suddenly fractions don't make any sense. Radicals are a disaster. Sine, cosine, tangent? Forget about it. That's why a few years ago, I decided to stop allowing calculators when we reach the trig unit. I give each student a copy of the trig table (also available in their textbook) and we work through it "old school." My hope has always been that this will underscore that sine/cosine/tangent are just names given to very specific ratios in right triangles, but I don't know to what extent that message has come across.

This year is no different, so as we went over yesterday's investigation, I talked about how we could create a list of every possible ratio, from 1 deg to 89 deg, for a total of 88 * 3 entries. The kids immediately freak out thinking that will be the next assignment, draw a 1 deg right triangle, measure, divide, repeat. Good - I want them to panic. They need to appreciate what the table is (and later, what the calculator is doing when you hit that 'sin' button).

I downloaded a pdf of a trig table and fit it onto a half-sheet of paper, then laminated and cut out each one (yes, it really takes as long as it sounds). The first thing I heard as I passed them out was "Oh, it's laminated. That means we don't get to keep it." That was a real eye-opener to me. We (at least in my district) have taught the students "If it's nice, it's not yours. Give it back." Ugh. In any event, they were ecstatic when I told them it was in fact theirs to keep and the lamination should serve as an indication of how important it is. "You will NOT be able to solve ANY problems without this table!" I bellow to each class.

We didn't get much past identifying opposite & adjacent sides and setting up the three ratios, but I'm optimistic as always.

I don't know what it's really like nationally, but I get the feeling that students are handed calculators as early as 4th grade and they never look back. In 6th grade they get a fancy scientific calculator, and in 9th grade they get an even fancier graphing calculator. My district has always been on the lookout for grants and giveaways so we can get more graphing calculators and put them in everyone's hands as soon as possible. But why? How much quadratic graphing are they doing in 9th grade algebra? Even if they do cover it, shouldn't we be reinforcing the underlying theory by manually graphing the functions?

My students have very poor number sense. Take away their calculators and suddenly fractions don't make any sense. Radicals are a disaster. Sine, cosine, tangent? Forget about it. That's why a few years ago, I decided to stop allowing calculators when we reach the trig unit. I give each student a copy of the trig table (also available in their textbook) and we work through it "old school." My hope has always been that this will underscore that sine/cosine/tangent are just names given to very specific ratios in right triangles, but I don't know to what extent that message has come across.

This year is no different, so as we went over yesterday's investigation, I talked about how we could create a list of every possible ratio, from 1 deg to 89 deg, for a total of 88 * 3 entries. The kids immediately freak out thinking that will be the next assignment, draw a 1 deg right triangle, measure, divide, repeat. Good - I want them to panic. They need to appreciate what the table is (and later, what the calculator is doing when you hit that 'sin' button).

I downloaded a pdf of a trig table and fit it onto a half-sheet of paper, then laminated and cut out each one (yes, it really takes as long as it sounds). The first thing I heard as I passed them out was "Oh, it's laminated. That means we don't get to keep it." That was a real eye-opener to me. We (at least in my district) have taught the students "If it's nice, it's not yours. Give it back." Ugh. In any event, they were ecstatic when I told them it was in fact theirs to keep and the lamination should serve as an indication of how important it is. "You will NOT be able to solve ANY problems without this table!" I bellow to each class.

We didn't get much past identifying opposite & adjacent sides and setting up the three ratios, but I'm optimistic as always.

## Monday, January 7, 2013

### Day 75: Trig Ratio Investigation

Ok, back from break well rested and ready to finish off the semester. We have two weeks left and we need to cover trigonometric ratios and interior & exterior angles of triangles. That should get us to roughly 25 standards for the first semester, about 5 per unit (and also per month). I'm very happy with how the pacing of this new curriculum has worked out.

The intro activity I used today is the same as I've used for years to start this unit (because I like it, not necessarily because I'm lazy). Have students work in groups of 4-5 (this is key, they NEED multiple data sets to draw conclusions) and draw a right triangle with a common angle (between 20 and 70 deg to make life easier). The students all thought that common angles make congruent triangles, so I'm not sure if I should spend more time on similarity != congruence beforehand next year, but for now we're on a time crunch.

The idea is to get proportional triangles, but the students don't really see that coming. Then they each measure the sides of their own triangle (another key point, they each need their own triangle) and fill in a data table with both their, and their group members' data. They finish by evaluating set ratios, and then answer some leading questions about what they found.

I know it's not the most student led investigation in the world, but it does tie in ideas of ratio & similarity to expand on what we've already talked about with very specific right triangles to help branch out into a broader world.

The intro activity I used today is the same as I've used for years to start this unit (because I like it, not necessarily because I'm lazy). Have students work in groups of 4-5 (this is key, they NEED multiple data sets to draw conclusions) and draw a right triangle with a common angle (between 20 and 70 deg to make life easier). The students all thought that common angles make congruent triangles, so I'm not sure if I should spend more time on similarity != congruence beforehand next year, but for now we're on a time crunch.

The idea is to get proportional triangles, but the students don't really see that coming. Then they each measure the sides of their own triangle (another key point, they each need their own triangle) and fill in a data table with both their, and their group members' data. They finish by evaluating set ratios, and then answer some leading questions about what they found.

I know it's not the most student led investigation in the world, but it does tie in ideas of ratio & similarity to expand on what we've already talked about with very specific right triangles to help branch out into a broader world.

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