## Tuesday, December 21, 2010

## Monday, December 20, 2010

### Fumble

Student: Did you grade the tests over the weekend?

Me: Nope.

Student: Isn't it your job to grade them?

What I should've said: Sometimes I need a break over the weekend. Also, I was a soup kitchen yesterday and I came back exhausted. On Saturday night I was working on computer science for my independent study, and when I got back last night I worked on planning lessons.

What I said: What did you do this weekend?

Student: Hung out with friends?

Me: You have friends?

Other Student: Ha! He went there!

I've got a lot of patience to gain.

Me: Nope.

Student: Isn't it your job to grade them?

What I should've said: Sometimes I need a break over the weekend. Also, I was a soup kitchen yesterday and I came back exhausted. On Saturday night I was working on computer science for my independent study, and when I got back last night I worked on planning lessons.

What I said: What did you do this weekend?

Student: Hung out with friends?

Me: You have friends?

Other Student: Ha! He went there!

I've got a lot of patience to gain.

## Sunday, December 19, 2010

### Factoring Resources

If I have anything to add as a factoring resources, it's that my students get way more multiplying polynomials questions right if they multiply the polynomials in a repurposed Punnet Square. I don't feel at all guilty about this for two reasons:

1. My strongest students ignore the box since they can do the multiplication much more quickly without it.

2. Me weaker students LOVE it, since it keeps them from making sloppy mistakes.

I like it because it reinforces their geometric intuitions about area and makes a nice connection between algebra and geometry. The downside is that it doesn't really do a great job setting them up for factoring, but we'll see how that goes.

I'm not sure whether to teach them factoring by grouping or whether to just focus on getting them the ac/a+c intuition for trinomials where a=1. The advantage of teaching them grouping is that it'll make Alg2 much easier for them. And some teachers swear by the grouping.

Here's a rolling post on factoring resources:

CONTEXT

http://samjshah.com/2009/08/13/factoring-schmactoring/

DEFENSE OF FACTORING

http://mathmamawrites.blogspot.com/2010/12/my-math-alphabet-f-is-for-factoring.html

SOME TRICKS THAT MIGHT BE USED TO MOTIVATE IT

http://blog.mrmeyer.com/?p=8713#comments

USING GROUPING TO FACTOR TRINOMIALS

http://jd2718.wordpress.com/2007/09/28/trinomial-factoring-nice-site-and-last-detail/

SOME FACTORING GAMES/WORKSHEETS

http://algebra.mrmeyer.com/week26/handouts.zip

http://algebra.mrmeyer.com/week27/handouts.zip

E-TEXT WORKSHEETS

http://www.teacherweb.com/NY/Arlington/AlgebraProject/photo3.aspx

1. My strongest students ignore the box since they can do the multiplication much more quickly without it.

2. Me weaker students LOVE it, since it keeps them from making sloppy mistakes.

I like it because it reinforces their geometric intuitions about area and makes a nice connection between algebra and geometry. The downside is that it doesn't really do a great job setting them up for factoring, but we'll see how that goes.

I'm not sure whether to teach them factoring by grouping or whether to just focus on getting them the ac/a+c intuition for trinomials where a=1. The advantage of teaching them grouping is that it'll make Alg2 much easier for them. And some teachers swear by the grouping.

Here's a rolling post on factoring resources:

CONTEXT

http://samjshah.com/2009/08/13/factoring-schmactoring/

DEFENSE OF FACTORING

http://mathmamawrites.blogspot.com/2010/12/my-math-alphabet-f-is-for-factoring.html

SOME TRICKS THAT MIGHT BE USED TO MOTIVATE IT

http://blog.mrmeyer.com/?p=8713#comments

USING GROUPING TO FACTOR TRINOMIALS

http://jd2718.wordpress.com/2007/09/28/trinomial-factoring-nice-site-and-last-detail/

SOME FACTORING GAMES/WORKSHEETS

http://algebra.mrmeyer.com/week26/handouts.zip

http://algebra.mrmeyer.com/week27/handouts.zip

E-TEXT WORKSHEETS

http://www.teacherweb.com/NY/Arlington/AlgebraProject/photo3.aspx

### Aaaaaaaauuuuggggghhhh!!!!! (Or, my 3 opposing inclinations on how to teach exponential functions/logs)

I'm going to begin by stating some standard-issue frustrations, some non-standard-issue frustrations, then I'm going to reflect on three ways I could teach exponential functions and logs.

First, the personal baggage: This is my first year teaching, I just got out of college, I've never taken a class in education, and I'm teaching at a yeshiva high school in NYC with 3 preps.

What this means is that I'm feverishly thinking about the entire three-year math sequence in New York. I'm really committed to teaching this stuff in a way that reveals its true value and combats my students grumpiness about math. But because I'm teaching at a yeshiva that means that I also have way less time to teach the same curriculum as everyone else does. The average public school has 180 days for teaching math--I have 124.

That means that I'm going to do a worse job in the classroom, and there's just nothing that I can do about that. What's lost in those 56 days is so much of the context and the meaning behind math. (Or, to put this another way, Sam Shah spends 13 days on exponential functions?!)

What does that mean for me? It means that I can't spend three days on exponential functions, including continuous growth. NY State gives me one day, more or less.

Which brings me to the decision: how do I teach it? I have three opposing inclinations.

1) Teach them science. Teach them population growth, or--even better--simple differential equations. Teach them dynamics! Get them to understand how these things are actually used in the world everyday by people--that is, by scientists.

PRO: It's true. They'll believe it and appreciate it.

CON: It'll take too long, both for me to prepare and also for them to understand. It isn't a standards-efficient activity.

2a) Teach them finance. True, you don't need to understand exponential functions to operate in the real world. But it'll help you understand credit cards and APR . I could start with exponential growth in general and then move into continuous growth.

PRO: The kids will like that it has to do with the "real world." "Oh, this stuff is actually useful!" Also, the amazing blogo-verse has already provided me with worksheets ready to go. In addition, it's a bit more standards-efficient then what I would cook up.

CON: When kids say "This stuff is actually useful" they're talking about today's lesson, not Algebra II. And they mean "Unlike everything else that we've learned." We have to be careful to distinguish "real-world" and "everyday-life." An example is "real-world" if people use it to understand the world. By this test, almost everything in Alg2/Trig passes. Almost. (I'm looking at you absolute value inequalities...) But very few of the "real-world" applications are "everyday-life" applications.

2b) Teach them this stuff: http://www.mathalicious.com/?cat=98. I forgot about mathalicious.

3) Teach them problems. Rules. Methods. Algorithms. This is what they're used to, but they find it boring and I find it SUPER boring.

PRO: I won't fall farther behind the pace.

CON: *sigh*

And this is my choice every time I sit down to figure out what to do in the classroom. It's a fight between science, the everyday, and, *sigh*.

First, the personal baggage: This is my first year teaching, I just got out of college, I've never taken a class in education, and I'm teaching at a yeshiva high school in NYC with 3 preps.

What this means is that I'm feverishly thinking about the entire three-year math sequence in New York. I'm really committed to teaching this stuff in a way that reveals its true value and combats my students grumpiness about math. But because I'm teaching at a yeshiva that means that I also have way less time to teach the same curriculum as everyone else does. The average public school has 180 days for teaching math--I have 124.

That means that I'm going to do a worse job in the classroom, and there's just nothing that I can do about that. What's lost in those 56 days is so much of the context and the meaning behind math. (Or, to put this another way, Sam Shah spends 13 days on exponential functions?!)

What does that mean for me? It means that I can't spend three days on exponential functions, including continuous growth. NY State gives me one day, more or less.

Which brings me to the decision: how do I teach it? I have three opposing inclinations.

1) Teach them science. Teach them population growth, or--even better--simple differential equations. Teach them dynamics! Get them to understand how these things are actually used in the world everyday by people--that is, by scientists.

PRO: It's true. They'll believe it and appreciate it.

CON: It'll take too long, both for me to prepare and also for them to understand. It isn't a standards-efficient activity.

2a) Teach them finance. True, you don't need to understand exponential functions to operate in the real world. But it'll help you understand credit cards and APR . I could start with exponential growth in general and then move into continuous growth.

PRO: The kids will like that it has to do with the "real world." "Oh, this stuff is actually useful!" Also, the amazing blogo-verse has already provided me with worksheets ready to go. In addition, it's a bit more standards-efficient then what I would cook up.

CON: When kids say "This stuff is actually useful" they're talking about today's lesson, not Algebra II. And they mean "Unlike everything else that we've learned." We have to be careful to distinguish "real-world" and "everyday-life." An example is "real-world" if people use it to understand the world. By this test, almost everything in Alg2/Trig passes. Almost. (I'm looking at you absolute value inequalities...) But very few of the "real-world" applications are "everyday-life" applications.

2b) Teach them this stuff: http://www.mathalicious.com/?cat=98. I forgot about mathalicious.

3) Teach them problems. Rules. Methods. Algorithms. This is what they're used to, but they find it boring and I find it SUPER boring.

PRO: I won't fall farther behind the pace.

CON: *sigh*

And this is my choice every time I sit down to figure out what to do in the classroom. It's a fight between science, the everyday, and, *sigh*.

## Tuesday, December 7, 2010

### Functions and Uncountable Sets

So I've got no idea how to offer a scientific context for functions in general, which is what I'm working on now in my Algebra 2 class. So I gave myself a bit of a challenge. I spent today trying to get my class excited and confused about the idea that besides for the countable infinity there is a larger, uncountable infinity. Then I told them that once we learn functions they have everything that they need to understand the proof.

I might have dug myself into a hole here. My plan was that I could lay the groundwork for the proof as I introduce them to domain, range, one-to-one, onto and bijection. Then I figured at the end I'd devote a bunch of class time to trying to help them grasp Cantor's diagonalization proof. This is problematic, though, because I'm going to need to devote almost a full period to help them grasp the diagonalization argument, and the confusing parts aren't the Algebra 2 parts, and I'm already crunched for time with this curriculum.

But I'm a desperate guy. Almost all my students think that what we're learning is worthless. I need to do something!

UPDATE: This might help.

I might have dug myself into a hole here. My plan was that I could lay the groundwork for the proof as I introduce them to domain, range, one-to-one, onto and bijection. Then I figured at the end I'd devote a bunch of class time to trying to help them grasp Cantor's diagonalization proof. This is problematic, though, because I'm going to need to devote almost a full period to help them grasp the diagonalization argument, and the confusing parts aren't the Algebra 2 parts, and I'm already crunched for time with this curriculum.

But I'm a desperate guy. Almost all my students think that what we're learning is worthless. I need to do something!

UPDATE: This might help.

## Wednesday, December 1, 2010

### Thin Lens Equation/Rational Equations

This lesson went well with my honors students this week, and I'll see how it goes with my lower track students next week. The idea is that the only legitimate context for much of the material in Algebra II is its scientific context, and I used this to give my students this context while practicing solving rational equations.

In my mind this is a step up from the "average rate" questions that dominate most of the applications of rational equations that I've seen. This is also easier for students (and this teacher) to understand and explain than resistor and circuit problems.

Here's my power point presentation, which I used to tease them ("Ever wonder why objects in your sideview mirror are closer than they appear?"), leading into the worksheet.

They split up into pairs and worked on the worksheet together.

Thin Lens Worksheet

In my mind this is a step up from the "average rate" questions that dominate most of the applications of rational equations that I've seen. This is also easier for students (and this teacher) to understand and explain than resistor and circuit problems.

Here's my power point presentation, which I used to tease them ("Ever wonder why objects in your sideview mirror are closer than they appear?"), leading into the worksheet.

**Thin Lens--Rational Equations**

View more presentations from mpershan.

They split up into pairs and worked on the worksheet together.

Thin Lens Worksheet

## Monday, November 29, 2010

### Rational Equations in Physics, as I find them

This is a post that I'll update as I find more material. The idea is to take one of these topics, give a scientific introduction and then present rational equations as the way of solving a scientific problem.

Circuits:

http://kalamitykat.com/2010/02/21/solving-rational-equations-project/

http://samjshah.files.wordpress.com/2010/02/rationalcircuits1.jpg

http://samjshah.files.wordpress.com/2010/02/rationalcircuits2.jpg

Lensmaker's Equation:

http://en.wikipedia.org/wiki/Lens_%28optics%29#Lensmaker.27s_equation

Additive velocities and Special Relativity:

http://en.wikipedia.org/wiki/Velocity-addition_formula#Special_theory_of_relativity

Circuits:

http://kalamitykat.com/2010/02/21/solving-rational-equations-project/

http://samjshah.files.wordpress.com/2010/02/rationalcircuits1.jpg

http://samjshah.files.wordpress.com/2010/02/rationalcircuits2.jpg

Lensmaker's Equation:

http://en.wikipedia.org/wiki/Lens_%28optics%29#Lensmaker.27s_equation

Additive velocities and Special Relativity:

http://en.wikipedia.org/wiki/Velocity-addition_formula#Special_theory_of_relativity

## Sunday, November 28, 2010

### Algebra II troubles

The “take everyday stuff and bring it into the classroom” shtick just doesn’t work for me when I’m preparing Algebra II lessons. And I think it’s because by the time we get to Algebra II we’ve reached a new point in the education of our students. We’ve exhausted the material that we think everybody out on the street ought to know, and we’ve started introducing specialized mathematics that not everyone needs to know. That is, our broader goal in Algebra II isn’t to provide people with the math they need to be average working folks, but rather to make more specialized education in the maths and sciences both attractive and feasible. That is, we teach it so that we attract kids to more math and science, and also so that it’s possible for kids to be prepared for more math and science.

As far as I can tell there are two reasons why we teach kids stuff:

(1) We think that they need to know it, even as a non-specialist. So comfort with percentages, ratios, rates, averages, comfort with numbers, abstract thinking--these are all skills that we want our students to have no matter what they do in life.

(2) We want to recruit and prepare students for a specialty. We, as a society, need mathematicians, physicists, doctors, engineers, accountants and all sorts of other professions that require more mathematical comfort than your average citizen, and therefore need more training. If we don't teach higher math, then our students won't be prepared for the training requisite for these jobs. Further, part of our job is to make working with math enticing enough that we're able to recruit workers into fields that require a good deal of number work. So our job is dual when we're in this mode: to prepare and recruit.

Basically, I think that Algebra I mostly falls into Category 1, and Geometry is half and half, but Algebra 2 is firmly in Category 2. So much of that curriculum is either preparatory for Calculus or of application only to scientists. Which is NOT a knock on it. But it just means that we can't use the same approach to teach Algebra 2 as we do Algebra 1.

So we need to think about the best way to do Algebra 2. I think where we end up is what so many teachers are already doing: integrating scientific material into our Algebra 2 courses. This is difficult for me, since I don’t have a great physics background beyond mechanics. But I think that this is the direction where I’m heading: our job in Algebra 2 is to make math, and its applications to science, seem attractive while simultaneously preparing students for their future.

Not sure exactly how this shows itself day-to-day, but I think we need to present material with scientific motivations. For instance, maybe the proper way to introduce complex numbers isn't as most general solutions to polynomial equations, but rather scientifically. Maybe we integrate complex numbers into our trigonometry so that we can ask "How can we model trigonometric fluctuations algebraically?" or something. (Truth is, I'm just learning about complex integration now, so I don't understand the applications of complex numbers at a depth greater than wikipedia browsing).

As far as I can tell there are two reasons why we teach kids stuff:

(1) We think that they need to know it, even as a non-specialist. So comfort with percentages, ratios, rates, averages, comfort with numbers, abstract thinking--these are all skills that we want our students to have no matter what they do in life.

(2) We want to recruit and prepare students for a specialty. We, as a society, need mathematicians, physicists, doctors, engineers, accountants and all sorts of other professions that require more mathematical comfort than your average citizen, and therefore need more training. If we don't teach higher math, then our students won't be prepared for the training requisite for these jobs. Further, part of our job is to make working with math enticing enough that we're able to recruit workers into fields that require a good deal of number work. So our job is dual when we're in this mode: to prepare and recruit.

Basically, I think that Algebra I mostly falls into Category 1, and Geometry is half and half, but Algebra 2 is firmly in Category 2. So much of that curriculum is either preparatory for Calculus or of application only to scientists. Which is NOT a knock on it. But it just means that we can't use the same approach to teach Algebra 2 as we do Algebra 1.

So we need to think about the best way to do Algebra 2. I think where we end up is what so many teachers are already doing: integrating scientific material into our Algebra 2 courses. This is difficult for me, since I don’t have a great physics background beyond mechanics. But I think that this is the direction where I’m heading: our job in Algebra 2 is to make math, and its applications to science, seem attractive while simultaneously preparing students for their future.

Not sure exactly how this shows itself day-to-day, but I think we need to present material with scientific motivations. For instance, maybe the proper way to introduce complex numbers isn't as most general solutions to polynomial equations, but rather scientifically. Maybe we integrate complex numbers into our trigonometry so that we can ask "How can we model trigonometric fluctuations algebraically?" or something. (Truth is, I'm just learning about complex integration now, so I don't understand the applications of complex numbers at a depth greater than wikipedia browsing).

## Tuesday, November 23, 2010

### Virtual Filing Cabinet Virtual Filing Cabinet

**Virtual Filing Cabinets:**

http://samjshah.com/worksheets-projects/

http://bowmandickson.com/virtual-filing-cabinet-2/

http://myweb20journey.blogspot.com/p/algebra-1-links.html

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