The more this year goes on, the more I wonder about what I don't know about students' misunderstandings. My geometry students were practicing today for our Geometry EOC test. I found some great resources online from Texas and North Carolina (whose problems I liked better). One problem gave the coordinates of the 3 vertices of a triangle and asked for the equation of the line that's the perpendicular bisector of one triangle side.

A student calls me over and asks me to refresh her memory on parallel and perpendicular lines. Okay. I asked her what was the same and what was different about 2 parallel lines. She got that the slopes were the same, but hesitated on whether the y-intercepts were the same. So I had her draw an xy-plane and draw me 2 parallel lines. I got this:

And with some prodding she said that the y-intercepts were not the same. Great. So I asked her what was true about perpendicular lines. She got that the slopes were opposite reciprocals. Great. I asked her if the y-intercepts were the same or different, expecting this to be a gimme. Now this is where it started to get enlightening. She couldn't tell me if the y-intercepts of 2 perpendicular lines were the same or different. So I thought, "great! the drawing helped before, so it'll help again." I asked her to use one of the lines before and draw a line that's perpendicular to it. I got this:

Slight alarm bells go off. I ask, "what's true about perpendicular lines?". She quickly states that they form 90 degree angles. So then I thought the x and y axes were throwing her off. So I made her draw a tilted line without the axes and put the perpendicular line on it somewhere. I got this:

Then I thought, well what if we start over and the 2nd line is her pencil that she can just move around to get perpendicular. I got this:

Then we had a LONG discussion (note that there are 23 other people in class practicing and thankfully none was too needy or off task) ... also note that the quiet girl next to this student was surreptitiously eavesdropping like she needed this information, too ... which made me nervous because I started thinking how many OTHER quiet strugglers are there out there? ... Anyway, I got it down to that IF the perpendicular lines form 90 degree angles (which she got), and the angles were supplementary (which she got), then the 2 angles should LOOK congruent. So next I got this:

And then this:

So what's the point of this story? I guess I always have in the back of my mind that if the student is not understanding something, they'll ask for help. But what if what they are not understanding is such a basic concept that it does not even come up in the scope of the course other than something that's used as a tool for the course (i.e. how do perpendicular lines look like when you draw them? what makes a shadow and how do shadows work) that the kid doesn't think or know to ask. And what if there's so much information and so many students to teach that you're going to miss these critical gaps. ... I won't always have the luxury of spending 5-8 minutes with one student and multiply that by the number of students.

Maybe I need some EXTRA problems on homework or class practice that can be a diagnosis of basics for each concept, and then I have to go over them very carefully to probe kids that may be missing elementary things that are holding them back and remediate further.

## Monday, April 30, 2012

## Thursday, April 26, 2012

### Algebra Algebra Algebra

I don't know why I'm surprised. I knew that from teaching calculus a long time ago, that it always seemed to be the pesky algebra that mostly tripped kids up. I guess I thought it would be different with my current crop of precalculus kids. They're different, I told myself. They're more savvy, I said. They won't have the standard problems, I lied to me. No. They're not and not and they will. And I hesitate to say "algebra". More like "simplifying rational expressions".
We're going over algebraic manipulations of limits, and I gave them basic polynomials and basic "single numerator / single denominator" expressions. They did fine. They were all impressed with themselves that they could chat up limits like they owned the place. And then WHOA. We delved into fractions imbedded in fractions. Huh! What do you mean I have one over "w+4" minus 1/4 ALL over w? What would I do with that? What? Common denom whozits?
Oh! So to get the common denominator of "1 over 'w+4'" and "1/4", I can just add 4 to the top and bottom of 1/4! Who knew!
Or! Once I get that sorted out, when I then have to simplify dividing by w ...... oh! I can just bring it to the "top" and multiply by w, right?
Oy!
And in another problem, the numerator was "9 - v" and the denominator was "(v - 9)(.......)". The student didn't see the connection of how to easily and correctly manipulate the two. It seemed like cheating to her when I mentioned how.
I guess it's a never-ending refreshing of their memory and of practicing and going over concepts. They're so used to JUST dealing with fractions involving ONLY numbers that this may have been too much of a leap for them. Now I'm debating whether to delve into derivatives or to just sprinkle in more intensive algebra and unit circle refreshers in hopes that they'll have an easier time next year in calculus.

## Thursday, April 19, 2012

### The Limit Does Not Exist

I'm at that (one of many) fun part of precalculus where we talk about limits. We do a day on finding limits graphically. I have them choral chant and watch my bouncing finger run over the notation, "the limit of f of x as x approaches ____ is...". Then on the 2nd day we talk about finding limits algebraically. I usually have skeleton notes and am done with it. This year, I went a step further and made a final chart for them to paste in their notebooks.

It's also one of my favorite topics because I get to use my "spit in the ocean" phrase. When we talk about limits of f(x) as x approaches infinity, and I tell them to only look at the highest powered terms in each of the numerator and denominator, and we test examples on the graphing calculator ...... I describe the largest term like the ocean and everything else as spit. The spit does not change the volume of the ocean significantly.

And without fail, every year someone brings up "Mean Girls".

It's also one of my favorite topics because I get to use my "spit in the ocean" phrase. When we talk about limits of f(x) as x approaches infinity, and I tell them to only look at the highest powered terms in each of the numerator and denominator, and we test examples on the graphing calculator ...... I describe the largest term like the ocean and everything else as spit. The spit does not change the volume of the ocean significantly.

And without fail, every year someone brings up "Mean Girls".

## Sunday, April 15, 2012

### Perfecting Recipes...

Totally off topic, but one of my latest obsessions: the perfect chocolate chip cookie recipe.

I like to bake, and I like home-baked goodies (and I need to lose some pounds .... connection?). I'm a wee bit snobby about store-bought because when I see all those extra ingredients listed on the label, it does not sound appealing at all. And once you've tasted home-cooked, store bought just doesn't taste the same. Does that mean I'll turn my nose up at an Oreo? Heck no. I'm talking about those hard-plastic-encased "fresh made" stuff from the bakery section. I'll imbibe my extra calories the old-fashioned way, thank you very much.

Anyway, I thought I had a good thing going with my chocolate-chip cookie recipe. I like to add spices to it and increase the vanilla extract and put oatmeal flakes in the dough. I scooped them up with a tablespoon, and poof! we had delicious cookies.

Then a person had generously baked LARGE cookies and brought them to me twice. HUGE. I asked, and this person used a quarter cup measuring cup to put the dough on the cookie sheet. The texture was great, the size was great. Home-made cookies. Delicious.

Now this part is totally NOT sour grapes, but the flavor .... eh! It was a bit too bland and too sweet for me, but did I scarf them all down? Why, yes I did.

So a few weeks ago, I attempted to make these 1/4 cup sized cookies with one of my recipes, but I didn't know what temperature to bake them at or how long to cook them. The flavor came out great, but they were hard as rocks and the texture was too floury. I jokingly wrote a note to the above-mentioned person and nicely asked for her recipe. I figured I could tweak it with my flavorings and happily chomp away.

I was refused. Refused! What kind of person doesn't share a recipe? Maybe this is more common than I know. I'm always willing to share, and I'm happy to share and am flattered that someone wants a recipe. It's not like it's my own creation. It's not like my food will taste any less delicious if someone else is making them. Sheesh. The person was a bit apologetic and said, "I don't even share this recipe with my closest friends". Oy!

Game on girlie. I've since then been testing out recipes, and have finally found one that has the right sweetness level. I also like to add cinnamon and cardamom and cloves and nutmeg and oatmeal. I'm going to experiment with orange rind after a coworker mentioned she did that in a cookie that was delicious. I still don't like the texture of my latest batch, so I have to adjust that.

One interesting note. I'm wondering if everyone's vision of the "perfect" c.c. cookie is akin to everyone's idea of the perfect banana. You know how there's a tiny window when a banana's taste is JUST right? And how everyone has a different idea of when that window is? Well, after testing my cookies out on various people, I think this is the same with c.c. cookies. I've had conflicting suggestions on what would make the perfect texture/flavor/size from a variety of people. Everyone has installed their own window of "acceptability" of a c.c. cookie.

I do think that I have to not test these recipes but once a month or less. Unless I want to totally have to go buy new clothes. But I will create my cookie, and I will share.

I like to bake, and I like home-baked goodies (and I need to lose some pounds .... connection?). I'm a wee bit snobby about store-bought because when I see all those extra ingredients listed on the label, it does not sound appealing at all. And once you've tasted home-cooked, store bought just doesn't taste the same. Does that mean I'll turn my nose up at an Oreo? Heck no. I'm talking about those hard-plastic-encased "fresh made" stuff from the bakery section. I'll imbibe my extra calories the old-fashioned way, thank you very much.

Anyway, I thought I had a good thing going with my chocolate-chip cookie recipe. I like to add spices to it and increase the vanilla extract and put oatmeal flakes in the dough. I scooped them up with a tablespoon, and poof! we had delicious cookies.

Then a person had generously baked LARGE cookies and brought them to me twice. HUGE. I asked, and this person used a quarter cup measuring cup to put the dough on the cookie sheet. The texture was great, the size was great. Home-made cookies. Delicious.

Now this part is totally NOT sour grapes, but the flavor .... eh! It was a bit too bland and too sweet for me, but did I scarf them all down? Why, yes I did.

So a few weeks ago, I attempted to make these 1/4 cup sized cookies with one of my recipes, but I didn't know what temperature to bake them at or how long to cook them. The flavor came out great, but they were hard as rocks and the texture was too floury. I jokingly wrote a note to the above-mentioned person and nicely asked for her recipe. I figured I could tweak it with my flavorings and happily chomp away.

I was refused. Refused! What kind of person doesn't share a recipe? Maybe this is more common than I know. I'm always willing to share, and I'm happy to share and am flattered that someone wants a recipe. It's not like it's my own creation. It's not like my food will taste any less delicious if someone else is making them. Sheesh. The person was a bit apologetic and said, "I don't even share this recipe with my closest friends". Oy!

Game on girlie. I've since then been testing out recipes, and have finally found one that has the right sweetness level. I also like to add cinnamon and cardamom and cloves and nutmeg and oatmeal. I'm going to experiment with orange rind after a coworker mentioned she did that in a cookie that was delicious. I still don't like the texture of my latest batch, so I have to adjust that.

One interesting note. I'm wondering if everyone's vision of the "perfect" c.c. cookie is akin to everyone's idea of the perfect banana. You know how there's a tiny window when a banana's taste is JUST right? And how everyone has a different idea of when that window is? Well, after testing my cookies out on various people, I think this is the same with c.c. cookies. I've had conflicting suggestions on what would make the perfect texture/flavor/size from a variety of people. Everyone has installed their own window of "acceptability" of a c.c. cookie.

I do think that I have to not test these recipes but once a month or less. Unless I want to totally have to go buy new clothes. But I will create my cookie, and I will share.

## Wednesday, April 11, 2012

### Interviews...

This year provides another first for me. I'm on the "interviewer" side of conducting interviews for possible openings next year. I'm not the only one asking questions, thank goodness, because it's good to get varied feedback and reactions to candidates. But here are some surprising things that seem to be a running theme. And by surprising, I mean, "have you not gone to basic interview 101 school, people?"

When you are asked, "do you have any questions?", here possibly are not good responses:

* No, I really can't think of any.

* Um, what time is your school day over?

* No, but you know, people always seem to ask that.

* Do you have to tutor a lot?

When asked about your career goals, you may not want to mention that in a year or two you plan on doing something else.

When showing up, if you haven't brought any samples of your work, why not?

When asked why you want to work at this particular school, hopefully your answer shows that you've researched us and can say something beyond words that imply, "I want to work with an easy population". First off, there's what SEEMS to be true and then there's reality. And also you don't just say you want a change from where you are, and that's it.

Oh, and maybe you can show up on time.

And maybe you want to wear something other than your sloppy work clothes.

Just saying.

Anyway, very interesting.

When you are asked, "do you have any questions?", here possibly are not good responses:

* No, I really can't think of any.

* Um, what time is your school day over?

* No, but you know, people always seem to ask that.

* Do you have to tutor a lot?

When asked about your career goals, you may not want to mention that in a year or two you plan on doing something else.

When showing up, if you haven't brought any samples of your work, why not?

When asked why you want to work at this particular school, hopefully your answer shows that you've researched us and can say something beyond words that imply, "I want to work with an easy population". First off, there's what SEEMS to be true and then there's reality. And also you don't just say you want a change from where you are, and that's it.

Oh, and maybe you can show up on time.

And maybe you want to wear something other than your sloppy work clothes.

Just saying.

Anyway, very interesting.

## Sunday, April 01, 2012

### Triangle Inequalities

On Saturday we went to see a show by this company (amazing tap dancing) and here's some footage from a past show. And as I was sitting in my seat fidgeting with the ticket stub, I got what I think was a great idea. Then I thought, oh surely other people already do this, and I just missed the boat. And now, maybe I'll never sail on that boat again for a while because I may not be teaching geometry next year. And then I went back to fiddling and thinking with my ticket stub.

I was thinking about the lesson on the restrictions on the 3 side lengths that can potentially make a triangle - how any 2 sides of the triangle have to sum up to more than the length of the 3rd. I've taught it various ways. We always explore with a variety of things. BUT. I've never used anything I'm completely satisfied with:

- spaghetti: messy, hard to measure, the thickness may allow students to think it's still possible, hard to get the situation where 2 of the shapes EXACTLY add to the 3rd.

- linking cubes: too thick, and sometimes students would still think they could make a triangle when they couldn't.

- cut strips of paper: see above

- drawing the triangles using compasses set at various lengths: too cumbersome.

PAPER SHOW TICKET to the rescue! Here's my idea. Cut some long strips of thickish paper (cardboard/stock paper?), or have them cut them. Maybe give one or a few to each student. Then have the students experiment with laying it flat on the desk and folding in from the two ends. They can write down all the situations they tried and measure the three resulting sides in centimeters and see what sorts of situations allow them to make a triangle from the resulting 3 sections of the strip. You may even have guiding questions about: "what 3 lengths are not even in the running and why not?" .... "what 3 lengths ALMOST look like they'll make it, but don't?" ... something to that effect.

I think this will be much better than my old props: easy, able to try a variety of cases, cheap, effective.

I was thinking about the lesson on the restrictions on the 3 side lengths that can potentially make a triangle - how any 2 sides of the triangle have to sum up to more than the length of the 3rd. I've taught it various ways. We always explore with a variety of things. BUT. I've never used anything I'm completely satisfied with:

- spaghetti: messy, hard to measure, the thickness may allow students to think it's still possible, hard to get the situation where 2 of the shapes EXACTLY add to the 3rd.

- linking cubes: too thick, and sometimes students would still think they could make a triangle when they couldn't.

- cut strips of paper: see above

- drawing the triangles using compasses set at various lengths: too cumbersome.

PAPER SHOW TICKET to the rescue! Here's my idea. Cut some long strips of thickish paper (cardboard/stock paper?), or have them cut them. Maybe give one or a few to each student. Then have the students experiment with laying it flat on the desk and folding in from the two ends. They can write down all the situations they tried and measure the three resulting sides in centimeters and see what sorts of situations allow them to make a triangle from the resulting 3 sections of the strip. You may even have guiding questions about: "what 3 lengths are not even in the running and why not?" .... "what 3 lengths ALMOST look like they'll make it, but don't?" ... something to that effect.

I think this will be much better than my old props: easy, able to try a variety of cases, cheap, effective.

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