Saturday, March 21, 2015

Algebra Skills Needed for Calculus

Sam Shah posted his list here. I loved his list, but wanted to rewrite it a bit for myself. (Also, Sam finds it more effective to review the algebra ahead of time, while I think it's more effective to review once we see the need in our exploration of calculus.) I am posting this now, so it's available as an answer to this question on math educators stack exchange.

I teach my calculus course in an order that I think will help students learn. I have four units:
  • Unit 1 includes history, graphing functions, slopes of tangent lines by approximation, algebraically finding the derivative using the limit (which we do not carefully define yet), seeing the similarities between velocity, rate of change, and slope, average versus instantaneous velocity, derivative from a graph, (estimated) derivative from a table of values.
  • Unit 2 includes derivative properties needed for polynomials, graphing, limits and continuity, trig derivatives, and optimization.
  • Unit 3 includes chain rule, derivatives of exponential functions, implicit differentiation, derivatives of inverse functions (ln x, tan-1x), and related rates.
  • Unit 4 includes integration (finding area under the curve), anti-derivatives, fundamental theorem of calculus, and substitution method. If there is time we include volumes of rotation (which I think is a perfect ending for the course).


Algebra Skills needed for Unit 1 

Algebra 
  • Determine the equation of a line given two points, or a point and a slope, or a graph of a line, 
  • Find the average rate of change over an interval given a function or its graph, 
  • Clearly express what is happening to an object given a position versus time graph, 
  • Evaluate f(x+h) for any given function f(x), 
  • Rationalize the numerator (to find the derivative of the square root function) , 
  • Simplify complex fractions (to find the derivative of the 1/x function). 

Algebra with Calculus Concepts 
  • Approximate, using two points close to each other, the instantaneous rate of change at a point, given a function or its graph, 
  • Explain clearly why the procedure you used gives an approximation of the true instantaneous rate of change, 
  • Sketch a velocity versus time graph given a position versus time graph, 
  • Construct the formal definition of the derivative by modifying the definition of slope, 
  • Apply the formal definition of the derivative to simple polynomials and to simple square root functions.


Algebra Skills needed for Unit 2

Algebra
  • Multiply out the expression (x+h)n (necessary to understand the proof for the derivative of y=xn),
  • Identify the holes, vertical asymptotes, x- and y-intercepts, horizontal or slant asymptote, and domain of any rational function,
  • Sketch the basic shape of a rational function,
  • Identify an equation for a rational function given a sketch of the function,
  • Explain clearly what a hole and an asymptote are,
  • Construct the equation of a piecewise function given its graph,
  • Sketch the graph of a piecewise function given its equation,
  • Work with inequalities,
  • Give both triangle and circle definitions of sin x, cos x, and tan x, and explain how they’re related,
  • Evaluate sin x, cos x, and tan x at all multiples of  π/6 and  π/4, without a calculator,
  • Understand trigonometry identities, including and sin(x+h)=sin x cos h + sin h cos x,
  • Accurately graph y = sin x and y = cos x.

Algebra with Calculus Concepts
  • Graph a polynomial or rational function, showing its maximums, minimums, and inflection points,
  • Follow complicated logic (in the definition of limit).


Algebra Skills needed for Unit 3

Algebra
  • Understand composition of functions,
  • Use logarithm properties to “break apart” a single logarithmic expression into simple logarithms,
  • Understand properties of exponents,
  • Be able to graph exponential and logarithmic functions.

Algebra with Calculus Concepts
  • Think in terms of composition of functions to determine outer and inner functions, in order to use the chain rule.


Algebra Skills needed for Unit 4

Algebra
  • Work with summations.

Friday, March 13, 2015

Copy Number One of Playing with Math

At 3:30 this afternoon, UPS knocked on the door and delivered copy number one of Playing with Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers!

It is beautiful!

Now we put in the full order. Books coming soon...


If you haven't ordered your copy yet, you still can.

Friday, February 6, 2015

Linkfest for Friday, February 6

Before I share all the delicious goodies I've stumbled on, news of the book is in order:

Playing with Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers is just about done with page layout - and it's looking so beautiful! I am sending in the last proofreading corrections today, and will do the last fixes to page number mentions as soon as I've seen the final copy. Then it's off to the printers, then all the copies get shipped to the publisher, and finally get sent to the hundreds of people who ordered copies during the crowd-funding last summer. If you're eager for your own copy and weren't around for the crowd-funding, you can order now. (You know I'd be tickled if we sell out our first printing quickly!)



The Links
  • Two Truths and a Lie: Get calculus students to make up stories from their lives, using the idea of rate of change, and matching given graphs. Brilliant, Shireen!
  • I like this for a first day activity! (I just figured out how to link to this on my google calendar to remember to look at it in August!) Getting the students involved in discussing what education should be, and what productive failure might look like.
  • Explained Visually has animated graphics for trig functions, exponential growth, statistical processes, and more. Fun.
  • Beautiful teacher story. “You just listened, so then I could figure it out.” 
  • This post asks: Is there room for math that isn't hard? The post and comments are both interesting reading, and I'd enjoy seeing more comments. The blog is called Math Exchanges, and their more recent post, Over or Under, is great too.
  • About half a year ago, I joined in the crowd-funding for the math game Prime Climb. It arrived in early December (or was it in Novemeber?) and we played it at my holiday party. People definitely enjoyed it. Now I've heard about another game being crowd-funded. Three Sticks is a geometric game, developed in India. It looks fun. For a $35 contribution, you get the full set (and escape the very high shipping charges).
  • The math in the solutions may be too hard to follow, but this problem is charmingly simple: Your hallway is one meter wide, and turns a corner. What is the greatest base area of an object that can be carried flat through the corner?
  • I'm not so good at making things (origami, etc), but these pretty mathematical sculptures do look fun.
  • Every textbook I've seen that includes conic sections shows the conic, and then shows another definition, and never connects the two. This blog post makes some of the necessary connections. (Anything on Dandelin's spheres catches my eye.)
  • Tricky puzzle. (Do you like that sort of thing?) The 7 at the bottom is NOT a typo.
  • I'm always happy to hear about new math circles. Here's one in Santa Cruz, in the news.
  • Estimation questions are a great way to build number sense. And Andrew Stadel has a twitter feed just for that. This week included a few questions about these Lego Lions: How many legos? How long to build? How many legos tall?


A Question
I'm teaching Linear Algebra, and I find it a bit odd that linear transformations by definition don't include lines like y = mx+b (with b not 0). A student asked the significance of the word linear (she thought it was a silly question, and I assured her it definitely was not silly), so I started searching online. I noticed this site, which defines a linear transformation for statistics - differently from the linear algebra definition. It looks like the two definitions contradict one another. Any ideas about how standard this statistics definition is, or pointers to discussions of this difference in definition?



[Oops! I lost a few weeks on the #YourEduStory challenge. Maybe I can get back to it. My pre-calc class is going better than usual. My calculus students loved having all those handouts in a coursepack. And I love thinking about all the connections in linear algebra. This week's topic: Define "learning" in 100 words or less.]

Sunday, January 18, 2015

My Favorite Teachers and Me

The #YourEduStory blogging challenge question of the week:
How are you, or is your approach, different than your favorite teacher?

I don't have just one favorite teacher. I have lots. Long, long ago, before I started teaching, I made a list of my favorite teachers:
Mr. West, high school biology, and then anatomy and physiology
Ms. Purvins, high school Shakespeare teacher
Mr. A, high school poetry teacher
Mr. X, UM philosophy prof
Ms. Y, UM history of feminism prof
Gisela Ahlbrandt, EMU math prof
 There were probably more on the list at the time. These are the ones I still remember. (And I'm losing the names. Yikes!) When I made the list, I noticed something interesting. There were about equal numbers of men and women on the list, but they were very different sorts of teachers. The men were good performers, and the women were good facilitators. A few did both well (the poetry guy and Gisela). I wanted to do both well. I thought about taking some drama courses to improve my performance skills. I did that while teaching in Muskegon, and realized I needed a different sort of course. Performing in a play is a lot different than performing as a teacher. Improv might be good for me. Hmm... I also learned a lot about facilitation over the years.

I know now that the best performers make students happy to come to class, but that's not enough. We need to get students actively engaging with the material for them to learn much. (Mr. West did that in lab, even though I remember his great lectures.) If you don't know the research done by Eric Mazur on this, check it out.  (This video might include the best parts of the hour-long video I watched a few years ago.)

How is my approach different than theirs? I think it's only in the combination that I'm different. I try to pull in all my students (like my Shakespeare and history of feminism profs did). I ask them multiple times each class to show me with thumbs up, down, or sideways how well they understand what I've just explained. I call on students randomly. (Because teachers tend to call on male students more.) I come in as excited as my bouncy philosophy prof. I suggest my students try strange experiments, like my poetry prof did (he had us write at a cemetery and a mall). I try to be as accepting and as challenging as my best teachers were.

Math Circles at Nueva School

Nueva School, in Hillsborough, south of San Francisco, puts on a math night three times a year, with multiple math circles, along with a puzzle and game room. Nancy Blachman invited me to lead two math circles last night, one for 2nd and 3rd graders and another for 4th and 5th graders.


2nd and 3rd grade Circle
This circle met for just 30 minutes. I know that the Collatz conjecture is dependably fun for kids this age, so that was our main activity. I asked the kids what they thought mathematicians do, and got a reasonable answer, but saw that there wouldn't be time for useful discussion. So I said a bit about math being like a game for mathematicians, and how fun it was to come up with a new puzzle.

In 1937 (I just said it was about a hundred years ago), Lothar Collatz came up with this puzzle/game:
  • Pick a number.
  • If it's even, cut it in half. Write your new number.
  • If it's odd, triple it and add one. Write your new number.
  • (We drew an arrow from each number to the next.)
  • Repeat until you get back to a number you've already written.

Collatz conjectured (guessed) that the sequence would end up at 1, no mater what number you started with, but he couldn't prove his conjecture. Mathematicians have tried to  prove this for over 75 years, and it is still an open question. (It is very likely to be true. Using computers, people have tested every number up to and past 5 quintillion.)

As I expected, the kids loved it. At the end, I showed them a "mind reading" trick.
  • Pick a number from 1 to 31. Don't say it, just keep it in your brain.
  • (I pretend I'm sucking their thoughts over to my own head.)
  • Now show me which of these five cards it's on.
  • (I barely glance at the cards.)
  • Your number is ___.
After we did it a few times, I had the parents cover their ears and told the kids how it worked. I had  the five cards on the board, and half-size index cards for them to make their own cards. They loved it.


4th and 5th grade Circle
This circle met for an hour and a half. My plan was to analyze Spot It with them. (I've written at least 4 posts on using Spot It for math circles. Search on Spot It to find them.) We started out playing the game for about 15 minutes, which they all enjoyed.

The problem was, half of them had done this last year in their math class at Nueva! Luckily, one girl had come early and I had shown her the number trick. I asked her if she wanted to teach it to the others. She did.

I split the group in two, and she showed her group the number trick, while my group started thinking about the game. I had one boy who answered every question very quickly, and asking him to slow down didn't help. So, after we had figured out that there would be 57 different pictures, I got out the half-size index cards and suggested they make their own decks, with 4 pictures per card. Or, if they weren't into drawing pictures, 4 numbers per card. They worked hard at trying to make a deck where each card matched every other card on exactly one picture.  Towards the end, they wanted to play with the number trick too.

About halfway through the girl who led the other group came over and said, "The number trick is done." So I joined their group for a bit, and asked, "Why does it work?" A few parents were there, thinking about it with their kids. I should have asked them to work with all the kids (about 6 of them), but didn't think to say it. A few kids wandered away, to the puzzle room, no doubt.

The kids who stayed worked hard on the problems and had fun. I had a great time.

Friday, January 16, 2015

Days Three and Four

8am
Calculus. Wednesday: Circle area. Archimedes. Zeno. Started Boelkins' Velocity of a Ball activity. On Thursday, we got through most of the Velocity of a Ball activity. The students did not recognize that (s(b) - s(a)) / (b-a) is a slope. So We are working through the parts they need to review. I am goign slower than in other semesters. I hope I'm not going too slowly.


10am
Linear Algebra. Wednesday: Discussed differences between Echelon Form and Reduced Echelon Form. I started with: a matrix in Echelon Form, and got them to tell me the values of the variables. I explained that this way is quicker for computers. We talked about number of possible solutions, and drew examples in 2D and 3D. Quiz tomorrow. (Quiz made and copied.)

Thursday: Most of them aced the quiz. The ones who didn't will be in my office to retake. We finished 1.2. (I hate referring to book sections, instead of math topics. Basically, we are working on row reducing matrices. We've started to think about parametric representation of solutions, where there are free variables.)


11am
Pre-Calc. Wednesday: We practiced an arithmetic sequence (find the nth term) and a geometric sequence. We looked at a problem that used a recursive definition for a n. I mentioned the Fibonacci sequence, but didn't do much with it. Quiz tomorrow.

Thursday: Only a few aced the quiz. It was harder than what we had done in class. I'll give a retake on Tuesday in class. We reviewed lines. I walked them through my proof that perpendicular lines have slopes that are negative reciprocals. (It's different from the text's proof.) In the process, I also walked them through the proof that the angles in a triangle add up to 180 degrees. I love how the result suddenly pops out of the picture. I asked them to show me with their thumbs (up, down, sideways) how cool it was. They all gave it a thumbs up and I said they were being too nice. The bigger proof (for perpendiculars) gets an 80% coolness rating from me.

1pm
Calc III. (I am sitting in on this class.) Wednesday: Ed showed us how to connect the tops first and use dotted lines for hidden lines. I noticed that it felt like we were seeing the xz-plane from the back. Thursday: Over an hour of lecture. Ed is a good lecturer, but that's too long for me. I fell asleep. I woke up for the quiz. It included drawing 3D surfaces. I understand all of this, but how well did I draw? I'm not satisfied yet.


Tuesday, January 13, 2015

Day Two

8am
Calculus. I talked about what we had done yesterday with finding a line tangent to y=x2 at x=3. In algebra, we find the slope when we are given two points. We know one point, (3,9), and there is no other point that we know. [Last semester, at least one person used the y-intercept of the tangent line they had graphed as their second point. I liked that, but forgot to mention it today.]

I asked them to give their definitions of the word tangent.
First student definition of tangent: A line that touches the curve in one place only.
Sue's counter-example: I drew y=x3 and drew at tangent line at about x=1. They agreed that I had drawn a tangent. Then I extended the curve and the line. They cross at x = -2. I suggested that we could add the word nearby, and maybe this would work.

Second student definition of tangent: A line that touches the curve but doesn't cross it.
Sue's counter-example: I asked them what the tangent to y=x3 at x=0 would look like. They told me it would be horizontal. I drew it in. Hmm. (I told them that later we'll talk about concavity, and showed it with my hand curved. I said that I think the only time the tangent line crosses the curve is when it's tangent at an inflection point. Is that true? I should try to prove it.)

Third student definition of tangent: A line that determines the direction of the curve.
I think this one is about as good as we can get at this point, although it's hard to turn it into something precise. I talked about thinking of the curve as a road, and your point being a car driving along the curve. Its headlights make half the tangent line, and its taillights make the other half.

Talked just a bit about history of calculus, and gravity. Got some volunteers who will drop a heavy and a light object, and see what happens.

Then we started our circle activity. I had a picture of a circle of radius 10cm on the back of the handout. I asked for the radius, a rough estimate of the area, and a more careful estimate of the area. (I asked them to pretend they knew no formulas. Next I had them fold a round coffee filters in half through the middle over and over, then cut it into wedges, and play with them. Tomorrow we'll do the area formula from that. Today I gave the definition of pi (C/D), and talked about how C=2*pi*r comes easily from this definition. I got a few volunteers who will measure around a circle and across it, using string, and will bring in their string tomorrow. Area is different...



10am
Linear Algebra. I used a desk corner as the origin, drew the x and y axes with my finger along its edges, and the z axis coming up from the corner. I asked them to figure out (in groups of four) what the equation x+y+z=1 would look like. I heard someone say circle. It is not at all obvious to most of them yet that it will be a plane. But we got there.

Was that before or after we worked on the definition of a linear equation? Yesterday I had asked for their definitions from their heads. I got four volunteers today (yay!) to give me their definitions to put on the board. They were all different, and none matched the official definition. So, after I went over the official definition from our textbook, I asked them to use that to prove or disprove each of the statements given by students. I think this will help them with proofs and with what a linear equation is.

Next I continued with the problem we had done, algebra style (no matrix), yesterday. I talked about computers, and representing it with just the coefficients, and wrote the matrix. I showed them the matrix that would represent the solution, and said our steps will be similar to those we used yesterday, but our order will be different. We did our same problem matrix-style, and I identified the three elementary row operations as we used them. (We never used the swap rows operation, but I talked about when it would be needed, and how you'd never do that with the algebra-style method.)

I finished up with one book problem.


11am
Pre-Calc. Stamped their homework. Had them share with their group the list of 5 problems they couldn't do. Had them each pick a problem from their partner's list, that they would later explain to their partner. Some people working hard; others feeling unsure what to do. (Everyone willing to participate.)

Showed them y=mx+b on desmos, but got caught up in another problem. We'll come back to this tomorrow.

They worked on finding an, with the hint that it might be good to find a100 first, for the sequence 12, 17, 22, 27. (I got starting value and jump size from students. Good it was five - some people struggle with arithmetic.) We worked on that a while, and then did a problem from 12.1 (Stewart) that turned out to be geometric. It was good to see the similarities.

I loved my day. Now I'm off to the chiropractor.

Monday, January 12, 2015

First Day of Class

So I have this health problem. It's seems to be a GI problem, but no one has managed to diagnose it. It used to happen about once a year, for a week. Now it happens more often. Always before my tummy hurts, I notice that my back is tight. I've had lots of ultrasound scans - no gall stones. And a GI scope - no ulcer. It is not clear what's happening. The chiropractor seems to help, but it still lasts a week or so, just milder. The first few times, I was in the emergency room, in agony. This time, I made it through my first day of classes, able to ignore it, and am distracting myself by writing a blog post. Digestive enzymes may be helping. Stress is a factor: the last time it happened was the beginning of fall semester. The time before was when we were interviewing candidates for a position in our department. I think it's time to figure out how to notice stress, and not internalize it this way. Meditation? Yoga? Definitely, I need to get more active.

I was very pleased that I was well enough to ignore my body while I was teaching. I hate not being chipper on the first day.  I was probably less prepared, and less organized in some ways. But that meant that I did new things that I liked.


8am
Calc I. I had 42 students. I talked about math not being as much about memorizing as many students think. We listed some of the things that do need to be memorized, and then I talked about which things might not belong on the list. Then we did the tangent line activity I always start with. Time was a little tight, and I didn't finish attendance. (I'll have to attend to that tomorrow.)  ;^) Tomorrow we'll look at area of a circle, think a little more about today's activity, and start on the first activity from the Boelkins text. I've put together a coursepack with all the handouts for this first unit, so students won't feel so scattered. (We do very little from our Anton textbook in this first unit, and then use the textbook much more for the rest of the course.)

Even though the way I started wasn't anything exciting, I feel excited just because I did something new with the students. It feels like today was a good start.


10am
Linear Algebra. I used the same warmup activity from the last time I taught the course. I don't need to do anything new to be excited about this course. What's new it that I'll be grading their homework (with help from an assistant), instead of just stamping it.


In one of my classes, I talked about neuron development during learning. But I don't remember which class. I am nervous about having two classes in a row with no break between. I hope I can keep track of what I've done and want to do next.


11am
Pre-Calc. I only had 15 students. There is some chance the college will cancel the course. I think I have a great bunch of students, so I am already feeling very invested in doing this course. I have no control over whether it gets canceled, though...

This is the class I wanted to change up some. Our first day hadn't been very memorable in the past few semesters. I couldn't think of anything better than asking them to add up the numbers 1 to 100 (without adding one by one). They worked in groups of four. They looked very stuck for a long time, but seemed willing to keep at it. One student knew a formula, and had the answer written. I asked her to hide that, and try to figure it out without the formula. In another group, I heard something about adding up pairs. When I finally called them all together, I asked for volunteers and got none, and then asked M to explain. They  had started out by adding up the numbers one to ten. (I had suggested that might give them some ideas.) They did the sum one by one at first, so they knew what number it would be. When they noticed 1+10 = 11, and the total was 55, they looked for a reason to multiply by 5. Their reason didn't make much sense to me. (There are five ways to add two numbers to get 10, one of which is 5+5.) So I agreed that multiplying by 5 made sense, and asked if we could think again about that 1+10 pairing. I had to nudge more than I would have with a math circle, but they still did most of the work. They found the sum of one to a hundred in their groups. And then we came up with a 'formula' for summing the numbers one to n. I liked how it went.

I think this class may take off this semester...


1pm
I'm sitting in on Calc III. I loved Ed Cruz's talk, about how bad a student he was at first, and how he evolved to become a better student, and then a teacher. No math yet.

Tomorrow my 8am and 11am classes meet for an hour and a half. From 8am to 12:30pm, I'll only have half an hour outside the classroom. Yikes! But being done so close to noon sounded too great to pass up, so I didn't try to change this strange schedule. I see my chiropractor tomorrow afternoon.

Sunday, January 11, 2015

Share #YourEduStory Blogging Challenge 2015

I decided to join the Share #YourEduStory Blogging Challenge 2015 this morning. Last month was the first month I ever skipped completely on this blog, since I started in 2009, and last year I posted less than any other. So this is a good time to push myself a little.


One Word
The group that set this up (folks from edcamps) has created a list of topics for those who need them. The first week's topic (last week) was:
What is your "one word" that will inspire you in your classroom or school in 2015?

I'll choose 'Connect'. I want to connect with my students in the classroom and out. I want to connect with my colleagues as I work on creating a new course. I want to connect with friends more - I need more of that in my life.

I'd like to get photos of my classroom up, and videos of lessons. But I won't make a commitment to that. Just one new thing at a time...


A Better World
Week Two's topic question:
Inspired by MLK: How will you make the world a better place?

I hope my book, Playing with Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers, will help people to enjoy math, fear it less, and overcome the hurdles math requirements place in their way. It's coming out within the next month or two.


Join Us
I've been following hundreds of math teacher blogs, and, though I haven't pared down my list, I'm seeing very little writing from math teachers these days. Will you join the Share #YourEduStory Blogging Challenge 2015? Each week you write a post and tweet it, and read a post and comment. That's it.

Tuesday, January 6, 2015

Think: The Five-Letter Word Game

Linda taught me this game when I was visiting her in Muskegon. She just called it "the five-letter word game." I thought it needed a name, and Think is as good as any.

Preparation
Each person thinks of a five-letter word with all five letters different, and writes it down in a hidden spot.

Goal
Guess your opponent's word.

Play
The players take turns offering a five-letter word (all letters different), as guesses, and as clue-gathering. If the guess is not the right word, then the opponent tells how many letters are right.


The game is similar to mastermind, but there's no information about where the letters go. (So you could get all five letters right, and still have the wrong word!) It's not a math game, but it is a logic game, and I think logic and math are twins.


Example
I'll share my guesses:
STOMP 0 (No letters right. Great. I can cross all of those off my alphabet where I'm keeping track.)
BLAND 2
QUOTE 1
FLASH 2
DIRTY 1
TRUCK 0 (Now I know 9 letters that aren't in the word.)
VOWEL 2
EXITS 2
QUICK 1
LARDY 2
BINDS 1
NIGHT 2
LINER 3

Would you have picked other clue words? Do you have enough information from these clues to guarantee any of the letters in the word?
 
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