Caclculus II: Parametric Equations

Last week I realized how much hand-holding my calc II students needed so they could get started with parametric equations. Most of them are very weak in trig. Many of them aren't sure how to get started when they're doing something new with graphing.

So I slowed down. But I also wanted them to do more exploration and experimenting. On a whim I gave them the typical assignment to draw something using parametric equations (or polar graphs). It can be anything, be creative. I don't have a particular assignment yet.

Sometimes (for some students) less is more. One of my students emailed me with a question, I replied, we went back and forth with over a dozen emails, and he produced this loveliness. I've included a screenshot below, but what he did is animated, so click on over to Desmos.

Wow! I hope his enthusiasm inspires the rest of them!

All You Need Is ...

(from the facebook meme, pictures removed)

[.docx here.]

If We Knew How to Trust

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If We Knew How to Trust

we see the amazing effort

young kids put into learning to walk and talk

(this work is their play),

and we know they are capable

of miraculous feats of learning.

if only we knew how to trust

their ability to continue learning

just as powerfully and miraculously,

maybe we could build a school system

that would honor every child’s fierce desire

to master the world’s skills.

their differences would no longer sort them

into good, mediocre, and bad students,

but into artists, scientists, poets, musicians, mathematicians,

writers, inventors, leaders, organizers, and more.

and each child would be many of these,

their differences adding to their strength as a community,

their school an ecosystem of learning.

written by Sue VanHattum, inspired by Lisa Cooley, 2014

[And here's another poem on what schools could be.]

Math Mama's Gazette - Issue Number One

I am creating a two-page newsletter, aimed at community college math students, which I'll be handing out to students, both at our Math Jam program these two weeks before the fall semester, and at the math lab during the semester.

I'm happy to share it with others. I hope to have one issue for each week of the fall semester, 15 to 17 issues. If you use  it, you'll have to change the bits that refer to my college. And please include this line: "Math Mama is Sue VanHattum, who blogs at mathmamawrites.blogspot.com." My copy is two-column. You can see it here. (Let me know if that link isn't enough to get you an editable copy.)

Like it? Please let me know.






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Math Mama’s Gazette

Issue Number 1, August 4, 2014

Math isn’t news, why a newspaper?

Well, there’s lots about math that’s news to most folks, and a gazette sounded fun. I’m all about fun, so I decided to go for it.

This first issue and the next few will have lots of ideas (some surprising) about how people learn math. If you’ve never enjoyed math, or never done very well with it, try changing your perspective with some of these tips. You might like the results.

Every issue will include a not-so-traditional advice column, a puzzle, and a comic, newspa­per favorites. There will also be links to cool math stuff online.

A Few Math Myths

Myth #1: Learning math is learning how to follow procedures - there's a lot to memorize.

Myth #2: Some people have a 'math mind' and some don't. (A more unfortunate variant of this is: Men are better at math than women.)

Myth #3: Math requires simple logic; intuition and creativity have no place.

Myth #4: There is one right way to do math problems.

Myth #5: I don’t need to know math - I’ve got my calculator and the Internet.

Myth #6: Mathematicians do problems quickly, in their heads, by working alone until the problem is solved.

In this issue, we’ll address myth number one. (Keep coming back for more myth-busting.)

What is math? Is it procedures?

Most people think it’s adding, subtracting, multiplying, and dividing; knowing your times tables; knowing how to divide fractions; knowing how to follow the rules to find the an­swer.  These bits are one tiny corner of the world of math.

Math is seeing patterns, solving puzzles, using logic, finding ways to connect disparate ideas, and so much more. People who do math play with infinity, shapes, map coloring, tiling, and probability; they analyze how things change over time, or how one particular change will affect a whole system.

Math is about concepts, connections, patterns. It can be a game, a language, an art form. Everything is connected, often in surprising and beautiful ways.

What do you memorize?

I went into math because I have a bad memory. If I had trusted my memory to be up to it, I think I would have gone into science.

[continued on back]

Puzzle: Math Without Words

by James Tanton (jamestanton.com)

Math Mama’s Advice

Dear Math Mama, I am a math tutor at a small Los Angeles community college. The students I have who need the most help are older women who are back in college, or here for the first time, who have had unpleasant math experiences in their youth. Do you have any ideas for us?

- Paula

Dear Paula, I had quite a few older women in one class last fall, and I had them in mind as I thought about your question.

First, I think it's important to address their fears directly. I recommend Managing the Mean Math Blues, by Cheryl Ooten, or Overcoming Math Anxiety, by Sheila Tobias. You can get used copies online for $3 or $4. My favorite site for that is betterworldbooks.com.

I also recommend an audio track I created, called Math Relax. It's a guided meditation to help people overcome math anxiety. It works best if the student listens to it every night for a few weeks. (Go to mathmamawrites.blogspot. com, and look on the right-hand side for the Math Relax audio track. It’s free.)

I think helping them lead from their strengths might be even more important, though. I try to help each class become a community. Some groups take off with it, and others don't. The older students know what they want, and are ready to go with it.

This particular class became an amazing community. Most days they came in over an hour early (we were so lucky the classroom was empty before their class!) and studied together. One of the students led the group, and even though I like getting questions in class, they felt freer to ask questions in their group. They were each determined to ‘get it’, and kept at it until they did.

I asked my students what advice I might offer you, and they said that working together was key. They said keeping each other going when it got tough was the most important thing they did for each other.

If you tutor one-on-one, you could still help this dynamic along by introducing the students to each other. Have you heard that “the one do­ing the most work is the one doing the most learning”? That would mean that you learn more from tutoring than they do - unless you can get them helping each other.

Perhaps if you recommend some of your favor­ite online resources for them to check out, they'll discover things that excite them. Many of my students really like watching math vid­eos. Check out vihart.org, khanacademy.org, mathtv.com, or (my favorite) jamestanton.com.

Good luck, and thanks for writing.

- Math Mama

Have a question for Math Mama? Deliver it to Sue VanHattum, in AA-210, and Math Mama will answer it in the next issue!

What do you memorize?    [continued from front]

But I figured there was no way I could memorize all those bones and muscles, chemical reactions, and so on.  So I stuck with math.

You need to know your multiplication facts to be able to factor numbers and polynomials smoothly. (If you don’t know them, there are easy ways to commit them to memory now. Professor VanHattum has a handout on this.) You’ll want to know that the x-axis is horizon­tal, and the y-axis is vertical, for algebra. And in trigonometry you’ll need to memorize a few definitions. Most everything else is more about understanding the connections than about memorizing.

from xkcd.com

A Sloppy Computerized Test

My college is running a program called Math Jam for two weeks before the semester begins, and it sounds fabulous. I'll be teaching it for the first time, starting on Monday. We use MyMathLab, and our director said students in prior years (who loved Math Jam) found the program helpful. So I will use it with my students next week, along with lots of other, more interesting mathematical explorations.

I checked out the first test just now, and got below 90% in a Beginning Algebra, or perhaps pre-algebra, topic. Let's look at why. I had 3 questions at least partly wrong out of 22.

#1
On an equilateral triangle, they asked for the height. I found it, rounded to tenths as requested, and got that right. Then they wanted me to find the area using the rounded answer. I did not do that. I did what I teach my students to do: Use the exact answer in your calculations, and only round at the end. My answer did not match theirs.

I can't get back to their problem now, so I will make one that's similar. Suppose the sides are length 6in. Then the height is 6*√3, or 10.3923.... If they are asking us to round to hundredths, we'd report 10.39in for the height. Now they want area, and they ask me to use my 10.39 as the height. But the proper area (in sq.in.) is 62.3538... and by their method we'd get 62.34 sq.in. I put the proper answer of 10.35 sq.in. and got it wrong.


#2
I got 0.55 as an answer, which they asked me to round to tenths. Both 0.5 and 0.6 should be right, as 0.55 is exactly in the middle of these. But only 0.6 counts as right on this test, and I had (randomly) chosen 0.5.


#3
This one is the most interesting. They gave the diagram below, which looks a bit badly done to me. The right angle mark toward the left does not seem to coincide with the line below it, making it seem like the angle isn't really a right angle. Not a big problem, I can still assume a right angle there. But they have only marked two right angles. I do not believe I have enough information to determine the area of this figure unless I know more about the angles. I believe the one unmarked line segment has an unknown length. I think they meant to show two attached parallelograms (or a parallelogram attached to a rectangle), but that's not a given from this diagram.

What do you think?

I think they need to learn more about rounding, and more about what one can read from a figure. Hmm... I wonder if all their tests will be this sloppy.

Playing With Math: Crowd-funding Campaign Has 9 Hours Left...

... and we need almost $1000 to meet our stretch goal. I am hoping we estimated a little bit high so that the Spanish translation can still be done quickly. It would be great if Vi Hart saw my message on her Facebook page, and decided to check us out. But it doesn't look like that's going to happen.

We've raised $10,569 so far, and may raise a few hundred more by the end of the day. We surpassed our original goal by $3000. And more importantly, contributors have reserved almost 300 copies of the book. We are eager to know what they all think once they've had a chance to read it.

If you haven't contributed yet, you still have a few hours left. $25 will reserve you a copy of Playing With Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers. I think you'll be very happy to have this book in your home.

Playing With Math: Crowd-funding Campaign Ends on Saturday

I hope to post soon about my lovely adventures in math at the Math Circle Teacher Training Institute. But that post will have to wait until the crowd-funding campaign for Playing With Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers is over. We reached our original goal of $7500 on Sunday, July 6, after only 17 days, giving us the funds we need to publish this fabulous book. Our stretch goal is $4000 more, for the Spanish translation. We are about halfway there, but we only have 3 days left. Can we make it?

If you haven't reserved your copy yet, please do it now!

While I was visiting family and friends in Michigan, I spent some time this past weekend with my friend Chris (who has helped with the book from the very beginning). While our kids played in the pool, we discussed index entries, and which terms might need cross-referencing. For example, Jamylle Carter had students build their own inclinometers, and wrote about it in her chapter on the Oakland Math Circle. If you had read about them, forgotten the word, and wanted to re-read her description a year later, you might think of them as angle-measuring devices, and look up angle. So we’ll cross-reference her inclinometer description under angles, measurement.

Here's a picture of an inclinometer. (The boy was drawn with a photo of my son as the model. He's older now.)

Don't forget to get your copy of this fabulous book.

How I'm Playing With Math Today: Geometry

In between sessions of trying to prep the manuscript for page layout, I've been playing with Euclid: The Game.  I am loving it. It may be just the same constructions kids learn to do in geometry class. But geometry is my weakness in math, and I love trying to figure out how to do these constructions. Exactly a year ago, I posted about another construction game. The two are different enough that you might enjoy doing both. I'd love to hear what you think of them.


Here are a few links to other geometry construction tasks:

• Andrew Stadel's Name Tent Challenge
• Constructing a Heptagon (Is it possible?)
• Trisecting the angle (impossible with Euclidean axioms, possible with origami axioms)

More geometry links:

• James Tanton worked with Great Courses on a geometry video series. Cool!
• Geometry Tasks

Playing With Math: Almost to the Finish Line!

Campaign Update
I would love to be able to see what's causing our good days. I have no idea what made July 1^st our best day for number of contributors since the beginning of the campaign, with 23 contributors, including ... a $1000 contribution from Nancy Blachman, founder of the Julia Robinson Mathematics Festival. Thank you, Nancy! We are now at 93% of our goal, with $6960 coming in from 212 contributors. Our thanks go out to each one of the 212. Every contribution makes a difference.

Yesterday was our lowest number of contributors yet, just 3. And today may be low, too, with everyone out having fun on the 4^th. So if you know someone who you think would like Playing With Math, please let them know about it.



The Book Reviews
Recently, Sam Shah and Beverly Baird have posted lovely reviews. And I began to be aware of something very cool. Each reviewer notices different things about the book, and uses different chapters when they mention their favorite parts of it.

Every chapter is special to me in one way or another, or it wouldn't have made it into the book.  But of course other people don't always love the same things I do. So it's great to hear the love coming in about so many different chapters.

I started the process of compiling the stories in this book as a story-lover and a math-lover, with very little interest in illustrations. I knew the book needed them to break up the text, but I didn't have much sense about what that would involve, what sorts of illustrations would be helpful, or even how to manage them on my computer. (I was saving lots of low-resolution images until half a year ago, which caused lots of trouble that I've finally taken care of.) I have come to love the illustrations we've pulled together.

Sam quotes Rodi Steinig's chapter, On Noticing and Fairness:

We began today’s math circle, the first of six sessions, sitting in an “ogre.” Not a circle, not an oval, but an ogre, the kids’ way of precisely describing the shape we made.

The kids were voting on the animals to be included in Zooman's private zoo, and an ogre sounds right at home in that discussion. Their first vote led to the tamandua (ant-eater) winning. Here's our tamandua...

 

Beverly mentions Julia Brodsky's interest in encouraging the children “to make mistakes and enjoy it.” As it turns out, that theme is repeated throughout the book. Here's an illustration that warms my heart, from Mary O'Keeffe's chapter, Agents of Math Circles.

I don't think anyone will buy the book for the illustrations, but if you find yourself reading it over and over, as I have, the illustrations will be one of the special graces of this heart-warming collection.



A Puzzle
There hasn't been enough math on this blog of late. So here's a puzzle-problem from James Tanton:

Give an example of a cubic polynomial and a quadratic whose three points of intersection form an equilateral triangle.

Playing With Math: Crowd-Funding Campaign Enters its Second Week

The Book

Did you know that that there are over 50 people working on Playing With Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers?! We are authors, puzzle & game-makers, artists, and editors. The most recent additions to the group are mathematical artists who agreed to share a piece of their work - the opening page of each section now features an amazing, thought-provoking work of art.

To see some of their work, you can visit these websites:

• The Math Circle Section piece comes from Paul Salomon (already in our group for his wonderful imbalance puzzles). Here's more: lostinrecursion.wordpress.com/art/stars-of-the-minds-sky
•  The Homeschoolers section piece comes from Erik and Martin Demaine. Here's more: erikdemaine.org/curved
• The Passionate Teacher section piece comes from Tom Banchoff and Davide Cervone. Here's more: http://www.math.brown.edu/TFBCON2003/art/welcome.html  and  http://www.math.union.edu/~dpvc/professional/art/welcome.html
• The Resource section piece comes from Henry Segerman. Here's more:  www.shapeways.com/shops/henryseg

Here's the last illustration that was made for our book. It was a community effort with original concept by me - inspired by Melanie Hayes' chapter (At the Eye of the Hurricane), drawing by Linda Palter, caption by a reader from this blog who I know only as Teach for Life, and lettering by Maria Droujkova. I love it, and I hope it goes viral some day...

​

The book is now ready to go to page layout. What a thrill! I am so proud of all of us.

The Campaign

Sol Lederman interviewed me last week, and posted the interview as part of his Inspired by Math series. Just before he posted it, Adrian Pumphrey, of MathEd Out, offered to interview me too. That happened on Wednesday, and is now posted on his site. If you like podcasts, listen to the interviews and let me know what you think. The two conversations definitely went in different directions.

As of this writing, we have over $5000 in contributions pledged, 67% of our $7500 goal, with 177 supporters. With your help, we can keep the momentum up. Maybe we'll be lucky enough to talk about our stretch goals toward the end of the month. Getting translations out quickly would be lovely.

Are you inspired enough to head over to the campaign site right now and make a $25 contribution (or more) to reserve yourself a copy? I hope so.