Prepping for Fall, Calc II: Lovely Arc Length Example

I'll be teaching Calc II for the first time in a few years. This is my first time starting out online with it. So I'm preparing my Canvas shell and thinking about how I want to explain each topic in Canvas. (I know the material well enough that I didn't have to prep this much when we were in person.) The extra prep before we start is so much work, but today it feels totally worthwhile.

 

 

For arc length I was excited to use "crinkle crankle walls" as an example. Isn't that a pretty wall? And you can actually use fewer bricks this way than for a straight wall, because one layer of bricks here is stronger than it would be straight (so the straight wall would need extra bricks for support). I'm thinking we'll try to prove that assertion in my Calc II class.

 

It turns out that arc length uses an integral which often has no "elementary solution", meaning there is no anti-derivative using the functions we are familiar with. 

 

The arc length for y=sin x is...

 

 

 

And this has no "elementary solution".

 

 

I often tell my students that we study infinite series to solve the integrals with no easier solution, but I just realized that that won't work here. (Can't do a square root of an infinite series!) 

 

Ok, no problem. I'm also teaching numerical integration. So I made a google sheet to do Simpson's method, and it turns out beautifully!! (Beautifully meaning that my answer matched the answers on Math SE that people explained in ways that were above my head. I don't know a thing about "elliptical integrals".)

 

I still need to remember how to explain Simpson's rule, but I'll get that back easily enough. 

 

If this wall follows a sine wave, then for 6.28 feet (2π feet) of straight distance covered, it has a length of 7.64 feet. That's just over 20% extra length. (Now to think with my students about whether that's better than the straight wall with supports.)

Technology Woes and Cheers: Venn diagram edition

I'm writing questions for my Discrete Math course that will be available to my students (and others) through MyOpenMath, a free online homework system. I'm not very good at programming in their environment, but I'm learning. The cool thing about MyOpenMath is that it uses random numbers in the questions so that each student might get a (slightly) different question.

I wanted a way to ask, for a random Venn diagram: What is the set notation for this?

First, I needed a way to make lots of Venn diagrams, all pretty, and all in the same style. I searched the internet for a free online Venn diagram maker. Nothing right showed up. I looked at over a dozen sites. Many wanted me to sign in. That should not be necessary and I skipped those. None of the others were even close to what I wanted, which is pretty simple. Really?! Isn't this something lots of people would want? 

I asked about it on Math Educators Stack Exchange. Within hours, Cameron Williams posted an answer. He made it on desmos for me. (How sweet is that?! Amazing.) I know desmos well, so I was able to modify his version to be exactly what I wanted, in less time than I had already spent searching. (I suggest you go play with it - it's lovely.) And, if you want orange shading instead of blue, it's very easy to modify this to get exactly what you want.

Then I made 17 screenshots of various combinations of the basic regions, and named them based on the set notation. So "(A un B) int not C.jpg" is the filename for ...

 

Next I went back to MyOpenMath, and wrote most of my multiple choice problem. I'm still stuck on how to get it to display a randomly chosen image file. I think the folks at the help forum there will help me out on that. Once I finish fixing it, I'll edit this post to show the question. MyOpenMath allows attached videos to explain how to answer the questions. I think I might do a video for this one. 

So if you want a free online Venn diagram maker, it's here.  I don't know how to help google move this up in the searches so people can find it. Do you?

Math Teachers at Play (aka Playful Math Education, Blog Carnival #157)

 

 

 

About once a year, I sign up to host this long-running blog carnival. Ever since Google Reader was snatched away, blogs seem to have fewer readers and less activity. Mine certainly has straggled along in recent years. (I guess I needed a very long rest after finishing my big book.) Today, I'm looking forward to exploring the new ideas I'll find online and gather here.

 

We start with cool facts about 157, and a puzzle...

 

Cool Little Facts

• 157 is the 37th prime number. (37 is prime too.)
• 157 is the largest known prime p for which ${\displaystyle {\frac {p^{p}+1}{p+1}}}$ is also prime (see OEIS: A056826).
  157 is a palindromic number in bases 7 (313₇) and 12 (111₁₂).
• 157 is the largest odd integer that cannot be expressed as the sum of four distinct nonzero squares with greatest common divisor 1. 
• 157 is the smallest three-digit prime that produces five other primes by changing only its first digit: 257, 457, 557, 757, and 857. [Opao] 
• 157 is the largest rating on the Saffir-Simpson Hurricane Wind Scale occurs at sustained winds of 157 mph or higher. 
• If we use the English alphabet code a = 1, b = 2, c = 3, … , z = 26, then número primo = 157. 

 

Puzzle

How many 3-digit numbers can we find where the last digit equals 2 times the first digit plus 1 times the second digit? 157 is one answer. How would you find the others without tediously checking each 3-digit number? (I use a spreadsheet when I want enough data to see patterns, but I worked hard to get the digits apart. Once you find the first few answers by hand, you might see the pattern...)

[Solution at bottom.]

Not Just Blogs...

I'm working on another book, much smaller this time. Althea and the Mystery of the Imaginary Numbers should be ready sometime next year. Since I'm working on a book, I've been thinking a lot about what makes a fun mathy book. It needs a good storyline. It needs interesting math. And if it's for young kids, it needs lovely illustration. 

There's a prize for good mathy books, called the Mathical Book Prize. It started in 2015 and doesn't seem to include small publishers like Natural Math (my publisher), so some of my favorites are missing. I think my favorite book on their list might be the picture book Which One Doesn't Belong, by math blogger Christopher Danielson.

Here are a few of my favorites that aren't on their list:

Quack and Count, by Keith Baker (for ages 2 to 7), a board book good for the youngest child who will sit and listen to a story. And it stays good because it's so luscious. Great illustrations, fun rhythm and rhyme, cute story, and good mathematics. 7 ducklings are enjoying themselves in every combination. “Slipping, sliding, having fun, 7 ducklings, 6 plus 1.” (And then 5 plus 2, 4 plus 3, 3 plus 4, and so on.) It would be great to have a book like this for each number, showing all the number pairs that make it.

How Hungry Are You? by Donna Jo Napoli and Richard Tchen (for ages 3 to 12), on equal sharing. The picnic starts with just two friends, rabbit is bringing 12 sandwiches and frog is bringing the bug juice. Monkey wants to come, "My mom just made cookies. I could take a dozen." They figure out how much of each goody each friend will get. In the end, there are 13 of them, and the sharing becomes more complicated. One of the delights of this book is the little icons showing who’s talking. It would make a good impromptu play. [There are lots of good books on equal sharing. Another lovely one is The Doorbell Rang, by Pat Hutchins.]

The Cat in Numberland, by Ivar Ekeland (for ages 5 to adult), starts when Zero knocks on the door of the Hotel Infinity. He’d like a room, but they’re all full (with the number One in Room One, and so on). Turns out that’s no problem. The cat who lives in the lobby gets confused - if the hotel is full, how can the numbers make room for zero just by all moving up one room? Things get worse when the fractions come to visit. This story is charming enough to entertain young children, and deep enough to intrigue anyone. Are you ready to learn about infinity with your 5 year-old?

 

The Man Who Counted, by Malba Tahan (for ages 6 to adult), was written in Brazil, and set in the Middle East. We follow the adventures of Beremiz, an accomplished mathematical problem-solver. He uses math to settle disputes, solve riddles and mysteries, and entertain his hosts. The series of 34 adventures, each with a math puzzle, is reminiscent of the Arabian Nights. If you read one chapter a night, your audience will be begging for more – and isn’t that the way it should be?

 

Carry On, Mr. Bowditch, by Jean Lee Latham (for ages 7 to adult), is a slightly fictionalized account of the life of Nathaniel Bowditch, who loved math, but had to leave school when his family needed his help. He was indentured to a ship chandlery for 9 years. Although that dashed his hopes of someday going to Harvard to study math, it was the right place to learn the mathematics behind navigation. When he finally went to sea, he invented a new way to ‘do a lunar’, and spent endless hours correcting errors in the tables used for navigation. Bowditch’s book, the American Practical Navigator, first published in 1902, is still regularly updated, and is carried on U.S. naval vessels to this day. 

 

And coming very soon ... Denise Gaskins' 2nd edition of Word Problems from Literature. (She'll be using kickstarter to raise some funds to get this out the door. Crowdfunding is how tiny publishers make it work!)

...And Now the Blogs (mostly geometry)

Denise Gaskins' How to Make Time for Exploration, in which Denise considers the benefits of Michelle's "Minimalist Math" curriculum, used along with games and books.

 

John Golden's Art, Math, and Geogebra Project, in which John has created a way for you to change a Kandinsky box to be new combinations of colors. Fun.

 

Daniel Scher's Euclid Walks the Plank on Geometric Construction, in which Daniel explores helping students to see the power of circles in building equal length line segments, using Geometer's Sketchpad for his online experiments. Once again, you get to play with the geometry.

Sam Shah on 3D Printing, in which Sam shares lots of cool 3D printing projects but decides they aren't really helping his students learn math. Do you have any 3D printing projects that help your students learn math?

Joann Sandford's Play, Persist, Prove on thinking about the angles in polygons. Can you use pattern blocks to prove what the angles are?

I adore Catriona Schearer's geometry puzzles, which she posts on twitter and elsewhere. Here's a video of her talking about them. (I recommend starting at about 8:30. They wait for participants and talk about Mathigon first.) Here's a lovely puzzle of hers. The big triangle that holds all the others is also isosceles. Find it on her twitter feed, and you'll see lots and lots of thoughts about it.

One more way to play with geometry ... this site gamifies geometric construction. I love it. 

 

Do you want more info on this blog carnival, or would you like to read old carnival posts? Denise Gaskins has got you covered.

 

 

 

 

Puzzle solution: There are 8 of these starting with 1: 113, 124, ..., 179, then 6 starting with 2, up to 2 starting with 4, for a total of 20.

Logic Puzzle, Supposedly from Einstein...

 ... but there's no evidence for that. The puzzle originally had folks smoking cigarettes. Yuck. I've changed that to eating candy.

The situation:

• There are 5 houses in five different colors.
• In each house lives a person with a different nationality.
• These five people drink a certain beverage, eat a certain candy, and keep a certain pet.
• No one has the same pet, eats the same kind of candy, or drinks the same beverage.  

 

The question is: Who owns the fish?

 

 

Hints:

• the Brit lives in the red house
• the Swede has a dog
• the Dane drinks tea
• the green house is on the left of the white house
• the green house's owner drinks coffee
• the person who snarfs M&Ms has birds
• the owner of the yellow house loves peanut butter cups
• the person living in the center house drinks milk
• the Norwegian lives in the first house
• the person who adores Heath bars lives next to the one who keeps cats
• the person who has a horse lives next to the peanut butter cup lover
• the person who eats Snickers bars drinks beer
• the German eats Almond Joys
• the Norwegian lives next to the blue house
• the person who eats Heath bars has a neighbor who drinks water

 

[There is one thing that's unclear: Is "the first house" the one on the left of the bunch? I assumed that. Apparently, you can assume that it's on the right end, and according to Wikipedia, you'll get the same answer. I haven't explored that.]

Still learning, after all these years...

This semester I'm teaching Calculus I and Linear Algebra. In each class, I've had a moment of discovery in the past week or so.

 

Calculus: Derivatives from Graphs

In calculus, I work with them on what the derivative graph of a function would look like, given just the graph of the function. So if the graph of f is this ...

... then what would f' look like? The activity (with 8 different graphs) went as it usually does. 

• Step 1: Find where the slope is 0, and give f' a value of 0 at that x.
• Step 2: Where the slopes of f are positive, highlight positive values for f' (and similarly for negative slopes). (Actually, the highlighting was new. I usually just draw dotted lines.)
  
• Step 3: Draw a curve that connects it all.

We had an absolute value curve and discussed where the derivative is undefined. (Which I marked with vertical red lines.) 

 

And then we got to this one ...

I said that w' looked like this ...

And a student asked how I knew the lines were straight.  Hmm, do I know that? "I'm not sure. Let's see..."

 

I thought about the curve given for w and said it looked like a bunch of parabola shapes (which I know have straight line derivatives), ... or like the absolute value of sine. I decided this was a fascinating question, and put both on desmos.

The red is y = |sin(π/2*x)|, and the blue is y = -(x+1) ² + 1 and y = -(x-1) ² + 1. To me, it looks like the original graph of w could be either one. But the derivative is the straight line segments only if w came from parabolas. If it came from a sine wave, then the derivative is curved (coming as it does from cosine). Using orange for the derivative of the sine graph and purple for the derivative of the parabolas graph, I got this in desmos...

Very different look to the derivatives, even though the original w could have been either of the original functions I put onto desmos. Fascinating!

 

 

Linear Algebra: Pivots vs Free Variables

We are using some fabulous activities from the Inquiry-Oriented Linear Algebra project, along with our textbook, Linear Algebra and Its Applications, by David Lay (we're using the 4th edition). We had just done part 3 of the Magic Carpet project the day before, and I was summarizing. We were talking about the span of a set of 3 vectors in ℝ³, and saw that the span made a plane through the origin. This was because there were 2 pivot columns. And then a student asked, "But don't we use the number of free variables to decide whether we have a line or a plane?"

 

To me that felt like a very deep question for a student to be asking this early in the semester. I said I'd answer the next day, since we were almost out of time. The next day I said, "We looked at the pivots because we were asking about span, which is all the linear combinations of the column vectors. Until we started considering span, we more typically asked about all the solutions to a set of equations, which is a different sort of question. For that, we look at how many free variables to determine if all our solutions create a line or a plane (or something more)."

I have never had a student ask a question like this, and was quite intrigued. I told them we'd explore somewhat similar questions in our 3rd unit (chapter 4 of Lay), when we will explore column space and null space. Once again, I was fascinated.

I've been teaching for over 30 years. I know calculus I inside and out. I've taught linear algebra often enough to feel like I'm a pretty solid expert on the basics. (I'd love to have more expertise on where this class might lead them.) Even so, I learn new things each semester. Even teaching beginning algebra, I have repeatedly seen it from a new perspective when prodded by some unique question a student was asking.

Yay for student questions.

Geometry Course for Homeschoolers, Spring 2022

 

I love geometry! (Well, I love a lot of math topics, but geometry feels especially like playing around.)

And I will be teaching a small course online for homeschoolers, starting in January. Here are the details:

Geometry Course

• Monday, January 10 to Thursday, May 26 (no class on Feb. 14)
  
• Mondays and Thursdays, 4:30 to 6pm CA time / 7:30 to 9pm East Coast time, on Zoom
  
• $800 for the course. (Please pay in advance. If you need sliding scale, please contact me to discuss.)
• 6 to 10 students
• Using Michael Serra's Discovering Geometry (4th edition, which is pretty reasonable used), along with (free) materials from Henri Picciotto. We will also use geogebra extensively (also free).
  
• Check out my site for more about the course and me.
  

Please contact me soon if interested. (Email suevanhattum@hotmail.com or mathanthologyeditor@gmail.com.) I'm happy to chat on the phone too, if you have any questions. You can text me at 510-367-8085, and we can talk at a time that works for us both.

Sizes of Infinity

 

I am floored. Here is a new mathematical result that sounds pretty important. I'm surprised I hadn't heard of it sooner. It was published online in April.

 

This Quanta article explains it pretty well. But if the article doesn't make sense to you, I can explain more. This is the field I had planned to go into when I was thinking I'd get a PhD. I loved my two logic courses at Eastern Michigan University.  But the one I took at UCSD was not fun. I think because it was too far above me, and I couldn't stay grounded.

 

The one problem with the article is that it made it sound like the big question was resolved. But it's not. I thought it was saying that the continuum hypothesis is false. The continuum hypothesis is about sizes of infinity. The smallest infinity is what you get when you count out all the infinite whole numbers (or all the fractions), and it is called the countable infinity. The continuum hypothesis says that the next size up is what you'd get "counting" the real numbers (like the number line). But there may be a size in between. 

 

I hope there is a way to get a meaningful example of that in-between size of infinity. (The are bigger and bigger infinities, but the two things grounded in numbers we know well, integers and real numbers, are the most interesting to me.)

 

A fun way to start thinking about infinity is a book that's accessible even to young kids. It's a five chapter picture book titled The Cat in Numberland. Sadly, it doesn't seem to be available (unless you want to pay ridiculous prices). My publisher, Natural Math, tried to help the author get it reprinted, but Cricket books (Carus publishing) wouldn't give up their rights, and won't republish. (Maybe we should look into that again...)

[The Quanta article links to the proof that was published online in April. I don't expect to understand that, but I'll try reading it. I might quit very quickly.]

More Tech: Sue Finally Learns How to do Screencasts

I broke my ankle a few months ago, and could no longer use my whiteboard. I asked my college for an iPad and got it within a week. I asked in the Math Mamas group on Facebook for software recommendations -  goodnotes and one other both got high recommendations. I went with goodnotes and fell in love.

Teaching online is significantly more work than teaching in person, and this just added to my workload. But I love that students can easily get my notes on Canvas. And this week I made my first screencast. And then my second. It took me a few hours to get the hang of it for the first one. I may have done the second one in under 20 minutes. Both of them are for a basic geometry course I'm teaching at my college, in which most of the students are high school students.

Indirect Proof (aka Proof by Contradiction)

A Direct Proof

 

I think I could do a few of these a week. Before posting on Youtube, I'd like to find a way to have my face in the corner if possible... Once I feel like I know what I'm doing, the Math Mama's channel gets underway!

LaTex, a curse and a blessing

I've been making teaching materials on computers for over 25 years. Maybe 15 years ago, I was introduced to MathType, and it made my equations so much nicer. Now it doesn't work with Word, and you have to pay a yearly fee. No thanks. It seems crazy to me that MS Word doesn't have a better equation editor. (I don't really remember what I don't like about it, but I think it has annoyed me lots over the years.)

I got a new computer in the Spring, and since then, whenever I need to make a formula, I've been using my old computer with an old version of Word, and my very old copy of MathType. Today I wondered if it was time to bite the bullet, and make a quiz using LaTex.

I've tried to learn a bit of Latex a number of times before, and it just felt overwhelmingly weird. I especially hated that I couldn't see what I was doing. This time was better in a number of ways. First, my colleague showed me overleaf, where I can see what I'm doing. You can choose split screen, and hit recompile after every little change.

The next thing that helped was that I got a bunch of materials from the author of the book I'll be using. (Oscar Levin, Discrete Mathematics: An Open Introduction.) I used those as templates for my own work. I deleted what I didn't want, and began to add what I did want. (If you want to learn LaTEx (or TEx), and you don't have a bunch of materials someone else made that you can modify, this quiz template might be helpful.)

The reason I was using LaTex was the equations, but that was one of the things I didn't know how to do. This site, codecogs, came to the rescue!

I also needed to include an image of a Venn diagram. I read up (googled latex image), tried to do what they said, and my image ended up in a weird place, next to the questions. I guessed, and added a line that I saw in other places in my documents from Levin (\vskip 1em). I figure that's a vertical skip. I have no idea what the 1em is. (I tried 5em for more space. Nope.) It worked!

But the image was still too big. Read up again, use [scale=0.5], put it in the wrong place, so it doesn't work. Figure out the right position, it works! And now the image doesn't look right hanging out on the left. I read up, use "the centered environment," and it is all just prefect!

Here's the centering:

\begin{center}
    \includegraphics[scale=0.5]{venn10}
\end{center}

That took me over an hour. (Maybe two.) I made a second version of that quiz in ten minutes.

 

I'm learning...

Summary

Does LaTex seem way too complicated, but it still might be the answer to your problems?

• Use a simple environment like overleaf where the split screen lets you see what you've done.
  
• Start with a template you can modify.
• Use something simple like codecogs to build your equations.
• google your questions.

Good luck!

Square & Triangular Numbers

 It's my vacation. And here I am, playing with math. Woo hoo.

If you've played with this problem before, perhaps this is boring and old hat. But I've seen the question many times, and never before have I followed up on it.

I just got a book I ordered. A Friendly Introduction to Number Theory, by Joseph Silverman. THe very first problem he asks the reader to attempt is:

Exercise 1.1. The first two numbers which are both squares and triangles are 1 and 36. Find the next one, and if possible, the one after that. Can you figure out a way to efficiently find triangle-square numbers? Do you think there are infinitely many?

I found the next one easily, by making lists on paper of the square and triangular numbers. It was about 6 times as big as 35 (which is about 6 times as big as 1). So I figured it would take too long to find another by hand. I wrote a Sage script. (It took me a few tries. I had lots more print statements until I was sure it was working.) I now have 7 of them. But more importantly, I've found a pattern. If you want to play with this, I would recommend not reading further.

.

.

.

.

.

The business about each one being about 6 times as big as the one before looked promising. So I checked. Let's call them m (for matching numbers), where the actual number is m².

m0 = 1, 

m1 = 6*m0=6,

m2 = 6*m1 - 1 = 35,

m3 = 6*m2 - 6 = 204,

m4 = 6*m3 - 35 = 1189.

At this point, it becomes clear that m(i) = 6*m(i-1) - m(i-2). And that's where I am now. I don't really know that this will continue to work forever. But it does continue for all the numbers I've found using Sage. And I just found one more to see if it continues further. It does.

Next step, proof. I will see if that's something I can do.

Edited to add:

I just found a closed form for the formula. It's ugly but it works. (I learned how to do that step from Oscar Levin's Discrete Mathematics: An Open Introduction, in 2.4, Solving Recurrence Relations. That's the book I'll be using to teach discrete math from this coming semester.) 

 

Now the next step is proof....