Wednesday, November 14, 2018

AMATYC 2018 IGNITE - Some of my favorite pictures in mathematics

Hello, welcome to the post that goes with my IGNITE talk for AMATYC 2018.  The slides and extra information and links are below.

At the AMATYC national conference, we have an IGNITE event on Friday night.  Each presenter gets 20 slides that auto advance every 15 seconds.  It is fun, exciting, and most speakers, myself included, get a little off in their timing due to the excitement, nerves, etc.  Anyway, I presented this time on “Some of My Favorite Pictures in Mathematics”

Those pictures, with comments and links, are below.  We also live-streamed the event, so if you want, you can watch it on the AMATYC Facebook Page.  (The audio and video may not be that great, as it was done “in house”) but that is what we have!

A general disclaimer, pictures are not proofs, but they sometimes give some real insight into a concept or problem.  In addition, we mathematicians often draw lots of pictures to help keep track of things, so I think this is an important idea.  Also, some of these are commercial images, so I have linked to the artists or people responsible.  Some of these I found cool enough to have a copy in my office.

The slides with comments are below.  Notice that the first and last slide were just slides pointing people here so if they wanted to follow up or see more.
Slide 2: This image was posted by Alice Proverbio, who is a cognitive neuroscience professor at the University of Milano-Bicocca in Italy.  See the following article for more information about how this static image is not really moving, even though it seems like it is.  This was to drive home the point that pictures are not proofs.
The article

Slide 3: Simon Beck Website is an artist who makes mathematical patterns in the snow.  He also uses graph theory as he makes the designs, by deciding where to back track to get to the other parts of the image and so forth.  On this image, Mr. Beck is in red in the middle.

Slide 4:  The Mandelbrot Set in mathematics is, in a sense, the basis of Chaos Theory.  This is a picture of the set, but made up like an old time map.  The poster version of this is hanging in my office.  Web store

Slide 5:  Robert J. Lang Store does all sorts of cool mathematical origami.  One of the things I like about this design is that it is a single sheet of paper that has been folded. 

Slide 6:  I love how these images can help visualize the topology that is at work here.  The coffee cup torus is from Wikipedia, but can be found in a lot of places.  The Klein Bottle is from Here

Slide 7:  some really neat books along these lines are the “Proofs Without Words” series published by the MAA.    There are a number of wonderful pictures here, and I mentioned a few in my talk.
Amazon site here  But the MAA has a nice “online” series that relates here.  

Slide 8:  Here are two of my favorite images from the first book.  I use these without the numbers in my Calculus 2 classes and ask students what sequence and/or series these are representing.  I then show the images with the numbers, and we talk about how you would calculate those series.  Finally, for a challenge, I ask the students to draw the series on the other shape.   i.e. the sum of reciprocal powers of three in a triangle, and reciprocal powers of 4 in the triangle.  Then have them defend their answers and show the area steps correspond to the partial sums.

Slide 9:  Yes, you can do integration by parts from the product rule, but the fact that you can tie it to areas is also cool.  Often, after we cover this technique in class, I just put the basic picture up, and have the students explain what the basic picture is trying to show.  A good discussion is generated, especially regarding function notation.


Slide 10: Okay, so maybe a little risque, but it generates some laughs.  I then have each student write a minute paper explaining the joke, but they have to keep it “G-rated”. 

Slide 11:  There are a lot of animated pictures of the Pythagorean relationship for a right triangle, but I like this one for the intermediate shapes that it shows.  For my classes, I often ask them to show at each stage why is the area the same.  
Friedrich A. Lohm├╝ller created this

Slide 12: I stumbled across this image via Twitter, and there are many neat things at this website.  There is even an animated version of this diagram that you can spend time with down towards the bottom of this page:

Slide 13: These two come via Wikipedia, but they are all over the internet.  I like this as a pair because of the very different looking shapes drawn simply by moving the circle inside or outside.  I use this to drive home the point that details matter in mathematics.

Slide 14:  If you haven’t visited, you should.  They have pages of different gifs and a lot of other resources.  The gifs start here

Slide 15:   This is a pretty graphic for the sum of the first n integers.  There are quite a lot of other really nice resources at the website

Slide 16: This is an Ulam spiral that is 3000 by 3000.  An Ulam spiral spirals the integers in counter clockwise direction, and colors the primes.  So you can see patterns, and gaps, and it looks cool.  In this case, I mentioned that my favorite prime is one of these dots:

Slides 17-19:   My favorite prime number.  Mainly because if you speak out the number, it is 8-6-7-5-3-0-9, a famous chorus from a pop song in the 1980’s.    Youtube video here
It turns out that 8675311 is also prime, so “Jenny has a twin” was the joke I finished with.

Tuesday, November 13, 2018

Lines and slopes and intercepts, oh MY!

Howdy, related to my posts about using Excel in Calculus, I have this project that I use on lines and slopes and intercepts.  Again, it is a gentle introduction to programming cells in Excel.  In this one, I have students vary each of the three parameters individually and examine what effect the changes have on the values of the slope and the intercepts. 

Again, the point of these is to start having students learn to use a spreadsheet, and to also examine the idea of playing with parameters. 
Again, DM me on twitter or elsewhere if you want the files.
 *** Instructions start here ***
  All of the lines we look at in this project will be of the form Ax + By = C 

1.       Replace the fake ID number with your own Blinn ID number.  (Purple boxes)
2.       In the yellow background boxes, we will keep B and C the same, and change the value of A.  Notice that the template will automatically give you the different values of A.  You will need to program Excel to compute the slope and the coordinates of the two intercepts.
a.       As you increased the values of A, did the slope increase or decrease?  Explain why this would have to be so.
b.       Which coordinates changed, and in which direction?
c.       Which coordinates stayed the same?
d.       Describe visually what in effect we were doing with the line.  (so, spinning it about a specific point, sliding it up or down, both, neither, etc)
3.       In the green boxes, we will keep A and C the same, and change B.  ** Same four questions

4.       In the blue boxes, we will keep A and B the same, and change C.   ** Same four questions

5.       So which change seems to have the biggest effect on the slope, and why does that work?

 The numbers from the first half.

The numbers from the other two sets of variations.

Thursday, November 1, 2018

Excel Projects in Calculus part 5

If you need background, see my first post:
First Excel in Calculus post.
** If you want the Word and Excel copies, @me (@robebymathdude) on Twitter **

The last of my Excel projects in Calculus posts.  I did five mainly because I also have the students do memos (see  ) and there are only so many weeks in the semester.  I made it a point when laying out the schedule to not have a memo or project due during an exam week, so that is one reason for having five.  The other is the grading load!

Anyway, in this project I wanted the students to explore Riemann Sums but in a way that involved more than just calculating the sums.  (If you didn’t know, sites like Wolframalpha will do a Riemann sum with ease.  See for example: Example Sum

I also wanted the students to try and go back to a function, so I had them do sums for 1/x, and then see that the basic log rules worked.  I had a few students who figured this out, but the majority just kinda shrugged on their answers.  I am going to have to think about how to get students to this point in a better / different way for next time.
Thanks for reading.