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.

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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....

 

Getting Better at Canvas

 I am not a Canvas expert, but I've learned a lot this past semester, and hope to keep learning more.

This post is a compilation of some of the things I've learned that make Canvas better for me and my students.

 

Images

I took a course offered by my employer (Contra Costa Community College District) called Becoming an Effective Online Instructor (BEOI). In the course they recommended using lots of pictures in our Canvas pages. I haven't gotten to the point of "lots" yet, but I'm trying to become more aware of what images will help students learn mathematical concepts, and also what mathematical images bring beauty to the screen. 

 

I love this image, titled Banded Torus, by Thomas Banchoff and Davide Cervone. I recently realized that part of its power for me was its black background. So I changed the cover images for my calculus and precalculus courses, to incorporate a black background. Both of these are done on desmos in reverse contract. The originals, with white background, were nowhere near as lovely.

For calculus, I wanted to show both slope and area.

For precalculus, I wanted to show all of the functions we study (along with the circle). I did leave out the rational functions, not wanting the image to look too busy.

Orientation

That BEOI course offered very specific ideas about how to set up an orientation module. (I had to do one their way for the course, and then I modified it to make it my own for my students.) One of the items in it is a quiz. I loved putting that together. I tell students where the answer to each question is (as part of the question), so they can look it up. Partly, it's a way to emphasize certain things from all of the pages I am hoping they will have read. (Yes, you can call me at home! But not after 8pm.), and it's also a chance to be silly (how many chickens does Sue have?). It also allows students to start out the semester with a perfect quiz score (hopefully!).

Zoom Recordings

I guess Zoom saves these already, but I wanted them listed in my modules. So I had a module with links to each day's recording. In a mid-semester survey, two students requested that the various topics covered be listed with timestamps. I don't have time to do that, but I figured out a way to allow students to do it for each other. I have one page in each unit where I link to each recording by date, and list the topics we covered underneath. I set that page so that students can edit it. (They didn't this semester, but if we start out this way, and they get a bit of extra credit for it, we might be able to jointly build a great resource.)

Quiz & Test Retakes

Until this semester, I did not use the Canvas grades function. I do my grading using Excel, and it has lots more flexibility for my crazy formulas that calculate the grade four different ways and take whichever is best for the student.  But everything was online this time. So that's where the grades were. I turned off the totals, so students wouldn't see the wrong scores that Canvas figured.

I allow students to take quizzes multiple times. (New version each time, of course.) And they get two chances on most tests. I started out building a new Canvas assignment for each retake. What a mess to figure grades! I finally realized that Canvas would accept multiple attempts on an assignment, and allow me to look at each one. That feature works great.

There is a "hide grades" feature that is supposed to hide the grades until I'm ready to post them. But it apparently doesn't hide my comments, which defeats the purpose. (Since I explain my grading in the comments.) Maybe there's a better way to do that, and I'll learn it soon. [Edit: After I wrote this post, I found out that there is indeed a better way. In the gradebook, go to the assignment, at the name of it, click on the three dots, choose 'Grade Posting Policy', and choose manually. Then remember to 'Post Grades' when you're done.]

 

Organizing Content

The Canvas "modules" serve as containers for each of my units. So each one starts with a "unit sheet", giving an introduction to the ideas they'll be learning about, objectives, and a schedule. That schedule is what I want my students to think of as their home base in my class. I add details to it daily, I highlight the current class session, and I link to pages and assignments in it. I add more detail to it when I'm prepping my next class. It works great for me, and I want it to work great for my students. I put a link to it on the Home page, so it's easy to get to.

 

Community Page-Building

Canvas pages start out as editable only by the teacher. But you can change that to allow students to edit a page. Our fist topic in our second unit (in trigonometry) was radians, and I wanted them to do something after our first test, before that next class session. So I created this page, and I told them to find the best videos online that explain radians. I think comparing video explanations was a great way for them to be thinking about whether they really understood the concept.

Next Semester

I am still thinking about how to get students to participate more, and will be looking for ideas to help with that. I know I should make a few videos where I explain some of the key concepts. But I seem to be resisting doing that.

What have you learned recently about how to use Canvas well?

Note-Taking & Learning Something New at 64

I've been teaching for over 30 years, almost all of it at the community college level. So I've gotten pretty used to what I do. (But not bored. I still discover new ideas every semester, and I still love connecting with students.)

That changed with quarantine. Before 2020, I was pretty sure that I never wanted to teach online. It looked like way more work, and it was clear to me that I wouldn't be able to have the same level of connection with my students in an online class. I was right about both things, but (amazingly, to me) I am enjoying teaching online. 

I meet my students in Zoom two days a week. Most of them won't turn their cameras on, and I want to respect that. (I offered extra credit for cameras on, and I get to see 2 to 5 faces each day. It's better than none.)

I have a light load this semester. Just two classes. And it still feels like full-time work. Next semester I'll have over twice as many units (in 3 classes). I'm starting to prepare ahead of time, so I don't drown.

I started taking notes for the Discrete Math book I'll be using, and after I wrote up some notes, I went back and wrote an introduction to note-taking. Tonight I described it to my bother (who's becoming a teacher), and realized that it was a bit of an epiphany for me.

I have terrible handwriting, and always thought I didn't know how to take good notes. I copy the board in a math class, just like everyone else. That's not really note-taking to my way of thinking. I highlight the good bits when I'm reading, and when I come to an example, I try to do it myself before looking at the author's steps. But notes? Nah, that just never seemed like one of my skills.

Well, I was a little excited as I finished up my notes for the first section of the textbook. I had set the Canvas page so that students could edit it too, and so I had purposely left some parts of my notes incomplete. As I looked at what I had written and did a bit of rearranging, I saw some patterns.

So I wrote this introduction:

How do you take notes when you read? My reading notes may surprise you. I see 4 types of things that I'm doing in my notes:

• The first, organizing by making lists, will be familiar to you. 
• But I am also trying to connect a new term to other meanings outside of math. 
• And I am reacting to what I read (surprise, and noticing how powerful something feels). 
• I also made up my own example.

That seemed kind of cool.

Then, when I talked to my brother, I realized that I had always thought I was no good at taking notes. (I didn't think I really needed to be any better at it, because I am good at most academics anyway. But...) I never thought I could teach students how to take better notes. And I realized that this one task I gave myself, to make some reading notes for the textbook, suddenly showed me that I know a lot about reading math and taking notes that I can share with students.

So that's my epiphany. I do know how to take good notes, and now I know how to describe that process to students.

What helps you conquer a text you're reading? Do you take "good notes"? What does that mean to you?

Division by 0

[Once again, I have written something for my class that I think will be valuable for others.]

Big question: What are the values of , , and ?

 

We want to be able to look at each of these fractions, know what it equals, and understand why. This becomes vital in calculus. [Note: Many students have trouble with this. It may be because elementary teachers are often uncomfortable with division, and teach it by memorization, instead of as something deep to understand. Or it may be that this is deep, and our brains need more time to really make sense of it.]

 

To help ourselves understand this, we tie it to something simpler that we understand better. Division is the inverse of multiplication (ie they undo each other). So it will help to explore how the two operations are connected.

We start with a very concrete and simple problem:

[Note: One notational problem with division is that it's written in different ways that place the numbers in opposite orders. , but these are also equal to. When I was young, I had trouble keeping track of which was which, so I would write down an easy problem, like this one, to help me remember.]

Now we consider the multiplication problem that goes with this division problem: , and we can say that 6 divided by 3 is 2 because .

 

Let's use T for top, B for bottom, and A for answer, and rewrite this equivalence of a division problem and its associated multiplication problem, in a way that will always be true:

In the fraction (or division), we have top over bottom gives answer, and that gives us a multiplication problem where the original bottom times the answer from the division gives us the original top.  [Note: I am purposely avoiding the proper terms: numerator or dividend, denominator or divisor, and quotient (for the answer). For anyone who gets those terms mixed up, it's easier just to focus on position for the moment.]

Now we are ready to consider each of the three original questions, using this correspondence.

1. Let's think about the multiplication associated with :

So what do we multiply 0 by to get 3? Hmm. It seems that nothing works. There is no number that can multiply with 0 and give us 3. So the division problem (or fraction) has no solution, and we say that is undefined.  This is why we say "division by 0 is undefined".

 

2. . Ahh, this one is easier. so the answer is 0.

 

3. . Hmm, this time A could be any number, and the multiplication would be correct. This is still division by 0, so it is still undefined, but it is very different from the first case. We call it indeterminate. We can see why by looking at a rational function example.

Example:

When x= -2 or 2, this function will be undefined (because we have division by 0). But the function's behavior for x values very close to -2 is very different from its behavior for x values very close to 2.

is a vertical asymptote for the graph. This means that as x approaches -2, the y values approach . (This can be written "as ".) You can verify this by trying these x values: -2.1, -1.9, -2.01, -1.99,... (You can also use desmos to view the function.)

What happens near ? We see that the y value does not depend on the factor , because it cancels. So, as long as , . At , this would equal 1/4. The function is not defined here, but now we can see that as .

So why was called indeterminate? Because the value associated with it in a particular function is determined by other parts of the function. Although is undefined, we saw that, in this particular function the value of the function got close to 1/4 as the x value got close to 2, which is the number that would give us . This concept goes with the concept of limits, one of the 3 major topics in calculus.

 

 

What sorts of things are impossible?

Here's an interesting article in QUanta, by David Richeson. I'll be thinking about what else I might add to this post...

Solving Application Problems (in Trigonometry)

I started this blog in 2009, was active for about 6 years, and then not so much for the past 5 years. I wrote two posts in the spring, both related to online teaching. We were all trying to learn how to teach well as we scrambled to do it while learning. I was happy to keep seeing my students online, and Zoom was our class. I used Canvas a little but not much.

Over the summer I learned a lot about effective online teaching. (I'm still not sure it can ever be nearly as effective as in-person, but...) I developed my Canvas shells for each course, and I started the semester readier than I had expected to be. My Canvas shells are not done. I created a "module" that orients students to online learning and my course. And I created a module for our first unit. The rest is still in progress.

Today I added a page for my trig students, on solving application problems. I want to share it here. (And I may share lots of my Canvas "pages" here, sometimes with modifications.)

Years ago, I modified George Polya's wonderful outline of problem solving steps. We start with that. It's a good idea to print it out, and turn to it whenever you're stuck. 

Draw a Diagram.

Always start by drawing a diagram. This step is vital, and is a major part of "Understanding the Problem".

Your diagram does not need to be artistically good. It does need to show relationships well. An artist might show my shadow going off at an angle. But for a math diagram, it is better to show the right angle involved, as a right angle.

In the diagrams on the right, the top drawing is prettier, and the shadow is more evocative, but the bottom drawing shows the right angle between a vertical object and its horizontal shadow, which is what will help you do your mathematical analysis.

Example (#22 in 2.4, page 93): If the angle of elevation of the sun is 63.4° when a building casts a shadow of 37.5 feet, what is the height of the building?

Draw your diagram now, labeling it with everything given and a variable for the value requested. (My drawing is below.)

 

 

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I labeled the height of the building h.

 

 

 

 

 

 

 

 

 

 

Write a Trig Equation.

In a simple problem, with only a few pieces of information this is all you need for the "Devising a Plan" step. We are given the value of the side adjacent (next to) the given angle, and we want to find the value of the side opposite the angle. (The hypotenuse is neither given nor asked for.) Which trig function uses adjacent and opposite? (Two of them do, but the one we use most of the time is...)

 

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... , and this gives us 

 

Do a bit of algebra.

This is the "Carry out the Plan" step. To solve for h, we multiply both sides of the equation by 37.5:

I pulled out my calculator for that last step (making sure it was in degree mode). Since our given length was given to tenths of a foot, I round, and give my final answer as 74.9 feet.

 

Check your Solution.

This is the "looking back" step on the handout. If we look at our diagram, does a height of about 75 feet seem reasonable? Well, the height seems bigger than the shadow, and maybe about twice as big, so yes, it seems reasonable.

 

 

Practice.

If you get stuck on application problems, a good way to practice is to re-do problems that you've watched someone else do (perhaps on youtube). Try not to look at your notes. If you need to, go ahead and look. Do as much of the problem on your own as you can. If you looked at your notes at all, do it again the next day.

Playful Math Education Carnival #139 (formerly known as Math Teachers at Play or MT@P)

"It’s like a free online monthly magazine of mathematical adventures." (Denise Gaskins)

 



Black Lives Matter. How does that idea and movement intersect with math and play?  It's hard to imagine play intersecting with the painful history of racism in the U.S.  We can collect data to show how pervasive anti-Blackness has been and is. We can discuss how math courses have been used to filter out students from desirable professions (doctors, engineers, lawyers).  We can discuss how Black people are more involved in the history of math than you'd guess from the Eurocentric naming. (Check out who knew Pascal's triangle before Pascal!) None of that is playful. But celebration can be playful. Let's celebrate Juneteenth!

139

Every number is cool.* Here are some ways 139 is cool:

• 139 is the sum of 5 consecutive prime numbers (19 + 23 + 29 + 31 + 37). 
• 139 is the smallest prime before a prime gap of length 10. 
• 137 and 139 form the 11th pair of twin primes. 
• 139 is the 34th prime number. 

Puzzle: The digit sum is the result after adding the digits repeatedly until you get down to one digit. 139’s digit sum is 4. If you write 139 in base two, you get 100 1011, which still has a digit sum of 4. Does this always happen? If not, does it happen in any other bases?




New Homeschoolers 
I have a hunch the quarantine has moved lots of families from school to homeschooling. If you’re new to homeschooling, get ready to have fun playing with math. Most mathematicians are in it at least partially for the fun of it. We like to play with numbers, shapes, and logic. The more you play with math with your kids, the more likely they are to enjoy it.

There are vast resources online to help you. Until 3rd grade, just play games, cook, measure, read mathy stories, and have fun with it all. If your kid wants a curriculum before that because they love math, then check out Beast Academy. It has levels 2 to 5 (topics correspond to grades 2 to 5, difficulty levels are a grade or two higher). Some families never use a curriculum; if you’re interested, you may want to explore unschooling. Math lovers eventually want to take classes, which you can do either through your local community college (I’ll be teaching trigonometry, pre-calculus, and calculus I online this fall) or Art of Problem Solving. There are lots of other great resources; these are just my personal favorites.

You might find ideas that work for you in my book, Playing with Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers. Or from other books from my publisher, Natural Math. I also highly recommend Denise Gaskins’ blog (especially this post on homeschooling math), website, and books.  Dan and Christopher have some good ideas about playing mathematically with kids too.




Talking Math With Your Kids (#TMWYK)

• Tracy Zager's daughter was thinking hard while mom sat there, exhausted.
• On the Muni, overheard by Breedeen Pickford-Murray.

Math & Language Play
One of my favorite math educators, Marilyn Burns, invented a game where students look for $1 words. A=1¢, B=2¢, etc. You could combine math and any other subject by making $1 phrases. Sometimes kids like the simplest games. This might be a craze at your house. (My son used to love Shut the Box, a simple dice game that did nothing for me. It sure was good number practice for him.)

π-ku, a competition, in which all their favorites will be posted at the Aperiodical blog. I'll try:

Three One Four.

Hmm.

Not very hard.

Games
So much of math is based on logic, any logic games you play will deepen your students' affinity for math. Here are a few others:

• Set Tic Tac Toe, described by Tanya Khovanova, invented by her students. You may want to play the basic game of Set for a few months before attempting this. But if I could figure out a way to do this at a distance, I'd love to try this out. 
• Planarity game. (This is connected to a field of math called graph theory.)
• Play with wallpaper symmetries.

Math History
Podcasts aren't my thing. Yet. But if this series is as good as it sounds, I'll just have to  figure this newfangled genre out. Opinionated History of Mathematics. With an interview and glowing review at Aperiodical.




Online Events
This summer Art of Inquiry is hosting free science webinars on space, astrobiology, and AI for school children and their families. The webinars are led by university professors and industry experts. You can register for the events on Eventbrite.  Here is their June-July 2020 schedule:

• Living Through a Revolution: Multi-messenger Astrophysics - Dr. Roopesh Ojha, GSFC NASA, June 26th 
• Figuring out the Earth from inside out - Dr. Kanani Lee, Lawrence Livermore National Laboratory, June 30th 
• Mars Rovers - Dr. Allan Treiman, Lunar and Planetary Institute, July 3rd 
• The search for life on Mars in XXI century - Dr. Alex Pavlov, GSFC NASA, July 10th 
• Where in the Universe did we come from? - Dr. Ethan Siegel, science author, "Starts with a Bang" Forbes contributor, July 23rd 
• Why we should build a Moon base - Dr. Ian Crawford, University of London, July 31st 

 If you know of other math-related online events, please mention them in the comments.




This series of blog carnivals was founded and is kept going by the fabulous Denise Gaskins. You can find out more at her blog. Last month's carnival was hosted by John Golden, the Math Hombre. Check it out!





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*Well, sometimes their coolness is in their bad reputation (sounds like a few people I knew in high school) ... 

The Math Teachers at Play Blog Carnival (aka Playful Math Education Carnival) will be a bit late this month.

I am looking for good posts now. If you can send me any links by Saturday, that would be great. I am hoping to put it together on Sunday.

Send your links to mathanthologyeditor@gmail.com, or post them here.

Want to know what a blog carnival is? Check out last month's, by my pal John Golden.

Corona Post #2: Teaching Online

[#2 because my previous post in March on my online math circle was due to people needing to take their math circles online when the shelter-in-place orders were just starting.]

I've been teaching online for 4 weeks now, two before our spring break and two after. At first I was just trying to learn how to manage teaching on zoom. I bought a whiteboard that's still sitting on two chairs in my living room, and I sit in a tiny chair while I write on it. Not ideal, but I get to see my students, and I feel like I'm still working with them where they are, not just lecturing.

(Some day I'll finally install it on my living room wall. I procrastinate with tasks like that. I'm not sure why it feels intimidating...)













A few weeks ago I made a google slides presentation for my Discrete Math course to explain a way of counting possibilities called Stars & Bars. I had fun doing it. You're welcome to modify it and use it in your teaching.








Just now I made another. This one is for Calculus II, on Taylor & Maclaurin Series (really just a Maclaurin series). I was motivated by knowing that there would be too much writing for my little whiteboard. This presentation has a handout to go with it.








I'm also teaching Calculus I. I haven't made any cool new materials for that course yet. But I will...

Online Math Circle: Pythagorean Triples

The Pythagorean theorem tells us that if a and b are the legs, and c the hypotenuse, of a right triangle, then a²+b² = c². Usually that makes at least one side something ugly like square root of 2. But a few combinations make all three sides whole numbers. Those are called Pythagorean triples. Here are a few of them: 3-4-5, 6-8-10, 5-12-13, 8-15-17, 20-12-29.


Are there patterns to this? Let's play, and see what we can figure out! (We will use some algebra.)

[This was my intro to the online circle, whose recording was included here. Sadly, zoom recordings expire, and it is no longer available.]



Way back in 2007, I read Bob and Ellen Kaplan's book, Out of the Labyrinth: Setting Mathematics Free, about the math circles they lead. It was such a discovery for me! I went to their first Summer Math Circle Teacher Training Institute, held at Notre Dame, and fell in love with this community. I kept going back for years, craving a discussion of math among equals, figuring out new ways of seeing. One summer we discussed Pythagorean triples, and that December I tried to rebuild what I had learned. I am blessed with a very bad memory, so what I did in December looked very different from what we had done in the summer.

I was also exploring online, and ended up putting together a book that collected some of the best resources I had found: Playing with Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers.

Our circle was prompted by Rodi Steinig's request for help learning how to use zoom for online math circles. I offered one of my favorite topics, and off we went. Participants came from as far away as Colombia (and farther?).

We proved a few things, and explored a bunch more. I hope some participants went home eager to prove more on their own.

Multiplication Chart with Pictures

In the story I'm writing, Althea remembers a multiplication chart that was posted in their bathroom. It had cool pictures around the edges for many of the facts.

• 2x3 was a 6-pack of soda.
• 2x6 was a carton of eggs.
• 8x8 was a chessboard.
• The fives were sometimes collections of nickels, but 5x12 was the 60 minutes on a clock, and 5x6 was time too.

I thought I knew of more iconic sets like these, but I can't think of any more as good as these. I'm hoping for help. Do you have images in your head for any of the multiplication facts?

Maybe threes will be 3-leaf clovers. 6 of them have 18 petals. That doesn't seem nearly as iconic as the ones above, though.

Fours could be legs on dogs. 6 dogs have 24 legs. Twos could be eyes on friends...


What are your favorite images for multiplication facts?