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?