Friday, March 4, 2022

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?
  • 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.]

Saturday, February 12, 2022

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 3, 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.

Thursday, December 16, 2021

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 or 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.

Friday, July 16, 2021

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.]

Friday, June 18, 2021

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!

Sunday, January 3, 2021

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:


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


I'm learning...


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!

Thursday, December 31, 2020

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....


Tuesday, December 15, 2020

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.



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.


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?

Thursday, November 12, 2020

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?

Sunday, September 20, 2020

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 LaTeX: \frac{3}{0}, LaTeX: \frac{0}{3}, and LaTeX: \frac{0}{0}?


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: LaTeX: \frac{6}{3}=2

[Note: One notational problem with division is that it's written in different ways that place the numbers in opposite orders. LaTeX: \frac{6}{3}=6\div3, 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: LaTeX: \frac{6}{3}=2\Longleftrightarrow3\cdot2=6, and we can say that 6 divided by 3 is 2 because LaTeX: 3\cdot2=6.


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: LaTeX: \frac{T}{B}=A\Longleftrightarrow B\cdot A=T

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 LaTeX: \frac{3}{0}:

LaTeX: \frac{3}{0}=A\Longleftrightarrow0\cdot A=3

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 LaTeX: \frac{3}{0} is undefined.  This is why we say "division by 0 is undefined".


2. LaTeX: \frac{0}{3}=A\Longleftrightarrow3\cdot A=0. Ahh, this one is easier. LaTeX: 3\cdot0=0 so the answer is 0.


3. LaTeX: \frac{0}{0}=A\Longleftrightarrow0\cdot A=0. 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: LaTeX: y=\frac{\left(x-1\right)\left(x-2\right)}{\left(x+2\right)\left(x-2\right)}

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.

LaTeX: x=-2 is a vertical asymptote for the graph. This means that as x approaches -2, the y values approach LaTeX: \pm\infty. (This can be written "as LaTeX: x\longrightarrow-2,\:y\longrightarrow\pm\infty".) 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 LaTeX: x=2? We see that the y value does not depend on the factor LaTeX: \left(x-2\right), because it cancels. So, as long as LaTeX: x\ne2, LaTeX: y=\frac{\left(x-1\right)}{\left(x+2\right)}. At LaTeX: x=2, this would equal 1/4. The function is not defined here, but now we can see that as LaTeX: x\longrightarrow2,\:y\longrightarrow\frac{1}{4}.

So why was LaTeX: \frac{0}{0} called indeterminate? Because the value associated with it in a particular function is determined by other parts of the function. Although LaTeX: \frac{0}{0} 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 LaTeX: \frac{0}{0}. This concept goes with the concept of limits, one of the 3 major topics in calculus.



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