Wednesday, September 16, 2020

Friday, September 11, 2020

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. 

tree with shadow, pretty vs helpful

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.)


















building with shadow, labeled



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...)














... LaTeX: \tan\theta=\frac{opp}{adj}, and this gives us  LaTeX: \tan63.4=\frac{h}{37.5}


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:

LaTeX: 37.5\cdot\tan63.4=37.5\cdot\frac{h}{37.5}\:\:\Longrightarrow\:\:h=37.5\cdot\tan63.4=74.8857...

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.




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.

Thursday, June 25, 2020

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!


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 booksDan and Christopher have some good ideas about playing mathematically with kids too.

Talking Math With Your Kids (#TMWYK)

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.
Not very hard.

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!

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

Thursday, June 18, 2020

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, or post them here.

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

Sunday, April 19, 2020

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...

Monday, March 23, 2020

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.)

Edited to add:

This online math circle happened on Friday, March 27, at 10am PDT (1pm EDT). [This link is to the zoom recording, along with its automatically produced (therefore hilariously bad) audio transcript.]

I promised to write up some of it here.

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.

Saturday, January 4, 2020

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?

Saturday, September 21, 2019

The History of Imaginary Numbers is a Soap Opera

I first read about this history in Journey Through Genius, by William Dunham, in the chapter titled Cardano and the solution of the cubic. It reads like a soap opera in which the math is done for glory, not for any possible connection to the real world. The people who came up with imaginary numbers thought they were fictitious elements in the process of solving a cubic, and never expected them to have any real meaning. Turns out they do. Imaginary numbers help scientists describe electrical current and probability distributions, among other things.

Years ago, on Living Math Forum, a mom wrote in to ask for help. Her son had asked: “The square root of 1 is 1, so what's the square root of -1 ?” That inspired me to write a math poem, Imaginary Numbers Do the Trick.

Recently I was having a lovely discussion with my editor, Maria Droujkova, and another author, about math storytelling. I realized this topic might possibly make for a good children's story. I'm working on it now. As I think about it, I'm not sure how to find the right age range. The math seems like it requires high school, but the story could interest younger kids, I think.

The History
Here's a short version:
  • Scipione del Ferro solves equations of the form  x3 + mx  = n (called depressed cubics). On his deathbed, in 1506, he passes his method on to his student, Antonio Fior.
  • Niccolo Tartaglia boasts that he can solve cubics of the form x3 + mx2  = n, so in 1535, Fior challenges him with 30 depressed cubics. (These challenges were a common feature of life as a mathematician in 1500's Italy, and provided a way for mathematicians to get more recognition and paying students.) Tartaglia's return problem list to Fior has a variety of problems. Tartaglia does not yet have a solution for the depressed cubic, and sweats it, working feverishly to try to figure it out. At the last moment, he succeeds, and solves all 30 problems. Fior does not do so well, and is humiliated.
  • Gerolamo Cardano comes to Tartaglia, asking him to disclose his method. He begs repeatedly, and Tartaglia, now Cardano's guest in Milan, finally concedes. Cardano takes an oath of secrecy. Tartaglia writes his solution in cipher, as a poem (!). 
  • Cardano takes on a brilliant student, Ludovico Ferrari, with whom he shares the secret. Together, they solve the general cubic, and then Ferrari goes on to solve the quartic. But all their work depends on reducing these to the depressed cubic, which Cardano has sworn not to tell about.
  • Cardano and Ferrari travel to Bologna, and are able to inspect the papers of ... Scipione del Ferro, where they find the solution. Cardano figures that relieves him of his oath and publishes, in his 1545 book, Ars Magna. He gives both del Ferro and Tartaglia credit, but Tartaglia is furious.
  • In the book, Cardano lays out the steps for solving the general cubic. But in doing so, he introduces a mystery. The depressed cubic x3 - 15x  = 4 clearly has solutions x = 4 and x = -2+-√3. And yet the formula found by Ferro, Tartaglia, Cardano, and Ferrari includes a √-121 for this equation. Cardano threw up his hands at the mystery. It was explored but not truly understood 30 years later by Rafael Bombelli. It took another almost two centuries for Euler to finally solve the mysteries of complex numbers.

    Here's a nice write-up I found online, but it suggests different facts than the version in Dunham's book. I will keep reading while I write, so I can hopefully get my facts right.

    My Request
    I'm looking for kids who would like to read my draft versions and tell me what parts they like. If you have kids who understand (at all) the notion of a square root and the idea of what a cubic equation is, would you ask them if they'd like to read my story? (I would, of course, mention them in my book if it gets published.) You, or your kids, can email me at mathanthologyeditor on gmail.

    Just a Beginning
    The Saga of the Imaginary Numbers
    “Mom, I’ve been thinking… If the square root of 1 is 1, what is the square root of -1?”

    “What a fun thing to think about, Althea! What have you figured out so far?”

    “I know that when I square 1 I get 1, and that’s why the square root of 1 is 1. But when I square negative 1, I get 1 too, so shouldn’t the square root of 1 be negative 1 too? But how can it be two things?”

    “Hmm, that’s a strange one, isn’t it? I think there are too many ones in this for me to keep track of things. Let’s switch to 3.

    "I’m going to try to say what you said, but with 3 and 9. 3 squared is 9, so the square root of 9 is 3. But negative 3 squared is still 9, So why isn’t the square root of 9 equal to negative 3 also? Is that basically the same question you asked?”

    “Yes. Except the square root of 9 can’t have two answers. Can it?”

    “Well, somebody a long time ago decided that there should be just one answer for the square root of a number. And so we say that there is the square root of 9 and also the negative square root of 9.”

    Sunday, August 25, 2019

    Another Semester, Starting with Good Intentions

    My handouts are copied, the piles are organized on my desk. My rosters are printed. (I entered student names into excel, so I can organize things my way.) I've looked over my computer folders and found a few more things to share tomorrow. And I'm getting better at using Canvas' features - I plan to have students evaluate the new activities online, to help me decide whether each activity stays, goes, or gets improved.

    Will I manage to set up a new student survey in Canvas for each new activity?
    Will I blog about my classes, like I'd like to?
    Will I do more activities and less lecture in each class?

    Once the semester gets rolling, it's hard for me to change things up. It's so much easier to do what I have done before. May my passion keep me improving, all through the term.

    Saturday, August 17, 2019

    First Day, Once Again

    I've seen some great advice for the first day of class.  (Here's the one I've read more than once. I've seen other great ideas, but I don't see them now.) I'd summarize my take on this article as:
    • (Community) Start learning names, and get them learning each other's names.
    • (Expectations) Don't spend much time on the syllabus; there are more important things to do. (Have them read it, and then you can quiz them on it the next day, or just ask for their questions.)
    • (Learning, Expectations) If you use group activities (which are a very effective way to help groups of students learn), then you want to introduce students to this on day one.
    • (Curiosity) An activity that helps them see what's coming in the course would be especially nice.

    Our classes start in ten nine days. I am getting ready...

    First day activity for Calculus I 
    I have them graph a parabola (y=x2), then draw a tangent to it at x=3. (Some don't know what that means, so I walk around checking.) And then estimate its slope. After they're done, I get to talk about what makes actually finding the slope hard - such a good intro to the course. And they've had time to review graphing a parabola.

    First day activity for Linear Algebra 
    I have them solve a system of 3 equations in 3 variables. I  ask them to:
    • Write down a description of the process, 
    • Solve the system, 
    • Now figure how to check whether your answer is correct. (Naysayers, has the group done enough to be sure that the answer is correct?) 
    • Extra: What does each equation represent geometrically? What does the solution represent geometrically? 
    Once again, a great introduction to the themes of the course.

    First day activity for Geometry
    I taught geometry earlier this summer for the first time. I had them draw a triangle (and make sure it was different than their neighbor's). Find the midpoints of each side (they could measure or fold). Connect each midpoint to the opposite vertex. I hoped most would do it well enough that the 3 connecting lines would intersect at one point. My goals were to highlight: vocabulary, shapes, construction (which we were not doing with straightedge and compass - yet), conjecture, and the possibility of proof.

    First day activity for Precalculus
    I've been thinking for the last few weeks about what I'd like to do for Precalculus. I have found exciting activities in the past that turned out to be way too hard, and intimidated the students. I have a lovely fractions activity, but that doesn't represent what we'll do going forward.

    I am working hard to create an activity that looks at functions (and circles too) from 4 perspectives: equations, graphs, tables of values, and stories. I have 7 types of relationships (linear, quadratic, polynomial, rational, exponential, periodic, and circles). I don't have stories yet for the polynomial and rational. (My eternal gratitude to anyone who can give me a story I like for either of these.) And I won't show an equation for the periodic. (A trig function wouldn't make sense yet. But we'll get to discuss that.) So that makes 25 "clue sheets".

    I have 40 students, who I'll put in 10 groups of 4. Each group will start with two clue sheets. [So 5 of the sheets will not be handed out at first. I can label those as graphs on the back, keep them at the front, and let student turn them over once they're pretty sure they didn't find a graph match to their set.] Each will describe a different type of function/relation from one perspective and ask them to do a few things. Then they pair up with a clue sheet for each pair, and go looking for the matching clue sheets (same function/relation, different perspective). They go back to their group and explain to each other what they found. (I'll have extras up front, so anyone done early can work on a 3rd function/relation.) When we're done, we'll have a summary of the function types we'll be studying all semester.

    I dreamed some of this up late last night. When I started working today, I worried that it would be too hard. (I make up some crazy stuff sometimes when I'm falling asleep.) So my goal as I put this together has been to scaffold it enough. I am assuming some comfort with linear functions, and some familiarity (but not comfort) with quadratics and exponentials. They may not have encountered the others. (And most will not know any trig.)

    I put my first second draft into a google doc here. Your suggestions may help me improve it. (I decided to leave out the rational function. 6 functions with 4 clues each would be 24. One story and one equation are left out. That's 22. The last two clues will sit up front.)

    Edited (8/17): This is a great activity, but too complex for day one. I will do it on day two. On day one, we will review linear functions in a similar, but much simpler way. Here is my handout.

    My Goals:
    • Review plugging values in for x to find y. (Do they remember that b0=1? What else might trip them up?)
    • Review graphing.
    • See functions/relations in the context of modeling a situation.
    • Identify functions/relations by type.
    • See precalculus as a place to strengthen their understanding of all of this.

    If you use this activity, please let me know what changes you decide to make and how it goes.

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