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.)

 

 

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building with shadow, labeled

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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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... 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.

 

 

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.

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!










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 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.
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) ... 

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 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.


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?
 
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