Sunday, October 25, 2015

Math Teachers at Play, #91

Number 91 feels like we're closing in on 100. The last time I hosted MT@P, we were at #71 and I managed to include 71 posts. I wasn't quite that ambitious this time. (Old math posts don't go stale. You might enjoy browsing through a bunch of the old Math Teachers at Play blog carnivals. And don't forget our partner carnival: the Carnival of Mathematics.)

If there are 14 people in a group, and each shakes hands with each other, there will be 91 handshakes. (Can you see why?)

91 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13
(which makes it triangular)


91 = 7 * 13
(the middle and last numbers in the sum above)

Will this always happen for triangular numbers?

Games & Puzzles

  • Shannon Duncan, a 6th grade math & science teacher, shares 4 Reasons to Promote Math Success through Games at the MIND Research Institute blog, illustrating her ideas with some of the games she has her students playing. I especially like the first point - making a mind-body connection.
  • John Golden (@mathhombre) shares Angle of Coincidence at his blog, Math Hombre, about an angle identification game he's developing. Ask your students to playtest it and give him feedback! John also wrote about the start of the semester, and included a game called In or Out?  that looks fun.
  • Jeff Trevaskis shares a Multiplication Tic-Tac-Toe Game at his blog, webmath. 
  • Carole Fullerton shares Number Tile Puzzles at  her blog, Mathematical Thinking. 
  • Gray Antonick interviewed Paul Salomon in the New York Times Numberplay column, about his Imbalance Puzzles, one of many puzzles and games featured in Playing with Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers (my book, published in April!).





  • Stephen Cavadino (@srcav) shares Parallelograms at his blog, cavmaths, on a student's creative way to find the area of a parallelogram.
  • Ioana I Pantiru (@LThMathematics) shares Playing with Paper Folding at her blog, Life Through a Mathematician's Eyes, showing the steps of an origami construction. In her post, Maths Class Everywhere, she asks readers to take her survey of math classes around the world. 
  • Curmudgeon shares Circles on a Lattice, at their blog, Math Arguments 180. I wonder if this would make a good problem for a math circle...  
  • Greg Blonder, a professor of manufacturing and product design, shares Trisecting the Angle With a Straightedge, at Plus Maths.
  • There have been lots of posts in the past few months about classifications of pentagons (here's one), because a new (15th) type of pentagon that will tile the plane was recently found. Here's a good background post, from before the discovery, from the Mathematical Tourist.


It's All Connected


Ideas for Learning ...

  • Kate Snow (@katesmathhelp) shares How to Teach Your Kids to Read Math  at her blog, Kate's Homeschool Math Help. I'm still trying to teach my college students how to read math, with some of the same tips.
  • Manan (@shalock) shares Becoming Mathematically Fluent at his blog, Math Misery.
  • Shecky (@sheckyr) shares True Deep Beauty ... at his blog, Math-Frolic, about the how our understanding of math deepens.
  • Chris Rime is making monthly math calendars (Algebra I, II, and Geometry), available as doc or pdf at his blog, Partially Derivative.

... And Teaching

  • Tom Bennison (@DrBennison) shares How to enjoy your NQT Year at his blog, Mathematics and Coding. [I had to look up NQT. It means newly qualified teacher, and in England and Wales, you are "inducted" in your NQT year, (generally) your first year of paid teaching.] I like his suggestion to make time for doing some math(s) yourself.


I'm going to the Joint Mathematics Meetings in January in Seattle. I'd love to connect with other bloggers who are going. There's a math poetry reading plus art exhibit on Thursday evening at 5:30. You can get all the details from JoAnne Growney's Intersections blog.

Friday, September 18, 2015

Joint Mathematics Meetings - Seattle in January

I think I'd like to present. I've never done that at the JMM. I'd like your help. Here's (my second draft of) what I've written for my proposed abstract: 
Have you seen your students disengage from your calculus class in the first week as they struggle with the technical topic of limits? They don’t see the point, get mired in the algebra and can become alienated. I will share why I save limits for later and start out with an exciting and historical approach using slope and velocity.

But perhaps your textbook, like mine, follows a traditional approach? I will also share how I used parts of two Open Education Resources (OER) by Matt Boelkins and Dale Hoffman, along with a few pages I created, to make a coursepack for my first unit. [Link to modifiable materials provided at talk, or by email.] Their materials gave my students the support they needed in our excursions off the traditional textbook’s beaten path.

I’ll help you see why there’s a better order to the topics. (It’s not just the limits.) And I’ll show you one way to make Calculus fun for yourself and your students.

You can use the experiences I share in my talk as inspiration to help you get started remixing OER to develop your own approach and materials. Using these materials in a coursepack alongside the required text may also be a way to show your reluctant department that they don’t need the $200-plus conventional textbooks.

  •  Have I said enough to make it clear what I have to offer?
  • What more should I say?
  • What should I change?
  • Would you come to my talk?

(My deadline is in 4 days.)  

Sunday, August 23, 2015

The algebra needed to read about climate change...

This article (at, on a lawsuit from a group of young people demanding that we do what it takes to recover from climate change, looks very interesting. One line seemed either wrong or surprising to me, though.

We must immediately commence carbon emissions reductions of 6% each year until the end of the century. Timing is crucial. If we wait until 2020 to begin emissions reductions the annual requirement is 15% per year.
Starting only 5 years earlier, they are saying that we can do 2/5ths as much reducing each year, for 85 years instead of 80, and get the same result. It seems too dramatic. I want to think about how to analyze it. I don't yet know what assumptions I can make.

  • Should I compare total emissions from now until 2100? (I think so.)
  • Should I assume emissions are growing exponentially from now until 2020 in the 2nd scenario? (I think so.)
  • What else would I need to know? (Are there other factors that make this more complicated?)
This seems like a perfect question for pre-calculus. Too bad I'm not teaching it this semester.

I think I got it. I think this assumes that we are currently increasing our carbon emissions at a rate of about 20% a year.  We are not. It's more like a tenth of that - about 2.5%. (Government source here.)

If you want to do some real math, think about what you would do before continuing.






I figured it like this. I count this year's carbon emissions as 1. If we decrease 6% a year, that means we have 94% of the previous year's emissions. So the total emissions from now until 2100 is
S=1+.94+.94^2+...+.94^84. This simplifies to S = (1-.94^85)/(1-.94). Note that the .94^85 is so close to 0 that we can ignore it. We Get S=1/.06 =  16.666. So the article is saying that for the next 85 years, we can emit 16 times this year's emissions.

If we increase until 2020, we would start with higher emissions, H. 15% decrease per year leaves 85% of the previous year's emissions. Our sum would be
S=H+.85H+.85^2H+...+.85^79H = H(1-.85^79)/(1-.85) = (almost) 1/.15 = 6.666.

16.666 - 6.666 = 10. So somehow we get 10 times this year's emissions within the next 5 years. If our emissions are currently increasing so that our emissions next year is r, then
S = 1 + r + r^2 + ... +r^5 = (1-r^6) / (1-r) = 10. I asked to solve  this and got r = 1.2, for a 20% increase per year.

I asked John Golden to check my work. He used a continuous increase model and got close to 9%, mush lower. But still not low enough to match what's happening.

So it seems that either the article has a typo, or my mathematical model is not including everything it should. Humanity seems to be at a tipping point. Can we change our ways of making decisions, from capitalism to something else, in time to save ourselves from our foolishness? I would like everyone to be able to do this sort of math.

Saturday, August 22, 2015

Linear Algebra Question

On Thursday we arrived at Theorem 1 in David Lay's Linear Algebra and Its Applications:
"Uniqueness of the Reduced Echelon Form
Each matrix is row equivalent to one and only one reduced echelon matrix."

The proof is in an appendix, which is a bummer, because this class feels like it could build from first principles nicely up to all its glory. The proof involves material from chapter 4, and I have to fight my way through it. Isn't he worried about being circular?

I was thinking out loud in class. I said (more or less):
If the system is consistent, it has a particular solution set. You can read the solution off from the reduced echelon form, so it can only give you one answer. [In class I wasn't thinking about free variables, and whether those could be different somehow. I was just thinking about problems with one unique solution.] We know it gives the right answer because
we've already shown that elementary row operations create row equivalent matrices, which have the same solution set.

What about an inconsistent system? I'm not sure about that. If you can break his theorem, I'll give you extra credit. 

Well, I just broke his theorem, I think. (I hope none of my students are reading my blog yet.) Given the system

Have I broken his theorem? Should he have said this instead?
"Each matrix representing a consistent system of equations is row equivalent to one and only one reduced echelon matrix."

Friday, August 21, 2015

Random Grouping Cards and Slips

I have just finished my first week of class.

I have finally used Myra Snell's Random Grouping Cards, to put students in groups. I've been wanting to do this for the past year, and finally got over my inertia problem. Research shows that putting students in visibly random groups gets them participating more. (Visibly means they don't wonder if the teacher made it non-random.)

Myra's cards work for a class of 32 students or (a bit) fewer. If you class is bigger or much smaller, you'll need something different. I couldn't figure out an easy way to get mine onto her format. So mine are Random Grouping Slips. I have sets for 16, 23, 32, and 48 students. You cut off the first column, and then slice apart the rows.

I was intrigued that I could not (easily) get 24 student slips. The last one would have put two people together in the last group who had been together before. The way I set it up was based on 16. There was no simple way to make it smaller.

I ended up with classes with 20, 40, and 28 students, so I've made those too now. They're organized a bit differently. I don't like the time it takes to cut them on the paper cutter. Hmm...

Some of the students complain, but I think I am already seeing more of a community forming among the whole class. I'll be watching for ways in which this changes classroom dynamics.

I have also finally begun to implement the Gallery Walk I learned about at the CAP (California Acceleration Project) conference from Myra. I hope to write about that soon.

All three of my classes seem to be going well.

Saturday, August 8, 2015

Links on Saturday (lots for First Day)

First Day 

 First Week

Other Good Stuff

Sunday, July 12, 2015

Playing with Math: Can you write a review?

Playing with Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers is on Amazon now! But we don't yet have any reviews. If you've gotten a copy of the book, can you write a review on Amazon? We would be so grateful.


Friday, July 10, 2015

Links on Friday

I'll be leading a Math Jam for eight days just before Fall semester starts, helping students prepare to succeed in Beginning Algebra. My eight topics:
  1. Number Sense
  2. Fractions
  3. Negatives
  4. Algebra
  5. Percents
  6. Graphing 
  7. Slopes
  8. Problem-Solving  

For fractions, I plan to do a bit with Egyptian Fractions. Here's a site that looks good for that. I looked at the Beast Academy site to see if they had anything good. I found 5 things I liked: one game and two puzzles using the area meaning of multiplication, one puzzle on ordering of decimals,  and one game like Taboo for communicating about shapes.

Thursday, July 2, 2015

Playing with Math: Inspiring Online Conversations

First sighting of a comment on a mathematical blog post that was inspired by seeing the content in my book...

Jonathan Halabi writes jd2718. His post, Puzzle: Who am I?, became one of the puzzles in Playing with Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers.

Today Lara H replied to his post:
I came across this puzzle in the book “Playing with Math.” I found a different solution based on a wrong assumption I made at the beginning of solving the puzzle. I was thinking that a number with 3 digits also has 2 digits so I made both of those statements true and came up with 4097, which works for all the other conditions.

I responded with:
I’d say ‘different interpretation’ instead of ‘wrong assumption’. I wonder how many solutions the puzzle has using your interpretation. (Pretty exciting to see my book has inspired new discussion on Jonathan’s blog post!)

We are hoping that the book will inspire online conversations. This is the first drop of what we hope will eventually become a deluge.  

Saturday, June 20, 2015

Book Review: The Archimedes Codex

I bought this book because I wanted to understand more about Archimedes' role in the ancient development of calculus ideas. When I got it, I was worried it would be another book I wouldn't want to wade through. I was so wrong!

The Archimedes Codex, by Reviel Netz and William Noel, is fascinating. Like much good science writing these days, The Archimedes Codex reads like a detective story. It is gripping! Netz writes chapters about Archimedes, his math, and translation issues. Noel writes chapters about the travels of the manuscript, and the attempts to use modern technology to get better images of Archimedes' writing.

In 1998 Christie's auctioned off this battered medieval manuscript which on its face was a prayer book, but also contained traces underneath of Archimedes work, which had been scraped off. It sold for two million dollars to an anonymous bidder. William Noel, of the Walters Art Museum in Boston, followed the story and emailed the agent of the buyer. The buyer agreed to work with the museum to attempt restoration of the manuscript. Most experts expected little from the work, since the manuscript was in such bad condition. But the project, which took years, brought to light previously unknown work by Archimedes.

Archimedes had explored the idea of infinity more carefully than had ever been realized. He also did work in combinatorics, which no one had even suspected. The math is pretty easy to follow, and it's amazing. I've dogeared about a dozen pages, so I can read passages to my calculus students.

This is perfect summer reading. Enjoy!

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