Sunday, August 23, 2015

The algebra needed to read about climate change...

This article (at occupy.com), 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.

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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 wolframalpha.com 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.

Warmly,
Sue

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!

Monday, June 1, 2015

Imbalance Abundance Puzzles (We're in the New York Times!!)

Paul Salomon posted some delightful puzzles a few years back, I got in touch with him about including them in the book, and now his puzzles are featured in the New york Times' Numberplay column!

I met Gary Antonick (who writes Numberplay) in person a month or two ago at a lovely meeting of math popularizers. We were both excited to meet each other*, and he asked if he could share some of the book's material in his column. Of course I said yes.

I knew the column was coming today, but forgot to look until I saw Mike South's Facebook post mentioning it. Mike writes great math explanations on Living Math Forum, but doesn't blog. I wanted to include something of his in the book, but didn't manage it. (Here's Mike on thinking about zero.)

Gary included a great photo that goes so well with the puzzles, I want to make up a new puzzle to go with it. Hmm.




If you don't already have your own copy of Playing with Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers, you can buy one here.





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*I finally got to meet the fabulous Fawn Nguyen in person, too! What an exciting day that was!

Wednesday, May 27, 2015

Preparing for the Fall Semester: How to Get Students to Participate More

Last summer, at a conference for the California Acceleration Project, Myra Snell used a cool way to set up random groups of participants/students. I wanted to use it in my classes, but just didn't get around to it. It seems, after almost 30 years of teaching, that it has become hard to change the way I run my classroom.

But I did change one thing this past semester. I noticed, while sitting in on a colleague's Calc III class, that I really appreciated the notices he wrote on the board at the beginning of each class. So I began to do it too. Maybe I could implement a few more good habits by watching other teachers during the second and third weeks of class.

Coming back to those random groups... I recently read research that found two effective strategies for getting students to participate more. One is visibly random groups. 'Visibly' means that they can't suspect the teacher of manipulating the group memberships. Myra's method is clearly random, looks easy to implement, and allows for up to four different groupings per class day. You have a slip for each student, with a number, a letter, an animal, and a food on it (for example). Those slips are set up so that no one is with any of the same other people more than once. I've asked Myra for her slips, but last night I was eager to think about it, and created my own. I don't know if this is the best way to do it, but I think it will work. Myra's slips had the 4 terms in a square and mine will be all in a row. I don't think that's a problem.

The second strategy which made a difference in student participation was student use of vertical whiteboards. The researcher(s?) compared paper and whiteboard, used vertically and horizontally. [Unfortunately, I can't find the research I originally read, which mentioned both the visibly random groups and the vertical whiteboards.] I'd like to try this out with the class I'll be teaching for the first time this fall, a compressed version of beginning algebra (first half of the semester) and intermediate algebra (second half of the semester). It's officially the same courses we've always taught, but I get to use a different curriculum, and will be using something project-based. I'm excited about implementing this.



 
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