Crewton Ramone teaches multiplication with manipulatives

December 16th, 2009

Crewton Ramone’s Blog of Math has a couple of videos that show the power of using manipulatives to do multiplication. I’ve always been pretty decent at multiplication yet I enjoyed seeing how manipulatives can really help students to get grounded into what multiplication really means. His videos demystify the cross-multiplication tricks that I and others present. And, Crewton does a great job of showing how to do visual multiplication using symbols so you don’t need need to even buy any manipulatives. For small kids, it’s great because they get to turn multiplication into counting! And, best of all, Crewton has a nice enthusiasm about teaching the basics to kids as you’ll see if you peruse his blog.

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Review: An Imaginary Tale

December 15th, 2009

An Imaginary Tale is an enjoyable historical and hands-on exploration of imaginary numbers. One Amazon reviewer, K. Jazayeri, characterized the book perfectly:

You may think of it as an “appetizer sampler” for the topic - “A Splash From The Complex Plane” - to quote the title of a diagram in the final pages of the book.

And a great sampler it is. Seven chapters, six appendices, and numerous illustrations provide a nice and deep introduction to the subject.

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Crazy paper thing that’s easy to make

December 15th, 2009

What fun thing can you make with card stock and scotch tape? Check out this video and find out.

Hat tip to Denise at Let’s Play Math.

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Very disturbing optical illusion

December 14th, 2009

I really like optical illusions. Until recently, my favorite was this one. But, I have a new favorite now.

Here’s one that bothers me. See if it bothers you.

Look at the image below. Which square is darker, A or B, or are they the same color?

A hat tip to John Cook for blogging about this illusion.

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Review: The Math Book

December 10th, 2009

I’ve been a fan of Clifford Pickover since I discovered his books a couple of years ago. Some of Pickover’s book are like Martin Gardner’s in that they present problems that inspire Math exploration. But, I must say that many of the problems that Pickover proposes are very difficult ones. Inspiring, yes. Rich in exploration values, yes. Easy, no. The Math Book is a different kind of book. It’s a survey of 250 milestones in the history of Math. It has beautiful full page color illustrations. It’s approachable for the mathematically curious crowd in the same way that Theoni Pappas’ books appeal to people who enjoy Math but are not hard core about it.


Primates count

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Review: Manga Guide to Calculus

December 2nd, 2009


No Starch Press contacted me and offered me a review copy of the Manga Guide to Calculus. I have a hard time turning down a free Math book so I accepted.

I had never heard of “manga” before so I read up on the subject at Wikipedia.

“Manga consist of comics and print cartoons (sometimes also called komikku コミック), in the Japanese language and conforming to the style developed in Japan in the late 20th century. In their modern form, manga date from shortly after World War II, but they have a long, complex pre-history in earlier Japanese art.”

So, manga is a style of comic book. The “Manga Guide to Calculus” aims to teach calculus in a comic book format. I was intrigued. I must be up front and say that I studied calculus in High School close to 30 years ago. I don’t know that I’m a good person to judge whether the book effectively teaches calculus to High School or College students. But, I can share my impressions of the book.

My first thought is that I like the idea of having pictures in a Math book. I find that Math can be a very creative endeavor and that illustrations help to reinforce the idea that the creative right brain can be every bit as engaged in solving Math problems as can the logical left brain. When I write business proposals I always think of ways to break up the monotony with graphs or other illustrations. So, the book gets a thumbs up for being heavy on the pictures.

A second element that I really like is that the Manga Guide to Calculus tells a story. I strongly believe in storytelling as a creative way to teach. Telling a story makes the subject more personal and more engaging, in my experience. And, the book tells a fun story:

Noriko is just getting started as a junior reporter for the Asagake Times. She wants to cover the hard-hitting issues, like world affairs and politics. But, does she have the smarts for it? Thankfully, her overbearing and math-minded boss, Mr. Seki, is here to teach her how to analyze her stories with a mathematical eye.

So, the plot is fun. More than that, though, the story gives some very powerful examples of Calculus being used to solve real-world problems. This takes Calculus out of the realm of merely solving abstract problems into solving problems that scientists and engineers really care about. As much as I love abstract Math I also very much appreciate it when the relevance of Math can be demonstrated.

I like this book. A lot. I found that the deeper I went into the chapters, the more I appreciated the teaching approach of using comics and weaving in engaging stories and those real-world examples. As you might expect, the book doesn’t go beyond very basic Calculus. But, it does do a very nice job of covering differentiation (including techniques and related theorems), integration, Taylor expansions, and partial differentiation. The Manga Guide to Calculus would make the perfect gift for a high school or college student who is curious about calculus or who, perhaps, is taking Calculus.

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History and trajectory of Mathematica

November 6th, 2009

Today’s Wolfram Alpha Blog is the first in a series of videos and transcripts from Stephen Wolfram’s keynote address at the International Mathematica User Conference 2009. From the Wolfram Blog entry:

Future Mathematica features, new directions for Wolfram|Alpha, and how the two Wolfram technologies will be integrated were highlights of Stephen Wolfram’s keynote address at the International Mathematica User Conference 2009.

This first video gives a quick history of Mathematica from its roots going back to 1981. I’m looking forward to hints about future features.

Disclosure: Wolfram Research gave me a review copy of Mathematica.

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MMM #38: We have a winner!

October 30th, 2009

Random.org selected Ken Lareau as the winner for MMM #38. Congratulations, Ken! Here’s what the problem was:

Prove or disprove: The product of any four consecutive integers is always one less than a perfect square.

Don’t assume the integers are all positive.

Here’s Ken’s solution:

The above conjecture is true. The domain can be simplified a bit by noting two things:

1) If there are both positive and negative integers in the product, then since they’re consecutive, 0 has to be present, meaning the product is 0, which is one less than 1, a perfect square.

2) If all the numbers are negative, their product is positive and equal to the product of the negation of each number (in other words, (-a) * (-b) * (-c) * (-d) = a * b * c * d).

Therefore we can restrict our validation to consecutive positive integers. From this, the proof is easy to see since if the first number is ‘n’, then we have for the product:

n * (n + 1) * (n + 2) * (n + 3)

which expanded is:

n^4 + 6n^3 + 11n^2 + 6n

One more than this is:

n^4 + 6n^3 + 11n^2 + 6n + 1

which happens to factor down to:

(n^2 + 3n + 1)^2

which is a perfect square. Hence the conjecture holds for all possible products.

I have one comment on the problems and on the solutions submitted. Many of the submissions ignored the part about not assuming all the integers to be positive. While it may be trivial to many of you that the signs of the numbers doesn’t affect the solution, if you ignore that part of the problem then you’re not solving the whole problem. Ken did a superb job of showing the signs don’t matter. Yes, I realize that the algebraic solutions don’t assume anything about the signs of the numbers but it’s important to make that point in your argument.

And, here’s a solution at Watchmath.com

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How much would you pay for a 146 year old Math book?

October 29th, 2009

Every now and then I like to buy old Math books. I buy them on Ebay. Most of the books I buy are from the early 1900’s or the tail end of the 1800’s. Yesterday, I happened upon this book:

Yes, it’s from 1863. Yes, I paid $6.50 for it and there was no shipping charge. What’s the book about? Beats me. I’m not quite sure what Analytical Arithmetic Antique College Math is. If you think you know, leave a comment. What condition is the book in? Dunno but for six and a half bucks I’m not complaining!

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‘Dismantling the calculus pyramid’ gets ton of views

October 29th, 2009

A couple of days ago I posted a very brief article, Dismantling the calculus pyramid. The post has been seen 2,746 times, thanks mostly to being noticed by ‘timwiseman’ who posted a link to the article at Y Combinator Hacker News. The posting at Y Combinator has gotten 45 comments while, here, it’s gotten four comments. The conversation at Y Combinator is quite lively. Check it out.

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