Reblog: Calculus Tidbits

[Feature photo above by Olga Lednichenko via Flickr (CC BY 2.0).]

This week I have a series of quotes about calculus from my first two years of blogging. The posts were so short that I won’t bother to link you back to them, but math humor keeps well over the years, and W. W. Sawyer is (as always) insightful.

I hope you enjoy this “Throw-back Thursday” blast from the Let’s Play Math! blog archives:

Finding the Limit

Eldest daughter had her first calculus lesson last night: finding the limit as delta-t approached zero. The teacher found the speed of a car at a given point by using the distance function, calculating the average speed over shorter and shorter time intervals. Dd summarized the lesson for me:

“If you want to divide by zero, you have to sneak up on it from behind.”

Harmonic Series Quotation

This kicked off my week with a laugh:

Today I said to the calculus students, “I know, you’re looking at this series and you don’t see what I’m warning you about. You look and it and you think, ‘I trust this series. I would take candy from this series. I would get in a car with this series.’ But I’m going to warn you, this series is out to get you. Always remember: The harmonic series diverges. Never forget it.”

—Rudbeckia Hirta
Learning Curves Blog: The Harmonic Series
quoting Alexandre Borovik

So You Think You Know Calculus?

Rudbeckia Hirta has a great idea for a new TV blockbuster:

Common Sense and Calculus


And here’s a quick quote from W. W. Sawyer’s Mathematician’s Delight:

If you cannot see what the exact speed is, begin to ask questions. Silly ones are the best to begin with. Is the speed a million miles an hour? Or one inch a century? Somewhere between these limits. Good. We now know something about the speed. Begin to bring the limits in, and see how close together they can be brought.

Study your own methods of thought. How do you know that the speed is less than a million miles an hour? What method, in fact, are you unconsciously using to estimate speed? Can this method be applied to get closer estimates?

You know what speed is. You would not believe a man who claimed to walk at 5 miles an hour, but took 3 hours to walk 6 miles. You have only to apply the same common sense to stones rolling down hillsides, and the calculus is at your command.

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Basel spiral

More Than One Way To Find the Center of a Circle

[Feature photo above by hom26 via Flickr.]

My free time lately has gone to local events and to book editing. I hope to put up a series of blog posts sometime soon, based on the Homeschool Math FAQs chapter I’m adding to the paperback version of Let’s Play Math. [And of course, I'll update the ebook whenever I finally publish the paperback, so those of you who already bought a copy should be able to get the new version without paying extra.]

But in the meantime, as I was browsing my blog archives for an interesting “Throw-Back Thursday” post, I stumbled across this old geometry puzzle from Dave Marain over at MathNotations blog:

Is it possible that AB is a chord but NOT a diameter? That is, could circle ABC have a center that is NOT point O?

Jake shows Jack a piece of wood he cut out in the machine shop: a circular arc bounded by a chord. Jake claimed that the arc was not a semicircle. In fact, he claimed it was shorter than a semicircle, i.e., segment AB was not a diameter and arc ACB was less than 180 degrees.

Jack knew this was impossible and argued: “Don’t you see, Jake, that O must be the center of the circle and that OA, OB and OC are radii.”

Jake wasn’t buying this, since he had measured everything precisely. He argued that just because they could be radii didn’t prove they had to be.

Which boy do you agree with?

  • Pick one side of the debate, and try to find at least three different ways to prove your point.

If you have a student in geometry or higher math, print out the original post (but not the comments — it’s no fun when someone gives you the answer!) and see what he or she can do with it.

Dave offers many other puzzles to challenge your math students. While you are at his blog, do take some time to browse past articles.

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Related posts on Let’s Play Math! blog:

photo by Carla216 via flickr

Reblog: The Case of the Mysterious Story Problem

[Feature photo above by Carla216 via flickr (CC BY 2.0).]

Seven years ago, I blogged a revision of the first article I ever wrote about homeschooling math. I can’t even remember when the original article was published — years before the original (out of print) editions of my math books.

I hope you enjoy this “Throw-back Thursday” blast from the Let’s Play Math! blog archives:

I love story problems. Like a detective, I enjoy sifting out clues and solving the mystery. But what do you do when you come across a real stumper? Acting out story problems could make a one-page assignment take all week.

You don’t have to bake a pie to study fractions or jump off a cliff to learn gravity. Use your imagination instead. The following suggestions will help you find the clues you need to solve the case…

[Click here to go read the original post.]

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Alexandria Jones and the Strange Attractor

[Feature photo above: Clifford Attractor by Yami89 (public domain) via Wikimedia Commons.]

Alexandria Jones collapsed onto the couch with a dramatic sigh. Her father, the world-famous archaeologist Dr. Fibonacci Jones, glanced up from his newspaper and rolled his eyes.

“I don’t even want to hear about it,” he said.

Alex’s brother Leonhard was playing on the floor, making faces at the baby. He looked up at Alex and grinned.

“I’ll take the bait,” he said. “What happened?”

“Mom called my bedroom a Strange Attractor.”

“Oh? What does it attract?”

“I don’t know. Mostly books and model horses. But what’s so strange about that?”

The Mathematics of Chaos

Animation of a double compound pendulum showing chaotic behaviour.

Dr. Jones laughed and put down his paper. “Strange attractor is a technical term from the branch of mathematics called dynamical systems analysis — often called chaos theory.”

“So my bedroom is a math problem?”

“No. I think Mom meant your bedroom was chaos.”

“Oh.” Alex looked like she might pout, then she shrugged. “I guess she’s right, at that. So what is a strange attractor, really?”

“Well, when scientists first drew graphs of classical, non-chaotic systems — like a planet’s orbit or the flight of a football — it was surprising how often they got an ellipse or parabola or some similar curve,” Dr. Jones explained. “For some reason, nature seemed to be attracted to the shapes of classical geometry.”

Click here to continue reading.

Ohio Jones 2

The Linear Inequality Adventures of Ohio Jones

Ohio Jones 1

Last week, Kitten and I reached her textbook’s chapter on graphing linear equations, and a minor mistake with negative numbers threw her into an “I can’t do it!” funk. It’s not easy teaching a perfectionist kid.

Usually her mood improves if we switch to a slightly more advanced topic, and luckily I had saved these worksheets on my desktop, waiting for just such an opportunity. Today’s lesson:

  • Some fun(ish) worksheets
    “For tomorrow, students will be graphing systems of inequalities, so I decided to create a little Ohio Jones adventure (Indiana’s lesser known brother)…”

I offered to give her a hint, but she wanted to try it totally on her own. It took her about 40 minutes to work through the first few rooms of the Lost Templo de los Dulces and explain her solutions to me. I’m sure she’ll speed up with experience.

So far, she’s enjoying it much more than the textbook lesson. It’s fascinating to me how the mere hint of fantasy adventure can change graphing equations from boring to cool. Thanks, Dan!

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Inflation & Gold

Pondering Large Numbers

[Feature photo above by Paolo Camera (CC BY 2.0) via Flickr.]

Half of our students were missing from this month’s homeschool teen math circle, but I challenged the three who did show up to wrap their brains around some large numbers. Human intuition serves us well for the numbers we normally deal with from day to day, but it has a hard time with numbers outside our experience. We did a simple yet fascinating activity.

First, draw a line across a page of your notebook. Label one end of the line $20 (the amount of money I had in my purse), and mark the other end as $1 trillion (rough estimate of the US government’s yearly overspending, the annual deficit):


  • Where on that line do you think $1 million would be?

Go ahead, try it! The activity has a much greater impact when you really do it, rather than just reading. Don’t try to over-think this, just mark wherever it feels right to you.

The kids were NOT eager to commit themselves, but I waited in silence until everyone made a mark.

  • Okay, now, where do you think $1 billion would be?

This was a bit easier. Once they had committed to a place for a million, they went about that much farther down the line to mark a billion.

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Math Playtime With Blocks

Feature video by Stuart Jeckel via youtube.

DO Try This at Home

And ask questions!

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Algebra for (Almost) Any Age


Fawn Nguyen’s Visual Patterns website just keeps getting better and better. Check it out:

In addition to the 115 puzzle patterns (as of this writing), the site features a Gallery page of patterns submitted by students. And under the “Teachers” tab, Fawn shares a form to guide students in thinking their way through to the algebraic formula for a pattern.

How can you use these patterns to develop algebraic thinking with younger students? Mike Lawler and sons demonstrate Pattern #1 in the YouTube video below.

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Logic: The Centauri Challenge

Another fun discovery from the #MTBoS Challenge: Brian Miller (@TheMillerMath) posted this interstellar puzzle on his blog today.

[Right-click image to download a pdf you can print for your students.]

More Logic Puzzles

If you liked the Centauri Challenge, you may also enjoy the following blog posts:

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Rectangle algebra

Puzzle: Algebra on Rectangles

Gordon Hamilton of Math Pickle recently posted these videos on how to make algebra 1 puzzles on rectangles. As I was watching, Kitten came in and looked over my shoulder. She said, “Those look like fun!”

They look like fun to me, too, and I bet your beginning algebra students will enjoy them:

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RIMG0186 Satellite dish

How To Master Quadratic Equations

G'Day Math logo

feature photo above by Junya Ogura via flickr (CC BY 2.0)

A couple of weeks ago, James Tanton launched a wonderful resource: a free online course devoted to quadratic equations. (And he promises more topics to come.)

Kitten and I have been working through the lessons, and she loves it!

We’re skimming through pre-algebra in our regular lessons, but she has enjoyed playing around with simple algebra since she was in kindergarten. She has a strong track record of thinking her way through math problems, and earlier this year she invented her own method for solving systems of equations with two unknowns. I would guess her background is approximately equal to an above-average algebra 1 student near the end of the first semester.

After few lessons of Tanton’s course, she proved — within the limits of experimental error — that a catenary (the curve formed by a hanging chain) cannot be described by a quadratic equation. Last Friday, she easily solved the following equations:

\left ( x+4 \right )^2 -1=80


w^2 + 90 = 22 w - 31

and (though it took a bit more thought):

4x^2 + 4x + 4 = 172

We’ve spent less than half an hour a day on the course, as a supplement to our AoPS Pre-Algebra textbook. We watch each video together, pausing occasionally so she can try her hand at an equation before listening to Tanton’s explanation. Then (usually the next day) she reads the lesson and does the exercises on her own. So far, she hasn’t needed the answers in the Companion Guide to Quadratics, but she did use the “Dots on a Circle” activity — and knowing that she has the answers available helps her feel more independent.

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photo by Sphinx The Geek via flickr

Homeschooling High School Math

photo by ddluong via flickr

photo by ddluong via flickr

Feature photo (above) by Sphinx The Geek via flickr.

Most homeschoolers feel at least a small tinge of panic as their students approach high school. “What have we gotten ourselves into?” we wonder. “Can we really do this?” Here are a few tips to make the transition easier.

Before you move forward, it may help to take a look back. How has homeschooling worked for you and your children so far?

If your students hate math, they probably never got a good taste of the “Aha!” factor, that Eureka! thrill of solving a challenging puzzle. The early teen years may be your last chance to convince them that math can be fun, so consider putting aside your textbooks for a few months to:

On the other hand, if you have delayed formal arithmetic, using your children’s elementary years to explore a wide variety of mathematical adventures, now is a good time to take stock of what these experiences have taught your students.

  • How much of what society considers “the basics” have your children picked up along the way?
  • Are there any gaps in their understanding of arithmetic, any concepts you want to add to their mental tool box?

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Quotable: Why Study Algebra?


[Photo by AlphaTangoBravo / Adam Baker via flickr.]

One reason to study algebra: because it’s a building block. And just as it was really hard at first to get those blocks to do what you wanted them to do, so also it can be really hard at first to get algebra to work. But if you persevere, who knows what you might build someday?

Algebra is the beginning of a journey that gives you the skills to solve more complex problems.

So, try not to think of Algebra as a boring list of rules and procedures to memorize. Consider algebra as a gateway to exploring the world around us all.

— Jason Gibson
Why Study Algebra?

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Nrich: Math Puzzles and More

Nrich recently updated their amazing website. I love exploring their backlog of puzzles and games — what a mother lode of resources for math club or a homeschool co-op class!

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Who Killed Professor X?

What a Fun Book!


Who Killed Professor X? is a work of fiction based on actual incidents, and its heroes are real people who left their mark on the history of mathematics. The murder takes place in Paris in 1900, and the suspects are the greatest mathematicians of all time. Each suspect’s statement to the police leads to a mathematical problem, the solution of which requires some knowledge of secondary-school mathematics. But you don’t have to solve the puzzles in order to enjoy the book.

Fourteen pages of endnote biographies explain which parts of the mystery are true, which details are fictional, and which are both (true incidents slightly modified for the sake of the story).

I ordered Who Killed Professor X? from The Book Depository (free shipping worldwide!), and it only took 5 days to arrive here in the middle of the Midwest. My daughter Kitten, voracious as always, devoured it in one sitting — and even though she hasn’t studied high school geometry yet, she was able to work a couple of the problems.

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Lockhart quadrilateral puzzle

Lockhart’s Measurement

After watching the video on the page, this book has jumped to the top of my wish list.

You may have read Paul Lockhart’s earlier piece, A Mathematician’s Lament, which explored the ways that traditional schooling distorts mathematics. In this book, he attempts to share the wonder and beauty of math in a way that anyone can understand.

According to the publisher: “Measurement offers a permanent solution to math phobia by introducing us to mathematics as an artful way of thinking and living. Favoring plain English and pictures over jargon and formulas, Lockhart succeeds in making complex ideas about the mathematics of shape and motion intuitive and graspable.”

If you take any 4-sided shape at all — make it as awkward and as ridiculous as you want — if you take the middles of the sides and connect them, it always makes a parallelogram. Always! No matter what crazy, kooky thing you started with.

That’s scary to me. That’s a conspiracy.

That’s amazing!

That’s completely unexpected. I would have expected: You make some crazy blob and connect the middles, it’s gonna be another crazy blob. But it isn’t — it’s always a slanted box, beautifully parallel.

WHY is it that?!

The mathematical question is “Why?” It’s always why. And the only way we know how to answer such questions is to come up, from scratch, with these narrative arguments that explain it.

So what I want to do with this book is open up this world of mathematical reality, the creatures that we build there, the questions that we ask there, the ways in which we poke and prod (known as problems), and how we can possibly craft these elegant reason-poems.

— Paul Lockhart
author of Measurement

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Rate × Time = Distance Problems

I love how Richard Rusczyk explains math problems. It’s a new school year, and that means it’s time for new MathCounts Mini videos. Woohoo!

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Sample from the Introduction to Mathematical Thinking Class

I’m really looking forward to Keith Devlin’s free Introduction to Mathematical Thinking class, which starts in mid-September. There are more than 30,000 nearly 40,000 students signed up already. Will you join us?

These days, mathematics books tend to be awash with symbols, but mathematical notation no more is mathematics than musical notation is music.

A page of sheet music represents a piece of music: the music itself is what you get when the notes on the page are sung or performed on a musical instrument. It is in its performance that the music comes alive and becomes part of our experience. The music exists not on the printed page but in our minds.

The same is true for mathematics. The symbols on a page are just a representation of the mathematics. When read by a competent performer (in this case, someone trained in mathematics), the symbols on the printed page come alive — the mathematics lives and breathes in the mind of the reader like some abstract symphony.

— Keith Devlin
Introduction to Mathematical Thinking

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Thinking (and Teaching) like a Mathematician

photos by fdecomite via flickr

Most people think that mathematics means working with numbers and that being “good at math” means being able to do (only slower) what any $10 calculator can do. But then, most people think all sorts of silly things, right? That’s what makes “man on the street” interviews so funny.

Numbers are definitely part of math — but only part, and not even the biggest part. And being “good at math” means much more than being able to work with numbers. It means making connections, thinking creatively, seeing familiar things in new ways, asking “Why?” and “What if?” and “Are you sure?”

It means trying something and being willing to fail, then going back and trying something else. Even if your first try succeeded — or maybe, especially if your first try succeeded. Just knowing one way to do something is not, for a mathematician, the same as understanding that something. But the more different ways you know to figure it out, the closer you are to understanding it.

Mathematics is not just memorizing and following rules. If we want to teach real mathematics, we teachers need to learn to think like mathematicians. We need to see math as a mental game, playing with ideas. James Tanton explains:

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Happy Birthday, Einstein (Part 4)

Albert Einstein’s birthday was a couple of weeks ago, but today we have a belated celebration. MinutePhysics has finally finished its series on Einstein’s “wonder year” discoveries of 1905. In the last video, we began learning about the Special Theory of Relativity. This time, we find out how that theory leads to the most famous equation in the world…

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Happy Birthday, Einstein (Part 3)

In 1905, when he was 26 years old, Albert Einstein rocked the scientific world with a series of papers that changed our understanding of the nature of the universe. At MinutePhysics, the celebration continues:

More Einstein Videos

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Happy Birthday, Einstein (Part 2)

Today would be Albert Einstein’s 133rd birthday. At MinutePhysics, the celebration continues:

More Einstein Videos

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Happy Birthday, Einstein!

March 14th is Pi Day, and it’s also Albert Einstein’s birthday. In honor of Einstein, MinutePhysics is posting a series of videos on his “wonder year” of 1905, when he published several papers that eventually earned him the Nobel Prize.

More Einstein Videos

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Super Bowl XLVI Math Worksheet and Football Comic

Lance Friedman of has posted two bits of fun in honor of Super Bowl XLVI. (Click the images to go to Lance’s site.) And if you’re a homeschooler, Currclick is offering a Super Bowl Mini-Helper free this week.

NFL Math Quiz

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Fibonacci Numbers and Plants

Have you ever wondered why so many plants grow in Fibonacci Numbers? Vi Hart offers a great explanation (with hands-on activities) in these three videos — and she introduces a new species called the slugcat, which my daughter thinks is adorable.

Spirals, Fibonacci, and Being a Plant [1 of 3]

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