# 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.

Don’t miss any of “Let’s Play Math!”:  Subscribe in a reader, or get updates by Email.

# Math Teachers at Play #52

[Photo by bumeister1 via flickr.]

Welcome to the Math Teachers At Play blog carnival — which is not just for math teachers! We have games, lessons, and learning activities from preschool math to calculus. If you like to learn new things and play around with mathematical ideas, you are sure to find something of interest.

Scattered between all the math blog links, I’ve included highlights from the Common Core Standards for Mathematical Practice, which describe the types of expertise that teachers at all levels — whether in traditional, experimental, or home schools — should seek to develop in their math students.

Let the mathematical fun begin…

## TRY THESE PUZZLES

By tradition, we start the carnival with a couple of puzzles in honor of our 52nd edition. Since there are 52 playing cards in a standard deck, I chose two card puzzles from the Maths Is Fun Card Puzzles page:

• A blind-folded man is handed a deck of 52 cards and told that exactly 10 of these cards are facing up. How can he divide the cards into two piles (which may be of different sizes) with each pile having the same number of cards facing up?
• What is the smallest number of cards you must take from a 52-card deck to be guaranteed at least one four-of-a-kind?

The answers are at Maths Is Fun, but don’t look there. Having someone give you the answer is no fun at all!

# Math Teachers at Play #46

Welcome to the Math Teachers At Play blog carnival — which is not just for math teachers! Here is a smorgasbord of ideas for learning, teaching, and playing around with math from preschool to pre-college. Some articles were submitted by their authors, others were drawn from the immense backlog in my blog reader. If you like to learn new things, you are sure to find something of interest.

# Quotable: From Calc 3

“We have 4 equations and only 4 unknowns so that gives us a fighting chance of actually solving it.”

— My daughter’s Calculus III teacher

“Of course, he was doing an easy problem compared to the homework. :P

— My daughter, Niner

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# Math Teachers at Play #39

Welcome to the Math Teachers At Play blog carnival — which is not just for math teachers! If you like to learn new things and play around with ideas, you are sure to find something of interest.

Several of these articles were submitted by the bloggers; others were drawn from my overflowing blog reader. Don’t try to skim everything all at once, but take the time to enjoy browsing. Savor a few posts today, and then come back for another helping tomorrow or next week.

Most of the photos below are from the 2010 MAA Found Math Gallery; click each image for more details. Quotations are from Mike Cook’s Canonical List of Math Jokes.

Let the mathematical fun begin…

# Math Teachers at Play #35

35 is a tetrahedral number

Welcome to the Math Teachers At Play blog carnival — which is not just for math teachers.

Do you enjoy math? I hope so! If not, browsing these links just may change your mind. Most of these posts were submitted by the bloggers themselves; others are drawn from my overflowing Google Reader. From preschool to high school, there are plenty of interesting things to learn.

Let the mathematical fun begin…

For my Calculus for Young People students: Beware! We studied a few infinite series that converge to a nice, tame sum — but not all series are so well behaved.

Check out this mind-blowing video from the author of Math Without Words:

[See also: Harmonic Series Quotation and For Niner: A Bit of Calculus Fun.]

Don’t miss any of “Let’s Play Math!”:  Subscribe in a reader, or get updates by Email.

# Planning a New Math Club

[Photo by Waponi.]

A few years ago, I had several (potentially) future engineers in our homeschool math club, and we enjoyed the challenge of MathCounts and AMC puzzles — but the current crop of local homeschool students is another story.

Last year’s contest-based club meetings dwindled to one student. Even before the recent MathCounts rule changes, I knew I needed a new plan. The final straw was Kitten, whose moaning complaint that she “hates math” has begun to drive me crazy.

So, what’s a homeschool math teacher to do?

# Math History Tidbits: The Battling Bernoullis

July 27th is Alex’s birthday. She shares it with Johann Bernoulli, an irascible mathematician from the late 17th century. This coincidence intrigued her enough that she wrote a research paper on Johann and his mathematical brother, titled “Jeering Jacob and Jealous Johann.”

Of course, to make the alliteration work, she had to mispronounce Johann’s name — but she figured he kinda deserved that. Read the historical tidbits below to find out why one writer said the Bernoulli brothers were “the kind of people who give arrogance a bad name.”*

# Math Teachers at Play #24

[Photo by internets_dairy.]

Welcome to the Math Teachers At Play blog carnival — which is not just for math teachers! If you like to learn new things and play around with ideas, you are sure to find something of interest. Let’s start the mathematical fun with an arithmetic card game in honor of our 24th edition and a few number puzzles:

# Math Teachers at Play #20

[Photo by shonk.]

Welcome to the Math Teachers At Play blog carnival — which is not just for math teachers! If you like to learn new things and play around with ideas, you are sure to find something of interest.

Let’s start the mathematical fun with a couple of puzzles in honor of our 20th edition: First, the shape to our right is an icosahedron, one of the Platonic solids. Each face is an equilateral triangle — can you count them? For more fun, make your own model.

# Math Teachers at Play #8

[Photo by jaaron.]

Welcome to the Math Teachers At Play blog carnival — which is not just for math teachers! We accept entries from anyone who enjoys playing around with math, as long as the topic is relevant to students or teachers of preK-12th grade mathematics.

Some articles were submitted by their authors, other were drawn from the back-log in my blog reader, and I’ve spiced it all up with a few math jokes courtesy of the Mathematical humor collection of Andrej and Elena Cherkaev.

Let the mathematical fun begin…

# Free: Calculus Student’s Best Friend

Considering how many fools can calculate, it is surprising that it should be thought either a difficult or a tedious task for any other fool to learn how to master the same tricks… Being myself a remarkably stupid fellow, I have had to unteach myself the difficulties, and now beg to present to my fellow fools the parts that are not hard. Master these thoroughly, and the rest will follow. What one fool can do, another can.

For years, I have recommended Calculus Made Easy as summer reading (and future reference) for high school or college students headed into a calculus course — and for the parents of those students, who may have studied calculus in ages past and now need to dredge out the dust bunnies of memory so they can help with homework.

[Hat tip to Sam and Michael for finding the Scribd version, which set me off searching for a clearer copy.]

Don’t miss any of “Let’s Play Math!”:  Subscribe in a reader, or get updates by Email.

# Math Teachers at Play #5

[Photo by Alex Kehr.]

Welcome to the Math Teachers At Play blog carnival — which is not just for math teachers! If you like to learn new things and play around with ideas, you are sure to find something of interest. Let the mathematical fun begin…

# Math Teachers at Play #2

[Photo by Sister72.]

Welcome to the second Math Teachers At Play blog carnival! Some articles were submitted by their authors, other were drawn from the back-log in my blog reader, and I’ve spiced it all up with a few of my favorite quotations.

Let the mathematical fun begin…

# Math Teachers at Play #1

[Photo by StuSeeger.]

Welcome to the inaugural edition of the Math Teachers At Play blog carnival! I hope you enjoy this collection of tips, tidbits, games, and activities for students and teachers of preschool-12th grade mathematics.

For this first carnival, I’ve drawn several recent posts from my blog reader as examples of the types of posts I’d love to include in future editions of Math Teachers at Play. I tried to find something for everyone, from multiplication drill for elementary students to advice for understanding high school math equations.

Let the mathematical fun begin…

# Rewriting the History of Math

Here are a couple of quick links to math in the news:

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# Secret Message

My algebra students could stand to hear this, too:

(2)(-4x2)n is not equal to (-8 )nx2n.

AAAAAARRRRGGGHH!!!!

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# Confession: I Am Not Good at Math

I want to tell you a story. Everyone likes a story, right? But at the heart of my story lies a confession that I am afraid will shock many readers. People assume that because I teach math, blog about math, give advice about math on internet forums, and present workshops about teaching math — because I do all this, I must be good at math.

Apply logic to that statement. The conclusion simply isn’t valid.

# Common Sense and Calculus

One more quote from W. W. Sawyer’s Mathematician’s Delight before I have to return the book to the library:

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.

Don’t miss any of “Let’s Play Math!”:  Subscribe in a reader, or get updates by Email.

# Harmonic Series Quotation

If you’d like to start your week with a laugh, here’s a great quote:

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

Don’t miss any of “Let’s Play Math!”:  Subscribe in a reader, or get updates by Email.

# Finding the Limit

[Rescued from my old blog.]

Eldest dd had her first calculus lesson last night: derivatives. 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.”

Of course, she understands you can’t really divide by zero, but I thought her tongue-in-cheek comment was a pretty good description of the process of finding the limit as delta-t approached zero.

Don’t miss any of “Let’s Play Math!”:  Subscribe in a reader, or get updates by Email.

Check out these related posts: My Family’s Adventures, Best Bits of Let’s Play Math! Humor

# So You Think You Know Calculus?

[Rescued from my old blog.]

Rudbeckia Hirta over at Learning Curves submits a great idea for a new TV blockbuster:

I’ve read that Charlotte Mason used oral exams like this with her students, though of course without the pressure and sarcastic comments. I’d love to hear how this works in “real life.”