# Reblog: The Case of the Mysterious Story Problem

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

Here’s a blast from the Let’s Play Math! blog archives: 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…

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…

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# Playing with Pinterest: New Math Boards

Do you like Pinterest? I’ve enjoyed exploring the site lately, so I set up a few boards where I can pin the goodies I find. It may take awhile before I get all the old games and posts from this blog loaded up, so save the links and come back often…

### Playful Math Games & Activities

As our children (and their parents!) play around with mathematical ideas and the relationships between them, we develop deep understanding that is strong enough to support future learning. Playful math links include math games, activities, and interesting lesson plans.

### Math Doodling

Making abstract math visual: Math doodles let us see and experiment with a wide range of mathematical structures — and even to feel them, if we include hands-on 3D doodles in clay or other media. Links include art projects, geometry constructions, and physical models to explore.

### Math Teaching Tips & Resources

A variety of math teaching ideas for homeschool families or classroom teachers. Learning mathematics is more than just answer-getting: help your students make conceptual connections. These links are more “schooly” than on the other boards, and they include conceptual lessons that build your own understanding of mathematics as well as that of your students. And math notebooking resources, too.

### MTaP Math Education Blog Carnival Archive

Since early 2009, the Math Teachers at Play (MTaP) blog carnival has offered tips, tidbits, games, and activities for students and teachers of preschool through pre-college mathematics. Now published once a month, the carnival welcomes entries from parents, students, teachers, homeschoolers, and just plain folks. If you like to learn new things and play around with ideas, you are sure to find something of interest.

### Math-Ed Quotes

Inspiration for homeschooling parents and classroom teachers. This is where I’m posting my Wednesday Wisdom quotes.

And that’s the end of my Pinterest boards (so far).

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# The Linear Inequality Adventures of Ohio Jones

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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# 2014 Mathematics Game

[Feature photo above by Artis Rams (CC BY 2.0) via flickr. Title background (right) by Dan Moyle (CC BY 2.0) via flickr]

Have you made a New Year’s resolution to spend more time with your family this year, and to get more exercise? Problem-solvers of all ages can pump up their (mental) muscles with the Annual Mathematics Year Game Extravaganza!

For many years mathematicians, scientists, engineers and others interested in mathematics have played “year games” via e-mail and in newsgroups. We don’t always know whether it is possible to write expressions for all the numbers from 1 to 100 using only the digits in the current year, but it is fun to try to see how many you can find.

## Rules of the Game

Use the digits in the year 2014 to write mathematical expressions for the counting numbers 1 through 100. The goal is adjustable by age: Young children can start with looking for 1-10, or 1-25.

• You must use all four digits. You may not use any other numbers.
• Solutions that keep the year digits in 2-0-1-4 order are preferred, but not required.
• You may use +, -, x, ÷, sqrt (square root), ^ (raise to a power), ! (factorial), and parentheses, brackets, or other grouping symbols.
• You may use a decimal point to create numbers such as .2, .02, etc., but you cannot write 0.02 because we only have one zero in this year’s number.
• You may use the overhead-bar (vinculum), dots, or brackets to mark a repeating decimal.
• You may create multi-digit numbers such as 10 or 201 or .01, but we prefer solutions that avoid them.
• You may use a double factorial, but we prefer solutions that avoid them. n!! = the product of all integers from 1 to n that have the same parity (odd or even) as n.

[Note to students and teachers: If you want to take part in the Math Forum Year Game, be warned that they do not allow repeating decimals.]

By Denise Gaskins Posted in Puzzles

# 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.

# Logic: The Centauri Challenge

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

## More Logic Puzzles

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

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# Math That Is Fun: Infinite Primes

Oh, my! Ben Orlin over at Math with Bad Drawings just published my new favorite math proof ever:

I had a fight with Euclid on the nature of the primes.
It got a little heated – you know how the tension climbs.

It started out most civil, with a honeyed cup of tea;
we traded tales of scholars, like Descartes and Ptolemy.
But as the tea began to cool, our chatter did as well.
We’d had our fill of gossip. We sat silent for a spell.
That’s when Euclid turned to me, and said, “Hear this, my friend:
did you know the primes go on forever, with no end?” …

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# How To Master Quadratic Equations

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:

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

and:

$w^2 + 90 = 22 w - 31$

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.

# Homeschooling High School Math

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?

# How to Recognize a Successful Homeschool Math Program

photo by Dan McCarthy (cc-by)

After teaching co-op math classes for several years, I’ve become known as the local math maven. Upon meeting one of my children, fellow homeschoolers often say, “Oh, you’re Denise’s son/daughter? You must be really good at math.”

The kids do their best to smile politely — and not to roll their eyes until the other person has turned away.

I hear similar comments after teaching a math workshop: “Wow, your kids must love math!” But my children are individuals, each with his or her own interests. A couple of them enjoy an occasional geometry or logic puzzle, but they never voluntarily sit down to slog through a math workbook page.

In fact, one daughter expressed the depth of her youthful perfectionist angst by scribbling all over the cover of her Miquon math workbook:

• “I hate math! Hate, hate, hate-hate-HATE MATH!!!”

Translation: “If I can’t do it flawlessly the first time, then I don’t want to do it at all.”

photo by Jason Bolonski (cc-by)

# 2013 Mathematics Game

feature photo above by Alan Klim via flickr

New Year’s Day

Now is the accepted time to make your regular annual good resolutions. Next week you can begin paving hell with them as usual.

Yesterday, everybody smoked his last cigar, took his last drink, and swore his last oath. Today, we are a pious and exemplary community. Thirty days from now, we shall have cast our reformation to the winds and gone to cutting our ancient shortcomings considerably shorter than ever. We shall also reflect pleasantly upon how we did the same old thing last year about this time.

However, go in, community. New Year’s is a harmless annual institution, of no particular use to anybody save as a scapegoat for promiscuous drunks, and friendly calls, and humbug resolutions, and we wish you to enjoy it with a looseness suited to the greatness of the occasion.

For many homeschoolers, January is the time to assess our progress and make a few New Semester’s Resolutions. This year, we resolve to challenge ourselves to more math puzzles. Would you like to join us? Pump up your mental muscles with the 2013 Mathematics Game!

By Denise Gaskins Posted in Puzzles

# 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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# 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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# How to Think like a Mathematician

Would you like to learn how to think like a mathematician? Stanford professor (and NPR “Math Guy”) Keith Devlin is teaching a free online course through Coursera. It starts in just a few weeks. I’ve signed up. Will you join us?

The prerequisite is to be taking or have finished high school math. If (like me) you took it so long ago that you can’t quite remember, don’t worry: The focus of the course is not on long-forgotten mathematical procedures, but on “learning to think in a certain (very powerful) way.”

Mathematical thinking is not the same as doing mathematics — at least not as mathematics is typically presented in our school system. School math typically focuses on learning procedures to solve highly stereotyped problems. Professional mathematicians think a certain way to solve real problems, problems that can arise from the everyday world, or from science, or from within mathematics itself.

The key to success in school math is to learn to think inside-the-box. In contrast, a key feature of mathematical thinking is thinking outside-the-box — a valuable ability in today’s world. This course helps to develop that crucial way of thinking.

— Keith Devlin
Introduction to Mathematical Thinking

# 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:

# 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…

# 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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# Purple Comet! Math Meet

The 2012 Purple Comet! Math Meet is a free, on-line, team competition for middle and high school students around the world. Every team needs an adult supervisor. Homeschoolers are welcome and should register under the Mixed Team category.

Register now. The contest will run Tuesday, April 17, through Thursday, April 26, 2012. That gives your team plenty of time for practice sessions between now and then.

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# What I’m Reading: Fermat’s Enigma

Homeschooling is much more than just doing school at home — it’s a lifelong lifestyle of learning. And thanks to the modern miracle of inter-library loan, even those of us who live in the middle of nowhere can get just about any book sent directly to our tiny home-town libraries.

As I mentioned in Math Teachers at Play 46, I’m trying to add more living books about math to our homeschool schedule, including my own self-education reading. So, a copy of Fermat’s Enigma: The Epic Quest to Solve the World’s Greatest Mathematical Problem finally showed up at my library, and I am thoroughly enjoying it.

# 2012 Mathematics Game

photo by Creativity103 via flickr

For our homeschool, January is the time to assess our progress and make a few New Semester’s Resolutions. This year, we resolve to challenge ourselves to more math puzzles. Would you like to join us? Pump up your mental muscles with the 2012 Mathematics Game!

## Rules of the Game

Use the digits in the year 2012 to write mathematical expressions for the counting numbers 1 through 100.

Bonus Rules
You may use the overhead-bar (vinculum), dots, or brackets to mark a repeating decimal.

You may use multifactorials:

• n!! = a double factorial = the product of all integers from 1 to n that have the same parity (odd or even) as n.
• n!!! = a triple factorial = the product of all integers from 1 to n that are equal to n mod 3

[Note to teachers: Math Forum modified their rules to allow double factorials, but as far as I know, they do not allow repeating decimals or triple factorials.]

By Denise Gaskins Posted in Puzzles

# Tau Day Limerick

So if working in radians you hate
(How can $\frac {\pi}{4}$ be really $\frac {pie}{8}$?),
Tau = τ = one turn around the circle = $\frac{C}{r}$ = 2π = 6.28318…