# 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.

## More Logic Puzzles

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

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# Things To Do with a Hundred Chart #30

Here’s one more entry for my 20+ Things to Do with a Hundred Chart post, thanks to David Radcliffe in the comments on Monday’s post:

(30) Can you mark ten squares Sudoku-style, so that no two squares share the same row or column? Add up the numbers to get your score. Then try to find a different set of ten Sudoku-style squares. What do you notice? What do you wonder?

Can you think of anything else we might do with a hundred chart? Add your ideas in the Comments section below, and I’ll add the best ones to our master list.

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# 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:

# Math Teachers at Play #62

by Robert Webb

Do you enjoy math? I hope so! If not, browsing this post just may change your mind. Welcome to the Math Teachers At Play blog carnival — a smorgasbord of ideas for learning, teaching, and playing around with math from preschool to pre-college.

Let the mathematical fun begin!

## POLYHEDRON PUZZLE

By tradition, we start the carnival with a puzzle in honor of our 62nd edition:

An Archimedean solid is a polyhedron made of two or more types of regular polygons meeting in identical vertices. A rhombicosidodecahedron (see image above) has 62 sides: triangles, squares, and pentagons.

• How many of each shape does it take to make a rhombicosidodecahedron?

Click for template.

My math club students had fun with a Polyhedra Construction Kit. Here’s how to make your own:

1. Collect a bunch of empty cereal boxes. Cut the boxes open to make big sheets of cardboard.
2. Print out the template page (→) and laminate. Cut out each polygon shape, being sure to include the tabs on the sides.
3. Turn your cardboard brown-side-up and trace around the templates, making several copies of each polygon. I recommend 20 each of the pentagon and hexagon, 40 each of the triangle and square.
4. Draw the dark outline of each polygon with a ballpoint pen, pressing hard to score the cardboard so the tabs will bend easily.
5. Cut out the shapes, being careful around the tabs.
6. Use small rubber bands to connect the tabs. Each rubber band will hold two tabs together, forming one edge of a polyhedron.

So, for instance, it takes six squares and twelve rubber bands to make a cube. How many different polyhedra (plural of polyhedron) will you make?

• Can you build a rhombicosidodecahedron?

And now, on to the main attraction: the 62 blog posts. Many of the following articles were submitted by their authors; others were drawn from the immense backlog in my blog reader. If you’d like to skip directly to your area of interest, here’s a quick Table of Contents:

# Logic Puzzle: Imbalance Problems

Kitten and I have been slogging through the decimals chapter in AoPS Pre-Algebra. She hates arithmetic, so I tried skipping ahead to the algebra puzzle in the exercises, but she refused to be taken in: a decimal problem with an x in it is still a decimal problem.

So I let her off early and pointed her toward these logical “algebra” puzzles instead:

Puzzle by Paul Salomon

By Denise Gaskins Posted in Puzzles

# Math That Is Beautiful

One of the sections in my book encourages parents to make beautiful math with their children. If you have trouble imagining that math can be beautiful, check out this video:

How many mathematical objects could you identify? Cristóbal Vila describes them all on his page Inspirations from Maths.

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# 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

# 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!

# Sample The Moscow Puzzles

Dover Publications is offering a free sample chapter from The Moscow Puzzles.

Cat and Mice
Purrer has decided to take a nap. He dreams he is encircle by 13 mice: 12 gray and 1 white. He hears his owner saying: “Purrer, you are to eat each thirteenth mouse, keeping the same direction. The last mouse you eat must be the white one.”

## More Free Math from Dover Publications

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# Skit: Knights and Knaves Logic Puzzles

photo by puuikibeach via flickr

Our homeschool co-op held an end-of-semester assembly. Each class was supposed to demonstrate something they had learned. I planned to set up a static display showing some of our projects, like the fractal pop-up card and the game of Nim, but the students voted to do a skit based on the logic puzzles of Raymond Smullyan.

We had a small class (only four students), but you can easily divide up the lines make room for more players. We created signs from half-sheets of poster board with each native’s line on front and whether she was a knight or knave on the flip side. In the course of a skit, there isn’t enough time to really think through the puzzles, so the audience had to vote based on first impressions — which gave us a fair showing of all opinions on each puzzle.

By Denise Gaskins Posted in Puzzles

# Raymond Smullyan Excerpts at Dover Publications

To celebrate their re-release of his classic puzzle books, the Dover Math and Science Newsletter featured an interview with Raymond Smullyan, as well as several extended excerpts from his books. (For my math club students: Professor Smullyan invented the Knights and Knaves puzzles.) Enjoy!

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# 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.

# 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

# Giveaway: Hexa-Trex Puzzle Book

Bogusia Gierus, host of this month’s Math Teachers at Play blog carnival, is offering to give away her First Book of Hexa-Trex Puzzles for just the cost of shipping. How generous!

My math club had fun with several of these puzzles a few years ago, and the “Easy” ones (like the sample shown here) were just right for my 4th-5th grade students. One girl enjoyed them enough that she took home extra copies to share with her father.

It’s a thin book, just the right size for a stocking-stuffer. To see the full range of difficulty levels, look over the puzzles on Bogusia’s Daily Hexa-Trex page. To get your own copy of the book, read the giveaway instructions on Bogusia’s blog.

Object of the Puzzle

The object of the puzzle is to find the equation pathway that leads through ALL the tiles.

Forming Equations

• Two or three (or four or five etc.) digit numbers are made up of the individual tiles in the particular order as the equation is read. For example 5 x 5 = 2 5 is correct, but read backwards 5 2 = 5 x 5 is incorrect.
• The equation must be continuous (no jumping over tiles or empty spaces).
• Each tile can be used ONLY ONCE.
• Order of operations is followed. Multiplication and division comes before addition and subtraction.
• The tile “-” can be used as both a subtraction operation or a negative sign in front of a digit, making it a negative number.

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# Math Carnival Update, and an Algebra Puzzle

Oops! I misread my calendar last week. The Math Teachers at Play blog carnival will be this Friday at Maths Insider. That means you still have today and tomorrow to send in your blog post submissions using the handy submission form. See you at the carnival!

In the meantime, let me share with you this monster algebra puzzle from the Well-Trained Mind forum. Simplify:

$[ \left ( {x}^{\frac {3}{2x}} \right )^{\frac{x}{9}} \times \left ( x^{\frac{9}{15}} \right )^{\frac{5}{18}}]^3$

How would you explain this problem to a beginning algebra student who has just learned the exponent rules? Or to his non-mathy mom?

## And Don’t Miss…

These other mathy blog carnivals:

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# Free Math from Dover Publications

I love Dover books, don’t you? They publish so-o-o-o-o many interesting titles at reasonable prices. I always have several Dover books on my wishlist, waiting for my next free gift card from Swagbucks.

But you don’t have to wait to enjoy free math from Dover books. Sign up for the Dover Sampler, and each week they will send an email with links to content from all sorts of books. Or try the Dover Children’s Sampler and Dover Teacher’s Sampler for coloring books, mazes, literature, and more. All the Dover samplers are completely free, and you can cancel at any time.

## From Last Week’s Sampler

Last week’s email included a section on “Exploring Mathematics”:

And that’s only the beginning. Below, I’ve listed a wide variety of math-related links collected from past samplers (though be warned: Dover does change its page links from time to time). Download, print, enjoy!

# 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…

# 2011 Mathematics Game

[Photo from Wikipedia.]

Two of the most popular New Year’s Resolutions are to spend more time with family and friends, and to get more exercise. The 2011 Mathematics Game is a chance to do both at once.

So grab a partner, slip into your workout clothes, and pump up those mental muscles!

## Here are the rules:

Use the digits in the year 2011 to write mathematical expressions for the counting numbers 1 through 100.

• All four digits must be used in each expression. You may not use any other numbers except 2, 0, 1, and 1.
• You may use the arithmetic operations +, -, x, ÷, sqrt (square root), ^ (raise to a power), and ! (factorial). You may also use parentheses, brackets, or other grouping symbols.
• You may use a decimal point to create numbers such as .1, .02, etc.
• Multi-digit numbers such as 20 or 102 may be used, but preference is given to solutions that avoid them.

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

You may use multifactorials:

• (n!)! = a factorial of a factorial, which is not the same as a multifactorial
• 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: The bonus rules are not part of the Math Forum guidelines. They make a significant difference in the number of possible solutions, however, and they should not be too difficult for high school students or advanced middle schoolers.]

By Denise Gaskins Posted in Puzzles

# A Football Puzzle

[Photo by rdesai.]

The MIT Mathmen got the ball on their own 20-yard line for the last drive of the game. They were down by 2 points, so they needed at least a field goal to win the game.

If quarterback Zeno and his offense advanced the ball halfway to the opposing team’s end zone on each play…

# Logic Games at Blogging 2 Learn

Image via Wikipedia

For the rest of NaBloPoMo (National Blog Posting Month), my other blog is featuring a logic game or puzzle every day. So far, I’ve shared three of my online favorites:

And there’s plenty more fun to come. Drop in every day until December to see a new puzzle or game:

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

# A Couple of Chess Puzzles

Image via Wikipedia

Chess is a favorite game for recreational mathematicians — not to play it, but to play around with it. Many puzzles and challenges are based on the moves of chess pieces.

Stretch your brain with these puzzles:

• Can you go on a Knight’s Tour? Start your knight on any square, and try to hop around to all the rest.
• Or, how many queens can you place on the board so that no queen can capture another?

# Lewis Carroll’s Logic Challenges

Image via Wikipedia

Symbolic Logic Part I was published in 1896. When Lewis Carroll (Charles Lutwidge Dodgson) died two years later, Part II was lost. Because they couldn’t find the manuscript, many people doubted that he ever wrote Part II. But almost eighty years after his death, portions of Part II were recovered and finally published. The following puzzles are from the combined volume, Lewis Carroll’s Symbolic Logic, edited by William Warren Bartley, III.

These puzzles are called soriteses or polysyllogisms. Carroll began with a series of “if this, then that” statements. He rewrote them to make them more confusing, and then he mixed up the order to create a challenging puzzle.

Given each set of premises, what conclusion can you reach?

# Cousin Sam’s 15 Challenge

Uncle Will drove in from the tree farm to drop off Alex’s cousin, Sam, so he could go to the Homeschool Math Carnival.

“Hey, Sam,” Alex said. “What’s in the sack?”

Sam smiled. “A secret puzzle.”

“Aw, c’mon,” Leon whined. “We’ll be busy with our own games at the carnival. Can’t you show us now?”

# Sept-Oct 2010 Math Calendars

As I was preparing for Wednesday’s Homeschool Math Club Games & Activities meeting, I remembered my old math calendars and thought, that would be a fun activity to offer. So I pulled up the files and discovered that the days of the week matched perfectly. What a cool coincidence!

So in case you missed the math calendars last year, or in case it’s been long enough that your children have forgotten, here are the “new” versions:

Umm Ahmad created an easier version for young students:

# How to Start an Argument: The Monty Hall Problem

[Photo by MontyPython.]

You can get a good argument going in almost any group of people with the infamous Monty Hall problem:

Imagine you are on a TV game show, and the host lets you choose between three closed doors. One of the doors hides a fancy sports car, and if you pick that door, you win the car.

You pick door #1.

The host opens door #3 to reveal a goat. Then he gives you a chance to switch your door for the unopened door #2.

Should you switch?

What if you say you’re going to switch, and then the host offers to give you \$5,000 instead of whatever is behind door #2?

Try the game for yourself at the Stay or Switch website.