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, other 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.

Living Books for Math

A child’s intercourse must always be with good books, the best that we can find… We must put into their hands the sources which we must needs use for ourselves, the best books of the best writers.
…
For the mind is capable of dealing with only one kind of food; it lives, grows and is nourished upon ideas only; mere information is to it as a meal of sawdust to the body.

— Charlotte Mason
Toward A Philosophy of Education

Princess Kitten and I took a longer than usual holiday break from homeschooling, but now I’m in plan-for-the-new-semester mode. I hope to include more living math in our schedule, so I decided to illustrate this edition of the MTaP carnival with a few of my favorite living math books. I’d love to hear more living book suggestions in the comments!

If you click on a book cover, the links take you to Amazon.com, where you can read reviews and other details (and where I earn a few cents through Skimlinks if you actually buy the book), but all of these books should be available through your public library or via inter-library loan.

Let the mathematical fun begin…

TRY THIS PUZZLE

image via WolframMathWorld

By tradition, we start the carnival with a puzzle in honor of our 46th edition. 46 is a centered triangular number.

• Study the diagram to see how centered triangular numbers are constructed. Every centered triangular number is equal to 1(mod 3). Can you see why?

• We start with n=0 for the single dot in the middle, n=1 includes the center + the smallest triangle, etc. Counting in this way, what value of n will have 46 dots?

• Can you find all the centered triangular numbers less than 100?

• Every centered triangular number 10 or greater contains three consecutive regular triangular numbers. Draw a few big centered triangles (n>3). Can you outline the three triangular numbers in each sketch?

ELEMENTARY CONCEPTS

12 Ways to Get to 11 is more than just a fun counting book. Introduce young children to number bonds and partitions — new ways to look at the world of numbers!

• Beanienus faces the challenge of teaching numbers to an active daughter in Rocket, soccer, musical chairs and Math.

• Kristen and her girls have fun with math in For the Love of the Rods. (And they reviewed 12 Ways to Get to 11, too.)

• Heather shows how to Make Your Own Math Games: Egg-O.

• MathWire offers a bit of seasonal fun in Snowman Probability and the Last Snowman Standing Game. Meanwhile, Haley creates beautiful, wintry symmetry in Paper Snowflakes 101.

• Jennifer and her son explore transformations in Cookie Cutter Geometry.

• While Mama makes dinner, Malke’s daughter creates Marshmallow Math: Solids & Sculpture. (And the story continues here, here, here, and here.)

ARITHMETIC

The Man Who Counted features classic mathematical puzzles in an exotic setting, following the fictional adventures of Persian sheep-herder turned mathematician Beremiz as he uses his wits to gain fame, fortune, and a beautiful wife.

• IMACS presents a clock-based game in An Introduction to Modular Addition and adds a couple new rules in More Modular Arithmetic.

• Gaurav asks, “Do you multiply this way?” My challenge: Don’t memorize this method as a trick, but study until you see why it works. How is it similar to the standard method?

• Phyllis demonstrates a variety of hands-on ways to teach Beginning Division.

• Ashley shares 5 Great Sites for Learning Math, and Jen shares her smart phone discoveries in Math Apps — Carnival of fun.

• Jen’s students add a “Types of Graphs” foldable to their math journals. Be sure to check out her other math journal ideas, too.

• Maria teaches all the basic fraction concepts in Free fraction videos online, part 1 and Part 2.

• Rebecca offers as series of counting challenges in her lesson plan, Combinatorics for Breakfast. What, no Spam?!

• David demonstrates a useful system for Combining Integers Instead of Adding and Subtracting Them.

BASIC ALGEBRA & GEOMETRY

Flatland: A Romance of Many Dimensions, a classic of mathematical fiction, follows the adventures of A. Square as he transcends his assumptions about reality and explores several geometric worlds.

• Mimi’s class has been Learning from Our (Algebra) Mistakes.

• Mrs. H. put together a Linear Function Review Booklet for her algebra 1 students.

• Kerry applies algebra to a variety of problems in her Mathspig Zombie Maths Series.

• Mr. Koh challenges us with an algebra puzzle about the Properties of Numbers.

• Ryan’s class uses basic geometry to solve the mystery of The Chopped Cubes.

• Guillermo shows how coordinate geometry can help simplify proofs in Diagonals of a Parallelogram. Check also his short Proof that log 2 is an irrational number.

ADVANCED MATH

The Language of Mathematics: Making the Invisible Visible (or the earlier version, Mathematics: The Science of Patterns) helps answer the question, “Why do we have to learn this?” If you’ve ever wondered what mathematicians mean when they say math is “beautiful” — read this book!

• Murray plays around with Biorhythm Graphs: “A bit of fun — a non-scientific application of composite sine curves!”

• Mr. Chase argues with his algebra 2 book in Inverse functions and the horizontal line test and then has a bit of fun in Function World.

• John shares a couple of problems from his preservice high school teachers’ final. How would you fare?

• Riley gives his students the run-around in What did you do to the x-axis?!? Using the most relevant context possible.

• Alexander explains the Star of David Theorems in Pascal Triangle.

• Robert opines that everyone should learn programming, and he offers several suggestions for where to start.

• Sue reviews Math Girls: A Novel Way to Learn Some Deep Math: “I would highly recommend this book for anyone at the level of pre-calculus and above who enjoys math.”

MATHEMATICAL PUZZLES

What Is the Name of This Book?: The Riddle of Dracula and Other Logical Puzzles is finally back in print. (And Alice in Puzzle-Land, too. Thank you, Dover Publications!) Try your hand at a few traditional brain teasers, and explore the Island of Knights and Knaves. Can you tell the difference between a sane human and an insane vampire?

• Gary shares a few puzzles that elementary-age children can understand but adults can enjoy exploring as well: Numberplay: Tanton Wordless. [Errata: Moving your mouse over the second picture will display solutions for four of the six puzzles shown, implying that two are impossible --- but really, only one is impossible. Can you find the missing solution?]

• MathCounts is a great source of problems for middle school and older students. Mrs. Lin offers Mathcounts competition preparation strategies I, along with a blogful of questions and solutions.

• My entry for the carnival is the 2012 Mathematics Game, a terrific puzzle for middle school and beyond.

• Fawn offers a series of challenges in Puzzles and Brainteasers, Set 3, Set 4, and Set 5.

• Can you solve Pat’s Interesting Rectangle Partitioning?

• John presents A Renaissance math puzzle. Intrigued, Thony digs into history in The story of a problem.

ABOUT TEACHING MATH

Martin Gardner inspired many of us to become mathematicians and math teachers. Now our children can enjoy 50 of his best “Mathematical Games” columns in one volume, The Colossal Book of Mathematics: Classic Puzzles, Paradoxes, and Problems. What a treasure!

• Bon hosts an interesting series of interviews with Mathematician Parents.

• Gisele offers several ideas for making Math Connections with other subjects.

• Erlina asks What are the uses of examples in teaching mathematics?, and answers the question with an example of well-chosen examples.

• Christopher argues that being a good teacher is not about problem-solving skills, It’s about understanding.

• Math Rules wonders, “How much of a class should be conceptual and how much should be procedural?“, and Kate shares her favorite Math Lesson Formula.

MORE MATH CARNIVALS

I love reading math biographies. Fermat’s Enigma: The Epic Quest to Solve the World’s Greatest Mathematical Problem is the “biography” of a math problem. It’s the next book on my library-loan list, and I can’t wait for it to arrive…

• Mathematics and Multimedia Blog Carnival 19

• Carnival of Mathematics 84

• Mathblogging.org Weekly Picks

• Carnevale della Matematica
  which is also available as a pdf ebook.

• Carnaval de Matemáticas

FINAL COMMENTS

That rounds out this edition of the Math Teachers at Play carnival. I hope you enjoyed the ride. The next installment of our carnival will open on February 17 at Math Hombre. If you would like to contribute, please use this handy submission form or email John directly. Posts must be relevant to students or teachers of preschool through precollege mathematics. Old posts are welcome, as long as they haven’t been published in past editions of this carnival.

Past posts and future hosts can be found on our blog carnival index page.

We need more volunteers. Classroom teachers, homeschoolers, unschoolers, or anyone who likes to play around with math (even if the only person you “teach” is yourself) — if you would like to take a turn hosting the Math Teachers at Play blog carnival, please speak up!

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

Have more fun on Let’s Play Math! blog:

• Math Teachers at Play #39
• Math Teachers at Play #35
• Math Teachers at Play #24
• Math Teachers at Play #8
• Math Teachers at Play #5

by thekirbster via flickr

Are you a math student or teacher (classroom or homeschool) or an independent learner? Do you blog about teaching or learning mathematics? If you have written anything playful about mathematics, now is the time to share!

The Math Teachers at Play blog carnival is seeking articles about learning, teaching, or just playing around with mathematics from preschool to precollege level (through the first year of calculus).

Everyone is welcome — you don’t have to be a teacher to join in the fun. You can submit your article online, but the blog carnival website has been giving us trouble again. So far, only one post has come through this month. A safer option is to email me directly to make sure I get your submission.

Current Mathy Carnivals

Math Teachers at Play will be posted January 20 here at Let’s Play Math! blog. Meanwhile, you may enjoy the other math carnivals:

• Last month’s Math Teachers at Play
• Carnival of Mathematics
• Mathematics and Multimedia
• Carnevale della Matematica
• Carnaval de Matemáticas

Support The Carnivals

The math carnivals are a great resource for all of us who enjoy reading and learning about mathematics, and especially for math bloggers who appreciate the wider audience the carnivals provide. But blog carnivals do not happen by themselves — the carnival hosts put in several hours of work every month to bring you the riches of the internet.

Here’s how you can help:

• Leave a comment to thank and encourage the host.
• Link to and promote the carnival on your blog or social network.
• Volunteer to host a future edition at your own blog.
• How to Host a Blog Carnival

Get all our new math tips and games:  Subscribe in a reader, or get updates by Email.

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.

• You must use all four digits. You may not use any other numbers.
• 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, .01, etc.
• You may create multi-digit numbers such as 10 or 202, but we prefer 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 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.]

How To Play

With only three distinct digits to work with this year, we will need every trick in the book to create variety in our numbers. Experiment with decimals, double-digit numbers, and factorials of all sorts. Remember that dividing (or using a negative exponent) creates the reciprocal of a fraction, which can flip the denominator up where it may be more helpful.

Use the comments section below to share the numbers you find, but don’t spoil the game by telling us how you made them. You may give relatively cryptic hints, but be warned: Many teachers use this puzzle as a classroom assignment, and there will always be students looking for people to do their homework for them.

• Do not post your solutions. I will delete them.

There is no authoritative answer key for the year game, so we will rely on our collective wisdom to decide when we’re done. We’ve had some lively discussions the last few years. I’m looking forward to this year’s fun!

Keeping Score

As players report their game results below, I will keep a running tally of confirmed results (numbers found by two or more players). Today is Kitten’s birthday, however, so I won’t spend much time at my computer. Also, I’ll be traveling a lot this month, so this tally will lag a few days behind the results posted in the comments.

Percent confirmed = 91%.

Reported but not confirmed =
57, 85.

Numbers we are missing =
77, 84, 86-88, 92-93.

And if you would like to join me in the “extended edition” game…

Made it with Math Forum rules = 72%.

• NOT Math Forum:
  33, 43, 55-57, 66-67, 69, 71, 78-79, 81-83, 89, 91.

Needed multi-digit numbers:
44, 67-68.

Could NOT keep the digits in order:
29, 31, 33, 37, 39, 41, 44, 55, 59, 65, 67, 69, 71, 76, 78, 89, 91, 95.

Math Forum will begin publishing student solutions after February 1, 2012. Remember, you may not submit answers with triple (or higher) factorials or repeating decimals to the Math Forum site.

Clarifying the Do’s and Don’ts

Finally, here are a few rules that players have found confusing in past years.

These things ARE allowed:

• . [See Dr. Math's Why does 0 factorial equal 1?]
• The only digits that you can use to build 2-or-more-digit numerals or decimals are the standard base-10 digits 2, 0, 1, 2.
• Unary negatives count. That is, you may use a “-” sign to create a negative number.
• You may use (n!)!, a nested factorial — a factorial of a factorial. Nested square roots are also allowed.
• The multifactorial = the product of all integers from 1 to n that are equal to n mod k. You may write the double factorial and triple factorial as !! and !!!, respectively, but for higher multifactorials BOTH n and k must be constructed from the year digits.

These things are NOT allowed:

• “0!” is not a digit, so it cannot be used to create a base-10 numeral.
• The decimal point is not an operation that can be applied to other mathematical expressions: “.(0!)” does not make sense.
• You may not use any exponent unless you create it from the digits 2, 0, 1, 2. You may not use a square function, but you may use “^2”. You may not use a cube function, but you may use “^(2+1)”. You may not use a reciprocal function, but you may use “^(-1)”.
• You have to “hit” each number from 1 to 100 exactly, without rounding off or truncating decimals. You may not use the integer, floor, or ceiling functions.

Helpful Links

• 2012 Mathematics Game Worksheet
  For keeping track of which numbers you’ve solved.

• 2012 Mathematics Game Manipulatives
  This may help visual or hands-on thinkers.

• 2012 Mathematics Game Student Submissions
  For elementary through high school students who wish to share their solutions.

For more tips, check out this comment from the 2008 game.

Heiner Marxen has compiled hints and results for past years (and for the related Four 4′s puzzle). Dave Rusin describes a related card game, Krypto, which is much like my Target Number game. And Alexander Bogomolny offers a great collection of similar puzzles on his Make An Identity page.

Get all our new math tips and games:  Subscribe in a reader, or get updates by Email.

Have more fun on Let’s Play Math! blog:

• 2011 Mathematics Game
• Puzzles for the New Year
• Head’s Up for e-Day
• Be My (Math) Valentine
• Planning Ahead: St. Patrick’s Day

Board games are a celebration of problem solving, and problem solving is at the heart of a quality mathematics education… The mathematics might be hidden, but I guarantee you that it will be there.

— Gordon Hamilton, MathPickle.com
Commercial Games 8

Games for Ages 8+

Games for Ages 12+

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The Math Teachers at Play Carnival is up at Virtual Math Tutor for your browsing pleasure. Articles range from preschool to high school level in math, and topics include puzzles, worksheets, games, teaching tips, ideas for the math/science lover on your Christmas gift list, and the cutest math monster I’ve seen in ages. Great fun!

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You can find just the song here: http://vihart.com/music/gauss12days.mp3.

Carnival Reminder

Send in your submission for the Math Teachers at Play blog carnival by Wednesday night. The blog carnival website has been a little funky (though several posts have come through), so your best option is to email Roman directly.

While you’re waiting for Friday’s carnival, check out the new Carnival of Mathematics.

Get all our new math tips and games:  Subscribe in a reader, or get updates by Email.

For Primary Students

For Secondary Students

Get all our new math tips and games:  Subscribe in a reader, or get updates by Email.

photo by Mike Licht, NotionsCapital.com

If you have written a blog post about learning or teaching (or just playing around with) mathematics from preschool to pre-college level, please share! We would love to see your post in the next Math Teachers at Play blog carnival, coming to Virtual Math Tutor on Friday, December 16.

After a few months of relatively glitch-free service, the blog carnival website is giving us trouble again. You can submit your article online, if you’re feeling brave, and I will try to make sure everything goes through. But your safer option is to email Roman directly to make sure he gets your submission.

Math Blog Carnivals = Fun!

While you’re waiting, take some time to visit the other math blog carnivals. They offer plenty of creative ideas for playing around with math:

• Carnival of Mathematics #83
• Math and Multimedia Blog Carnival #17
• Last month’s Math Teachers at Play
• Carnevale Della Matematica #43
• Carnaval de Matemáticas 2.8

Get all our new math tips and games:  Subscribe in a reader, or get updates by Email.

One way to visualize this is by asking:

• “Which values of x will make the equation equal to zero — that is, will make the graph cross the x-axis?”

I wish my algebra teacher had explained it like James Tanton does. It makes so much sense!

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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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by fdecomite via flickr

The new Math Teachers at Play blog carnival is up for your browsing pleasure:

• Math Teachers at Play – November Blog Carnival

If you think the carnival seems shorter than usual, you’re right. We had many more submissions, but it appears that the blog carnival website is having trouble again. If your article is missing (as mine is), I’m sorry! Please try again next month.

This is the first time I ever hosted a blog carnival so please bear with me.

While reading the posts submitted to this month’s Math Teachers at Play blog carnival, I was struck by how visualization is very important in teaching math, and just math in general. I was happy to read all the “visualization” posts since my recent interest is exactly in visual representations and how they help in learning, especially learning math.

For instance…

Go read the entire article at Nucleus Learning.

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It began with a humble list of 7 things to do with a hundred chart in one of my out-of-print books about teaching home school math. Over the years I added a few new ideas, and online friends contributed still more, so the list grew to its current length of 26. Recently, thanks to several fans at pinterest, it has become the most popular post on my blog:

• 20+ Things to Do with a Hundred Chart

Now I am working several hours a day revising my old math books, in preparation for publishing new, much-expanded editions. And as I typed in all the new things to do with a hundred chart, I thought of one more to add to the list:

(27) How many numbers are there from 11 to 25? Are you sure? What does it mean to count from one number to another? When you count, do you include the first number, or the last one, or both, or neither? Talk about inclusive and exclusive counting, and then make up counting puzzles for each other.

Share Your Ideas

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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by Bob Jagendorf via flickr

Are you a math student or teacher (classroom or homeschool) or an independent learner? Do you blog about teaching or learning mathematics? We would love to have you submit an article to this week’s Math Teachers at Play blog carnival, which will be posted next week at Nucleus Learning.

• Do you have a story about teaching math to young children?
• Could you share favorite math games, numerical investigations, or contest-preparation tips?
• Can you help readers understand why we never divide by zero, or what is wrong with canceling out the x in the expression ?
• Can you make advanced math topics stick in a student’s mind?
• What math do you play with, just for the fun of it?
• Give us your best insights about how to teach math.

Warning: The deadline is next Wednesday night! So go find your favorite blog post and fill out the handy submission form now.

While You’re Waiting for the Carnival

Check out these recent math blog carnivals:

• Last month’s Math Teachers at Play
• Carnival of Mathematics #83
• Mathematics and Multimedia Carnival #16
• Carnevale della Matematica #42
• Carnaval de Matemáticas edition 2.7

Support The Carnivals

A math carnival is like “the shop-front of the mathematics blogging world” — a place to browse and enjoy all the wide variety of mathematics on the web. Blog carnival hosts put in several hours of work every month to bring you the riches of the internet. The math carnivals are a great resource for all of us who enjoy reading and learning about mathematics, and especially for math bloggers who appreciate the wider audience the carnivals provide.

Blog carnivals do not happen by themselves — there is a lot of work involved. Here’s how you can help:

• Leave a comment to thank and encourage the host.
• Link to and promote the carnival on your blog or social network.
• Volunteer to host a future edition at your own blog.
• How to Host a Blog Carnival

To host Carnival of Mathematics on your blog, email Mike and tell him you’re available. Similarly, you can check on “future hosts” tab for upcoming editions of the Math Teachers at Play carnival, and let me know which month you’d like to host. Or contact Guillermo Bautista to volunteer for Mathematics and Multimedia.

Get all our new math tips and games:  Subscribe in a reader, or get updates by Email.

How Do You Count Infinity?

Tanton continues to play with triangular numbers in the next video, which is probably too abstract for Kitten, but I enjoyed it. The triangular numbers are countably infinite, as are the square numbers. But if you look at it another way…

Get all our new math tips and games:  Subscribe in a reader, or get updates by Email.

Have more fun on Let’s Play Math! blog:

• Puzzle: Figuring Out Figurate Numbers
• Visual Proof of the Pythagorean Theorem
• Euclid’s Geometric Algebra
• Math Club: Counting 101
• Sign Up for the MathCounts Club Program

by kennymatic via flickr

An Apollonian pumpkin patch and Sierpinski candy corn:

• Fall Fractal Fun
• Halloween Fractals

Great ideas for mathy porch decorations:

• How to Carve Fractals and Stars on Pumpkins

My favorite blood-thirsty online game from Murderous Maths:

• Urgum and Grizelda’s Head Hunt

And more:

• Halloween Math Games
• Pumpkin Dice Game (printable)
• Fall Problem Solving
• Pumpkin Coordinate Graphing

Do you know of more mathy fall fun? Share your links in the comments section!

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