In the land of Fantasia, where people communicate by crystal ball, Wizard Mathys has been placed in charge of keeping the crystal connections clean and clear. He decides to figure out how many different ways people might talk to each other, assuming there’s no such thing as a crystal conference call.

Mathys sketches a diagram of four Fantasian friends and their crystal balls. At the top, you can see all the possible connections, but no one is talking to anyone else because it’s naptime. Fantasians take their siesta very seriously. That’s one possible state of the 4-crystal system.

On the second line of the diagram, Joe (in the middle) wakes up from siesta and calls each of his friends in turn. Then the friends take turns calling each other, bringing the total number of possible connection-states up to seven.

Finally, Wizard Mathys imagines what would happen if one friend calls Joe at the same time as the other two are talking to each other. That’s the last line of the diagram: three more possible states. Therefore, the total number of conceivable communication configurations for a 4-crystal system is 10.

For some reason Mathys can’t figure out, mathematicians call the numbers that describe the connection pattern states in his crystal ball communication system *Telephone numbers*.

**Can you help Wizard Mathys figure out the Telephone numbers for different numbers of people?**

T(0) = ?

T(1) = ?

T(2) = ?

T(3) = ?

T(4) = 10 connection patterns (as above)

T(5) = ?

T(6) = ?

and so on.

**Hint: **Don’t forget to count the state of the system when no one is on the ~~phone~~ crystal ball.

- Printable version: Crystal Ball Connection Patterns.

Feature photo at top of post by Christian Schnettelker (web designer) and wizard photo by Sean McGrath via Flickr. (CC BY 2.0) This puzzle was originally featured in the *Math Teachers At Play* (MTaP) math education blog carnival: MTaP #76.

Get monthly math tips and activity ideas, and be the first to hear about new books, revisions, and sales or other promotions. Sign up for my Tabletop Academy Press Updates email list.

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The monthly *Math Teachers at Play* (MTaP) math education blog carnival is almost here. If you’ve written a blog post about math, we’d love to have you join us! Each of us can help others learn, so in a sense we are all teachers.

Posts must be relevant to students or teachers of school-level mathematics (that is, anything from preschool up to first-year calculus). Old posts are welcome, as long as they haven’t been published in past editions of this carnival.

Have you noticed a new math blogger on your block that you’d like to introduce to the rest of us? Feel free to submit another blogger’s post in addition to your own. Beginning bloggers are often shy about sharing, but like all of us, they love finding new readers.

**Don’t procrastinate:** *The deadline for entries is this Friday, May 22.* The carnival will be posted next week at ZenoMath.

Thank you so much to the volunteer bloggers who have stepped up to carry this MTaP math education blog carnival through the summer! I would never be able to keep the carnival going on my own.

If you’d like to join in the fun, we have openings for the 2015-2016 school year. Read the instructions on our Math Teachers at Play page. Then leave a comment or email me to let me know which month you’d like to take.

While you’re waiting for next week’s *Math Teachers at Play* carnival, you may enjoy:

- Browse past editions of the
*Math Teachers at Play*blog carnival - Carnival of Mathematics
- Carnaval de Matemáticas

Get monthly math tips and activity ideas, and be the first to hear about new books, revisions, and sales or other promotions. Sign up for my Tabletop Academy Press Updates email list.

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Teresa Gaskins’s new ebook *Hunted: The Riddled Stone ~ Book Two* is available now at Amazon. The paperback should follow within the next couple days, and the other online retailers will come along whenever their automated systems get caught up.

You can download the first five chapters here:

To celebrate the release of *Hunted,* the ebook version of *Banished*—the first book in the Riddled Stone series—will be on sale for 99 cents for the next few weeks.

Get monthly math tips and activity ideas, and be the first to hear about new books, revisions, and sales or other promotions. Sign up for my Tabletop Academy Press Updates email list.

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**Math Concepts:** sorting by attribute (card suits), counting up, counting down, standard rank of playing cards (aces low).

**Players:** two or more, best with four to six.

**Equipment:** one complete deck of cards (including face cards), or a double deck for more than six players. Provide a card holder for young children.

Deal out all the cards, even if some players get more than others. The player to the dealer’s left begins by playing a seven of any suit. If that player does not have a seven, then the play passes left to the first player who does.

After that, on your turn you may lay down another seven or play on the cards that are already down. If you cannot play, say, “Pass.”

Once a seven is played in any suit, the six and the eight of that suit may be played on either side of it, forming the fan. Then the five through ace can go on the six in counting-down order, and the nine through king can go on the eight, counting up. You can arrange these cards to overlap each other so the cards below are visible, or you can square up the stacks so only the top card is seen.

Players do not need to wait for both the six and eight of a suit to be played before they begin building the fan up or down.

The first player to run out of cards wins the game.

If you want to keep score, count the cards remaining in your hand after one player goes out. After everyone has had a turn as dealer, whoever has the lowest total score is the champion.

In some traditions, play always begins with the seven of diamonds, so whoever has that card goes first.

The player to the dealer’s left may lead any card, and then all the suits must start with that number (instead of with seven) and build up and down from there.

When the dealer gets to the end of the deck and there aren’t enough cards to give every player one more, the last few cards are turned face up and may be played by anyone as needed. The suit of the last card becomes the trump suit, and cards of that suit may be played on any of the fans, with the card they replace going on the trumps fan. In this case, the cards must be laid out in overlapping rows, not stacked up, so everyone can see where the trumps have gone.

For instance, if spades are trump, then a nine of spades could be played on the eight of hearts, which would leave the nine of hearts without a home—so it has to go on the spades fan.

*Exceptions:* The seven of the trump suit starts its own fan, like any other seven, and the last card dealt (the one that named the trump suit) must also be played to the trumps fan when its turn comes.

Deal only seven cards to each player, and set the rest of the deck out as a draw pile. The first player who cannot play must draw one, which he may play if possible. If not, and the next player also cannot play, she must draw two. If neither of those cards will play, and the next player has nothing to play, he must draw three, and so on, each player drawing one more card than the last person. When one of the players is finally able to lay down a card, this resets the draw count back to zero.

In Crazy Tan, players are allowed to lay down a *run *(playing several cards in a row of the same suit on a single turn). Or they may play *parallel cards* (cards of the same rank in different suits, all played in the same turn). Or a player may even lay down parallel runs, if the cards happen to work out that way.

Fan Tan may also be called Crazy Sevens. Like any folk game, it is played by a variety of rules around the world. If you search for it on the Internet, you may run into an unrelated Chinese gambling game called Fan Tan, which is similar to Roulette.

This post is an excerpt from my book *Counting & Number Bonds: Math Games for Early Learners*, available now in bookstores all over the Internet.

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And remember: Newsletter subscribers are always the first to hear about new books, revisions, and sales or other promotions.

Math Snack: Two Math Games

Playful, no-preparation math activities for all agesMath games pump up mental muscle, reduce the fear of failure, and develop a positive attitude toward mathematics. Through playful interaction, games strengthen a child’s intuitive understanding of numbers and build problem-solving strategies. Mastering a math game can be hard work, but kids do it willingly because it is fun.

The easiest no-prep strategy game for all ages is the finger game Chopsticks. You don’t need any equipment, so it’s a great game to keep in mind for when you and your kids are stuck in a line or waiting room. Math concepts: counting up to five, thinking ahead.

And one of my all-time favorite games is…

Feature photo at top of post by marcello (CC BY 2.0) via Flickr.

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**Math Concepts:** multiples, factors, composite numbers, and primes.

**Players:** only two.

**Equipment:** pair of 6-sided dice, 10 squares each of two different colors construction paper, and the game board (click the image to print it, or copy by hand).

On your turn, roll the dice and make a 2-digit number. Use one of your colored squares to mark a position on the game board. You can only mark one square per turn.

- If your 2-digit number is prime, cover a PRIME square.
- If any of the numbers showing are factors of your 2-digit number, cover one of them.
- BUT if there’s no square available that matches your number, you lose your turn.

The first player to get three squares in a row (horizontal, vertical, or diagonal) wins. Or for a harder challenge, try for four in a row.

Feature photo at top of post by Jimmie via flickr (CC BY 2.0). This game was featured in the *Math Teachers At Play* (MTaP) math education blog carnival: MTaP #79. Hat tip: Jimmie Lanley.

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Do you enjoy math? I hope so! If not, browsing the articles linked in this post just may change your mind.

Welcome to the 85th edition of the ** Math Teachers At Play** math education blog carnival—a smorgasbord of links to bloggers all around the internet who have great ideas for learning, teaching, and playing around with math from preschool to pre-college.

By tradition, we start the carnival with a short puzzle or activity. But if you would like to jump straight to our featured blog posts, click here to see the Table of Contents.

Let the mathematical fun begin!

In honor of our 85th edition, I present: the *centered triangular numbers.*

You can build centered triangles with stones in a sandbox, or with any small manipulative that won’t roll away. Like all figurate numbers, the centered triangles start with the number one: a single stone. Imagine this as a triangle with a stone at each corner and sides of length zero.

Around this, you build the next triangle, which has 2 stones on each side. The sides are one unit long. Four stones in all, so the second centered triangular number is 4.

Then build a 3-stones-per-side triangle centered around that. Each new side is 2 units long, and we’ve used a total of 10 stones so far. The third centered triangular number is ten.

Keep building triangles centered around each other, each with one more stone per side. Each triangle’s sides are one unit longer than the sides of the triangle just inside it.

- 85 is a centered triangular number. How many triangles will you need to use up 85 stones? (Don’t forget to count the first stone as a triangle.)
- Can you find a pattern in the numbers?
- What other centered polygon shapes can you build?
- High school students: Can you find an equation to fit the pattern?

And now, on to the main attraction: the blog posts. Many articles were submitted by their authors; others were drawn from the immense backlog in my rss reader. If you’d like to skip directly to your area of interest, click one of these links.

- Early Learning Activities
- Elementary Exploration and Middle School Mastery
- Adventures in Basic Algebra and Geometry
- Advanced Mathematical Endeavors
- Puzzling Recreations
- Teaching Tips

And since I’ve been in a bookish mood lately, each section includes a link to one of my favorite under-appreciated (5 reviews or fewer) math book. The covers link to Amazon.com, where I get a few cent’s commission if you actually buy something—but you should be able to borrow all these books through your local library or library loan system.

**Click to tweet:** Math Teachers at Play #85: a smorgasbord of great ideas for learning, teaching, and playing around with math.

*Math from Three to Seven: The Story of a Mathematical Circle for Preschoolers* by Alexander Zvonkin

This book is a captivating account of a professional mathematician’s experiences conducting a math circle for preschoolers in his apartment in Moscow in the 1980s—what he tried, what worked, what failed, but most important, what the kids experienced.

- Phil Rowlands (@Help_Your_Child) explains that play is fundamental to ensuring young children learn effectively and discusses the Cuisenaire rod maniulatives.

- Thomas Hobson (@TheTeacherTom) slows down to model “safe and proper woodworking procedures” while the children keep track—debating, frequently recounting, always rearranging, stacking, building, making patterns.

- Tracy Zager (@TracyZager) creates an opportunity for kindergarten kids to play with body-scale number lines, records what happens, and wonders how to make it better. Lots of great comments here.

- Julie (@jmommymom) finds patterns and shapes in a circular grid: Creative Math Art for Kindergarten.

- Pam Odd (@pameladonnis) introduces her children to one of her childhood favorite craft activities in Reinforcing Math Concepts with Picture Pie.

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*More Math Games & Activities from Around the World* by Claudia Zaslavsky

Math, history, art, and world cultures come together in this delightful book for kids, even for those who find traditional math lessons boring. More than 70 games, puzzles, and projects encourage kids to hone their math skills as they calculate, measure, and solve problems.

- Joe Schwartz (@JSchwartz10a) and students puzzle over how to count from 6 to 8.

- Joshua Greene (@JoshuaGreene19) and son find math in the trash, with a nudge or two from Grandma.

- Kristin Gray (@MathMinds) seeks understanding after her students uncover something she never learned.

- Julie (@jmommymom) and family read the math fairy tale book
*The Man Who Counted: A Collection of Mathematical Adventures*and try their hand at the Four 4s challenge.

- Spencer Olmsted’s class is on fire with math patterns. “It’s a wonderful thing to connect a physical model, the ordered pairs that describe it, and a graph—it’s practically poetry.”

- And for my own (@letsplaymath)contribution to the carnival, here’s a math-and-strategy card game from British mathematician Henry Dudeney’s classic book,
*The Canterbury Puzzles.*

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Mathematical Cavalcade by Brian Bolt

This collection of puzzles, games and activities is designed to stimulate and challenge people of all ages who enjoy puzzles with a mathematical flavor. The second part of the book contains a commentary giving hints and solutions.

- Stephen Cavadino (@srcav) says, “I love it when my student talk maths well, and this post looks at an interesting discussion my year 9s had on perimeter.”

- Tina Cardone (@crstn85) gets a seasonal reminder that extended wait time and letting kids ask us for help rather than continuing the conversation as soon as they have responded really does work.

- Bridget Dunbar (@BridgetDunbar) uses Fawn Nguyen’s Visual Patterns website to help her students understand finding the constant rate of change between two points.

- Bryan Anderson (@Anderson02B) begins a series of linear pattern challenge posts: Linear Patterns 180.

- Mrs. E (@MrsETeachesMath) diagrams the relationships in the Quadrilateral Family Tree. “Just like all families, they have some issues, however, they all get along and are happy together.”

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*A Decade of the Berkeley Math Circle: The American Experience* by Zvezdelina Stankova, Tom Rike

A wide variety of enticing mathematical topics: from inversion in the plane to circle geometry; from combinatorics to Rubik’s cube and abstract algebra… Also features 300 problems, ranging from beginner to intermediate level, with occasional peaks of advanced problems and even some open questions.

- Manan Shah (@shahlock) relates what he really learned in Calculus class, aside from Calculus, in Pigeonholed: What’s Your Major?

- Dan MacKinnon (@mathrecreation) uses The Geometer’s Sketchpad to construct patterns through iteration. Working through how to build iterations helps teach basic principles of geometric construction as well as more advanced ideas (self similarity, limits).

- Bob Lochel (@bobloch) develops a new perspective on imaginary numbers. “The bulbs have gone off. I GET this now! What I appreciate most here is that we don’t need to wait until deep into algebra 2 to think about the imaginary unit.”

- Michael Fenton (@mjfenton) offers a Polar Graphing Sorting Activity to help students establish some connections between polar and Cartesian forms.

- Have you ever wondered where Euler’s Formula comes from? See how arithmetic, geometry, trigonometry, and calculus dance together on the complex plane to create mathematical beauty.

- And don’t miss the Carnival of Mathematics #121.

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Puzzles and paradoxes from Lewis Carroll, the author of Alice in Wonderland, whose interests ranged from inventing new games like Arithmetical Croquet to important problems in symbolic logic and propositional calculus. Written by Carroll expert and well-known mathematics author Martin Gardner.

- Joel David Hamkins shares a new booklet of Math for eight-year-olds: graph theory for kids. Download, print, explore, and enjoy!

- Sam Blanco (@SamBlancoBCBA) reviews a strewable puzzle resources for upper-elementary and middle school kids (and adults!): Visual Brainstorms.

- Samantha Oestreicher (@SamanthaOestrei) applies the measuring rod of mathematics to the world of books and their movies: An Unexpectedly Long Journey.

- Mike Lawler (@mikeandallie) and sons investigate how scaling affects area and perimeter in a variation of The “rope around the Earth” problem.

- Mario Livio (@Mario_Livio) leads NOVA viewers on a mathematical mystery tour—an exploration of math’s astonishing power across the centuries. Is math a human invention or the discovery of the language of the universe?

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*Common Core Math For Parents For Dummies* by Christopher Danielson

Many new teaching methods are very different from the way most parents learned math, leading to frustration and confusion as parents find themselves unable to help with homework or explain difficult concepts. This book cuts the confusion and shows you everything you need to know to help your child succeed in math.

- Stephen Cavadino (@srcav) points out how a new textbook nearly ruins a nice puzzle: A missed opportunity?

- Sue VanHattum (@suevanhattum) reports the first sighting of her book,
*Playing with Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers*. Have you ordered your copy yet?

- Jennifer Smith (@4mulaFun) discusses several techniques you can use to help children review and understand what they write in their notebooks.

- Several authors at Heinemann Publishing (@HeinemannPub) unpack, examine, and reflect on the Standards for Mathematical Practice and how they can help students grow into confident, proficient mathematicians.

- Ben Orlin (@benorlin) ponders The Math Ceiling: Where’s your cognitive breaking point? and sets off a storm of thought-provoking comments.

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And that rounds up this edition of the ** Math Teachers at Play** carnival. I hope you enjoyed the ride.

The next installment of our carnival will open sometime during the week of May 25-29 at ZenoMath. If you would like to contribute, please use this handy submission form. Posts must be relevant to students or teachers of preK-12 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 information page.

We need volunteers for the fall semester. 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!

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Are you tired of flashcards and repetitive worksheets? Now your children can practice their math skills by playing games.

Math games pump up mental muscle, reduce the fear of failure, and develop a positive attitude toward mathematics. Through playful interaction, games strengthen a child’s intuitive understanding of numbers and build problem-solving strategies. Mastering a math game can be hard work, but kids do it willingly because it is fun.

*Counting & Number Bonds* features 21 kid-tested games, offering a variety of challenges for preschool and early-elementary learners. Young children can play with counting and number recognition while they learn the basic principle of good sportsmanship, to respond gracefully whether they win or lose. Older students will explore place value, build number sense, and begin practicing the math facts.

Buy now at:

- Amazon.com
- Amazon.uk
- Amazon.ca
- Amazon.au
- and all the other Amazons worldwide

*Addition & Subtraction* features 22 kid-tested games, offering a variety of challenges for elementary-age students. Children will strengthen their understanding of numbers and develop mental flexibility by playing with addition and subtraction, from the basic number facts to numbers in the hundreds and beyond. Logic games build strategic thinking skills, and dice games give students hands-on experience with probability.

Buy now at:

- Amazon.com
- Amazon.uk
- Amazon.ca
- Amazon.au
- and all the other Amazons worldwide

You don’t need a Kindle device to read Amazon ebooks. Click here to download the Kindle program for your computer, phone, or tablet.

For those of you who prefer to buy ebooks from iTunes, Barnes & Noble, Kobo, etc.—those versions are coming soon! The epub book format takes a bit more work, but I’m hoping for time to finish it up within a week or so.

Paperback editions are also in the works.

Featured photo above by Richard Riley via Flickr (CC BY 2.0).

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Lay out the ace to six of each suit in a row, face up and not overlapping, one suit above another. You will have one column of four aces, a column of four twos, and so on—six columns in all.

The first player flips a card upside down and says its number value. Players alternate, each time turning down one card, mentally adding its value to the running total, and saying the new sum out loud. The player who exactly reaches thirty-one, or who forces the next player to go over that sum, wins the game.

For a shorter game, use only the ace to four of each suit. Play to a target sum of twenty-two.

Thirty-One comes from British mathematician Henry Dudeney’s classic book, *The Canterbury Puzzles*. According to Dudeney, “This is a game that used to be (and may be to this day, for aught I know) a favourite means of swindling employed by card-sharpers at racecourses and in railway carriages.”

Dudeney challenges his readers to find a rule by which a player can always win: “Now, the question is, in order to win, should you turn down the first card, or courteously request your opponent to do so? And how should you conduct your play?”

Dudeney, H. E. *The Canterbury Puzzles,* Thomas Nelson and Sons, 1919 (originally published 1907); available at Project Gutenberg or the Internet Archive.

http://www.gutenberg.org/ebooks/27635

https://archive.org/details/canterburypuzzle00dudeuoft

This post is an excerpt from my book *Addition & Subtraction: Math Games for Elementary Students*, available now in bookstores all over the Internet.

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**Math Concepts:** counting up to five, thinking ahead.

**Players:** two or more.

**Equipment**: none.

Each player starts with both hands as fists, palm down, pointer fingers extended to show one point for each hand. On your turn, use one of your fingers to tap one hand:

- If you tap an opponent’s hand, that person must extend as many extra fingers on that hand (in addition to the points already there) as you have showing on the hand that tapped. Your own fingers don’t change.

- If you force your opponent to extend all the fingers and thumb on one hand, that makes a “dead hand” that must be put behind the player’s back, out of the game.

- If you tap your own hand, you can “split” fingers from one hand to the other. For instance, if you have three points on one hand and only one on the other, you may tap hands to rearrange them, putting out two fingers on each hand. Splits do not have to end up even, but each hand must end up with at least one point (and less than five, of course).

- You may even revive a dead hand if you have enough fingers on your other hand to split. A dead hand has lost all its points, so it starts at zero. When you tap it, you can share out the points from your other hand as you wish.

The last player with a live hand wins the game.

**House Rule:** Do you want a shorter game? Omit the splits. Or you could allow ordinary splits but not splitting fingers to dead hands.

**Nubs:** All splits must share the fingers evenly between the hands. If you have an odd number of points, this will leave you with “half fingers,” shown by curling those fingers down.

**Zombies:** (For advanced players.) If a hand is tapped with more fingers than are needed to put it out of the game, it comes back from the dead with the leftover points. For instance, if you have four fingers out, and your opponent taps you with a two-finger hand, that would fill up your hand with one point left over. Close your fist, and then hold out just the zombie point. In this variation, the only way to kill a hand is to give it exactly five points.

Finger-counting games are common in eastern Asia—and they must be contagious, since my daughters caught them from their Korean friends at college. Middle school teacher Nico Rowinsky shared Chopsticks (which is simpler than the version my daughters brought home) in a comment on the “Tiny Math Games” post at Dan Meyer’s blog.

This post is an excerpt from my book *Counting & Number Bonds: Math Games for Early Learners*, available now in bookstores all over the Internet.

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The April “Let’s Play Math” newsletter went out Monday morning to everyone who signed up for Tabletop Academy Press math updates. If you’re not on the mailing list, you can still join in the fun:

And remember: Newsletter subscribers are always the first to hear about new books, revisions, and sales or other promotions.

Math Snack: Math Treks

Playful, no-preparation math activities for all agesCreated by Maria Droujkova, a Math Trek is a “virtual reality” game, played at the intersection between the real world and your imagination. Participants explore their towns and communities, start noticing mathematics everywhere, and grow their math eyes.

Math Treks are like scavenger hunts for math. Gather a group of friends, choose a topic, and go for a walk to see how much math you can find. Take pictures to share and compare. My math club families have enjoyed a Multiplication Trek (looking for things in groups and arrays) around our local library and a Symmetry Trek through the woods.

If you have time for a little preparation, Maria posted several Math Trek game sheets you can download and print. …

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Six years ago, my homeschool co-op classes had fun creating this April calendar to hand out at our end-of-semester party. Looking at my regular calendar today, I noticed that April this year starts on Wednesday, just like it did back then. I wonder when’s the next time that will happen?

A math calendar is not as easy to read as a traditional calendar — it is more like a puzzle. The expression in each square simplifies to that day’s date, so your family can treat each day like a mini-review quiz: “Do you remember how to calculate this?”

The calendar my students made is appropriate for middle school and beyond, but you can make a math calendar with puzzles for any age or skill level. Better yet, encourage the kids to make puzzles of their own.

**At home:**

Post the calendar on your refrigerator. Use each math puzzle as a daily review “mini-quiz” for your children (or yourself).

**In the classroom:**

Post today’s calculation on the board as a warm-up puzzle. Encourage your students to make up “Today is…” puzzles of their own.

**As a puzzle:**

Cut the calendar squares apart, then challenge your students to arrange them in ascending (or descending) order.

If you like, you may use the following worksheet:

Submission details here: Kids’ Project — More Math Calendars?

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**But Before You Go…**

I’m running out of carnival hosts! Would you like to volunteer? It’s a bit of work, but great fun, too. Leave a comment here, or send me an email.

**Excerpt:**

Welcome to the 84th Math Teachers at Play Blog Carnival!

84 is a portentous number. It’s the sum of twin primes (What’s the previous sum of twin primes? Next?). It’s thrice perfect, twice everything. It’s positively Orwellian. It’s even a town in Pennsylvania.

84 puzzler 1:

Number the intersections of these five circles with the integers 1 to 20 so that the points on each circle sum to the same.It was a good month for math reading related posts …

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- the Happy Birthday, Einstein! video series
- Happy Birthday, Einstein (Part 2)
- Happy Birthday, Einstein (Part 3)
- Happy Birthday, Einstein (Part 4)
- Albert Einstein’s math biography
- Math-related quotes from Albert Einstein

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