Math Teachers at Play #76

76[Feature photo (above) by U.S. Army RDECOM. Photo (right) by Stephan Mosel. (CC BY 2.0)]

On your mark… Get set… Go play some math!

Welcome to the 76th 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 puzzle in honor of our 76th edition. But if you would like to jump straight to our featured blog posts, click here to see the Table of Contents.


K4 matchings

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.

[Wizard photo by Sean McGrath. (CC BY 2.0)]


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

Tweet: Math Teachers at Play #76: a smorgasbord of great ideas for learning, teaching, and playing around with math.

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  • Amy Tanner offers Four Super Simple Counting Games that help your child build number sense, learn to estimate, begin to think about addition and subtraction, and practice counting backward.
    [My favorite perk of hosting the MTaP carnival is discovering yummy new blogs. This one definitely belongs in my rss list.]
  • There may not be any numbers, but there’s a whole lot of math going on in Teacher Tom’s post, A Current Of Curiosity.
  • Joshua Greene tells how a simple pattern led to deep and interesting questions — and it only took “5 minutes in between other play”: Pattern Blocks (mini follow-up).
  • Sarah Dees adapted an activity from the Curious George PBS show in Composing Music with Math Activity for Kids. “Seriously, this was a lot of fun. The boys wrote many compositions, and couldn’t wait to perform them for Dad when he came home from work!”

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  • Margo Gentile suggests practicing the math facts with Picnic Time Multiplication. If I were to modify it, I’d skip saying the equations and add the ABCs back in: “I’m going on a picnic, and I’m going to bring 3 apples, 6 buffalo, 9 candy canes, and…”
  • Stephen Cavadino’s class stumbles on what should have been an easy review problem, and he responds with “Aaargh Ruddy BIDMAS!
  • Bryan Anderson’s class creates a variety of graphs to compare different data sets in Human Histogram.

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  • Fawn Nguyen’s students have fun investigating the relationship between a circle’s diameter and circumference in Friday Bubbles.
  • Sue VanHattum takes a break from book editing to explore Euclidean geometry in How I’m Playing With Math Today. “Geometry is my weakness in math, and I love trying to figure out how to do these constructions.”
  • Don Steward posts a grand collection of geometry puzzles in angle proofs. Each image can be printed landscape-orientation on a regular sheet of paper or added to PowerPoint for sharing with students.

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  • John Golden discusses how help students understand complex numbers in Complex Instruction, with a little help from GeoGebra. “One of the morals of the capstone class was that if mathematicians labeled a theorem as Fundamental, it’s worth your focus and understanding…”
  • Tina Cardone tweaks some Parametric Functions lessons to work on Desmos. “It was important to begin graphing by hand so students had an understanding of how parametrics work. Some students were concerned that the t value wasn’t showing up on the graph and tried to include it in some rather creative ways…”
  • Rebecka Peterson steals a favorite lesson and refuses to feel guilty because “this magic should be shared.” And so she does: Slope Field Activity.
  • As I’ve put my energy into working on my math books, my blogging has suffered. So I’ve started dipping into the past and bringing up oldy-but-goody articles to reblog. I especially enjoyed The Calculus Tidbits Collection.

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  • Fran Wisniewski shares one of my all-time favorite puzzle games: Tangrams. Print and cut out a set of pieces, or play online.
  • Shecky Riemann challenges us to try a Li’l Game From Martin Gardner. “Whoever does this gets all the money played, in cases of draws (no winner) you each take your money back. The question is, is there any strategy by which you could be assured a win?”
  • Julie’s family folds up some beautiful 3-dimensional math in Origami Icosahedron. “When the faces of solid figures protrude to form more complex solids, the shapes become star-like and are known as stellations. The icosahedron we created is the small triambic icosahedron…”
  • The Math Curmudgeon’s MathArguments180 is still going strong, bringing us some cool recreational puzzles to debate. What would your students do with 187: Spiral or 191: Walking the Labyrinth?
  • One of the great puzzles of mathematics is how to think about infinity. Along this line, Yelena McManaman and her son read the book Really Big Numbers in Infinity Is Farther Than You Think. And Vi Hart posts the latest in “a potential infinity of spinoff videos” in Transcendental Darts.

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  • Donna Boucher takes a look at one of my favorite elementary math curricula in What is Singapore Math? “Singapore Math is really a philosophy for mathematics instruction — it’s as much about how to teach as it is what to teach.”
  • Stephen Cavadino asks some important questions about assessment: “What is the big picture? What are we testing for? Should we be doing it?”
  • A friend asks, “I am doubtful that he will actually be able to solve this problem he’s puzzling through. What does a good teacher do in such a situation? You have a student who is really interested in this problem, but you know that it’s far more likely that he will hit a wall (or many walls) that he really doesn’t have the tools to work through.” Ben Blum-Smith offers wise advice in Hard Problems and Hints.

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I found the pretty pictures at Creative Commons. John Riordan wrote about Telephone numbers in Introduction to Combinatorial Analysis.

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

The next installment of our carnival will open sometime during the week of August 25-29 at Math = Love. 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.

You can explore all our past MTaP carnival posts on our blog carnival Pinterest 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 want to take a turn hosting the Math Teachers at Play blog carnival, please speak up!

[Photo by Bob Jagendorf.]

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

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Playing With Math — the Book


There are only a few days left to reserve your copy of Playing With Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers. I don’t have time to finish the review I hoped to write, so instead I’ll share some of my favorite quotes from the book:

What do mathematicians do? We play with math. What are little kids doing when they’re thinking about numbers, shapes, and patterns? They’re playing with math. You may not believe it yet, but you can have fun playing with math, too.

— Sue VanHattum, editor

We had a discussion at the end of the club on how we are all confused now, but pleasantly so, and how important it is to rejoice in confusion and to be comfortable with it. Adults often strive very hard to get rid of any and all possible traces of confusion for kids, making things dreadfully boring.

— Maria Droujkova, after a math circle exploration of infinity

All it talkes to do mathematics is opportunity, a frustrating problem, and a bit of stubbornness.

— Ellen Kaplan, math circle leader

Our own school experiences can make it hard for us to teach without being tempted to “help them master” a concept that they may or may not be ready to master. What we never learned in school was the concept of playing around with math, allowing ideas to “percolate,” so to speak, before mastery occurs, and that process may take time.

— Julie Brennan, homeschooler

4 hurricane learning lp md

The most profound learning often takes place silently and invisibly, in between activities and away from prying eyes. It is here that all those pieces of information, having been shaved from actual experience, are pulled inward to jostle against one another in various combinations and arrangements until gradually, or sometimes suddenly, a new understanding emerges.

— Holly Graff, unschooler

I do a mean T. Rex impression and the class was convulsed in giggles — the perfect way to enter a “hard” math lesson. I chucked the planned lesson for the day, and we went with the dinosaurs, and eventually various other creatures with different numbers of digits. I asked the class how the T. Rex would count. After all, it has only three fingers. I’ll admit to a lot of roaring and stomping as I, the T. Rex, became more and more frustrated trying to write a note to my mother in which I wanted to tell her that I had eaten those four velociraptors.

— Michelle Martin, elementary teacher


It is the process of sharing — of not only creative and insightful problem-solving approaches, but also memorable moments filled with camaraderie, generosity, and incomparable joy. That is why I love math.

— Luyi Zhang, math major

Remember that joy and passion lead to more learning than duty ever did.

— Sue VanHattum, editor

This book truly captures the joy and passion of learning (and teaching) mathematics. Whether you love math and want to share it with your kids, or whether you fear and loathe math and need help getting over that hurdle so you won’t pass your fear on, Playing With Math will encourage you to see math more deeply and play with it more freely.

Excerpts from Playing With Math

If I’m reading the website aright, the crowdfunding campaign for Playing With Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers ends in the early wee hours of Sunday morning. Be sure to place your order by Saturday, July 19th!

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


Quotable: Math Connections


It turns out that the people who do well in math are those who make connections and see math as a connected subject. The people who don’t do well are people who see math as a lot of isolated methods.

— Jo Boaler
Math Connections

If you or your children struggle with math, Boaler’s non-profit may help you recover your joy in learning.

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


Playing With Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers


There’s a problem: Most people don’t like math. Why is that? Perhaps it has something to do with the way math is taught in school. As a teacher to my own kids and mentor to homeschooling parents, I’ve been fighting math anxiety for decades.

This book is one part of the solution.

Playing With Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers features more than thirty authors who tell delightful stories of learning to appreciate math and of sharing their enthusiasm with their communities, families, or students. After every chapter is a puzzle, game, or activity to get you and your kids playing with math, too.

You can read a couple of excerpts at

Whether you love math and want to share it with your kids, or whether you fear and loathe math and need help getting over that hurdle so you won’t pass your fear on, Playing With Math will encourage you to see math more deeply and play with it more freely.

I’ve been waiting for this book for years, and I’m thrilled to see it finally come together. As I read the advance copy (review coming soon!), I am amazed at how many different ways there are to think about math. Each writer has a new perspective and unique insight, and my quotes journal is filling up with inspiration.

A Word from the Editor

The idea of crowd-funding may be new to you. Here’s how it works:

Today is the first day of our crowd-funding campaign. For a contribution of $25, we’ll send you a book as soon as it’s printed.

You can contribute anything from $1 to $5000 (with rewards at each contribution level) to help us pay for our illustrators, editors, page layout person, and printing. This is our way of asking for community support for this book as part of the production process. We hope to build lots of energy around the ideas in the book through this campaign.

Besides contributing, here’s another way you can help: Think of five friends who would enjoy this book.

  • Do you have friends who get frustrated helping their kids with math homework?
  • Or who teach young kids but don’t feel comfortable with math themselves?
  • Do you have friends who enjoy math?
  • Or who want ideas to share with the kids in their lives?
  • Do you know someone who might want to start a math circle?

Would you send them a quick message, to let them know we’re here?

I’m hoping for the power of exponential growth with this. Our outrageous goal is to change the way people all over this country, and maybe even the world, think about math. If you each send this to five friends who might enjoy the book, and each of them sends it to five friends, and each of them … Well, pretty soon we cover the world, right?

In fact, if we kept it going through eleven steps, that would make 5 to the 11th power, or over 40 million people. Does Sue dream big? Yep.

Sue VanHattum

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Reblog: The Handshake Problem

[Feature photo above by Tobias Wolter (CC-BY-SA-3.0) via Wikimedia Commons.]

Seven years ago, our homeschool co-op held an end-of-semester assembly. Each class was supposed to demonstrate something they had learned. I threatened to hand out a ten question pop quiz on integer arithmetic, but instead my pre-algebra students voted to perform a skit.

I hope you enjoy this “Throw-back Thursday” blast from the Let’s Play Math! blog archives:

If seven people meet at a party, and each person shakes the hand of everyone else exactly once, how many handshakes are there in all?

In general, if n people meet and shake hands all around, how many handshakes will there be?


1-3 narrators
7 friends (non-speaking parts, adjust to fit your group)


Each friend will need a sheet of paper with a number written on it big and bold enough to be read by the audience. The numbers needed are 0, 1, 2, 3, … up to one less than the number of friends. Each friend keeps his paper in a pocket until needed.

[Click here to go read Skit: The Handshake Problem.]

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Reblog: In Honor of the Standardized Testing Season


[Feature photo above by Alberto G. Photo right by Renato Ganoza. Both (CC-BY-SA-2.0) via flickr.]

Quotations and comments about the perils of standardized testing, now part of my book Let’s Play Math.

I hope you enjoy this “Throw-back Thursday” blast from the Let’s Play Math! blog archives:

The school experience makes a tremendous difference in a child’s learning. Which of the following students would you rather be?

I continued to do arithmetic with my father, passing proudly through fractions to decimals. I eventually arrived at the point where so many cows ate so much grass, and tanks filled with water in so many hours. I found it quite enthralling.

— Agatha Christie
An Autobiography


“Can you do Addition?” the White Queen asked. “What’s one and one and one and one and one and one and one and one and one and one?”

“I don’t know,” said Alice. “I lost count.”

“She can’t do Addition,” the Red Queen interrupted. “Can you do Subtraction? Take nine from eight.”

“Nine from eight I can’t, you know,” Alice replied very readily: “but—”

“She can’t do Subtraction,” said the White Queen. “Can you do Division? Divide a loaf by a knife — what’s the answer to that?”

[Click here to go read In Honor of the Standardized Testing Season.]

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

Natural Math Multiplication Course


This April, the creative people at Moebius Noodles are inviting parents, teachers, playgroup hosts, and math circle leaders to join an open online course about multiplication. My preschool-2nd grade homeschool math group is eager to start!

Each week there will be five activities to help kids learn multiplication by exploring patterns and structure, with adaptations for ages 2-12.

The course starts April 6 and runs for four weeks.

Preliminary Syllabus

Week 1: Introduction.
What is multiplication? Hidden dangers and precursors of math difficulties. From open play to patterns: make your own math. 60 ways to stay creative in math. Our mathematical worries and dreams.

Week 2: Inspired by calculus.
Tree fractals. Substitution fractals. Multiplication towers. Doubling and halving games. Zoom and powers of the Universe.

Week 3: Inspired by algebra.
Factorization diagrams. Mirror books and snowflakes. Combination and chimeras. Spirolaterals and Waldorf stars: drafting by the numbers. MathLexicon.

Week 4: Times tables.
Coloring the monster table. Scavenger hunt: multiplication models and intrinsic facts. Cuisenaire, Montessori, and other arrays. The hidden and exotic patterns. Healthy memorizing.

Sounds like lots of fun!

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Pi Day Roundup


[Feature photo above by Nicolo' Canali De Rossi.]

Math holiday alert: March 14th is Pi Day. And why limit ourselves to a single day? As Tyler Jarvis pointed out, March 2014 (3/14) is Pi Month! Here are some ideas to help you celebrate…

Pi Day Posts on Let’s Play Math! Blog


And Did You Know?

Pi Day is also Albert Einstein’s birthday.

A Pinch of Pi Day Humor


Favorite Pi Day Posts on Other Blogs


And if that’s not enough to keep you busy, Education World offers another huge round-up of Pi Day lesson plans and activity ideas.

Do you have any favorite activities or blog posts about Pi Day? Please share in the comments!

This post is featured in the Carnival of Homeschooling {Edition # 428}.

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photo by Carla216 via flickr

Reblog: The Case of the Mysterious Story Problem

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

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 this “Throw-back Thursday” blast from the Let’s Play Math! blog archives:

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…

[Click here to go read the original post.]

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

Pinterest boards

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

Middle-High School

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

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

Math Teaching Tips

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

MTaP 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

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).

What are some of your favorite Pinterest sites? Please share a link in the comments!

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Math Teachers at Play #70

800px-Brauchtum_gesteck_70_1[Feature photo above by David Reimann via Bridges 2013 Gallery. Number 70 (right) from Wikimedia Commons (CC-BY-SA-3.0-2.5-2.0-1.0).]

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

Welcome to the 70th edition of the Math Teachers At Play math education blog carnival — a smorgasbord of 42+ links to bloggers all around the internet who have great ideas for learning, teaching, and playing around with math from preschool to pre-college. Let the mathematical fun begin!

By tradition, we start the carnival with a puzzle in honor of our 70th edition. But if you would like to jump straight to our featured blog posts, click here to see the Table of Contents.

Click here to continue reading.

Multiplication Matching Cards

Multiplication Models Card Game

[Poster by Maria Droujkova of This game was originally published as part of the Homeschooling with a Profound Understanding of Fundamental Mathematics Series.]

Homeschooling parents know that one of the biggest challenges for any middle-elementary math student is to master the multiplication facts. It can seem like an unending task to memorize so many facts and be able to pull them out of mental storage in any order on demand.

Too often, we are tempted to stress the rote aspect of such memory work, which makes our children lose their focus on what multiplication really means. Before practicing the times table facts, make sure your student gets plenty of practice recognizing and using the common models for multiplication.

To help your children see what multiplication looks like in real life, explore the multitude of Multiplication Models collected at the Natural Math website. Or try some of the hands-on activities in the Family Multiplication Study.

You may want to pick up this poster and use it for ideas as you play the Tell Me a (Math) Story game. Word problems are important for children learning any new topic in math, because they give children a mental “hook” on which to hang the abstract number concepts.

And for extra practice, you can play my free card game…

Click here to continue reading.


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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Land or Water 2

Maze Game: Land or Water?

This was a fun activity from Moebius Noodles for our PK-1st grade Homeschool Math in the Park group. The children take turns making a maze and setting a dinosaur inside. Then the other dinosaurs (parents or siblings) try to guess whether their friend is on the land or in the water.

Draw the maze

Player #1

(1) First, draw a big circle on the white board. This is your lake.

(2) With a finger or a bit of cloth, erase a small section of the circle to create the opening for your maze.

(3) Starting at one edge of the opening, draw a random squiggle inside the circle. Make your squiggle end at the other edge of the opening.

Looks like Land

(4) Set your dinosaur anywhere inside the maze.

Player #2

(1) Now it’s your turn to guess. Is the dinosaur standing on the land? Is it swimming in the water?

(2) How will you figure out if you guessed right?

(3) Check by jumping across the lines of the maze. Each jump takes you across a boundary: Splash! (Into the water.) Thump! (Back on the land.) Splash! Thump! … Until you reach the dinosaur inside.

(4) Or go to the maze entrance and walk your dinosaur along the path. Can you find your way?

land or water

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Related posts on Let’s Play Math! blog:

Sierpinski Cookies-16

Happy Math Storytelling Day

Feature photo (above) by L. Marie. Math comic by davidd. Both via flickr (CC BY 2.0).
Math-ter and Commander

Hooray for September 25th — it’s Math Storytelling Day!

Celebrate Math Storytelling Day by making up and sharing math stories. Everyone loves a story, so this is a great way to motivate your children to play around with math. What might a math story involve? Patterns, logic, history, puzzles, relationships, fictional characters, … and yes, even numbers.

For inspiration, visit:

Have you and your children created any math stories? We’d love to hear! Please share your links in the comments section below.

Continue reading

Talking Math with Your Kids

Danielson-Talking Math

Christopher Danielson, one of my favorite math bloggers, has a new book out that is perfect for parents of preschool and elementary-age children:

It’s a short book with plenty of great stories, advice, and conversation-starters. While Danielson writes directly to parents, the book will also interest grandparents, aunts & uncles, teachers, and anyone else who wants to help children notice and think about math in daily life.

You don’t need special skills to do this. If you can read with your kids, then you can talk math with them. You can support and encourage their developing mathematical minds.

You don’t need to love math. You don’t need to have been particularly successful in school mathematics. You just need to notice when your children are being curious about math, and you need some ideas for turning that curiosity into a conversation.

In nearly all circumstances, our conversations grow organically out of our everyday activity. We have not scheduled “talking math time” in our household. Instead, we talk about these things when it seems natural to do so, when the things we are doing (reading books, making lunch, riding in the car, etc) bump up against important mathematical ideas.

The dialogues in this book are intended to open your eyes to these opportunities in your own family’s life.

— Christopher Danielson
Talking Math with Your Kids

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Related posts on Let’s Play Math! blog:

Carnival Parade in Aachen 2007

Math Teachers at Play #66

[Feature photo above by Franz & P via flickr. Route 66 sign by Sam Howzit via flickr. (CC BY 2.0)]
Route 66 Sign

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.

By tradition, we start the carnival with a couple of puzzles in honor of our 66th edition.

Let the mathematical fun begin!

Puzzle 1

how crazy 66

Our first puzzle is based on one of my favorite playsheets from the Miquon Math workbook series. Fill each shape with an expression that equals the target number. Can you make some cool, creative math?

Click the image to download the pdf playsheet set: one page has the target number 66, and a second page is blank so you can set your own target number.

Continue reading


WOW – Multiplication! An Open Online Course for Parents and Teachers


Once again, the authors of Moebius Noodles are teaming up to offer a free course for families, math clubs, playgroups, and others who want to explore adventurous mathematics with kids of any age.

This short course will last only two weeks, and the topic is multiplication:

Both of my homeschool math circles (one with preschool-1st grade, and one with teens) thoroughly enjoyed the month-long problem solving course this summer, and we expect the new one to be just as much fun. Will you join us?

As with most of the Moebius Noodles courses, Maria and Yelena have adapted the activities for all ages from toddlers to adults. Where young ones go on a scavenger hunt for pretty snowflakes and cool truck wheels, older kids build bridges from multiplication to symmetry, spatial transformations, and proportions.

Visit the registration page to sign up no later than September 8. The main course activities will happen September 9th through 22nd. Expect to spend about two hours a week.

Continue reading


Summer Problem Solving for the Young, the Very Young, and the Young at Heart

Here is yet another wonderful summer math opportunity for homeschoolers or anyone who works with kids: a free, 3-week mini-course on math problem solving for all ages.

The course is being organized by Dr. James Tanton, Dr. Maria Droujkova, and Yelena McManaman. The course participants include families, math clubs, playgroups, and other small circles casually exploring adventurous mathematics with kids of any age.

Would you like to join us? Check out the mpsMOOC13 home page for instructions. The deadline for joining is July 7 July 3.

And then the real fun begins!

Continue reading

What Do You Notice? What Do You Wonder?

Drawing Star

If you want your children to understand and enjoy math, you need to let them play around with beautiful things and encourage them to ask questions.

Here is a simple yet beautiful thing I stumbled across online today, which your children may enjoy:

It reminds me of string art designs, but the app makes it easy to vary the pattern and see what happens.

  • What do your students notice about the patterns?
  • What questions can they ask?

I liked the way the app uses “minutes” as the unit that describes the star you want the program to draw. That makes it easier (for me, at least) to notice and understand the patterns, since minutes are a more familiar and intuitive unit than degrees, let alone radians.

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Summer School for Parents, Teachers: How to Learn Math

Here’s an interesting summer learning opportunity for homeschooling parents and classroom teachers alike. Stanford Online is offering a free summer course from math education professor and author Jo Boaler:

Boaler’s book is not required for the course, but it’s a good read and should be available through most library loan systems.

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Youth Sports Baseball Camp

Quotable: Learning the Math Facts

feature photo above by USAG- Humphreys via flickr (CC BY 2.0)

During off-times, at a long stoplight or in grocery store line, when the kids are restless and ready to argue for the sake of argument, I invite them to play the numbers game.

“Can you tell me how to get to twelve?”

My five year old begins, “You could take two fives and add a two.”

“Take sixty and divide it into five parts,” my nearly-seven year old says.

“You could do two tens and then take away a five and a three,” my younger son adds.

Eventually we run out of options and they begin naming numbers. It’s a simple game that builds up computational fluency, flexible thinking and number sense. I never say, “Can you tell me the transitive properties of numbers?” However, they are understanding that they can play with numbers.

photo by Mike Baird via flickr

photo by Mike Baird via flickr

I didn’t learn the rules of baseball by filling out a packet on baseball facts. Nobody held out a flash card where, in isolation, I recited someone else’s definition of the Infield Fly Rule. I didn’t memorize the rules of balls, strikes, and how to get someone out through a catechism of recitation.

Instead, I played baseball.

John Spencer
Memorizing Math Facts

Conversational Math

The best way for children to build mathematical fluency is through conversation. For more ideas on discussion-based math, check out these posts:

Learning the Math Facts

For more help with learning and practicing the basic arithmetic facts, try these tips and math games:

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RIMG0186 Satellite dish

How To Master Quadratic Equations

G'Day Math logo

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


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.

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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!


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 full-size template.

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:

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photo by Sphinx The Geek via flickr

Homeschooling High School Math

photo by ddluong via flickr

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?

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Moebius Noodles: New Must-Read Math Book


Homeschoolers, after-schoolers, unschoolers, or anyone else: if you’re a parent with kids at home, you need this book. If you work with children in any way (grandparent, aunt/uncle, teacher, child care, baby sitter, etc.) you need this book. Or if you hated math in school and never understood how anyone could enjoy it, you need this book!

Moebius Noodles is a travel guide to the Math Universe for adventurous families (and it has lots of beautiful pictures, too!) featuring games and activities that draw out the rich, mathematical properties of everyday objects in ways accessible to parents and children:

  • A snowflake is an example of a fractal and an invitation to explore symmetry.
  • Cookies offer combinatorics and calculus games.
  • Paint chips come in beautiful gradients, and floor tiles form tessellations.

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Happy Square of a Square Day

3-19 Happy Square of a Square Day sign

In Response To

Make your own “Happy Math Day” sign:

Here’s a fun activity for any age that will encourage your children to play with numbers:

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