*[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.

## PUZZLE: CRYSTAL BALL CONNECTION PATTERNS

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.

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

## TABLE OF CONTENTS

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:

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

## EARLY LEARNING ACTIVITIES

- 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.]*

- Casey Rutherford’s son notices, wonders, and draws a logical conclusion — and then modifies it after further investigation: Experience Driving Misconceptions.

- 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).

- Casey Rutherford reminds us of something we all know, but too easily forget: We Underestimate What Kids Can Do.

- 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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## ELEMENTARY EXPLORATION AND MIDDLE SCHOOL MASTERY

- Amy Mascott plays 4 sums in a row: a quick & easy math game to keep her son’s addition skills and strategic thinking sharp over the summer.

- 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…”

- Simon Gregg’s students make a hands-on proof of “a curious and wondrous fact” in A Square of Cubes in Year 4. See also the related post: Successive cubes summed.

- Problem solving can be as much about the journey as the destination. Mike Lawler’s lesson didn’t go quite the way he’d planned in A bit of a struggle with estimation.

- 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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## ADVENTURES IN BASIC ALGEBRA & GEOMETRY

- Fawn Nguyen’s students have fun investigating the relationship between a circle’s diameter and circumference in Friday Bubbles.

- A couple of year ago, Elizabeth Statmore put together a fantastic game for beginning algebra students to practice a too-often-neglected skill, turning Words into Math. This summer, with the help of Twitter friends, she added a Geometry version: DIY Geometry Vocabulary Game, courtesy of the MTBoS (a collaborative effort).

- Both of the above are based on Maria Anderson’s tic-tac-toe style Block games. More advanced algebra students will enjoy Exponent Block and Factor Pair Block.

- 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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## ADVANCED MATHEMATICAL ENDEAVORS

- Anna Blinstein and Kate Nowak are collecting a set of deep questions that drive parts of high school mathematics. How would your students answer these Essential Questions for Geometry & Algebra 2? Or these Essential Questions for Algebra 2? What questions would you add to the lists?

- William Wu serves up a couple of important proofs suitable for high school students: Why is e irrational? and How to prove square root of 2 is irrational (Constructive Approach)

- 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…”

- Neil Irwin and Kevin Quealy give a warning to all statistics students: “Human beings, unfortunately, are bad at perceiving randomness.” Read How Not to Be Misled by the Jobs Report.

- 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.

- And don’t miss the 112th Carnival of Mathematics!

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## PUZZLING RECREATIONS

- Fran Wisniewski shares one of my all-time favorite puzzle games: Tangrams. Print and cut out a set of pieces, or play online.

- I love Sian Zelbo’s puzzle blogs, since being targeted at kids puts them right at my level. Here are a couple of my recent favorites: Subtraction Snakes and The Painted Tetrahedron.

- 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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## TEACHING TIPS

- Christina Tondevold warns us that “too often we use the concrete manipulatives incorrectly” and encourages us to give students room to think problems through for themselves in Stop Using Base 10 Blocks To ‘Teach’ The Algorithms!

- 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.”

- Monica Utsey reviews another curriculum I love: Beast Academy Comic Book Math. Meanwhile, Claire discovers a UK curriculum I’ve never heard of (
*Galore Park*): Finally…a maths program which works for us!

- Lucinda Leo explains How my autodidactic 9 year old is learning maths without a curriculum and Why we love Edward Zaccaro more than Khan Academy.

- So, you’ve collected your students’ responses to a rich mathematical task. Now what? David Wees experiments with Categorizing Student Strategies.

- Cindy Smith examines The Power of Specific, Non-Graded Feedback.

- 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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## GIVING CREDIT WHERE IT’S DUE

I found the pretty pictures at Flickr.com 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.]*