10 Ways to Celebrate World Tessellation Day

Guest post by Emily Grosvenor.

June 17 marks the first-ever World Tessellation Day, a holiday I created to bring awareness to the fun of finding and making tessellations.

Will you celebrate with us?

Here are 10 great ways to play with tessellations, learn about them, and introduce your children to a math concept that opens a variety of creative learning opportunities.

1) Learn about tessellations with your kids.

A tessellation is a tiled mosaic pattern of the same shape laid out over and over again, repeating into infinity. Tessellations can be found in nature, or they can be created by people. Learn more at these websites:

1WorldTessellationDayExcept where otherwise noted, graphics and photos copyright ©2016 Emily Grosvenor. All rights reserved.

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Math Teachers at Play #97 Blog Carnival

Did you know 97 is an emirp?
Did you know 97 is an emirp? It’s prime both forward and backward! What other emirps can you find?

Welcome to the 97th edition of the Math Teachers At Play math education blog carnival: a monthly 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.

A few articles were submitted by their authors, but most were drawn from the immense backlog in my rss reader. If you’d like to see your blog post featured next month, be sure to send it in yourself. Our hosts are busy parents and teachers who have limited time to scour the Internet for goodies.

To add a bit of color, I’ve thrown in several favorites from my newly updated Math with Living Books pages. Some (affiliate) links go to Amazon.com, where you can read descriptions and reviews — but there’s no need to buy. Most of these books should be available through your local library.

Table of Contents

If you’d like to skip directly to your area of interest, click here:

Please: If you enjoy the carnival, would you consider volunteering to host sometime this year? 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, please speak up!

And now, let the mathematical fun begin!


Pinczes-A Remainder of One

When the queen of her bugs demands that her army march in even lines, Private Joe divides the marchers into more and more lines so that he will not be left out of the parade.

Talking Math with Kids

  • Crystal Wagner (@Tri_Learning) shares several Math Games to Play in the Car: “Or maybe you are waiting in line at the grocery store or doctor’s appointment. Turn these times of waiting into learning opportunities.”
  • Christopher Danielson (@Trianglemancsd) shows how The sequence machine can launch math conversations with older students: “Now you can generate number sequences, without being distracted by the multiplication facts.”

richman-bykids

Help inspire your kids to try writing their own unique problems. Includes a wide range of math topics and concepts: money and time, fractions, percentages, geometry, logic, and multi-step problem solving.

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New Picture Book: Tessalation!

When I run a math circle or co-op class, I love starting with a picture book. This new beauty from Emily Grosvenor will be perfect.

You could say that Tessalation is a book about tessellations (repeating tiled patterns), but it is really a children’s picture book about discovering order in a chaotic world.

— Emily Grosvenor

Seeing Math in the World

In taking a playful approach to mathematics, I hope to open children’s eyes to math in their world. Schooly math lessons have led many of my math group kids to think a “pattern” has to be a strictly repeating (and rather boring) series of shapes or colors.

But in the real world, patterns are so important that American mathematician Lynn Arthur Steen defined mathematics as the science of patterns.

“As biology is the science of life and physics the science of energy and matter, so mathematics is the science of patterns,” Steen wrote. “We live in an environment steeped in patterns — patterns of numbers and space, of science and art, of computation and imagination. Patterns permeate the learning of mathematics, beginning when children learn the rhythm of counting and continuing through times tables all the way to fractals and binomial coefficients.”

Tessa Truman-Ling’s delight in patterns is contagious. And it will provide a wonderful jumping-off point for a variety of math activities.

Visit Grosvenor’s Kickstarter page to find out more about her lovely book:

Further Exploration


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2016 Mathematics Game

PostcardNewYearsResolutionSoapBubbles1909

[Feature photo above from the public domain, and title background (below) by frankieleon (CC BY 2.0) via Flickr.]

2016-math-game

Have you made a New Year’s resolution to spend more time with your family this year, and to get more exercise? Problem-solvers of all ages can pump up their (mental) muscles with the Annual Mathematics Year Game Extravaganza. Please join us!

For many years mathematicians, scientists, engineers and others interested in math have played “year games” via e-mail. We don’t always know whether it’s possible to write all the numbers from 1 to 100 using only the digits in the current year, but it’s fun to see how many you can find.

Math Forum Year Game Site

Rules of the Game

Use the digits in the year 2016 to write mathematical expressions for the counting numbers 1 through 100. The goal is adjustable: Young children can start with looking for 1-10, middle grades with 1-25.

  • You must use all four digits. You may not use any other numbers.
  • Solutions that keep the year digits in 2-0-1-6 order are preferred, but not required.
  • 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, .02, etc., but you cannot write 0.02 because we only have one zero in this year’s number.
  • You may create multi-digit numbers such as 10 or 201 or .01, but we prefer solutions that avoid them.

My Special Variations on the Rules

  • You MAY use the overhead-bar (vinculum), dots, or brackets to mark a repeating decimal. But students and teachers beware: you can’t submit answers with repeating decimals to Math Forum.
  • You MAY use a double factorial, n!! = the product of all integers from 1 to n that have the same parity (odd or even) as n. I’m including these because Math Forum allows them, but I personally try to avoid the beasts. I feel much more creative when I can wrangle a solution without invoking them.

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Math Teachers at Play #92

MTAP-92

Welcome to the 92nd edition of the Math Teachers At Play math education blog carnival‌—‌a monthly 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.

Let the mathematical fun begin!

By tradition, we start the carnival with a couple of puzzles in honor of our 92nd edition…

Puzzle #1

Pentagonal numbers92 is a pentagonal number, so I was delighted when Lisa Winer‘s (@Lisaqt314) carnival submission came in. Her class spent some time playing around with figurate number puzzles‌—‌including pentagonal numbers‌—‌and collaborated on a blog post about their discoveries.

Click here to find Winer’s own notes about the lesson, along with all the puzzle handouts.

What fun!

Puzzle #2

Or, try your hand at the classic Queen’s Puzzle:

  • What is the maximum number of queens that can be placed on an chessboard such that no two attack one another?

Spoiler: Don’t peek! But the answer is here‌—‌and the cool thing is that there are 92 different ways to do it.


Table Of Contents

The snub dodecahedron is an Archimedean solid with 92 faces.

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.

Along the way, I’ve thrown in some videos in honor of the holiday season.

Please: If you enjoy the carnival, would you consider sending in an entry for next month’s edition? Or volunteering to host sometime in 2016?


Early Learning Activities

  • Kids can enjoy making up math problems, but sometimes they can get a bit carried away. Just ask A. O. Fradkin (@aofradkin) about her daughter’s Gruesome Math.
  • Nancy Smith (@nancyqsmith) notices her students struggling with the equal sign in Equality. Strong opinions, and even a few tears. It will be interesting to hear what tomorrow brings…

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Elementary Exploration And Middle School Mastery

  • Joshua Greene (@JoshuaGreene19) offers some great ways to tweak an already-wonderful multiplication game in Times square variations. “It was really interesting to see the different strategies that the students took to determining what would go on their boards.”
  • For my own contribution to the carnival, I’ve posted a couple of hands-on arithmetic explorations in A Penny for Your Math.

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Adventures in Basic Algebra & Geometry

  • Tina Cardone (@crstn85) experiments with Bar Models in Algebra to help her students think about linear equations. “I did not require students to draw a model, but I refused to discuss an incorrect equation with them until they had a model. Kids would tell me ‘I don’t know how to do fractions or percents’ but when I told them to draw a bar, and then draw 4/5, they could do that without assistance…”

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Advanced Mathematical Endeavors

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Puzzling Recreations

  • Pradeep Mutalik challenges readers to “infer the simple rule behind a number sequence that spikes up and down like the beating of a heart” in Be Still My Pulsating Sequence.

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Teaching Tips

  • How can we get a peek at how our children are thinking? Kristin Gray (@mathminds) starts with a typical set of 1st Grade Story Problems and tweaks them into a lively Notice/Wonder Lesson. “When I told them they would get to choose how many students were at each stop, they were so excited! I gave them a paper with the sentence at the top, let them choose a partner and sent them on their way…”
  • Tracy Zager (@tracyzager) talks about her own mathematical journey in The Steep Part of the Learning Curve: “The more math I learn, the better math teacher I am. I keep growing as a learner; I know more about where my kids are headed; and I understand more about what building is going on top of the foundation we construct in elementary school.”
  • And finally, you may be interested in my new blog post series exploring what it means to understand math. Check out the first post Understanding Math: A Cultural Problem. More to come soon…

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Credits

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

The December 2015 installment of our carnival will open sometime during the week of December 21-25 at Math Misery? blog. 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 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!


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A Penny for Your Math

You know you’re a math teacher when you see a penny in the parking lot, and your first thought is, “Cool! A free math manipulative.”

penny

My homeschool co-op math students love doing math with pennies. They’re rather heavy to carry to class, but worth it for the student buy-in.

This month, I’m finishing up the nearly 150 new illustrations for the upcoming paperback edition of my Let’s Play Math book. I’m no artist, and it’s been a long slog. But a couple of the graphics involved pennies‌—‌so when I saw that penny on the ground, it made me think of my book.

And thinking of my book made me think it would be fun to share a sneak peek at coming attractions…

The Penny Square: An Example of Real Mathematics

Real mathematics is intriguing and full of wonder, an exploration of patterns and mysterious connections. It rewards us with the joy of the “Aha!” feeling. Workbook math, on the other hand, is several pages of long division by hand followed by a rousing chorus of the fraction song: “Ours is not to reason why, just invert and multiply.”

Real math is the surprising fact that the odd numbers add up to perfect squares (1, 1 + 3, 1 + 3 + 5, etc.) and the satisfaction of seeing why it must be so.

Did your algebra teacher ever explain to you that a square number is literally a number that can be arranged to make a square? Try it for yourself:

  • Gather a bunch of pennies‌—‌or any small items that will not roll away when you set them out in rows‌—‌and place one of them in front of you on the table. Imagine drawing a frame around it: one penny makes a (very small) square. One row, with one item in each row.
  • Now, put out three more pennies. How will you add them to the first one in order to form a new, bigger square? Arrange them in a small L-shape around the original penny to make two rows with two pennies in each row.
  • Set out five additional pennies. Without moving the current four pennies, how can you place these five to form the next square? Three rows of three.
  • Then how many will you have to add to make four rows of four?
pennies
Twenty-five is a square number, because we can arrange twenty-five items to make a square: five rows with five items in each row.

Each new set of pennies must add an extra row and column to the current square, plus a corner penny where the new row and column meet. The row and column match exactly, making an even number, and then the extra penny at the corner makes it odd.

Can you see that the “next odd number” pattern will continue as long as there are pennies to add, and that it could keep going forever in your imagination?

The point of the penny square is not to memorize the square numbers or to get any particular “right answer,” but to see numbers in a new way‌—‌to understand that numbers are related to each other and that we can show such relationships with diagrams or physical models. The more relationships like this our children explore, the more they see numbers as familiar friends.

The Penny Birthday Challenge: Exponential Growth

Penny-Challenge-1

A large jar of assorted coins makes a wonderful math toy. Children love to play with, count, and sort coins.

Add a dollar bill to the jar, so you can play the Dollar Game: Take turns throwing a pair of dice, gathering that many pennies and trading up to bigger coins. Five pennies trade for a nickel, two nickels for a dime, etc. Whoever is the first to claim the dollar wins the game.

Or take the Penny Birthday Challenge to learn about exponential growth: Print out a calendar for your child’s birthday month. Put one penny on the first day of the month, two pennies on the second day, four pennies on the third day, etc. If you continued doubling the pennies each day until you reach your child’s birthday, how much money would you need?

Warning: Beware the Penny Birthday Challenge! Those pennies will add up to dollars much faster than most people expect. Do not promise to give the money to your child unless the birthday comes near the beginning of the month.

A Penny Holiday Challenge

The first time I did pennies on a calendar with my homeschool co-op class was during December, so we called it the Penny Christmas Challenge:

  • How many pennies would you need to cover all the days up to the 25th?

I told the kids that if their grandparents asked what gift they wanted for Christmas, they could say, “Not much. Just a few pennies…”


LPM-ebook-300The Penny Square, Dollar Game, and Penny Birthday Challenge are just three of the myriad math tips and activity ideas in the paperback edition of Let’s Play Math: How Families Can Learn Math Together and Enjoy It. Coming in early 2016 to your favorite online bookstore…


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Spirolateral Math Doodles

This is not a math book coverInterrupt your regular math programming to try this fantastic math doodling investigation, and you might even win a prize!

Anna Weltman wrote a math/art book, and Dan Meyer is offering a classroom-size set of them to the winner of his fall contest (deadline Tuesday, October 6, and homeschoolers are welcome, too).

Even if you don’t want to enter Dan’s contest, spirolateral math doodles—‌or “loop-de-loops”—‌make a great mathematical exploration.

loop-de-loops1

How to Get Started

To make a spirolateral, you first pick a short series of numbers (1, 2, 3 is a traditional first set) and an angle (90° for beginners). On graph paper, draw a straight line the length of your first number. Turn through your chosen angle, and draw the next line. Repeat turning and drawing lines, and when you get to the end of your number series, start again at the first number.

Some spirolaterals come back around to the beginning, making a closed loop. Others never close, spiraling out into infinity—‌or at least, to the edge of your graph paper.

loop-de-loops2

For Further Reading

Articles by Robert J. Krawczyk:

Anna Weltman appeared on Let’s Play Math blog once before, with the game Snugglenumber. And she’s a regular contributor to the wonderful Math Munch blog.


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