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.

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 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.

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

92 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.

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

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.

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.

Sarah Dees (@FrugalFun4Boys) comes up with a way to do Sorting, Counting, and Graphing for Preschoolers. “We were able to get in lots of counting practice and conversations about more and less and which group was the greatest. And, it kept him busy for quite awhile, which is always a good thing.”

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…

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.”

How often have you found that students are taught “tricks” to remember math rules—but they still make mistakes, because the rules don’t make sense? Ellie Nix (Bloglovin’)explains Why I’m Not Teaching Decimal Operations “Rules”.

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…”

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.

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…

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.

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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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.”

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?

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

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…”

The 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…

Claim your two free learning guide booklets, and be one of the first to hear about new books, revisions, and sales or other promotions.

Interrupt 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.

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.

For Further Reading

Mike Lawler and sons explore Loop-de-Loops: Part 1, and Part 2.

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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Math Concepts: division as equal sharing, naming fractions, adding fractions, infinitesimals, iteration, limits Prerequisite: able to identify fractions as part of a whole

This is how I tell the story:

We have a cake to share, just the two of us. It’s not TOO big a cake, ‘cuz we don’t want to get sick. A 8 × 8 or 16 × 16 square on the graph paper should be just right. Can you cut the cake so we each get a fair share? Color in your part.

How big is your piece compared to the whole, original cake?

But you know, I’m on a diet, and I just don’t think I can eat my whole piece. Half the cake is too much for me. Is it okay if I share my piece with you? How can we divide it evenly, so we each get a fair share? How big is your piece?

How much of the whole, original cake do you have now? How can you tell?

I keep thinking of my diet, and I really don’t want all my piece of cake. It looks good, but it’s still just a bit too big for me. Will you take half of it? How big is that piece?

Now how much of the whole, original cake do you have? How could we figure it out?
[Teaching tip: Don’t make kids do the calculation on paper. In the early stages, they can visualize and count up the fourths or maybe the eighths. As the pieces get smaller, the easiest way to find the sum is what Cohen does in the video below—identify how much of the cake is left out.]

Even for being on a diet, I still don’t feel very hungry…