How to Make a Flexagon Christmas Card

tetra-tetraflexagonHere’s how Alex created tetra-tetraflexagon Christmas cards to send to her friends:

1. Buy a pack of heavy paper at the office supply store. Regular construction paper tears too easily.

2. Measure and divide the paper into fourths one direction and thirds the other way. Fold each line backward and forward a few times.

3. Number the front and back of the paper in pencil, lightly, as shown. Then carefully cut a center flap along the dotted lines.

4. Fold the paper along the dark lines as shown, so the center flap sticks out from underneath and the right-hand column shows all 2’s.

5. Fold the flap the rest of the way around to the front and fold the right-hand column under again. (Shown as dark lines on the diagram.) This makes the front of the flexagon show 1’s in every square.

6. Carefully, tape the flap to its neighbor on the folded column. Don’t let the tape stick to any but these two squares.

7. Gently erase your pencil marks.

Find All the Faces

A tetra-tetraflexagon has four faces: front, back, and two hidden. It is shaped like a tetragon — better known as a rectangle.

Here’s how to flex your tetra-tetraflexagon card:

  • Face 1 is easy to find. It’s on top when you make the card.
  • Turn the card over to find Face 2.
  • Face 3 is hidden behind Face 2. Fold your flexagon card in half (vertically) so that Face 1 disappears. Unfold Face 2 at the middle, like opening a book. Face 3 should appear like magic.
  • Face 4 is hidden behind Face 3. Fold the card (vertically) to hide Face 2, then open the middle of Face 3. Face 2 vanishes, and Face 4 is finally revealed.

When Faces 2 and 3 are folded to the back, you will notice that any pictures you drew on them will look scrambled. What happened?

Add Your Designs

Alex wrote a holiday greeting on Face 1. Then she drew Christmas pictures on the other three faces of her card.

To Be Continued…

Read all the posts from the December 2000/January 2001 issue of my Mathematical Adventures of Alexandria Jones newsletter.


“Happy Holidays” photo by Mike Brand via Flickr (CC BY 2.0). Video by Shaireen Selamat of DynamicEducator.com.

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Alexandria Jones and the Magic Christmas Cards

The Jones family sat around the dining table performing a traditional holiday ritual: the Christmas card assembly line.

First, Dr. Fibonacci Jones (the world-famous mathematical archaeologist) signed for himself and his wife. He handed the card to Alex, who signed for herself and baby Renée. Then Alex’s younger brother Leon added his own flourish. Finally, Mrs. Jones wrote a personal note on the cards going to immediate family and close friends.

One-year-old Renée sat in her high chair, chewing the corners of an extra card.

Alex Poses a Problem

Alex dropped her pen and shook out her tired fingers.

“I’m stumped,” she said. “I’d like to send a special Christmas card to some of my friends from camp last summer. But I can’t think of anything that seems good enough.”

Leon leaned his chair back in thought.

Then he snapped his fingers. “I’ve got it! We’ll throw a handful of sand in each of their envelopes. You know, to make them remember all the fun you guys had digging up old stuff.”

Alex humphed. “How would you like to get sand in your Christmas present?” she asked. “Besides, it wasn’t stuff. It was artifacts.”

“You should not make such a display of your ignorance, young man,” Dr. Jones said. “Stuff, indeed!”

Mrs. Jones put her hand to her forehead and sighed dramatically. Then she turned to Alex. “Have you considered doing a jigsaw puzzle card? They sell them at the hobby store.”

“I’ve tried those before,” Alex said, “but the ones I had always warped. The puzzles didn’t go back together very well.”

Dad Gets an Idea

Dr. Jones got an out-of-focus, “I’m thinking” look in his eyes. He stood up, tapped his chin with his pen, and walked away. He almost ran into the wall, but he caught himself. Shaking his head, he disappeared into his study.

Mrs. Jones put down her pen and picked up Renée.

“Why don’t you two address those envelopes while we wait for your dad’s inspiration to reveal itself? I need to put a little one down to S-L-E-E-P.”

Alex laughed. “If you keep that up, Renée will learn to spell before she’s out of diapers!”

Leon thumbed the stack of envelopes and groaned. “C’mon, sis. Back to work!”

Before long, Mrs. Jones came back and chased the kids away from the table. “I’ll finish this,” she said.

Unfolding the Magic

Alex and Leon ran to the study. They found Dr. Jones at his desk, playing with a piece of paper.

“Ah, there you are,” he said. “Here, Alex. What do you think?”

“Well,” she said, “it looks like a regular piece of paper that’s been folded over on itself.”

Dr. Jones nodded. “Now you know a sheet of paper has two faces—that is, it has a front and a back.”

Leon reached for the paper and flipped it over. “Is that why you put red stripes on one side and blue stripes on the other?”

“Observe,” Dr. Jones said.

He took the piece of paper and folded it in half. Then he unfolded it and handed it to Alex.

“Hey, how’d you do that?” she asked. “Now there are blue polka-dots on this side.”

“Cool! It’s magic,” Leon said.

“It is called a tetra-tetraflexagon,” Dr. Jones said, “and it has one more hidden face. Can you find it?”

Alex folded the paper this way and that. Then she held it up in triumph.

“Look, red dots—I did it!”

She gave her dad a tremendous hug. “Thanks, Dad! I’ll make magic flexagons. They’ll be the best Christmas cards ever!”

To Be Continued…

Read all the posts from the December 2000/January 2001 issue of my Mathematical Adventures of Alexandria Jones newsletter.


“Christmas Window” photo by slgckgc via Flickr (CC BY 2.0). Video by Shaireen Selamat of DynamicEducator.com.

howtosolveproblemsWant to help your kids learn math? Claim your free 24-page problem-solving booklet, and sign up to hear about new books, revisions, and sales or other promotions.


Christmas with Alexandria Jones

Alexandria JonesAlexandria Jones and her family are fictional characters from my old Mathematical Adventures newsletter. Their stories appear sporadically as I find time to transcribe them from the back-issues. You can find them all on this blog page.

Here are all the Alexandria Jones stories Christmas stories, with activity and craft ideas…

Alexandria Jones and the Christmas Present Quandary

Alex designs tessellation wrapping paper, hunts for the perfect Christmas tree, and comes up with a lively present for her brother. We meet the rest of Alex’s family — her father was introduced in an earlier issue — along with historical figures Maria Agnesi and Leonhard Euler, and we take a brief glance at mathematics from China.

Alexandria Jones and the Christmas Gifts

Most of this issue focuses on other topics — but the Jones family has a new baby, so Alex makes two gifts.

And New This Year: Alexandria Jones and the Magic Christmas Cards

Dr. Jones suggests a way to make the “best Christmas cards ever” (according to Alex), and the Jones children create geometric gifts to celebrate the holiday.


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Count Up to Christmas

secondary-starBack when we were still homeschooling, I always dropped the “regularly scheduled program” in December. School plus holiday prep added up to one stressed-out mom.

Instead, we read plenty of library books. And we played around with informal activities like the NrichMaths Advent Calendars:

For older students and adults, the online Plus Magazine offers a calendar of daily tidbits from their “Maths in a minute” series, explaining important mathematical concepts in just a few words”

And for still more winter fun, check out the links in my old Christmas Math Puzzles and Activities post.

And a Question for You

How do you handle schoolwork during this busy season? I’m collecting new links for an updated Holiday Math post next month. I’d love to hear your ideas!


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Advent Math Calendars from Nrich

Once again, Nrich.Maths.org and Plus.Maths.org are offering a selection of activities to encourage your students’ mathematical creativity, one for each day in the run-up to Christmas. Click one of the images below to visit the appropriate Advent Math Calendar page.

For Primary Students

Easier activities for elementary and middle school. When you get to the Nrich website, click a number to go to that day’s math.

primary_650px

For Secondary Students

Activities for middle and high school. When you get to the Nrich website, click a number to go to that day’s math.

secondary

For Teens and Adults

“Wild Maths” puzzles and articles for teens and up. When you get to the +Plus Magazine website, you can tell which links are live by the drop shadow under the picture. One link becomes live each day — so come back tomorrow and discover something new!

plusmaths advent


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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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