Math Debates with a Hundred Chart

Euclid game
Wow! My all-time most popular post continues to grow. Thanks to an entry from this week’s blog carnival, there are now more than thirty great ideas for mathematical play:

The latest tips:

(31) Have a math debate: Should the hundred chart count 1-100 or 0-99? Give evidence for your opinion and critique each other’s reasoning.
[Hat tip: Tricia Stohr-Hunt, Instructional Conundrum: 100 Board or 0-99 Chart?]

(32) Rearrange the chart (either 0-99 or 1-100) so that as you count to greater numbers, you climb higher on the board. Have another math debate: Which way makes more intuitive sense?
[Hat tip: Graham Fletcher, Bottoms Up to Conceptually Understanding Numbers.]

(33) Cut the chart into rows and paste them into a long number line. Try a counting pattern, or Race to 100 game, or the Sieve of Eratosthenes on the number line. Have a new math debate: Grid chart or number line — which do you prefer?
[Hat tip: Joe Schwartz, Number Grids and Number Lines: Can They Live Together in Peace? ]


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Math Teachers at Play #80 via Triumphant Learning

The new Math Teachers at Play math education blog carnival is up for your browsing pleasure. Each month, we feature activities, lessons, and games about math topics from preschool through high school. Check it out!

[Photo by Paul Downey. (CC BY 2.0 via Flickr)]

PaulDowney80

Welcome to the 80th Edition of the Math Teachers at Play (MTaP) blog carnival.

Before we dive into some math posts from around the web, let’s see what is special about the number eighty.

80 is…

  • how long it took Phileas Fogg to travel around the world in Jules Verne’s novel Around the World in Eighty Days.
  • 4 scores.
  • commonly used in the “80:20 rule,” which originated from Vilfredo Pareto, an Italian economist…

…And more! Click here to read the whole carnival, featuring 17 posts of mathy awesomeness!


Don’t miss any of our math tips and games:  Subscribe in a reader, or get updates by Email.


Boots - 2352

Roadmap to Mathematics: 3rd Grade

[Feature photo (above) by Phil Roeder. (CC BY 2.0 via Flickr)]

roadmap3

A frequently-asked question on homeschooling forums is, “Are my children working at grade level? What do they need to know?”

The Council of the Great City Schools has published a handy 6-page pdf summary of third grade math concepts, with suggestions for how parents can support their children’s learning:

Whether you are a radical unschooler or passionately devoted to your textbook — or, like me, somewhere in between — you can help your children toward these grade-level goals by encouraging them to view mathematics as mental play. Don’t think of the standards as a “to do” list, but as your guide to an adventure of exploration. The key to learning math is to see it the mathematician’s way, as a game of playing with ideas.

The following are excerpts from the roadmap document (along with a few extra tips) and links to related posts from the past eight years of playing with math on this blog…

What Your Child Will Learn in 3rd Grade Math

In grade three, students will continue to build their concept of numbers, developing an understanding of fractions as numbers. They will learn the concepts behind multiplication and division and apply problem-solving skills and strategies for multiplying and dividing numbers up through 100 to solve word problems. Students will also make connections between the concept of the area of a rectangle and multiplication and addition of whole numbers.

Activities in these areas will include:

  • Multiplying one-digit numbers by multiples of 10 (such as 20, 30, 40).
  • Relating the measurement of area to multiplication and division.
  • Measuring weights and volumes and solving word problems involving these measurements.
  • Representing and interpreting data.

scan_1

Tip: Encourage your children to practice their creative math skills by making up problems for you to solve. This is from my youngest daughter, Kitten Poses a Puzzle.

Helping Your Child Learn Math

  • Use everyday objects to allow your child to explore the concept of fractions. For example, use measuring cups to have students demonstrate how many 1⁄3’s are in a whole, how many 1⁄4 cups you need to make 11⁄4 cups, and how many times you have to refill a ½ cup measure to make 1½ cups.

Yay

[Photo by tanakawho. (CC BY 2.0 via Flickr)]

“The way we taught students in the past simply does not prepare them for the higher demands of college and careers today and in the future. Your school and schools throughout the country are working to improve teaching and learning to ensure that all children will graduate high school with the skills they need to be successful.

“In mathematics, this means three major changes. Teachers will concentrate on teaching a more focused set of major math concepts and skills. This will allow students time to master key math concepts and skills in a more organized way throughout the year and from one grade to the next. It will also call for teachers to use rich and challenging math content and to engage students in solving real-world problems in order to inspire greater interest in mathematics.”

Council of the Great City Schools
Parent Roadmaps to the Common Core Standards- Mathematics

Additional Resources

  • For creative ways to build a love for mathematics, follow Moebius Noodles.

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


Templeton Elementary field day beach ball games

Roadmap to Mathematics: 2nd Grade

[Feature photo (above) by Loren Kerns. (CC BY 2.0 via Flickr)]

roadmap2

A frequently-asked question on homeschooling forums is, “Are my children working at grade level? What do they need to know?”

The Council of the Great City Schools has published a handy 6-page pdf summary of second grade math concepts, with suggestions for how parents can support their children’s learning:

Whether you are a radical unschooler or passionately devoted to your textbook — or, like me, somewhere in between — you can help your children toward these grade-level goals by encouraging them to view mathematics as mental play. Don’t think of the standards as a “to do” list, but as your guide to an adventure of exploration. The key to learning math is to see it the mathematician’s way, as a game of playing with ideas.

The following are excerpts from the roadmap document (along with a couple of extra tips) and links to related posts from the past eight years of playing with math on this blog…

Continue reading

First Grade Outdoor Education

Roadmap to Mathematics: 1st Grade

[Feature photo (above) by woodleywonderworks. (CC BY 2.0 via Flickr)]

roadmap1

A frequently-asked question on homeschooling forums is, “Are my children working at grade level? What do they need to know?”

The Council of the Great City Schools has published a handy 6-page pdf summary of first grade math concepts, with suggestions for how parents can support their children’s learning:

Whether you are a radical unschooler or passionately devoted to your textbook — or, like me, somewhere in between — you can help your children toward these grade-level goals by encouraging them to view mathematics as mental play. Don’t think of the standards as a “to do” list, but as your guide to an adventure of exploration. The key to learning math is to see it the mathematician’s way, as a game of playing with ideas.

The following are excerpts from the roadmap document, along with links to related posts from the past eight years of playing with math on this blog…

Continue reading

Education Bloggers: Share Your Post!

photo by Omar Omar via flickr

photo by Omar Omar via flickr

If you are a homeschooler or classroom teacher, student or independent learner, or anyone else who writes about math, now is the time to send in your favorite blog post for next week’s Math Teachers at Play (MTaP) math education blog carnival.

Tweet: Education Bloggers: Share Your Post! Math education blog carnival open for submissions. http://ctt.ec/4zYQA+Don’t procrastinate: The deadline for entries is this Friday. The carnival will be posted next week at Triumphant Learning.

Click to tweet about the carnival:
(No spam, I promise! You will have a chance to edit or cancel the tweet.)

If you haven’t written anything about math lately, here are some ideas to get your creative juices flowing…

Need an Idea-Starter?

  • Elementary Concepts: As Liping Ma showed, there is more to understanding and teaching elementary mathematics than we often realize. Do you have a game, activity, or anecdote about teaching math to young students? Please share!
  • Arithmetic/Pre-Algebra: This section is for arithmetic lessons and number theory puzzles at the middle-school-and-beyond level. We would love to hear your favorite math club games, numerical investigations, or contest-preparation tips.
  • Beginning Algebra and Geometry: Can you explain why we never divide by zero, how to bisect an angle, or what is wrong with distributing the square in the expression \left(a + b \right)^2 ? Struggling students need your help! Share your wisdom about basic algebra and geometry topics here.
  • Advanced Math: Like most adults, I have forgotten enough math to fill several textbooks. I’m eager to learn again, but math books can be so-o-o tedious. Can you make upper-level math topics come alive, so they will stick in my (or a student’s) mind?
  • Mathematical Recreations: What kind of math do you do, just for the fun of it?
  • About Teaching Math: Other teachers’ blogs are an important factor in my continuing education. The more I read about the theory and practice of teaching math, the more I realize how much I have yet to learn. So please, fellow teachers, don’t be shy — share your insights!

Would You Like to Host the Carnival?

We need more hosts! Hosting the blog carnival can be a lot of work, but it’s fun to “meet” new bloggers through their submissions. And there’s a side-benefit: The carnival usually brings a nice little spike in traffic to your blog.

If you think you’d like to join in the fun, read the instructions on our Math Teachers at Play page. Then leave a comment or email me to let me know which month you’d like to take.

Explore the Other Math Carnivals

While you’re waiting for next week’s Math Teachers at Play carnival, you may enjoy:


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


SAKURAKO - Kindergarten Sports Festival

Roadmap to Mathematics: Kindergarten

[Feature photo (above) by MIKI Yoshihito. (CC BY 2.0 via Flickr)]

RoadmapK

A frequently-asked question on homeschooling forums is, “Are my children working at grade level? What do they need to know?”

The Council of the Great City Schools has published a handy 6-page pdf summary of kindergarten math concepts, with suggestions for how parents can support their children’s learning:

Whether you are a radical unschooler or passionately devoted to your textbook — or, like me, somewhere in between — you can help your children toward these grade-level goals by encouraging them to view mathematics as mental play. Don’t think of the standards as a “to do” list, but as your guide to an adventure of exploration. The key to learning math is to see it the mathematician’s way, as a game of playing with ideas.

The following are excerpts from the roadmap document, along with links to related posts from the past eight years of playing with math on this blog…

Continue reading

factor game

Math Teachers at Play #79

79

[Feature photo above by Jimmie, and “79” image (right) by Steve Bowbrick via flickr (CC BY 2.0).]

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

Welcome to the 79th edition of the Math Teachers At Play (MTaP) 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.

Let the mathematical fun begin!

By tradition, we start the carnival with a puzzle, game, or trivia tidbits. If you would like to jump straight to our featured blog posts, click here to see the Table of Contents.

Since I’ve been spending all my free time working on my upcoming Math You Can Play book series, I’m in the mood for games. So I found a few games featuring prime and nonprime numbers [which category is #79 — do you know?], and I’ll sprinkle some of my best-loved math game books throughout the carnival.

TRY THESE NUMBER GAMES

Students can explore prime and non-prime numbers with two free classroom favorites: The Factor Game (pdf lesson download) or Tax Collector. For $15-20 you can buy a downloadable file of the beautiful, colorful, mathematical board game Prime Climb. Or try the following game by retired Canadian math professor Jerry Ameis:

Factor Finding Game

FactorFindingGame

Math Concepts: multiples, factors, composites, and primes.
Players: only two.
Equipment: pair of 6-sided dice, 10 squares each of two different colors construction paper, and the game board (click the image to print it, or copy by hand).

On your turn, roll the dice and make a 2-digit number. Use one of your colored squares to mark a position on the game board. You can only mark one square per turn.

  • If your 2-digit number is prime, cover a PRIME square.
  • If any of the numbers showing are factors of your 2-digit number, cover one of them.
  • BUT if there’s no square available that matches your number, you lose your turn.

The first player to get three squares in a row (horizontal/vertical/diagonal) wins. Or for a harder challenge, try for four in a row.

Hat tips: Jimmie Lanley.



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.

Tweet: Math Teachers at Play #76: a smorgasbord of great ideas for learning, teaching, and playing around with math. http://ctt.ec/fU9Z2+

Click to tweet: Share the carnival with your friends.
(No spam, I promise! You will have a chance to edit or cancel the tweet.)

Continue reading

640px-Horseshoes_game

Horseshoes: A Place Value Game

[Feature photo above by Johnmack161 via Wikimedia Commons (CC BY 2.5).]

I first saw place value games on the old PBS Square One TV show (video below). Many teachers have posted versions of the game online, but Snugglenumber by Anna Weltman is by far the cutest variation. Anna kindly gave me permission to use the game in my upcoming Math You Can Play book series, and I added the following variation:

Horseshoes

snugglenumber

Math Concepts: place value, strategic thinking.
Players: two or more.
Equipment: one deck of playing cards, or a double deck for more than three players.

Separate out the cards numbered ace (one) through nine, plus cards to represent the digit zero. We use the queens (Q is round enough for pretend), but you could also use the tens and just count them as zero.

Shuffle well and deal eleven cards to each player. Arrange your cards in the snugglenumber pattern shown here, one card per blank line, to form numbers that come as close to each target number as you can get it.

Score according to horseshoes rules:

  • Three points for each ringer, or exact hit on the target.
  • One point for each number that is six or less away from the target.
  • If none of the players land in the scoring range for a target number, then score one point for the number closest to that target.

For a quick game, whoever scores the most points wins. Or follow tradition and play additional rounds until one player gets 21 points (40 for championship games) — and you have to win by at least two points over your closest opponent’s score.

But Who’s Counting?


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Math Teachers and Homeschool Bloggers: We Want You!

chica usando ordenador

[Photo by Olga Berrios via flickr.]

Do you have a favorite blog post about math activities, games, lessons, or hands-on fun? The Math Teachers at Play (MTaP) math education blog carnival would love to feature your article!

We welcome math topics from preschool through the first year of calculus. Old posts are welcome, as long as they haven’t been published in past editions of this carnival.

Don’t procrastinate: The deadline for entries is Monday, October 20. The carnival will be posted soon afterwards at — well, we don’t have a host for this month yet. Would you like to volunteer?

Would You Like to Host the Carnival?

Hosting the blog carnival is fun because you get to “meet” new bloggers through their submissions. And there’s a side-benefit: The carnival often brings a nice little spike in traffic to your blog. If you think you’d like to join in the fun, read the instructions on our Math Teachers at Play page. Then leave a comment or email me to let me know which month you’d like to take.

Explore the Other Math Carnivals

While you’re waiting for next week’s Math Teachers at Play carnival, you may enjoy:

MathART Projects


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


Teddy Bear Hospital

Math Storytelling Day: The Hospital Floor

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

Have you ever heard of Math Storytelling Day? On September 25, people around the world celebrate mathematics by telling stories together. The stories can be real — like my story below — or fictional like the tale of Wizard Mathys from Fantasia and his crystal ball communication system.

Check out these posts for more information:

My Math Story

tiles2

My story begins with an unexpected adventure in pain. Appendicitis sidewhacked my life last week, but that’s not the story. It’s just the setting. During my recovery, I spent a lot of time in the smaller room of my hospital suite. I noticed this semi-random pattern in the floor tile, which made me wonder:

  • Did they choose the pattern to keep their customers from getting bored while they were … occupied?
  • Is the randomness real? Or can I find a line of symmetry or a set of tiles that repeat?
  • If I take pictures from enough different angles, could I transfer the whole floor to graph paper for further study?
  • And if the nurse finds me doing this, will she send me to a different ward of the hospital? Do hospitals have psychiatric wards, or is that only in the movies?
  • What is the biggest chunk of squares I could “break out” from this pattern that would create the illusion of a regular, repeating tessellation?

I gave up on the graph paper idea (for now) and printed the pictures to play with. By my definition, “broken” pattern chunks need to be contiguous along the sides of the tiles, like pentominoes. Also, the edge of the chunk must be a clean break along the mortar lines. The piece can zigzag all over the place, but it isn’t allowed to come back and touch itself anywhere, even at a corner. No holes allowed.

I’m counting the plain squares as the unit and each of the smaller rectangles as a half square. So far, the biggest chunk of repeating tiles I’ve managed to break out is 283 squares.

tiles1

What Math Stories Will You Tell?

Have you and your children created any mathematical stories this year? I’d love to hear them! Please share your links in the comments section below.


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


Math Teachers at Play #78 via 1001 Math Problems

blog carnival 78 graphic

Math Teachers at Play is a traveling collection of math tidbits — games, lesson ideas, and more — from around the Internet. It moves around from month to month, and the September edition is now posted at 1001 Math Problems blog. What a fun list of math posts to browse!

Welcome to the 78th edition of the Math Teachers At Play math education blog carnival, which I am thrilled to be hosting this month in celebration of my soon-to-be-released book, Camp Logic. What is the blog carnival? It is a monthly snapshot of some interesting recent ideas and activities posted by math education bloggers all over the internet.

By tradition, I begin with a fun fact about the number 78:

Seventy-eight is the 12th triangle number, which means that it is the sum of the integers from 1 to 12. Therefore, it is also the total number of gifts given on the last day in the song “The Twelve Days of Christmas!” …

Click here to go read the whole carnival.


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


Calling All Math Teacher Bloggers and Homeschoolers: Carnival Time!

by Bob Jagendorf via flickr

by Bob Jagendorf via flickr

The monthly Math Teachers at Play (MTaP) math education blog carnival is almost here. If you’ve written a blog post about math, we’d love to have you join us! Each of us can help others learn, so in a sense we are all teachers.

Posts must be relevant to students or teachers of school-level mathematics (that is, anything from preschool up to first-year calculus). Old posts are welcome, as long as they haven’t been published in past editions of this carnival.

Don’t procrastinate: The deadline for entries is this Friday. The carnival will be posted next week at 1001 Math Problems blog.

We Need More Hosts to Fill Out this School Year

Help! I can’t keep the carnival going on my own. Would you volunteer to host the MTaP math education blog carnival some month this year? Hosting the carnival can be a lot of work, but it’s fun to “meet” new bloggers through their submissions. And there’s a side-benefit: The carnival usually brings a nice little spike in traffic to your blog.

If you think you’d like to join in the fun, read the instructions on our Math Teachers at Play page. Then leave a comment or email me to let me know which month you’d like to take.

Continue reading

Quotable: I Do Hate Sums

MrsLaTouche

I’ve been looking for quotes to put at the beginning of each chapter in my math games books. I found a delightful one by Mrs. LaTouche on the Mathematical Quotations Server, but when I looked up the original source, it was even better:

I am nearly driven wild with the Dorcas accounts, and by Mrs. Wakefield’s orders they are to be done now.

I do hate sums. There is no greater mistake than to call arithmetic an exact science. There are Permutations and Aberrations discernible to minds entirely noble like mine; subtle variations which ordinary accountants fail to discover; hidden laws of Number which it requires a mind like mine to perceive.

For instance, if you add a sum from the bottom up, and then again from the top down, the result is always different.

Again if you multiply a number by another number before you have had your tea, and then again after, the product will be different. It is also remarkable that the Post-tea product is more likely to agree with other people’s calculations than the Pre-tea result.

Try the experiment, and if you do not find it as I say, you are a mere sciolist*, a poor mechanical thinker, and not gifted as I am, with subtle perceptions.

Of course I find myself not appreciated as an accountant. Mrs. Wakefield made me give up the book to [my daughter] Rose and her governess (who are here), and was quite satisfied with the work of those inferior intellects.

— Maria Price La Touche
The Letters of A Noble Woman
London: George Allen & Sons, 1908

*sciolist: (archaic) A person who pretends to be knowledgeable and well informed. From late Latin sciolus (diminutive of Latin scius ‘knowing’, from scire ‘know’) + -ist.


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Capture

Math Teachers at Play # 77 via Math = Love

MTaP77

Math Teachers at Play is a traveling collection of math tidbits — games, lesson ideas, and more — from around the Internet. It moves around from month to month, and the August edition is now posted at Math = Love blog. What a fun list of math posts to browse!

Welcome to the 77th edition of the Math Teachers at Play Blog Carnival! I’m super excited to be hosting this carnival because I’ve been reading it for years! Yes, I am that crazy person who started reading math teacher blogs as a high school junior. I think you are going to enjoy going through the submissions. I know I found several new-to-me blogs to add to my RSS reader! …

Click here to go read the whole carnival.


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


Do You Blog About Math?

by Mike Licht, NotionsCapital.com

It’s carnival time again. Activities, games, lessons, hands-on fun — if you’ve written a blog post about math, we’d love to have you join our Math Teachers at Play (MTaP) math education blog carnival.

Posts must be relevant to students or teachers of school-level mathematics (that is, anything from preschool up through first-year calculus). Old posts are welcome, as long as they haven’t been published in past editions of this carnival.

Don’t procrastinate: The deadline for entries is this Friday. The carnival will be posted next week at Math = Love.

Would You Like to Host the Carnival?

Hosting the blog carnival can be a lot of work, but it’s fun to “meet” new bloggers through their submissions. And there’s a side-benefit: The carnival usually brings a nice little spike in traffic to your blog. If you think you’d like to join in the fun, read the instructions on our Math Teachers at Play page. Then leave a comment or email me to let me know which month you’d like to take.

Explore the Other Math Carnivals

While you’re waiting for next week’s Math Teachers at Play carnival, you may enjoy:

Sawyer--Vision in Elem Maths


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


dividing sausages

Fractions: 1/5 = 1/10 = 1/80 = 1?

[Feature photo is a screen shot from the video “the sausages sharing episode,” see below.]

Fractions: 1/5 = 1/10 = 1/80 = 1?

How in the world can 1/5 be the same as 1/10? Or 1/80 be the same as one whole thing? Such nonsense!

No, not nonsense. This is real-world common sense from a couple of boys faced with a problem just inside the edge of their ability — a problem that stretches them, but that they successfully solve, with a bit of gentle help on vocabulary.

Here’s the problem:

  • How can you divide eight sausages evenly among five people?

Think for a moment about how you (or your child) might solve this puzzle, and then watch the video below.

What Do You Notice?

Continue reading

playing cards

Fraction Game: My Closest Neighbor

[Feature photo above by Jim Larrison, and antique playing cards below by Marcee Duggar, via Flickr (CC BY 2.0).]

I missed out on the adventures at Twitter Math Camp, but I’m having a great time working through the blog posts about it. I prefer it this way — slow reading is more my speed. Chris at A Sea of Math posted a wonderful game based on one of the TMC workshops. Here is my variation.

Math concepts: comparing fractions, equivalent fractions, benchmark numbers, strategic thinking.

Players: two to four.

Equipment: two players need one deck of math cards, three or four players need a double deck.

How to Play

three-eighths-of-clubs

Deal five cards to each player. Set the remainder of the deck face down in the middle of the table as a draw pile.

You will play six rounds:

  • Closest to zero
  • Closest to 1/4
  • Closest to 1/3
  • Closest to 1/2
  • Closest to one
  • Closest to two

In each round, players choose two cards from their hand to make a fraction that is as close as possible (but not equal) to the target number. Draw two cards to replenish your hand.

The player whose fraction is closest to the target collects all the cards played in that round. If there is a tie for closest fraction, the winners split the cards as evenly as they can, leaving any remaining cards on the table as a bonus for the winner of the next round.

After the last round, whoever has collected the most cards wins the game.


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


You may enjoy these related posts:

StudentTeam

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.

PUZZLE: CRYSTAL BALL CONNECTION PATTERNS

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.

TheWizardBySeanMcGrath-small

  • 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)]


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:

Continue reading

300_stars_in_orbit_1

Playing With Math — the Book

body_Book_cover_for_upload

Update: The crowdfunding campaign is now closed and the book is in the final stages. It should be headed to the printer soon. Check the Playing With Math homepage for publication and ordering information.


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

Continue reading

Education Bloggers: Share Your Post!

photo by Omar Omar via flickr

photo by Omar Omar via flickr

If you are a homeschooler or classroom teacher, student or independent learner, or anyone else who writes about math, now is the time to send in your favorite blog post for next week’s Math Teachers at Play (MTaP) math education blog carnival.

Don’t procrastinate: The deadline for entries is this Friday extended through the weekend. The carnival will be posted next week at Let’s Play Math.

If you haven’t written anything about math lately, here are some ideas to get your creative juices flowing…

Need an Idea-Starter?

  • Elementary Concepts: As Liping Ma showed, there is more to understanding and teaching elementary mathematics than we often realize. Do you have a game, activity, or anecdote about teaching math to young students? Please share!
  • Arithmetic/Pre-Algebra: This section is for arithmetic lessons and number theory puzzles at the middle-school-and-beyond level. We would love to hear your favorite math club games, numerical investigations, or contest-preparation tips.
  • Beginning Algebra and Geometry: Can you explain why we never divide by zero, how to bisect an angle, or what is wrong with distributing the square in the expression \left(a + b \right)^2 ? Struggling students need your help! Share your wisdom about basic algebra and geometry topics here.
  • Advanced Math: Like most adults, I have forgotten enough math to fill several textbooks. I’m eager to learn again, but math books can be so-o-o tedious. Can you make upper-level math topics come alive, so they will stick in my (or a student’s) mind?
  • Mathematical Recreations: What kind of math do you do, just for the fun of it?
  • About Teaching Math: Other teachers’ blogs are an important factor in my continuing education. The more I read about the theory and practice of teaching math, the more I realize how much I have yet to learn. So please, fellow teachers, don’t be shy — share your insights!

Would You Like to Host the Carnival?

We need more hosts! Hosting the blog carnival can be a lot of work, but it’s fun to “meet” new bloggers through their submissions. And there’s a side-benefit: The carnival usually brings a nice little spike in traffic to your blog.

If you think you’d like to join in the fun, read the instructions on our Math Teachers at Play page. Then leave a comment or email me to let me know which month you’d like to take.

Explore the Other Math Carnivals

While you’re waiting for next week’s Math Teachers at Play carnival, you may enjoy:


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


JoBoalerProportionalRelationships

Quotable: Math Connections

ConnectedGearsJoBoaler

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 YouCubed.org may help you recover your joy in learning.


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Sconic Sections

Math(s) Teachers At Play #75 via CavMaths

[Feature photo above “Sconic Sections” by Lenore Edman and “75” by R/DV/RS via Flickr (CC BY 2.0).]
75

The monthly math education blog carnival Math Teachers at Play features games, lessons, puzzles, activities, and teaching tips from classroom teachers, homeschoolers, and self-educated learners around the Internet world. Check out the 20 posts of mathematical fun in the June edition:

Math(s) Teachers At Play #75 via CavMaths

Hello, and welcome to the 75th issue of the Math(s) Teachers at Play Blog Carvinal! For those of you who are unaware, a “blog carnival” is a periodic post that travels from blog to blog and has a collection of posts on a certain topic.

This is the first time I’ve hosted a carnival and there were some excellent submissions. I enjoyed reading them all and have discovered some new blogs. I have also input some posts I’ve seen this month which I thought were excellent too…

Click here to go read Math(s) Teachers At Play #75 via CavMaths.


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body_maria_with_kids_1200x971

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

body_Book_cover_for_upload

Update: The crowdfunding campaign is now closed and the book is in the final stages. It should be headed to the printer soon. Check the Playing With Math homepage for publication and ordering information.


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 PlayingWithMath.org:

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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Math Teachers and Homeschool Bloggers: We Want You!

chica usando ordenador

[Photo by Olga Berrios via flickr.]

Do you have a favorite blog post about math activities, games, lessons, or hands-on fun? The Math Teachers at Play (MTaP) math education blog carnival would love to feature your article!

We welcome math topics from preschool through the first year of calculus. Old posts are welcome, as long as they haven’t been published in past editions of this carnival.

Don’t procrastinate: The deadline for entries is this Friday, June 20. The carnival will be posted next week at CavMaths blog.

Would You Like to Host the Carnival?

Hosting the blog carnival is fun because you get to “meet” new bloggers through their submissions. And there’s a side-benefit: The carnival often brings a nice little spike in traffic to your blog. If you think you’d like to join in the fun, read the instructions on our Math Teachers at Play page. Then leave a comment or email me to let me know which month you’d like to take.

Explore the Other Math Carnivals

While you’re waiting for next week’s Math Teachers at Play carnival, you may enjoy:

MathART Projects


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


calculus

Reblog: Calculus Tidbits

[Feature photo above by Olga Lednichenko via Flickr (CC BY 2.0).]

This week I have a series of quotes about calculus from my first two years of blogging. The posts were so short that I won’t bother to link you back to them, but math humor keeps well over the years, and W. W. Sawyer is (as always) insightful.

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


Finding the Limit

Eldest daughter had her first calculus lesson last night: finding the limit as delta-t approached zero. The teacher found the speed of a car at a given point by using the distance function, calculating the average speed over shorter and shorter time intervals. Dd summarized the lesson for me:

“If you want to divide by zero, you have to sneak up on it from behind.”


Harmonic Series Quotation

This kicked off my week with a laugh:

Today I said to the calculus students, “I know, you’re looking at this series and you don’t see what I’m warning you about. You look and it and you think, ‘I trust this series. I would take candy from this series. I would get in a car with this series.’ But I’m going to warn you, this series is out to get you. Always remember: The harmonic series diverges. Never forget it.”

—Rudbeckia Hirta
Learning Curves Blog: The Harmonic Series
quoting Alexandre Borovik


So You Think You Know Calculus?

Rudbeckia Hirta has a great idea for a new TV blockbuster:


Common Sense and Calculus

Sawyer-MathDelight

And here’s a quick quote from W. W. Sawyer’s Mathematician’s Delight:

If you cannot see what the exact speed is, begin to ask questions. Silly ones are the best to begin with. Is the speed a million miles an hour? Or one inch a century? Somewhere between these limits. Good. We now know something about the speed. Begin to bring the limits in, and see how close together they can be brought.

Study your own methods of thought. How do you know that the speed is less than a million miles an hour? What method, in fact, are you unconsciously using to estimate speed? Can this method be applied to get closer estimates?

You know what speed is. You would not believe a man who claimed to walk at 5 miles an hour, but took 3 hours to walk 6 miles. You have only to apply the same common sense to stones rolling down hillsides, and the calculus is at your command.


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6260501380_7dbe7a7aed_z

Reblog: Patty Paper Trisection

[Feature photo above by Michael Cory via Flickr (CC BY 2.0).]

trisection2

I hear so many people say they hated geometry because of the proofs, but I’ve always loved a challenging puzzle. I found the following puzzle at a blog carnival during my first year of blogging. Don’t worry about the arbitrary two-column format you learned in high school — just think about what is true and how you know it must be so.

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


trisection

One of the great unsolved problems of antiquity was to trisect any angle using only the basic tools of Euclidean geometry: an unmarked straight-edge and a compass. Like the alchemist’s dream of turning lead into gold, this proved to be an impossible task. If you want to trisect an angle, you have to “cheat.” A straight-edge and compass can’t do it. You have to use some sort of crutch, just as an alchemist would have to use a particle accelerator or something.

One “cheat” that works is to fold your paper. I will show you how it works, and your job is to show why …

[Click here to go read Puzzle: Patty Paper Trisection.]



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old warrior dragon

Reblog: Solving Complex Story Problems

[Dragon photo above by monkeywingand treasure chest by Tom Praison via flickr.]

Dealing with Dragons

Over the years, some of my favorite blog posts have been the Word Problems from Literature, where I make up a story problem set in the world of one of our family’s favorite books and then show how to solve it with bar model diagrams. The following was my first bar diagram post, and I spent an inordinate amount of time trying to decide whether “one fourth was” or “one fourth were.” I’m still not sure I chose right.

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


Solving-Complex-Story-Problems

Cimorene spent an afternoon cleaning and organizing the dragon’s treasure. One fourth of the items she sorted was jewelry. 60% of the remainder were potions, and the rest were magic swords. If there were 48 magic swords, how many pieces of treasure did she sort in all?

[Problem set in the world of Patricia Wrede’s Enchanted Forest Chronicles. Modified from a story problem in Singapore Primary Math 6B. Think about how you would solve it before reading further.]

How can we teach our students to solve complex, multi-step story problems? Depending on how one counts, the above problem would take four or five steps to solve, and it is relatively easy for a Singapore math word problem. One might approach it with algebra, writing an equation like:

x - \left[\frac{1}{4}x + 0.6\left(\frac{3}{4} \right)x  \right]  = 48

… or something of that sort. But this problem is for students who have not learned algebra yet. Instead, Singapore math teaches students to draw pictures (called bar models or math models or bar diagrams) that make the solution appear almost like magic. It is a trick well worth learning, no matter what math program you use …

[Click here to go read Solving Complex Story Problems.]



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mtap74

Math Teachers at Play #74 via Triumphant Learning

74 by Stephan Mosel

photo by Stephan Mosel (CC BY 2.0)

The new Math Teachers at Play math education blog carnival is up for your browsing pleasure. Each month, we feature activities, lessons, and games about math topics from preschool through high school. Check it out!

Here’s a peek at a few of the entries:

Origami
Learn how to make Origami Stars, Tessellation Stars, and Chaotic Stars at Math Munch. I think once your students or children see this, you will find Transforming Ninja Stars littering your house and classroom!

Pi
Here’s a fun activity to explore other ways to get the number Pi on the calculator from William Wu at Singapore Maths Tuition.

Math Games
Math Hombre shares a coordinate grid game that also calculates area of rectangles. And all you need is some grid paper and dice.

…And much more!

Click here to go read the entire blog carnival.

Would You Like to Host the Carnival?

Hosting the blog carnival can be a lot of work, but it’s fun to “meet” new bloggers through their submissions. And there’s a side-benefit: The carnival usually brings a nice little spike in traffic to your blog. If you think you’d like to join in the fun, read the instructions on our Math Teachers at Play page. Then leave a comment or email me to let me know which month you’d like to take.


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


Basel spiral

More Than One Way To Find the Center of a Circle

[Feature photo above by hom26 via Flickr.]

My free time lately has gone to local events and to book editing. I hope to put up a series of blog posts sometime soon, based on the Homeschool Math FAQs chapter I’m adding to the paperback version of Let’s Play Math. [And of course, I’ll update the ebook whenever I finally publish the paperback, so those of you who already bought a copy should be able to get the new version without paying extra.]

But in the meantime, as I was browsing my blog archives for an interesting “Throw-Back Thursday” post, I stumbled across this old geometry puzzle from Dave Marain over at MathNotations blog:

Is it possible that AB is a chord but NOT a diameter? That is, could circle ABC have a center that is NOT point O?

Jake shows Jack a piece of wood he cut out in the machine shop: a circular arc bounded by a chord. Jake claimed that the arc was not a semicircle. In fact, he claimed it was shorter than a semicircle, i.e., segment AB was not a diameter and arc ACB was less than 180 degrees.

Jack knew this was impossible and argued: “Don’t you see, Jake, that O must be the center of the circle and that OA, OB and OC are radii.”

Jake wasn’t buying this, since he had measured everything precisely. He argued that just because they could be radii didn’t prove they had to be.

Which boy do you agree with?

  • Pick one side of the debate, and try to find at least three different ways to prove your point.

If you have a student in geometry or higher math, print out the original post (but not the comments — it’s no fun when someone gives you the answer!) and see what he or she can do with it.

Dave offers many other puzzles to challenge your math students. While you are at his blog, do take some time to browse past articles.


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Related posts on Let’s Play Math! blog: