Playing with Math Shapes

Playing-with-shapesI love it when a plan — or rather, a series of math thoughts — comes together.

On Monday, Emily Grosvenor (author of the Tessalation! picture book) asked me how parents who are insecure in math could help their children learn through play, and I responded with this quote from my Let’s Play Math book:

If you are intimidated by numbers, you can look for patterns of shape and color. Pay attention to how they grow. Talk about what your children notice.

But I wasn’t entirely satisfied with that answer. So many adults have come away from their own school experience thinking math is only numbers. Even with shapes, isn’t it the numbers about them — how many sides, what size of angles, calculate the the area or perimeter — that are important? That’s what school math tends to focus on.

Those of us who are comfortable with math know that there are many more things to notice and think about than just numbers. We know that it’s this noticing, thinking, and wondering that is at the heart of math. And that just playing with shapes can build a powerful foundation for future math learning.

And then yesterday, Malke Rosenfeld posted a beautiful article about a paper manipulative created by Paula Krieg. Which included this video:

The ability to create, and maintain, and manipulate shapes mentally — that’s the goal. Just like kids who can put numbers together in their heads, kids who can rotate, flip, and think of how shapes fit together in their heads have a powerful tool to analyze not only simple shape puzzles, but dividing up an area that’s a more complex room shape … to look at a piece of artwork … or look at a building … For these kids, all the world around becomes a playground to use mathematical ideas.

— Doug Clements
Problem Solving Development: Composing Shapes

Of course, pattern blocks are good for much more than just filling in worksheet pictures. But I love this peek into how a child’s understanding grows, in bits and spurts — without any numbers at all — until the world itself becomes a playground for mathematical ideas.

Want more?

You know what? Children like mathematics. Children see the world mathematically … When we do a puzzle, when we count things, when we see who’s got more, or who’s taller … Play and mathematics are not on opposite sides of the stage.

— Doug Clements
Why Early Childhood is the Right Time to Start Learning Math


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Hotel Infinity: Part Five

Hotel Infinity1Tova Brown concludes her exploration of the Hilbert’s Hotel Paradox with a look at the cardinality of the real numbers.

You run a hotel with an infinite number of rooms. You pride yourself on accommodating everyone, even guests arriving in infinitely large groups — but some infinities are more infinite than others, as it turns out.

Tova Brown
Hotel Infinity: Part Five

Check out Tova Brown’s growing collection of videos that explore advanced math concepts through story-telling.


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Hotel Infinity: Part Four

Hotel Infinity1Tova Brown dives deeper into Hilbert’s Hotel Paradox, considering the difference between rational numbers and reals.

You run an infinitely large hotel, and are happy to realize that you can accommodate an infinite number of infinite groups of guests.

However, a delicate diplomatic situation arises when a portal to another universe opens, introducing a different kind of guest, in a different kind of group.

Can you make room for them all?

Tova Brown
Hotel Infinity: Part Four

Click here to read Part Five…


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Hotel Infinity: Part Three

Hotel Infinity1Tova Brown continues to examine Hilbert’s Hotel Paradox, pondering infinite sets of infinite sets.

As the proprietor of an infinitely large hotel, you pride yourself on welcoming everyone, even when the rooms are full. Your hotel becomes very popular among infinite sports teams, as a result.

Recruitment season presents a challenge, however, when many infinite teams arrive at once. How many infinite teams can stay in a single infinite hotel?

Tova Brown
Hotel Infinity: Part Three

Click here to read Part Four…


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Hotel Infinity: Part Two

Hotel Infinity1Tova Brown explores the second part of Hilbert’s Hotel Paradox. What’s infinity plus infinity?

Running an infinite hotel has its perks. Even when the rooms are full you can always find space for new guests, so you proudly welcome everyone who appears at your door.

When two guests arrive at once, you make room. When ten guests arrive, you accommodate them easily. When a crowd of hundreds appears, you welcome them all.

Is there no limit to your hospitality?

Tova Brown
Hotel Infinity: Part Two

Click here to read Part Three…


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Hotel Infinity: Part One

Hotel Infinity1Tova Brown’s introduction to Hilbert’s Hotel Paradox, a riddle about the nature of infinity…

Once upon a time, there was a hotel with an infinite number of rooms. You might be thinking this is impossible, and if so you’re right. A hotel like this could never exist in the real world.

But fortunately we’re not talking about the real world, we’re talking about math. And when we do math we can make up whatever rules we want, just to see what happens.

Tova Brown
Hotel Infinity: Part One

Click here to read Part Two…


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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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