Playful Math Carnival #142: Math Art Edition

Posted on by Denise Gaskins

Welcome to the 142nd edition of the Playful Math Education Blog Carnival — a smorgasbord of delectable tidbits of mathy fun. It’s like a free online magazine devoted to learning, teaching, and playing around with math from preschool to high school.

Bookmark this post, so you can take your time browsing.

Seriously, plan on coming back to this post several times. There’s so much playful math to enjoy!

By tradition, we start the carnival with a puzzle/activity in honor of our 142nd edition. But if you’d rather jump straight to our featured blog posts, click here to see the Table of Contents.

Activity: Planar Graphs

According to the OEIS Wiki, 142 is “the number of planar graphs with six vertices.”

What does that mean?

And how can our students play with it?

A planar graph is a set of vertices connected (or not) by edges. Each edge links two vertices, and the edges cannot intersect each other. The graph doesn’t have to be fully connected, and individual vertices may float free.

Children can model planar graphs with three-dimensional constructions using small balls of playdough (vertices) connected by toothpicks (edges).

Let’s start with something smaller than 142. If you roll four balls of playdough, how many different ways can you connect them? The picture shows five possibilities. How many more can you find?

Sort your planar graphs into categories. How are they similar? How are they different?

A wise mathematician once said, “Learning is having new questions to ask.” How many different questions can you think of to ask about planar graphs?

Play the Planarity game to untangle connected planar graphs (or check your phone store for a similar app).

Or play Sprouts, a pencil-and-paper planar-graph game.

For deeper study, elementary and middle-school students will enjoy Joel David Hamkins’s Graph coloring & chromatic numbers and Graph theory for kids. Older students can dive into Oscar Levin’s Discrete Mathematics: An Open Introduction. Here’s the section on planar graphs.

[“Geöffneter Berg” by Paul Klee, 1914.]

Click here for all the mathy goodness!

FAQ: I’ve Ruined My Daughter

Posted on by Denise Gaskins

My daughter is only eleven, but I’m afraid I’ve ruined her chance of getting into college because she is so far behind in math. We’ve tried tutors, but she still has trouble, and standardized testing puts her three years below grade level. She was a late reader, too, so maybe school just isn’t her thing. What else can I do?

Standardized tests are not placement tests. They cannot tell you at what level your daughter should be studying. They aren’t designed that way. The “placement” they give is vague and general, not indicative of her grade level but rather a way of comparing her performance on that particular test with the performance of other students.

There can be many different reasons for a low score. I’ve listed a few of them in my post In Honor of the Standardized Testing Season.

Continue reading FAQ: I’ve Ruined My Daughter

To Badger a Child

Posted on by Denise Gaskins

Here’s the full quote:

Audrey seemed, for once, at a loss for words. She was thinking about the question.

I try to stay focused on being silent after I ask young children questions, even semi-serious accidental ones. Unlike most adults, they actually take time to think about their answers and that often means waiting for a response, at least if you want an honest answer.

If you’re only looking for the “right” answer, it’s fairly easy to gently badger a child into it, but I’m not interested in doing that.

—Thomas Hobson
Thank You For Teaching Me

CREDITS: “Pismo Beach, United States” photo by Tim Mossholder on Unsplash.

Math for Star Wars Day

Posted on by Denise Gaskins

May the Fourth be with you!

Here is a math problem in honor of one of our family’s favorite movies…

Han Solo was doing much-needed maintenance on the Millennium Falcon. He spent 3/5 of his money upgrading the hyperspace motivator. He spent 3/4 of the remainder to install a new blaster cannon. If he spent 450 credits altogether, how much money did he have left?

Stop and think about how you would solve it before reading further.

Continue reading Math for Star Wars Day

How to Homeschool Math

Posted on by Denise Gaskins

Far too many people find themselves suddenly, unexpectedly homeschooling their children. This prompts me to consider what advice I might offer after more than three decades of teaching kids at home.

Through my decades of homeschooling five kids, we lived by two rules:

Do math. Do reading.

As long as we hit those two topics each day, I knew the kids would be fine. Do some sort of mathematical game or activity. Read something from that big stack of books we collected at the library.

Conquer the basics of math and reading, then everything else will fall into place.

Continue reading How to Homeschool Math

Morning Coffee – 4 November 2019

Posted on by Denise Gaskins

Morning Coffee image

One of the best ways we can help our children learn mathematics (or anything else) is to always be learning ourselves.

Here are a few stories to read with your Monday morning coffee:

“Games aren’t just about practice and fluency. My favorite games create opportunities for learning, too. They spark discourse, promote the use of strategies, and allow students to dig into the mathematics.”

—Jenna Laib
The Simple-but-High-Leverage Game Collection: Making Games Routine

  • Have you read Pam Harris’s Development of Mathematical Reasoning series? Check out Part 1, Part 2, and Part 3. Well worth your time!

“The teacher’s role is to help students change the way they think, in increasingly sophisticated ways. The goal is not answers. The goal is development. We don’t need students who can just answer a multiplication question, we need students who can reason multiplicatively.”

—Pam Harris
The Development of Mathematical Reasoning

CREDITS: Feature photo (top) by Kira auf der Heide via Unsplash. “Morning Coffee” post format inspired by Nate Hoffelder at The Digital Reader.

Morning Coffee – 28 October 2019

Posted on by Denise Gaskins

Morning Coffee image

One of the best ways we can help our children learn mathematics (or anything else) is to always be learning ourselves.

Here are a few stories to enjoy with your Monday morning coffee:

“When you’re working every day, you’re not thinking, ‘What impact is this going to have on the world?’ You’re thinking, ‘I’ve got to get this right.’”

—Gladys West
quoted in Dr. Gladys West: The Black Woman Behind GPS Technology

  • I like to keep a quick game in reserve for spare time in my homeschool co-op class. Kent Haines explains Sprouts and suggests ways to launch math discussions.

“I don’t get irritated by these mistakes. I desperately wait for such mistakes. Yes! Because I think it is a golden opportunity for the teacher to spot a student thinking this way. It presents just the right context and time for driving an enriching mathematical conversation in the whole class.”

—Rupesh Gesota
Part-2: Re-learning and Enjoying Polynomial Division with students

“To teach students SSS congruence without pointing out why this is so interesting is harmful for two reasons. First of all, this is an amazing result. It is the our job to point out amazing results! Triangles are rigid figures in a way that other polygons are not.”

—Rachel Chou
Teaching the Distributive Property

CREDITS: Feature photo (top) by Kira auf der Heide via Unsplash. “Morning Coffee” post format inspired by Nate Hoffelder at The Digital Reader.

Morning Coffee – 4 October 2019

Posted on by Denise Gaskins

Morning Coffee image

One of the best ways we can help our children learn mathematics (or anything else) is to always be learning ourselves.

Here are a few stories to read with your Friday morning coffee:

  • In the spirit of cracking eggs to make omelets, Michael Pershan cracks open some of the ideas around Equations and Equivalence and relational thinking.

“My experience is that when I have vague hope that children will learn something from an activity that is related to the mathematics I want them to learn, they usually don’t.”

—David Wees
Hands on or minds on?

CREDITS: Feature photo (top) by Kira auf der Heide via Unsplash. “Morning Coffee” post format inspired by Nate Hoffelder at The Digital Reader.

Morning Coffee – 30 September 2019

Posted on by Denise Gaskins

Morning Coffee image

One of the best ways we can help our children learn mathematics (or anything else) is to always be learning ourselves.

Here are a few stories to read with your Monday morning coffee:

  • Kathy Iwanicki’s students explore an unsolved math puzzle: the Hailstone Sequence. I think my co-op class might enjoy this activity, too.
  • If you want to understand how number concepts develop through the school years, Graham Fletcher’s Progression Videos are a great place to start. Watch, re-watch, and learn.

“Learning occurs when we get something wrong and have to correct it. This is analogous to the much better known fact that when we subject our bodies to physical strain, say by walking, jogging, or lifting weights, the muscles we strain become stronger — we gain greater fitness.
…
“Indeed, the learning is better if the correction occurs some time after the error is made. Stewing for a while in frustration at being wrong, and not seeing how to fix it, turns out to be a good thing. Cracking your ego is an unavoidable part of learning.”

—Keith Devlin
On making omelets and learning math

CREDITS: Feature photo (top) by Kira auf der Heide via Unsplash. “Morning Coffee” post format inspired by Nate Hoffelder at The Digital Reader.

Morning Coffee – 23 September 2019

Posted on by Denise Gaskins

Morning Coffee image

One of the best ways we can help our children learn mathematics (or anything else) is to always be learning ourselves.

Here are a few stories to read with your Monday morning coffee:

  • David Butler invented a challenging new game that can spark plenty of mathematical thinking: Digit Disguises.
  • If you liked James Tanton’s video on the area model in last week’s post, you may enjoy his in-depth discussion of The Astounding Power of Area.
  • On a lighter note, I’m sure any classroom or homeschool teacher can think of several ways to use Sara VanDerWerf’s collection of Math Fails. Scroll down for links to earlier collections, too.

“I told them that actually what they did was exactly what maths is — reasoning things out using the information you have and being able to be sure of your method and your answer. Just because there’s no symbols, it doesn’t mean it’s not maths.”

—David Butler
The Seven Sticks and what mathematics is

 

“I am not willing to teach mindless math. It leads to mindless adults. Thinking is not an add-on once they have memorized. Thinking is the basic tool to negotiate the world.”

—Geri Lorway
Teaching division?… Do you know the “basics”?

CREDITS: Feature photo (top) by Kira auf der Heide via Unsplash. “Morning Coffee” post format inspired by Nate Hoffelder at The Digital Reader.