Playing with a Hundred Chart #36: Cover 100 Squares

Patrick Vennebush shared this puzzle from his new book, One-Hundred Problems Involving the Number 100:

It’s easy to cover a hundred chart with 100 small squares: 10 rows of 10 squares = 100.

It’s easy to cover a hundred chart with one big square: one 10×10 square = 100.

But can you cover the chart with 20 squares? Or with 57 squares? The squares do NOT have to be all the same size.

If we only consider squares with whole-number sides, so they exactly fit on the grid, then:

  • What numbers of squares work to cover the chart?
  • What numbers don’t work — and can you prove it?

Click to read the original puzzle along with some teaching tips at Patrick’s blog:

Covering 100 Squares

If you’d like some printable hundred charts for coloring in squares, download my free Hundred Charts Galore! file from my publisher’s online store:

Hundred Charts Galore!

And discover more ways to play with these printables in my classic blog post: 30+ Things to Do with a Hundred Chart.

Playful Math Education 142

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!

Parallel and Perpendicular Art

I love this easy-but-beautiful math art project!

1. Print a page of dotty or lined graph paper for each student. You’ll also need a ruler and a large assortment of markers or colored pencils.

2. Students draw a line across the page, lining up their ruler with the grid points. The first line can be vertical, horizontal, or diagonal.

3. Keep drawing lines, but NEVER cross a line you’ve already drawn. Following the grid will create many lines parallel or perpendicular to each other. What angles can you identify?

4. Color as desired. For a stained-glass effect, outline the colored areas with a black Sharpie marker.

Look for more math art ideas in my FREE new book Geometric Coloring Designs 2: Create Your Own Art. Visit my publisher’s online store and click the “Free Books” button.

CREDITS: I saw this project at Cindy’s Love2Learn2Day blog. She got the idea from Zachary‘s MathActivities site.

The Best Math Game Ever

The Substitution Game features low-floor, high-ceiling cooperative play that works with any age (or with a mixed-age group) — and you can use it while distance learning, too. It’s great for building algebraic thinking.

Excerpted from my upcoming book, Prealgebra & Geometry: Math Games for Middle School, scheduled for publication in early 2021. Sign up for my newsletter to get updates.

The Substitution Game

Math Concepts: addition, subtraction, multiplication, division, order of operations, integers, fractions, equivalence and substitution.

Players: any number (a cooperative game).

Equipment: whiteboard and markers (preferred) or pencil and paper to share. Calculator optional.

Continue reading The Best Math Game Ever

Journaling Pages

This afternoon, I’ve been working on the printable pdf math activity booklets I’ll be sending out as stretch goals to the backers of my Math You Can Play Kickstarter campaign.

Some of the booklets include dot grid pages for student journaling.

I love dot grid pages for writing because I can start a line anywhere on the page, and the dots help me keep things in line. (They’re also great for doodling.)

As students wrestle their thoughts into shape and create explanations, they do the same sort of work that mathematicians do every day. It’s difficult for children (or anyone) to pin down a thought and put it into words. But it’s great practice for life.

Journaling is a great practice for adult learners, too — and don’t we all want to be lifelong learners?

So I thought I’d share the journaling pages with you all, in case you’d like to get your children writing about math. There are three styles, ranging from plain to ornate parchment. Enjoy!

Download the Journaling Pages

UPDATE: The Kickstarter deals have ended, but my playful math books are still available through your favorite online store or by special order at your local bookshop. (Except for the Prealgebra & Geometry Games book, scheduled for publication in early 2021. Sign up for my email list to get the latest news.)

How Mathematics Works

The full quote, as it appears in my new book:

Make a conjecture. A conjecture is a statement that you think might be true.

For example, you might make a conjecture that “All odd numbers are…” How would you finish that sentence?

Make another conjecture. And another. Does thinking about your conjectures make you wonder about math?

Can you think of any way to test your conjectures, to discover if they will always be true?

This is how mathematics works. Mathematicians notice something interesting about certain numbers, shapes, or ideas. They play around and explore how those relate to other ideas. After collecting a set of interesting things, they think about ways to organize them. They wonder about patterns and connections. They make conjectures and try to imagine ways to test them.

And mathematicians talk with one another and compare their ideas. In real life, math is a very social game.

—Denise Gaskins
Prealgebra & Geometry: Math Games for Middle School

Excerpted from my upcoming book, Prealgebra & Geometry: Math Games for Middle School, scheduled for publication in early 2021. Sign up for my newsletter to get updates.

CREDITS: “Three girls counting” photo by Charlein Gracia on Unsplash.

Math Conversation Starter

(Click for larger image.)

What do you see?

Does it make you wonder?

How is perspective art similar to the isometric drawing in yesterday’s post? How is it different?

You may also enjoy:

CREDITS: William Hogarth – “The importance of knowing perspective” (Absurd perspectives), Engraving on paper. From Wikimedia Commons.

How to Draw Minecraft Blocks

Running out of time on my Math You Can Play Kickstarter, so I better get to work on that Kickstarter Special Edition math-art book I promised to all the backers as a bonus reward.

Today I’m working on the Isometric Drawing and Impossible Figures section, because my co-op math classes had so much fun learning how to draw those.

Here’s a starter image on how to draw Minecraft blocks. At first I called them “isometric blocks” — but changing the name to “Minecraft” made the students really excited to learn. I’m not sure whether I like the pencil sketch, or if I should remake the illustrations on the computer…

Key steps:

  1. Make a Y.
  2. Turn it into an M.
  3. Slant down for the bottom.
  4. Slant up for the top.
Student drawings from my co-op classes.

The most common problem for beginners is that they try to make the base straight. They know a block can sit on a table, so the bottom has to be flat, right? But once students get a feel for how it goes, they can really take off and have fun.

UPDATE: The Kickstarter deals have ended, but my playful math books are still available through your favorite online store or by special order at your local bookshop. (Except for the Prealgebra & Geometry Games book, scheduled for publication in early 2021. Sign up for my email list to get the latest news.)

Moving Patterns Game on Kickstarter

We all know kids like to move. But did you know you can harness all of that innate energy into developing a conceptual understanding of mathematics?

The Moving Patterns Game is an active, self-directed game featuring patterns, footwork, friends, and math. Dancing makes life fun, and math makes the dancing more interesting!”

—Malke Rosenfeld

Continue reading Moving Patterns Game on Kickstarter