Numberless Word Problems

As I mentioned yesterday, my new book includes links to online resources to help you play with word problems. So this week, I’m sharing a few of my favorites.

Today we examine a time-tested method to help kids reason about math: Leave out the numbers.

First up, there’s Brian Bushart’s numberless problem bank for young students. Then we’ll look at Farrar Williams’s modern revision of a math teaching classic with problems for upper-elementary and middle school students.

Have fun thinking math with your kids!

Word Problem Bank

Word problems are commonplace in mathematics classrooms, and yet they regularly confound students and lead to frustrated teachers saying things like:

• “They just add all the numbers! It doesn’t matter what the problem says.”
• “They don’t stop to think! They just start computing as soon as they’re done reading the problem.”

Brian Bushart offers a collection of ready-to-go slide presentations that walk through the steps of making a word problem make sense.

Math With No Numbers

Discover Farrar Williams’s book Numberless Math Problems: A Modern Update of S.Y. Gillian’s Classic Problems Without Figures, available in ebook or paperback.

Williams writes: “In order to answer the question, they’ll have to explain it, because the problem doesn’t give you anything to calculate with. The only way to answer is by explaining your process. See how sneaky a numberless problem is? It makes students really think about the process of solving the problem.”

“When students face a word problem, they often revert to pulling all the numbers out and “doing something” to them. They want to add, subtract, multiply, or divide them, without really considering which operation is the right one to perform or why.

“When you don’t have numbers, it sidesteps that problem.

“For students who freeze up when they see the numbers, this can be a really good way to get them to think about their process with math.”

—Farrar Williams, Math With No Numbers

CREDITS: Feature photo (top) by saeed karimi via Unsplash.com.

Problem Solving with James Tanton

At the back of my new Word Problems from Literature book, I’ve included an appendix with links to recommended online resources.

So I thought this week, I’d share some of my favorites with you. First up: Problem Solving Tips from James Tanton.

You may know Tanton from the popular Exploding Dots and other activities at the Global Math Project website. But he’s been busy for decades sharing the delight and the beauty of the subject. He currently serves as the Mathematician-at-Large for the Mathematical Association of America.

Read on to discover several of Tanton’s best problem-solving tips for middle school and older students.

Have fun exploring math with your kids!

How to Think like a School Math Genius

In this 4-video series, Tanton presents five key principles for brilliant mathematical thinking, along with loads and loads of examples to explain what he means by each of them. A call for parents and teachers to be mindful of the life thinking we should foster, encourage, promote, embrace and reward — even in a math class!

Two Key — but Ignored —Steps to Solving Any Math Problem

Every challenge or problem we encounter in mathematics (or life!) elicits a human response. The dryness of textbooks and worksheets in the school world might suggest otherwise, but connecting with one’s emotions is fundamental and vital for success — and of course, joy — in doing mathematics.

MAA AMC Curriculum Inspirations

Essays and videos showing how to approach math puzzles in a way that a) is relevant and connected to the curriculum, and b) revels in deep, joyous, mulling and flailing, reflection, intellectual play and extension, insight, and grand mathematical delight.

Think Puzzles and Think Cool Math

Here are some essays illustrating astounding tidbits of mathematical delight. And here are some purely visual puzzles to surprise.

“The true joy in mathematics, the true hook that compels mathematicians to devote their careers to the subject, comes from a sense of boundless wonder induced by the subject.

“There is transcendental beauty, there are deep and intriguing connections, there are surprises and rewards, and there is play and creativity.

“Mathematics has very little to do with crunching numbers. Mathematics is a landscape of ideas and wonders.”

—James Tanton

CREDITS: Feature photo (top) by Ian Stauffer via Unsplash.com.

For details on using a Minimalist Math Curriculum, see my earlier post How To Make Time for Exploration.

If you’re teaching or homeschooling students in 6th grade, here’s the new minimalist curriculum link:

As I mentioned in my earlier post, a minimalist curriculum can be a great way to free up time for playing math games and exploring enrichment activities (like these).

Prealgebra & Geometry Games Now Available

Publication Day!

Prealgebra & Geometry: Math Games for Middle School hits the online bookstores today.

You can prepare your children for high school math by playing with positive and negative integers, number properties, mixed operations, algebraic functions, coordinate geometry, and more. Prealgebra & Geometry features 41 kid-tested games, offering a variety of challenges for students in 4–9th grades and beyond.

A true understanding of mathematics requires more than the ability to memorize procedures. This book helps your children learn to think mathematically, giving them a strong foundation for future learning.

And don’t worry if you’ve forgotten all the math you learned in school. I’ve included plenty of definitions and explanations throughout the book. It’s like having a painless math refresher course as you play.

Math Puzzle from the Ancient Kingdom of Cats

It may look like Cimorene has lain down on the job, but don’t be fooled! She’s hard at work, creating a math investigation for your students to explore.

Cats know how important it can be for students to experiment with math and try new things. Playing with ideas is how kittens (and humans!) learn.

Cimorene wants you to know that the Make 100 Math Rebels Kickstarter offers a great way for human children to learn math through play. She encourages you to go watch the video and read all about the project.

Too often, school math can seem stiff and rigid. To children, it can feel like “Do what I say, whether it makes sense or not.” But cats know that kids are like kittens — they can make sense of ideas just fine if we give them time to play around.

So Cimorene says you should download the free sample journaling pages from the Math Rebels Kickstarter page. The beautiful parchment design makes doing math an adventure.

[The free download will always be there, even after the Kickstarter project ends.]

Cimorene’s Puzzle Challenge

Cimorene’s math puzzle is a classic geometry problem from the ancient Kingdom of Cats: Squaring the Circle.

Draw a circle on your journal page. Can you draw a square (or rectangle) that has the same area?

How would you even begin such a task?

Notice Cimorene’s hint in the photo above: Try drawing the square that just touches the edges of your circle. (We call those just-touching lines “tangents” to the circle.)

• What do you notice? Do the square and the circle have the same area? How close are they?

The tangent square sets an upper limit on the area of the circle. You can see that any square that exactly matches the circle would have to be smaller than the tangent square.

• Can you find a square that sets a lower limit on the area of the circle? That is, a square that must have less area than the circle?
• What’s the biggest square you can draw inside your circle? Can you find a square that has all four corners on the circle?

We call that biggest-inside square “inscribed” in the circle. Any polygon whose corners all sit on the circle is an inscribed polygon.

• Play around with circles and squares. How close can you get to matching their size?

Further Exploration

After you have explored for awhile on your own, Cimorene has one more twist in her puzzle.

In the ancient Kingdom of Cats, the wise ones estimated the area of a circle this way:

Divide the width of the circle in thirds, and then in thirds again. (That is, cut the diameter into nine parts.) Draw a square with sides measured by eight such parts.

You can try this on your journaling page by drawing a circle that is nine squares wide. Then draw a square overlapping it, with sides that are eight squares in length.

• How closely do the areas match?

Playing with Pi

Here’s a surprise: Cimorene’s puzzle isn’t really about squares, but about calculus.

The problem of Squaring the Circle is really a much bigger question: Finding the area of a square, rectangle, or other polygon is relatively easy, but how can we discover the area of a curved shape?

For a circle, the area is related to the number pi, which is the number of times you would have to walk across the circle to equal the distance of one time walking around it.

So the problem of Squaring the Circle is really the same as asking, “What is the value of pi?”

• Can you figure out what approximate value for pi matches the 8/9 square used in the ancient Kingdom of Cats?

If you’d like to learn more about pi, get ready for a celebration: Pi Day is coming soon! Every year, millions of children celebrate math on March 14th, because if you write the date as 3/14, it’s the same as the first three digits of pi.

Find out more about playing with pi in my Pi Day Round-Up post.

You may also enjoy:

New Printable Puzzle Books: Diffy Inception

The best way to practice math is to play with it—to use the patterns and connections between math concepts in your pursuit of something fun or beautiful.

Diffy Inception puzzles have their own symmetric beauty, but mostly they are just plain fun. Students can practice subtraction and look for patterns in the difference layers.

I just published four new activity books to our online store:

Notes to the teacher include puzzle instructions, game variations, journaling prompts, and more. Plus answers for all puzzles.

Available with 8 1/2 by 11 (letter size) or A4 pages.

My publishing company runs this online store, so you can find all my playful math books there — including an exclusive pre-publication ebook edition of my newest title, Prealgebra & Geometry: Math Games for Middle School. Click here to browse the Tabletop Academy Press store.

FAQ: Playful Math for Older Students

My students are so busy that time-consuming math projects are a luxury. How is it possible for older kids to play with mathematics?

Too often, the modern American school math curriculum is a relentless treadmill driving students toward calculus. (Does this happen in other countries, too?)

But that’s definitely not the only way to learn. For most students, it’s not the best way, either.

Here are a few ideas to get your older children playing with math…

More Math War with Special Decks

Just updated my blog post Math Game: War with Special Decks to add a couple of games I missed the first time around:

And…

If you’d like more ways to play with math from preschool to high school, check out My Favorite Math Games.

CREDITS: “Red playing cards” photo by José Pablo Iglesias via Unsplash.com.

Prealgebra & Geometry Preorders

My book Prealgebra & Geometry: Math Games for Middle School is scheduled for release to regular bookstores in February, 2021. Because no publisher wants to send a new book into the world during such hectic, unsettled times as a big election, the winter holidays, or during inauguration season.

But preorder links are beginning to appear at several of the major online booksellers. And more stores will join them, as the information filters through their website systems.

The paperback will also be up for preorder, whenever the sites catch that update.

And remember: If you don’t favor a particular bookstore, you can buy the early-release ebook right now at my publisher’s webstore — and get a 10% discount if you order before 15 October.

Math Game: Hit Me

I believe this was the first math game I ever invented. Of course, ideas are common currency, so I’m sure other math teachers thought of it before I did. But to me, it was original.

I’ve blogged about the game before, but here’s the updated version as it appears in my new book Prealgebra & Geometry: Math Games for Middle School — scheduled for publication in early 2021. Sign up for my newsletter to get updates.

Hit Me

Math Concepts: integer addition, absolute value.

Players: two or more.

Equipment: playing cards (two decks may be needed for a large group).