Reasoning about Ratios and Proportions

photo of sailboat on the Nile river

“The value of a problem is not so much coming up with the answer as in the ideas and attempted ideas it forces on the would-be solver.”

—I. N. Herstein

Whenever I give a problem or puzzle in an Alexandria Jones story, I’ll try to post the answer soon afterward.

But don’t peek!

In the old days of my snail mail newsletters, the answers always came in the next issue, so students had two months to let the puzzle ruminate without temptation.

After all, if you only read the answer I give, you miss out on the fun of solving the puzzle. Follow the Math Adventurer’s Rule:

  • Figure it out for yourself!

Give the problem a good workout on your own. See what you can notice about the puzzle. Wonder about how it connects to the larger world of mathematics.

Play with the ideas and find out where they may lead.

Then check the answer just to prove you got it right, and maybe to see whether we did it the same way. There’s always more than one way to approach any math problem, and each method offers its own insights.

Continue reading Reasoning about Ratios and Proportions

Conversion Factors: How Old Are You in Nanoseconds?

birthday cupcake with sparkler

Homeschool Memories

[Based on a problem I made up for my co-op students, once upon a time…]

Conversion factors are special fractions that contain problem-solving information. Why are they called conversion factors?

  • “Conversion” means change, and conversion factors help you change the numbers and units in your problem.
  • “Factors” are things you multiply with. So to use a conversion factor, you will multiply it by something.

Continue reading Conversion Factors: How Old Are You in Nanoseconds?

New Book: Farzanah and the 17 Camels

illustration from Farzanah and the 17 Camels

Now’s your chance to promote creative math for kids, with Sue Looney’s new book on Kickstarter: Farzanah and the 17 Camels.

You can order the book in digital or paperback format, get multiple copies for a classroom or book club, or just donate to support the project.

Visit the Kickstarter

Farzanah and the 17 Camels celebrates the excitement and the rewards of solving a challenging and intriguing math problem. Set against the backdrop of the ancient Silk Road, with bustling markets, stunning carpets, fun characters, and camels, the story draws readers into the magic of Farzanah’s surroundings. 

Continue reading New Book: Farzanah and the 17 Camels

Playful Math: Getting Students To Write Their Own

To wrap up our week of exploring the resources from Word Problems from Literature, let’s talk about getting students to write their own math.

Check in on the Kickstarter

First up, I’m sharing an excerpt from the Word Problems Student Workbook. The “Story Problem Challenge” is one of my favorite math club activities.

Following that, you’ll find an amazing online mathemagical adventure for middle school: The Arithmetiquities. It’s great fun, and a great inspiration for students to create their own math stories.

Have fun writing math with your kids!

The Story Problem Challenge

What do you get when you cross a library book or favorite movie with a math worksheet? A great alternative to math homework!

The rules are simple:

(1) Choose a worksheet calculation to be the basis for your word problem.

(2) Solve the calculation.

(3) Consider where these numbers could make sense in your book or movie universe. How might the characters use math? What sort of things would they count or measure? Do they use money? Do they build things, or cook meals, or make crafts? Do they need to keep track of how far they have traveled? Or how long it takes to get there?

(4) Write your story problem.

To make the game easier, you may change the numbers to make a more realistic problem. But you must keep the same type of calculation. For example, if your worksheet problem was 18÷3, you could change it to 18÷6 or 24÷3 or even 119÷17 to fit your story, but you can’t make it something like 18−3.

Remember that some quantities are discrete and countable, such as hobbits and fireworks. Other quantities are continuous, such as a barrel of wine or a length of fabric. Be sure to consider both types when you are deciding what to use in your problem.

Then share your problem with friends, and you try their problems. Can you stump each other?

A Note about Copyright and Trademarks

Old books are in the public domain, so you can always use characters like Robin Hood, Sherlock Holmes, or Winnie-the-Pooh (but not the newer Disney version with the red jacket). But most books and movies are the protected intellectual property of their authors or estates, or of the company who bought those rights.

When you write problems for your own private use, feel free to use your favorite characters from any story. That’s like fan fiction, secret, just for your own pleasure.

But if you decide to share your creation beyond your own home or classroom, then be sure to “genericize” it first. Change or remove the proper names, using general descriptions instead.

For example, if you love the Harry Potter series, you might want to use Harry or Hermione in your story problems. Instead, write about “the boy wizard destined to fight an evil sorcerer.” Or “the bright young witch who can master any spell.”

Or if you like the Star Wars movies, you might write about “an interstellar justice warrior with an energy sword.” Or “an alien master of martial arts training a cocky but inexperienced apprentice.”

The Arithmetiquities

When the world of Sfera is threatened by the machinations of a malevolent sorcerer, it will be up to a band of unlikely heroes to become the brightest light in the darkness.

The adventurers fan out across the land to find and retrieve the Arithmetiquities, a set of ancient mathemagical artifacts.

The Arithmetiquities is a fantasy adventure story told through a sequence of 36 mathematical puzzles.

Join the Adventure

“Though it is still before sunrise, Lumparland Harbor is already bustling. Sailing ships moor at the misty docks, bringing travelers and goods to the seaside town. Three dwarves disembark from different ships, each adventurer returning home from some faraway locale. The three women gather at the end of the pier.

    “The strangers discover that they all live along the main road that leads from the harbor, so they decide to split the cost of a wagon. Egga lives 10 miles away, Floora lives 20 miles away, and Greeta lives 30 miles away. The wagon ride costs $1.50 per mile regardless of the number of passengers.

      “How much should each of the adventurers pay so that each one has a fair fare?”

      —Jason Ermer, “Lumparland Harbor,” The Arithmetiquities Chapter I

       
      * * *

      Are you looking for more creative ways to play math with your kids? Check out all my books, printable activities, and cool mathy merch at Denise Gaskins’ Playful Math Store. Or join my email newsletter.

      This blog is reader-supported. If you’d like to help fund the blog on an ongoing basis, then please join me on Patreon (or choose the paid level on Substack) for mathy inspiration, tips, and an ever-growing archive of printable activities.

      “Playful Math: Getting Students To Write Their Own” copyright © 2022 by Denise Gaskins. Image at the top of the post copyright © Hannah Olinger via Unsplash.com.

      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.

      Visit the Kickstarter

      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.

      Visit the Site

      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.”

      Find Out More

      “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.

          Buy It Now

          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. He says it’s “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!”

          Watch the Videos:

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

          How many degrees in a Martian circle?
          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.

          Read the Article

          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.

          Scroll down and start with the Ten Problem-Solving Strategies.

          Download the Puzzles

          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.

          Explore and Enjoy

          “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.

              Why Word Problems?

              Wow! My Word Problems from Literature Kickstarter is just barreling along. I love seeing how many people are interested in a playful approach to teaching math.

              Check It Out

              But you might wonder: Why do I care so much about word problems?

              In many textbooks, word problems are an afterthought tacked on to the end of a math lesson.

              For me, it’s just the opposite. Word problems are the key part of a lesson, because that’s where children come face-to-face with the meanings of math concepts.

              The Key to Learning Math

              If we want our children to learn real math, we need to offer them plenty of problems to solve. A child may work through several pages of number calculations by rote, following memorized steps, but a good problem demands more thought.

              A story problem puts flesh on the abstract bones of arithmetic. Word problems encourage children to ponder what it means for one thing to be bigger than another, or smaller, or faster, or slower, or made up of several parts.

              Word Problems from Literature will feed your child’s mathematical imagination with story problems inspired by classic books, from 2nd-grade stories based on Mr. Popper’s Penguins to prealgebra stumpers inspired by The Lord of the Rings.

              And when you finish my puzzles, I’ll show you how to create your own word problems from literature, using your children’s favorite story worlds.

              The Trouble with Word Problems

              Most young children solve math problems by the flash-of-insight method: They hear the problem, and they know by instinct how to solve it.

              This is fine for simple problems like “Four kittens played with a yarn ball. Two more kittens came to join the fun. Then how many kittens were playing with the yarn ball?”

              When problems grow more difficult, however, that flash of insight becomes less reliable, so we find our children fidgeting with their paper or staring out the window. They complain, “I don’t know what to do. It’s too hard.”

              Too often, the frustrated child concludes, “I’m just not good at math.”

              But the truth is that nobody is good at math, if you define “good at math” to mean they can see the answer instantly. Here’s a more useful definition: You’re good at math if you have problem-solving tools and know how to use them.

              And that is something everyone can learn.

              Word Problems from Literature and the Word Problems Student Workbook will show you how. Order your copies today!

              Visit the Kickstarter

               
              * * *

              This blog is reader-supported.

              If you’d like to help fund the blog on an on-going basis, then please head to my Patreon page.

              If you liked this post, and want to show your one-time appreciation, the place to do that is PayPal: paypal.me/DeniseGaskinsMath. If you go that route, please include your email address in the notes section, so I can say thank you.

              Which I am going to say right now. Thank you!

              “Why Word Problems?” copyright © 2022 by Denise Gaskins. Photos copyright © TarasMalyarevich, ArturVerkhovetskiy, Wavebreakmedia / Depositphotos.

              Sample My New Playful Word Problems Book

              I’m so excited to announce that on August 9th, I’m launching my newest playful math book, Word Problems from Literature (2nd Edition).

              And the prelaunch page is now live! You can sign up to get an email from Kickstarter as soon as the campaign launches.

              Visit the Prelaunch Page
              (free Kickstarter account required)

              This is one of my all-time favorite books, and I’ve had so much fun with this new edition — adding stories, writing make-your-own-problem prompts, sneaking a few teaching tips into the worked-out solutions, creating an almost-magical guide to helping kids reason their way through math problems.

              I’ve put together a free 14-page printable (pdf) “Word Problems Sampler” to give you a taste of things to come, including three ready-to-use student workbook pages.

              Check it out:

              Download the Sample Pages

              Help Your Kids Learn Math the Creative Way

              • Are you a parent trying to help your child learn math?
              • Or a teacher looking for creative ideas for your classroom?
              • Or a homeschooling parent hoping to enrich your student’s understanding?

              Then you’ll love Word Problems from Literature and the Word Problems Student Workbook because they prepare your children for school success, providing the tools they need to solve tough math problems.

              Feed your child’s mathematical imagination with puzzles inspired by classic books, from 2nd-grade stories based on Mr. Popper’s Penguins to prealgebra stumpers inspired by The Lord of the Rings.

              Hit the button to visit the prelaunch page and sign up for notifications:

              Get Notified

              Math for Star Wars Day

              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

              Math Makes Sense — Let’s Teach It That Way

              I had forgotten this video, and then rediscovered it yesterday and loved it just as much as ever. Perhaps you’ll enjoy it, too — especially if you think of yourself as “not a math person.”

              Annie Fetter is talking to classroom teachers, but her message is just as important for homeschoolers. Math is all about making sense. Let’s help our kids see it that way.

              “Sense-making is the first mathematical practice for a reason. If we don’t do this one, the rest of them don’t matter. If we’re not doing this, our children are not going to learn mathematics.”

              —Annie Fetter
              Sense Making: It isn’t Just for Literacy Anymore

              You can download the notes for Fetter’s updated session on sense-making and find several links to wonderful, thought-provoking posts on her blog:

              How Can We Encourage Sense-Making?

              Here are some ideas from Fetter’s updated notes, which expand on her comments in the video above:

              • Get rid of the question. Literally.
              • Ask students “What could the question be?”
              • Get rid of the question and the numbers.
              • Give the answer.
              • Or give several answers.
              • Ask about ideas, not answers.
              • Ask “Why?” or “How did you know?” or “How did you decide that?” or “Tell me more about that.”
              • Use active reading strategies.

              Get this free downloadable poster from Teacher Trap via Teachers Pay Teachers.

              A Few Resources to Practice Sense-Making

              In no particular order…

              “I implore you, stop ‘cracking the math code.’ Make sense-making the focus of every single thing you do in your math classroom.”

              —Annie Fetter
              Sense Making: It isn’t Just for Literacy Anymore

              And if you haven’t seen it before, don’t miss Annie Fetter’s classic video “Ever Wonder What They’d Notice?”

              CREDITS: “Building a rocket ship” photo by Kelly Sikkema via Unsplash. “Reading is thinking” poster by Teacher Trap via Teachers Pay Teachers.