What Is Multiplication, Anyway?

At some point during the process of teaching multiplication to our children, we really need to come to terms with this question:

What IS multiplication?

Did your device hide the video? Find it on YouTube here.

“What’s my answer? It’s not one that society’s going to like. Because society expects — demands, even — that mathematics be concrete, real-world, absolute, having definitive answers.

    I can’t give a definitive answer.

      Multiplication manifests itself in different ways. So maybe the word ‘is’ there is just too absolute. And it’s actually at odds with what mathematicians do.

        Mathematicians do attend to real-world, practical scenarios — by stepping away from them, looking at a bigger picture.”

        —James Tanton, What is Multiplication?

        For Further Study

        You may also enjoy these posts from my blog archive:

        Memorizing the Times Table: A Life Skills Approach

        Continuing on my theme of times table facts, here’s the inimitable James Tanton:

        Did your device hide the video? Find it on YouTube here.

        “If our task is to memorize this table, please make it about mathematics — about thinking your way through a challenge, and what can I do to make my life easier.”

        —James Tanton, Making Memorising Multiplication Facts (if one really must) a meaningful Life Skill Lesson

        For Further Study

        You may also enjoy my blog post series about working through the times tables, paying attention to mathematical relationships (and a bit of prealgebra) along the way.

        Times Tables Series

        Click the button to see the whole series. Scroll down to the first post to go through it in order.

        Only Three Facts to Memorize

        A comment from a friend got me playing around with multiplication. I found a few videos from some of my favorite math people, so I’ll be sharing over the next few days.

        Here’s one from Sonya Post of Learning Well at Home. Also, Sonya just hosted Playful Math Education Carnival #143, which is well worth your time to explore!

        Did your device hide the video? Find it on YouTube here.

        “When students have to drill multiplication facts, it’s frustrating, unproductive and it makes them hate math. A better way to master the multiplication table is work on the skills that allow students to multiply quickly and efficiently.”

        —Sonya Post, Why We Don’t Drill Multiplication Facts – What We Do Instead

        Doubling and Halving

        Making doubles and halves are a great foundation for all sorts of math.

        Do you ever play the doubling game with your children? One player picks a starting number, and then you take turns doubling it until your mental math skills run out. How far can you go?

        Or try the halving game: One player chooses a starting number, and you take turns cutting it in half. How tiny can you go?

        As Sonya demonstrated, these skills help your child master their multiplication facts. And they are fantastic preparation for exponents and logarithms, too!

        Prime Factor Art on a Hundred Chart

        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.

        So this art project is a great way to practice multiplication. Use the prime factors of numbers from one to one hundred to create a colorful design.

        Start with a Hundred Chart

        First, download this printable file of hundred charts in non-photo blue (or light gray, if you’re printing in grayscale). The file includes:

        • Line-by-line traditional chart, counting from top to bottom.
        • Line-by-line bottom’s-up chart, counting from bottom to top.
        • Ulam’s Spiral chart, spiraling out from the center.
        • Blank grids for making your own patterns.

        Download the Printable Charts

        Continue reading Prime Factor Art on a Hundred Chart

        New Book: Multiplication & Fraction Games

        It’s here! My long-awaited upper-elementary Math You Can Play games book has finally hit the online bookstores.

        Multiplication & Fractions features 25 kid-tested games, offering a variety of challenges for school-age students. Children master several math models that provide a sturdy foundation for understanding multiplication and fractions. The games feature times table facts and more advanced concepts such as division, fractions, decimals, and multistep mental math.

        Free Online Preview

        Click here to find Multiplication & Fractions at your favorite bookstore.
        170 pages, ebook: $5.99, paperback: $17.99.

        Multiplication & Fraction Games

        multfrac-300Maybe you never really understood what multiplication means or what fractions are? As long as you start with an open mind and are willing to engage playfully, the activities in the book can help you as you help your kids.
        Anecdotally, these two areas are the first major stumbling point for students in their math studies. The sequencing in the book will help kids develop a strong foundation.
        Kids (and parents!) find these games fun. I’ve been field testing math games for the last 18 months and keep seeing how engaged kids get when playing math games.

        — Joshua Greene
        Multiplication & Fractions Math Games from Denise Gaskins (a review)

        Chapters include:

        • Mathematical Models: Learn the basic pictures that help support your child’s comprehension.
        • Conquer the Times Tables: Enjoy practicing the math facts until correct answers become automatic.
        • Mixed Operations: Give mental muscles a workout with games that require number skills and logical thinking.
        • Fractions and Decimals: Master equivalent fractions, work with decimal place value, and multiply fractions and decimal numbers.

        If you are a parent, these games provide opportunities to enjoy quality time with your children. If you are a classroom teacher, use the games as warm-ups and learning center activities or for a relaxing review day at the end of a term. If you are a tutor or homeschooler, make games a regular feature in your lesson plans to build your students’ mental math skills.

        So what are you waiting for? Clear off a table, grab a deck of cards, and let’s play some math!

        Check It Out

        It starts with models that are visual explanations of the concepts. Gaskins also breaks learning these concepts into comfortable steps that emphasize patterns and relationships, the real ideas that are behind properly understanding multiplication and fractions (indeed, math generally).
        The sequence of games in each section starts by building familiarity and then fluency (speed) to solidify all of that work.

        — Joshua Greene
        Multiplication & Fractions Math Games from Denise Gaskins (a review)

        Multiplication & Fraction Printables

        Multiplication & Fraction Printables

        Most of the Math You Can Play games use materials you already have around the house, such as playing cards or dice. But this book introduces multiplication and fractions with several games using two special mathematical model card decks.

        Click here to download the Multiplication & Fraction Printables, featuring all the math model cards, hundred charts, and game boards you will need for any game in the book.

        Multiplication Is Not Repeated Addition: Update

        Multiplication Is Not Repeated Addition: Update[Photo “Micah and Multiplication” by notnef via Flickr (CC-BY 2.0).]

        Some Internet topics are evergreen. I noticed that my old Multiplication Is Not Repeated Addition post has been getting new traffic lately, so I read through the article again. And realized that, even after all those words, I still had more to say.

        So I added the following update to clarify what seemed to me the most important point.

        I’d love to hear your thoughts! The comment section is open down below . . .


        Language Does Matter

        Addition: addend + addend = sum. The addends are interchangeable. This is represented by the fact that they have the same name.

        Multiplication: multiplier × multiplicand = product. The multiplier and multiplicand have different names, even though many of us have trouble remembering which is which.

        • multiplier = “how many or how much”
        • multiplicand = the size of the “unit” or “group”

        Different names indicate a difference in function. The multiplier and the multiplicand are not conceptually interchangeable. It is true that multiplication is commutative, but (2 rows × 3 chairs/row) is not the same as (3 rows × 2 chairs/row), even though both sets contain 6 chairs.

        A New Type of Number

        In multiplication, we introduce a totally new type of number: the multiplicand. A strange, new concept sits at the heart of multiplication, something students have never seen before.

        The multiplicand is a this-per-that ratio.

        A ratio is a not a counting number, but something new, much more abstract than anything the students have seen up to this point.

        A ratio is a relationship number.

        In addition and subtraction, numbers count how much stuff you have. If you get more stuff, the numbers get bigger. If you lose some of the stuff, the numbers get smaller. Numbers measure the amount of cookies, horses, dollars, gasoline, or whatever.

        The multiplicand doesn’t count the number of dollars or measure the volume of gasoline. It tells the relationship between them, the dollars per gallon, which stays the same whether you buy a lot or a little.

        By telling our students that “multiplication is repeated addition,” we dismiss the importance of the multiplicand. But until our students wrestle with and come to understand the concept of ratio, they can never fully understand multiplication.

        For Further Investigation

        nunes-doingmathIf you’re interested in digging deeper into how children learn addition and multiplication, I highly recommend Terezina Nunes and Peter Bryant’s book Children Doing Mathematics.

        To learn about modeling multiplication problems with bar diagrams, check out the Mad Scientist’s Ray Gun model of multiplication:

        And here is an example of the multiplication bar diagram in action:

        Memorizing the Math Facts

        Central City Times Tables[Photo by dsb nola via flickr.]

        The most effective and powerful way I’ve found to commit math facts to memory is to try to understand why they’re true in as many ways as possible. It’s a very slow process, but the fact becomes permanently lodged, and I usually learn a lot of surrounding information as well that helps me use it more effectively.

        Actually, a close friend of mine describes this same experience: he couldn’t learn his times tables in elementary school and used to think he was dumb. Meanwhile, he was forced to rely on actually thinking about number relationships and properties of operations in order to do his schoolwork. (E.g. I can’t remember 9×5, but I know 8×5 is half of 8×10, which is 80, so 8×5 must be 40, and 9×5 is one more 5, so 45. This is how he got through school.) Later, he figured out that all this hard work had actually given him a leg up because he understood numbers better than other folks. He majored in math in college and is now a cancer researcher who deals with a lot of statistics.

        Ben Blum-Smith
        Comment on Math Mama’s post What must be memorized?

        The entire discussion (article and comments) is well worth reading:

        You may also enjoy:

        Most Difficult Math Fact in the Whole Times Table

        7-8 sign

        Happy Multiplication Day!

        For help learning the Times Table facts, check out my multiplication blog post series:

        Encourage your family to play with math every day:


        howtosolveproblemsWant to help your kids learn math? Claim your free 24-page problem-solving booklet, and sign up to hear about new books, revisions, and sales or other promotions.


        Review Game: Once Through the Deck

        [Feature photo above by Shannon (shikiro famu) via Flicker (CC BY 2.0).]

        Math Concepts: basic facts of addition, multiplication.
        Players: one.
        Equipment: one deck of math cards (poker- or bridge-style playing cards with the face cards and jokers removed).

        The best way to practice the math facts is through the give-and-take of conversation, orally quizzing each other and talking about how you might figure the answers out. But occasionally your child may want a simple, solitaire method for review.

        Continue reading Review Game: Once Through the Deck