Mental Math: Advanced Multiplication, Part 1

Posted on by Denise Gaskins
Mother and daughter working mental math together

Mental math is the key to algebra because the same principles underlie them both.

As our children learn to do calculations in their heads, they make sense of how numbers work together and build a strong foundation of understanding.

Remember that while mental math is always done WITH the mind, reasoning our way to the answer, it doesn’t have to be only IN the mind. Make sure your students have scratch paper or a whiteboard handy to jot down intermediate steps as needed.

Besides, math is always more fun when kids get to use colorful markers on a whiteboard.

Continue reading Mental Math: Advanced Multiplication, Part 1

Mental Math: Early Multiplication

Posted on by Denise Gaskins
mother and daughter talking math together

Children learn best through interaction with others, and mental math prompts can lead to fascinating conversations, listening as our kids apply their creativity to the many ways numbers interact.

With mental math, students master the true 3R’s of math: to Recognize and Reason about the Relationships between numbers.

And these 3Rs are the foundation of algebra, which explains why flexibility and confidence in mental math is one of the best predictors of success in high school math and beyond.

Let’s Try an Example

Multiplication involves scaling one number by another, making it grow twice as big, or three times as much, or eightfold the size. Multiplication by a fraction scales the opposite direction, shrinking to half or a third or five-ninths the original amount.

The key friendly numbers for multiplication and division are the doubles and the square numbers. As with addition and subtraction, students can estimate the answer using any math facts they know and then adjust as needed.

How many ways might children think their way through the most-missed multiplication fact, 8 × 7?

Continue reading Mental Math: Early Multiplication

If Not Methods: Multi-Digit Multiplication

Posted on by Denise Gaskins
Mother helping her daughter with math homework

As we’ve seen in earlier posts, there are more ways to solve any math problem than most people realize. Teaching children to follow memorized steps and procedures actually cripples their understanding of number relationships and patterns.

But what if our children get stumped on a multi-digit multiplication calculation like 36 × 15?

Continue reading If Not Methods: Multi-Digit Multiplication

Math Musings: Lies My Teacher Told Me

Posted on by Denise Gaskins

I mentioned last time that the common phrase “Multiplication is repeated addition” is a mathematical lie we tell our children. And it’s not the only one.

Did you ever say, “Subtraction means take-away”? Or how about “Division is sharing”? I know I have, but both of those statements are also mathematical lies.

One of the reasons I like Cuisenaire rods so much is that they can help us avoid lying to our children about math.

Continue reading Math Musings: Lies My Teacher Told Me

What Is Multiplication, Anyway?

Posted on by Denise Gaskins

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

        Posted on by Denise Gaskins

        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

        Posted on by Denise Gaskins

        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

        Posted on by Denise Gaskins

        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

        Posted on by Denise Gaskins

        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.

        Click here to find Multiplication & Fractions at your favorite bookstore.

        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.