Mental Math: Advanced Multiplication, Part 2

Posted on by Denise Gaskins
Father and son celebrate a mental math answer

The methods in last week’s Advanced Multiplication post only work for certain numbers, but we have another, more powerful multiplication tool: We can always use a ratio table to make sense of any multiplication.

Ratios are the beginning of proportional thinking. We can systematically alter the numbers in a ratio to reach any quantity required by our problem.

Students begin working with ratios in story problems that help them visualize and make sense of a proportional relationship.

Continue reading Mental Math: Advanced Multiplication, Part 2

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