Mental Math: Advanced Multiplication, Part 1

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.

The Any-Order Principle

Multiplication is both associative and commutative.

That means we can shift factors around any way we like without changing the final product.

As numbers get bigger, we tend to have more factors to work with, thus many more options for solving any calculation. And the more we play with numbers, the more we learn about the web of relationships that we call mathematics.

Double and Half

Doubling is the first multiplication children learn, and it is so powerful that at least one ancient culture based their whole system around doubling.

We can double-and-half whenever at least one of our factors is an even number.

For example:

31 × 12
= 62 × 6
= 124 × 3

Then finish with funny numbers:

124 × 3
= three-sixty-twelve = 372

Double-and-half works with any even factor, but it can be especially impressive with the powers of two:

32 × 63
= 16 × 126
= 8 × 252
= 4 × 504
= 2 × 1008
= 2016

Why It Works

Double-and-half relies on the associative property of multiplication.

We take a factor of two, associated with one of our numbers, and move it over to connect to the other number instead.

Like this:

Number A × Number B
= Number A × (2 × Half B)
= (Number A × 2) × Half B
= Double A × Half B

Flexible Factoring

Double-and-half is only one example of a wider principle: We can move factors around however we like because multiplication works in any order.

This can be useful in many situations where we recognize an easy sub-product inside our problem.

For example, it’s simple to multiply by ten:

45 × 18
= (9 × 5) × (2 × 9)
= (9 × 9) × (5 × 2)
= 81 × 10
= 810

Twenty-one has small digits that are fun to work with:

24 × 7
= (8 × 3) × 7
= 8 × (3 × 7)
= 8 × 21
168

And twenty-five is always nice because it’s a quarter of one hundred:

36 × 75
(9 × 4) × (3 × 25)
(9 × 3) × (4 × 25)
27 × 100
2700

Practice Builds Fluency

It takes time and practice to develop the skill of looking at a problem and seeing the factors hidden inside. But it’s worth the effort, because factors are the true building blocks of numbers.

Learning to think in terms of factors is more important to your child’s mathematical success than the ones-tens-hundreds of early-elementary place value.

Of course, we all know that kids must learn place value. We spend uncountable lessons, worksheets, and quizzes on the topic over the course of several years. Let that underscore how vital it is for our children to gain fluency and flexibility working with factors!

Spend some time practicing this week:

• Take turns making up double-and-half problems for each other to solve.
• Think of some big numbers to multiply and see how many ways you can move around the factors.
• Try some of the multiplication challenges at Pam Harris’s #MathStratChat archive. How many ways can you solve each problem? Did the online commenters think of any ways you missed?

“When you own more connections, and relationships, and strategies, there’s more to play with, and math can be more creative, and we can be more flexible.

“Be mathy. Ask people what they think about.

“Stay in the problems that feel like they would be tricky, and try some things, and ask somebody for an idea if you get stuck.”

—Pam Harris, “Playing with Mathematical Relationships in Life”

Read the Whole Series

Check out all the posts in my Mental Math Series:

• Mental Math Is the Key to Algebra
• Three Basic Principles
• Early Addition
• Early Subtraction
• Early Multiplication
• Early Division
• Advanced Addition
• Advanced Subtraction
• Mental Math Do’s and Don’ts
• Advanced Multiplication, Part 1
• Coming Soon: One more tool for advanced multiplication…