Mental Math: Advanced Addition

Posted on by Denise Gaskins
photo of kids having fun with math

Mental math is doing calculations with our minds, and perhaps with the aid of scratch paper or a whiteboard to jot down notes along the way.

But we cannot simply transfer the standard pencil-and-paper calculations to a mental chalkboard. That’s far too complicated.

We still want to follow our basic strategies of using friendly numbers, estimating, and adjusting the answer. So how can we help children do math in their heads as the numbers get bigger and the problems more challenging?

How might kids figure out a multi-digit addition like 87 + 39?

Here are three useful strategies…

Continue reading Mental Math: Advanced Addition

Mental Math: Early Division

Posted on by Denise Gaskins
Boy doing mental math calculation

Mental math is doing calculations with our minds, though we can use scratch paper or whiteboards to make notes as we work.

Doing mental math, children use the basic principles of arithmetic to simplify problems so they can think about number relationships, mastering the basic structures of how numbers work, the same structures that underlie algebraic reasoning.

As always, we rely on two key mental-math strategies.

  • Use friendly numbers.
  • Estimate, then adjust.

Division is the mirror image of multiplication, the inverse operation that undoes multiplication, which means we are scaling numbers down into smaller parts. Important friendly numbers include halves, thirds, and tenths, plus the square numbers and any multiplication facts the student happens to remember.

Continue reading Mental Math: Early Division

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

FAQ: How To Start a Homeschool Math Club

Posted on by Denise Gaskins
Denise Gaskins reading math with preschoolers

The question hits my inbox whenever parents start planning for a new school year:

“Hello! I am on the board of a homeschool co-op. We have had requests for a math club and wondered if you have any tips for starting one. We service children from K-10th and would need to try to meet the needs of as many ages as possible.”

There are several ways you might organize a homeschool math club, depending on the students you have and on your goals. I think you would have to split the students by age groups — it is very hard to keep that wide of a range of students interested. Then decide whether you want an activity-oriented club or a more academic focus.

When I started my first math club, I raided the math shelves in the children’s section at my library (510-519) for anything that interested me. I figured that if an activity didn’t interest me, I couldn’t make it fun for the kids. Over the years we have done a variety of games, puzzles, craft projects, and more — always looking for something that was NOT like whatever the kids would be doing in their textbooks at home.

Let’s look at the possibilities by grade level…

Continue reading FAQ: How To Start a Homeschool Math Club

Mental Math: Early Subtraction

Posted on by Denise Gaskins
mother and child doing math homework

By doing mental math, we help our children use the basic principles of arithmetic to simplify problems so they can think about number relationships, mastering the basic structures of how numbers work.

And the more our children practice these structures in mental math, the better prepared they will be to recognize the same principles in algebra.

The basic idea of subtraction is finding the difference between two quantities: comparing a larger amount to a smaller one, figuring out what’s left when you remove a part, or finding the distance between two measurements (or two points on the number line).

When you work with young children learning subtraction, remember our two key mental-math strategies.

  • Use friendly numbers.

For early subtraction with numbers less than 20, the most important friendly numbers are 5 and 10, the pairs of numbers that make 10, and the doubles.

  • Estimate, then adjust.

When children apply their creative minds to reasoning about math, they can use friendly numbers to get close to an answer, and then tweak the result as needed.

Continue reading Mental Math: Early Subtraction

Mental Math: Early Addition

Posted on by Denise Gaskins
child counting on fingers

From the very beginning of a child’s experience with math, we want to focus on reasoning, making sense of numbers, thinking about how they relate to each other and how we can use these relationships to solve problems.

The basic idea of addition is putting like things together: combining parts to make a whole thing, putting together sets to make a collection, or starting with an original amount and adding the increase as it grows. Connecting two numbers in relationship with a third number we call the sum.

When you work with young children learning addition, remember the two key mental-math strategies I mentioned in the previous post.

  • Use friendly numbers.

For early single-digit addition, the most important friendly numbers are 5 and 10, the pairs of numbers that make 10, and the doubles.

  • Estimate, then adjust.

When children apply their creative minds to reasoning about math, they can use friendly numbers to get close to an answer, and then tweak the result as needed.

Continue reading Mental Math: Early Addition

Tell Children Interesting Things

Posted on by Denise Gaskins
quote by John Conway

“You don’t educate people by telling them useful things; you educate people by telling them interesting things.”

— John Conway

If you want help educating your children with interesting things about math, check out Denise Gaskins’ Playful Math store.

We’re currently running a huge back-to-school sale on ALL of my playful math ebooks, problem-solving activities, math journaling task cards, and math art projects.

So many great ways to play with math!

The 20% discount will automatically apply when you check out. No discount code required.

Check it out:

Back to School Sale 2025

Morning Coffee: That Moment of Epiphany

Posted on by Denise Gaskins
Morning Coffee Lifelong Learning for Parents

One of the best ways we can help our children learn mathematics (or anything else) is to be lifelong learners ourselves.

Here are a few stories to read as you sip your morning brew. . .

Download your printable Morning Coffee journal

This week’s rabbit hole started with a thought-provoking article from Dan Meyer…

“It would have been quite easy, nothing at all really, to share the epiphany with students, to share the short-cut, to tell my kid that these are all the even numbers and here is where you’ll find them…”

—Dan Meyer

Read more about the value of taking the harder long-cut in this fifth installment of professional development for homeschooling parents.

 
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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 on-going basis, then please join me on Patreon for mathy inspiration, tips, and an ever-growing archive of printable activities.

“Morning Coffee: That Moment of Epiphany” copyright © 2025 by Denise Gaskins. Image at the top of post copyright © Kira auf der Heide / Unsplash.

Happy Pythagorean Triple Day!

Posted on by Denise Gaskins
Pythagorean Theorem demonstrated with tangrams

Thursday is Pythagorean Triple Day, one of the rarest math holidays.

The numbers of Thursday’s date: 7/24/25 or 24/7/25, fit the pattern of the Pythagorean Theorem: 7 squared + 24 squared = 25 squared.

Any three numbers that fit the a2 + b2 = c2 pattern form a Pythagorean Triple.

Continue reading Happy Pythagorean Triple Day!

A Poet Completes the Square

Posted on by Denise Gaskins
photo of quill pen and books for a math poet

Sue VanHattum and I were chatting about her young adult math books.

[Sue would love to get your help with beta-reading her books. Scroll down to the bottom of this post for details.]

In the first book of the series, Althea and the Mystery of the Imaginary Numbers, Althea learns that Tartaglia came up with a formula to solve cubic equations and wrote about it in a poem.

Sue had discovered an English translation of that poem and shared it with me. (You can read it on JoAnne Growney’s blog.) Then we wondered whether we could come up with a simpler poem, something an algebra student might be able to follow.

Perhaps you and your kids would enjoy making up poems, too. An algebra proof-poem might be too difficult for now, but check out my blog for math poetry ideas.

Continue reading A Poet Completes the Square