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

New Membership Tiers: One-on-One Mentorship

Posted on by Denise Gaskins
Two women chatting over tea

Over the years, in my public writing and in personal correspondence, I’ve tried to help younger parents by sharing the things I wish someone would have told me forty years ago. Oftentimes, I will answer a question but never hear back what the person ended up doing or how their family got along.

It leaves me feeling as though I could have done more to encourage them along the way.

In the past few years, I’ve watched a professional writer I admire organize several more formal mentorships for younger indie authors to grow their writing and publishing careers. I think a similar program could be helpful for homeschooling families.

So now I’m offering two new Patreon membership tiers that give you a direct one-on-one connection with me through email or email-plus-video.

My goal is to help you think about math as a creative, playful investigation where you and your children explore concepts together using the math program or curriculum of your choice. This is not a tutoring arrangement. I will help you understand and teach math, but I will not do the teaching for you.

Continue reading New Membership Tiers: One-on-One Mentorship

Hints for the Patty Paper Trisection

Posted on by Denise Gaskins
drafting tools

No peeking! This post is for those of you who have given the trisection proof a good workout on your own.

If you have a question about the proof or a solution you would like to share, please post a comment here.

But if you haven’t yet worked at the puzzle, go back and give it a try.

When someone just tells you the answer, you miss out on the fun. Figure it out for yourself — and then check the answer just to prove that you got it right.

Continue reading Hints for the Patty Paper Trisection

Puzzle: Patty Paper Trisection

Posted on by Denise Gaskins
student using drafting tools

One of the great unsolved problems of antiquity was to trisect any angle, to cut it into thirds with only the basic tools of Euclidean geometry: an unmarked straight-edge and a compass.

Like the alchemist’s dream of turning lead into gold, this proved to be an impossible task. If you want to trisect an angle, you have to “cheat.” A straight-edge and compass can’t do it. You have to use some sort of crutch, just as an alchemist would have to use a particle accelerator.

One “cheat” that works is to fold your paper.

I will show you how it works, and your job is to show why.

Continue reading Puzzle: Patty Paper Trisection

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

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

Playful Math 181: The Symmetry Carnival

Posted on by Denise Gaskins
Playful Math Carnival 181

If you’re looking for entertainment to while away the winter (or summer, for those of us up north!) — or if you’re just curious about how learning math could possibly be fun — you’ll definitely want to check out the latest edition of the Playful Math Carnival.

It’s a collection of awesome blog posts curated by Johanna Buijs and published on the Nature Study Australia website:

The whole point of the carnival is to show that math doesn’t have to be tedious or repetitive. Through a bunch of fun and engaging posts, we celebrate math that’s playful, creative, and totally relevant to everyday life.

Because what could be more relevant than having fun while we learn?

Continue reading Playful Math 181: The Symmetry Carnival