The farther we go in math, the more division disappears. It ceases to exist as a separate concept.
Instead, we learn to see division as:
Each of these perspectives offers us a new way to think about and make sense of our calculations.
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 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
In this game, players practice multiplication facts and use strategic planning as they build 2-D shapes to block their opponent.
Many parents remember struggling to learn math. We hope to provide a better experience for our children.
And one of the best ways for children to enjoy learning is through hands-on play.
Math Concepts: multiplication, area, 2-D shapes.
Players: two players.
Equipment: square graph paper (lined or dotty).
The question came up on a homeschool math forum:
“My first grader and I were playing with equivalent expressions. We were trying to see how many ways we could write the value ‘3.’
“He wrote down 10 – 2 × 3 + 1.
“When I tried to explain the problem with his calculation, he got frustrated and didn’t want to do math.
“How can I help him understand order of operations?”
[If you think this sounds like too complex of a math expression for a first grader, you may want to read my blog post about math manipulatives and big ideas.]
Order of operations doesn’t matter in this instance. What matters is communication.
The mother didn’t know how to read what her son wrote.
He could help her understand by putting parentheses around the part he wanted her to read first.
He doesn’t need to know abstract rules for arbitrary calculations, or all the different ways we might possibly misunderstand each other. He just needs to know how to say what is in his mind.
I’ve been getting questions about my Math Journaling Adventures books:
“I’m so excited to try math journaling! We bought your Logbook Alpha, and my 11-year-old math-averse son is trying to be a math rebel at every turn.
“But I feel uncomfortable with the idea of rebellion. Doesn’t he need to learn how to solve math problems the right way?”
One of my favorite things about math is that there really is no “right” way to solve math problems.
As I pointed out in my ongoing Mental Math series, even a problem as basic as 6+8 can be approached from many directions. So perhaps I should say, the “right” way is however the student wants to make sense of the problem.
In math, sense-making and reasoning are always the most important things.
Over the course of this series, we’ve seen how mental math relies on a child’s own creative ways of thinking. In mental math, children develop understanding of how numbers interact with each other in many ways.
In this way, they learn the true 3R’s of math: to Recognize and Reason about the Relationships between numbers.
And the principles that underlie mental calculation are also fundamental to algebra, so that flexibility and confidence in mental math is one of the best predictors of success in high school math and beyond.
But as we went through the various example problems, did you find the written-out calculations hard to follow?
Don’t force your children to write down their mental math. It looks dreary when I write the calculations out step by step, but that’s not how it works in a child’s mind. With regular practice, this sort of thinking becomes second nature.
Ever since the school year started, I’ve been getting questions about how to use my new Math Journaling Adventures logbooks.
[SIDE NOTE: These logbooks are included in this month’s Thanksgiving Sale! You’ll get an automatic 10% discount off all print books, applied at checkout, no special code required.]
“I love the way your math books get my children thinking.
“Finally, they are having fun with math!
“But sometimes I have no idea what the journaling prompt is all about or how to teach it. Where can I buy a solutions manual?”
Um, that’s not how math journals work.
The cool thing about journaling prompts is that they have no “right” answer. They are explorations into different parts of the world of math, nature walks in the land of numbers, shapes, and patterns. Springboards into whatever our children want to investigate, whatever sparks their interest.
A few of the problem-solving prompts may have specific answers, but it really doesn’t matter if our kids find the exact solution a math professional might give. If they write what makes sense to them, they’ve accomplished the goal.
If later, they think of something they hadn’t noticed, or they want to change their answer — well, that is mathematical thinking, too.
As our children grow and develop their math skills, the mental math strategies grow with them.
The basics of mental math don’t change:
But we have new ways to help children do math in their heads as the numbers get bigger and the problems more challenging.
For example, how might kids figure out a multi-digit subtraction like 67 − 38?
First, we need to adjust our mindset…
I’ve been updating my old book files to a new publishing program, and in the process discovering anew how often websites change and disappear. So frustrating!
Thankfully, the Internet Archive keeps some things that we can look back at. It doesn’t always work, but today it came to the rescue and let me recover the wonderful workbook from the Metropolitan Museum of Art, Islamic Art And Geometric Design, plus some of the pattern pages at the old School of Islamic Geometric Design.
I reference these pages in my Geometric Coloring Designs series (available at my Playful Math Store), so I was glad to find they weren’t lost forever.
Math Art = a great way to spark energy when your students hit the mathematical doldrums.
If you’d like to play around with math art patterns, each image below links to Eric Broug’s step-by-step instructions for creating the design. Also, Broug made printable pages to get students started, which you can download here. No compass construction required!
There are two more patterns on the old SIGD site, but I can’t get those instructions to load. That might be my antique rural internet connection, however, so feel free to do some exploring on your own.
And have fun drawing math with your kids!
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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.
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“Geometric Math Art, Recovered” copyright © 2025 by Denise Gaskins. Image at the top of the post copyright © Denise Gaskins.
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