My hundred chart list began many years ago as seven ideas for playing with numbers. Over the years, it grew to its current 30+ activities.
Now, in preparing the new book, my list has become a monster. I’ve collected almost 70 ways to play with numbers, shapes, and logic from preschool to middle school. Just yesterday I added activities for fraction and decimal multiplication, and also tips for naming complex fractions. Wow!
Gonna have to edit that cover file…
In the “Advanced Patterns” chapter, I have a section on math debates. The point of a math debate isn’t that one answer is “right” while the other is “wrong.” You can choose either side of the question — the important thing is how well you support your argument.
Here’s activity #69 in the current book draft.
Have a Math Debate: Adding Fractions
When you add fractions, you face a problem that most people never consider. Namely, you have to decide exactly what you are talking about.
For instance, what is one-tenth plus one-tenth?
Well, you might say that:
of one hundred chart
+ of the same chart
= of that hundred chart
But, you might also say that:
of one chart
+ of another chart
= of the pair of charts
That is, you started off counting on two independent charts. But when you put them together, you ended up with a double chart. Two hundred squares in all. Which made each row in the final set worth of the whole pair of charts.
So what happens if you see this question on a math test:
+ = ?
If you write the answer “”, you know the teacher will mark it wrong.
The posts on my Let’s Play Math blog are, for the most part, first-draft material. Of course, I’ve proofread each post — many times! because I’m a perfectionist that way, and yet I still miss typos — but these articles haven’t gotten the sort of feedback that polishes a book manuscript.
Well, now I’m taking some of the best of my old blog posts, expanding them with a few new games or activities, and giving them that book-quality polish. Let me introduce my newest series, the Playful Math Singles.
Under Construction …
The Playful Math Singles from Tabletop Academy Press will be short, topical books featuring clear explanations and ready-to-play activities.
I’m hoping to finish up two or three of these this year. Watch for them at your favorite online bookstore.
The first one is done …
Word Problems from Literature: An Introduction to Bar Model Diagrams
You can help prevent math anxiety by giving your children the mental tools they need to conquer the toughest story problems.
Young children expect to look at a word problem and instantly see the answer. But as they get older, their textbook math problems also grow in difficulty, so this solution-by-intuitive-leap becomes impossible.
Too often the frustrated child concludes, “I’m just not good at math.”
But with guided practice, any student can learn to master word problems.
Word Problems from Literature features math puzzles for elementary and middle school students from classic books such as Mr. Popper’s Penguins and The Hobbit.
For each puzzle, I demonstrate step by step how to use the problem-solving tool of bar model diagrams, a type of pictorial algebra. For children who are used to playing with Legos or other blocks — or with computer games like Minecraft — this approach reveals the underlying structure of a math word problem. Students can make sense of how each quantity in the story relates to the others and see a path to the solution.
And when you finish the puzzles in this book, I’ll show you how to create your own word problems from literature, based in your children’s favorite story worlds.
“I found this method really clarified for me what was going on visually and conceptually. Particularly when it came to more complex questions, for which I would normally write out an equation, I felt that thinking about what was going on with the bars actually made more sense … This is a wonderful book for those who want to support their children in finding better ways to work on word problems.”
But as with any paperback book, these have one problem. How do I use them without cracking the spine?
When we exercise, we need to warm up our bodies with a bit of stretching to prevent injury. In the same way, we need to warm up a new book to protect it. The process is called “breaking it in.”
It only takes a few minutes to break in a paperback book:
Step by Step
Never force the book but help it limber up gradually, and it will serve you well.
Because my journals are working books, I take the breaking-in process a bit further than shown in the video:
(1) Set the book on its back and follow the process above. Press down each cover, but not completely flat — let it bend at the fold line, about 1 cm from the actual spine. Then press a couple pages at a time, alternating front and back, down flat on each cover.
(2) Flip through the pages of the book forward and backward to limber them up.
(3) Repeat the steps of the video. This time, gently lean the main part of the book away from the part you are pressing down. Aim for a 130–140 degree angle.
(4) Flip through the pages again. Even roll the book back and forth a bit — curving the cover and pages as if you’re trying to fold the book in half — to encourage flexibility.
(5) Repeat the breaking-in process one more time. This time, fold each section back as close to 180 degrees as it will go.
And you’re done!
The pages will still curve in at the fold line, where they connect to the spine of the book. You want that because it makes the book strong. But now they’ll also open up to provide a nice, wide area for writing or math doodling.
Doodling gives our minds a chance to relax, wander, and come back to our work refreshed. And though it goes against intuition, doodling can help us remember more of what we learn.
Math doodles let us experiment with geometric shapes and symmetries. We can feel our way into math ideas gradually, through informal play. Through doodles, our students will explore a wide range of mathematical structures and relationships.
Our own school experiences can make it hard for us to teach. What we never learned in school was the concept of playing around with math, allowing ideas to “percolate,” so to speak, before mastery occurs, and that process may take time.
I like to doodle on dotty grid paper, like the pages in my math journals, but there’s No Purchase Necessary! You can design your own printable dot page at Incompetech’s PDF generator, or download my free coloring book (which includes several pages of printable dot and graph paper).
Patterns in Shape and Angle
To make a faceted mathematical gemstone, start with any shape you like. Then build other shapes around it. What do you notice? Does your pattern grow outward from its center? Or flow around the corner of your page? How is each layer similar, and how is it different?
Arbitrary constraints can lead to mathematically interesting doodles. For instance, create a design out of 45-45-90 triangles by coloring exactly half of every grid square. How many variations can you find?
Play a symmetry puzzle game. Draw a line of symmetry and fill in part of the design. Then trade with a partner to finish each other’s doodles.
Make more complex symmetry puzzles with additional reflection lines.
Math Doodle Links
Who can talk about mathematical doodling without mentioning Vi Hart? If you’ve never seen her “Doodling in Math Class” video series, you’re in for a treat!
My newest book project began with a few simple coloring pages for my homeschool co-op kids. You may recall when I collected those into a downloadable coloring book last December. Well, I kept tinkering with the designs into January. And then it was time to buy a new planner…
The problem is, I’m not a naturally organized person. I like making lists and plans, but sticking to them is tougher. And I’ve never found a planner or organizational system that I could follow for longer than two weeks at a go. That is until I heard of bullet journaling.
But journaling requires a journal — a notebook of some sort. And I couldn’t find any that I liked. Either the pages were too narrow and felt cramped, or the thing didn’t fit even in my oversized purse. Or the fancy, hardcover binding made it heavy to lug around. Or there weren’t enough pages to last more than a few weeks. Or the lines were too dark, or too widely spaced.
Never quite what I wanted.
So I decided to make my own.
I started with dot-grid pages for flexible layouts and for doodling. I scattered some of my favorite math and education quotations through each book. And then I added several of my most flexible geometric coloring pages (based on Islamic tessellation designs).
And I had so much fun I couldn’t stop with just one. So let me introduce my Dot Grid Notebook with Coloring Pages series:
Also available through:
With 170 roomy pages, each book gives you plenty of space to record memories, plan projects, and keep track of tasks. The dot grid makes it easy to draw graphs or diagrams. Take notes, jot down ideas, copy your favorite quotations, or doodle to your heart’s content.
Light gray dots at 5 mm spacing provide guidance for flexible page layouts.
11 geometric coloring pages allow a multitude of artistic possibilities.
31 favorite quotes offer a vision for creative math education.
6 × 9 inch (about 15 × 23 cm) pages are wider than many journals, yet still fit comfortably into a large purse or bag.
Paperback binding makes the journal sturdy but lightweight. Carry it anywhere!
The ebook edition features all 124 quotations (31 from each journal) about mathematics, education, and problem solving. Read through for your own pleasure, post them by your workspace, or use them as writing prompts for yourself or your students.
Yes, all of the ebooks are the same, so there’s no point in buying more than one. And at Amazon, if you buy a paperback journal, you can download the companion ebook for free!
But, what can you DO with all those nice, dotty pages?
Of course, you can use them for bullet journaling. That’s why I originally created the books, because I couldn’t find planners that fit my personal style. My bullet journal is basically an anthology of To-Do lists, bound together so they don’t get lost in the clutter. It’s the only planner system I’ve been able to stick with for more than two weeks at a go.
Or you could use the dotty pages for a commonplace book. That’s my favorite kind of journaling. Like a magpie, I collect shiny tidbits from books, websites, conversations overheard, and more. Passages. Definitions. Poems. Recipes. Proverbs. Things I’m wondering about. Cute kid sayings. It all goes into the mix.
And math puzzles, of course! Below, I’m playing my way through Paul Lockhart’s Measurement. I use the cloud-like labels in the outer margins of each page for keywords that identify what I’m writing, because someday I’ll need to skim back and find an old note.
But where dot grid pages really excel is at doodling — I’m sure you noticed the faceted design filling the lower half of my journal page above and the gem almost overrunning my February calendar. So watch for tomorrow’s blog post featuring a variety of ways to create your own mathematical doodles.
Best wishes, and happy mathing!
P.S.: Do you have a blog? If you’d like to feature a Dot Grid Journal review and giveaway, I’ll provide the prize. Leave a comment below, and we’ll work out the details.