Math Debate: Adding Fractions

Cover image by Thor/ geishaboy500 via Flickr (CC BY 2.0)

I’ve been working on my next Playful Math Singles book, based on the popular Things to Do with a Hundred Chart post.

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?

1/10 of 100

Well, you might say that:

\frac{1}{10}  of one hundred chart
+ \frac{1}{10}  of the same chart
= \frac{2}{10}  of that hundred chart

But, you might also say that:

\frac{1}{10}  of one chart
+ \frac{1}{10}  of another chart
= \frac{2}{20}  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 \frac{1}{20}  of the whole pair of charts.

So what happens if you see this question on a math test:

\frac{1}{10}  + \frac{1}{10}  = ?

If you write the answer “\frac{2}{20}”, you know the teacher will mark it wrong.

Is that fair? Why, or why not?

CREDITS: Feature photo (above) by Thor/geishaboy500 via Flickr (CC BY 2.0). “One is one … or is it?” video by Christopher Danielson via TED-Ed. This math debate was suggested by Marilyn Burns’s blog post Can 1/3 + 1/3 = 2/6? It seemed so!

A Beautiful Puzzle

This lovely puzzle (for upper-elementary and beyond) is from Nikolay Bogdanov-Belsky’s 1895 painting “Mental Calculation. In Public School of S. A. Rachinsky.” Pat Ballew posted it on his blog On This Day in Math, in honor of the 365th day of the year.

I love the expressions on the boys’ faces. So many different ways to manifest hard thinking!

Here’s the question:

No calculator allowed. But you can talk it over with a friend, as the boys on the right are doing.

You can even use scratch paper, if you like.

Thinking About Square Numbers

And if you’d like a hint, you can figure out square numbers using this trick. Think of a square number made from rows of pennies.

Can you see how to make the next-bigger square?

Any square number, plus one more row and one more column, plus a penny for the corner, makes the next-bigger square.

So if you know that ten squared is one hundred, then:

… and so onward to your answer. If the Russian schoolboys could figure it out, then you can, too!

Update

Simon Gregg (@Simon_Gregg) added this wonderful related puzzle for the new year:

Check Out These Cool Math Sales

I’ve been following Sonya’s Arithmophobia No More blog for a couple of years, and I love the work she is doing. But this month, she’s teamed up with Lacy at Play, Discover, Learn (another great blog to follow!) to offer a humongous bundle of playful math.

You get math journaling pages, games, creative task cards, thought-provoking worksheets, and video training resources to help you build your child’s understanding of math from arithmetic to early algebra. Wow!

These activities are perfect for homeschooling families or anyone looking to supplement their child’s current math curriculum with effective discovery-based activities. If you’ve ever wondered what to do with those Cuisenaire rods you picked up on sale way back when, this bundle is for you.

I’m so looking forward to using some of these ideas with my elementary homeschool co-op kids next year!

Sale price is $30 from December 2-15.

Cuisenaire Rod Activities Blowout Bundle

But Wait, There’s More

If you’ve been reading my blog for very long, you’ve probably seen how much I love the blog, books, and classes available from the Natural Math folks.

Their newest book is just off the presses — Funville Adventures, a math adventure chapter book.

And until December 20, they’re having a holiday sale. Make your own bundle of any Natural Math books. Playful algebra, calculus for 5-year-olds, inquiry problems and more: Great deal!

Natural Math Book Sale

Stock Up on My Playful Math Books

Finally, if you’ve been wanting to pick up a paperback copy of Let’s Play Math or some of my game books, or maybe a set of dot-grid math journals, I’m currently offering a discount on bulk orders.

Bundle ANY assortment of titles. Stock up on books for your family, friends, or homeschool group.

  • 2–4 books: 15% discount off retail prices
  • 5–9 books: 25% discount
  • 10–19 books: 35% discount
  • 20+ books: 35% discount, and free Continental U.S. standard shipping or the equivalent discount off other shipping options

Bulk Order Playful Math Paperbacks

(US customers only: We’re sorry we can’t offer bulk discounts for our international readers, but the complexities of international duties and tax laws are too much for this very small family business.)

Do You Know of Any Math Deals?

Apollonian greetings from my homeschool co-op kids, and best wishes for a grace-filled holiday season.

If you’ve seen a great deal or holiday price on a math resource you love, please share!

Add your deal to the comment section below, so we can all take advantage of the math joy this season.

Playful Math Education Carnival 106

Do you enjoy math? I hope so! If not, browsing this post just may change your mind.

Welcome to the 106th edition of the Math Teachers At Play math education blog carnival — a smorgasbord of links to bloggers all around the internet who have great ideas for learning, teaching, and playing around with math from preschool to pre-college. Let the mathematical fun begin!

By tradition, we start the carnival with a puzzle in honor of our 106th edition. But if you would like to jump straight to our featured blog posts, click here to see the Table of Contents.

Try This Puzzle

If you slice a pizza with a lightsaber, you’ll make straight cuts all the way across. Slice it once, and you get two pieces.

If you slice it five times, you’ll get a maximum of sixteen pieces. (And if you’re lucky you might get a star!)

  • How many times would you have to slice the pizza to get 106 pieces?

Click here for all the mathy goodness!

Dot Grid Doodling

What can you DO with a page full of dots?

Yesterday, I mentioned my new series of paperback dot grid notebooks, and I promised to share a few ideas for mathematical doodling.

But first, let me share my new dot-grid journals for kids. Aren’t they pretty?

Click Here for More Information

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.

—Julie Brennan

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?

Symmetry Challenge

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!
  • See if you can draw a rotational-symmetry design, like Don’s “Order 4” graphs.
  • Or experiment with the more flexible rules in John’s “Knot Fun” lesson.
  • And my latest obsession: the “ultimate” tutorial series on Celtic Knotwork, which explores the link between knots and their underlying graphs.
My favorite knot doodle so far.

Inspirations: A Recreational Mathematics Journal
Reflections: A Math Teacher’s Journal
Explorations: A Math Student’s Journal
Contemplations: A Homeschooler’s Journal

Also available through:
Barnes-Noble-logo the_book_depository_logo

Before you start doodling: How to Break In Your New Math Journal.

Feature photo (top): Sommermorgen (Alte Holzbrücke in Pretzfeld) by Curt Herrmann, via Wikimedia Commons. [Public domain]

March 2016 Math Calendars

Once again, a few of my favorite bloggers have come through with math calendars for our students to puzzle over. Check them out:

algebra calendar

Things to Do with a Math Calendar

At home:
Post the calendar on your refrigerator. Use each math puzzle as a daily review “mini-quiz” for your children (or yourself).

In the classroom:
Post today’s calculation on the board as a warm-up puzzle. Encourage your students to make up “Today is…” puzzles of their own.

As a puzzle:
Cut the calendar squares apart and trim off the dates. Then challenge your students to arrange them in ascending (or descending) order.

Make up problems to fill a new calendar for next month.
And if you do, please share!