Well, that’s what I thought until I tried using pizzas to teach equivalent fractions. Sure, it’s easy to use a pizza to represent quarters, sixths, eighths. Thirds aren’t even too bad.

But just try dividing a circle into five equal pieces. Or seven. Or ten. Even though I’m a grown-up and a professional math educator, it’s really hard for me to consistently do this.

And as for showing visually how two fifths equals four tenths? Unless I do an impossibly immaculate job of dividing those pizzas into pieces, forget about it.

Are they really the same?

Fortunately, I’ve found a much better way to represent fractions visually: the Math-U-See fraction overlays. If we’re going to use a food metaphor, they’re flat and square, like brownies from a square pan.

And like brownies, they are much easier to slice evenly than pizzas. In fact, because they’re reusable transparent overlays, you don’t even have to draw (or slice) anything. You just arrange them on top of each other.

Check it:

Awwww, yeah! Those fractions are DEFINITELY equivalent!

You can use the overlays to represent basic fractions with divisors from two through seven, or combine multiple overlays to build other denominators.

Not only do they make it really easy to see how equivalent fractions work, but using them is way easier than drawing pizzas and hoping the slices come out even.

The fraction overlays make the concepts extremely clear, and because they’re tactile, visual, and kinesthetic, they make fractions feel like a game instead of work. Whenever I’ve used them with my students, either in person or during online tutoring sessions with a webcam, they’ve react the same way: “This is fun!”

Now that’s how I want my students to feel about fractions!

Related Posts:
Math student’s bill of rights
The best algebra book in the world?
Gallon man to the rescue!
Five fun ways to help your kids learn math this summer

I’m totally psyched! I have a new guest post up on mashable about five of my favorite fun technologies to help kids learn math over the summer — technologies which involve incorporate hip-hop, wailing rock guitar solos, talking calculators, brain science, and pirates.

So if you’ve ever dreamed of helping your kids learn their times tables by rapping, or just want to find some fun new ways to help your kids learn math this summer, click on over and check it out!

Related Posts:
Tips for how to help your kid with their math homework
The best algebra book in the world?
How to find a good math tutor
Be yourself, do what you want, wear what you love (Ada Lovelace/Coder Barbie Follow-Up)

Here’s how to figure it out. Draw a gallon man!

First, draw a really big capital G. (This is the gallon.)

Inside the G, draw four big Qs. (These are the quarts.)

Inside each Q, draw two Ps. (These are the pints.)

Inside each P, draw two cs. (These are the cups.)

For the final flourish, draw an arrow to one of the cs and write “8 ounces.” (There are eight ounces in every cup.)

When one of my students, a fifth grader, taught me about Gallon Man, I thought, I wish I had learned about this in fifth grade! My entire life, I’ve had to look up each of the conversions and never really internalized how they all fit together.

Since I’ve been introduced to Gallon Man, I’ve gleefully shared him with a fourth grade tutoring student (online), a friend who is a professional organic farmer (in person), innocent bystanders (at a restaurant), and most recently, my Mom (over the phone…”first, draw a really big G…”)!

They’ve all found Gallon Man helpful. Responses have included: “Can I take that drawing home with me?”, “Oh…I get it!”, and “I’m going to hold onto this.”

Gallon Man is totally visual and works for many learning styles. You can SEE how many quarts are INSIDE a gallon. Gallon Man is intuitive for all grade levels (unlike dimensional analysis, you don’t have to worry about the numerators or denominators). Gallon Man is practical. You can use it in your kitchen or in the grocery store. Gallon Man is easy to remember. And Gallon Man is fun to draw!

Gallon Man has recently gotten some airtime from other math bloggers, including Sam J Shah, who pointed out that it really helped him to see someone drawing Gallon Man. Here’s Sam’s post and video.

Yay for mnemonic devices!

Related posts:
The best algebra book in the world?
An easy way to remember how logarithmic notation works
Tips for how to help your kid with their math homework

To celebrate 3.14 – PI DAY – here is a beautiful music video of an awesome song about PI !!

It’s a delightful way to learn or remember some important facts about this lovely number, 3.14159….

Here are the lyrics. I would love to know who wrote the song (couldn’t figure it out on google) and who made the gorgeous video (the info was all in Russian).

(Thanks to Sue Van Hattum for posting this on *her* blog today, and to Maria D for sharing it on her Natural Math google group!!)

This book doesn’t have a ton of frills—there are barely any pictures or “extras.” But
what makes this book exceptional is its GREAT sequencing. It does an excellent job of breaking the math down without dumbing it down. The problems get harder very incrementally. There are so many practice problems to choose from that you can really practice until each procedure becomes second nature. And the book only introduces new concepts once you’ve already mastered the prerequisite skills.

For example, when this book introduces factoring trinomials, it introduces each pattern that you might encounter one at a time. You practice that pattern extensively before facing a new pattern. Once you’ve practiced all the different patterns separately, THEN it mixes all the different patterns together in one problem set. But by now you know how to recognize the different patterns and what to do differently for each pattern. So when faced with a page full of different types of factoring patterns, you can just think, “OH—difference of squares!” or “OH—perfect squares!” instead of having to do trial and error until you erase a hole in your paper!!

The students I’ve used this book with acquire very, very strong algebra skills without getting bored or frustrated. And I think it’s because the sequencing forces students to learn how to “chunk,” a concept I learned from Daniel T. Willingham’s book, Why Don’t Students Like School?

For example, take two algebra students. One is still a little shaky on the distributive property, the other knows it cold. When the first student is trying to solve a problem and sees a(b + c), he’s unsure whether that’s the same as ab + c, or b + ac, or ab + ac. So he stops working on the problem and substitutes small numbers into a(b + c) to be sure he’s got it right. The second student recognizes a(b + c) as a chunk and doesn’t need to stop and occupy working memory with this subcomponent of the problem. Clearly the second student is more likely to complete the problem successfully. (p 31)

Thank you, thank you to those who wrote this book so chunk-fully: Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, and William L. Cole!

I have the right to learn at my own pace and not feel put down or stupid if I’m slower than someone else.
I have the right to ask whatever questions I have.
I have the right to need extra help.
I have the right to ask a teacher or tutor for help.
I have the right to say I don’t understand.
I have the right to not understand.
I have the right to feel good about myself regardless of my abilities in math.
I have the right not to base my self-worth on my math skills.
I have the right to view myself as capable of learning math.
I have the right to evaluate my math instructors and how they teach.
I have the right to relax.
I have the right to be treated as a competent person.
I have the right to dislike math.
I have the right to define success in my own terms.

One of my best friends found this in high school and shared it with me, and I remember thinking it was amazing and wishing I had known about it earlier. While these rights now seem basic to me, if I had read them in middle school or high school, I think they would have been a revelation.

I really want my students to know that they have the right to ask whatever questions they have. (I’m still shocked to hear how some teachers will tell their students that they won’t answer their questions because if a student has a question it must be because the student wasn’t “paying attention.”) If a student is discerning enough to know what they have a question about, and courageous enough to actually ask it, that should be encouraged!

I also think it’s important that students realize that they can evaluate their teachers and how they teach. For teachers, this might be the scariest aspect of the Math Bill of Rights.

I can speak from experience on this. The one time I gave a copy of the Rights to a student, I was fearful that I might not be the right person to help her. But I thought the Rights might help her, and I didn’t want to not share a resource with her just because reading it might cause her to realize that she should be working with another tutor. And I realized that my ultimate goal was to make sure that she got the help she really needed, even if it wasn’t from me.

I’m not sure if the Rights were at all responsible, but shortly after I gave them to her we had a huge breakthrough and her understanding really improved. But it was one of the scarier things I’ve done as an instructor, and I can see why teachers might not want to plaster this everywhere…

But I wish that each math student in the world could have their own personal copy emblazoned on their binder or even their math book cover.