The Playful Math Carnival is a collection of blog posts and articles from around the internet, putting lots of goodies in one place for your enjoyment. The theme for this issue is fractions and division. Why are division and fractions so much harder than what came before? And how can we explore them in playful, delightful, engaging ways? This carnival includes lots of perspectives, and approaches the topic from many levels, elementary to college.
A puzzle for 174: What are all the factors of 174? Learning how to find factors goes hand in hand with division and fractions. It's
easy to see that 2 is a factor of 174. Can you see any others before
you divide by 2? There's a "trick" for 3 (and 9), but everything in math
has a reason. Do you know why that "trick" works? I see that 1 plus 7
plus 4 is 12, and I know that 3 goes into 12. Why would that tell me
something about whether or not 3 is a factor of 174? [Solutions at end.
Hint: It's got to do with 10 being 9 plus 1.]
Before I started teaching, I had no idea that fractions might be hard. Part of what makes fractions difficult for some students is how many meanings fractions can have: a fraction of one whole, a fraction of some collection, a fraction of a measurement, etc.
My own troubles with division came from a slight case of (undiagnosed) dyslexia. Why is it that we write a / b, but then we have b going into a, with the numbers in the opposite order? The way we write it made no sense to me. And I got confused if the numbers were big ones. Because of this challenge for me, I learned one of my first problem-solving lessons: Make a simpler problem with the same structure. If I saw 158 ÷ 79, I could think to myself, "That's like 6 ÷ 3." And then I knew what to do - find out how many 79s in 158. Aha, it's 2, just like 6 ÷ 3!
I used to hate the words divisor and dividend. I could not keep them straight. And I still don't know which is which (but if I care, the internet is my friend). And I, my friends, am a math professor. I tell my students often that my bad memory has helped me learn math, because I always tried to make sense out of it, instead of memorizing.
My personal favorite division issue now is why division by 0 is undefined. I wrote about it in my forthcoming book, Althea and the Mysteries of Triangles, Circles, and Pi. I'll share that passage at the end of this post. It's written at about high school level. We can go even higher level with the math and explore 0 / 0, an important concept for calculus that took mathematicians 150 years to come to terms with, which I did in a post a few years back.
John Golden is the Math Hombre.
- He wrote Divide and Conquer in 2010. It's still golden.
- John loves games and works with future teachers. This post, Fraction Reaction, written mostly by one of his students, really gives you a feel for how anyone can create a new game.
- When we make fractals, we can explore what fraction of the area is shaded. Here's something John made in geogebra to allow students to play with fractals.
Denise Gaskins writes at Let's Play Math!
- John and Denise have something in common ... What is it? Games, of course! Check out a perennial favorite from Denise: The Game That's Worth 1,000 Worksheets. (It's just variations on the card game of war, but 'just' is entirely the wrong word for how much you can do with that! And there is a Fractions War.)
- Her Dividing Fractions article helps you see how to think about dividing fractions so that you're not tempted to fall into using a procedure that might make no sense to you.
- Besides writing articles, collecting math games, and publishing books with all of that goodness, Denise has also written a lovely series of stories in which Alexandria Jones solves some sort of problem her archeologist dad, Dr. Fibonacci Jones, encounters. Here's a small taste, where the two of them are puzzling out Egyptian Fractions.
- If that story entices you to want to learn more, David Reimer has written a lovely book, Count Like An Egyptian. (Used copies are available at biblio.com.)
Who the heck is Professor Smudge?
- Here's what they say in their twitter bio: "Wrote Maths Medicine. Rumoured to be called Sigi, after Sigismund (hello surds, hello pi) Arbuthnot, and to have personated Dietmar Küchemann on occasions." Hmm, that's a bit mysterious. Anyway, I found some good looking fraction puzzles on their twitter feed, and you can probably find lots more.
- Here's one:
- And here's another:
Shayla Heavner (aka SJ Bennett) created MathBait, and is the author of Marcos the Great and the History of Numberville. She brings us two factoring games.
Maria Droujkova, founder of Natural Math (my publisher), is conducting a crowdfunding campaign for a lovely book, Farzanah and the 17 Camels, by Dr. Sue Looney, which tells the story of an ancient math puzzle. One part of that puzzle asks: How can we possibly give one heir half of the 17 camels? Join that campaign here (your donation is basically an advance order of the book).
Do you want more?! The Ontario Math Links blog is updated weekly. Browse to your heart's content. (That's where I found Professor Smudge.)
The Math Teachers at Play Blog Carnival was created in 2009. Its name changed to Playful Math Carnival along the way, and it's been going strong for 15 years! (15 years online feels like a century anywhere else.) Links to all past posts available here. I used to include dozens of bloggers in my posts. This one only includes 5 people. (When Google evilly got rid of Google Reader, it really devastated the "math blogosphere".) If you have written something you think we'd like to see, please add a comment.
Puzzle Solutions:
The factors of 174 are 1, 2, 3, 6, 29, 58, 87, and 174. (There are 8 of them. Do all numbers have an even number of factors, or do some have an odd number of factors? Which are which?)
Understanding that factoring "trick" for 3 and 9: Add the digits of your number. If 3 or 9 goes into the sum, then it goes into the original. Why? Let's consider 174. The sum of the digits is 12, and 3 goes into 12. Hmm. 174 means 1*100 + 7*10 + 4, and that can be written 1*(99+1) + 7* (9+1) + 4. If I distribute, I get 1*99 + 1 + 7*9 +7 + 4. 99 and 9 are multiples of 3. So we have 1*99 + 7*9 + (1+7+4). Each term is a multiple of 3. The last term is that sum of the digits we looked at. After reading this, could you explain to someone else why the 3 and 9 factoring "tricks" work?
p.s Here's that ...
Sneak Preview from Althea and the Mysteries of Triangles, Circles, and Pi:
Sofia nods. “I messed up. I had 1 over 0, so I wrote 0 for
my final answer. I’m not really sure why it’s supposed to be undefined instead.
Can you explain that? It did feel kind of tricky to me.”
Mom says, “That’s a great question. And to answer it, we
actually need to go back to some basics. The problem is that division doesn’t
always work. It turns out that dividing by 0 doesn’t make sense. But to see
why, we have to go back and look at how we define division."
She starts writing on the whiteboard and
explaining at the same time. “We know that 6 over 2 is 3, because 2 times 3 is
6. I want to take that relationship and write it in a more generic way. I’m
going to use T for top, B for bottom, and A for answer. When I was younger, I
think I had trouble remembering numerator and denominator. That might be why I
like saying top and bottom. Or maybe I just like shorter words. Anyway, now I
can look at multiplication to help me think about weird division problems.
I look at Sofia. It seems like she’s deep in thought.
Kiara’s taking notes, even though she seemed to know this. I think Mom has
shown me this before.
“So 0 over 5 equals A becomes 5 times A equals 0. So what’s
A?”
Sofia says, “It can only be 0.”
Mom nods. “And there are other good ways to think about this
one. But for the one that tripped you up, this is the only way I know of to
make it really make sense. So now 5 over 0 equals A. What does that become?”
Aiden starts to talk, but Sofia gives him a look. She says,
“I’m the one who doesn’t get this, so let me try. It turns into 0 times A equals 5. But Miss
Annie, you can’t get 5. If you have 0 times anything, you’ll get 0.”
Mom nods and waits.
Sofia continues, “So there is no A that works in this one,
and that means there’s no A for the first one. So it has no answer, and that’s
why they say undefined?”
Mom nods again.
Kiara says, “Whoa! I just knew it was supposed to be
undefined, but I definitely did not know why. And until this moment, I would
not have known I was missing something.”
Aiden is nodding too. “I always knew one of those was
undefined, but sometimes I mix up which one is which. I don’t think I’ll have
that problem anymore.”
Sofia looks at him. “So you didn’t get it either?”
Aiden says, “I had number 5 right, but I think it was a
lucky guess.”
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