## Friday, August 23, 2024

### Farzanah and the 17 Camels

My publisher, Natural Math, uses crowdfunding to support each book they publish. I'm excited about this book.

Check it out, and contribute (which is really the same as making an advance order). I think you'll enjoy it.

Farzanah and the 17 Camels celebrates the excitement and the rewards of solving a challenging and intriguing math problem. Set against the backdrop of the ancient Silk Road, with bustling markets, stunning carpets, fun characters, and camels, the story draws readers into the magic of Farzanah's surroundings.

As Farzanah searches for an unusual approach, a way of solving the problem that no one else could think of, she follows the wise advice of her mother:

"My dear Farzanah, don't be discouraged,” said Mama. “Sometimes, being stuck is exactly where you need to be. I find the best thing I can do is to step away. I free my mind to think about other things. It is in that space that the magic happens. I am able to look at things from a different perspective. With wait time and wishful thinking comes the solution.”

## Wednesday, August 14, 2024

### Playful Math Carnival #174 (the June, July, and now August edition): On Fractions & Division

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.
• 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.”

## Sunday, February 18, 2024

### Free Online Math Circle Has a Few Spots Open Still

We picked a time. We're meeting for nine weeks, each Saturday from March 2 to April 27, for an hour, at 3pm PT / 6pm ET. We still have a few spots open. We'll be playing with Triangles, Circles, and Pi, along with the fictional Althea and her friends. Participants will get an introduction to geometry, proof, and trigonometry.

I'm writing a new book series, Althea's Math Mysteries. In four young adult novels, Althea and her friends explore some of the mysteries of mathematics. The first two books are nearing publication, and the second book needs folks to test it out. In Althea and the Mysteries of Triangles, Circles, and Pi, Althea and friends, with the help of Althea’s mom, explore geometry and proof in order to then learn the basics of trigonometry.

We'd like to find some eager math students to join us in an online math circle, led by me, to explore along with Althea and her friends. Students will participate in 9 weeks of lively small-group sessions: in part a deep and friendly math course, and also a unique book club, with the author refining the story based on student reactions.

Do you know any students who enjoy math, know a bit of algebra, and would enjoy "user testing” Althea and the Mysteries of Triangles, Circles, and Pi? We're looking for a few more young people to try out the activities in this book together.

Commitments:

• Attend 9 weeks of 60-minute live online sessions from March 2 to April 27, each Saturday at 3pm PT / 6pm ET.
•  Read and comment on 1 to 3 chapters of the book each week.
• Keep an informal math journal during this time.

For all who stay the course:

• You'll learn the foundations of geometry and trigonometry (and will get a certificate for completing the course).
• You'll get a signed copy of the published book.
•  Your name or alias will appear in the book's acknowledgements, and you will receive a letter of appreciation for your help with this STEM project. (If you’d like a letter of recommendation later, we will be happy to write one for you.)
•  You’ll get to build community with math friends and mentors.

## Wednesday, February 7, 2024

### Openings Now in Free Online Math Circle

Join an online math circle for students ages 12 to 15 in March and April, exploring geometry, proof, and the basics of trigonometry.

As most of you know, I'm writing a new book series. In four young adult novels, Althea and her friends will be exploring some of the mysteries of mathematics. The first two books are nearing publication at Natural Math.

In Althea and the Mysteries of Triangles, Circles, and Pi, Althea and friends, with the help of Althea’s mom, explore geometry and proof in order to then learn the basics of trigonometry. My publisher and I would like to find some eager math students to join me in an online math circle, exploring some math mysteries along with Althea and her friends. Participants will join lively small-group sessions: in part a deep and friendly math course, and also a unique book club, allowing me to refine the story based on student reactions.

Althea and the Mysteries of Triangles, Circles, and Pi is a fictional story set in the present, in which the characters discuss math, with Mom throwing in a few true stories from the past. Like The Number Devil and Math Girls, this book gives you more the more you put into it by doing the math yourself.

Do you know any students who enjoy math, know a bit of algebra, and would enjoy user testing Althea and the Mysteries of Triangles, Circles, and Pi? We’re looking for 5 to 8 young people to try out the activities in this book together. If interested, please add your name and information here.

Commitments:

• Attend 9 weeks of 60-minute live online sessions in March and April. Times to be determined, most likely 4 p.m. EST / 1 p.m. PST, on Saturday or a weekday (whichever works for more students).
• Read and comment on 1 to 3 chapters of the book each week.
• Keep an informal math journal during this time.

For all who stay the course:

• You’ll learn the foundations of geometry and trigonometry (and will get a certificate for completing the course).
• You’ll get a signed copy of the published book.
• Your name or alias will appear in the book’s acknowledgements, and you will receive a letter of appreciation for your help with this STEM project. (If you’d like a letter of recommendation later, we will be happy to write one for you.)
• You’ll get to build community with math friends and mentors.

## Monday, February 5, 2024

### The Storytellers of Math

Anyone here reading knows that I'm working on my series of young adult novels with math at the center - Althea's Math Mysteries.

But did you know that this drive to tell math stories is growing among budding storytellers across the lands?

Sue in California (me!) is writing Althea's Math Mysteries. Four of them!

Shayla (aka SK Bennett) in New Mexico is writing the next book after Marco the Great and the History of Numberville. (I'm loving this one. I'm so glad there will be another.)

Sarah in Washington has written some wonderful fairy tales about physics and math. I'm reading Newton's Laws: A Fairy Tale right now.  (Currently free.)

And of course there are about a dozen lovely stories from the authors who work with Natural Math.

Who else is out there, writing tales of mathjoy that I haven't discovered yet?!

## Friday, December 8, 2023

### Illustrating Althea

I'm not much good at drawing, but most of the illustrations in Althea and the Mysteries of Triangles, Circles, and Pi are math work. I can do those. So I've put my own illustrations into the manuscript as placeholders. There will be a professional illustrator, later.

This past week, I was looking for where more illustrations are needed. I decided Althea would draw a map of California while thinking about their summer trips. So I drew it. Their home is in Berkeley, they go to camp in Quincy, and they're planning a trip to San Diego to visit Legoland (because her younger brother Rudy would love that, and their moms met in San Diego).

I had fun drawing the map. First I downloaded a map of California into goodnotes, as the template for my document. Then I outlined it, and then changed the template to make the original fancy map go away. Finally, I got to add the places of interest to Althea.

As one friend on facebook pointed out, it would help to make the line weights different for the outline versus the routes she's imagining. The professional illustrator can either take care of that, or show me how it will work best for the published book.

## Sunday, November 26, 2023

### Althea's Math Mysteries

This blog may not be as active as it used to be, but it's a good way for me to remember some things. My first post about my Althea stories was in September of 2019, so I've been working on the first two books in this series for four years now. I'm hoping we'll be able to publish them in about a year.

I have a very hopeful timeline that puts publication in October. But we all know that things never go as well as we hope. (And I'm wishing we could do it just a bit faster than that, so they'd come out in time for Math Storytelling Day, September 25, Maria's and my birthday.)

I have pretty complete drafts done of Althea and the Mystery of the Imaginary Numbers and Althea and the Mysteries of Triangles, Circles, and Pi. Soon I'll be asking for folks to read the manuscripts and comment on them. (Email me at mathanthologyeditor@gmail.com if you'd like to be one of our readers.) After that's done, we'll do our usual (Natural Math publishing's usual) crowdfunding campaign. And then there will be illustration, copy editing, proofreading, page layout, and books!

Here's my mock-ups of the covers, and lots of information that goes with the books.

At a few points, I've really wanted to see what these would look like as actual paperback books. lulu.com made that easy. Two books cost me under \$25. I've done that 3 times, while I've polished up the books. What you see in the photo above is me holding the 3rd printed draft copy of Althea and the Mysteries of Triangles, Circles, and Pi.

In other news, I retired on May 20 from my full-time job teaching math at a community college. Teaching online was way too much work, and less satisfaction than teaching in person. I'm still covid-cautious, so I also didn't want to go back to teaching in person. Retirement has been wonderful. I'm working hard on the books, visiting Michigan where I help my dad (who's 90), cleaning up my house a bit, and working on my yard.

All that was plenty for about the first five months. When I noticed that I sometimes felt like I had nothing to do, I posted in a Beast Academy group on facebook that I was thinking of offering a class. Someone suggested that I apply for a position with AOPS. (Art of Problem Solving is an amazing online resource, providing great math textbooks and classes, and they wrote the fabulous Beast Academy curriculum.) I did that, and I'll start teaching for them soon!

I will definitely be blogging more over the next year, to let anyone still reading my blog know what's up with these books. (If you're out there reading this, I'd love to hear from you.)

## Monday, July 31, 2023

### Playful Math Blog Carnival #166

This blog carnival has been around for 14 years. Almost every month for 14 years, someone has added a post to this collection. That's quite a long life for an internet phenomenon. (Congratulations, Denise, for keeping this going!) If you'd like to see any of the previous posts in this series, check them out here. For many years, blogs were a big part of my time online. But not so much lately.

When our number (166 now) was in the 20s, 30s, or 40s, I'd make sure to have that many links. Nope, I don't have time to find 166 great links (and you'd get tired just looking through them). But they're out there. I have learned so much from bunny hopping around the web of math bloggers over the years. And even though blogs aren't the popular thing now, most of the old ones are still out there, waiting for you to find them and get excited.

The 166 puzzle: It turns out that 166 is a 'centered triangular number'. If you start with a dot, and then you put a triangle around that, and a bigger one around that, etc, you get up to 166. How many triangles did you use?

I have just run out of envelopes. How should I make myself one? (a puzzle from Fawn Nguyen) What shape of paper will you use?

Online Mathy games

Geometry Puzzles

Find the blue area

This one stumped me (no trig required).

What fraction is shaded? Catriona Shearer (@Cshearer41) made this, along with gobs more, mostly pretty challenging, which she posts on twitter. And here's a collection of over 300 of them.

Beyond the Games & Puzzles

Searching for more? Some good hashtags are:  #MTBoS #ITeachMath  #Elemmathchat #MSmathchat

If you've seen some good math pedagogy out in the wilds of the internet, add a comment.

## Friday, February 24, 2023

### Althea's Math Mysteries

I've been working for a few years on Althea and the Mystery of the Imaginary Numbers. It's almost ready for the illustrator. But I wanted to dive deeper into the characters, and started working on the second book in the series, Althea and the Mysteries of Pi. I'm about 80 pages in on my first (very rough) draft. It has been a blast writing this, because I pretty much know where I'm headed. (Although sometimes I worry that there's too much math, and not enough character development. And then I back up and think about Althea, Kiara, Sofia, and Aiden some more.)

Today I wanted a good place to put links that the book refers to, so I made a temporary website for all the books. It's a google site (for now). And I made mock-ups for the book covers. It helps me to organize my thoughts, and it is super exciting to see. So even though the books won't be published for another year (or 2?), maybe this will tantalize you. Here's the site for Althea's Math Mysteries.

When this book is pretty much done, I'll start working on the third one, Althea and the Mysteries of Infinity. I have lots of ideas for that one, but they have no structure. I have no idea where I'll start or end.

When I'm all done, and these 3 books are published, maybe I'll have realized that there are more books in the series. For now, it's looking like just the three.

## Thursday, September 8, 2022

### What does it mean when we feel we "understand" something?

On facebook, I'm in a group for people who use Beast Academy (even though I'm not using it), because Beast Academy fascinates me. I love most of what they do.

A parent today posted that she was confused about the BA way of multiplying 59*59. They have you draw a 60 by 60 box, and then take off one row (of 60) and one column (which is now 59). Your box is now 59 by 59, and its area is 60*60 - 60 - 59. Cool.

She wasn't seeing it, so she taught her kid the standard algorithm. Lots of people were giving her flak for that. (We each do our best, so I don't see why folks would jump on her.) She replied to them that she thought learning it multiple ways was a good thing.

I wrote: "Sure it's great to do things multiple ways, but does he really understand the algorithm you showed him? (Do you really understand why it works?) I think that's why you're getting pushback here."

She said they both understood it. I replied that I'd have trouble explaining to a young kid why you "put a 0". She wrote: "
I just told him we put it to show that the one number is done. I don’t know if it’s accurate but he understood it. I don’t really remember it ever being explained in school."

So what she originally meant when she said he understood it, was that he could follow the steps and get it right. Not that he understood why it worked.

I think this is common with math.  People think 'understand' means the same as 'can follow the steps'. But I'm afraid that doing math without really seeing why each step makes sense is part of why a lot of people don't like math. It's surely why we easily forget how to do those things.

Here's an article by Richard Skemp, written back in 1978, about why the deeper understanding, which he calls "relational understanding" is a better way to approach math. (He calls being able to follow the steps "instrumental understanding".) I wrote about this topic and this article ten years ago here, but people's ideas about math haven't changed much in that time.

Of course, this parent can still explain to her son why the standard algorithm works, so she hasn't somehow wrecked the beauty of Beast Academy, as some people seemed to feel. And that's what got me writing - I want to see how well I can explain the standard algorithm.

I figure that the standard algorithm packs in a lot with as little writing as possible. (Maybe when we didn't have calculators, and had to do lots of by-hand multiplication, writing as little as possible was considered an important goal for the way we write things down?) So I figured that it needs to be unpacked a little. That's what I tried to do here.

The first calculation is adding up all 4 areas. The one to the far right is the standard algorithm. The first number in the standard algorithm (531) is the 81 and the first 450 added together (with carrying), and the second number (2950) is the other 450 and the 2500 added together. It's surely as little writing as possible, but it hides so much! Does my unpacking on the left help?

It all makes sense to me, but the Beast way feels more fun. (And I don't have to write anything that way. I can hold it all in my head!) What do you think?