# Another Blog Post About Fraction Division

Person 1 mentioned on Twitter tonight that there aren’t enough blog posts out there about fraction division.

Person 2 recommended using rectangles to model fraction division.

I decided to help Person 1 using Person 2’s suggestion. Though the meat of this post is in this PDF I made and not in the blog post itself:

Fraction Division (1/29/2015 Stacked all fractions and made a cover page.)

Good enough, I say. I made a lot of examples fairly quickly, so I apologize if there are some errors here and there. Let me know and I can easily fix them and re-post the PDF.

And now there are n + 1 posts on fraction division on the internet. Woot!

# Suck It, Gauss!

I’m a little bit embarrassed to share this story, but damn it, I’m so proud of myself I’m going to do it anyway. Apparently an extra hour of sleep on Sunday morning is all I needed to figure out the answer to a math problem that left me feeling dumb back when I was in middle or high school.

For the last hour or so that I was in bed, I dozed. At one point as I was going in and out of random dreams, an image flashed across my mind. It was a clever solution to a math problem I saw posted on Twitter a week or two ago. I can’t even remember what the problem was, just that the visual solution had this shape:

There were numbers in the squares, maybe counting how many squares are in each column? All I remember is that this visual represents how the answer to the problem is 42.

Clearly my memory of this problem is not what I’m proud of. The important thing is that this random image triggered the line of thinking I continued on.

So after thinking about this image for a few seconds, I started to think about a math problem that has plagued me since I was in school: How can you quickly find the sum of all the numbers from 1 to 100?

I have no clue why this question popped in my mind, but it did, and since I was only semi-conscious, I just went where my thoughts took me.

I decided to try a simpler version of the problem to see if any insights struck me. I also tried thinking of a visual to see if it would lead me to a clever solution like the 42 image I was just thinking about.

I started with finding the sum of all the numbers from 1 to 5, and I thought of it like stair steps:

Then I played around with the squares in my mind to see if reconfiguring them would lead to an epiphany. My first thought was to redistribute the 5 squares to make columns of 4 which led me to this:

Having three 4s and a 3 made finding the sum a lot faster, but it didn’t seem generalizable. I decided to try summing all the numbers from 1 to 6 to see how my redistributing strategy would work.

Now I ended up with four 5s and a 1, which is easy to compute, but clearly showed me I wasn’t going to get anywhere satisfying with this strategy.

Feeling a bit frustrated, I started thinking about what it would look like if I made stair steps of all the numbers from 1 to 100. That’s when it hit me. The shape of the numbers from 1 to 100 is a jaggedy right triangle that looks kind of like this:

I thought to myself, if I duplicate that triangle, I can rotate it and fit it with the original triangle to make a rectangle. So long as I can find the number of squares in the rectangle, then all I have to do is half it to find the number of squares in the original triangle. All that was left was determining the dimensions of my rectangle.

I knew the length had to be 100 because I was finding the sum of the numbers from 1 to 100, so it is 100 units long. From there I had to think about the width of the rectangle if the tallest portion (100) was set on top of the shortest portion (1), which got me 101. I tested out the next few columns that would line up to make sure I wasn’t making a mistake. The 99 column would rest on top of the 2 column to get 101, and the 98 column would rest on top of the 3 column to also make 101.

I felt pretty confident that the height of my rectangle would be 101. Then it was just a matter of multiplying 101 by 100 and halving the product to find the number of squares in the numbers 1 to 100.

I had no idea if 5,050 is the correct answer, so I decided to go back to testing out smaller numbers to see if this idea was solid. I thought to myself that the dimensions of my rectangle for anything I try have to be n and n + 1, where n is the final number in my series.

So going back to the sum of the numbers from 1 to 5, I would be creating a rectangle that is 5 by 6, and then halving it to get 15. Then I tested out summing the numbers from 1 to 6. This makes a rectangle that is 6 by 7, which gives me 21 when it is halved.

Success!…followed by grabbing my phone and consulting the internet to make sure I was actually correct. When I saw that I had derived the same equation that I had been shown when I was a student years and years ago, I was beyond elated.

To give you a bit of back story, back when I was in school, I remember our teacher presenting us this exact problem, and I didn’t have a clue how to approach it. You may as well have asked me to sum all the numbers from 1 to 1,000,000. I hated challenges like this because I always ended up feeling stupid. Even after giving up and waiting for the teacher to share the solution with us, I still didn’t really get it.

The whole experience reinforced my fixed mindset thinking that I wasn’t actually intelligent at math. I may have been a whiz at learning procedures and following them, but when it came to doing “real” math, I was terrible at it. I think hearing the story about how Gauss solved this problem really quickly in elementary school didn’t make me feel any better about it either. Instead of being inspired about the power of looking for patterns, it only made me feel that much worse about myself.

It’s funny how many people see me today and hear what job I do and they automatically think of me as a “math person.” It’s true, today I am a “math person,” but only because I’ve put a ton of effort into relearning how to think about and do math for most of my adult life.

And finally, after all that hard work, here I am lying in bed at the age of 37, casually entertaining mathematical thoughts as I wake up, and I finally figured out that damn problem all by myself!