Alpha's Adventures in Mathematics: Book 1

Recap

When we last left Alpha he was calculating time zones trying to figure out how he could take off and land at exactly the same (local) time... And before he knew it he was being called to board the plane and travel across the world's biggest puddle to find his forever home!

Come join Alpha on his adventure through the sky...

Chapter 6: Over The World's Biggest Puddle (Gulp!)

14 hours is a long time to be on a plane and me and the other dogs had to travel in cages. So it didn't take me long to start to get very, VERY bored.

But instead of letting the boredom get to me, I used my imagination and guessed where the plane passengers were heading.

Then I heard some people on the plane talking...

This is my first time on a plane. I'm kind of nervous.

No need to worry. Planes are super safe and I'd know. It's my job to design and test them.

I felt the same way as the nervous passenger, so I listened carefully as the woman explained that she was an aerospace engineer who designed, built, and tested airplane engines. Casually, she mentioned that it all starts with calculations that are all about how the engines work. This includes calculating how fast the plane has to go; how much the plane, with and without passengers, weighs; and how much power it needs to take off, slow down, and move safely through the skies.

But the fun part was when she started talking about how they test plane engines for safety.

We freeze them.

We heat them.

We soak them in water.

We zap them with electricity.

We even throw things into them!

We also look at what happens when they fail and investigate what went wrong and why, so that we can prevent it from happening when the plane is in flight. 

But that's very rare.

We collect all the data from these tests, plot it onto graphs, and look for patterns so that we can fix things.This way, every engine on every plane is as safe as possible. All the passengers and crew, and even people on the ground, are kept safe.

But how does the pilot know where to go? It’s not like there’s road signs in the sky.

Well, that's actually really interesting too...

This lady knew that plotting a course on a globe is much different than on flat surfaces like a map. Planes travel in a curve while cars and other land vehicles travel in a relatively straight line. This curve changes the geometry quite a bit and creates new set of rules to follow.

Pilots use this fascinating geometry, along with calculations of how much fuel they need or if they need to refuel, what the weather is like, wind patterns, and the direction they are flying to plot a course. The math allows them to gets passengers to their destination the quickest or the most efficient way.

From Pam

Alpha wants to learn about how plotting a course on a globe is a tricker than plotting a course on a flat map.

This is because a globe is curved and geometry works differently on a sphere. For example right angles on a sphere are different than on a flat surface.

On a flat piece of paper a triangle can only have one right angle:

But on a sphere a triangle can actually have 3 right angles:

This is because on a sphere lines curve.

It's also because in 2-dimensions we only have an x- and y- axis to draw shapes on (length and width), but a sphere is drawn in 3-dimensions using x-, y- and z- axises (length, width and depth).

When x- and y- meet they make one 90 degree angle:

But when we have three axises (x, y and z), we have three places where a 90 degree angle can form:

If we look at this on a sphere (or a globe) it's where the lines from the equator to the pole meets.

Neat huh!

So this means, when a plane is flying from one place to another, they are not following the geometry rules of 2-dimensional shapes so they have to plot their courses differently than what they might look like on a map.

You can see what Alpha's route from Seoul to Toronto looks like on a map here (notice it's not a straight line).

Wow! I never realized that flying was so interesting. Thank you for explaining this. I feel much better about flying.

You're very welcome!

It made me feel better too and soon I found myself listening to the super safe engines and trusting we were going to make it to Toronto in one piece. Dreaming of what waited for me when we landed, I feel asleep.

When I woke from my nap, the plane was landing and I heard the flight attendant telling the passengers all about Toronto...

Welcome to Toronto, home of the CN Tower, which, at 553m or 1,815ft tall, is one of the tallest free-standing structures in the world. 

It also has a population of over 2.8 million people. Making it the 4th largest city in North America.

Wow! I knew Toronto was big but didn’t realize it was huge. Paju only had a population of about 420,000. My mind started tinkering with numbers, calculating how much bigger the population of Toronto was than Paju. This meant I had to take the population of Toronto and subtract it from the population of Paju.

 

2,800,000 - 420,000 = 2,380,000

That’s 2.38 million MORE people in Toronto than Paju.

If Toronto is that big, how does it compared to other cities around the world?

Help Alpha

The biggest city in the world by population is Tokyo with 37.5 million people.

How much bigger is Tokyo than Toronto or Paju.

How about where you live?

How much bigger (or smaller) is where you live from Toronto, Paju or Tokyo?

 

Try At Home

The CN Tower is one of the world's tallest free-standing structures in the world.

Making a super stable free-standing structure isn't as easy as you may think and involves a lot of math.

One of the trickiest parts is making sure the base of the building is strong and stable enough that it can support not only the weight of the rest of it but it can handle things like wind. (Neat fact, if you stand at the top of the tower on even a calm day you can see the tower moving back and forth with the wind)

So engineers need to build these towers to be super stable, even in high winds. 

So here's your challenge...

Gather the following materials:

thick spaghetti noodles (thin ones will work but it will make it harder)
marshmallows
that's it

What you want to do is try to build the tallest tower you can using just noodles and marshmallows (the marshmallows are to connect the pieces of pasta together).

You can also use white glue to connect the pasta pieces together and you can use straws instead of noodles - be creative with the materials you choose.

Bonus: you can use a fan or a hairdryer (careful with the heat) to blow on your tower and see if it still stands.

Experiment with different types of bases (a round one like the CN tower, or a square one like the Eiffel Tower in Paris). Play around with different ways to connect the noodles (or straws or pipe cleaners) to each other.

The most important thing to remember though is you can't have anything outside the tower supporting it. So no strings attaching it to the ground/table or noodles propping up the sides. The tower has to be completely free-standing.

Get a good tower design you want to share? Send a picture of it to lookmath@vedavox.ca (with your parents permission of course) and we'll put it in our student gallery.

Happy building!

Before I had too much time to get distracted in the math I was unloaded from the plane and we all left the airport for the next part of our journey.

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