Math 10 Honours

Math Inquiry: The Math and Appeal Behind Animations


(Not all animations of pictures entering are supported on PowerPoint Online)

 

I started this inquiry with the question: How does Math affect Animations? But then as I was working, my project slowly changed into the appeal of an animation (Mostly 2D Animations). So from there, I worked my project around the 12 Principles of Animation by Frank Thomas and Ollie Johnston. From there, I was able to pinpoint math concepts within three of the 12 principles.

 

My curiosity continued during this project which made me think:
Why do we find human like components appealing?
and
Is it possible to create something appealing without doing math?

With this, I think that I can go further and start finding out what other things we as humans find appealing. I found other presentation that people made about Phi and found that interesting. I was able to see how a number like Phi is used in so many ways to make something seem ‘pretty’. I will also be able to look into abstract art and see if there are any math concepts to be found within the work of art.

Desmos Art

 This is the Santa Face I made on Desmos!

I stared by doing as many circles as I could (Head, nose, eyes) because then later on, it would be easier to put things over them. To change where I wanted the circles, I added or subtracted value from x and y which moved the circle around.  For Santa’s smile, I used a parabola (x²=y) and changed the y intercept and made it so y<-1 so it would stop the parabola from going into infinity. Then I started making a line across his head to start his hat. I made the hat with straight lines, restricted where the lines went, filled in the shape (using > or< instead of =) then restricted where the shape would finish filling. I added a little pompom on the top of his head and some points to make it look like a few stitches. The last part was his beard and I used two parabolas, one for the top and one for the bottom to fill up to the top. I had both a black and a orange one to try making a brown colour since Desmos doesn’t have a white fill.

That’s how this Santa was made!

 

Algebra Tiles Review Math 10H 2017

Factoring two Binomials

When expanding two binomials, you create a chart so that the binomials are on the outside, and you multiply them inwards to get your answer on the inside. In this model, it shows:

(x+2)(x-2)= x²-4

As the red sides are negative and the positive sides are green, blue and yellow. You can see that there are an equal amount of positive and negative x’s in the result. Because of that, the x’s cancel each other out because your adding 2x, then subtracting 2x, which leaves you with 0.

Simplifying a Trinomial

 In this trinomial, x²-x²+3x-3x (or 0), it can be simplified back into the binomial/trinomial it originated from. When you start, I find it easier to try to figure out one side first. If we start with the left side, you can tell that it has to be 2x because you need and x to get a x². To coninue we can assume that the left side is (x-x). On the top side we can use the x²s to figure out the first x. We can guess that it’s -x because a double negative is a positive and a positive and a negative is a negative.

We can then look and see that since on the far right side, they are all single x’s. With that we know that they are 1’s/ Since our first x is positive, we can see that it’s 2-1. So in the end, we simplified it to (x-x)(-x+2-1).

Surface Area of a Sphere

Partner: Micole

In this activity, we had cut an orange in half (the top and bottom, not symmetrically), we used to those halves to trace circles on a piece of paper. With those circles, we started peeling the two halves and filled the circles with the orange peel so that you could only see orange. In the end, we filled exactly 4 circles with orange peel and found that the surface area of a sphere is 4πr².

Garibaldi Lake Task

How to Estimate the amount of Water in the Lake

Since we already know the surface area of the lake (9.94 km², as provided by Wikipedia), we can use the average depth of the lake to estimate the amount of water that the lake holds:

SA= 9.94 km²
Average Depth= 119m

9.94 km²  =  \frac{1000 m}{9.94} =     9940000 m²

9940000 m² • 119 m = 1 182 860 000 m³

 

This shows that the volume on the lake is around 1 182 860 000 m³. We know that one cubed meter of fresh water at 4°C is 1000 kg. Considering the lake’s high altitudes, we can assume that the water’s pretty cold. Since there is 1 182 860 000 m³ of water in the lake, we multiply it by 1000 kg.

 

1 182 860 000 • 1000 kg = 1 182 860 000 000 kg of water.

 

Since 1 L of water is equal to 1 kg of water,this shows that there is roughly 1 182 860 000 000 L of water in Garibaldi Lake.

What would happen if the Barrier faulted?

I think that if the barrier were to break down, the water (theoretically) will flow out. The results of the water coming out, may be similar to what happened at the Hope Slide (but on a grander scale). The highway below would be washed out and the amount of water may also flood out in the valley. It would definitely affect the provincial parks in the area and Squamish. Depending on how much water flows down south than flows up north after it washes up against Cloudburst Mountain, the water from Garibaldi Lake could reach Howe Sound and even effect the lower regions of Whistler. The water that also washes up against Cloudburst Mountain can possibly take out parts of the rock, reforming the mountain completely from it’s previous state.

Sources

“The Barrier.” Edited by Volcanguy, Wikipedia, Wikimedia Foundation, 11 Oct. 2017, en.wikipedia.org/wiki/The_Barrier.

Enns, Andrew. “Garibaldi Lake.” Wikipedia, Wikimedia Foundation, 9 Oct. 2017, en.wikipedia.org/wiki/Garibaldi_Lake.

DigitalGlobe. “Garibaldi Lake.” Google Maps, Google, 2017, www.google.ca/maps/place/Garibaldi+Lake/@49.8779355,-123.2091781,16868m/data=!3m1!1e3!4m5!3m4!1s0x54871de627a17cc9:0x913f7329f4571920!8m2!3d49.9366437!4d-123.0272101.

Miller, Derek K. “Aerial Photos of the Hope Slide, Hope, B.C.” The Hope Slide – Hope-Princeton Highway #3, British Columbia – Aerial Photos, Penmachine.com, 12 Aug. 2004, www.penmachine.com/photoessays/2004_08_aerial2/hopeslide.html.

Math 10 Honors Numbers Summary

A lot of the stuff that we did on Numbers was a good review. Some new things that learned were:

  1. The Lowest Common Multiple:
    Finding all the prime factors of the two or more numbers, then multiplying it all together. Ex. The LCM of 15 and 35 is 105. Common prime factors of 15 and 35 are 3, 5 and 7. So you multiply the three and get: 3•5•7= 105.
  2. Finding Perfect Squares and Perfect Cubes using Prime Factors
    You’re able to find if a number is a perfect square or a perfect cube by looking at the prime factors of the number. So if the prime factors come in pairs, ex. 4= 2•2, then it’s a perfect square. If the prime factors come in triplets, ex. 9= 3•3•3, then it’s a perfect cube. If they come in both, then it’s a perfect square and a perfect cube, ex. 64= 2²•2²•2²/2⁶.
  3. How to make Repeating Decimals a Fraction

  4. Mixed Radicals and Entire Radicals
    You’re able to simplify radicals so that they are smaller numbers, but have a coefficient, or just keep it as a whole. An example of a Mixed Radical is 5√2 and an entire radical is √50.
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