Line Drawings for Children - In-Progress

My current mini-project that I'm doing with/for the kids at work is a coloring sheet with numbers - sort of a color-by-numbers idea, except the key for coloring is math based. I haven't added the numbers to the line art yet, so it's just a bunch of straight lines right now.

This is a falcon, when colored in properly.

This is a little more obvious - it's a cheetah! Running! There's supposed to be green grass on the ground, blue sky, yellow cat, and black spots.

Math with Construction Paper

Chris explained the proof for the sum of angles identities (cosine and sine) to me at breakfast. He used colors, because I get lost when there are more than two triangles overlapping, even when all the points are labeled. Also, colors are pretty and pleasant and non-threatening, which helps me get through math that would otherwise intimidate me.

We decided it would be awesome to post a picture of the proof, but Chris wanted a cleaner version than the one he'd used at breakfast. So he and I made a proof out of construction paper pieces (one piece for each of the four triangles that matter for the proof).

Here it is! I'm going to try and explain what is happening in each picture as I go. I think it's surprisingly pretty and comprehensible! Some of the pictures ended up oriented incorrectly (oops!!!).

Start with two right triangles, with the second (yellow) triangle's base equal to the length of the first (orange) triangle's hypotenuse.

The triangle that we use to find the Sum of Angles (the point of the whole exercise) is white here. The base runs along the bottom/adjacent side of the orange triangle, and the opposite side is a vertical line dropped from the highest point of the yellow triangle. This makes the corner of the white triangle which covers the corners of the orange and yellow triangles equal to the Sum of (their) Angles.

The last triangle we need is the green one. It is created from having a line perpendicular to the white triangle split the yellow triangle. I'm not sure if I'm describing it very well, but hopefully the picture is fairly clear.

Alright! Now that we know the general layout of our four triangles, let's start actually defining the lengths of the sides! An important note: all of the triangles we are working with a RIGHT triangles, so we know ahead of time that one of the angles is equal to 90 degrees.

We start with the yellow triangle, and say that the hypotenuse equals 1. We name the angle we're interested in beta (I'll use "b" when I'm typing, but the symbol in the picture is a Greek letter). Using info from the Unit Circle, (and at least for me, the mnemonic "Soh Cah Toa"), we label the adjacent side as having a length equal to cos(b) and the opposite side as having a length equal to sin(b).

Next we label the white triangle. We know that hypotenuse is 1, the same as the yellow triangle. The angle we're interested in is alpha plus beta, or a+b. Again, using the Unit Circle derived information (or my cheating Soh Cah Toa method) we label the adjacent side as having length cos(a+b) and the opposite side having length sin(a+b). But actually calculating cos or sin of two angles added together is a pain. So a formula explaining how to manage it would be clever.

The orange triangle comes next (for me, at least). We call the interesting angle alpha, or a. Unlike the nice hypotenuse of the yellow triangle, the orange triangle is not 1. Laying it next to the yellow triangle, we can see that it turns out to equal cos(b).

Once we know the hypotenuse of the orange triangle, we can state the lengths of the other two sides. First we say that cos(some angle)=(length of adjacent)/(length of hypotenuse), and with some algebra we get (length of adjacent)=(length of hypotenuse)cos(some angle). Plugging in the info we already know about our hypotenuse and our angle, we get (length of adjacent)=cos(b)cos(a). Then we use the same process for figuring out the length of the opposite side. Written out, it looks like: sin(some angle)=(length of opposite)/(length of hypotenuse) and with some algebra we get (length of opposite)=(length of hypotenuse)sin(some angle). Again, plugging in the information we already know, it turns out that (length of opposite)=sin(a)cos(b). Ta-da! That's our orange triangle!

Our white triangle shows us that the length of cos(a+b), or the length of the adjacent side of the white triangle, is less than the length of cos(b)cos(a), or the length of the adjacent side of the orange triangle. Also, the length of sin(a+b), or the length of the opposite side of the white triangle, is greater than the length of cos(b)sin(a), or the length of the opposite side of the orange triangle. But exactly how much more (for opposite) or less (for adjacent)?

Here is where our green triangle is going to come in handy. One side of the green triangle is the length to add to the orange triangle find the total length of sin(a+b) and another side of the green triangle is the length needed to subtract from the orange triangle to find the total length of cos(a+b). The last side of the green triangle is its hypotenuse.

Here is the length of the hypotenuse of the green triangle. It ends up equaling the length of the opposite side of the yellow triangle, or sin(b).

And here is a better view.

To find the desired angle in the green triangle, start by pretending there is a rectangle that is made from the orange triangle (the grey arrows in this picture). Because the top and bottom of the rectangle are parallel, the diagonal line (the blue arrow) creates the SAME ANGLES as it crosses the top and bottom lines. Therefore the part of the yellow triangle showing below the green triangle and above the orange triangle forms an angle of a (or alpha). The corner of the green triangle directly above that completes the right angle of the yellow triangle, and therefore equals (90-alpha). Knowing two of the angles in the green triangle gives us the last angle: (sum of all angles in a triangle is 180)-(90 from right angle)-(90-alpha)=(180)-(90)-(90)+(alpha)=alpha.

Now that we know the angle and the length of the hypotenuse, we can find the remaining lengths using that same method as we used on the orange triangle. Shortcutting a little bit, let's go (length of opposite)=(length of hypotenuse)sin(some angle) which ends up equaling sin(b)sin(a); then the (length of adjacent)=(length of hypotenuse)cos(some angle) which ends up equaling sin(b)cos(a). And there's our finished green triangle!

Here are all of our triangles on top of each other.

Length of the adjacent side of the green triangle plus the opposite side of the orange triangle equals the opposite side of the white triangle. The formula looks like: sin(a+b)=[sin(b)cos(a)]+[cos(b)sin(a)].

Length of the adjacent side of the orange triangle minus the opposite side of the green triangle equals the adjacent side of the white triangle. The formula looks like: cos(a+b)=[cos(b)cos(a)]-[sin(b)sin(a)].

The End. <3

Rose Red and Snow White as a POEM

Ok, so this is a first draft and super rough. But I think it's pretty adorable. I'm planning on illustrating it after I polish it some. Enjoy!

Rose Red and Snow White

Once upon a time there lived a widow with her two daughters.

One daughter was named Snow White and the other was Rose Red.

One night, in the coldest part of winter, there was a knock on the family's door. It was a bear. "Let me in. It is cold out and your home looks warm," the bear said.

The widow and her daughters let the bear come in and lay by the fire. Night after night, the bear returned.

Finally spring came and the bear stopped coming every night. He said he had treasure to protect since the than had softened the ground.

The widow and her daughters said a sad farewell to the bear. Now that the weather was warmer, Rose Red and Snow White spent hours roaming the countryside.

One day, Snow White and Rose Red found a small man with a long beard. His beard was stuck beneath a tree which he had just chopped down. "My beard! My beard!" he called out. "Someone help me!" "Here, this should do the trick," Snow White said. She pulled a pair of scissors out of her day basket and snipped the end of the man's beard, freeing (him) from the tree.

"Oh, my beard! My beard!" the small man cried. "You evil girls, look what you have done to my beautiful beard!" He scooped up a bag full of jingling and jangling treasure and ran off.

The second time the girls met the small, mean man, they found him caught in his own fishing line. After failing to untangle the hook from his beard, Snow White once again got out her scissors and snipped the beard.

"Oh, my beard! My beard!" the small man cried. "You evil girls, look what you have done to my beautiful beard!" He scooped up a bag full of jingling and jangling treasure and ran off.

The third time the girls met the small, mean man, he was being carried off by an eagle. Snow White and Rose Red each grabbed one of the man's legs and pulled him the the eagle's talons.

The man's coat was ripped and torn from the eagle. "You evil girls, look what you have done to my beautiful coat!" He scooped up a bag full of jingling and jangling treasure and ran off.

The fourth time the girls met the little man, his boot was stuck in the rocks of a steep hill. The girls tugged and pulled and finally freed the man from the rocks by pulling his foot out of his boot.

Just then, the bear from the winter ran down the hill and hit the small man in the head with his paw.

The small man fell down dead and, free of the small man's curse, the bear turned in a prince.

The prince invited Snow White, Rose Red, and their mother to live with him at his castle. They lived together happily ever after.

Political and Philosophical Questions

Some interesting questions that have really been changing the way I think about politics lately:

Are poor people to blame for their monetary situation?
Are rich people to blame for their monetary situation?
If different reactions, why?

Is it the government's job to bail out large companies?
Is it the government's job to bail out individuals?
If different reactions, why?

What counts as a bailout? (Social Security? Medicaid? Pro-business regulations and subsidies?)

Do you agree or disagree with the following statement? "A healthy society involves the use of no coercive force."

(For following examples, "freedoms" or "rights" are assumed to stop where another person's begins.)
Coercive force applied to freedom of speech is acceptable/unacceptable. When one vs the other? (For instance, is it one person's job to PAY for another person to have the ability to speak freely).
Coercive force applied to freedom of movement is acceptable/unacceptable. When one vs the other? (For instance, is it one person's job to PAY for another person to have the ability to move freely).
Coercive force applied to freedom of property (the right to claim ownership of something) is acceptable/unacceptable. When one vs the other? (For instance, is it ok to take money? What about land or a house? A car? Who has the right to take money/land/a car?).

Who has the right to use coercive force? (In most places, the reality is that it is more or less the monopoly of the government.)

What is a government? (should be vs actually is)
Who is part of a government? (should be vs actually is)
Who controls a government? (should control vs actually controls)

Ok, so I have some replies to these things, but I don't want to just post them here. I want some people to actually think about these questions. It'd be AWESOME if someone typed up their thoughts, but so long as some people read this and kind of think about their ideas, that's good.

Something I know I care about is having a consistent approach to the world. I try not to overdo it - I think trying too hard to be conscientiously consistent can lead to indecisiveness of a ridiculous extent. But I think these questions made me think about things from a slightly different perspective, and reevaluate how consistent my beliefs actually were.

Looking forward to some interesting discussions with whoever reads this!