The math we should have been taught in high school

I love The Oatmeal and they have a funny little bit about what they think should have been taught in high school. I just posted the math for obvious reasons but you can check out the whole strip that covers all the subjects.

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Khan Academy

If you haven’t heard about Khan Academy, then you probably aren’t in the field of education. There has been tremendous criticism of Khan Academy by educators because it ultimately just lectures to students and many educators – at least the ones I keep up with – prefer having students approach math in a different way. Math is often seen as something that you learn and then practice. With more practice, you’re supposed to – with the traditional thought behind mathematics education – get better at math but I don’t think the point of math is to be able to memorize and practice problems over and over until you’ve mastered the skill. Although this can be one way people learn math, I don’t think they’ll truly understand it this way. Many other teachers really like Khan Academy because it does the lecturing for them. The program does have its benefits as it does allow students to work and learn at their own pace. Unfortunately, I’ve witnessed teachers just have their entire class watch a lesson in lieu of them teaching it, then pass out a worksheet for them to practice this new concept that they have just “learned.” While I may not be its biggest fan, I know Khan Academy does help some students. So for those of you that may get some benefit from it, check it out. Here’s a sample of what the videos are like:

Wait, What?

Sometimes math can be used for trickery. There are card tricks that utilize math in order to amaze others. Then there are just some calculations that appear to hold true but really “hide” a small truth from people who may not see it.

Example:

a = x

(add a to both sides)

a + a = a + x

(combine like terms)

2a = a + x

(subtract 2x from both sides)

2a – 2x = a + x – 2x

(combine like terms)

2a – 2x = a – x

(factor out a 2)

2 (a – x) = a – x

(divide both sides by a – x)

2 = 1

Hmmm… That doesn’t make sense. There’s a mathematical fact that isn’t explicit here but with knowledge of it, you won’t be fooled by the conclusion here that 2 equals 1. Do you know what it is?

While you’re working on that, check out Math-Fail.com for some math-related funnies.

NASA Has Space Math

Here’s just a quick heads up on the awesomeness that is NASA. They have a page dedicated to math! My favorite are the “Problem Archives” where teachers can present real data, pictures, graphs, etc. from NASA to their students in an already lesson-friendly PDF. There’s nothing better than supplementing a textbook with this kind of stuff!

Space Math @ NASA

Please Excuse My Dear Aunt Sally

The reason your math teacher made you (most likely) remember PEMDAS, or “Please Excuse My Dear Aunt Sally,” is because without it we wouldn’t be able to properly read mathematics. For example, you can read the sentence, “The quick brown fox jumps over the lazy dog,” and picture what it is trying to convey. There’s a dog, he or she is lazy, maybe laying down. A fox, which is the color brown, jumps over the dog and is doing this at a fast speed. Everyone reads this sentence and understands it in the exact same way. We know which adjectives describe which nouns and what the verb is referring to. We know the fox jumped, not the dog. We know that the word “brown” describes the color of the fox. We learned the rules of English in order to be able to read, write, and communicate so that everyone knows exactly what we are trying to say. The order of operations is the equivalent of this, but for mathematics.

Without the order of operations, everyone would do calculations however he or she wanted and everyone could get different answers. If I say to you, “First add two and eight, then divide your sum by five,” I could write it out as: (2+8)÷5 or (2+8)/5. If you follow the order of operations properly, you will get the solution I was referring to. If you don’t, you don’t understand what I’m trying to say. I also use the knowledge of the order of operations to write those expressions. I couldn’t write 2+8÷5 because the answer you would get, following the correct procedures according to the order of operations, would not be the one I was referring to. The order of operations allows everyone to be able to communicate mathematics correctly, without further explanation or confusion.

PEMDAS stands for: Parentheses, Exponents, Multiplication, Division, Addition, Subtraction and that’s the order you’re supposed to do the mathematics (sort of). You’d think math teachers would love PEMDAS, and I’m sure you thought they did, but most math teachers have a huge problem with it. It makes their students think that multiplication comes before division and addition before subtraction when in fact multiplication and division are essentially the same operation, as is addition and subtraction. With two small examples, let me explain. 2-1 (two minus one) is mathematically equivalent to 2+(-1) (two plus negative one). 4×2 (four times two) is mathematically equivalent to 4÷½ (four divided by one half). If we have a longer expression for which we will use PEMDAS to simplify, like 4÷½-2+4×2, and we don’t use PEMDAS correctly, then when I rewrite the expression but keep it mathematically equivalent, like 4×2+(-2)+4÷½, you’d get a different solution even though both expressions are mathematically equivalent and give you the same solution if you follow the order of operations correctly.

Ok, that was a lot of boring math stuff and I apologize, so here’s an awesome graphic design I found on shirt.woot! to lighten up this post a little. Enjoy!

You can use the order of operations to read this expression!

Mathy FoxTrot

Since Borders is going out of business, I had to use a gift card I had so I purchased Bill Amend’s themed FoxTrot collection:

In general, this is a pretty amazing little collection. I highly recommend it to those that are especially fond of math, science, and/or computer programming. As you can see in the above photo, I took my new gift to myself outside and decided to share some particularly mathy gems in one of the least-tech ways possible, by taking pictures of them. Please excuse the quality, I just don’t have a scanner. Enjoy.

Fermat’s Last Theorem

Yesterday Google honored Pierre de Fermat with a doodle of the Google logo remembering his last theorem. When you left your mouse over the doodle, it read, “I have discovered a truly marvelous proof of this theorem, which this doodle is too small to contain.”

PCWorld did a nice short article on the doodle, Fermat, and this theorem. Fermat was known for being confident in his mathematical ability, which wasn’t unwarranted, and for writing in the margin of Arithmetica, written by the mathematician Diophantus, that the theorem xn + yn = zn, where x, y, z, and n are non-zero integers, has no solution with n greater than 2. He then went on that he could prove this but that there wasn’t enough room to write the proof.

I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.

There are arguments as to whether or not Fermat actually did have a proof, but it is unlikely he did since it took hundreds of years, many mathematicians, and very complicated mathematics to prove it.

When Will I Ever Use This?

It’s the question that makes math teachers cringe: When will I ever use this?

It’s true that many of the exercises that teachers make students go through may never be used outside of math class. What I hope is that students see math as a different way of thinking about problems and how to solve those problems.

“There are people who say, ‘I’ll never need this math. These trig identities from 10th grade, or 11th grade.’ Or maybe you never learned them. Here’s the catch: Whether or not you ever again use the math that you learned in school, the act of having learned the math established a wiring in your brain that didn’t exist before, and it’s the wiring in your brain that makes you the problem solver.”

-Neil deGrasse Tyson