Existential Math

The Absolute Value of Humans

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This week I taught 5th graders about integers.  We discussed the number line they were used to, which started with zero and only had positive numbers, and then I added in negative numbers.  We practiced getting used to this number line by doing the integer dance.  It closely resembled the electric slide, but the point was to get the students used to moving positive and negative directions.  After all, number lines can be tricky.

We are taught in primary grades that zero is a starting place.  Eventually we get fluent enough in math to amend our previous thoughts about zero and the number line to include negative numbers.  So, now our number line increases to show that really, zero is the middle of a big scale with infinite integers on each side. All numbers on both sides gain their identity from the zero, or the origin.  So, you could say the value of a number is dictated by how far away a number is from the origin, or zero.  This is the absolute value. This week while teaching I wondered “what if we saw people with absolute values, instead of only positives and negatives?”

Zero is the only integer that is neither positive nor negative.  In theory we all want to be greater than zero.  No one wants to be a negative number.  Theoretically being a negative number means you are worse than when you started at the origin.  Zero technically means no objects are present.  If you offer a child zero popsicles, zero pieces of candy, or zero trips to the zoo it might seem to them that zero is a negative, but really, it isn’t.  It’s just unrealized potential.  Zero of something just means nothing has been added or taken away.

Life is just a giant number line.  It’s a series of positives and negatives.  We take steps forward, and we take steps backward. Sometimes we are way ahead of the origin.  Sometimes we are behind the origin.  There are times we tend to feel our value is less than zero when more bad than good happens.  I was encouraged when I thought about absolute value.  We can be -6 or 6 from zero, and the absolute value of both of these is still 6.  There are no negatives in absolute value.  So, even when we have terrible things happen, our value is never negative.  We are always just so many spaces away from where we started, and knowing that can help us get back on the right path, which is right back up the number line.  As we take steps up and down the number line, instead of focusing on the negatives, it is a lot more fun to pretend we are just doing the electric slide.

Existential Geometry: Don’t Be An Asymptote

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Everything is made up of lines. As humans we spend our days drawing lines, crossing lines, walking lines, or signing on dotted lines. We drive inside of lines, we stand in lines, we argue over property lines, and lines of code make up how you see this blog entry. Lines are important. Lines give our world meaning it would not have without them. We live our lives in relationship to lines.

Today I was teaching geometry, which at first seems harmless enough, but when people wrote the book on analytic geometry something tells me what goes on in my brain isn’t exactly what they had in mind. Today when explaining parallel lines to a student it occurred to me how sad parallel lines would be if they had human characteristics. We know that parallel lines are always the same distance apart. They will never touch. They never get any closer than they were the day before. They point in the same direction. The old joke goes, “Parallel lines have so much in common! It is a shame they will never meet!”

The better existential question is do the parallel lines know each other exists? Do they whisper across the plane? Do they know just how congruent they are? They are always moving in the same direction, but will never share a common point. How many people do we pass in the halls at work or on the road in our cars and we might all be driving toward a common location, but we will never meet? We might be listening to the same radio station, eating the same breakfast, or drinking the same brand of coffee, all while never knowing this, and will never come to a point in our lives where we do know this. I think of this kind of line cinematically as The Lake House. In order for those two lines to meet, they had to move to an entirely different plane. I’m glad they did too, because what a frustrating plot otherwise!

There are also lines that intersect. You can have intersecting lines, and some of those intersecting lines happen to be perpendicular. I know I have met a lot of people that I have come into contact with at some point in my life, and I have no idea where they are right now. The geometrical reason for this is we weren’t perpendicular lines. Perpendicular lines are interesting, because they aren’t as tragic. Two lines are perpendicular when they are right angles to each other. Perpendicular lines meet. They meet at one spot. We know that when they meet at this one point, they form a right angle. There is hope there, right? They meet! And when they do, they form not just an angle, but a right angle. Intersecting lines are the Casablanca of the movie world, while perpendicular lines are more When Harry Met Sally. You do have to realize though that even Harry and Sally knew when the time was right to be perpendicular. For a while we were all on the edge of our seats.

There is but one winner for the saddest existence when it comes to lines. It goes to the asymptote. Asymptote derives from Greek meaning “not falling together.” These lines come as close as you can get and never intersect. They appear as though they might intersect, and as soon as they have you persuaded they are going to intersect, they run parallel all the way to infinity. I think of this kind of line as the movie Lost In Translation. I’ve also never forgiven the movie writers for that ending, but I digress.

Archimedes said, “The shortest distance between two points is a straight line,” but once you grow up you realize it is a lot more complicated than that. You see that rarely are lines straight, rarely are they what you imagine them to be, and you tend to want to color outside the lines just a bit. You also realize that the best of friends know how to read between your lines, and most importantly the most critical line is your smile. As long as you still have that, you have everything.

Adventures In Algebra: Existentially Solving For Why

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Math Atheist

I’ve always been what Calvin of Calvin and Hobbes calls a math atheist.  I understand why numbers exist, and I comprehend the importance of basic math.  What does not seem to compute with me is when we start having to deal with fractions of numbers, decimals, and the rest of the concepts that start the downward spiral toward adding in the letters of the alphabet. That’s right; I’m talking about algebra.

As a math atheist, I believe the science involved in math is somehow made up in order to confuse those that have an undying affection for all things word related and none of the things number related. But it would stand to reason if I love words, then I would find comfort in the kind of math where they start adding in letters.   I not only do not find comfort in it, I honestly think that there is not a time I have ever used it after the year I graduated high school and passed the math in college.

So, why talk about math if I hate it so immensely? It’s one of those things in life that makes you realize God has a sense of humor. If someone had asked me this time last year what I would be teaching this fall, I probably wouldn’t have said math.  I probably wouldn’t have said high school.  I know I definitely would not have said high school math, but I love teaching, so when an opportunity opened to teach Algebra I to 9th graders for a maternity leave I decided to give it my best shot.  Going straight to high school from 2nd grade was a lot like being dumped at a dance contest with two left feet and chronic knee pain, but I was determined to make this work.

Things have been going well, and I’ve been reflecting on the first three weeks of teaching. I’ve learned a lot by teaching algebra, and I have a much different take away on the things I have taught in the first few weeks of school.  I already told you I wasn’t endowed with a math brain, so you probably won’t be shocked that while the things I’ve learned had to do with math, they weren’t all that mathematical.

Most people know that all numbers fall into specific categories.  Two of these categories you can place numbers into are rational and irrational.  People can also fit nicely into those two categories.  This is applicable no matter who you are or what you do.  The biggest thing to remember is that with people and numbers the rational and irrational both fit into a larger category: real.  It’s okay to be irrational, as long as you remember to stay real. And who among us hasn’t parked themselves for a few days on the side of  irrational?

In order to solve any problem in algebra, you have to remember how to combine like terms.  You can’t add things up that aren’t supposed to be together.  How can you tell if numbers should be together?  You have to examine them closely.  Variables can be a pain. Adding 4r to 2b is still 4r + 2b.  It’s never going to add to up 6.  And, just like in algebra when you have people in your life, sometimes it is best to keep the ones that are alike together.  They will always add up correctly that way.  I’ll be the first to say I’ve tried to combine myself with people that were unlike me and tried to make us add up.  In order to have a successful relationship with those people I have to remember to see individual parts, and not all jumbled up together.  This helps me when solving problems in both math and with people.

The equation is another concept we address immediately in algebra.  What a great concept! When we think of an equation we can think of a scale.  The things on one side have to equal what is on the other side of the equal sign.  Sure, there are different ways of writing them on each side, but they will add up.  That variable is just an unknown on one side.  We all have unknowns when trying to make things add up in our life.  We are sometimes given puzzles with missing pieces. We can think of the task of working equations as practice for the real world.  There’s always going to be a y (or a why???) on one side of the problem.  There is always a solution.  All you have to do is remember to keep a balance when you are adding and removing things in your life to make sure the scale doesn’t tip to one side.

Inequalities have also gotten me thinking.  You solve an inequality like equations for the most part. Sometimes things aren’t equal on both sides in math or in real world relationships.  Sometimes one side is greater than the other side. The hard part for so many students is figuring out after you solve the problem whether you have to flip the sign the opposite way.  The rules say if you divide or multiply by a negative you need to flip the sign.  So, that’s also a great life lesson, right?  If you are multiplying by a negative, you are going to have to change something.  Rarely do negatives do anything good in our lives if nothing changes.

So I am not teaching what I thought I would this fall.  The students are doing a fabulous job putting up with an algebra teacher that is doing her best to teach the rules while questioning them herself.  For now you can find me  reminding students to show their work while I  glance at a clock every day with a sign on it placed by the classroom teacher that says,“Time will pass, will you?”  What a great question! I hope I pass.