Opposites Attack

Some people don't learn from visuals like the ones I posted last night, so I want to give a couple of examples of the normally-written math problems that you may come across in your everyday life. But first, I want to give a couple of vocabulary words that may help with your understanding.

integer: a whole number, either positive or negative

absolute value: 

• Well, here's the Google definition: the actual magnitude of a numerical value or measurement, irrespective of its relation to other values.
• Here's the plain-speak definition: The distance of a number from zero. 

When you're adding integers (one negative and one positive), you want to look at the absolute value of the negative number. 

 As you can see, the absolute value of the ten is higher than the absolute value of the seven, because 10 is greater than 7. Even though negative 10 is lower than positive 7, the absolute value of 10 is greater! Here's another reminder of the definition of absolute value:

So now let's take a look at this problem:

Because the negative number has a greater absolute value, the answer ends up being negative!

On this one, the absolute value of the positive number is greater, so the answer ends up positive.

It always works!*

I hope you all have a great day, and that this helps someone out there in the ether.


*As long as you're adding one negative with one positive.**

**Actually, it could also work if you're adding two positives.***

***Technically, I guess it would also work if you are adding two negative numbers.

Flagpoles

Dear Mr. C,

The other day, I was trying to explain to someone how to add a positive integer with a negative one. That's when I realized I had no idea what I was talking about! Can you shed some light on this subject, because I'm thinking I may just have a conniption if I can't figure it out soon.

Love,

Florence Jean Castleberry

Dear Flo,

I feel your pain. It can be hard to think about positive and negative numbers without going crazy. Let's see if I can help you out, though.

First off, it may help to think of positive numbers, such as five, as a five foot hill. And also, think of a negative number, like four, as a four foot hole in the ground.

This is a four-foot hole. The bottom of it can be thought of as -4.

Now, let's say we have a ten-foot long flagpole. I know this isn't exactly normal, but let's say we dug that four-foot hold to plant the ten-foot flagpole. 

If you sink the flagpole down to the bottom of that hole, you know the entire ten feet are not going to be sticking up into the air. We know that only six feet of it will be sticking out of the ground. 

We can think of this like the mathematical expression -4 + 10. And then we know that if you make it into an equation, you can say that -4 + 10 = x, where x = 6. 

Similarly, let's set up another flagpole. This time, we don't want the high winds to take down the flag and flagpole. So we are going to dig a nine-foot hole this time! Crazy! 

So we already know that we are starting this math problem out with a -9.

Sadly, the first time we went shopping for a flagpole, the one we picked out was only seven feet tall. Needless to say, it sunk all the way into the ground, and the top of it still doesn't reach the top.

We can think of this as the mathematical problem -9 + 7 = x. If we solved this problem, we would find that the answer comes to -2. It's still two feet underground!

So we head back to the flagpole store. This time, we're smart enough to get one that is long enough. Let's see, the hole we dug is nine feet deep, so we are going to need a nine foot flagpole!

Uh-oh. We didn't take it into consideration this time that we have a math problem that looks like -9 + 9 = x. A negative nine plus a positive nine will just level out to zero! -9 + 9 = 0! You're flat on the ground!

So on our third trip to the flagpole store, we get an 18-footer. Here, we can discover that -9 + 18 = 9! Positive nine!

I'll post some more examples of this tomorrow without the visual aids, plus a helpful tip when it comes to adding a negative with a positive.

I hope this helped out out, Flo. Don't forget to know your negatives, but to always remain positive.

What Do Owls Dream?

Dear Sensei:

Can you please explain this whole coordinate plane thing? I'm completely baffled by all these negatives and positives. I can't stand it!

Sincerely, 

Buddy Lembeck

Sure thing, Buddy. It's not as bad as it seems. You see, the coordinate plane, which tells you exactly where to put points on a flat surface, is divided into two dimensions, x and y, which we talked about last time.

 

From these two lines, it is divided into four sections, called quadrants. For some reason, they are always noted with Roman numerals, one through four.

It may seem a little strange to start in the upper right hand corner, and then work your way around, but it may help if you think of drawing a big C. You can remember this because the C stands for "cool", because Math is cool! (Yeah, no one ever buys that. I always try, though.)

From here, you can remember that point of origin is (0,0) from the last post. To the left of that are the negative numbers, and to the right are the positive numbers. On the y axis, everything above the point of origin are positive numbers, and everything below it are negative.

Here are some examples using (6,3). You'll notice how things change when you turn either one or both negative. 

You can always remember that anything with two positive coordinates will be in the first quadrant. And then the others follow suit from there. (-,+) is in the second quadrant, (-,-) is in the third, and (+,-) is in the fourth. 

(-,-) looks like the owl is sleeping. That one's my favorite.

I hope this helps clear a few things up for you, Buddy. Have a great day, and don't forget to learn something today.

Location, Location, Location

Dear Jeffrey, 

How are we supposed to know which number to use first when they give us coordinates? If they give you (5,7), are you supposed to go over 5 and up 7, or over 7 and up 5? Is it always the same? How is anyone supposed to know, anyway? 

I'm baffled, and I don't have time for this. 

Signed,

Ethel Beavers

Ethel, I hope you haven't given up, because you will be relieved to know that it's not so bad. First off, it helps if you can remember that the coordinates are always written in the form of (x,y). You can just remember this because they are in the order that x and y are in the alphabet.

The x-axis is the line that is horizontal, or side-to-side. You can remember this because "x goes across" and an x is a cross.

You can remember that the y-axis goes up and down. I always teach that y goes up because of "Wise up, fool!", which sounds exactly like y's up, fool! True story: I had a student who told me years after being in my class that he still said "Wise up, fool" in his head every time he had to use a coordinate plane. So a coordinate plane looks like this:

See that (0,0) in the corner? Those are the coordinates for that spot. This is also called the Point of Origin. I always think the coordinates for the point of origin look like an owl.

See?

Okay, how about now? Now do you see it?

Anyway, once you know those facts, you are ready to find any coordinates on the plane. For today, we're only going to focus on the positive numbers.

For (7,9), you would simply slide over 7, and then move up 9. You can also think about it like moving a ladder. You have to move the ladder before you can climb it. If you try to climb it and then move it, you're going to have a pretty rough time.

Then you can figure out how to find other coordinates like (0,7) by moving over zero spaces and up 7, and then (14,0) by sliding over 14 spots and then moving up no spots.

I hope this clears things up, and that you feel like you are a little wiser than before. Don't worry, we'll get into the negative numbers another time. But for now, I hope you have an easier time finding your spots when given two coordinate numbers.

Now, Where Do I Begin?

Mr. C,

I have a question that I hope you can answer.  How can you tell exactly where the line goes on the graph? I get everything about how you can tell what the slope is, but I can't figure out where to put it on the coordinate plane. 

Can you please help me?

Best,
Sweet Lou

Good question, Lou. The best thing about this is that there is a formula. Once you know and understand this formula, you can tell not only how steep the slope is and which direction it goes, but you can tell where the slope crosses through the y-axis. This will help you know where it goes on the graph.

The formula is called slope intercept form, and it looks like this.

In this formula, you leave the x and the y. You'll understand why in a minute.

The m stands for the slope. This is usually a fraction, but sometimes it's a whole number (remember yesterday's post, where the slope was 3, and stands for 3/1). There are a few theories about why they use m for the slope, but no one really has it nailed down. Just know: m = slope!

The b stands for the y-intercept. This is very important. This tells where the line will pass through the y-axis.

So with a slope like this one...

The line would begin at the 3 on the y-axis, and from there, it would go up 4 spots and over 5 spots. Like this:

See? It's fun! Knowing your slopes doesn't have to be an uphill battle. Now you know how to chart a simple line given the slope-intercept formula.

We'll go over more slopes in future days, but for now, I hope this clears up your worries, and you can get started trying slopes. It will also become clear how these are actually useful in everyday life. (More or less.)

I hope everyone has a great day, and an amazing adventure through the wild world of mathematics.

Take This To the Bank

The Shortest Distance Between Two Points

Dear Mr. C, 

I'm trying to figure out how I can tell the slope if someone just gives me two coordinate points. I asked my math teacher, but I three of my friends Snapchatted me while she was answering, so I had to read them, and then she got all mad because I was looking at my phone instead of listening, which is totally unfair and dumb, because she doesn't understand about Snapchat. 

Whatever,

Jen Snodgrass

Well, Jen, I sure hope you're ready to read something other than Snapchat messages, because I'm about to lay some knowledge on you.

Let's say that your teacher gives you these two points: (2,1) and (4,7). Simply plot those points.

Then, draw the line between them, and then see how far across you have to go, and then how far up you have to go in order to get from one to the other.

You can see that this one went over 2 spaces, and up 6 spaces. Since slope is always written as rise over run, this one goes up 6, and over 2. Therefore, the slope is 6/2.

Of course, we always want to simplify our fractions, so 6/2 is equal to 3/1, and that simplifies all the way down to 3.

The slope of this line is 3!

Now, let's say your two points head in a downward direction. You will remember from yesterday's post that this would indicate a negative slope.

If they gave you (3,8) and (8,6), you would still put the dots in their place and draw the line between them.

You can see that this one goes across 5 spaces, and down 2 spaces. So, "rise" over run would indicate that this one has a slope of -2/5.

In this case, "rise" over run was actually "drop" over run, but it still works the same. It's just negative!

So there you have it, Jen. Hopefully you will put down the phone and listen, because your teacher is most likely a pretty smart lady.

Until tomorrow, I hope everyone has a fantastic day and that you will find that math is all around you. It helps to speak the language!

What To Do When You Forget Your Lines

Greetings, Mr. Carter

Sometimes being honest can have its downsides. My son, Tad, asked me if I could help him remember what the name of a line's slope is if it goes straight up and down, and what it's called if the line goes straight from left to right without going up and down at all.

I remember one of them is a 0-slope, and the other one is called no slope, but I had to confess to him that I did not know which is which. 

Can you help me, so that I can tell him? He has a quiz this Thursday. 

Mathematically, 

A. Lincoln


Hello, Mr. Carter

I cannot tell a lie. I also struggle with this.

Respectfully, 

G. Washington

I understand the struggle, fellows.

Luckily, just a couple years ago, I ran across a very handy way to tell these apart.

First off, you can remember that a 0-slope always is a horizontal line, or from side to side, but using this visual reminder:

Then the up-and-down slope is often called "undefined," although for many years I knew it as "no slope". Luckily this one works either way:

I'll go into how to name these lines on a coordinate plane tomorrow, but for now, I think you'd have to admit that it's easier to remember than you thought!

I hope you gentlemen have a fantastic Presidents' Day, and that everyone comes back soon for more mathematical fun at Mr. Carter's Dojo. 

Wicked Inclinations

Dear Sensei:

Hello! I was wondering if you could tell me how I can tell if a slope is positive (+) or negative (-). I can't stand it when I get the numbers on the slope correct, but I get the answer wrong because it was negative! Help me!

Your friend, 

George Glass

George, I wish I could have saved you from this mathematical head-scratcher earlier. But the good news is that once you learn, you will hopefully never forget.

When you are reading a sentence, you read it from left to right (assuming you're reading in English or other European languages, that is).

It's the same thing when you are looking at a slope. You need to begin on the left, and move to the right in order to tell if it's positive or negative.

If the line goes uphill, it's positive! You're going uphill.

If the line goes down, it's negative. You're going downhill, so it's going in a negative direction.

So if you look at the lines above, the one on the left could be 2/3, but the one on the right would be -2/3. 

I hope this helps, and that you have a positive week, and stay tuned for more mathematical know-how here at Mr. Carter's Dojo.

Precious

Happy Valentines Day, folks.

Uphill Battles

Shoot! I meant to have this to you before now. Apologies that I'm just now getting it out there, as if any of you are just hanging on this blog to learn how to do slopes. 

If you find a problem in your homework that says something like this...

Draw a line with a slope of 4/5 that passes through point (1, 4). Give the coordinates for another point on this line.

...let me show you what to do.

First, find the point on the coordinate plane. Put a dot there.

Now that you know where to start, simply move up four (rise over run) and to the right five. Our slope is 4/5, so that's what we do.

Don't forget, a line goes in both directions for infinity, so it's best to draw the line longer than just a few centimeters. Don't be lazy.

For another point that goes through this line, I would just use the second point that you found. In this case, it's (8, 6). Of course, if you did draw the line (using a straightedge) in both directions, you could also use any coordinates where the line passes through. But you have to make sure that the line passes through an exact coordinate, so it's best to find a good spot for that.

I'll try hard not to wait so long before the next lesson!

Hit the Slopes

It's time to talk about slopes.

No, this isn't going to be a profound post about the downward slope society is in, or the slippery slopes that lead to bad habits. This is quite literally just about slopes in math. I have worked with a number of students lately who are working on this right now, so I thought maybe I would just post here and then I can tell them to refer to this blog if they have any need for review if there is anything they have forgotten.

First off, a slope is written as a fraction that tells how steep a slope is. It is written as rise over run:

So if you had a fraction, for instance, of two fifths, the slope would rise up by two for every five across. A 2/5 slope would look like this on a graph:

There are a couple more things I want to talk about here. First, you often here that a slope has to be a fraction, but they don't give you a fraction. They just give you a regular number. Well, in this case, the number is a fraction (in fact, nearly all numbers are). The number is simply over one. So if the slope is 7, that means the slope rises up seven and over one.

Lastly, let's say you have a slope that is a negative number. In this case, reading left to right (just like in a sentence), the slope will go downhill.

In this low-quality photograph, the top slope is 2/3. The bottom slope is -2/3. Left to right. If it goes up, it's positive. If it goes down, it's negative.

That's all for today. Next time, I want to talk to you about a slope that passes through a certain point on the coordinate plane.

Having the Answers; Still Failing the Test

Please click here to see the article that ran on CBS WTTV 4.

This story makes me sick, but it's not surprising at all. It's been going on for years. The State Superintendent of Education, Jennifer McCormick, is stepping down because all she was being asked to do was follow the money. She wasn't asked to look at problems with education within the state of Indiana.

For years, the education system has been forced to follow (or in some cases with administrators, followed willingly) the desires of curriculum companies. These are the very curriculum companies that own the news agencies. It's a fact.

It has nothing at all to do with the greater good of the students. Teachers are helpless to do anything about it other than run for office and try to change it from there. And between McCormick and the Glenda Ritz term, you can see how well that turns out. Tony Bennett's lying and cheating a few years ago with the Christel House Academy is an example of the corruption, but a pretty minor one in the grand scheme of things.

And the fact that some members of the administration follow them so eagerly just shows that there is a high probability that they are receiving compensation from said companies. You don't even have to do any more than draw the lines yourself--you don't even have to "follow the money", but it's pretty safe to say that if you did, you would find a whole mess of scandal and dirty money.

Now that is sad. 

Happy For Now

He didn't see his shadow!

The bad news is that he's historically known to be wrong. In fact, he's correct only 39% of the time. But look how happy he is, so that's something.

Hopefully everyone gets a chance to watch a good Bill Murray movie this weekend in addition to the Superbowl as you enjoy the thaw.