The Ordered Pair

We know from an earlier post that a number shows where you are on a number line. For example, we know the number 7 tells we are 7 units to the right of 0. The number -9 tells us we’re nine units to the left of zero.

What happens if we need to locate a point against two number lines, though. Let’s call our normal number line the “x-axis”. Then, let’s create another number line that intersects the x-axis at zero on both lines, and call this new vertical number line the “y-axis”.

Now we need to describe a point in relation to both axes. Let’s start with that point 7 units away from 0 on the x-axis. That point could be on the x-axis, or it could be somewhere above or below the line. That’s where the y-axis comes in. It tells us how many units above or below the x-axis the point is. For example, our point could be located 3 units below the x-axis, making its location on the y-axis -3.

So our point is at 7 on the x-axis and -3 on the y-axis. That’s a lot of words to describe where this point is located, so we can use a shorter notation called the ordered pair.

The basic pattern for an ordered pair is (x, y).

Since we know our values for both axes, we can plug the 7 in for x and the -3 in for y to come up with the ordered pair (7, -3)

The important thing to remember with an ordered pair is that the first number always describes where you are in relation to the x-axis, and the second describes where you are in relation to the y-axis.

Basic Trigonometry Functions

On the bulletin board that plays home to things the students give me, there is one large drawing that will never come down. His name is George, and the band on his hat reads “SOHCAHTOA”. George was a student’s means of remembering basic trigonometric functions.

While none of my other students have met George, most of them have a page to remind them what SOHCHTOA stands for. This mnemonic will help you keep the trig functions straight. In order, SOHCAHTOA means:

sine = cosine = tangent =

That’s great, but what does it mean?

Opposite and adjacent describe the legs of the triangle in relation to the angle you’re working with. The leg not touching the angle is the “opposite” leg. The leg touching the angle is the “adjacent” leg. The hypotenuse will always be the longest side of the triangle. To solve, find the formula for the function you’re working with and plug in the appropriate numbers.

If you have the measures of two sides, but need the third, you can use the Pythagorean Theorem to find the missing side.

The Pythagorean Theorem

If you’ve been doing math for any period of time, you’ve probably run into a formula that looks like this:

a² + b² = c²

This very useful bit of math is called the Pythagorean Theorem, named after Greek mathematician Pythagoras. Put into words, the above equation tells us that the sum of the square of the two legs of a right triangle equals the square of the hypotenuse (the longest side of a right triangle).

This formula has many potential uses. If you know the length of both legs, or one leg and the hypotenuse, of a right triangle, then you can solve for the missing side using the Pythagorean Theorem.

For example, we are given that a = 3 and b = 4. Let’s solve for c.

3² + 4² = c²

9 + 16 = c²

25 = c²

= c

5 = c

You can also use the Pythagorean Theorem to prove whether or not a triangle is a right triangle. For this example, let’s have a = 5, b = 10, and c= 13.

5² + 10² = 13²

25 + 100 = 169

125 = 169

Wait a second! 125 does not equal 169. Therefore, a triangle with sides 5, 10, and 13 is not a right triangle.

Let’s try it again with a triangle with the sides 7, 24, and 25.

7² +24² = 25²

49 + 576 = 625

625 = 625

This triangle is a right triangle!

In a bind where a right triangle is involved? Try out the Pythagorean Theorem and see if it helps!

Tips for memorizing multiplication facts

Last month, I encouraged everyone to learn their facts. Brian Foley responded by encouraging the teaching of memorization, but he failed to include a link to his own post on memorizing multiplication facts.

Brian’s post is a great start, but I’d actually change it slightly. I’d actually make myself flash cards for each “family”. One card would read 0 * 3. The next would read 1 * 3. And so forth and so on.

Once I felt I honestly knew my facts in order, I’d shuffle the cards and try saying them to myself out of order. After completing cards for  each family (1 – 12), put them into one big deck and practice them. (You can also blend smaller groups of cards together if you want to work on a specific set of numbers.) Rote memory only works if we can take it out of context.

This would also work for addition, subtraction, and division if you’re struggling with any of those.

Keep a math journal

My students are finally through the state’s math test, and I think I’m just as grateful as they are. I’ve spent the past two weeks helping them review formulas and processes. They’ve been so afraid of not being able to remember what they need at the right moment, and there’s only so much I can do at the last minute to help them.

In the middle of the testing, I found this article on creating a math journal, and I realized it would probably help a number of math students be more successful in their studying from a technical point of view.

Some math teachers have their students develop “toolkits” for their math class. The teachers provide a packet where the students fill in grid squares with the formulas and diagrams the teacher defines as important. Then the packet is dropped into a backpack, never to be seen again. (Yes, I’ve watched this one happen far too often.)

What if instead of packets for each class, students were encouraged to develop a math notebook to follow them through school? When they learn a new school, they write it into the notebook. They draw whatever pictures they need. They define the process in their own words. They write an example problem and its step-by-step process. If there is a theorem involved, the student should copy it verbatim, and then write it in their own words underneath.

Why would a math journal help students? First, as the article points out, it provides the student a chance to reflect on what they’ve learned, to put their learning into their own words. Second, if students were encouraged to use the same notebook throughout school (or change them only when the notebook is full), then when they encounter a forgotten skill, they can find it in the notebook and review it. It would also make reviewing for tests easier because everything would be in a simple-to-read format (if written correctly) in a concise space.

Perhaps I should start encouraging students to try out the math journal. I work mostly with high schoolers, but it could be very helpful for them.

Facing the open-ended response

My students are panicking this week as they prepare for the state math exam next week. Posts are going to be a bit sparse this week while I work on talking them down off ledges and reminding them that they can, in fact, pass this test.

I can understand their nerves. For many of my high schoolers, next week’s test will set the course of the rest of their high school career. They have to be able to solve problems, and they have to be able to explain their work. Some of them don’t have a clue about math vocabulary, so they’re trying to learn a whole new language in less than a week.

My best advice for them, and for anyone who has to take a math test where there are open-ended responses, can be summed up in what I told one of my sophomores last night.

1. Read the question carefully.
2. Answer the question being asked. (It doesn’t hurt to underline the question.)
3. Write something, even if you aren’t sure.
4. Show your work.
5. Label your work.
6. Double check your math. (Or…at least, ask yourself if the answer seems reasonable)
7. Don’t panic!

Compound inequalities

Not all inequailities are simple. Some have a third part to the equation, or have a separate equation altogether.  Regardless of what it looks like, it’s still solved the same way as a simple inequality.

Let’s look at this first example:

3 < x – 4 < 8

Although there are three parts to this inequality, we’re still going to get x by itself. In this case, we’ll add 4 to all three sides of the equation. We add it to the middle to cancel out the -4, and then we add to both the front and back terms to keep our equation balanced. The new equation looks like:

7 < x < 12

This means that x could only be a number between, but not including, 7 and 12.

Even if the compound inequality has you multiplying or dividing by a negative number, you’ll still perform the operation on all three terms. For example:

4 < x/-2 < 6

resolves to:

-8 > x > -12

when you multiply each term by -2. If you think about the number line, you’ll realize it makes more sense for -8 to be greater than -12.

Not all compound inequalities are in a line like this, though. Some compound inequalities describe two different lines.  For example:

x > 3 or x < -3

If you have to solve for a variable in this type of compound inequality, all the inequality rules apply.

 

Simple inequalities

Continuing on this path of solving equations with a variable, let’s take a look at solving simple inequalities.

At this point, you’re comfortable with solving for a variable on one or both sides of the equation, right? Honestly, solving simple inequalities isn’t much different. There’s one small detail you have to watch out for, but otherwise it’s the same process.

Let’s take a look at the following inequality:

3x + 5 < 26

Just like we did when we were solving for a variable on one side of an equation, we’re going to start by grouping all of our non-x terms together.  We’re going to subtract 5 from both sides, which will leave us with:

3x < 21

Now, we’ll divide both sides by 3 (again, we’re always working to keep our equation or inequality balanced) to get:

x < 7

It’s fairly straightforward, right? Well, let’s take a look at the next example.

4x + 4 > 6x + 8

We’ll start by grouping our x terms together. We’re going to subtract 6x from both sides. This gives us the following:

-2x + 4 > 8

Now we’ll subtract 4 from both sides to group our non-x terms together. The inequality now looks like this:

-2x > 4

We now divide both sides by -2 to get:

x < -2

Wait a minute! The inequality just changed direction! What gives?  Well, what gives is that whenever you multiply or divide both sides by a negative number, the inequality changes its direction. That’s the unique fact you have to remember when working with inequalities.

Solving for a variable on both sides of the equation

On Monday, we looked at solving an equation where the variable was on one side of the equation. Today, we’re going to look at solving an equation where the variable is on both sides.

Actually, if you’ve got the hang of solving for a variable on one side, then solving for a variable on both sides of the equation is fairly simple. It just involves one extra step. Remember when I said that the most important thing in solving for a variable was getting the variable on one side of the equation and everything else on the other side? This is why that’s so important.

Like we did on Monday, we’re going to start with an equation:

8x – 3 = 2x + 3

Let’s start by moving all of our x terms to the left. This means we’re going to subtract 2x from both sides of the equation, again keeping our equation balanced. The equation now looks like this:

6x – 3 = 3

Now, we’ll group the non-x terms to the right. For this step, we’ll add 3 to both sides, still working to keep the equation balanced. This leaves us with this:

6x = 6

Now we can divide both sides by 6 to get:

x = 1

See? It’s just like solving an equation with the variable on one side, with an additional step of moving the variable terms to be on the same side.

 

Solving for a variable

One of the most basic skills (and the one that will get you through every upper level math) is the ability to solve for a missing number in any equation.

The key to finding a missing number is to undo every operation in the sentence. What does that mean? Basically, it means that you’re going to add where you see subtraction (or vice versa) and multiply where you see division (or vice versa).

For example, let’s solve for x in the following equation:

3x – 6 = 9

The first step we need to do is group all of the non-x terms on the same side of the equation.  To do that, we add 6 to both sides of the equation. We add it to the left side to cancel it from that side, and we add it to the right side to keep our equation balanced. The result looks like this:

3x = 15

Remember that 3x is the same as saying 3 * x, so we can now divide both sides by 3.  Again, we divide the left side by 3 to cancel it from that side, and we divide the right side by three to keep our equation balanced. This leaves us with:

x = 5

We now know the value of the variable.

The most important thing to remember when solving for a variable is that we need to group like terms so the variable we’re solving for ends up alone on one side of the equation. This makes it possible for us to find the value of that variable.