Revenge of the Math Tips!

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Prove it!

“But why?” was the question my math teachers hated to hear most. But it’s a perfectly good question, and any decent math teacher should be able to answer it. If you don’t believe a mathematical fact, the good news is you probably don’t have to accept it on faith. Besides the definitions and axioms, we don’t have to accept anything as true in mathematics unless there’s a proof for it. The proof will usually explain exactly why something is true.
A proof is a step-by-step chain of logical reasoning which takes you from the premises (the given information that we are accepting as true because it is an axiom or it has already been proved) to the conclusion (the statement of the theorem, the thing we are trying to prove).

For example, suppose I don’t understand why it’s true that in order to multiply two exponents with the same base, I can just add the exponents. (In other words, why it’s true that for any numbers a, x, and y, .) I can ask for a proof, and it might go something like this:

If you can’t find the proof in your textbook and you don’t have the luxury of having a tutor go through the proof with you, the good news is you can find many proofs of theorems online.

If you’re confused or have a question about something in math, chances are the explanation is in the proof.

To Be Continued…?

Oh, no! Could this be the end of the daring Math Tips? Tune in next week to find out, true believer! Same math-time, same math-channel! And now, a word from our sponsors:

Do you need a math tutor in Tampa, FL?

Call (813)-563-MATH! (563-6284)

We also do online tutoring in our virtual classroom, no matter where you are in the world!

The Math Tips Strike Back!

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Your textbook is your best friend

If all of our students had a good relationship with their textbook, MathBusters might be almost out of business.

Okay, we’re pretty awesome so we’d probably do just fine. But you will save a lot of money on tutoring if you learn to use your textbook.

Your textbook has all the definitions and theorems clearly marked. When you’re studying for a test, go through the chapters that are covered on the test and write down all the definitions. Write down any theorems that you can’t derive easily from the definitions and axioms.

Your textbook has an index. Can’t remember what a math word means? Look it up in the index.

Chances are, your textbook (or notes that your instructor has given you or put on their website) already contains everything you need to know for your class. Did you miss a day of class? Most likely whatever you missed, it’s in your textbook. Did you not understand something the instructor said in class? 8 times out of 10, your textbook has a clearer explanation, one that you can read again and again until it makes sense.

An excellent study habit is to find out on which days your instructor is covering which chapters in your textbook, and then read that chapter the day before your class. Make notes of what you had trouble understanding. Wait for your teacher to get to that point in the lesson. See if their explanation clears things up. If not, raise your hand and ask them to explain more. Because you’ve already seen this in the book and had a chance to think about what it was that was confusing you, your question will be clearer and the teacher or professor will be better able to answer you so that you understand.

Do you need a math tutor in Tampa, FL?

Call (813)-563-MATH! (563-6284)

We also do online tutoring in our virtual classroom, no matter where you are in the world!

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Know the definitions

Math has its own language with its own vocabulary. You wouldn’t expect to do well on a cell biology test if you don’t know what a ribosome or an endoplasmic reticulum is. And chances are, you won’t get the best grade you can in your geometry class if you never learn the difference between corresponding angles and alternate interior angles. Most math textbooks make it very clear what their definitions are. They are usually inside a big colored rectangle and have the word Definition at the top. If you’re not understanding what a problem is asking, ask yourself if you know what all the words mean. If you don’t, and it seems like it might be a math word, go back through your book chapter and look through the definitions to see if it appears there. Or look up the word in the index at the back of the book, maybe it was in an earlier chapter and you just forgot about it. Make sure you periodically review the definitions you’ve learned so far to make sure you have them memorized, or at least can remember them with a little bit of thought.

Understand the notation

“Notation” is just a fancy word meaning “the way we write math.” Understanding the standard way of writing things in math is obviously important if you’re going to get anywhere. Imagine trying to read a book without knowing the alphabet!

Misunderstanding notation actually can get you the wrong answer sometimes. Imagine if I was asked to find . I remember that my teacher said is just funny mathematical notation that means the same thing as . That must mean that means the same thing as which means the same thing as 1/sin(1), right? I plug that into my calculator and get 57, which is the wrong answer. It’s the wrong answer because in this context, the power of -1 is not actually an exponent at all, but really just means “the inverse sine function,” and so gives me the angle between -90 degrees and +90 degrees that I would need to put into sin(x) to get 1. (in this case, the answer would be 90 degrees).

“But that’s dumb! Why does a power of negative 1 mean one thing sometimes, and another thing a different time?” Well, it’s all about context. Mathematicians have adopted a convention (an agreed upon way of doing things) to interpret -1 in this way because it’s suggestive. and (which can also be written ) are inverses because they “undo” each other. If I multiply 5 by and then multiply that by I will get back 5, the thing I started with. Similarly, and are inverses because and . In other words plugging x into the sin function and then plugging that into the function, we get back x, the thing we started with.

Do you need a math tutor in Tampa, FL?

Call (813)-563-MATH! (563-6284)

We also do online tutoring in our virtual classroom, no matter where you are in the world!

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Math is different from other subjects

At MathBusters, we have a lot of experience teaching and tutoring math. And you know what? Some of us weren’t always good at it. Some of our tutors learned the hard way that they kept making the same kinds of mistakes over and over. We’ve seen our students make these mistakes over and over again, too. The following is a list of math tips for students who are struggling.

Math is learned by doing

You may be used to doing well in many classes such as history, social studies, even biology by simply paying attention in class and absorbing the information through osmosis. Or you may learn better by reading. You read your book, the information goes in your memory bank, and you spit the information back out on the test.

Unfortunately this will probably not help you succeed very well in most math classes. Maybe when you were learning your multiplication tables in elementary school this worked, but in order to learn any math beyond basic arithmetic (algebra and up) you’ll need to be more actively engaged with the material. That means it is not enough just to pay attention in class or read the chapter. You have to practice, practice, practice. That means doing lots and lots of problems. At MathBusters, our experience has taught us that learning math is is like learning to ride a bike or play the guitar. You can watch someone ride a bike or play the guitar till the cows come home, but that doesn’t mean you will know how to do it yourself. The only way you will learn is by actually practicing.

Try to see the big picture

When you’re solving a math or physics problem, you may forget what it is you were even trying to do in the first place. Always keep asking yourself, “What am I being asked to do?”

Are you being asked to find x? Are you being asked to find the perimeter of a shape? Are you being asked to prove a statement like “A square is a rhombus?” Write your goal in big letters at the top of the paper you’re working on and put big circle around it.

When you feel lost or like you’re not sure what to do next, take a look at that big WANT. You may find yourself surprised at how much time and frustration having this simple reminder will save you.

Divide and conquer!

Break harder problems up 2 or more easier problems. Break longer problems up into smaller problems.

For example, I might be trying to find the area of a parallelogram on a high school geometry test. I know there’s a formula for the area of a parallelogram, but for the life of me I can’t remember it. That means I just have to skip that problem and get 0 points, right?

Nope! I can break that parallelogram up into two triangles, find the area of each one using the formula and then add the two areas together to get the area of the whole parallelogram. Many, many, many hard problems in math can be broken up into easier problems like this.

Work backwards

Many times you will be asked a question that you can’t answer until you first answer a question that wasn’t even asked! The rule of divide and conquer is sometimes optional, but in a situation like this it’s mandatory.  

For example, a question on a physical science homework assignment might say

“Ted started jogging at 2pm and finished jogging at 4pm. Find Ted’s average speed in miles per hour if he jogged a total distance of 10 miles.”

Since this is a physical science class, let’s assume I’ve already learned that I can divide the total distance travelled by the time spent travelling to find the average speed. The problem is that I don’t know what the time spent travelling is. So now I have a new sub-problem: find the time spent travelling. Fortunately this is pretty easy to do, I just subtract the time from the ending time to  find that I spent 2 hours jogging.  Now that my sub-problem has been answered, I can use the answer to my sub-problem to find the answer I really want. (Divide 10 miles by 2 hours to get 5 miles per hour)

Whenever you are asked to find something and you have no idea how you’re supposed to do that, ask yourself whether you need to solve a sub-problem first. Write down on your paper something like this:

Once you’ve found Y, You can use Y to get X. And then you’re done!

Do you need a math tutor in Tampa, FL?

Call (813)-563-MATH! (563-6284)