prealgebra Articles

Since I made a serious, adult student laugh the other day, I thought I would share the tip that did the trick. We were reviewing math for the GMAT (which is the same math as the SAT and GRE, by the way). When we got to the different types of averages, I decided to tell a riddle to help him remember the difference between mean and median, which are two kinds of "averages." Here is the riddle:

Tutor: Did you know that the average salary of geography majors at UNC is over $700,000 per year?

Student: Really?!

Tutor: Yes. That's because Michael Jordan was a geography major at UNC, and there aren't that many geography majors. His enormous multi-million-dollar salary pulled up the average because it was calculated as a mean (everyone's salary added together, then divided by the number of geography majors). However, if you calculated the median instead, by listing all of the salaries in order and finding the middle one, you would get a much more representative number to report as the "average." Medians are very useful when you have a few extreme values that skew the mean to one side. We often hear about median incomes or median home values instead of the means for that reason.

Source: I originally got this riddle/story from Ruth Curran Nield at the Univ. of Pennsylvania, who taught me statistics. She had a nice graph showing the average salaries of different majors at UNC.

"Fractions are funky fiends, which can disguise themselves as dicey divisions (decimals), or perky parts (percents)."

If you ask the average American what gave them the most trouble in school, the answer is usually math, and if you were to ask what aspect of math, I think the general consensus would reply, "Ooo... fractions. I hated them."

People have a general fear and dislike of fractions. But fractions are merely what they are, a fraction of a whole. When you cut an apple into 6 slices, each slice is 1/6 of the whole. When you receive a pizza delivery and you promptly eat one of the eight slices, you have 7/8 left. And if you have 4 cookies to share with eight friends, you use fractions to decide that either you need to cut the cookies in half or lose some friends.

Even though fractions are used in everyday life ALL the time, people would much rather work with whole numbers, decimals, or percents when it comes to doing math. Why?

"There are too many rules with fractions... All that adding the numerator, but not the denominator stuff," "It’s easier to avoid them," "I don't have a calculator," or "They just aren't normal."

BUT... decimals, percents, and even whole numbers are just fractions in disguise. "Percent" is actually a phrase meaning, "for every (divided by) one hundred," and Decimal is a number written in divisions of ten (i.e. tenths, hundredths, thousandths...).

EX: 50% = 50/100; 4% = 4/100; 645% = 645/100; 2.5%= 2.5/100 or 25/1000; .5 = 5/10 or 1/2; 0.45 = 45/100; 0.565 = 565/1000; 0.0001 = 1/10000; 2 = 2/1

I am not a fraction-lover myself, but I appreciate that there are different tools I can use to solve a mathematical problem.

That’s right, Big Fat One. The idea here is to understand equivalency and how often the number one is used in mathematics. Consider the following:

1 = 1/1 = 5/5 = 2356/2356

Notice 1 can be written as a fraction and this fraction has many different "number names". If you understand the example above then you already understand equivalency which can apply to numbers other than one such as:

1/4 = 2/8 = 3/12

1/2 = 2/4 = 3/6

3/2 = 6/4 = 9/6

However, back to the number 1. We all know that 5 x 1 = 5 and 7/7 x 1 = 7/7 but when students look at the following:

8 x 3/3 = 8

4/5 x 9/9 = 4/5

they get confused. It is mostly because they have forgotten the ease of multiplying by 1 and that 3/3 or 9/9 (or millions of other number names) is the same thing. Sure, 8 x 3/3 is actually 24/3 first but then simplifies to 8/1 then 8 which is what we started with anyway. The idea is for you to recognize the use of one when working with numbers.

Common denominator, simplify, reduce. Have you heard any of the following math terms before? Well, they all revolve around the use of the number 1. But where’s the "big fat" part, you ask? Simple. Notice that 1 is pretty skinny as numbers go. If I exchange 1 for a number name like 17/17, for example, then we call it "big fat one" (BFO) to recognize that the numbers are wider YET we are still talking about the same old number: 1. This is way easier to show you than to explain in words:

1/3 + 1/4 = 1/3 x 4/4 (BFO) + 1/4 x 3/3 (BFO) = 4/12 + 3/12 = 7/12

You see? Adding 4/12 and 3/12 is really the exact same as 1/3 + 1/4 because we used BFO to find a common number name for both fractions. And since BFO is 1 we know that nothing has really been changed except the name. This is CRITICAL for you to understand as one of the foundational concepts in mathematics.

Now look at how it applies to simplifying 24/30:

24/30 ÷ 2/2 (BFO) = 12/15

12/15 ÷ 3/3 (BFO) = 4/5

Notice, it took two steps of using BFOs to get the fraction simplified. Is that O.K.? Of course! Don’t let anyone tell you different! Now, are there more efficient ways to do this? Again, of course. Think about it. If I had used 6/6 from the beginning, I could have simplified in just one step:

24/30 ÷ 6/6 = 4/5

However, the only thing this does is save time which, I’ll admit, is important EVENTUALLY but, regardless, gets the same answer either way. FYI: using 6/6 instead of 2/2 then 3/3 is called using the greatest common factor (heard of it?) but this is a much better way to understand how it is found than other methods I’ve seen. Remember, the number of steps it takes to simplify is NOT IMPORTANT, what is important is that you do it correctly and completely.

Math rocks!

The idea of absolute value is simple enough: distances must be positive values therefore we are only concerned with distance from zero. Blah, blah, blah. I call this idea Reality Sticks for some simple reasons. First, absolute value symbols look like sticks. Simple enough. Second, the reality part is the difference between positive and negative numbers. Think of it as the difference between non-fiction and fiction books. Fiction books and negative numbers aren’t real. They may appear similar to something real and maybe have some real parts to them but they are ultimately fake (don’t say imaginary here for risk of confusing this with the math term "imaginary" which you learn in algebra). Non-fiction books and positive numbers can, and do, exist in the real world. So, here’s the skinny: absolute value is called "reality sticks" because it changes "fake" numbers into real ones BUT it doesn’t affect numbers that are already "real". Here’s an example:

|-5| = 5
|5| = 5

You see? The absolute value of a "fake" number like -5 is 5 because it is five spaces from zero. However, the absolute value of +5 is also 5 because it, too, is five spaces from zero. In other words, absolute value changes fake numbers into real ones but leaves real numbers alone. That’s all it is! Bear in mind, though, that this is only a basic understanding and does not cover trickier expressions. Here’s a few examples. Try to figure out how the answer is determined but don’t sweat it if you don’t. Check other articles of mine for more specifics on absolute value.

-|-4| = -4
|-(-3)| = 3
-|-(-8)| = -8

Math rocks!

This entry covers one of the most overlooked aspects of mathematics and is a major reason why students struggle from the beginning. It is also named Convenience Notation referring to the fact that we write this way to save time. Writing 1 instead of +1 is a Hidden Secret; writing 7 instead of 7/1 is a Hidden Secret; writing 3(4) instead of 3 x 4 or 3 • 4 is, you guessed it, a Hidden Secret. Including those above, here is a list of the most common:

1 instead of +1
7 instead of 7/1
3(4) instead of 3 x 4 or 3 • 4
5 instead of 5+1
x instead of +1x

If you are an Algebra student this would translate an equation like:
24 = (2x)(x + 6)
back into an equation like:
+24+1 = (+2x+1) • (+1x+1 + +6).

This is a critical concept for all math students to grasp early on in school yet it is the most overlooked. What makes it worse is the way some teachers will discover a student who struggles with this idea then blame them for not understanding the idea of Convenience Notation as if they should have known it already. Worse yet if the teacher believes that it is just something students "pick up" along the way.

You need to practice translating from one form to the other and back again until it happens mentally when you look at numbers from now on. It may take some getting used to but remember that you’re making up for mistakes of the past and breaking old habits is difficult at first but totally worth the results.

Math rocks!