Math Puzzles and Fun

This is just a short little post. If you have an interest in math puzzles and things like this, it might be worth the trip for you to check out http://wildaboutmath.com/. I've been linking to their site for a long time now, and I just thought I would try to introduce some of my more recent followers to that site. There are math puzzles and all sorts of interesting bits of info over there. It's really well done. Check them out!

Which Measure of Central Tendency to Use? Mode, Mean, or Median?

A concept which may need a bit more explanation is: which average is appropriate for a given question? What is the best measure of central tendency? When would you use a median instead of a mean, or perhaps use a mode instead?

For any data set, you can perform the analysis to come up with a value for each average. However, here are a few basic guidelines to help you choose the most appropriate form of central tendency to describe your data.  If this is helpful, it would be great if you could please hit the +1 button to share it!

1. For a normal, random distribution of data (evenly distributed), the mean is preferred.
2. For a skewed data set, a median is more appropriate than a mean. The skewed data set (ie. extreme data points) will cause the mean value to be much more extreme than the median, and therefore less central.
3. The mode can be used for non-numerical data. Eg. hair colour in a classroom.

Here are a few examples of where each would be appropriate:

Mean:
1) students' heights in a classroom
2) temperature over a length of time

Median:
1) income of a group of people
2) test scores for a group of students

Mode:
1) finding the most common hair colour in a room
2) finding the most common car in a parking lot

Hopefully these guidelines will help you to determine which is the most appropriate measure of central tendency to report for your data set.

Again, if this was helpful, please share and hit the +1 for me! :)

Perfect Squares

This post should have been put up when I posted about square roots (here, here, and here), because it is the exact opposite of a square root!

Where a square root of a number "x" is some number "y" that, when multiplied by itself, gives "x", a (perfect) square of a number is the result of multiplying a number by itself. That is to say, the square of "y", by multiplying "y" by "y", is "x". You can also talk about "squaring" a number, which is to find what the square is. (It can be both a noun and a verb.)

For example: the square of 4 is "4 x 4" = 16. Also, if you square 4, you get 16.

If you want to think of a visual representation of it, "what is the square of 5" is essentially the same as asking "what is the area of a square with a length of 5?" (Of course, all sides have equal lengths in a square.) As you can probably figure out already, to find the area of a square (or, in general, a rectangle) you multiply the length by the width. So here, it is obviously 5 x 5 and the area, or the square of 5, is 25.

This concept of perfect squares can also be extended to polynomials. For example, let's look at the following case:

(x+1)*(x+1) is the square of (x+1). It can also be written as (x+1)^2. You can do the visual trick i just described above if you want, using x+1 as the side length.

If you multiply (FOIL) these binomials, you get (x^2 + 2x + 1). As it is equal to our original expression, you can also say that this product is a perfect square, just as you can say that 16 or 25 is a perfect square. To find out what the square root of this is expression is, it is the same as asking what the square root of (x+1)*(x+1). Over time you will see patterns and be able to quickly notice that the square root of (x^2 + 2x + 1) is (x+1).

This shouldn't be too difficult of a concept to understand. I will do a few short examples, but post comments if you require clarification:

Find the square of 12:
12 x 12 = 144

Find the square of 25:
25 x 25 = 625

Find the square of (x-1):
(x-1)*(x-1) = (x-1)^2 = (x^2 - 2x + 1)

Find the square of (2x+3):
(2x+3)*(2x+3) = (2x+3)^2 = (4x^2 + 12x + 9)

It can also be noticed, and should be kept in mind, that squares will always be positive. Try it to see: plus x plus = plus... negative x negative = plus. A plus times a negative is NOT a square! Squares multiply the SAME number (sign and all!).

See my previous post about "completing the square" for some more stuff relevant to this post.

Completing the Square

"Completing the square" is a method of expressing a quadratic function, and it is an especially useful form for graphing the expression.

If we have a function:

f(x) = (x+1)^2 + 3,

from the rules of graphing (1, 2, 3) that I have posted already, we can tell that this is a second degree polynomial (upright parabola), shifted to the left 1 unit, and up 3 units. However, if I were to ask you to graph:

f(x) = x^2 + 2x + 4,

that appears to be much more complicated. However, it describes the exact same curve, it is just written differently. "Completing the square" is the process used to convert this complicated and nasty form into the simpler form above.

To do this, you basically have to ignore the constant number (4, in this case) at first. So, to start, we look at x^2 + 2x. Now, we essentially do FOIL backwards to get it in the form (x+y)(x+y), or (x + y)^2... a little tricky, but with practice, you will get better at it and recognize certain patterns.

I will write as step by step below, and hopefully that demo will explain what I mean with all of this:

f(x) = x^2 + 2x + 4... re-writing to focus what we're doing:

f(x) = [x^2 + 2x] + 4... now reverse FOIL, determine the constant required in the square brackets to make it a perfect square. To keep things balanced, whatever number you add, you have to also take away:

f(x) = [x^2 + 2x + 2] + 4 - 2... simplify:
f(x) = (x+1)^2 +2

To get an x^2 and a 2x, we can see that the First term has to be just x, and the Outside and Inside terms have to be the same (because it's a perfect square) and add to 2x, which is 1x plus 1x, and which means that the y term in the general statement has to be 1. So, to get the x^2 + 2x correct, we have determined that is corresponds to (x+1)^2. Then, as noted, whatever number we have added to the expression to make it a perfect square, we also have to subtract from the number that we started with.

Hopefully this makes some sense to you. Let's try another example:

f(x) = x^2 + 10x + 37
f(x) = [x^2 + 10x] + 37
f(x) = [x^2 + 10x + 25] + 37 - 25
f(x) = (x+5)^2 + 12

Much easier to graph this than when it started. :) Here's another:

f(x) = x^2 + 14x + 20
f(x) = [x^2 + 14x] + 20
f(x) = [x^2 + 14x + 49] + 20 - 49
f(x) = (x+7)^2 -29

Let's try one more, a bit harder this time. Same logic applies.

f(x) = 4x^2 + 4x + 19
f(x) = [4x^2 + 4x] + 19
f(x) = [4x^2 + 4x + 1] + 19 -1
f(x) = (2x + 1)^2 + 18

Introduction to Statistics - Mode

The third concept of introductory statistics that I will explain here is the mode. (Refer to my previous posts about mean and median.)

The mode is also a measure of average (central tendency), but again, different from how you have likely thought of an average before.

The mode is simply the value in a data set that is represented the most times.

Again, let's refer to my ongoing test grade example:

Example:
The grades received for a test in a math class, composed of 13 students, were:
65, 98, 92, 43, 76, 64, 69, 72, 75, 85, 96, 90, 90

When you rearrange them from lowest to highest, you can quickly identify which value appears the most times:
43, 64, 65, 69, 72, 75, 76, 85, 90, 90, 92, 96, 98

So, for this data set of math scores, the mode was 90... that is, more students scored 90 than any other single grade achieved.

Hopefully this series of posts has helped to explain the calculation of median, mean, and mode. As you will see in your studies, each has its own applications and are useful in their own ways. It is important to realize though that they are all related as forms of average, and they all describe the centeredness of a data set. See my post here for tips on how to choose which of these measures of central tendency to use.

As always, leave a comment if you need clarification or more information on the topics I've posted. As well, as I have done with this series of posts, I will try to address any requests students may have as well in future posts. :)