I hear so many people say that it's important to understand what teachers mean when they tell you math is all about looking for patterns, because when you really start seeing patterns instead of just a bunch of numbers, it's like you've found the key to all kinds of questions, even really hard ones. Well, I don't really understand how all the pattern stuff can help so much, especially when the question never even talks about patterns at all. I mean, when I think of patterns, I think mostly of the stuff kids do in kindergarten and first grade with different colored shapes and blocks, or repetitions of measures in music. I know that's not it, so can you please explain what everybody means when they talk about the patterns in math, and do you really think they're that important?
Get my joke?
Get my joke?
But seriously, can you please explain to me what I'm not understanding?
For many, many reasons, I'm quite certain that patterns are 'that important'. In fact, if I tried to come up with a complete list of the reasons, it would be so long that you'd get bored before I even got to the examples (or "tricks") that I know you would much rather learn. Still, I will tell you that learning some patterns and learning how to spot new patterns and learning how to use them is probably one of the quickest and easiest ways to go from being a "non-math" person to a becoming a serious and passionate mathematician. Sometimes the mere ability to appreciate the importance of patterns is truest when the problem doesn't even mention patterns, which is one of the things you asked. Think about it: Of course most problems aren't going to point you right to the particular pattern that will lead you right to the answer, because if every math question was written so the answer pops right out, it would be so easy it would get boring pretty fast. Instead, when you have to think a little to notice the pattern for yourself, things get exciting. There are, as you can imagine, many different kinds of patterns, but to keep it simple, I break them all up into just two categories: The first category, which students are often introduced to in kindergarten or first grade, is the stagnant kind, in which the goal is the identification of a repeating sequence, or the continuation of a repeating sequence, or even an elaboration of a – yup, you guessed it – a repeating sequence. The second category is what I think of as the group of "operational patterns"; this is the collection that leads to all the formulas that fill math textbooks to help us figure out the solutions to problems without having to force us all to try everything out in the real world or even on paper. In other words, we have developed lots and lots of procedures and formulas that we can apply to new questions, and, if we do it right, they give us much of the information we need before we take out our saw and start cutting through some block of wood – and (surely making mistakes along the way) wasting trees and wasting time and energy only to find out that something doesn't work. For example, each type of wood has a range of flexibility and a range of density and range of what's called "tolerances" for changes in climate. BUT once you KNOW the particular range of flexibility, for example, of different kinds of wood (information which is usually based on precisely recorded experiments), it takes a lot of the guesswork out and people can often use a much thinner veneer of an expensive wood than they might have thought to cover another wood which would be selected for its greater strength rather than appearance.
Believe it or not, it often seems to me that Math is PRIMARILY the study of patterns, so that we can analyze them to make predictions to help us solve or avoid future problems.
Okay, but you probably don't want to hear about wood (unless you're about to build a guitar), and I'd guess you're not overly focused on future problems – with one giant exception....
You would like "tricks" to help you get the right answer quickly and easily on your next math test anywhere else q question comes up. So...
Here's a few of those magical tricks that I have selected as randomly as I could form the list that's gotten stuck in my own head of my top favorite six or seven hundred such 'tricks'. Just keep in mind, they all are results of people noticing a pattern in a group of answers, or someone taking a look at a well-known pattern and finding a new way to benefit from its certainty.
Here they are:
If you have to add up a giant group of consecutive numbers (that just means they come one after another, (like 1,2,3,4,5), then all you have to do is this:
Take the average of the numbers and multiply that average by the number of numbers.
So, the first trick is knowing a REALLY good trick for averaging a bunch of consecutive numbers. I DO! It's this:
You just have to take the average of the first and last numbers of the consecutive series, and you're DONE with that part!
(By the way, I am going to explain the workings behind each and every trick I describe, because it is very important to understand what you are doing, even if you don't fully understand the procedure each and every time you use it. However, those explanations will not be included here. I will describe them in coming weeks, so, for that part, stay tuned to Adler-n-Subtract.com. )
So, getting back to the question, if someone asks me (or you) to add up all the numbers from 1 – 100, it would go like this:
The first number is 1, and the last number is 100.
To find the average, we add them together and then divide that sum by two.
So the sum is 101, which when divided by 2 is 50.5.
Then we multiply that average by the number of numbers. Here that number of numbers is 100. So 50.5 x 100 = 5,050.
That's the answer to the question: What is the sum of all the numbers from 1 to 100: 5,050.
And now, we relax and go back to some very basic patterns that most of us learned when we were so young that we might occasionally forget them. (But DON'T! They can really help a lot when you are checking over your work and hunting down errors.)
Some numbers are "odd" and some are "even". The even ones are the ones that can be divided by two and give a quotient that is a whole number with no remainder and no extra parts. The odd numbers are the ones that cannot be divided by two to give a quotient that's a whole number without a remainder.
Here's what happens when you perform addition:
Even + even = even
Odd + odd = even
Even + odd = odd
Odd + even = odd
How to picture it:
One afternoon, it's raining hard, so the school has indoor recess after lunch. Two teachers have an idea: instead of going to the gymnasium or P.E. Center, their classes would have more fun just doing an activity in one of the two classrooms. The activity selected is one that requires everyone being partnered up, such as arm wrestling. (And the students have to be partnered up at the same time, because it's not fair to make one person arm wrestle a second time, when his arm is already tired out just so that someone without a partner gets to finally have one. If there are an even number of students (such as 18) in one class, and an even number of students in the other class (and to make this REALLY easy to picture, let's say this even number is ALSO 18), everything works out evenly and no one is left out of the partnering. If each class has an even number, and those even numbers are different, such as 16 and 18, then everything still works out EVENLY because they could already have paired up with students in their own class before the other class is ADDED TO their classroom.
If one class has an odd number, such as 19 students, and the other class also has the same odd number, 19, then it is easy to imagine that when the classes are ADDED together everyone gets a partner and it works out EVENLY. And if they have different odd numbers, such as nineteen and 21, then each class can partner up before being ADDED together, AND EACH CLASS WILL THEN HAVE ONE EXTRA PERSON, WHO WHEN THE TWO GROUPS COME TOGETHER, THOSE TWO FORM A NEW PAIR AND everything works out EVENLY.
BUT: If one class has an even number, and the other class has an odd number, and they pair up before being added together, there will be one class with no extra people and one class with one extra person, so after they're mixed together in one room, the activity does NOT work out evenly because there is only one extra person. He (or she) might feel a little ODD while the others are playing.
Here's what happens when you multiply numbers:
Odd x odd = odd
Odd x even = even
Even x odd = even
Even x even = even
To picture these, here's a completely different method (or "trick") that could be used in many different areas of math: WHAT IS TRUE FOR SMALL NUMBERS IS TRUE FOR ALL NUMBERS. So, all you have to do is remember that 1 (one) is odd, and that any other number that you multiply by one remains the same number. So now:
2 is an even number.
1 x 2 = 2 (odd x even = even)
1 x 3 = 3 (odd x odd = odd)
2 x 4 = 8 (even x even = even)
And how do you know if a really long giant number is odd or even? ANSWER: All even numbers on planet earth (no matter how huge they are), have a 0 or a 2 or a 4 or a 6 or an 8 as its last digit. So if the last digit is one of the other numbers (1 or 3 or 5 or 7 or 9), then the entire huge long number is ODD.
EXAMPLE: 222,000,000,022,222,444,446,463 is an ODD number. Why? Because the last digit is a 3, and three is an ODD number.
Another EXAMPLE: 113,131,111,353,313,333,355,535,755,537,774 is even. Why? Because the last digit is 4, and 4 is an even number (it's a 0 or a 2 or a 4 or a 6 or an 8!), and that's the rule that comes from the PATTERN that numbers form.
Even - even = even
Even - odd = odd
Odd - even = odd
Odd - odd = even
Let's step things up:
(Average) x (number of members of the group) = the total EXACTLY.
Look: If you bowled ten games one summer, and you calculate your average to be 93, then it is a fact that the total number of points you scored that summer bowling is 93 x 10, which equals 930.
(This "average x members of the group" one surprises many people, probably because we are trained not to think of an average as an exact of anything. But that's only true as a predictor for future events. You may have a 93 average in bowling after you've bowled 1000 games, but that does not mean you will score 93 exactly on your next game. In fact, it doesn't even mean you will score close to a 93. It could be the first day of your summer vacation, and you are so excited that on your first try, you might slip and send the ball flying up into the air and then catch it on your head, conking yourself out and waking up the day before school begins again, in which case you will probably score a zero or close.
Or you could get lucky and bowl a perfect game, which happens to be a score of 300!
For now, I am NOT going to explain why averages ARE exact numbers when you do the reverse trick I mentioned above. For now, TRY TO FIGURE IT OUT YOURSELF! (In a week or two I'll reveal that little secret).
Multiplying numbers with "POWERS OF TEN":
If you multiply a number by 10, 100, 1,000, 10,000, 100,000, or any other number with a 1 as its first digit and only zeros in the places following that one, then simply take the other number of your multiplication example and attach the number of zeroes that the "power of ten" number had. Example: 67 x 1000 = 67,000
As you probably can tell by now, the whole subject of "PATTERNS" in mathematics is an infinite topic, but for now I am going to draw the line after just one more, which happens to be one of my two or three hundred all-time favorites! Here it is:
To generate (come up with) a Pythagorean triple, stick any integer into this set of formulas:
a = 2n + 1
b = 2n(n+1),
c = 2n(n+1)+1
And you will find that your new a, b, and c fulfill this magic and very famous equation:
a2 + b2 = c2
Until Next Time,