| Math Mojo - Making Math Meaningful |
A Math Miracle which is more than just a Trick
The lesson below may seem like a lesson about how to square some large numbers easily. It is really a lesson about looking at any math operation in new ways, and opening your mind to creative math.
The lesson is hard to read on a computer. If you print it out, you will probably be able to follow it more easily.
This is a great way to get into learning speed-math. Speed-math is a quicker way to do math than most people do it. It is also more accurate, and helps you get a deeper understanding of how math works by getting you curious as to why it works.
The thing we are going to learn this time is how to square any two-digit number which ends in the digit 5.
That means how to raise 15, 25, 35, 45, 55, 65, 75, 85, or 95 to the second power.
Do you remember what it means to raise something to a power? A lot of people (even very smart ones!) make the mistake of thinking it means multiplying the number by the power. For example:
They think 452 means 45 X 2. Which gives them 90. This is not right! 452 is not 90.
What does 452 really mean? It means multiplying the root (the 45) by itself. It means you do a multiplication where 45 appears two times. That means 45 X 45. Which gives you 2,025. That is a lot more than 90!
So what is the way that they show you how to do it in school? They teach you the old way of multiplying
45 X 45, which is:
| 1 | 2 | ||
| 1 |
2 |
||
| 4 | 5 | ||
| x | 4 | 5 | |
| 1 |
|||
| 2 | 2 | 5 | |
| 1, | 8 | 0 | 0 |
| 2, | 0 | 2 | 5 |
This is fine, but it is not quick or thoughtful. It is the easy way for teachers to show you how to do it, but it is not the easiest way for you to do it. You need to be able to take what you are taught in school, and look further.
How about if you tried squaring a number which ends in 5 this way?:
Why does this work?
If you look carefully at the old way to do the above example, you will discover that these multiplications are involved:
at it:5 in the units column x 5 in the unitscolumn 4 in the tens column x five in the units column another 4 in the tens column x five in the units column 4 in the tens column x 4 in the tens column
So what can we notice from this? The multiplications in black are pretty clear. We can deduce from that that part of the trick to squaring any 2-digit number ending in 5 will be to add 25 to the square of the tens column.
So far, in the above example, it means adding 1,600 + 25.To get the rest, you must notice something about the multiplications in the green boxes. They are always going to be the number in the tens column times 5, twice.
Now is the time to remember that math is about finding relationships between numbers. Well, "5, twice", is always 10, isn't it? So we can take the two green steps, and combine them. That makes the missing part of the trick the following rule - "Add the number in the tens column multiplied by 10."
That would mean, in this case, 40 x 10, which would be 400.Now, if you look closely, you see that you are really adding 40 x 40 + 40 x 10 + 25.
You can shorten that to 40 x 50 + 25. Which brings us to the shortened form of the rule:
Multiply the digit in the tens column by the next highest digit. Now tack 25 behind that number, and you have your answer.
Compare this rule (which you do in your head) with all the wasteful writing and multiplying in the original example (in light blue) above. It’s now up to you to try it with all the other two-digit numbers which end in five. After you try them all and understand how it works (which should take you about five or ten minutes) you will now be able to do this kind of problem quicker than a friend with a calculator. Really!
I’ll try just one more with you. Let’s try 74^2.
First, multiply 7 times the next higher digit, which is 8. That gives you 56. Now tack 25 on to the end, and you get 5,625. That’s all! That’s the answer. Check it the old way of multiplying 75 X 75. (That will take much longer, and you may even make some multiplying or adding mistakes on the way!)
I hope you try this method and learn it. There are millions of other easy, thoughtful methods of doing all sorts of math problems, which you probably never learned in school. If you really learn this well, and want to learn more, check your library for books about speed-math or lightning-calculating. Books by Martin Gardner are also very good.
Besides practicing the numbers already given, at least once a day for the next five days (which will lock in what you learned, so you won’t forget it when you need it), there is one assignment. That is to try to see if you can discover some rules for squaring any three-digit number which ends in five.
One hint is this – it is probably going to be a little more complicated than a two-digit number. You may need to use a pencil and paper, but it will certainly be easier than actually multiplying, say, 365 X 365! It will save you at least one step.By the way, an algebraical way to represent any 2-digit number which ends in 5 is: 10n+5 (where n is a single digit). Can you see why that is? Can you think of a way to represent a 3-digit number which ends in 5?
I am not going to give you the answer or any more hints. This is what math is about – setting a problem for yourself and seeing if you can find a solution, or at least learn something while you are searching for a solution. Remember, trust your brain! Don’t only look for solutions from someone else. Their brain is not any better than yours. Really.
One last thing:
Never, ever, ever write anything down (besides the answer) when you are doing this kind of problem! Don't use pencil and paper as a crutch for what your mind can easily handle. And for sure, don't even think of using a cursed calculator for anything like this. Calculators were invented by vampires to suck your brains out!
copyright Brian Foley 2001- Feel free to copy and reproduce this page for students. Under no circumstances may this material be sold by anyone but the copyright holder.
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