Still Here in Spite of Weather

I realize that it has been a long time since the blog was updated. Unfortunately I haven't had the time to reflect and ponder on the everyday wonders of math and life since I've been inredibly busy reminding people of the simple strategies we should use in solving problems.

Some of these strategies are (in no particular order):

Draw a picture or diagram
Make a table
Find a pattern
Make an organized list
Pick an operation (+,-,x,/)
Use logic
Guess, check, revise
Work backward
Underline
Use your formula sheet.

Using these strategies won't guarantee you'll be the next mathematical superhero (Formula Girl?), but they can definitely help you save time and increase your accuracy.

Stay radical.

DON'T TRY THIS AT HOME!

I REPEAT - DON'T TRY THIS AT HOME!

DON'T JUMP!

That sure is some crazy math- assuming you read the rest of the page after the video.

Stay radical, but don't make an error in your calculations. Splash and Splat are mighty close to each other- in more than the dictionary

... And a Product of 2 Primes Later...

Dear Mathman,

You said "next week". What calendar are you on? It's been a month and 4 days since you last posted! Come on, your fan can't wait!

Sincerely,
Your Fan


Dear Fan,

Thanks for paying attention. I have no good calendar explanation other than it's been the first prime number shy of the 6th perfect square since I've posted. The perfect square is the perfect vehicle for our next topic anyways: Roots.

Roots are the BFFs of the exponents. I know some may say "BFF"- that's SOOOO 2007 I can't believe you'd use that term Mathman; After all, you are incredibly hip and relevant. Well, most people still know what it means so we're going with it. With me? Groovy.

Let's review exponents: shortcut for repeated multiplication where the base is the number that is repeatedly multiplied and the exponent is the little number high and to the right that tells us how many times we repeatedly multiply.

Ex 4^2 (the ^ means raise that number) = 4x4 = 16. Now you may have heard that 4^2 can be read as "4 squared." NWDTCF? (Now Where Did That Come From) If you remember finding the area of a square, you remember you multiply the side by itself (since they're all the same it doesn't really matter which side you pick). When we get the answer, we measure it in square units (which are tiny squares that fill up the space). Extending this, we get the "Perfect Squares" (made up of squares with side lengths that are counting numbers):

1^2 = 1x1=1
2^2 = 2x2=4
3^2 = 3x3=9
4^2 = 4x4=16
5^2 = 5x5=25
6^2 = 6x6=36

1,4,6,16,25 and 36 are the first Perfect Squares.

Next awesome math term: Square Root. Simply put, that's what the length of the side would be for a square with a given area. Looks like:
___
V

So the easy ones are the square roots of 1,4,9,16,25,36,49,64,81,100 - their square roots are 1,2,3,4,5,6,7,8,9,10 respectively.

Let's tie the whole room together: Pick a positive whole number (no fraction or decimal). Square it (multiply it by itself). Now find the square root of that result. You should get your original number. So the square root is the inverse to ^2.

Take this further: any number raised to a power has a root that "undoes" what it just did. Trying to show this on this blog is next to impossible since I get a "typewriter" interface. And you thought BFF was an old term...

Exponents and Roots - BFFs.

Stay radical (or at least learn a bit more about them)

- Mathman.

Yes, Supersize Those Fries.

Last entry we had determined that: 1. PEMDAS as a saying doesn't make a whole lot of sense, 2. The MDAS is ok, but ignores the "I'll write my name with <3 on the back of my hand" relationship between Multiply and Divide (also Add and Subtract, but not Multiply and Subtract), and 3. Sheesh Mathman, you're pretty loquacious.

What we didn't deal with was: What about the PE a.k.a. Parenthesis and Exponents part of the Order of Operations?

First, the Parenthesis are always first. DO IT NOW! is their mantra. Don't ask questions. Just Do It. (Actually just saying Parenthesis are the only bossy symbols is not a correct statement. Brackets also mean the same thing.)

I don't know about you, but I'm not a fan of people constantly bossing me around. The grouping symbols, ( ), [ ], and { }, kind of have their own thing going on. They live in their own little dreamworld and don't really have anyone else to hang out with since they're so full of themselves and don't really care about others- unlike Multiply <3 Divide or Subtract <3 Add. Ahh, special relationships...

So the pressing question is now, what about E? I mean, are Exponents to be left forever by themselves? In PEMDAS they're right after the bossy ones and before the googly eyed pairs. Don't they have any prospects for a long term relationship?

First up, Exponents deserve to be right after the bossy ones. Exponents are a shortcut for repeated multiplication which was at the forefront of the MDAS (but may be second to divide). An Exponent- like 2^5 is a shorter way of writing 2*2*2*2*2, just like 2*5 is a shorter way of writing 2+2+2+2+2 (multiplication is a shortcut for repeated addition). I guess that makes exponents kind of the big brother to multiplication and the even bigger brother to addition. Does that mean exponents are repeated repeated addition?

But what about their prospects for BFFs or eternal love? They're at the top (not counting the brats) of the Order of Operations, but should they have to be alone?

In short, no.

Exponents do have a special relationship going on, but they're so secure in it that they don't really advertise it on the back of their hand or with a sketchy tattoo. They've got roots. No really, they've got roots.

Tune in next week to find out more.

Until then: Stay Radical. (Ooh, That's the perfect tagline in this case.)

Would You Like Fries With That Order?

Dear Mathman,

I've been learning about the "Order of Operations" in Math class. I don't like following orders. Do I really need to? Why?

Thanks,
Ivanna Havitall


Dear Ivanna,

Order is a fabulous concept. I mean without orders, how would you get what you want at a restaurant?

Oh wait, I just reread your question. I guess that wasn't the proper answer. Let's try again:

First a bit of history. Actually, a bit of your history. What was the first thing that you learned in math class after you learned how to count. (Please don't say "flush" or "wash hands"). I hope you said "Add". Those were the days... It wasn't too long before the next thing to do showed up on the scene: Subtract. Now you probably didn't like subtract as much as you liked add, because we're all a bit selfish and don't want things taken away. Wasn't it better to say, "You have 3 m and m's (not the rapper) and I give you two more - How many do you have?" rather than "You have 5 m and m's and I take away two." Simple so far, eh?

The next amazing thing you learned to do was multiply. I don't know about you, but when I was back in school I found multiplication to be a fantastic shortcut when I had to repeatedly add a collection of numbers. Example: 3+3+3+3+3+3+3+3+3+3+3+3 = 3 x 12 (the 3+3 was hard to even type -try it!). Just after getting pretty good at multiplication, division came along and ruined the whole party. I mean seriously, subtraction was hard enough - now equal groups?! Come on!

Let's recap: Divide, Multiply, Subtract, Add- those are the operations in reverse order we learned them. If you learned a saying for the "Order of Operations" it probably was PEMDAS (which isn't even a word) or "Please Excuse My Dear Aunt Sally." Personally, I'm not a fan of either. PEMDAS sounds like a contagious disease (Did you hear about Joe? He didn't wash his hands and came down with a nasty case of PEMDAS.) And "Please Excuse My Dear Aunt Sally?" doesn't make any sense to me at all. I don't have an Aunt Sally, and if I did, why would she need continually excused? Is she uber-rude? Does she have a gatrointestinal disorder? I might as well say, "Prickly Elvises Maim Deer And Studebakers" or even "Portly Elephants Make Doilies After September." All nonsense.

Not really, but the saying is incomplete. It's missing something. We don't always do things in the order of Multiply, Divide, Add, and Subtract. If we take the "big 4" and slightly switch the order we get the MDAS of the end of that saying. I personally think (in regards to the saying) we think Multiply before Divide because we prefer to Multiply first. Likewise with Add and Subtract. (It does make sense that the Multiply comes before Add because Multiplication is the "shortcut" for repeated addition.)

In reality, sometimes Divide comes before Multiply and Subtract before Add. You see, Multiply and Divide are in a "special relationship". They are equal and opposite. It's like they're "going out"- but permanently (unlike typical junior high "special relationships" which may not even outlast a sock change). When you encounter Multiply and Divide in a problem remember they <3 each other. Do the operation that comes first when you read the problem normally (left to right).

Add and Subtract have the same thing going on between them. This is getting too mushy so I'll stop there in explaining their situation.

So (hopefully) we've got the MDAS covered. We haven't dealt with the PE part of the saying (Note: PE doesn't stand for gym class). Since this post is getting a bit long, we'll save PE for another time.

Ivanna, hopefully this starts to answer your question a bit.

Stay radical.

Have it Your Way?

Sheesh, I take the summer off and the Inbox is full. Time to get a bit caught up. Here's a good one:

Dear Mathman,

I have an iPod shuffle. I really like the thing as it allows me to ignore everyone around me and get exactly what I want when I want it- except for when I use the shuffle mode. Half the time I do it just ruins my day as I get stuck listening to a mix that just Dysons. Seriously, why doesn't this thing just give me what I want?

Sincerely,
Cody H.

Dear Cody,

Bigger isn't necessarily better. Especially when it comes down to doing the iPod shuffle. The more songs you have on that iPod, the greater the chances that the song you want to hear won't be coming up next. It makes sense if you think about it, but did you ever stop to think what the chances are you'll get a mix you want?

Let's think back to a time before the mp3 format. Let's go back to when "Don't Stop Believin'" first came out on 8 track, cassette, or LP. At that time you would have to buy the entire album of 10 or so songs in the order that the band intended it to be. You couldn't change the order that the songs were in unless you did some type of physical work. The "shuffle" option (or "random" in the CD world) didn't exist. You were forced to listen to the album or you had to physically change the media- you couldn't just push a button to access another band's work (unless you bought the K-Tel compilations).

The point of this? There was only one easy choice to hear the album (2 if you count side A and side B).

When CDs popped into the mix, things began to change. Now you could hit the "random" function and you could hear the songs in a bunch of different orders. This was good and bad since it's like reading a book by skipping around from chapter to chapter not necessarily staying in order.

How many orders could you hear all of the songs? Get in the math zone.

If you had a single, it would typically have 3 or 4 songs on it. (So how's that a single?) Let's figure out how many orders you could listen to the songs in with various numbers of songs: (Let's just say the first song starts with "A", the second with "B", the third with "C", ...)

One song : One way
Two songs : Two ways (A then B or B then A)
Three songs : Six ways-
ABC
ACB
BAC
BCA
CAB
CBA
Four songs : 24 ways. Think of it as three songs with a different starting song each time. Ex: DABC DACB DBAC DBCA DCAB DCBA. Note that the D comes first, but there are 6x4 ways since A,B, or C could have been as first as well.
Five songs: 5 x 24 = 120
Six songs: 6 x 120 = 720
Seven songs: 7 x 720 = 5,040
Eight songs: 8 x 5,040 = 40,320
Nine songs: 9 x 40,320 = 362,880
Ten songs: 10 x 362,880 = 3,628,800 ways

Whoa.

That's just for a 10 song CD placed on "random". Imagine an iPod with 500 songs on it. Yikes!

Let's look at the pattern again to figure out what is happening:
1 song : 1 way
2 songs : 2 ways you could pick the first, the second is decided; 2x1=2
3 songs : 3 ways to choose the first, 2 for the second; 3x2x1=6
4 songs : 4 ways for the first then hit me baby one more time with the results from the 3 song calculations; 4x3x2x1
5 songs : 5x4x3x2x1
6 songs : 6x5x4x3x2x1
and on and on.

There is a term for this, it's called the factorial. It looks like this: ! It means, HOLY COW THAT'S A WHOLE LOT!

so 10 songs is 10! = 10x9x8x7x6x5x4x3x2x1.

With over 3 million ways to listen to just 10 songs, no wonder why you've got a good chance for a bad mix!


stay radical.

Ahh, Summer...

Greetings Citizens of the Junior High. Hope your summer is going well. Since it is summer and people look forward to nice days, I've been asked: "Mathman, can you change the weather?"

In short, no. I, however, can predict the weather:

There is a 100% chance that it will either rain or not rain.

Stay radical.

Better Than Pencil Tapping

Radical.

The Chocolate Calculator?

Dear Mathman,

I got this in my email. You should try it. It's amazing. How in the world could chocolate know my age?

Check it out below.

Sincerely,
Cal Orie


YOUR AGE BY CHOCOLATE MATH:

Work this out as you read .
Be sure you don't read the bottom until you've worked it out!

1. First of all, pick the number of times a week that you would like to have chocolate (more than once but less than 10)

2. Multiply this number by 2 (just to be bold)

3. Add 5

4. Multiply it by 50 -- I'll wait while you get the calculator

5. If you have already had your birthday this year add 1759 ..
If you haven't, add 1758.

6. Now subtract the four digit year that you were born.


You should have a three digit number

The first digit of this was your original number
(i.e., how many times you want to have chocolate each week).

The next two numbers are YOUR AGE! (Oh YES, it is!!!!!)

THIS IS THE ONLY YEAR (2009) IT WILL EVER WORK, SO SPREAD IT AROUND WHILE IT LASTS.



Dear Cal,

It's not really magic- unless you consider algebra and the distributive property to be magic. If you do, you probably haven't read this far...

Here's each step with the algebra behind it:

1. First of all, pick the number of times a week that you would like to have chocolate (more than once but less than 10)

We're going to call this "n" and it has to be between 1 to 9 because multiplying by 10 doesn't keep the place value consistent.

2. Multiply this number by 2 (just to be bold)

Now we have 2n

3. Add 5

2n+5

4. Multiply it by 50 -- I'll wait while you get the calculator

50(2n+5)= 100n+250 This step is key because the n will now be the hundreds place in a three digit number (100, 200, 300, etc.)

5. If you have already had your birthday this year add 1759 ..
If you haven't, add 1758.

100n+250+1759= 100n+2009 - this is the key move where you gave away your age according to this year. It is 2008 for those who haven't yet had your birthday since you haven't had it in 2009 yet so you need to keep the year straight. Think about it- I'm having a hard time explaining it right now.

6. Now subtract the four digit year that you were born.

100n+2009-the year you were born leaves you with:

100n+your age (the adjustment for 2008 makes more sense here).

So what you're left with is the number you started with in the hundreds place and the other 2 digits being your age (in the tens and ones place) since you subtracted the year you were born from 2009 (or 2008 if you haven't had your birthday yet this year).

THIS IS THE ONLY YEAR (2009) IT WILL EVER WORK, SO SPREAD IT AROUND WHILE IT LASTS. We'll it will work other years, you just have to adapt the +1759 part.

Hopefully this didn't melt your brain like chocolate in the sun on a 30 degree Celsius day...

Stay radical Cal,
Mathman

Origami Saves Lives?

It's no secret that origami is popular at BJHS. The males work on perfecting the triangle, the females get more creative and fold notes in interesting ways. What you may not realize is that origami may save lives.

How?

Watch this:



So math is good for more than just keeping score during a paper football game.

Stay radical.

Binary (Base 2)- Either You Get it or You Don't.

Dear Mathman,

I heard this joke in math class the other day:

“There are only 10 types of people in this world- those who understand binary and those who don’t.”

I’m one of the types of people who doesn’t get it. Please help.

Thanks,
Sly T. Lee Confused



Dear Sly,

Thanks for asking for help. That is a great skill more of us need to use. BTW, you sure know how to tell a joke. I just had to pick myself up off the floor. I laughed myself right out of my chair. Keep them coming.

Anyways, this joke revolves around understanding the binary system. You might not realize it, but we both needed the binary system to even see this blog entry. Huh? As a matter of fact, you’re using the binary system anytime you use anything powered by electricity. Anything.

The binary system is based on the prefix “bi”. Do you know what number “bi” is referring to? Hint: think bicycle. I hope you figured out “Bi” means “two”. So the binary system is counting system based on the number 2. A bit lost? Let’s go back to the electricity example for a second. . Think about the common light switch. There are only two options, on or off. The electricity flows or it doesn’t (we’re ignoring dimmer switches to keep it simpler).

Any electronic device is broken down into a whole lot of light switches that are on or off. Fortunately for us, many of these switches have been shrunken down to a microscopic level. If they hadn’t, this computer that I’m typing on would take up a fair amount of real estate here in Mathopolis. Ah, the wonders of modern science.

Enough of science, let’s get back to math:

We’re not used to the simplicity of the binary system, we’re used to a much more complex system of keeping track of numbers. Our system is base 10 and is called the decimal system. It’s worth it to think about this for a minute so we can understand and appreciate the binary system a bit more. The decimal system revolves around 10 possibilities for each digit (single number). You know these possibilities as 0,1,2,3,4,5,6,7,8, and 9 (that was fun to type from the top of the keyboard). We know the place values in our system as “ones”, “tens”, “hundreds”, “thousands”, “ten thousands”, and so on. Think of a number to keep this straight:

67,890 is 6 ten thousands, 7 thousands, 8 hundreds, 9 tens, and 0 ones.

In our decimal system the number 10 is made up of 1 tens and 0 ones. When we write a number we start with the lowest value for a digit, 0, and work our way through the digits until we get to the highest value, 9. If we have to go higher than 9 we add the next place value and use a 0 to hold the first place value. We’re so used to doing this automatically we don’t think about it anymore.

The binary system works the same way. We start with a 0 in the first place value and count up until we hit the largest digit, then we add the next place value and start over. So, the number 0 in binary is 0. The number 1 in binary is 1. The number 2 in binary exceeds the highest digit for first place value so we need to add the second place value.

In base 2 (binary), the number 2 is written as 10. The place values go ones, twos, fours, eights, and so on. In the decimal system the place values are: 10^0=1, 10^1=10, 10^2=100, 10^3=1000, and so on. In the binary (base 2) system, the place values are: 2^0=1, 2^1=2, 2^2=4, 2^3=8, and so on. You count the same: 1,2,3,4,5,6,7,8,9,10,11, etc. but the numbers are written differently. This is like a foreign language for math.

Now, take a moment and reread the joke. Is it funny yet? If we wrote it without the 10 in binary, which really is the counting number 2, it wouldn’t be nearly as funny:

“There are only 2 types of people in this world- those who understand binary and those who don’t.”


Stay radical,
Mathman

Whoa, Math AND Science

Reuben Margolin definitely has mathematical and scientifical (?) superpowers. He probably won his school's science fair as a student.



Stay radical.