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.)
Tuesday, September 22, 2009
Monday, September 14, 2009
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.
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.
Friday, September 4, 2009
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.
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.
Friday, July 10, 2009
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.
In short, no. I, however, can predict the weather:
There is a 100% chance that it will either rain or not rain.
Stay radical.
Tuesday, May 26, 2009
Thursday, May 7, 2009
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
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
Friday, May 1, 2009
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.
How?
Watch this:
So math is good for more than just keeping score during a paper football game.
Stay radical.
Friday, April 24, 2009
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
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
Wednesday, April 8, 2009
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.
Stay radical.
Friday, April 3, 2009
Being Careless Can Stink.
Recently a lot of math classes have been dealing with percents and sale prices. For some reason, many people have problems with percent problems. I guess they are called “problems” for a reason, but really they’re not that hard to solve- IF you read carefully. Most of the time errors occur it is due to careless mistakes in either not writing down the work, or reading the problem incorrectly.
We can deal with the first situation rather quickly: show your work on paper. Percent problems go with proportions like cheese goes with pizza. Be careful to match the original amount to the 100 in the denominators of each side of the proportion, and keep the percent over the 100 and you’re set. If you don’t know what I’m talking about when I say, “proportion”, look at the previous blog entries below.
The second case is actually the first thing that should happen when you’re starting the problem. Just like my last blog entry said, take your time to read carefully the first time and you’ll save time in the long run. Even with careful reading though, some people still struggle. Most of the time I’ve found this is due to confusion over one little letter. This letter is crucial to the meaning of the problem and the work involved in finding its solution.
How can one stinking letter be so powerful you ask? What is this powerful letter?
A: “f”
The letter “f” has the power to change “of” to “off”. If the problem is “10% of the original price”, the answer will turn out to be less than ½ of the original amount. If the letter is “10% off the original price”, the answer will end up being more than ½ of the original price. People will flock to a sale of "10% of" as it can also be interpreted as "90% off".
One little letter = one big difference.
If you aren’t careful with that one little letter “f” then “art” takes on an entirely new meaning.
That stinks. Read carefully.
Stay radical.
We can deal with the first situation rather quickly: show your work on paper. Percent problems go with proportions like cheese goes with pizza. Be careful to match the original amount to the 100 in the denominators of each side of the proportion, and keep the percent over the 100 and you’re set. If you don’t know what I’m talking about when I say, “proportion”, look at the previous blog entries below.
The second case is actually the first thing that should happen when you’re starting the problem. Just like my last blog entry said, take your time to read carefully the first time and you’ll save time in the long run. Even with careful reading though, some people still struggle. Most of the time I’ve found this is due to confusion over one little letter. This letter is crucial to the meaning of the problem and the work involved in finding its solution.
How can one stinking letter be so powerful you ask? What is this powerful letter?
A: “f”
The letter “f” has the power to change “of” to “off”. If the problem is “10% of the original price”, the answer will turn out to be less than ½ of the original amount. If the letter is “10% off the original price”, the answer will end up being more than ½ of the original price. People will flock to a sale of "10% of" as it can also be interpreted as "90% off".
One little letter = one big difference.
If you aren’t careful with that one little letter “f” then “art” takes on an entirely new meaning.
That stinks. Read carefully.
Stay radical.
Thursday, April 2, 2009
Girls (and Boys) Just Wanna Have Fun...
Recently, I’ve come to the conclusion that many students secretly love doing math work. This is exciting. I can’t say, “Everybody loves doing math work” because that would imply 100% of the students love doing math work. It is very difficult to account for 100% of anything, let alone everybody, but there is a lot of evidence to support my stance.
Let me explain:
If you like doing something, you’ll spend time doing it. The more you like it, the more you’ll make it a priority. For instance, if you like playing video games, you’ll often trade sleep for the experience of saving the princess or ridding the planet of aliens. If you like inequalities, you’ll write them on the back of your hand ( 1<3 …). As a side note, these inequalities usually express a commitment to a unit rate of humans. For example: 1<3 Clarence. Also interesting is that what follows the <3 should, according to proper English, be plural but seldom is. Of course, some people prefer 1<3 ?, which as far as plurality goes, is ambiguous. This has been an interesting tangent, but now back to our regularly scheduled program…
If spending time doing something = fun then a lot of students enjoy some things that are a bit puzzling to me. For instance, some students enjoy being yelled at by their parents. These students must enjoy it because they invest time in being yelled at. They know their parents will nag them to do something that they are required to do on a regular basis. They don’t do what their parents ask them to do the first time, or the second, or the third, or the nth time (when n>3). What most of these students don’t realize is that each time their parents nag them it takes more of the student’s personal time that could be spent on video games or inequalities (or getting better at pencil tapping by practicing to YouTube videos). Since it’s taking up their time, and they could have made a choice to not be nagged by doing what they knew needed done in the first place, these students must view being nagged as fun.
Now this is bizarre behavior, but it’s not as strange as those students who secretly love math. On the surface these students may say, “I hate math”, but deep down their actions show otherwise. They could save a bundle of time if they used a bit of common sense, but they’re having fun doing (and redoing) things the hard way.
I hear the question now, “Mathman, what in the 2nd prime numbered planet from the sun do you mean?”
A: These students think that they’re taking the easy way out by saying: “I did it in my head” or “I did it on my calculator” or “I used another piece of scratch paper, but my little sister feels the need for more fiber in her diet so she ate it.” What he (used as the indefinite pronoun) doesn’t realize is that, in the end, math teachers are a stubborn lot and WILL require him to “Show your work” or “Explain your answer”. So the person who answered “IDK”, “?”, or “ITL” will, in fact, be spending more time on the problem then necessary. He could have put the effort into solving the problem correctly the first time, but instead chose the path that required not just solving the problem, but the path of extra writing by putting down the wrong answer in the first place. What’s the clincher in the whole deal though is that he also must like being nagged by his math teacher. Why? The student could have saved time by reading the directions and solving the problem correctly in the first place. Instead he chose to have the math teacher tell him to do it again (and again, and again, …).
Some people sure have strange ideas of what constitutes fun.
Maybe we need shirts:
1 <3 Math.
Or
1<3 Redoing things that I could have done right the first time.
Stay radical.
Let me explain:
If you like doing something, you’ll spend time doing it. The more you like it, the more you’ll make it a priority. For instance, if you like playing video games, you’ll often trade sleep for the experience of saving the princess or ridding the planet of aliens. If you like inequalities, you’ll write them on the back of your hand ( 1<3 …). As a side note, these inequalities usually express a commitment to a unit rate of humans. For example: 1<3 Clarence. Also interesting is that what follows the <3 should, according to proper English, be plural but seldom is. Of course, some people prefer 1<3 ?, which as far as plurality goes, is ambiguous. This has been an interesting tangent, but now back to our regularly scheduled program…
If spending time doing something = fun then a lot of students enjoy some things that are a bit puzzling to me. For instance, some students enjoy being yelled at by their parents. These students must enjoy it because they invest time in being yelled at. They know their parents will nag them to do something that they are required to do on a regular basis. They don’t do what their parents ask them to do the first time, or the second, or the third, or the nth time (when n>3). What most of these students don’t realize is that each time their parents nag them it takes more of the student’s personal time that could be spent on video games or inequalities (or getting better at pencil tapping by practicing to YouTube videos). Since it’s taking up their time, and they could have made a choice to not be nagged by doing what they knew needed done in the first place, these students must view being nagged as fun.
Now this is bizarre behavior, but it’s not as strange as those students who secretly love math. On the surface these students may say, “I hate math”, but deep down their actions show otherwise. They could save a bundle of time if they used a bit of common sense, but they’re having fun doing (and redoing) things the hard way.
I hear the question now, “Mathman, what in the 2nd prime numbered planet from the sun do you mean?”
A: These students think that they’re taking the easy way out by saying: “I did it in my head” or “I did it on my calculator” or “I used another piece of scratch paper, but my little sister feels the need for more fiber in her diet so she ate it.” What he (used as the indefinite pronoun) doesn’t realize is that, in the end, math teachers are a stubborn lot and WILL require him to “Show your work” or “Explain your answer”. So the person who answered “IDK”, “?”, or “ITL” will, in fact, be spending more time on the problem then necessary. He could have put the effort into solving the problem correctly the first time, but instead chose the path that required not just solving the problem, but the path of extra writing by putting down the wrong answer in the first place. What’s the clincher in the whole deal though is that he also must like being nagged by his math teacher. Why? The student could have saved time by reading the directions and solving the problem correctly in the first place. Instead he chose to have the math teacher tell him to do it again (and again, and again, …).
Some people sure have strange ideas of what constitutes fun.
Maybe we need shirts:
1 <3 Math.
Or
1<3 Redoing things that I could have done right the first time.
Stay radical.
Wednesday, March 18, 2009
Reason # Nineninenine to own a Calculator
I know you've been working hard this week and reading a lot so I'll keep this brief. Here's one final PSSA style question:
Why in Mathopolis would you get one of these?
a. You need a clock.
b. You've got mad math skills and you want to practice them.
c. You like Nines.
d. You are so cool your friends need to wear a sweater when they hang out with you.
This one is from the Triple Nine Society (if you search for that you can find out more info.)
Stay radical!
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