Sit down because it is a long one!
A tribe of Native Americans generally referred to their woman by the animal hide with which they made their blanket. Thus, one woman might be known as Squaw of Buffalo Hide, while another might be known as Squaw of Deer Hide. This tribe had a particularly large and strong woman, with a very unique (for North America anyway) animal hide for her blanket. This woman was known as Squaw of Hippopotamus hide, and she was as large and powerful as the animal from which her blanket was made.
Year after year, this woman entered the tribal wrestling tournament, and easily defeated all challengers; male or female. As the men of the tribe admired her strength and power, this made many of the other woman of the tribe extremely jealous. One year, two of the squaws petitioned the Chief to allow them to enter their sons together as a wrestling tandem in order to wrestle Squaw of the Hippopotamus hide as a team. In this way, they hoped to see that she would no longer be champion wrestler of the tribe.
As the luck of the draw would have it, the two sons who were wrestling as a tandem met the squaw in the final and championship round of the wrestling contest. As the match began, it became clear that the squaw had finally met an opponent that was her equal. The two sons wrestled and struggled vigorously and were clearly on an equal footing with the powerful squaw. Their match lasted for hours without a clear victor. Finally the chief intervened and declared that, in the interests of the health and safety of the wrestlers, the match was to be terminated and that he would declare a winner.
The chief retired to his teepee and contemplated the great struggle he had witnessed, and found it extremely difficult to decide a winner. While the two young men had clearly outmatched the squaw, he found it difficult to force the squaw to relinquish her tribal championship. After all, it had taken two young men to finally provide her with a decent match. Finally, after much deliberation, the chief came out from his teepee, and announced his decision. He said...Y
"The Squaw of the Hippopotamus hide is equal to the sons of the squaws of the other two hides"
Wednesday, March 18, 2009
Triangle Calculator
If you know three pieces of information about your triangle, this will give you the others.
Monday, March 16, 2009
This is a great site for playing with conic sections
You know what conic sections are: circles, parabolas, ellipses, and hyperbolas.
Draw them here and get a feel for what all those numbers really mean.
Draw them here and get a feel for what all those numbers really mean.
Tips and trick for teaching money
Before you even begin teaching money, teach the little ones to count by 5's and 10's.
Practice starting at random places. Start at 30 the next is 40 and then 50. Or start at 45 the next is 50. Don't forget to learn to count by 25 too!
Then when you introduce the coins, they know just how to add things up in their heads.
Practice starting at random places. Start at 30 the next is 40 and then 50. Or start at 45 the next is 50. Don't forget to learn to count by 25 too!
Then when you introduce the coins, they know just how to add things up in their heads.
Friday, March 13, 2009
Exponents
What is an exponent?
An exponent is the lazy way of saying, "multiply something against itself a bunch of times."
a*a*a*a*a = a^5 (there were 5 a's so the exponent is 5)
See isn't that shorter?
But along with writing something shorter, you have to learn to use the notation.
Let's learn how you deal with
a^3 * a^4
In long hand that is:
a*a*a * a*a*a*a = a^7
So in other words
a^3 * a^4 = a^(3+4)
Generically that is:
a^x * a^y = a^(x + y)
Multiplying a terms with similar variables, means you need to ADD exponents.
Now let's look at negative exponents.
1/a * 1/a * 1/a = = 1/(a*a*a) = 1/(a^3)
If we look at this example
(a^5)/(a^3) it looks like this in long form:
(a*a*a*a*a)/(a*a*a) = a*a = a^2
So in other words:
a^5/a^3 = a^(5 - 3) = a^2
Dividing terms with similar exponents, is the same as SUBTRACTING the exponents.
Generally speaking:
a^x/a^y = a^(x-y)
Now for the fun part!
If x is smaller than y, the exponent would be NEGATIVE!
a^3/a^5 = a^-2 = 1/a^2
So what does it mean if the exponent is zero????
Let's look at this example.
a^x/a^x = 1
a^(x - x) = 1
a^0 = 1
And that is true for ANY a!
An exponent is the lazy way of saying, "multiply something against itself a bunch of times."
a*a*a*a*a = a^5 (there were 5 a's so the exponent is 5)
See isn't that shorter?
But along with writing something shorter, you have to learn to use the notation.
Let's learn how you deal with
a^3 * a^4
In long hand that is:
a*a*a * a*a*a*a = a^7
So in other words
a^3 * a^4 = a^(3+4)
Generically that is:
a^x * a^y = a^(x + y)
Multiplying a terms with similar variables, means you need to ADD exponents.
Now let's look at negative exponents.
1/a * 1/a * 1/a = = 1/(a*a*a) = 1/(a^3)
If we look at this example
(a^5)/(a^3) it looks like this in long form:
(a*a*a*a*a)/(a*a*a) = a*a = a^2
So in other words:
a^5/a^3 = a^(5 - 3) = a^2
Dividing terms with similar exponents, is the same as SUBTRACTING the exponents.
Generally speaking:
a^x/a^y = a^(x-y)
Now for the fun part!
If x is smaller than y, the exponent would be NEGATIVE!
a^3/a^5 = a^-2 = 1/a^2
So what does it mean if the exponent is zero????
Let's look at this example.
a^x/a^x = 1
a^(x - x) = 1
a^0 = 1
And that is true for ANY a!
Wednesday, March 11, 2009
Math fun
This website had mathcaching which is like geocaches but with math. What fun!
http://mathbits.com/Caching/BasicOpenCache1.html
http://mathbits.com/Caching/BasicOpenCache1.html
Teaching addition to kids: part 2
I'm starting with part 2 because, I'm teaching this to my 7yr old right now. She is not a math-natural and it seems to be working like a charm. This part is about numbers whose sum is greater than 10. Part 1 is about numbers whose sum is less than or equal to 10.
In part 1, which isn't yet posted--sorry!, we emphasized "magic friends add to ten." The short story here is that every number has a friend that will sum to ten.
1 and 9
2 and 8
3 and 7
4 and 6
5 and 5
The kids should know this very well before proceeding.
Step 1 to this process is teaching kids what 10 + another number is. They should know 10+1 is 11. They don't necessarily need to understand ones place and tens place to get through this though it is helpful. If they can count to twenty, they can recognize that 10 + 2 is twelve.
In order to teach this, I start at the top.
10 + 9 is 9 teen. Emphasize the teen here.
10 + 8 is 8 teen.
Do this down to 16. They'll hear the pattern.
Next, tell them that 13 and 15 are very similar but we just pronounce them funny.
5 teen is fifteen
3 teen is thirteen
Don't forget to mention 14.
When you get to 12 and 11, mention that these numbers are got named differently since 12 is such an important number, i.e. 12 months in a year, 12 hours in a day/night, 12 in a dozen.
2 teen is twelve
1 teen is eleven
Practice adding 10 + another number until this is easy for them to do. No paper is necessary for this work.
(If the child doesn't, yet, understand that 20 + another number is just twenty and the other number, now is a good time to alert them to that fact as well. We teach that
twenty = twoty
thirty = threety
fifty = fivety
etc.
Notice, the strange naming practices end at the same spots. Twenty and Twelve have odd spelling when compared to sixty and ninety. Also, look up twelve and eleven in wiki and you'll see some interesting stuff to make math come alive for the linguists among us.)
The second step has two sub-steps and that is the end, I promise.
2a)the smaller of the two numbers in need of addition is decomposed into two smaller yet numbers (one of which is the magic friend of the larger number)
2b)the resulting 3 numbers are added pointing out that two of them are magic friends.
And example will clarify what I mean.
7 + 8
Take 7 and decompose it into 2 and 5. Then since the 2 is the magic friend of 8, the 2 and 8 make a 10. Now just add the 5 to 10 yielding five teen--15.
The key to this is knowing the magic friend of 8. And then being able to decompose 7 into that magic number and another number.
That is the concept, but now to how to teach it to a child.
First, start with the problem, I'll use a different example. 6 + 9
Write it out with lots of space below. Ask the child to recognize which of the numbers is bigger (9) and draw a line straight down from it and rewrite 9.
Now, from the smaller number, draw two lines that angle away like you would use for a family tree. Construct these lines such than when you write a number at the end of the lines, they will be level with the 9. We will be creating a new equation with the numbers beneath the lines.
Now, ask the child to give you the "magic friend" of 9. Write that number in the position that is nearest the 9. In our problem, we'll have a 1 in that position.
Ask the child what number is needed with the 1 to sum to 6 and of course that is a 5. Write that beneath the line that is empty. Put + signs between all the numbers.
So now in our new equation we have 5 + 1 + 9. Circle the 1 and the 9. Point out that this sums to 10 and then ask them to sum 5 and 10 which is of course five teen or 15.
(Note: I always find the magic friend of the larger number because 1) decomposing smaller numbers is easier than larger numbers and 2) the magic friend is smaller as well. You don't have to do it this way, but I find it works more easily.)
After they have the idea and can do this on paper easily, have them practice in their heads. Any math that can be done in the head, should be done in the head.
In part 1, which isn't yet posted--sorry!, we emphasized "magic friends add to ten." The short story here is that every number has a friend that will sum to ten.
1 and 9
2 and 8
3 and 7
4 and 6
5 and 5
The kids should know this very well before proceeding.
Step 1 to this process is teaching kids what 10 + another number is. They should know 10+1 is 11. They don't necessarily need to understand ones place and tens place to get through this though it is helpful. If they can count to twenty, they can recognize that 10 + 2 is twelve.
In order to teach this, I start at the top.
10 + 9 is 9 teen. Emphasize the teen here.
10 + 8 is 8 teen.
Do this down to 16. They'll hear the pattern.
Next, tell them that 13 and 15 are very similar but we just pronounce them funny.
5 teen is fifteen
3 teen is thirteen
Don't forget to mention 14.
When you get to 12 and 11, mention that these numbers are got named differently since 12 is such an important number, i.e. 12 months in a year, 12 hours in a day/night, 12 in a dozen.
2 teen is twelve
1 teen is eleven
Practice adding 10 + another number until this is easy for them to do. No paper is necessary for this work.
(If the child doesn't, yet, understand that 20 + another number is just twenty and the other number, now is a good time to alert them to that fact as well. We teach that
twenty = twoty
thirty = threety
fifty = fivety
etc.
Notice, the strange naming practices end at the same spots. Twenty and Twelve have odd spelling when compared to sixty and ninety. Also, look up twelve and eleven in wiki and you'll see some interesting stuff to make math come alive for the linguists among us.)
The second step has two sub-steps and that is the end, I promise.
2a)the smaller of the two numbers in need of addition is decomposed into two smaller yet numbers (one of which is the magic friend of the larger number)
2b)the resulting 3 numbers are added pointing out that two of them are magic friends.
And example will clarify what I mean.
7 + 8
Take 7 and decompose it into 2 and 5. Then since the 2 is the magic friend of 8, the 2 and 8 make a 10. Now just add the 5 to 10 yielding five teen--15.
The key to this is knowing the magic friend of 8. And then being able to decompose 7 into that magic number and another number.
That is the concept, but now to how to teach it to a child.
First, start with the problem, I'll use a different example. 6 + 9
Write it out with lots of space below. Ask the child to recognize which of the numbers is bigger (9) and draw a line straight down from it and rewrite 9.
Now, from the smaller number, draw two lines that angle away like you would use for a family tree. Construct these lines such than when you write a number at the end of the lines, they will be level with the 9. We will be creating a new equation with the numbers beneath the lines.
Now, ask the child to give you the "magic friend" of 9. Write that number in the position that is nearest the 9. In our problem, we'll have a 1 in that position.
Ask the child what number is needed with the 1 to sum to 6 and of course that is a 5. Write that beneath the line that is empty. Put + signs between all the numbers.
So now in our new equation we have 5 + 1 + 9. Circle the 1 and the 9. Point out that this sums to 10 and then ask them to sum 5 and 10 which is of course five teen or 15.
(Note: I always find the magic friend of the larger number because 1) decomposing smaller numbers is easier than larger numbers and 2) the magic friend is smaller as well. You don't have to do it this way, but I find it works more easily.)
After they have the idea and can do this on paper easily, have them practice in their heads. Any math that can be done in the head, should be done in the head.
Factoring tests
There are ways to easily check to see if a large number is divisible by a smaller number.
1: All numbers are divisible by 1, even prime numbers.
2: Even numbers are divisible by 2. An even number is any number that ends in 0, 2, 4, 6, or 8.
3: When you sum the digits of your large number, if that sum is divisible by 3, then the original number is divisible by 3. Here is an example.
417
4+1+7 = 12
1+2 = 3
3 is divisible by 3 so 417 is divisible by 3.
5: If the number ends in 0 or 5 (excluding 0 itself) then the number is divisible by 5.
9: This is similar to 3. Sum the digits of the larger number, if they sum to a multiple of 9, then the original number is divisible by 9. Here is an example:
459
4+5+9 = 18
1+8 = 9
9 is divisible by 9 therefore 459 is divisible by 9.
10: If the number ends in 0 (excluding 0 itself) then the original number is divisible by 10.
Advanced hint:
When summing digits to check for 3 and 9 divisibility, you can skip adding in the 9's. One day I'll post why that is but until then, give it a try and see for yourself that you can just skip over the 9's.
1: All numbers are divisible by 1, even prime numbers.
2: Even numbers are divisible by 2. An even number is any number that ends in 0, 2, 4, 6, or 8.
3: When you sum the digits of your large number, if that sum is divisible by 3, then the original number is divisible by 3. Here is an example.
417
4+1+7 = 12
1+2 = 3
3 is divisible by 3 so 417 is divisible by 3.
5: If the number ends in 0 or 5 (excluding 0 itself) then the number is divisible by 5.
9: This is similar to 3. Sum the digits of the larger number, if they sum to a multiple of 9, then the original number is divisible by 9. Here is an example:
459
4+5+9 = 18
1+8 = 9
9 is divisible by 9 therefore 459 is divisible by 9.
10: If the number ends in 0 (excluding 0 itself) then the original number is divisible by 10.
Advanced hint:
When summing digits to check for 3 and 9 divisibility, you can skip adding in the 9's. One day I'll post why that is but until then, give it a try and see for yourself that you can just skip over the 9's.
Tuesday, March 10, 2009
Order of operations
Why do we have order of operations? Well it is pretty simple--so we can all read the same expression in the same way. It is all about communicating to one another effectively in the language of mathematics.
if What wrote a in sentence words the I of order any?
It would be really hard to understand what I meant--but not impossible. I bet with some effort you could figure out my sentence. And that is because English has some rules that we "just know" for the most part. The rest we call grammar.
We learned the basics of grammar long, long ago when we were just little kids. And mathematics is the same way. Unfortunately, we don't "speak" much math to little kids so there isn't much in the way of the basics of the mathematical language in our heads. We have to learn it the "formal way."
But that is ok. The rules are very simple--way more simple than the rules of the English language. And PEMDAS sums it all up. I have a post explaining what that means and how to use it here.
But the long and the short of it is that PEMDAS is a set of simple rules that keep us all speaking the same language.
Here is an example.
in the box
That is a nice prepositional phrase. We never see it written "box the in." There is an order that we MUST follow here. But after we do, we can put that phrase in many locations relative the thing that is "in the box."
Put the sweater in the box.
Put the sweater that I love in the box.
In the box goes the sweater.
Going in the box is the sweater.
Some sentences are better than others but you can understand the meaning of all of them.
Now, in mathematics.
(3 + 4)
This is the same as the phrase "in the box." I can move it around in an expression.
8*(3+4)
(3+4)*8
(3+4)/8
(3+4)^5
(3+4) + (5+6)
When we follow know the order of operations, we can construct very complex and terribly organized equations, yet all will be able to read it. Of course we strive to make our equations simple and beautiful--but that is were experience comes into play.
The order of operations is simply part of the grammar of mathematics making a jumble of symbols and numbers make coherent concepts.
if What wrote a in sentence words the I of order any?
It would be really hard to understand what I meant--but not impossible. I bet with some effort you could figure out my sentence. And that is because English has some rules that we "just know" for the most part. The rest we call grammar.
We learned the basics of grammar long, long ago when we were just little kids. And mathematics is the same way. Unfortunately, we don't "speak" much math to little kids so there isn't much in the way of the basics of the mathematical language in our heads. We have to learn it the "formal way."
But that is ok. The rules are very simple--way more simple than the rules of the English language. And PEMDAS sums it all up. I have a post explaining what that means and how to use it here.
But the long and the short of it is that PEMDAS is a set of simple rules that keep us all speaking the same language.
Here is an example.
in the box
That is a nice prepositional phrase. We never see it written "box the in." There is an order that we MUST follow here. But after we do, we can put that phrase in many locations relative the thing that is "in the box."
Put the sweater in the box.
Put the sweater that I love in the box.
In the box goes the sweater.
Going in the box is the sweater.
Some sentences are better than others but you can understand the meaning of all of them.
Now, in mathematics.
(3 + 4)
This is the same as the phrase "in the box." I can move it around in an expression.
8*(3+4)
(3+4)*8
(3+4)/8
(3+4)^5
(3+4) + (5+6)
When we follow know the order of operations, we can construct very complex and terribly organized equations, yet all will be able to read it. Of course we strive to make our equations simple and beautiful--but that is were experience comes into play.
The order of operations is simply part of the grammar of mathematics making a jumble of symbols and numbers make coherent concepts.
Monday, March 9, 2009
Book suggestion
13 1/2 Lives of Captain Bluebear
Science fiction hidden inside an action adventure story. Plus who can resist a title with a fraction?
Reading level: 7th grade (all ages would love it though!)
Science fiction hidden inside an action adventure story. Plus who can resist a title with a fraction?
Reading level: 7th grade (all ages would love it though!)
PEMDAS--order of operations
P= Parenthesis
E= Exponent
MD = Multiply and Divide
AS = Add and subtract
This is the order of operations.
Look at your mathematical expression and if you see any parenthesis (since it is the first item) work with that first. If there are no parentheses, look for exponents next. Continue down the list until you find the operation you need.
After you do anything to your expression, start at the top again. Check for P, then E, then MD, finally AS.
Here is an example:
(5 + 3 * 4 - 8)^(16/4)
Do we have parentheses? Yes. Let's pick one and look within it.
5 + 3 * 4 - 8 I'll look here first.
Is there a parenthesis? No.
Is there an exponent? No.
Is there multiplication or division? Yes. Let's do that.
5 + 12 - 8
Starting at the top again, but staying with this term in the first parenthesis.
Is there a parenthesis? No.
Is there an exponent? No.
Is there multiplication or division? No.
Is there addition or subtraction? Yes. Let's do that.
5 + 12 - 8 = 9
Ok, we are down to a single number let's put that in the place of the parenthesis we chose to start with.
(5 + 3 * 4 - 8)^(16/4) = 9^(16/4)
Now we start again.
Is there a parenthesis? Yes. Let's work on it now.
16/4
Is there a parenthesis? No.
Is there an exponent? No.
Is there multiplication or division? Yes. Let's do that.
16/4 = 4
Ok, we are down to a single number. Let's put that back into the problem.
9^(16/4) = 9^4
Starting again.
Is there a parenthesis? No.
Is there an exponent? Yes. Ok, let's do that.
9^4 = 9*9*9*9 = 6561
We are down to a single number so we are done.
(5 + 3 * 4 - 8)^(16/4) = 6561
E= Exponent
MD = Multiply and Divide
AS = Add and subtract
This is the order of operations.
Look at your mathematical expression and if you see any parenthesis (since it is the first item) work with that first. If there are no parentheses, look for exponents next. Continue down the list until you find the operation you need.
After you do anything to your expression, start at the top again. Check for P, then E, then MD, finally AS.
Here is an example:
(5 + 3 * 4 - 8)^(16/4)
Do we have parentheses? Yes. Let's pick one and look within it.
5 + 3 * 4 - 8 I'll look here first.
Is there a parenthesis? No.
Is there an exponent? No.
Is there multiplication or division? Yes. Let's do that.
5 + 12 - 8
Starting at the top again, but staying with this term in the first parenthesis.
Is there a parenthesis? No.
Is there an exponent? No.
Is there multiplication or division? No.
Is there addition or subtraction? Yes. Let's do that.
5 + 12 - 8 = 9
Ok, we are down to a single number let's put that in the place of the parenthesis we chose to start with.
(5 + 3 * 4 - 8)^(16/4) = 9^(16/4)
Now we start again.
Is there a parenthesis? Yes. Let's work on it now.
16/4
Is there a parenthesis? No.
Is there an exponent? No.
Is there multiplication or division? Yes. Let's do that.
16/4 = 4
Ok, we are down to a single number. Let's put that back into the problem.
9^(16/4) = 9^4
Starting again.
Is there a parenthesis? No.
Is there an exponent? Yes. Ok, let's do that.
9^4 = 9*9*9*9 = 6561
We are down to a single number so we are done.
(5 + 3 * 4 - 8)^(16/4) = 6561
Mathematics for everyone
This blog has been created to explore math, to learn math, to share my love of math. Shall we begin?
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