- Babylonian System
- Started with 2 symbols
- Around same time as Egyptian system, but much earlier than Roman system
- Introduced notion of place value
- Heres a link to see a picture of the Babylonian System http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Babylonian_numerals.html
- ¡Egyptian System
- 3400 BC
- Hieroglyphics
- Based on 10
- Additive
- No Place Values
- Heres a link to see a picture of the Egyptian System https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi4HLO_G_c2V_fIo3xFQHUhApGyA9MwTdx3yrvrpAJHP_ZBRBrE148q8XWQVvsoqYcvl0RogncQfw6cvqiLGFFPwMibgQeaG4z8kfrc2Px3ybhrr4hiWjruPd4OAUm5Wc07pP29qzcROf5_/s1600-h/egyptian%20number%20system%5B3%5D.png
- Roman System
- Based on 10
- Originally only Additive, later added Subtractive, and even later added Multiplicative
- No Place Values
- Heres a link to show you a picture of the Roman System http://mathsyear7.wikispaces.com/Roman+Numerals
- ¡Mayan System
- 3 symbols – includes zero
- Based on 20 and 18 x 20
- Place values
- Heres a link to show you a picture of the Mayan System http://picsdigger.com/keyword/mayan%20number%20system/
- Indo-Arabic System
- Also called the decimal system
- Base is ten
- 0,1,2,3,4,5,6,7,8,9 are the digits used
- Positional system: take the digit and multiply it by the value based on its position
Sunday, October 24, 2010
Numeration System
Greatest Common Factors & Least Common Multiples
Methods for finding Greatest Common Factors
List the Factors:
Find GCF (16,56)
Factors of 16: 1,2,4,8,16
Factors of 56: 1,2,4,7,8,14,28,56
Now you look and find which number is a factor of both that is the highest or greatest number!
So in this case the highest number is 8... So the GCF (16,56) = 8
Linear (Cuisenaire rods)
Find GCF (6,12)
Find GCF (4,8)
Find GCF (4,9)
And for this all you do is put together the blocks that would make up the number your looking for. And then you will find the Great Common Factor for both.
Prime Factorization
Find GCF (30,42)
30= 2 x 3 x 5
Now how you find these prime factors
3 x 5 = 15
15 x 2 = 30
2,3,5 are all prime numbers
42= 2 x 3 x 7
SAME as the top
3 x 7 = 21
2 x 21 = 42
2,3,7 are all prime numbers they can't be divide down anymore.
Methods for finding Least Common Multiples
List the Multiples
Find LCM (3,5)
Multiples: 3,6,9,12,15,18,21
Multiples:5,10,15,20,25,30
And your least common multiple is 15 and for this one you can stop as soon as you find a common factor for both because your finding the lowest number.
Linear
Find LCM (4,5)
You can use the number line to help you with this type
And all you have to do is add the number to itself everytime.
Prime Factorization
Find LCM (8,20)
8: 2 x 2 x 2
20: 2 x 2 x 5
And this is how you get these numbers.
2 x 2 = 4
4 x 2 = 8
2 x 5 = 10
10 x 2 = 20
You always keep dividing until you can't divide anymore.
List the Factors:
Find GCF (16,56)
Factors of 16: 1,2,4,8,16
Factors of 56: 1,2,4,7,8,14,28,56
Now you look and find which number is a factor of both that is the highest or greatest number!
So in this case the highest number is 8... So the GCF (16,56) = 8
Linear (Cuisenaire rods)
Find GCF (6,12)
Find GCF (4,8)
Find GCF (4,9)
And for this all you do is put together the blocks that would make up the number your looking for. And then you will find the Great Common Factor for both.
Prime Factorization
Find GCF (30,42)
30= 2 x 3 x 5
Now how you find these prime factors
3 x 5 = 15
15 x 2 = 30
2,3,5 are all prime numbers
42= 2 x 3 x 7
SAME as the top
3 x 7 = 21
2 x 21 = 42
2,3,7 are all prime numbers they can't be divide down anymore.
Methods for finding Least Common Multiples
List the Multiples
Find LCM (3,5)
Multiples: 3,6,9,12,15,18,21
Multiples:5,10,15,20,25,30
And your least common multiple is 15 and for this one you can stop as soon as you find a common factor for both because your finding the lowest number.
Linear
Find LCM (4,5)
You can use the number line to help you with this type
And all you have to do is add the number to itself everytime.
Prime Factorization
Find LCM (8,20)
8: 2 x 2 x 2
20: 2 x 2 x 5
And this is how you get these numbers.
2 x 2 = 4
4 x 2 = 8
2 x 5 = 10
10 x 2 = 20
You always keep dividing until you can't divide anymore.
Wednesday, October 20, 2010
Prime Factorization
Methods for finding Prime Factorization
1. Increasing Primes - Divides the numbers by increasing prime numbers
Ex: 74
(2) / 74 = (37)
2 / 37 = 18.10
3 / 37 = 12.1
you can keep trying higher numbers to see if 37 divides into anything. But the answer would be 2 x 37 sense nothing else divides into 37 b/c its a prime number.
another example is 84: (2) / 84 = 42 (2) / 42 = 21 (3)/ 21 = (7)
so your answer is 2 x 2 x 3 x 7
2. Two Factors - Keep writing numbers as a product of Two Factors
Ex 84: 2 x 42
now break up 42
6 x 7
so your answer now will be 2 x 6 x7 = (84)
3. Factor Tree - this goes hand and hand with Two Factors
1. Increasing Primes - Divides the numbers by increasing prime numbers
Ex: 74
(2) / 74 = (37)
2 / 37 = 18.10
3 / 37 = 12.1
you can keep trying higher numbers to see if 37 divides into anything. But the answer would be 2 x 37 sense nothing else divides into 37 b/c its a prime number.
another example is 84: (2) / 84 = 42 (2) / 42 = 21 (3)/ 21 = (7)
so your answer is 2 x 2 x 3 x 7
2. Two Factors - Keep writing numbers as a product of Two Factors
Ex 84: 2 x 42
now break up 42
6 x 7
so your answer now will be 2 x 6 x7 = (84)
3. Factor Tree - this goes hand and hand with Two Factors
Divisibilty Test
In class we learned how to find divisible numbers between 2-11
By 2. A number is divisible by 2 if the ones digit is divisible by 2
example: 3,846,711,439,28(4) / 2 = Divisible by 2
By 3 or 9. A number is divisible by 3 or 9 if the sum of the digits is divisible by 3 or 9
example: 3,846,711,4(39),284 / 3 = Divisible by 3 the sum of its digit is 60...... 60 / 3 = 20
example: 3,846,711,439,284 / 9 = Not divisible by 9..... 60 / 9 = 6.6666
By 4. A number is divisible by 4 if the last two digits are divisible by 4
example: 3,846,711,439,2(84) / 4 = Divisible by 4
By 5. A number is divisible by 5 if the ones digit is a 5 or 0
example: 3,846,711,439,28(4) / 5 = Not divisible by 5 b/c no 0 or 5 in the ones place
By 6. A number is divisible by 6 if its divisible by both 2 and 3
example: 3,846,711,439,284 / 6 = Divisible by 6 b/c up we have proven that 2 and 3 is divisible
By 7. A number is divisible by 7 if you can group up the numbers and if they are divisible by 7 or 11
example: (003),846,(711),439,(284) / 7 = Divisible by 7........... 287 / 7 = 41
Not divisible by 11...... 287 / 11 = 26.09
003 846 1285 - 998 = 287
711 + 439 = 1285
+284 = 998
By 8. A number is divisible by 8 if the last three digits are divisible by 8
example: 3,846,711,439,(284) / 8 = Not divisible by 8..... 284 / 8 = 35.5
Here is the link to show you the divisiblility rules 1-10. And if you don't understand what was typed up top.
By 2. A number is divisible by 2 if the ones digit is divisible by 2
example: 3,846,711,439,28(4) / 2 = Divisible by 2
By 3 or 9. A number is divisible by 3 or 9 if the sum of the digits is divisible by 3 or 9
example: 3,846,711,4(39),284 / 3 = Divisible by 3 the sum of its digit is 60...... 60 / 3 = 20
example: 3,846,711,439,284 / 9 = Not divisible by 9..... 60 / 9 = 6.6666
By 4. A number is divisible by 4 if the last two digits are divisible by 4
example: 3,846,711,439,2(84) / 4 = Divisible by 4
By 5. A number is divisible by 5 if the ones digit is a 5 or 0
example: 3,846,711,439,28(4) / 5 = Not divisible by 5 b/c no 0 or 5 in the ones place
By 6. A number is divisible by 6 if its divisible by both 2 and 3
example: 3,846,711,439,284 / 6 = Divisible by 6 b/c up we have proven that 2 and 3 is divisible
By 7. A number is divisible by 7 if you can group up the numbers and if they are divisible by 7 or 11
example: (003),846,(711),439,(284) / 7 = Divisible by 7........... 287 / 7 = 41
Not divisible by 11...... 287 / 11 = 26.09
003 846 1285 - 998 = 287
711 + 439 = 1285
+284 = 998
By 8. A number is divisible by 8 if the last three digits are divisible by 8
example: 3,846,711,439,(284) / 8 = Not divisible by 8..... 284 / 8 = 35.5
Here is the link to show you the divisiblility rules 1-10. And if you don't understand what was typed up top.
ENJOY!!!!!
Wednesday, October 13, 2010
Models of Multiplication
In class we learned that there was Three Models of Multiplication. You hve Rectangular Array, Tree Diagram, Partial Products. All three of these ways are very useful and easy to understand; here are some examples of each to give you an understanding of all of them.
Rectangular Array 4x6
Tree Diagram
23x57
rewrite it like this to make it easier to visualizes it
23
x 57
now you write each number out thats in different place values
20 + 3
50 +7
now you can use cross multiplication
7 x 3= 21
20 x 7= 140
50 x 3= 150
20 x 50= 1000
now add all your end solutions
1311
and that is your Partial Products.
Friday, October 1, 2010
Addition Algorithms
In class Thursday, we learned about Addition Algorithms and that there are three of them Left to Right, Partial Sums, and Right to Left (aka the traditional algorithm). I found that the Left to Right and the Partial Sums algorithm way wasn't to hard if you got to understand it and the way it works. So heres a link that shows the Partial Sums and Left to Right algorithm http://www.youtube.com/watch?v=mvdVjwbim6w.
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