Using different bases makes it easier to see how someone might feel when learning arithmetic for the first time. A base determines how many numbers it will include. For example, base 5 has the numbers 0, 1, 2, 3, and 4. Numbers five or greater cannot be used. A good way to learn to use a different base is by using manipulatives or base pieces, or other hands on materials. Another method is to use "scoops" (circling marks on the page to show how much of something is needed. The sharing method involves making a table for division & "sharing" the pieces among the group.
As you can see, there are many different methods for finding an answer. I usually find one method that works the best, but it's also important to try all the different methods.
This site explains how the numbers in base 5 work. It gives a lot of information.
http://www.basic-mathematics.com/base-five.html
This site is very descriptive about arithmetic in other bases
http://mathforum.org/library/drmath/view/55727.html
Monday, October 25, 2010
Greatest Common Factor
The Greatest Common Factor of a number is a number that is a factor of both a and b, while the least common multiple is the smallest number that is a multiple of both a and b.
So, let's find the GCF & LCM of (6,24)
to find the GCF, first we have to find the factors of each number. We can do this in a variety of ways. First, we can use the "List Factors" method.
Factors of 6: 1, 2,3 , 6
Factors of 24: 1,2,3,4,6,8,12, 24
We then list the factors both numbers have in common: 1, 2, 3, and 6 Since 6 is the greatest number, 6 is the greatest common factor of 6 and 24.
Another way to do this is by using cuisenaire rods or another linear model. We represent each factor by using colored rods, and determine which is the largest rod in both numbers that is the same color. http://webcache.googleusercontent.com/search?q=cache:JGWSHn386nsJ:www.cerritos.edu/mnikdel/Folder%2520for%2520Word%2520files/Cuisenaire%2520Rods.doc+greatest+common+factor+using+cuisenaire+rods&cd=1&hl=en&ct=clnk&gl=us
This website is a very good visual representation of the method, and also explains how to use it.
We can also use prime factorization again, along with venn diagrams.
Prime Factorization of 6: 2x3
Prime Factorization of 24: 2x2x2x3
Then we draw a venn diagram and write the numbers in the circles based on how many times they appear in either or both numbers. http://www.learner.org/courses/learningmath/number/session6/part_a/index.html
The same methods can be applied to finding the Least Common Multiple, but by substituting multiples for factors.
So, let's find the GCF & LCM of (6,24)
to find the GCF, first we have to find the factors of each number. We can do this in a variety of ways. First, we can use the "List Factors" method.
Factors of 6: 1, 2,3 , 6
Factors of 24: 1,2,3,4,6,8,12, 24
We then list the factors both numbers have in common: 1, 2, 3, and 6 Since 6 is the greatest number, 6 is the greatest common factor of 6 and 24.
Another way to do this is by using cuisenaire rods or another linear model. We represent each factor by using colored rods, and determine which is the largest rod in both numbers that is the same color. http://webcache.googleusercontent.com/search?q=cache:JGWSHn386nsJ:www.cerritos.edu/mnikdel/Folder%2520for%2520Word%2520files/Cuisenaire%2520Rods.doc+greatest+common+factor+using+cuisenaire+rods&cd=1&hl=en&ct=clnk&gl=us
This website is a very good visual representation of the method, and also explains how to use it.
We can also use prime factorization again, along with venn diagrams.
Prime Factorization of 6: 2x3
Prime Factorization of 24: 2x2x2x3
Then we draw a venn diagram and write the numbers in the circles based on how many times they appear in either or both numbers. http://www.learner.org/courses/learningmath/number/session6/part_a/index.html
The same methods can be applied to finding the Least Common Multiple, but by substituting multiples for factors.
Finding Factors
Here are some different methods that can be used to find factors of numbers.
One of these methods is called the "Curtain Method". Start with the smaller number on the left, and put the larger numbers on the right
ex: Factors of 12: 1 2 3 4 6 12
You can also use prime factorization
this involves finding the prime factors of any given number ex: 14: 2x7
Another way to find factors is by using a Factor Tree
The link below has a game that uses a factor tree.
http://www.mathgoodies.com/factors/prime_factors.html
More info about finding factors
http://www.math.com/school/subject1/lessons/S1U3L2GL.html
One of these methods is called the "Curtain Method". Start with the smaller number on the left, and put the larger numbers on the right
ex: Factors of 12: 1 2 3 4 6 12
You can also use prime factorization
this involves finding the prime factors of any given number ex: 14: 2x7
Another way to find factors is by using a Factor Tree
The link below has a game that uses a factor tree.
http://www.mathgoodies.com/factors/prime_factors.html
More info about finding factors
http://www.math.com/school/subject1/lessons/S1U3L2GL.html
Mental Calculations
It is important to be able to do mental calculations and apply different strategies for calculating. Mental calculations can make it easier to find the product without using a calculator or pencil & paper.
One method that can be used is the Compatible Numbers property.
For example: 4x2x25x5 - this problem looks fairly difficult but it can be easy if we use the method.
1st, find numbers that are easy to multiply together such as 4x25, and 2x5. Since we know 4x25 is 100 and 2x5 is 10, we are left with 100x10
2nd, multiply the remaining numbers. 100x10=1000
Subtraction/Distribution property
ex: 96x30
1) 100 is an easier number to work with than 96, so we can write 96 as 100-4.
2) We are left with (100-4)x30 which is the same as 30(100-4)
3) Distribute the numbers: 30x100= 3000 and 30 x-4 = -120
4) We are left with 3000-120 which equals 2880.
These are the mental calculations I found the easiest to use and the most helpful. However, there are a lot more methods and this website describes many more of them. http://www.cut-the-knot.org/arithmetic/rapid/rapid.shtml
Also, i've included a video that I found very interesting. It teaches a method of multiplying that uses the base itself (base 10) to mentally calculate the product.
One method that can be used is the Compatible Numbers property.
For example: 4x2x25x5 - this problem looks fairly difficult but it can be easy if we use the method.
1st, find numbers that are easy to multiply together such as 4x25, and 2x5. Since we know 4x25 is 100 and 2x5 is 10, we are left with 100x10
2nd, multiply the remaining numbers. 100x10=1000
Subtraction/Distribution property
ex: 96x30
1) 100 is an easier number to work with than 96, so we can write 96 as 100-4.
2) We are left with (100-4)x30 which is the same as 30(100-4)
3) Distribute the numbers: 30x100= 3000 and 30 x-4 = -120
4) We are left with 3000-120 which equals 2880.
These are the mental calculations I found the easiest to use and the most helpful. However, there are a lot more methods and this website describes many more of them. http://www.cut-the-knot.org/arithmetic/rapid/rapid.shtml
Also, i've included a video that I found very interesting. It teaches a method of multiplying that uses the base itself (base 10) to mentally calculate the product.
Monday, October 18, 2010
Bases Part I
Today we are going to start learning about using different bases for couting, addition, subtraction, multiplication, division, etc. We use base ten everyday without thinking about it, so it's easy to take for granted how hard it is to learn these functions.
So, what's a base?
An example of a base we are all familiar with is base ten- it uses the numbers 0,1,2, 3,4,5,5,7, 8, and 9. Once we need a bigger number than 9, we exchange/borrow for the next larger manipulative piece, since we want to use the least amount of pieces as possible. A useful way to illustrate this is by using manipulatives. Below is a video that demonstrates using manipulatives for addition in base ten.
Here is another example: 19
+ 7
____
This means that we have one long and nine units, and we are adding seven units. First we add the units together. 9+7 = 16. Since 16 is larger than 9, we have to exchange some of the units for a long. A long uses 10 units, so we take away ten units out of the 16 we have. This leaves us with 6 units (on the ones side), and another long to put on the tens side. Now we have:
1
19
+ 7
___
6
Next, we have to add the longs together. There are only 2, so no exchanges need to be made. This means we have 2 longs and 6 units which means we have 26.
19
+ 7
____
26
So, what's a base?
An example of a base we are all familiar with is base ten- it uses the numbers 0,1,2, 3,4,5,5,7, 8, and 9. Once we need a bigger number than 9, we exchange/borrow for the next larger manipulative piece, since we want to use the least amount of pieces as possible. A useful way to illustrate this is by using manipulatives. Below is a video that demonstrates using manipulatives for addition in base ten.
Here is another example: 19
+ 7
____
This means that we have one long and nine units, and we are adding seven units. First we add the units together. 9+7 = 16. Since 16 is larger than 9, we have to exchange some of the units for a long. A long uses 10 units, so we take away ten units out of the 16 we have. This leaves us with 6 units (on the ones side), and another long to put on the tens side. Now we have:
1
19
+ 7
___
6
Next, we have to add the longs together. There are only 2, so no exchanges need to be made. This means we have 2 longs and 6 units which means we have 26.
19
+ 7
____
26
Sunday, October 3, 2010
Ancient Numerical Systems
Last week in class, we started learning about Numeration Systems. I found it interesting to learn about the Mayan, Egyptian, Babylonian, and Roman numerical systems and their similarities and differences, along with how different they are from the number system we use today (the Indo -Arabic system). It was also interesting to see how the systems developed and progressed into the system we use today.
Below in some information about some of the earlier Numeration Systems.
Egyptian System
*The egyptian system was created in 3400 B.C.
*This system used hierogyphics as symbols for each number.
*This system has no place value (contains no zero).
Below is a visual representation of the Eygyptian System.
*base 10
http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Egyptian_numerals.html
* The "Roman Numeration System" is actually the Roman Numerals that are still occasionally used today- Think of book indexes, movie and television show copywrite numbers, chapter numbers in books, outlines, etc.
* There are many rules involved in using the Roman System.
-> powers of 10 (I, X, or C) can be subtracted from another number (symbol), and only a single letter can be subtracted from a single numeral.
->Don't subtract a letter from another letter that is more than ten times greater.
*base 10
* 1=I 6=VI 40=XL
2=II 7 = VII 50=L
3=III 8=VIII 100= C
4 =IV 9=VIV 10,000= X (with a line over it)
5=V 10=X
Mayan System
*The Mayan system includes 3 symbols, including zero.
*The symbol for zero can vary- it can look like an eye or a conch shell.
*base 20 & 18x20
* http://www.basic-mathematics.com/mayan-numeration-system.html This is a helpful website that shows the mayan symbols and gives examples on how to use them.
Babylonian System
*includes 3 symbols: a "martini glass" and a "boomerang" shape. The third symbol is a combination of the other two.
* base 60
*does not include zero (no place holder)
*Although this doesn't seem specifically related to these systems, i found this to be a very useful hint, which I was reminded of in class last week: the word number refers to the value of the numeric symbol being used. (e.g. the number 7 is greater than the number 4). The word numeral refers to the symbol itself. (e.g. the numeral 4 is greater than the numeral 7). *
Below in some information about some of the earlier Numeration Systems.
Egyptian System
*The egyptian system was created in 3400 B.C.
*This system used hierogyphics as symbols for each number.
*This system has no place value (contains no zero).
Below is a visual representation of the Eygyptian System.
*base 10
http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Egyptian_numerals.html
Roman System
* The "Roman Numeration System" is actually the Roman Numerals that are still occasionally used today- Think of book indexes, movie and television show copywrite numbers, chapter numbers in books, outlines, etc.
* There are many rules involved in using the Roman System.
-> powers of 10 (I, X, or C) can be subtracted from another number (symbol), and only a single letter can be subtracted from a single numeral.
->Don't subtract a letter from another letter that is more than ten times greater.
*base 10
* 1=I 6=VI 40=XL
2=II 7 = VII 50=L
3=III 8=VIII 100= C
4 =IV 9=VIV 10,000= X (with a line over it)
5=V 10=X
Mayan System
*The Mayan system includes 3 symbols, including zero.
*The symbol for zero can vary- it can look like an eye or a conch shell.
*base 20 & 18x20
* http://www.basic-mathematics.com/mayan-numeration-system.html This is a helpful website that shows the mayan symbols and gives examples on how to use them.
Babylonian System
*includes 3 symbols: a "martini glass" and a "boomerang" shape. The third symbol is a combination of the other two.
* base 60
*does not include zero (no place holder)
*Although this doesn't seem specifically related to these systems, i found this to be a very useful hint, which I was reminded of in class last week: the word number refers to the value of the numeric symbol being used. (e.g. the number 7 is greater than the number 4). The word numeral refers to the symbol itself. (e.g. the numeral 4 is greater than the numeral 7). *
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