Greatest Common Factors and Least Common Multiples

I vaguely remember when I learned about factor trees and prime factorization for the first time, but they are definately important to finding the Greatest Common Factor (GCF) and the Least Common Multiple (LCM).  Though there is only one prime factorization of a number, there are 3 method for finding the prime factorization.
Remember a prime number has only 2 factors!
One method is increasing primes.  In this method you divide the number by increasing prime numbers.
Example:  84=2•2•3•7
               84÷2=42       42÷2=21      21÷3=7

There is the 2 factors method, where you keep writing numbers as a product of 2 factors.
Example:  84= 2•42
                       2•6•7
                       2•2•3•7
Then there is the factor tree, my favorite method.

Image from:  schoolworkout.co.uk

Above is an example of a factor tree for the number 72.  72 is broken down into 2 factors, 9 and 8, then 9 and 8 are broken down into 2 factors, and so on, until the numbers that are left are the prime numbers.  The numbers that are circled are the prime numbers. 

Once you understand prime factorization you can explore GCF and LCM.

GCF

GCF(a,b)= the largest number that is a factor of both a and b

Example:  GCF(7,14)=7

LCM

LCM(a,b)= the smallest number that is a multiple of both a and b

Example:  LCM(3,5)=15

Check out methods for finding the GCF and LCM at http://www.purplemath.com/modules/lcm_gcf.htm

Then you can test out your skills at http://www.math-play.com/Factors-and-Multiples-Jeopardy/Factors-and-Multiples-Jeopardy.html

The difference between multiples and factors

I was looking for a video on YouTube to make factors and multiples more fun, and this video made me giggle a little.  I like rhymes and songs that help me remember, even if they are a little juvenile.



          

Factors and Multiples

If a and b are whole numbers and a is not 0, then a is a factor of b, if and only if there is a whole number c such that a•c=b.  We can also say that a divides b or that b is a multiple of a.

Here is an example of  the 6 sentences that represent this:

• 2 is a factor of 20, since 2•10=20

• 10 is a factor of 20, since 10•2=20

• 2 divides 20, since 2•10=20

• 10 divides 20, since 10•2=20

• 20 is a multiple of 2, since 2•10=20

• 20 is a multiple of 10, since 10•2=20

A site that my help is http://www.kwiznet.com/p/takeQuiz.php?ChapterID=732

In class, we also covered divisibility tests.  I can't believe that I have never learned these tricks.

By 2:  a number is divisible by 2 if the ones digit is divisible by 2

By 5:  a number is divisible by 5 if the ones digit is a 5 or 0

By 3 or 9:  a number is divisible by 9 if the sum of its digits is divisible by 9, a number is divisible by 3 if the sum of its digits is divisible by 3

By 6:  a number is divisible by 6 if it is divisible by both 2 and 3

By 4:  a number is divisible by 4 if the last 2 digits are divisible by 4

By 8:  a number is divisible by 8 if the last 3 digits are divisible by 8   

I like the way that the divisibility tests are demonstrated on http://www.mathgoodies.com/lessons/vol3/divisibility.html

I also found the prime number test really helpful.  A number is a prime number if it has exactly 2 factors.  Once again, I can't believe that I have never learned this before.  The prime number test:

Suppose n is a whole number and k is the smallest whole number such that k•k>n (k²>n).  If there is no prime less than k that is a factor of n, then n is a prime number.

Example

51:  The largest prime you would need to check as a factor is the one that is ≤ √51≈7.1.  The largest prime ≤ 7.1 is 7.  So you would only need to check divisibility by 2,3,5,7.  It is divisible by 3, so 51 is not prime.

Until later. 

Candy and Math

Since I love chocolate, I loved the activity that we did in math class the other day.  I think that anytime a teacher brings multiple bags of candy to class, the students eyes light up, no matter what their ages are.  We used the candy to demonstrate one of the concepts of division of whole numbers.

The definition of division of whole numbers is:  For any whole numbers r and s, with s not equal to 0, the quotient of r divided by s, written r÷s, is the unique whole number k, if it exists, such that r=s×k.
                                          For example:  15÷3=5 because 3×5=15.
                                15 is the dividend, 3 is the divisor, and 5 is the quotient
Check out this link:  http://www.ltcconline.net/greenl/courses/187/a/WholeNumberDivision.htm 

We used candy bars to demonstrate the sharing/partitive concept of division.  Lets say that you have 20 candy bars and you want to share them equally with 4 of your friends.  How many does each friend get?  You may start by taking the 20 candy bars and dividing them into 4 equal groups.  Soon you will realize that each friend gets 5 candy bars.  So, 20÷4=5.

Then we examined the measurement concept of division.  In this concept you use repeated subtraction, each time "scooping" away an equal part.
Example:  15÷3=5  This is equal to 5 "scoops" of 3.

Personally, I like the sharing/partitive concept.  Which do you like better?
 

Multiplication Tables For Adults

In about the third grade, I learned my multiplication tables.  Just the other day, I learned them again, only this time the multiplication tables were a little different.  This time they were in base-five.  It was fun watching the other students in my math class taking their time to carefully calculate the multiplication table in base-five.  They looked like many of the students in my third grade math class, learning multiplication for the very first time.  One thing is for sure, the rules and properties of multiplication stay the same whether in base-five or base-ten.  The difference is definately in grouping the numbers.  As you might remember from one of the earlier posts, base-five is groups of five, while base-ten is groups of ten.

An easy way to look at multiplication of whole numbers is by remembering them as repeated addition.  An example of repeated addition is 4x5, that is 5+5+5+5.  So, 4x5=5+5+5+5=20.  Shown in the picture below from http://elko.k12.nv.us/webapps/vmd/mathdictionary/htmldict/english/vmd/full/m/multiplication.htm.

  In class we also learned about multiplication models.  The three models that we went over are the rectangular array, tree diagram, and partial products.  I really like the tree diagram, even though it may be hard to work with when you are using large numbers.  It really is pretty fun.  Here's a website to see how tree diagrams work:  http://www.regentsprep.org/Regents/math/ALGEBRA/APR4/PracTre.htm

I also want to include some information about the rectangular array and partial products.  The picture below is from another blog that I found interesting:  understandingmultiplication.blogspot.com/2007/03/third.html.

A video that may be helpful to understand partial products is:


Until next time...

Math and Shopping

Sometimes, I can't believe just how much math that I use on a regular basis.  I use it at work, at school, and most importantly to me, at the store.  Since I'm a mom, I work, and I go to school, I'm constantly trying to stay on budget with my money.  A lot of times I use math without even thinking about it, as I imagine that many of us do.  When I'm in the grocery store, I use rounding, and compatible numbers.  I tend to round up to the nearest dollar, and then add the compatible numbers.  To find more on compatible numbers, go to:  http://www.northstarmath.com/sitemap/CompatibleNumber.html .

I suppose, that if I really thought about it, I use a lot of mathematical properties while shopping as well.  For example, when you go to the grocery store, and you are adding the milk, cheese, and bread together, you can do it in any sort of order and the price of all three stays the same, right? Milk+ cheese+ bread= bread+ cheese+ milk  Even if you add the milk and the cheese and and get a certain price, only to later remember the bread and then add it in, it is the same as if you added the cheese and the bread, and then remembered the milk and quickly added it into the total. (Milk+ cheese)+ bread= (cheese+ bread)+ milk  The first example is the Commutative Property for Addition, and the second example is the Associative Property and the Commutative Property together.

 Image:  rjdposters.com

The Cross Between Math and Ancient History

Recently, I found myself thinking about one of my math lessons in Anthropology class.  We had been talking about how experts in many areas of science and math come together to help anthropologists put pieces of the ancient puzzle together.

The numeration systems that we had gone over in math class came to my mind.  It's interesting to think about the symbols that we use for numbers.  They vary from the Egyptians, to Babylonians, to the Romans, to our own set of numerals, the Hindu-Arabic numeration system.

In class, we did an activity to find out what symbol represented what number in various ancient numeration systems.  This was helpful to understand that there are different ways to group objects, or numbers, other than in groups of ten.  Certain cultures group numbers by different bases.  What I mean by base, is how many numbers or objects that they place in a group.  Our numeration system is a base-ten system.  That means that we group our numbers in groups of ten.  So, the symbol that we use to represent the number eleven is 11, one group of ten, plus one more unit.  To understand base-ten, you can use base-five pieces.  This is grouping of five, instead of grouping of ten.  An example of this would be:  the symbol that represents the number six is 6 in base-ten.  In base-five the symbol is 11;  that is one group of five and one unit.  Learning this helps to understand carrying numbers in addition by making exchanges.  Some examples of base-five and base-ten can be found at this website http://www.basic-mathematics.com/base-five.html