Prime Factorization

In class today we worked on prime factorization.  Prime factorization is a factor string that contains only prime numbers.  See below:

If we start with 100, we can decompose 100 into 4 x 25 (decomposing is breaking a number into a known factor pair).  If the numbers are composite (meaning more than 2 factors), we can decompose those numbers further.  The prime factorization of 100 is 2 x 2 x 5 x 5 as you can see above.

There are two methods we can use to find the prime factorization of a number.  Method 1 is dividing.

As you can see, we divide using only prime numbers.  We start with 360 and divide by 2.  We end up with 180.  180 is divisible by 3, which leaves use 60.  60 divided by 3 gives us 20, then 20 divided 2 equals 10.  10 divided by 2 gives us 5, our last prime number.  We want to put our factor string in order from least to greatest so 2 x 2 x 2 x 3 x 3 x 5 is the prime factorization of 360.

The second method is the factor tree method.  You begin by writing 360, then using "branches" you decompose into two known factors.

We decomposed 360 into 36 x 10.  Obvious right?  Since 36 and 10 are composite, we can"branch" off of both of those.  36 can decompose to 6 x 6 and 10 can decompose to 5 x 2.  5 and 2 are prime numbers and therefore can no longer decompose.  6 is composite and decomposes to 3 x 2.  Prior to rearranging our factor string, we end up with 3 x 2 x 3 x 2 x 5 x 2.  We rearrange it to be in order from least to greatest leaving us with 2 x 2 x 2 x 3 x 3 x 5.

A step further would be to use exponents, which we did not cover in class yet.  We can rewrite our factor string by counting the number each time occurs and using an exponent to represent each number.  Since there are 3 2's and 2 3's we can rewrite the string to look like this.  (^ is a symbol used to denote exponents)

2^3 x 3^2 x 5

Prime Time Quiz B Study Guide

Determining Equal Sharing Using Factors

Our problem yesterday was to determine the maximum number of snack packs that could be made with 2 or 3 different items.  First we used 24 apples and 36 bags of trail mix.  We have to find out the common factors first.  We did this by making a Venn diagram (we have used these before).

As you can see, 1, 2, 3, 4, 6, and 12 are the common factors of 24 and 36.  So:

Packs                       Apples                              Bags of Trail Mix

1                                        24                                             36

2                                        12                                             18

3                                         8                                              12

4                                         6                                               9

6                                         4                                               6

12                                       2                                               3

We use fact families to determine this, but we usually do this in our heads by now.  It would look like this:

1x24=24                      1x36=36

2x12=24                      2x18=36

3x8 = 24                      3x12=36

4x6 = 24                      4x9  =36

6x4 = 24                      6x6  =36

12x2=24                     12x3=36

The ones in bold are the ones that are common factors and the number of packs we can have so we use the other number of the factor pair to determine what amount of apples and trail mix can go in each pack.  

Least Common Multiple

We continued our work with least common multiples (LCM).  The problem today included determining when 13-year and 17-year cicadas would both emerge from the ground.  The way you would find out is much like what we did earlier.

13x1=13        17x1=17
13x2=26        17x2=34
13x3=39        17x3=51

We would do this until we found multiples that match.  In this case:

13x17=221    17x13=221

When the two numbers or factors of the two numbers are relatively prime, we can see that the numbers multiplied together often represent the least common multiple.  This is true 95% of the time.  Here are two counterexamples:

2x1=2            4x1=2
2x2=4            4x2=8
2x3=6            4x3=12
2x4=8            4x4=16

Here we see that 2x4 is 8, but the cycle would happen sooner, at an interval of 4.  Let's look at 4 and 6 now.

4x1=4            6x1=6
4x2=8            6x2=12
4x3=12          6x3=18
4x4=16          6x4=24
4x5=20          6x5=30
4x6=24          6x6=36

As you can see, 24 is a multiple of 6 and 4 (4x6) but the interval that would happen sooner would be 12 which is the LCM of 4 and 6.

When finding LCM, it is helpful to list in order each number times 1-? to determine the LCM.  Yes it is work but barring any mathematical errors, this method will always work.

2-5-3-7 L Method for GCF and LCM

This is a new method in finding GCF and LCM.  You begin first by listing your two numbers side by side.  We chose 24 and 36.  Then, draw an upside down division house.

Start with 2.  If 2 is a factor of both, write it outside the house, then divide.  Put the quotient underneath your number.  For our example, 2 is a factor of both 24 and 36, so we put 12 and 18 under them, respectively.  Continue this process until you have only prime numbers OR 2-5-3-7 are no longer factors of what you have left.

The numbers on the left side represent the prime factor string GCF.  Multiply them together to find the GCF.

The "L" (left side and bottom) represent a factor string for LCM.  Multiply all of them together.

As you can see, 12 is the GCF for 24 and 36. 72 is the LCM for 24 and 36.

Here is another example with 30 and 50.  This time we have a 5.

This method can be done with 3 numbers as well.  Just place the 3rd number to the right.  However, you need to continue until at least two numbers are prime, even if you cannot decompose one of the 3 numbers left.  Just drop it down as is and continue.

LCM and GCF-Listing and Venn Diagram

Today is class we took a quiz.  After I graded the quiz and went over the quiz with them, we began work on least common multiple.  To find least common multiples, we take each number and multiply it by 1, 2, 3, etc., until we find a multiple that they have in common.  For example, 20 and 30 have common multiples of 60, 120, and 180, with 60 being the least (or lowest).  Multiples are also known as products, and must always be equal to or larger than the number itself.  See the example below.

We have been finding factors, but we haven't compared the factors of two numbers.  Below we look at 20 and 30 again.  I used a Venn Diagram to organize my work.  The factors of 20 go into the circle under 20 and they are 1, 2, 4, 5, 10, and 20.  The factors for 30 go into the circle under 30 and they are 1, 2, 3, 5, 6, 10, 15, and 30.  In the area where the circles overlap we put the factors they have in common.  1, 2, 5, and 10 are all the common factors of 20 and 30.  When we find the greatest common factor, we use the largest one inside the overlap of the circles.  In this case, the greatest common factor of 20 and 30 is 10.  See the example below.

This is what is covered in the homework.  For 2, 4, 6, and 8, follow the first example (Least Common Multiple) for 16, 18, 20, and 22 follow the second example (Greatest Common Factor)

Below is a picture of the listing method for GCF.  

Rectangles and Factors

In class today we looked at the relationship between rectangles and factors.  If we look at the number 12 in terms of area of a rectangle, we know that area is equal to length times width, or a = l x w.  So for an area of 12 we can see that

 12x1  1x12  2x6  6x2  3x4  4x3

are possibilities that would give us an area of 12.  Two numbers multiplied together are called a factor pair.  This means each individual number in a factor pair is a factor.  In this case, the factors of 12 are:

1, 2, 3, 4, 6, and 12.

The dimensions of an rectangle are also the factors of the area of that rectangle.  This is important in helping us find both factors and area of numbers.

In describing factors, we use a few terms to determine what type of number we are dealing with which also allows use to make generalizations and "rules" so to speak.

Even-numbers that have 2 as a factor.
Prime-numbers that have only 2 factors, 1 and itself
Square-also called perfect squares because the have a factor pair that makes a square in terms of area; numbers who have an odd number of factors (ex: 4, 9, 16, 25.  factors of 25 are 1, 5, and 25)
Composite-numbers that have more than 2 factors (3 or more factors make a number composite.
Abundant-"more than enough"; numbers whose proper factors when added together are greater than the number itself (ex: 30, with the proper factors being 1, 2, 3, 5, 6, 10, and 15 adding up to 42)
Deficient-:"not enough"; numbers whose proper factors when added together are less than the number itself (ex: 15, with proper factors of 1, 3, and 5 adding up to 8)
Perfect-"exactly right"; numbers who proper factors when added together equal the number itself (ex: 28, with proper factors of 1, 2, 4, 7, and 14 adding up to 28)





Tomorrow in class we will have a quiz.  I went over what to expect on the quiz.  Each student should have written down a study guide like information sheet.  This picture goes with it.

Finding Factors

One helpful hint in finding factors is to think about the number your pick, and start going through each whole between

1 and N/2.  Here is an example using the number 40.


40

1 x 40              40/2=20 therefore....
2 x 20              3 is not a factor of 40, but 4 is.  40/4 = 10
4 x 10              5 is a factor of 40 because 40/8 = 5
5 x 8              

6 and 7 are not factors of 40, so when factors repeat, we have found all the factors of a number.

The factors of 40 are 1, 2, 4, 5, 8, 10, 20, and 40.

Finding Factors

Finding factors is very important in math.  It will develop number sense, help with multiplication and division, and be beneficial when finding things like least common multiple and greatest common factor.  When we get into fractions, this will be invaluable in finding common denominators.

There are many methods in finding factors and factor pairs.  I have included the standard algorithm below.  In order to determine if you have found all the factor pairs, you have to try every number between 1 and N ÷ 2, where N represents your original number.  See the example below.

As you can see, we chose 18.  So 18 ÷ 2 equals 9, so we will try all whole numbers between 1 and 9.  1, 2, 3, 6 and 9 are all called proper factors.  We count 18 as a factor because of the factor pair 1 X 18.  Proper factors include all whole numbers except for the number itself.

Below is an example for 36.

36 ÷ 2 = 18, so we have to try all whole number between 1 and 18.  36 is unique as it is a perfect square.  As you can see 6 X 6 is a factor pair.  All perfect squares will have an odd number of factors for this reason.

Factors and Divisors

Today in class we discussed factors.  We came up with definitions and labeled the parts of an equation.

Here is a bulleted list of the all the terms you will need to know, including the ones in the picture.

• Factor-numbers multiplied together to get a product
• Product-answer to a multiplication problem
• Dividend-the number being divided (first/top number in a division problem)
• Divisor-the number doing the dividing (second/bottom number in division problem)
• Quotient-answer to a division problem
• Prime number-number that has only 2 factors, 1 and itself (ex: 13 has two factors, 13 and 1)
• Composite number-number that has more than 2 factors (ex: 12 has six factors, 1, 2,, 3, 4, 6, and 12)

After we defined factor, product, divisor, dividend, and quotient, we played a factor game.  A copy of this along with the rules can be found in your child's binder.

Here is a chart related factors, divisors, products and quotients.

A question we had in class was are factors of a number also a divisor?  The answer is YES!

Welcome!

Welcome to Mr. German's Math Class.  I will post pictures, videos, worksheets and homework help here!  Please feel free to access this site frequently as I will try to post several times a week.

Here is a little bit about me.  I graduated from Arkansas Tech University in 2007 and completed my master's degree from Arkansas State University in 2013.  This is my 8th year to teach and I have taught a variety of subjects including driver's ed, ESL, health, physical education, 7th and 8th grade science, physical science, and physics.  I have 6 years of coaching experience; 3 as a head coach (2 in baseball, 1 in softball) and 3 years as an assistant (2 years softball, 1 year basketball).  I am married and I have 1 dog named Chino.

You can contact me at the school or via my email.  This info can be found at cs.conwayschools.org.

Hope to have a great school year!