Algorithm's For Multiplication and Division Of Whole Numbers

Multiplication of Algorithms

Multiplication of Algorithm is a method to multiply two numbers.

Partial-Product Algorithm: is where each factor is thought of as a sum of ones, tens, hundreds and so on.  Then each part of one factor is multiplied by each part of the other factor, and all of the resulting parital products are added together.   An example of this is 

Modified Standard Algorithm: is where you multiply ones and tens separately and add everything.

An example of this is 

Division of Algorithms

If a and b are whole numbers with b not being zero, then there are unique integers q and r such that a=bq+r. q is the quotient and r is the remainder of a divided by b.

Long-Division Algorithm: is where you divide the dividend by the divisor.  Dividend 425 divided by divisor 25.  An example of this is 

Short-Division Algorithm: is like long-division but shorter and the divisor is less than 10.  Dividend 500 divided by divisor 5.   An example of this is 

Greatest Common Divisors and Least Common Multiples

Greatest Common Divisors

Let m and n be natural numbers.  The greatest natural number d that divides both m and n is called their greatest common divisor and we write d=GCD (m,n),

The Greatest Common Divisor by Intersection of Sets: in this method you list all of the factors of each number, then list the common factors and choose the largest one.  An example of this is, find GCF(36,54).

Solution.

Let F36 denote the set of factors of 36. Then F36 = {1,2,3,4,6,9,12,18,36}

Similarly, Thus,

F54 = {1,2,3,6,9,18,27,54}

F36 ∩F54 ={1,2,3,6,9,18}.   So, GCF(36,54) = 18

The Greatest Common Divisor from Prime Factorization: This method you list prime factors, then multiply the common prime factors (including zero).  An example of this is, find GCF(36,54) using prime factorization.

Solution.

Writing the prime factorization of both 36 and 54 we find 36 = 2×2×3×3

54 = 2×3×3×3

Notice that the prime factorizations of 36 and 54 both have one 2 and two 3s in common. So, we simply multiply these common prime factors to find thegreatestcommonfactor. That is,GCF(36,54)=2×3×3=18

Greatest Common Divisor from the Euclidean Algorithm: This method a and b is any two natural numbers and a is greater than or equal to b.  You divide a by b, we obtain a quotient q and a remainder r. a=bq+r, 0 is less than or equal too r, r is less than b.  Then, GCD(a,b)=GCD(b,r).   An example of this is, find the greatest common factor of 36 and 54 using the division algorithm.

54 = 36×1+18 GCF(54,36)=GCF(36,18) 36 = 18×2+0 GCF(36,18)=GCF(18,0)

Thus, GCF(54,36) = GCF(36,18) = GCF(18,0)=18. 

Hence, to find the GCF of two numbers, apply the above theorem repeatedly until a remainder of zero is obtained. The final divisor that leads to the zero remainder is the GCF of the two numbers.

Least Common Multiples

The smallest (non-zero) number that is a multiple of two or more numbers.

Least Common Multiples by Intersection of Sets: This method you list the non-zero multiples of each number until a first common multiple appears.  This number is the LCM(a,b).  An example of this is, find LCM(12,8).

Solution.

Let M8 and M12 denote the set of nonzero multiples of 8 and 12 respectively. Then

and Thus, LCM(8,12) = 24.

M8 = {8,16,24,···} M12 = {12,24,···}

Least Common Multiples from Prime Factorization: This method to find the LCM you have to find the prime factorization.  You would take the primes that are factors of either of the given numbers.  The product of these primes is the LCM and each of these are raised to the greatest power.  An example of this is, find LCM(2520,10530).

Solution.

Writing the prime factorization of each number we find 2520 = 23·32·5·7

10530 = 2·34 ·5·13 So LC M (2520, 10530) = 23 · 34 · 5 · 7 · 13 = 294, 840.

Find LCM(731,952).

Least Common Multiples by Using the Euclidean Algorithm: This method the LCM is found by dividing the product a*b by the GCF.  An example of this is, find LCM(731,952).Solution.

Using the Euclidean algorithm one will find that GCF(731,952) = 17. Thus, by the above theorem

LCM(731,952)= 731×952 =40,936. 17

Algorithms for adding and subtracting whole numbers

Adding Algorithms

Algorithm of adding is a step by step solution to a problem by combining the two groups together, so you would combine the place values one at a time. 

Short Standard Algorithm: this is where you add from right to left, one place-value at a time, regrouping when necessary.  An example of this is 

additional examples if you go to this web address: http://www.mathsisfun.com/numbers/addition-column.html

Partial-Sums Algorithm: is where you work one place value at a time and adding all the sums to get the total.  You can also do this algorithm mentally.

An example of this is 

Subtracting Algorithms

Algorithm of subtracting is a step-by-step process in taking away a subset of objects from a given set.

Right to left(Standard) Algorithm: is where you subtract from right to left one place-value at a time.

An example is 

Left-to-right Subtraction Algorithm: is where you have a number life 356 and the sum is 300+50+6. Then you subtract the sum one at time, starting at the hundreds.

An example is 

Still Focusing On The Basics

Multiplication and Division Of Whole Numbers

 The basics of multiplication is repeated addition.  There are 6 conceptual models for multiplying two numbers.  The six conceptual models are: multiplication as repeated addition, the array model for multiplication, the rectangular area model for multiplication, the skip-count model for multiplication, the multiplication tree model, and the Cartesian product model.  I will be going over these models more in a bit.  The basics of division is splitting into equal parts or groups, also known as "fair sharing."  There are three conceptual models for division of whole numbers.  The three conceptual numbers of division are: the repeated-subtraction model, the partition model, and the missing-factor model.  These models will also be explained in more details.

The six models of multiplication: each model provides useful conceptual and visual representation of the multiplication operation.

The first model is multiplication as repeated addition, is when a and b are any two whole numbers and the product is found from them.  (a*b=b+b+...+b when a doesn't equal 0, when 0*b=0) An example of this is, if we are combining 7 groups with 4 objects in each group we can arrive at the same answer with addition.  4+4+4+4+4+4+4=28 is equivalent to 7*4=28. 

The second model is the array model for multiplication, is finding the product when items like numbers or objects are arranged in rows or columns. 

The third model is skip-count model for multiplication, is when you skip by the number b exactly a times.  An example is five times 3, start with 0 and skip to 3 as our first position, then skip 3 more to 6 our second position.  We then skip 6 to 9 third position, 9 to 12 fourth position, and 12 to 15 fifth and final position.  We have skipped 3 five times to get 15.

The fourth model is the rectangular area model of multiplication, is when you use the dimensions of the rectangle to correspond to the factors and the area of the rectangle corresponds to the value of the product. 

The fifth model is the multiplication tree model,  is a way to show solutions for counting problems. An example of this isif there are 4 flavors of ice cream—chocolate, vanilla, strawberry, and mint—and 2 types of cones, sugar and waffle, how many different choices of ice-cream cones are possible?

The sixth model is the Cartesian product model of multiplication, is an ordered pair (a,b) of objects, where the first component a of the ordered pair indicates the type of object and the second component b of the ordered pair indicates the other object.  An example is with two shirts and three pairs of pants, you could have 2 • 3, or 6, different shirt-pant combinations.

The three conceptual models for division: the division a divided by b of a whole number a by a nonzero whole number b.

The first model is the repeated subtraction model of division, is realized easily with physical objects and is also called division by grouping.  An example is 28/4=7 28-4=24-4=20-4=16-4=12-4=8-4=4-4=0

The second model is the Partition model of division, is also realized with physical objects and also called division by sharing.  An example is 

The third model is the missing-factor model of division, is based on inverse operations.

An example is 28/4=7 28=4*c or 28=c*4

Starting With The Basics

Addition and Subtraction of Whole Numbers

When you start out with the basics of math its easier to move forward and work towards harder problems.  Addition of whole numbers, is adding numbers together to get a final value.  Some terms to remember are addend which is adding two numbers together and sum which is the final answer to a problem.  Subtraction of whole numbers, is the inverse operation of adding whole numbers.  Instead of adding numbers to get a sum, you are removing one number from another to get the difference of them.  We will start with examples of adding whole numbers.

http://youtu.be/3bgZpTKx_nw

Properties of whole number addition: is any three whole numbers when you add two of them together the third becomes the sum of the two.

First property is closure property, is the sum of any two real numbers equals the sum of another real numbers. (a+b=c)
Second property is commutative property, is the changing the order of the three numbers which does not change the sum.    (a+b=b+a)
Third property is associative property, is when three or more numbers are added, the sum is the same, regardless of the order of addition. (a+b)+c=b(a+c)
The fourth property is additive identity property, is when you add zero to any number the number stays the same.  (a+0=a)

Four models of whole number subtraction: teaches kids to advance in math courses and think abstractly.

First model is take-away model, is the basic concept that teaches kids to take away some objects from a set resulting in fewer objects in the set.  An example, is suppose you have 9 barbies and give 6 away.  How many do you have left? 9-6=3
Second model is missing addend model,  this is when you have two numbers, you count up from the smallest number to the bigger number and get your final answer.  An example, is suppose you have 6 turtles and you need to have 9 turtles. How many more turtles are needed?  We need to add 3 turtles to the 6 turtles to get a total of 9 turtles.
Third model is comparison model, is where you compare one collection to another to determine the difference.  An example is suppose you have 9 frogs and someone else has 6 frogs.  How many more frogs do you have than the other person?  How to solve this is line up 9 frogs in one row and in another row line up 6 frogs.  By doing this it gives us the answer by showing us how many more frogs there are.
The fourth model is the number-line model, is when you mark the higher number on the number line and than mark the lower number on the number line.  By doing this you will find the answer by counting how many lines are between these numbers to get your answer.  An example is

Math Is Hard To Understand and Scary

Do you cringe when you have to do math work or help someone with math work?  Do you try to avoid math by giving any excuse you can so you don't have to even look at it?  Do you want to find a cave to hide in when someone talks to you about math or wants help?  You don't need to panic and hide anymore there is techniques to calm you down and relax when math work needs to be done.  

For those that hate math and try to avoid it its time to open your mind to new techniques that will benefit your style of learning and enjoy doing math.  I'm going to with some visual ways of learning math for those that are visual learners.

http://media.pearsoncmg.com/aw/aw_long_mathelem_6/video/classroom/LMT06_classroom.html.
Click on equations.

This video shows how the teacher explains the problem but also uses small blocks to help visually see the problem that needs to be worked out.  This helps when someone is struggling with certain mathematical problems to eventually help them understand these problems without the blocks.  These visual help aids help for the future because when you don't use these objects to help with math problems anymore you can visually see it in your mind.  I also loved her idea of using things that they are interested in and plugging them into a problem.  That really does keep a kids focus on the problem at hand.

Small groups are so helpful when doing math because two heads can always be better than one.  Everyone has different techniques and ways of learning math.  There are those that excel in math and help those that are having more of a tough time because their techniques may be beneficial for that person struggling.  The teacher may have a different technique to a problem so having others to talk to can be very helpful in learning math and learning new techniques you may like.  Always keep an open mind when learning math and be ready to try new techniques to help you understand math without needing to be afraid of it and avoid it.