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