Friday, December 18, 2015

Whole Number: Division

I. Intro
Division is the operation of separating objects into equal sized groups (by the divisor). Another simple definition is to say that division is the process of calculating how many times one quantity (the divisor) is contained in another (the dividend).
                                                       
The symbols for division are ÷, /, ⟌ , ⎯

The number that gets divided is called the dividend. The number that the dividend is divided by is called the divisor. The answer is called the quotient. The following are various ways of presenting the problem 32 divided by 4 equals 8.

Verbally Expressing Division Problems
There are many ways of saying division problems. Here are some ways of saying 32÷4:
  • Thirty two divided by four
  • How many times does four go into thirty two?
  • Share thirty two among four groups
  • Split 32 into four groups

II. Algorithms
1. Partial Quotients Division Algorithm
Generally the first method of division currently taught to children in the 4th grade. 

2. Standard Division Algorithm (Long Division)
The following is the basic long division algorithm (procedure for solving a problem) taught in the United States. The process involves the following basic steps: 1) Divide, 2) Multiply, 3) Subtract, 4) Drop down the next digit.

Two digit divisor example: Divide 9303 by 25
               
 25) 9303

Divide the divisor, 25, into the left digits of the dividend. Do this by choosing the smallest part of the dividend (moving from left to right) the divisor goes into. Through trial and error we determine that 25 goes into 93 three times. Place the 3 above the 3 of the hundreds column of the dividend. Multiply 3 x 25 = 75 and place the number under 93.

         3    
 25) 9303
       75

Subtract 93 - 75

         3    
 25) 9303
        75
        18

Bring down the next digit which in this case is the 0.

         3    
 25) 9303
        75
        180

Determine the most number of times 25 goes into 180 which is 7. Write the 7 above the 0 in the tens place of the dividend. Multiply 7 x 25 = 175 and write the answer below the 180.

         37   
 25) 9303
        75
        180
        175

Subtract 180 - 175

         37   
 25) 9303
        75
        180
        175
            5

Bring down the next digit which in this case is the 3.

         37   
 25) 9303
        75
        180
        175
            53

Determine the most number of times 25 goes into 53 which is twice. Write the 2 above the 3 of the ones place of the dividend. Multiply 2 x 25 = 50 and write the answer below the 53.

         372 
 25) 9303
        75
        180
        175
            53
            50
              
Subtract 53 - 50

         372 
 25) 9303
        75
        180
        175
            53
            50
              3

There are no more numbers to bring down (unless we get into decimals which will be looked at in a later post) so the 3 is the remainder. Therefore the answer is 372 r 3.


III. Conceptual Understanding
1. Two interpretations of division are the partitive approach and the quotative approach.

Partitive Approach

With the partitive approach (aka sharing model), the divisor represents the number of parts (or groups) the objects will be distributed among. The number of groups is known (the divisor) and the quantity of objects in each group is unknown. So the question being asked is "how many items will there be in each group?" 

For example:

If I have 20 cookies and want to sort them into 5 bags, how many cookies go in each bag? or "How many will there be in each group?"


Here we know the number of groups (5 bags) and need to figure out how many of the 20 cookies will go into each bag. Dealing the cookies out one at a time into each bag results in 4 cookies in each bag. 20 / 5 = 4.

Here's another simple example

8 ÷ 2

Using the partitive approach we are dividing 8 into 2 groups.

**Partitive - Divisor is the number of groups (parts). Quantity in each group is unknown.**


Quotative Approach

With the quotative approach (aka measurement/grouping model), the divisor represents the number of objects in each group. The quantity in each group is known but the number of groups is unknown. So the question being asked is "how many groups will there be?"

For example:

If I have 20 cookies and want to put 5 cookies in each bag, how many bags will I get? or "How many groups of 5 with there be?" or "How many 5's are in 20?" We are asking how many times will he divisor go into the dividend.


Here we know the number of cookies to go into each bag (5 per bag) and need to figure out how many bags (groups) there will be. Putting 5 cookies at a time into a bag results in a total of 4 bags.
20 / 5 = 4

Another example

8 ÷ 2

Using the quotative approach we are dividing 8 into groups of 2. How many groups of 2 are there?


**Quotative - Divisor is the quantity in each group. Number of groups (parts) is unknown**


2. Using a number line and base ten blocks to conceptually better understand division.
Division on a Number Line
See:
Division on a Number Line: Cuemath

Division using base ten blocks
Long Division Using Base Ten Blocks: Lafountaine of Knowledge

3. Division Based on Divisor-Dividend Size

1) Divisor smaller than dividend (e.g. 12÷3)
  • The quotient is greater than 1, because you are asking “how many 3s fit into 12?”, and more than one copy fits.​
2) Divisor equal to dividend (e.g. 12÷12)
  • The quotient is exactly 1: one copy of 12 fits into 12.
  • Conceptually, you are putting the whole into exactly one group, so nothing really changes.
3) Divisor larger than dividend (e.g. 3÷53÷5 or 1÷41÷4)
  • The quotient is between 0 and 1, because less than one copy of the divisor fits into the dividend.​

IV. Properties of Division

Identity Property
Any non-zero number divided by 1 equals itself (e.g., 5 ÷ 1 = 5).

Zero Property 
Zero divided by any non-zero number equals zero (e.g., 0 ÷ 6 = 0).

Division by Zero
Dividing any number by zero is undefined or meaningless in mathematics.

V. Inverse Operation
Inverse operations are opposite operations that undo each other. Division and multiplication are inverse operations. For example:

6 ÷ 2 = 3 then multiply 3 x 2 = 6

            
VI. Miscellaneous
Multiplication & Division Problem Structures (Word Problems)

 
VII. Practice:
Multi Digit Multiplication and Division: Khan Academy

https://www.mathisfunforum.com/viewtopic.php?id=19826
https://www.themathpage.com/Arith/division.htm
https://www.llcc.edu/learning-center/helpful-resources/math-signal-words
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