Saturday, November 28, 2015

Whole Number: Subtraction

I. Intro
The operation of subtraction is the process of finding the difference between two numbers.

The symbol for subtraction is -.

The number to be subtracted from is called the minuend, the number to be subtracted is the subtrahend. The answer is called the difference.



With whole number subtraction the subtrahend must be less than or equal to the minuend since whole numbers don't include negative numbers. 

Horizontal vs Vertical Form
When subtracting with whole numbers we can write the problem horizontally or vertically.

Horizontal Form
4 - 3  = 1
minuend - subtrahend = difference

Vertical Form
  4
- 3
   1

   minuend
-  subtrahend
= difference

Verbally Expressing Subtraction Problems
There are many ways of saying subtraction problems. Here are a few common ways of saying 4 - 3:
  • Four minus three
  • The difference between four and three
  • Four takeaway three
  • Three less than four
  • Subtract three from four
II. Standard Subtraction Algorithm
The standard subtraction algorithm has three simple steps. 
1) line up the numbers being subtracted by their place value. As the commutative property does not apply to subtraction, it is important to write them in the correct order with the minuend above the subtrahend.
2) From right to left, subtract the digits in each corresponding column. 
3) Borrow (regroup) when subtracting a larger number from a smaller number within a place value column.

Borrowing (Regrouping)
Regrouping can be thought of as the rearranging of a number into different groups to make it easier to work with. Regrouping when doing subtraction used to be referred to as borrowing.

For instance, subtract 26 from 73.

 73
-26

Since the 6 in the ones column is bigger than the 3, we must regroup by borrowing 1 ten from the 7 in the tens column. Then subtract 6 from 13 in the new ones column.

 6 13
 73
-26
  7

Complete by subtracting 2 from 6 in the tens column.

 6 13
 73
-26
 47

III. Conceptual Understanding
Subtraction can be illustrated with the use of a number line. For example, to solve 11 - 4 we start on the number 11 on the number line and move 4 units to the left to finish on the number 7.




IV. Inverse Operation
Inverse operations are opposite operations that undo each other. Subtraction and addition are inverse operations. For example:

10 - 3 = 7 then add to get back to where we started 7 + 3 = 10

Note that the commutative and associative properties do not apply to subtraction.

V. Miscellaneous


Reference
Prealgebra: Alan Tussy, Diane Koenig
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Saturday, November 21, 2015

Whole Numbers: Addition

I. Intro
One of the four elementary mathematical operations, addition is the process of combining two or more numbers to find their total.

The symbol for addition is +.

The numbers being added are called addends and the answer is called the sum.

Horizontal vs Vertical Form
When adding with whole numbers we can write the problem horizontally or vertically.

Horizontal Form
4 + 3  = 7

Vertical Form
  4
+3
  7

Verbally Expressing Addition Problems
There are many ways to say addition problems. Here are a few common ways of saying 4 + 3:
  • Four plus three
  • Four added to three
  • Add three to four
  • Four and three
  • The sum of four and three
II. Standard Addition Algorithm
The standard algorithm for addition has three simple rules. 
1) line up the numbers being added vertically by their place value. 
2) From right to left, add the digits in each corresponding column. 
3) If the total of the digits in any place value column produces a sum greater than 9, carry to the next place value (now known as regrouping).

Carrying (Regrouping)
Regrouping is essentially the rearranging of a number into different groups to make them easier to work with. Regrouping when doing addition used to be referred to as carrying.

For instance, add together the numbers 45 and 17.

  45
+17

First we add the numbers in the ones place: 5+7=12. As only one number can go into each place column, we regroup the 12 into one 10 and two 1's. We then put the two 1's (the number 2) in the ones place and carry the one 10 to the tens place. This is represented by writing a small 1 over the tens place column.

  1
  45
+17
    2

Finally, we add the numbers in the tens place: 1+4+1=6

  1
  45
+17
  62

III. Conceptual Understanding
Addition can be illustrated with use of a number line. To calculate 4 + 3, we begin at 0 then move 4 units to the right. This represents 4. To add in the number 3, from where we left off, we move 3 more units to the right. This leaves us at seven on the number line, the total of 4 + 3.

Khan: Adding using ten frames



IV. Properties of Addition
The operation of addition has several important properties.

Commutative Property
Addition (as well as multiplication) is commutative, meaning the order of the numbers being added doesn't affect the sum. For instance,

3 + 6 = 9 is the same as 6 + 3 = 9.

Associative Property
Addition (and multiplication) is associative, meaning that when adding more than two numbers, the order in which addition is performed does not matter. For example:

(3 + 6) + 2 = 9 + 2 = 11 is the same as 3 + (6 + 2) = 3 + 8 = 11

Additive Identity Property
Additive identity property states that adding 0 to any number results in the number itself.

5 + 0 = 5  

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

7 + 3 = 10 then subtract to get back to where we started 10 - 3 = 7

VI. Miscellaneous
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Thursday, November 12, 2015

Whole Numbers: Base Ten Positional Number System & Place Value

I. Whole Number
Whole numbers are the set of natural/counting numbers but with the inclusion of zero.
{0, 1, 2, 3, 4...}


II. Understanding Base Systems
A long time ago, before base systems, there was no convenient way of writing big numbers. So to count one, you'd (using the established symbol of the time) write:

1

To write five, you'd write

11111

And if you wanted eight, would have to write eight notches

11111111

and so on.

To deal with this problem, humans invented number systems. The number system we use is referred to as the Hindu-Arabic numeral system. This system uses as it's base the numerals 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Since the number of unique digits in this system is ten, it is referred to as a base 10 system.

As a side note, other societies in history have used other base systems. For instance, the Mayans used a base 20 system and the Babylonians used a base 60 system. 

III. Place Value
All numbers in our number system are written using the digits 0 through 9 with the position of the digit in the number determining the value of the digit. Each place has a value 10 times the place to its right. More technically, the value of a digit in a number is a function of its position or place in the numeral. For example, when we write the number 2,364 we mean the sum of

2 Thousands + 3 Hundreds + 6 Tens + 4 Ones

With this number, we find that there are 2 units of a thousand plus 3 units of a hundred plus 6 units of ten plus 4 units of one. Moving from right to left, each unit increases by a power of ten (ten times the previous unit).

Another way of expressing it would be:

2 x 1000 + 3 x 100 + 6 x 10 + 4 x 1

This presentation of the number expresses the important idea that the quantity represented by a digit is the product of its face value and its place value.

Khan: Abacus
Video

IV. Periods
To make large whole numbers easier to read, commas are used to separate the digits into groups of three referred to as periods. Each period has a name such as ones, thousands, millions, billions, and so on.

Place Value & Periods Chart

IV. Misc.
Expanded Form

Rounding Numbers
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Thursday, November 5, 2015

Operations and Equal Sign

I. Operations
In mathematics, an operation is a calculation (process) from input values (called "operands") to an output value.

Types of operation
There are two common types of operations: unary and binary. Unary operations involve only one input value, such as negation and absolute value. Binary operations, on the other hand, involve two input values and include addition, subtraction, multiplication, division, and exponentiation.

Operations can involve mathematical objects other than numbers. For instance logical values true and false can be combined using logic operations, such as "and", "or", and "not".

An operation may or may not have certain properties, for example it may be associative, commutative, anticommutative, idempotent, and so on.

The values combined are called operands, arguments, or inputs, and the value produced is called the value, result, or output. Operations can have fewer or more than two inputs.


Basic Arithmetic Operations
The four basic arithmetic operations are addition, subtraction, multiplication and division.


II. Equal Sign
The equal sign is represented as = . We use this symbol to show when two or more quantities are exactly the same.





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