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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Wednesday, December 9, 2015

Whole Number: Multiplication

I. Intro
The operation of multiplication is generally thought of as either repeated addition or scaling. Generally, multiplication is taught as repeated addition when learning about whole numbers and integers. This model starts to fall apart when multiplying fractions, which is when scaling is sometimes taught. As such, multiplication as repeated addition is shown here. Multiplication as scaling is introduced in the section multiplication of fractions.

The symbols for multiplication are ×, ∙, *, ()().

The numbers to be multiplied are called the factors (multiplicand and multiplier) and the answer the product.

The multiplicand is the number which is multiplied by the multiplier. Generally the multiplier is written first, then the multiplicand (Some people have learned the opposite). See here 

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

Horizontal Form
4 x 3 = 12

Vertical Form
   4
x 3
 12

Verbally Expressing Multiplication Problems
There are many ways to say multiplication problems. Here are some common ways of saying 4 x 3:
  • Four times three
  • Four multiplied by three
  • Four times as much as three
  • Four groups of three
  • The product of four and three
Multiplication Tables
The basic multiplication table for digits one through ten.



II. Algorithms
1. Area Model (Box) Method Multiplication
Often the first method of multiplication currently taught to children around 4th grade. It is believed that it better helps children conceptually understand multiplication.

1. Partial Products
Often the second method of multiplication currently taught to children around 4th grade.


2. Standard Multiplication Algorithm
The standard method of multiplication involves multiplying every digit in the first number by every digit in the second and adding the results. It has five steps: 
1) line up the numbers being multiplied by their place value putting the larger number on top. 
2) Multiply the ones digit of the bottom number by each digit of the top number, starting from the right. 
If a single-digit multiplication results in a two-digit number, write down the ones digit and carry over the tens digit to add to the next multiplication’s product in the next column. This carried value is added to the partial product of the next digit’s multiplication before writing it down.
3) Multiply the tens digit of the bottom number by each digit of the top number, starting from the right and add a zero placeholder since you are now multiplying by tens place 
4) Repeat similarly for hundreds, thousands, etc., adding more zeros accordingly.
5) Add all the products calculated from the steps above for the final answer.

Example: Multiply 28 x 7

  28
x  7

Multiply 8 x 7 = 56. Regroup the product placing the 6 ones below and the 5 tens over the 2.

  5
  28
x  7
    6

Multiply 2 x 7 = 14. Add the 5 tens resulting in 19 tens.

  5
  28
x  7
196

Example: Multiply 846 x 12

    846
x    12

Multiply the ones digits. 6 x 2 = 12. Regroup the product carrying the 1 ten.

     1
   846
x   12
       2

Multiply 4 x 2 = 8, then add the 1 ten resulting in 9 tens.

     1
   846
x   12
     92

Multiply 8 x 2 =16. Write below to complete the partial product aligned with the ones digit of the bottom number.
 
   846
x   12
 1692

Now we will multiply the tens digit of the bottom number with each digit of the top number. Multiply 6 x 1 = 6. Make sure the answer is aligned under the tens column below. You can first put a zero in the ones column to avoid mistakes.
 
   846
x   12
 1692
     60

Then multiply 4 x 1 = 4
 
   846
x   12
 1692
   460

Then multiply 8 x 1 = 8. Write below to complete the partial product aligned with the tens digit of the bottom number.
 
   846
x   12
 1692
 8460

Finally, add the two partial products for the total product.
 
   846
x   12
 1692
 8460
10152


IV. Conceptual Understanding
Repeated addition
When dealing with whole numbers, multiplication is often taught as repeated addition. Repeated addition is the process of  repeatedly adding the same number.  There are various ways this can be illustrated.

Set Model Approach

O O O      O O O      O O O
 O O          O O         O O

Above are 3 groups of 5 circles which can be written as 5 + 5 + 5. The number 5 is repeated 3 times which would be written 3 x 5. This is read as "three times five".

So 3 x 5 = 3 groups of 5 = 5 + 5 + 5 = 15.

Number Line Approach
Multiplication can also be illustrated with the use of a number line. On the above example 3 x 5 is shown as three movements of five units ending on the number fifteen.

Area Model Approach
Multiplication can also be thought of in terms of an area model. The above area model shows three rows of five units or 3 x 5. This also demonstrates the commutative property of addition in that you could alternatively interpret it as five columns of three units or 5 x 3. Both give the same answer of 15. 

Array Model Approach
Here's a nice video from Khan Academy Multiplication with arrays


V. Properties of Multiplication
Commutative Property
Multiplication (as well as addition) is commutative, meaning the order of the numbers being multiplied doesn't matter. For instance,

2 x 7 = 14 is the same as 7 x 2 = 14

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

(5 x 2) x 4 = 10 x 4 = 40  is the same as  5 x (2 x 4) = 5 x 8 = 40

Multiplicative Identity Property
The multiplicative identity property states that any number multiplied by 1 results in the number itself.

1 x 18 = 18

Distributive Property
The distributive property states that the sum of two numbers times a third number is equal to the sum of each addend times the third number. 

Stated with variables, the distributive property states that  x(y+z) = x(y) + x(z)

In arithmetic, this property can make it easier to solve a large multiplication problem by splitting it into two smaller ones and adding the products. For example:

17 x 101

Can be turned into 

17(100 + 1)

Then multiply (distribute) the number outside the parentheses by each number inside the parentheses, and then add the products.

1700 + 17 = 1717

So when coming across a problem in the form of 

a(b + c)

There are two ways you can go about it. You can use the normal order of operations and first add (b + c) and then multiply the sum by a. The other method is to use the distributive property and multiply (a x b) then (a x c) and then add the two products together. Why not just use the normal order of operations? Much of this is preparation for dealing with variables in Algebra. 


VI. Inverse Operations
Inverse operations are opposite operations that undo each other. Multiplication and division are inverse operations. For example:

4 x 5 = 20 then divide to get back to where we started 20 ÷ 5 = 4

VII. Miscellaneous

Do You Know the Five Meanings of Multiplication?
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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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Tuesday, October 27, 2015

Number, Set, & Number Sets

I. Set, Subset, Number, Numeral
Before defining what a number is we first have to look at what a set is.

A. Set
In mathematics, a set is a collection of distinct objects (generally referred to as elements) for which we can decide whether or not a given object belongs. A set can have any group of items, be it a collection of numbers, days of a week, types of vehicles, and so on. For example, the set of days of the week is a set that contains 7 objects: Mon., Tue., Wed., Thur., Fri., Sat., and Sun..

Sets are exhibited by providing a list of it's elements inside two curly brackets. Sets are named and represented using capital letters. For instance, the collection of even natural numbers less than 10 can be represented in the form of a set, A = {2, 4, 6, 8}, which is a finite set.. The order in which the objects of a set are written doesn't matter. Sets can be finite or infinite. 

B. Subset
One set is a subset of another set if every object in the first set is an object of the second set as well. For instance, the set of weekdays is a subset of the set of days of the week, since every weekday is a day of the week.

C. Number/Numeral
The mental concept of number is of a mathematical object used to count, measure and label. Therefore, when we speak of numbers, we are referring to a particular set of numbers (sometimes called number systems).

A numeral is a symbol that represents a number. In common usage, number and numeral are used interchangeably.



II. Number Sets
The following are the most common sets of numbers, each increasing in complexity.
  • Natural Numbers (counting numbers) - The set of numbers {1, 2, 3, 4, ...}
  • Whole Numbers - The set of numbers {0, 1, 2, 3, 4, ...}
  • Integers - The set of numbers {...,-3,-2,-1,0,1,2,3,…}
  • Rational Numbers - A number that can be written in fraction form such as a/b where a and b are integers, but b is not equal to 0. This includes integers, terminating decimals, and repeating decimals as well as fractions.
  • Irrational Numbers - A number whose decimal form is nonterminating and nonrepeating. Irrational numbers cannot be written in the form a/b, where a and b are integers (b cannot be zero). So all numbers that are not rational are irrational. π is a well know irrational number.
  • Real Numbers - All the rational and irrational numbers; that is, all of the numbers that can be expressed as decimals.
  • Complex Numbers - Numbers in the form of a+bi, where a is the real part of the number and bi is the imaginary part of the number.






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