Friday, January 29, 2016

Adding & Subtracting Fractions

I. Standard Algorithms 
A. Adding and Subtracting Like Fractions
Like fractions are fractions that have the same denominator. To add or subtract like fractions, simply add or subtract the numerators and place the answer over the common denominator.

Examples:

3/6 + 2/6 = 5/6

12/15 - 3/15 = 9/15


B. Adding and Subtracting Unlike Fractions
(Learn to multiply fractions first)
To add or subtract fractions with different denominators, they first must be changed so that they have the same denominators. To do this:

1. Find the Least Common Denominator (LCD): Determine the smallest common multiple of the denominators. (To find the Least Common Denominator, simply list the multiples of each denominator until you find the smallest number that appears in each list.)

2. Convert each fraction: Rewrite each fraction as an equivalent fraction with the LCD as its new denominator. To do this, multiply both the numerator and the denominator by the same number that makes the denominator equal to the LCD.

3. Add or subtract the new numerators: Once both fractions have the same denominator, add or subtract their numerators as you would with like fractions.

4. Simplify (if necessary): Reduce the resulting fraction to its simplest form

Example:

1/5 + 1/6

First, find the least common denominator.

Multiples of 5: 5, 10, 15, 20, 25, 30, 35, ...
Multiples of 6: 6, 12, 18, 24, 30, 36, ...

Now convert the original problem into fractions with the common denominator. To do this multiply each original fraction by a fraction that has the same numerator and denominator, which will result in a new fraction that has the common denominator we seek.

To convert 1/5 so that it has a denominator of 30, multiply the fraction by 6/6.

1/5 x 6/6 = 6/30

To convert 1/6 so that it has the denominator of 30, multiply the fraction by 5/5.

1/6 x 5/5 = 5/30

Now simply add the new like fractions.

6/30 + 5/30 = 11/30

II. Adding and Subtracting Mixed Numbers
There are two methods of adding and subtracting mixed numbers.

Adding/Subtracting the Whole Numbers and Fractions Separately
1. If dealing with unlike fractions, find the least common denominator, then increase the terms of each fraction so that the denominator of each equals the LCD (see adding and subtracting unlike fractions above).
2. Add or subtract the fractions and add or subtract the whole numbers.

*The biggest weakness with this method is that sometimes you have to borrow. In a problem like 5 1/4 - 2 3/4, you can't just subtract the fractions. You have to "borrow" from the 5, turning it into 4 5/4. This is where students get confused. 

Converting the Mixed Numbers to Improper Fractions and Adding/Subtracting
1. Convert the mixed numbers into improper fractions.
2. If they are unlike fractions, find the least common denominator, then increase the terms of each fraction so that the denominator of each equals the LCD (see adding and subtracting unlike fractions above).
3. Add or subtract the fractions.
4. Convert your answer back into a mixed number.

*This is the method I generally use as it avoids issues with borrowing. 

Adding Mixed Numbers with Regrouping - Khan Academy Video
Subtracting Mixed Numbers with Regrouping - Khan Academy Video


III. Conceptual Understanding
Area Model (Like Fractions)
Adding and subtracting like fractions can be illustrated using an area model.

Example: 2/6 + 3/6


Example: 3/4 - 2/4



Area Model (Unlike Fractions)
An area model can be a useful representation of the process used to add or subtract unlike fractions.

Example: 2/3 + 1/4


Visually you can see that an individual piece from the first square is larger than an individual piece from the second square, so you can't treat them as if they were the same and just add them together. 

We proceed by finding the least common denominator of the two fractions, which in this case is 12. To convert 2/3 to a fraction with 12 as the denominator, we multiply it by 4/4 (We can do this because of the identity property of multiplication which states that any number multiplied by 1 remains unchanged). To convert 1/4 to a fraction with a denominator of 12, it is multiplied by 3/3. You can see this conversion illustrated in the middle two squares. 

Now that the fractions have the same denominators, they can be easily added.

IV. Miscellaneous
A. Simplifying Before You Add
While it's important to simplify your final answer, it is also sometimes easier to solve a problem by simplifying the initial fraction first.

Example:
5/10 + 2/8

Instead of finding a common denominator for 10 and 8 (which is 40), you could reduce them to 1/2 + 1/4, making the common denominator smaller (4).




Practice:
IXL: Add and subtract fractions with like denominators
IXL: Add and subtract mixed numbers with like denominators
IXL: Add and subtract fractions with unlike denominators
IXL: Add mixed numbers with unlike denominators
IXL: Subtract mixed numbers with unlike denominators
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Friday, January 22, 2016

Fractions

I. Fundamentals
A fraction represents a part of a whole which is expressed in the form a/b.

The number on the bottom of the fraction is called the denominator. It represents how many equal parts the whole is divided into. The number on the top of the fraction is called the numerator. It represents how many of the parts we have.

For example, in the fraction 3/4, the numerator tells us the fraction represents 3 equal parts, and the denominator tells us that 4 parts make up a whole.

From this we can see that whenever the numerator and denominator are the same number, the fraction is equal to 1. For instance, 4/4 = 1.

Whole numbers can also be written in fraction form. To do so, place the the whole number over 1. For example, the whole number 5 written as a fraction is 5/1.

II. Conceptual Understanding
Number Line
We can use the number line to help conceptually make sense of fractions. Again using 3/4 as our example, the number line below divides the number one into 4 parts (as represented in the denominator). Starting at zero, we move 3 fractional units (as represented in the numerator) to the right ending at 3/4.


Area Model
Fractions can also be graphically represented through the use of area models. The area models below show a whole can be divided into various fractional parts.

III. Decompose Fractions


IV. Proper Fractions, Improper Fractions & Mixed Numbers
1. Definitions
A proper fraction is one where the numerator is smaller than the denominator. Examples would include 2/5, 3/4 and 1/6.

A improper fraction is one where the numerator is greater than or equal to the denominator. Examples include 5/2, 7/3 and 6/6.

A mixed number is one that has both a whole number and a fraction. Examples include 3 3/4, 15 2/5 and 17 1/4.

2. Converting Improper Fractions to Mixed Numbers
To convert an improper fraction into a mixed number, simply divide the numerator by the denominator. The answer becomes the whole number and the remainder becomes the numerator of the new fraction. The denominator of the new fraction is the same as the denominator of the old fraction.

Example: Convert 9/4 to mixed number

9/4 = 9 ÷ 4 = 2 R1 

Write the whole number part 2. Then take the remainder and write it over the original denominator

2 1/4

3. Converting a Mixed Number to an Improper Fraction
Method 1
To convert a mixed number into an improper fraction, multiply the whole number by the denominator and add it to the numerator. This becomes the numerator of the improper fraction; the denominator of the new fraction is the same as the original denominator.

Example: Convert 2 3/4 to an improper fraction

Multiply whole number by denominator 

2 x 4 = 8 

Then add the answer to the numerator  

8 + 3 = 11

Now write the sum on top of the original denominator

11/4 

Method 2
This method is really the same as the above method, just worked out. First, convert the whole number into a fraction. Then add the fractions. 

Example Convert 6 1/2 to an improper fraction.

Convert the whole number to a fraction

6 = 6/1

Then add the fractions

6/1 + 1/2 = 6/1 x 2/2 + 1/2 = 12/2 + 1/2 = 13/2


V. Equivalent Fractions
Equivalent fractions are fractions that have the same value. They are different ways of representing the same fractional part of a whole. Though they look different each fraction represents the same number. For instance, 1/2 = 2/4 = 3/6 = 4/8.


And here are some of the same equivalent fractions represented on number lines.



VI. Reducing Fractions to Lowest Terms (Simplifying Fractions)
Reducing fractions to their lowest terms involves finding the lowest equivalent fraction. This means that there is no number except 1 that can be divided evenly into both the numerator and the denominator.

To do this, divide both the numerator and denominator by a number that divides evenly into both (common factors). Repeat the process until the fraction is in it's lowest terms.

To do this in one step, divide both the numerator and denominator by the largest number that divides evenly into both (greatest common factor).

Example: Reduce 4/8

The number 2 divides evenly into both 4 and 8.

4/8 / 2/2 = 2/4 

The number 2 divides evenly into both 2 and 4.

2/4 / 2/2 = 1/2

Alternatively you could do this in one step by using the greatest common factor, which in this case is 4.

4/8 / 4/4 = 1/2

A more precise algorithm for simplifying fractions would be:

Step 1: Write the factors of numerator and denominator.
Step 2: Determine the highest common factor of numerator and denominator.
Step 3: Divide the numerator and denominator by their highest common factor (HCF). The fraction so obtained is in the simplest form.

VII. Common Fraction
A common fraction is a fraction where both the numerator and denominator are both integers (Technically where the denominator is a non-zero integer). For instance, 1/5 is a common fraction but 1.5/2 is not a common fraction since the numerator contains a decimal.

To write fractions that contain a decimal number in the numerator or denominator, we first have to make both numbers into integers.

For example with the fraction 1.5/2 we can see that moving the decimal in the numerator in 1.5 to the right would make it an integer. To do this we multiple both the numerator and denominator by 10.

1.5/2 x 10/10 = 15/20

We then simplify as normal

15/20 = 3/4


VIII. Converting Fractions to Decimals & Converting Decimals to Fractions
See Decimals

IX. Converting Fractions to Percents & Converting Percents to Fractions
See Percentages

X. Multiple Meanings of a Fraction
See Five Meanings of a fraction


Practice:
IXL: Convert between improper fractions and mixed numbers
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Monday, January 18, 2016

Real Numbers: Intro & Decimal Place Value

Introduction
Real numbers are the set of numbers that include both the rational and irrational numbers.

The rational numbers are numbers that can be written in the form of a fraction 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 are numbers 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 known irrational number.



Decimal Place Value 
The decimal place value chart shows the place value of decimal numbers. We know that place value is the numerical value represented by a digit in a number. Therefore, just like the whole number place value chart, decimal place value charts are used to identify the correct position of all the digits in a decimal number. This chart indicates the place value of the digits that are given before the decimal point and after the decimal point.

The place value to the right of the decimal point represents the fractional part of the number. For example, the number 0.56 is made up of 5 tenths and 6 hundredths. This can also be written as 0.56 = 0.5 + 0.06. In other words, it means, 0.56 = 5/10 + 6/100.






Reference:
Cuemath:Representation of Real Numbers on Number Line
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Wednesday, January 13, 2016

Integers: Multiplication & Division

I. Algorithm for Multiplying or Dividing Integers

Use the standard algorithm for multiplying whole numbers
1) When multiplying or dividing  integers that have the same sign the result is always positive. Simply multiply or divide the absolute value of each number and make the answer positive.

-Positive x Positive = Positive
-Negative x Negative = Positive

2) When multiplying or dividing integers that have different signs the result is always negative. Multiply or divide the absolute value of each number and make the answer negative.

-Positive x Negative = Negative
-Negative x Positive = Negative

II. Conceptual Models
1. Number Line
We can use the number line to help conceptually understand the rules for multiplying and dividing integers.

positive x positive = positive
Multiplying positive integers is no different than multiplying whole numbers. For example, 3 x 4 can be thought of as moving 4 units to the right 3 times.


Since division is the inverse operation of multiplication, we can reverse the above problem. Starting at 12, we move in groups of 4 to the left until we reach 0. We then count the number of groups or movements to reach the answer of 3.


positive x negative = negative
Multiplying 3 x (-4) can be thought of as "3 groups of -4." Starting at 0 on the number line, we move 4 units to the left 3 times, or as your teacher might say, you take 3 jumps of -4 to arrive at -12.


 negative x positive = negative
With (-3) x 4 the 4 is our value (multiplicand) and the -3 is our action (multiplier). The negative sign in the multiplier essentially means do the opposite (go in the opposite direction). So we are“−3 groups of 4” means the opposite movement: start at 0, move 4 units to the left 3 times to arrive at -12..


 negative x negative = positive
As a negative number is the opposite of a positive number, the opposite of a negative number is positive. Hence, a negative times a negative equal a positive. For instance, to solve (-3) x (-4) we can start by using the number line above. As with the problem 3 x (-4) we can view this problem as repeated addition where we are going to move 4 units to the left 3 times leaving us at -12. But we haven't accounted for the negative sign in front of the 3 yet. To do so, we take the opposite of -12 (a negative of a negative is a positive) which leaves us at positive 12 for our answer.


2. Using Color Counters
This video does a good job of explaining ways of conceptualizing integer multiplication and division.
Mathispower4u Video

3. Division Based on Divisor-Dividend Size
In the whole number division section I included some rules on how the quotient of a division problem would result based on the size of the divisor compared to the dividend. Those rules where:

1) Divisor smaller than dividend - The quotient is greater than 1.
2) Divisor equal to dividend - The quotient is 1
3) Divisor larger than dividend - The quotient is between 0 and 1.

These rules work for integers if you change all the values to absolute values. So the revised rules would be:

1) Absolute value of Divisor is smaller than absolute value of the dividend - The absolute value of the quotient is greater than 1.
(e.g. -12 ÷ 3 becomes |-12| ÷ |3| so |quotient| is greater than 1

2) Absolute value of the Divisor is equal to the absolute value of the dividend - The absolute value of the quotient is 1

3) Absolute value of the Divisor larger than absolute value of the dividend - The absolute value of the quotient is between 0 and 1.
(e.g. -3  ÷ 12 becomes |-3| ÷ |12| so |quotient| is between 0 and 1


III. Properties



Practice:
IXL: Multiplying and Dividing Integers

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Sunday, January 10, 2016

Integers: Addition and Subtraction



I. Rules for Adding and Subtracting Integers
By conceptually understanding the addition and subtraction of integers as movements on the number line, the reasoning for the following rules should make sense (see below):
1) When adding integers that have the same sign, add the absolute value of each number and keep the sign.
2) When adding integers that have different signs, subtract the smallest absolute value from the largest and keep the sign of the number with the largest absolute value.
3) When subtracting integers, one method is to convert it into an addition problem then follow the rules for addition of integers. To do this (1) change the subtraction sign into an addition sign, (2) take the opposite of the number that immediately follows the newly placed addition sign.

Example: positive + positive:

15 + 3

This of course is simply the addition of two whole numbers but following the rules above, you simply add the absolute values of each number (15 and 3) and keep the sign, which here is positive.

15 + 3 = 18

Example: negative + negative:

-15 + (-3)

Again, add the absolute values of each number (15 and 3) and keep the sign, which here is negative.

-15 + (-3) = -18

Example: positive + negative:

8 + (-10)

Following the rules for adding numbers with different signs, we subtract the smallest absolute value (8) from the larger (10). Then use the sign of the number with the largest absolute value, which in this case is negative.

8 + (-10) = -2

Example: negative + positive:

-5 + 7

Again, following the rules for adding numbers with different signs, subtract the smallest absolute value (5) from the larger (7). Then use the sign of the number with the largest absolute value.

-5 + 7 = 2

Example: subtracting integers:

-3 - (-9)

Convert the subtraction problem into an addition problem by replacing the subtraction sign with an addition sign. Then convert the number that follows the newly placed addition sign to it's opposite (-9 become 9).

-3 + 9

Then follow the rules for adding integers.

-3 + 9 = 6

II. Conceptual Model
Number Line
Addition and subtraction of integers can be illustrated with use of a number line. For instance, to calculate -1 + 4, we start at 0 then move one unit to the left. This represents -1. From here, move 4 units to the right. This leaves us at 3 on the number line. -1 + 4 = 3.


III. Properties of Addition
Commutative Property
Addition (as well as multiplication) is commutative, meaning the order of the numbers being added doesn't matter.

x + y = y + x

Example:
4 + (-6) = -2 is the same as -6 + 4 = -2

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:

x + (y + z) = (x + y) + z

Example:
1 + (2 + (-3)) = 0 is the same as (1 + 2) + (-3) = 0
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Friday, January 8, 2016

Integers: Intro

I. Intro

Integers are the set of numbers that can be written without a fractional component that consist of positive numbers, zero and negative numbers: {-4, -3, -2, -1, 0, 1, 2, 3, 4}.

Integers can be illustrated using a number line such as this.



 Absolute Value
The absolute value of a number is the distance between zero and the number on the number line. As such, the absolute value is always positive.

For instance, the absolute value of 6 is 6 since it is 6 units to the right of 0 on the number line. The absolute value of -6 is 6 since it is 6 units to the left of 0 on the number line. 

Negation (Opposites)
The negation of a number a is the number that, when added to a, yields zero. Negation is also known as the additive inverse or opposite of a number.

For example, the negation of 5 is -5 since 5 + (-5) = 0. Of course this also means that the negation of -5 is 5 since -5 + 5 = 0.

Negation and Subtraction
Subtracting a number is the same as adding it's opposite.

For example 5 - 3 = 5 + (-3) and 5 - (-2) = 5 + 2.

Negation and Multiplication
Negation is the same as multiplication by -1.

For example, -3 = (-1) x 3 and 2 = (-1) x (-2)

Opposite of Opposite
The opposite of the opposite of a number is the number itself.

For example, -(-3) = 3

II. Conceptualizing Integers

Real world representations of integers include:
1. Temperature (temperature above 0 degrees is positive, below 0 is negative)




4. Elevation (Above sea level is positive, below sea level is negative)




3. Money (Money received or owed to you is positive. Money paid or owed to others is negative)




Practice


Reference
https://www.splashlearn.com/math-vocabulary/integers

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