Equivalent Fractions 6-2
Fractions can be pictured on the number line.
If you multiply the numerator and the denominator of a fraction but the same nonzero number the resulting fraction is equivalent to the original fraction
1/2 (3/3) = 3/6
Properties
For any whole numbers a, b, and c, with b≠ 0 and c≠ 0
a/b = a(c)/b(c) and a/b = (a÷c) / (b÷c)
A fraction is in lowest terms if its numerator and denominator are relatively prime If their Greatest Common Factor (GCF) is 1
Fractions and Mixed Numbers 6-3
You know that ½ + ½ + ½ = 3/2
A fraction such as 3/2 (whose numerator is greater than or equal to its denominator) is called an improper fraction. Every improper fraction is greater than or equal to 1. A proper fraction is a fraction whose numerator is less than its denominator.
Proper fractions
1/4 2/3 5/9 10/12
Improper fractions
5/2, 8/3. 11/9 , 18/15
You can express any improper fractions as the sum of a whole number and a proper fraction
3/2 = 1 + ½ or 1½
A number such as 1½ (that is expressed as the sum of a whole number and a fraction) is called a mixed number.
If the fractional part of a mixed number is a proper fraction in lowest terms, the mixed number is said to be in simplest form.
To change an improper fraction to a mixed number in simple form, we divide the numerator by the denominator and express the remainder as a fraction
14/3 = 14 ÷ 3 = 4 2/3
30/4 = 30 ÷ 4 = 7 2/4 = 7 ½
To change a mixed number to an improper fraction, rewrite the whole number part as a fraction with the same denominator as the fractional part
1 3/8 = 8/8 + 3/8 = 11/8
Thursday, February 12, 2009
Math 6 H Periods 1, 6 & 7 (Tuesday)
Fractions 6-1
Can you think of some familiar expressions that include fractions?
Notice that the symbol ¼ can mean several things:
It means one divided by four
It represents one out of four equal parts
It is a number that has a position on a number line.
If an object is divided into 8 equal parts, each part is one eight of the whole.
1/8 means 1 divided by 8 or 1 ÷ 8.
If an object is divided into eight parts and three of these parts are being considered, then the fraction that represents the parts is 3/8
A fraction consists of two numbers.
The denominator tells the number of equal parts into which the whole has been divided.
The numerator tells how many of these parts are being considered.
Properties
For any whole numbers a, b, and c, with b ≠ 0
(a/b)(c) = ac/b
Examples:
2/7 + 2/ 7 + 2/7 = 6/7 which is really (3) 2/7 = 6/7
Can you think of some familiar expressions that include fractions?
Notice that the symbol ¼ can mean several things:
It means one divided by four
It represents one out of four equal parts
It is a number that has a position on a number line.
If an object is divided into 8 equal parts, each part is one eight of the whole.
1/8 means 1 divided by 8 or 1 ÷ 8.
If an object is divided into eight parts and three of these parts are being considered, then the fraction that represents the parts is 3/8
A fraction consists of two numbers.
The denominator tells the number of equal parts into which the whole has been divided.
The numerator tells how many of these parts are being considered.
Properties
For any whole numbers a, b, and c, with b ≠ 0
(a/b)(c) = ac/b
Examples:
2/7 + 2/ 7 + 2/7 = 6/7 which is really (3) 2/7 = 6/7
Algebra Period 3 (Wednesday)
How do you determine whether a given number is a solution?
Plug it in, plug it in, plug it in! Do this carefully. Use ( ) when you plug in a value for x and for y.
How do you find a solution to an equation yourself?
Plug in for x and find y!
You can use ANY number for x
Then plug in your number and find y
How can you graph a linear equation?
Make an x/y table of values and then graph the coordinates.
You only need 3 coordinates to make a good line!
(The 3rd coordinate serves as a "check" for the other two...in case you made a mistake!)
I always try x = zero and y = zero first because it's usually easy. Then pick another easy x value!
If this doesn't work well (you get a fraction as an answer and that's not easy to graph),
then try setting x equal to 1, then 2, then 3
Linear equations 7-3
What do they look like ( and what is not a linear equation?)
The variable is to the 1 power - like x, or y, or a, or b
What is not a linear equation? the variable is not to the 1 power - like x2, x3, etc, or 1/x (x-1)
2 ways to graph:
1) 3 points using a table (like Ch 7-2)
EXAMPLE: 2x - 3y = -6
x y
0 2
3 4
-3 0
2) 2 points using the y and x intercepts (where the line intersects the y and x axis)
Standard form of a linear equation: Ax +By = C
A, B and C should not be fractions
A should be positive (y will be positive or negative)
We won't be using this form to look at the slope of the line!
This is a good format for finding the x and y intercepts!
If it's in standard form, this way works great if both the x and y coefficients are factors of the constant on the other side of the equal sign.
EXAMPLE: 2x - 3y = -6
If x = 0, y = 2
If y = 0, x = -3
Special linear equations:
Ones that are parallel to either the x or the y axis:
Lines parallel to the y axis are vertical lines:
They end up as the form x = with no y variable in the equation at all!
EXAMPLE: x = 4 ends up as a vertical line at x = 4
Still don't get this???
Pick of few points with the x value of 4:
(4, 0) (4, 2) (4, -3)
Graph those and join them in a line.
What do you get???
A vertical line!
Lines parallel to the x axis are horizontal lines:
They end up as the form y = with no x variable in the equation at all!
EXAMPLE: y = 4 ends up as a horizontal line at y = 4
Still don't get this???
Pick of few points with the y value of 4:
(0, 4) (2, 4) (-3, 4)
Graph those and join them in a line.
What do you get???
A horizontal line!
Plug it in, plug it in, plug it in! Do this carefully. Use ( ) when you plug in a value for x and for y.
How do you find a solution to an equation yourself?
Plug in for x and find y!
You can use ANY number for x
Then plug in your number and find y
How can you graph a linear equation?
Make an x/y table of values and then graph the coordinates.
You only need 3 coordinates to make a good line!
(The 3rd coordinate serves as a "check" for the other two...in case you made a mistake!)
I always try x = zero and y = zero first because it's usually easy. Then pick another easy x value!
If this doesn't work well (you get a fraction as an answer and that's not easy to graph),
then try setting x equal to 1, then 2, then 3
Linear equations 7-3
What do they look like ( and what is not a linear equation?)
The variable is to the 1 power - like x, or y, or a, or b
What is not a linear equation? the variable is not to the 1 power - like x2, x3, etc, or 1/x (x-1)
2 ways to graph:
1) 3 points using a table (like Ch 7-2)
EXAMPLE: 2x - 3y = -6
x y
0 2
3 4
-3 0
2) 2 points using the y and x intercepts (where the line intersects the y and x axis)
Standard form of a linear equation: Ax +By = C
A, B and C should not be fractions
A should be positive (y will be positive or negative)
We won't be using this form to look at the slope of the line!
This is a good format for finding the x and y intercepts!
If it's in standard form, this way works great if both the x and y coefficients are factors of the constant on the other side of the equal sign.
EXAMPLE: 2x - 3y = -6
If x = 0, y = 2
If y = 0, x = -3
Special linear equations:
Ones that are parallel to either the x or the y axis:
Lines parallel to the y axis are vertical lines:
They end up as the form x = with no y variable in the equation at all!
EXAMPLE: x = 4 ends up as a vertical line at x = 4
Still don't get this???
Pick of few points with the x value of 4:
(4, 0) (4, 2) (4, -3)
Graph those and join them in a line.
What do you get???
A vertical line!
Lines parallel to the x axis are horizontal lines:
They end up as the form y = with no x variable in the equation at all!
EXAMPLE: y = 4 ends up as a horizontal line at y = 4
Still don't get this???
Pick of few points with the y value of 4:
(0, 4) (2, 4) (-3, 4)
Graph those and join them in a line.
What do you get???
A horizontal line!
Algebra Period 3 (Tuesday)
Graphing Ordered Pairs 7-1
Review of x y Coordinate Plane Graphing from Pre-Algebra
Cartesian plane: Named after French mathematician Descartes.
PLANE: a two dimensional (across and up/down) flat surface that extends infinitely in all directions. It’s 2-D
QUADRANT: 2 perpendicular lines called axes split the plane into 4 regions....quad means 4
quadrant names: begin in the top right (where you normally write your name!) and go counterclockwise in a big "C" (remember it for "C"oordinate)
They are named I, II, III, IV in Roman Numerals
COORDINATE - A coordinate is the position of a point in the Cartesian plane
coordinate = "co" means goes along with (COefficient, COworker, CO-president, CO-champions)
"ordinate" means in order
So coordinate means numbers that go along with each other in a certain order
The numbers are the x and y values and the order is that the x always comes first
Also called an ordered pair (x y "ordered" and they are a "pair" of numbers)
Ordered pairs are recognized by the use of ( x , y) format
origin = (0, 0) the center of the graph (its beginning or origin)
When you count the coordinate' s position, you count from the origin.
x comes before y in the alphabet so the order is (x, y)
always go right or left first, then up or down
the x axis is the horizontal axis (goes across)
Remember that because the number line also is horizontal and you learn that first
(the pattern to remember is x is always first and the number line is before going up and down)
NOW LET'S GET TO WHAT YOU ACTUALLY DO!!!
1) Count your x value:
positive x, count right from origin (positive numbers are to the right of zero on number line)
negative x, count left from origin
2) Count your y value:
positive y value, count up from where your x value was (up is the positive direction)
negative y value, count down from where your x value was (down is the negative direction)
EXAMPLE:
(3, 5) Count 3 to the right from the origin, then 5 up
(3, -5) Still count 2 to the right, but now count 5 down
(-3, 5) Count 3 to the left from the origin, then count 5 up
(-3, -5) Again count 3 to the left, but now count 5 down
BUT WHAT HAPPENS WHEN
ONE OF THE VALUES IS ZERO?
If the y value is zero it means that you move right or left, but don't go up or down:
SO YOUR POINT WILL BE ON THE x AXIS........x axis is where y = 0
Example: (3, 0) is a point on the x axis, 3 places to the RIGHT
Example: (-3, 0) is a point on the x axis, 3 places to the LEFT
If the x value is zero it means that you don't move right or left, you just go up or down.
SO YOUR POINT WILL BE ON THE y AXIS...........y axis is where x = 0
Example: (0, 3) is a point on the y axis, 3 places UP
Example: (0, -3) is a point on the y axis, 3 places DOWN
Review of x y Coordinate Plane Graphing from Pre-Algebra
Cartesian plane: Named after French mathematician Descartes.
PLANE: a two dimensional (across and up/down) flat surface that extends infinitely in all directions. It’s 2-D
QUADRANT: 2 perpendicular lines called axes split the plane into 4 regions....quad means 4
quadrant names: begin in the top right (where you normally write your name!) and go counterclockwise in a big "C" (remember it for "C"oordinate)
They are named I, II, III, IV in Roman Numerals
COORDINATE - A coordinate is the position of a point in the Cartesian plane
coordinate = "co" means goes along with (COefficient, COworker, CO-president, CO-champions)
"ordinate" means in order
So coordinate means numbers that go along with each other in a certain order
The numbers are the x and y values and the order is that the x always comes first
Also called an ordered pair (x y "ordered" and they are a "pair" of numbers)
Ordered pairs are recognized by the use of ( x , y) format
origin = (0, 0) the center of the graph (its beginning or origin)
When you count the coordinate' s position, you count from the origin.
x comes before y in the alphabet so the order is (x, y)
always go right or left first, then up or down
the x axis is the horizontal axis (goes across)
Remember that because the number line also is horizontal and you learn that first
(the pattern to remember is x is always first and the number line is before going up and down)
NOW LET'S GET TO WHAT YOU ACTUALLY DO!!!
1) Count your x value:
positive x, count right from origin (positive numbers are to the right of zero on number line)
negative x, count left from origin
2) Count your y value:
positive y value, count up from where your x value was (up is the positive direction)
negative y value, count down from where your x value was (down is the negative direction)
EXAMPLE:
(3, 5) Count 3 to the right from the origin, then 5 up
(3, -5) Still count 2 to the right, but now count 5 down
(-3, 5) Count 3 to the left from the origin, then count 5 up
(-3, -5) Again count 3 to the left, but now count 5 down
BUT WHAT HAPPENS WHEN
ONE OF THE VALUES IS ZERO?
If the y value is zero it means that you move right or left, but don't go up or down:
SO YOUR POINT WILL BE ON THE x AXIS........x axis is where y = 0
Example: (3, 0) is a point on the x axis, 3 places to the RIGHT
Example: (-3, 0) is a point on the x axis, 3 places to the LEFT
If the x value is zero it means that you don't move right or left, you just go up or down.
SO YOUR POINT WILL BE ON THE y AXIS...........y axis is where x = 0
Example: (0, 3) is a point on the y axis, 3 places UP
Example: (0, -3) is a point on the y axis, 3 places DOWN
Algebra Period 3 (Review)
FACTORING CHECKLIST
Look for a GCF of all terms
2. Binomials - look for difference of two squares
both perfect squares - double hug - one positive, one negative - square roots of both terms
2. Trinomials - look for Trinomial Square (factors as a binomial squared)
first and last must be perfect squares - middle must be double the product of the two square roots
SINGLE hug - square roots of both terms - sign is the middle sign
3. Trinomials - last sign positive - double hug with same sign as middle term - factors that multiply to last and add to middle
4. Trinomials - last sign negative - double hug with different signs, putting middle sign in first hug - factors that multiply to last and subtract to middle - middle sign will always be with the bigger factor
5. Trinomial with "a" coefficient - Use T chart - multiply first to last to get new product - then find factors that multiply to that new produce and either add or subtract to the middle term (use trinomial rules above) - replace middle term with these two factors and place appropriate signs so they will add to the original middle term - proceed as if you have factoring by grouping (see 6 below)
6. 4 term polynomial - factor by grouping - pair of the first 2 terms and then the second 2 terms by placing parentheses around them - make sure you always have a plus sign between the 2 pairs (you may need to double check) - factor out the GCF of each pair - if it factors, there should now be a new GCF - factor that out in front parentheses and place what ever is left in the second parentheses
REMEMBER:
FACTORING WILL NEVER CHANGE THE ORIGINAL VALUE OF THE POLYNOMIAL SO YOU SHOULD ALWAYS CHECK BY MULTIPLYING BACK!!!!
Look for a GCF of all terms
2. Binomials - look for difference of two squares
both perfect squares - double hug - one positive, one negative - square roots of both terms
2. Trinomials - look for Trinomial Square (factors as a binomial squared)
first and last must be perfect squares - middle must be double the product of the two square roots
SINGLE hug - square roots of both terms - sign is the middle sign
3. Trinomials - last sign positive - double hug with same sign as middle term - factors that multiply to last and add to middle
4. Trinomials - last sign negative - double hug with different signs, putting middle sign in first hug - factors that multiply to last and subtract to middle - middle sign will always be with the bigger factor
5. Trinomial with "a" coefficient - Use T chart - multiply first to last to get new product - then find factors that multiply to that new produce and either add or subtract to the middle term (use trinomial rules above) - replace middle term with these two factors and place appropriate signs so they will add to the original middle term - proceed as if you have factoring by grouping (see 6 below)
6. 4 term polynomial - factor by grouping - pair of the first 2 terms and then the second 2 terms by placing parentheses around them - make sure you always have a plus sign between the 2 pairs (you may need to double check) - factor out the GCF of each pair - if it factors, there should now be a new GCF - factor that out in front parentheses and place what ever is left in the second parentheses
REMEMBER:
FACTORING WILL NEVER CHANGE THE ORIGINAL VALUE OF THE POLYNOMIAL SO YOU SHOULD ALWAYS CHECK BY MULTIPLYING BACK!!!!
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