Rise Up... Run out.....
Thursday, February 9, 2012
Tuesday, February 7, 2012
Algebra Honors (Period 6 & 7)
Functions Defined by Equations 8-7
A relation is a set of ordered pairs such as
{ (2,3), (3,5), (-4, 0), (5, 0)}
it is also a function because no repeating of the x value,
Domain of a relation is the set of 1st coordinates
the x values
Range is the 2nd coordinates
the y values
So in the example above
Domain is {2, 3, -4}
Range {3, 5, 0}
A relation that assigns to each value in the domain exactly one value in the range is called a FUNCTION
{ (2,3), (3,5), (-4, 0), (5, 0)} is a FUNCTION, whereas,
{ (2,3), (2,5), (-4, 0), (5, 0)} is NOT a function
values of domains ( x's) each are paired with only one element in the range.
Several ways to check... we looked at T- tables to compare the x values, we looked at mapping and we looked at graphs. Notice the vertical line test.
WE then compared the notation for functions
first we looked at y = 3x + 4 vs f(x) = 3x + 4
solve for x = 5
For y = 3x + 5
y = 3(5) + 4
y= 15 + 4
y = 19
Now, what was x again
Oh yeah... x = 5
so the ordered pair is (5, 19)
With f(x) = 3x + 4 however we have
f(5) = 3(5) + 4
f(5) - 15 + 4
f(5) = 19
and you can see what x was originally
(5, 19) is the ordered pair
a
Two ways to show functions
f(x) ... and we noted tht it could be g(x) or h(t) etc
or
f:x-->
g:x-->4 + 3x - x2
if the domain D = { -1, 0, 1, 2}
g:-1--> g(-1) = 4 + 3(-1) - (-1)2= 0
g:0-->g(0) = 4 +3(0) -0 = 4
g:1--> g(1) 4 + 3(1) - (1)2 = 6
g:2-->4 + 3(2) - (2)2
Range = {0. 4. 6} Notice that we list each number once ( even if there is a repeated number)
f:x --> x2 - 2x for the set of all REAL numbers
find
f(4) = (4)2 -2(4) = 8
f(-3) = (-3) 2 -2(-3) = 15
f(2) = (2)2 -2(2) = 0
so the Range is {8, 15, 0}
Relations & Functions
RELATIONS: Set of ordered pairs where the x values are the DOMAIN and the y values are the RANGE.
FUNCTIONS: Relations where there is just one y value for each x value IN OTHER WORDS----YOU CAN'T HAVE TWO y VALUES for the SAME x value!!!
If you see x repeated twice, it's still a relation, but it's not a function.
In the real world, there are excellent examples....pizza prices.
A restaurant can't have two different prices for the same size cheese pizza.
If you charge $10 and $12 on the same day for the same pizza, you don't have a function.
But, you certainly can charge $10 for a cheese pizza and $12 for a pepperoni pizza.
VERTICAL LINE TEST: When you graph a function, if you draw a vertical line anywhere on the graph, that line will only intersect the function at one point!!!!
If it intersects at 2 or more, it's a relation, but not a function.
So a horizontal line function, y = 4, is a function, but a vertical line function, x = 4 is not.
Any line, y = mx + b, is a function.
INPUTS: x values
OUTPUTS: y values
f(x) means the value of the function at the given x value
You can think of f(x) as the y value
Finding the value of a function: Plug it in, plug it in!
f(x) = 2x + 7
Find f(3)
f(3) = 2(3) + 7 = 13
The function notation gives you more information than using y
If I tell you y = 13 you have no idea what the x value was at that point
But if I tell you f(3) = 13, you know the entire coordinate (3, 13)
Domain of a function = all possible x values (inputs) that keep the solution real
Range of a function = all possible y values (outputs) that result from the domain
EXAMPLE:
f(x) = x + 10 has the domain of all real numbers and the same range because every value will keep the answer f(x) a real number
EXAMPLE:
f(x) = x2 has the domain again of all real numbers, BUT the range is greater than or = to zero
because when a number is squared it will never be negative! So f(x) will always be 0 or positive
EXAMPLE:
f(x) = absolute value of x has the domain of all real numbers, but again the range will be greater than or equal to zero because absolute value will never be negative
EXAMPLE:
f(x) = 1/x has a domain of all real numbers EXCEPT FOR ZERO because it would be undefined if zero was in the denominator. The range is all real numbers except zero as well.
This function will approach both axes but never intersect with them.
The axes are called asymptotes which means that they will get very close but never reach them
EXAMPLE:
f(x) = (x - 10)/x + 3
Domain is all real numbers EXCEPT -3 because -3 will turn the denominator into zero (undefined)
What is the range?
A relation is a set of ordered pairs such as
{ (2,3), (3,5), (-4, 0), (5, 0)}
it is also a function because no repeating of the x value,
Domain of a relation is the set of 1st coordinates
the x values
Range is the 2nd coordinates
the y values
So in the example above
Domain is {2, 3, -4}
Range {3, 5, 0}
A relation that assigns to each value in the domain exactly one value in the range is called a FUNCTION
{ (2,3), (3,5), (-4, 0), (5, 0)} is a FUNCTION, whereas,
{ (2,3), (2,5), (-4, 0), (5, 0)} is NOT a function
values of domains ( x's) each are paired with only one element in the range.
Several ways to check... we looked at T- tables to compare the x values, we looked at mapping and we looked at graphs. Notice the vertical line test.
WE then compared the notation for functions
first we looked at y = 3x + 4 vs f(x) = 3x + 4
solve for x = 5
For y = 3x + 5
y = 3(5) + 4
y= 15 + 4
y = 19
Now, what was x again
Oh yeah... x = 5
so the ordered pair is (5, 19)
With f(x) = 3x + 4 however we have
f(5) = 3(5) + 4
f(5) - 15 + 4
f(5) = 19
and you can see what x was originally
(5, 19) is the ordered pair
a
Two ways to show functions
f(x) ... and we noted tht it could be g(x) or h(t) etc
or
f:x-->
g:x-->4 + 3x - x2
if the domain D = { -1, 0, 1, 2}
g:-1--> g(-1) = 4 + 3(-1) - (-1)2= 0
g:0-->g(0) = 4 +3(0) -0 = 4
g:1--> g(1) 4 + 3(1) - (1)2 = 6
g:2-->4 + 3(2) - (2)2
Range = {0. 4. 6} Notice that we list each number once ( even if there is a repeated number)
f:x --> x2 - 2x for the set of all REAL numbers
find
f(4) = (4)2 -2(4) = 8
f(-3) = (-3) 2 -2(-3) = 15
f(2) = (2)2 -2(2) = 0
so the Range is {8, 15, 0}
Relations & Functions
RELATIONS: Set of ordered pairs where the x values are the DOMAIN and the y values are the RANGE.
FUNCTIONS: Relations where there is just one y value for each x value IN OTHER WORDS----YOU CAN'T HAVE TWO y VALUES for the SAME x value!!!
If you see x repeated twice, it's still a relation, but it's not a function.
In the real world, there are excellent examples....pizza prices.
A restaurant can't have two different prices for the same size cheese pizza.
If you charge $10 and $12 on the same day for the same pizza, you don't have a function.
But, you certainly can charge $10 for a cheese pizza and $12 for a pepperoni pizza.
VERTICAL LINE TEST: When you graph a function, if you draw a vertical line anywhere on the graph, that line will only intersect the function at one point!!!!
If it intersects at 2 or more, it's a relation, but not a function.
So a horizontal line function, y = 4, is a function, but a vertical line function, x = 4 is not.
Any line, y = mx + b, is a function.
INPUTS: x values
OUTPUTS: y values
f(x) means the value of the function at the given x value
You can think of f(x) as the y value
Finding the value of a function: Plug it in, plug it in!
f(x) = 2x + 7
Find f(3)
f(3) = 2(3) + 7 = 13
The function notation gives you more information than using y
If I tell you y = 13 you have no idea what the x value was at that point
But if I tell you f(3) = 13, you know the entire coordinate (3, 13)
Domain of a function = all possible x values (inputs) that keep the solution real
Range of a function = all possible y values (outputs) that result from the domain
EXAMPLE:
f(x) = x + 10 has the domain of all real numbers and the same range because every value will keep the answer f(x) a real number
EXAMPLE:
f(x) = x2 has the domain again of all real numbers, BUT the range is greater than or = to zero
because when a number is squared it will never be negative! So f(x) will always be 0 or positive
EXAMPLE:
f(x) = absolute value of x has the domain of all real numbers, but again the range will be greater than or equal to zero because absolute value will never be negative
EXAMPLE:
f(x) = 1/x has a domain of all real numbers EXCEPT FOR ZERO because it would be undefined if zero was in the denominator. The range is all real numbers except zero as well.
This function will approach both axes but never intersect with them.
The axes are called asymptotes which means that they will get very close but never reach them
EXAMPLE:
f(x) = (x - 10)/x + 3
Domain is all real numbers EXCEPT -3 because -3 will turn the denominator into zero (undefined)
What is the range?
Math 6 Honors ( Periods 1, 2, & 3)
Ratios 7-6
In our textbook, the example given involves the number of students --at what I called a mythical middle school --as well as the number of teachers. There are 35 teachers and 525 students. We can compare the number of teachers to the number of students by writing a quotient
number of teachers
number of students
35
525
1/15
The quotient of one number divided by a second number is called the ratio of the first number to the second number.
We can write a ratio in the following ways:
1/15 OR 1:15 OR 1 to 15
All of these expressions are read one to fifteen.
If the colon notation is used the first number is divided by the second. A ratio is said to be lowest terms if the two numbers are “relatively prime.”
You do not change an improper fraction to a mixed number if the improper fraction represents a ratio
There are 9 players on a baseball team. Four of these are infielders and 3 are outfielders. Find each ratio in lowest terms.
a. infielders to outfielders
b. outfields to total players
# of infielders
# of outfielders
= 4/3 or 4:3 or 4 to 3
# of outfielders
# total of players
= 3/9 = 1/3 or 1:3 or 1 to 3
Some ratios compare measurements. In these cases we must be sure the measurements are expressed in the same units
It takes Kiana (or Helen or Emme) 4 minutes to mix some paint. It takes her 3 hours to complete painting her room. What is the ratio of the time it takes Kiana (or Helen or Emme) to mix the paint to the time it takes her to paint her room?
Use minutes as a common unit for measuring time. You must convert the hours to minutes first
3h = 3 • 60min = 180 min
The ratio is :
min. to mix
min. to paint
= 4/180 = 1/45 or 1:45
Some ratios are in the form
40 miles per hour or 5 pencils for a dollar
“ I want my… I want my…. I want my … MPG!!”
These ratios involve quantities of different kinds and are called rates. Rates may be expressed as decimals or mixed numbers. Rates should be simplified to a per unit form. When a rate is expressed in a per unit form, such a rate is often called a unit rate.
I know you will be driving in a blick of an eye... so Justin, in his Lamborghini(and Shane in his Corvette and Nick in his yellow Lamborghini) went 258 miles on 12 gallons of gas. Express the rate of fuel consumption in miles per gallon.
The rate of fuel consumption is
258 miles
12 gallons
= 21 1/2 miles per gallon
Some of the most common units in which rates are given are the following:
mi/gal or mpg miles per gallon
mi/h or mph miles per hour
km/L kilometers per liter
km/h kilometers per hour
Page 229
1 What is the cost of grapes in dollars per kilogram if 4.5 kg of grapes costs $7.56?
$7.56/4.5 kg divide carefully and you discover it is $1.68/kg
2. THe index of refraction of a transparent substance is the ratio of the speed of light in space to the speed of light in the substance.
Using the table from the textbook (look at page 229) Find the index of refraction of
a) glass
300,000/200,000 straight from the chart, which can simplify to 3/2
b) water
300,000/225,000 again from the chart, which can simplify to 4/3
3. The mechanical advantage of a simple machine is the ratio of the weight lifted by the machine to the forse necessary to lift it.
What is the mechanical advantage of a jack that lifts a 3200 pound car with a force of 120 pounds?
3200/120 = 80/3
4. The C string of a cello vibrates 654 times in 5 seconds. How many vibrations per second is this?
654 vibrations/5seconds... divide carefully and you find... 130 4/5 vibrations per second
5. A four-cubic-foot volume of water at sea level weights 250 pounds. What is the density of water in pound per cubic foot?
250 pounds/4 cubic ft ... divide carefully and you find 62 1/2 lb/ft3
6. A share of stock that costs $88 earned $16 last year. What was the price to earnings ratio?
88/16 = 11/2
7. we did in our spiral notebooks this year... please check
In our textbook, the example given involves the number of students --at what I called a mythical middle school --as well as the number of teachers. There are 35 teachers and 525 students. We can compare the number of teachers to the number of students by writing a quotient
number of teachers
number of students
35
525
1/15
The quotient of one number divided by a second number is called the ratio of the first number to the second number.
We can write a ratio in the following ways:
1/15 OR 1:15 OR 1 to 15
All of these expressions are read one to fifteen.
If the colon notation is used the first number is divided by the second. A ratio is said to be lowest terms if the two numbers are “relatively prime.”
You do not change an improper fraction to a mixed number if the improper fraction represents a ratio
There are 9 players on a baseball team. Four of these are infielders and 3 are outfielders. Find each ratio in lowest terms.
a. infielders to outfielders
b. outfields to total players
# of infielders
# of outfielders
= 4/3 or 4:3 or 4 to 3
# of outfielders
# total of players
= 3/9 = 1/3 or 1:3 or 1 to 3
Some ratios compare measurements. In these cases we must be sure the measurements are expressed in the same units
It takes Kiana (or Helen or Emme) 4 minutes to mix some paint. It takes her 3 hours to complete painting her room. What is the ratio of the time it takes Kiana (or Helen or Emme) to mix the paint to the time it takes her to paint her room?
Use minutes as a common unit for measuring time. You must convert the hours to minutes first
3h = 3 • 60min = 180 min
The ratio is :
min. to mix
min. to paint
= 4/180 = 1/45 or 1:45
Some ratios are in the form
40 miles per hour or 5 pencils for a dollar
“ I want my… I want my…. I want my … MPG!!”
These ratios involve quantities of different kinds and are called rates. Rates may be expressed as decimals or mixed numbers. Rates should be simplified to a per unit form. When a rate is expressed in a per unit form, such a rate is often called a unit rate.
I know you will be driving in a blick of an eye... so Justin, in his Lamborghini(and Shane in his Corvette and Nick in his yellow Lamborghini) went 258 miles on 12 gallons of gas. Express the rate of fuel consumption in miles per gallon.
The rate of fuel consumption is
258 miles
12 gallons
= 21 1/2 miles per gallon
Some of the most common units in which rates are given are the following:
mi/gal or mpg miles per gallon
mi/h or mph miles per hour
km/L kilometers per liter
km/h kilometers per hour
Page 229
1 What is the cost of grapes in dollars per kilogram if 4.5 kg of grapes costs $7.56?
$7.56/4.5 kg divide carefully and you discover it is $1.68/kg
2. THe index of refraction of a transparent substance is the ratio of the speed of light in space to the speed of light in the substance.
Using the table from the textbook (look at page 229) Find the index of refraction of
a) glass
300,000/200,000 straight from the chart, which can simplify to 3/2
b) water
300,000/225,000 again from the chart, which can simplify to 4/3
3. The mechanical advantage of a simple machine is the ratio of the weight lifted by the machine to the forse necessary to lift it.
What is the mechanical advantage of a jack that lifts a 3200 pound car with a force of 120 pounds?
3200/120 = 80/3
4. The C string of a cello vibrates 654 times in 5 seconds. How many vibrations per second is this?
654 vibrations/5seconds... divide carefully and you find... 130 4/5 vibrations per second
5. A four-cubic-foot volume of water at sea level weights 250 pounds. What is the density of water in pound per cubic foot?
250 pounds/4 cubic ft ... divide carefully and you find 62 1/2 lb/ft3
6. A share of stock that costs $88 earned $16 last year. What was the price to earnings ratio?
88/16 = 11/2
7. we did in our spiral notebooks this year... please check
Thursday, February 2, 2012
Monday, January 30, 2012
Math 6 Honors ( Periods 1, 2, & 3)
Multiplication of Fractions 7-3
If a rectangle is divided into 4 equal parts, each part is ¼ of the whole. If each of these parts is then divided into 3 parts, that is into thirds, then there are 12 equal parts and each is 1/(3 ∙4) or 1/12 of the whole.
That is 1/3 of 1/4 is 1/(3 ∙4) or 1/12 and 1/3 ∙ 1/4 = 1/12 is
so another example 2/3 of 4/5 is 2∙4 /(3∙8) or 2/3 ∙4/5 = 8/15
Notice, that the numerator of the product, 8, is the product of the numerators 2 and 4. The denominator of the product, 15, is the product of the denominators 3 and 5
Rule
If a, b, c, and d are whole numbers with b ≠ 0 and d ≠ 0 , then
a/b(c/d) = a∙c/(b∙d)
When multiplying two fractions, you can simplify the multiplication by dividing either of the numerators and either of the denominators by common factors
6/35 ( 7/3) we can simplify first because both 6 and 3 are divisible by 3
2/35 (7/1) and then both 35 and 7 are divisible by 7 so 2/5 (1(1) = 2/5
Try the following
25/6 ( 42/5) What can we do there?
7/8(20/21) How about with these two sets of fractions?
19/20 ( 25/38) … and these fractions?
What happens when you have
15/2(7/8- 5/24)
What must we do first?
PEMDAS... in my classroom...
15/2( 21/24 - 5/24)
= 15/2(16/24)
= 15/2(2/3)
then simplify to
15/1(1/3)
= 5
What about
8/9∗ 15/32∗ 9/10 = 3/8
or 16/11 × 33/20 × 5/3 = 4
Division of Fractions 7-4
Certain numbers when multiplied together have the product 1
5 X 1/5 = 1
3/4 X 4/3 = 1
Two numbers whose product is 1 are called reciprocals of each other.
Thus 3/4 is the reciprocal of 4/3.
Zero does not have a reciprocal
Look at the following:
We know 18 = 3 X 6 and we know 18 ÷ 6 = 3 as well as 18 X 1/6 = 3
Dividing a number by a fraction is the same as multiplying the number by the RECIPROCAL of the fraction
a/b ÷ c/d = a/b ÷ d/c
Remember- you are using the reciprocal of the divisor... that is , as students want to say "You FLIP the 2nd number!!"
42/ 55 ÷ 36/11
you must rewrite the problem using the reciprocal of the 2nd number
42/55 X 11/36
Now using your skills of observing GCF simplify before you multiply ( MUCH EASIER and FASTER)
42/ 5 X 1/36 which becomes 7/5 X 1/ 6 = 7/30
If a rectangle is divided into 4 equal parts, each part is ¼ of the whole. If each of these parts is then divided into 3 parts, that is into thirds, then there are 12 equal parts and each is 1/(3 ∙4) or 1/12 of the whole.
That is 1/3 of 1/4 is 1/(3 ∙4) or 1/12 and 1/3 ∙ 1/4 = 1/12 is
so another example 2/3 of 4/5 is 2∙4 /(3∙8) or 2/3 ∙4/5 = 8/15
Notice, that the numerator of the product, 8, is the product of the numerators 2 and 4. The denominator of the product, 15, is the product of the denominators 3 and 5
Rule
If a, b, c, and d are whole numbers with b ≠ 0 and d ≠ 0 , then
a/b(c/d) = a∙c/(b∙d)
When multiplying two fractions, you can simplify the multiplication by dividing either of the numerators and either of the denominators by common factors
6/35 ( 7/3) we can simplify first because both 6 and 3 are divisible by 3
2/35 (7/1) and then both 35 and 7 are divisible by 7 so 2/5 (1(1) = 2/5
Try the following
25/6 ( 42/5) What can we do there?
7/8(20/21) How about with these two sets of fractions?
19/20 ( 25/38) … and these fractions?
What happens when you have
15/2(7/8- 5/24)
What must we do first?
PEMDAS... in my classroom...
15/2( 21/24 - 5/24)
= 15/2(16/24)
= 15/2(2/3)
then simplify to
15/1(1/3)
= 5
What about
8/9∗ 15/32∗ 9/10 = 3/8
or 16/11 × 33/20 × 5/3 = 4
Division of Fractions 7-4
Certain numbers when multiplied together have the product 1
5 X 1/5 = 1
3/4 X 4/3 = 1
Two numbers whose product is 1 are called reciprocals of each other.
Thus 3/4 is the reciprocal of 4/3.
Zero does not have a reciprocal
Look at the following:
We know 18 = 3 X 6 and we know 18 ÷ 6 = 3 as well as 18 X 1/6 = 3
Dividing a number by a fraction is the same as multiplying the number by the RECIPROCAL of the fraction
a/b ÷ c/d = a/b ÷ d/c
Remember- you are using the reciprocal of the divisor... that is , as students want to say "You FLIP the 2nd number!!"
42/ 55 ÷ 36/11
you must rewrite the problem using the reciprocal of the 2nd number
42/55 X 11/36
Now using your skills of observing GCF simplify before you multiply ( MUCH EASIER and FASTER)
42/ 5 X 1/36 which becomes 7/5 X 1/ 6 = 7/30
Algebra Honors (Period 6 & 7)
Points, Lines, and Their Graphs 8-2
We reviewed graphing or plotting an ordered pair as a point on a coordinate plane.
Horizontal axis is the x-axis
vertical axis is the y-axis
origin is at (0,0)
an ordered pair (3,2) lists the coordinates of a point. In this instance we called the Point A
3 is the x-coordinate also know as the abscissa of A
2 is the y-coordinate also known as the ordinate of A
the x- and y-axes are also called coordinate axes and the number plane is often called the coordinate plane. The coordinate axes separate a coordinate plane into four quadrants identified by Roman Numerals. See page 354 for details.
Points on the coordinate axes are NOT considered to be in any quadrant.
The graph of an equation in two variables consists of all the poins that are the graphs of the solutions of the equations.
x + 2y = 6 has the following ordered pairs:
(0,3)
(2,2)
(4,1)
(6,0)
There are infinite number of solutions-- such as
(-2,4)
(1, 2.5)
The graph of all the solutions lie on the straight line that is drawn when the points are connected.
x + 2y = 6 is a linear equation because its graph is a line.
All linear equations in the variables x and y can be written in the form
ax + by = c
or
Ax + By = C
where a, b, and c are real numbers with a and b noth both zero. If a, b, and c are integers, then the equation is said to be in standard form.
2x -5y = 7 and 4x + 9y = 0 and y = 3 are examples of linear equations in standard form
(1/2)x + 4y = 12 is not
y = 3x -1 is not
neither is x2y + 3y = 4
nor xy = 6
Although you only need two points to determine a line, I suggest you plot 3-- whenever possible to guard against mistakes.
The easiest solutions to find are those where the line crosses
the x-axis ( y = 0) and
the y-axis ( x = 0)
We reviewed graphing or plotting an ordered pair as a point on a coordinate plane.
Horizontal axis is the x-axis
vertical axis is the y-axis
origin is at (0,0)
an ordered pair (3,2) lists the coordinates of a point. In this instance we called the Point A
3 is the x-coordinate also know as the abscissa of A
2 is the y-coordinate also known as the ordinate of A
the x- and y-axes are also called coordinate axes and the number plane is often called the coordinate plane. The coordinate axes separate a coordinate plane into four quadrants identified by Roman Numerals. See page 354 for details.
Points on the coordinate axes are NOT considered to be in any quadrant.
The graph of an equation in two variables consists of all the poins that are the graphs of the solutions of the equations.
x + 2y = 6 has the following ordered pairs:
(0,3)
(2,2)
(4,1)
(6,0)
There are infinite number of solutions-- such as
(-2,4)
(1, 2.5)
The graph of all the solutions lie on the straight line that is drawn when the points are connected.
x + 2y = 6 is a linear equation because its graph is a line.
All linear equations in the variables x and y can be written in the form
ax + by = c
or
Ax + By = C
where a, b, and c are real numbers with a and b noth both zero. If a, b, and c are integers, then the equation is said to be in standard form.
2x -5y = 7 and 4x + 9y = 0 and y = 3 are examples of linear equations in standard form
(1/2)x + 4y = 12 is not
y = 3x -1 is not
neither is x2y + 3y = 4
nor xy = 6
Although you only need two points to determine a line, I suggest you plot 3-- whenever possible to guard against mistakes.
The easiest solutions to find are those where the line crosses
the x-axis ( y = 0) and
the y-axis ( x = 0)
Wednesday, January 25, 2012
Algebra Honors (Period 6 & 7)
Fractional Exponents
In chapter 4 we reviewed the law of exponents:
am ⋅an = am+n
Thus you know
24⋅25= 29
What do you notice? What would be the value of n in the equation
2n⋅2n = 2
Using what we know from above,
2n⋅2n = 2n+n = 22n
The bases are equal ( and NOT -1, 0 or 1). Therefore the exponents must be equal.
That says
2n = 1
n = 1/2
and you have
21/2⋅21/2=2
Because √2⋅√2 = 2 and (-√2)(-√2) = 2 we note that 21/2 as either the positive or negative square root of 2
Selecting the positive or principal square root we define,
21/2 = √2
Radicals are not restricted to square roots. The symbol ∛ represents the third ( or cube) root, ∜ represents the fourth root and so on...
As you have learned the root index is omitted when n = 2
Just as the inverse of squaring a number is finding the square root, the inverse of cubing a number is finding the cube root. Since 23 = 8
∛8 ( read the cube root of 8) is 2.
Likewise (-2)3 = -8
∛(-8) = -2
BE CAREFUL---> While ∛-8 is a real number √-8 is not
In general, you CAN find ODD roots of negative numbers but not EVEN Roots!!
Solve
4n⋅4n⋅4n= 4
43n = 4
Since the bases are EQUAL ( that's the KEY), the exponents are also!!
so 3n = 4
n = 3/4
You know that ∛7 = 7 1/3 So How would you write (∛7) 2 in exponential form?
(∛7) 2 = (71/3)2 = 7(1/3)2 = 72/3
Simplify:
163/4
First write as
∜163
Now change 16 into 24 Why?
You end up with ∜(24)3
Looking at just ∜24 you realize you have 2
and so you are left with
23 = 8
In chapter 4 we reviewed the law of exponents:
am ⋅an = am+n
Thus you know
24⋅25= 29
What do you notice? What would be the value of n in the equation
2n⋅2n = 2
Using what we know from above,
2n⋅2n = 2n+n = 22n
The bases are equal ( and NOT -1, 0 or 1). Therefore the exponents must be equal.
That says
2n = 1
n = 1/2
and you have
21/2⋅21/2=2
Because √2⋅√2 = 2 and (-√2)(-√2) = 2 we note that 21/2 as either the positive or negative square root of 2
Selecting the positive or principal square root we define,
21/2 = √2
Radicals are not restricted to square roots. The symbol ∛ represents the third ( or cube) root, ∜ represents the fourth root and so on...
As you have learned the root index is omitted when n = 2
Just as the inverse of squaring a number is finding the square root, the inverse of cubing a number is finding the cube root. Since 23 = 8
∛8 ( read the cube root of 8) is 2.
Likewise (-2)3 = -8
∛(-8) = -2
BE CAREFUL---> While ∛-8 is a real number √-8 is not
In general, you CAN find ODD roots of negative numbers but not EVEN Roots!!
Solve
4n⋅4n⋅4n= 4
43n = 4
Since the bases are EQUAL ( that's the KEY), the exponents are also!!
so 3n = 4
n = 3/4
You know that ∛7 = 7 1/3 So How would you write (∛7) 2 in exponential form?
(∛7) 2 = (71/3)2 = 7(1/3)2 = 72/3
Simplify:
163/4
First write as
∜163
Now change 16 into 24 Why?
You end up with ∜(24)3
Looking at just ∜24 you realize you have 2
and so you are left with
23 = 8
Math 6 Honors ( Periods 1, 2, & 3)
Addition & Subtraction of Mixed Numbers 7-2
To add or subtract mixed numbers we could first change the mixed numbers to improper fractions and then use the method from 7-1 .
1 4/9 + 3 1/9 = 13/9 + 28/9 = 41/9 = 4 5/9 but that was 5th grade….
In the second method, and the one I prefer, you work separately with the fractional and whole number parts of the given mixed numbers.
STACK THEM!!
3 4/9
1 7/9
4 11/9 = 5 2/9
If the fractional parts of the given mixed numbers have different denominators, we find equivalent mixed numbers whose fractional parts have the same denominator, usually the LCD.
5 3/10 + 7 7/15
Stack
5 3/10
+7 7/15
Draw a line separating the fractional part from the whole numbers Find the LCM of the denominators the LCD and add…
9 5/9 - 4 13/15
To add or subtract mixed numbers we could first change the mixed numbers to improper fractions and then use the method from 7-1 .
1 4/9 + 3 1/9 = 13/9 + 28/9 = 41/9 = 4 5/9 but that was 5th grade….
In the second method, and the one I prefer, you work separately with the fractional and whole number parts of the given mixed numbers.
STACK THEM!!
3 4/9
1 7/9
4 11/9 = 5 2/9
If the fractional parts of the given mixed numbers have different denominators, we find equivalent mixed numbers whose fractional parts have the same denominator, usually the LCD.
5 3/10 + 7 7/15
Stack
5 3/10
+7 7/15
Draw a line separating the fractional part from the whole numbers Find the LCM of the denominators the LCD and add…
9 5/9 - 4 13/15
Tuesday, January 24, 2012
Algebra Honors (Period 6 & 7)
Simple Radical Equations 11-10
Solving equations involving radicals are solved by isolating the radical on one side of the equals sign and then squaring both sides of the equation.
140 = √2(9.8)d all under the √
140 = √19.6d
(140)2 = (√19.6d)2
19600 = 19.6d
1000=d
The solution set is {1000}
Solve
√(5x+1) + 2 = 6
√(5x+1) = 4
[√(5x+1)]2 = (4)2
5x + 1 = 16
5x = 15
x = 3
The solution set is {3}
When you square both sides of an equation, the new equation may NOT be equivalent to the original equation Therefore, you must CHECK EVERY POSSIBLE ROOT IN THE ORIGINAL EQUATION to see whether it is indeeed a root.
Solve
√(11x2 -63) - 2x = 0
√(11x2 -63) = 2x
√(11x2 -63)2 = (2x)2
11x2 -63 = 4x2
7x2 = 63
x2 = 9
x = ± 3
Now we need to check for BOTH + 3 and - 3
Rewrite the original equation
√(11x2 -63) - 2x = 0
√(11(3)2 -63) - 2(3) = 0
√99-63 - 6 = 0
√36 - 6 = 0
6-6 = 0
That's true
Now for x = -3
√(11(-3)2 -63) - 2(-3) = 0
√(99 -63) + 6 = 0
√36 + 6 = 0
12 ≠ 0
So -3 is NOT a solution
Solving equations involving radicals are solved by isolating the radical on one side of the equals sign and then squaring both sides of the equation.
140 = √2(9.8)d all under the √
140 = √19.6d
(140)2 = (√19.6d)2
19600 = 19.6d
1000=d
The solution set is {1000}
Solve
√(5x+1) + 2 = 6
√(5x+1) = 4
[√(5x+1)]2 = (4)2
5x + 1 = 16
5x = 15
x = 3
The solution set is {3}
When you square both sides of an equation, the new equation may NOT be equivalent to the original equation Therefore, you must CHECK EVERY POSSIBLE ROOT IN THE ORIGINAL EQUATION to see whether it is indeeed a root.
Solve
√(11x2 -63) - 2x = 0
√(11x2 -63) = 2x
√(11x2 -63)2 = (2x)2
11x2 -63 = 4x2
7x2 = 63
x2 = 9
x = ± 3
Now we need to check for BOTH + 3 and - 3
Rewrite the original equation
√(11x2 -63) - 2x = 0
√(11(3)2 -63) - 2(3) = 0
√99-63 - 6 = 0
√36 - 6 = 0
6-6 = 0
That's true
Now for x = -3
√(11(-3)2 -63) - 2(-3) = 0
√(99 -63) + 6 = 0
√36 + 6 = 0
12 ≠ 0
So -3 is NOT a solution
Math 6 Honors ( Periods 1, 2, & 3)
Addition and Subtraction of Fractions 7-1
Most of you already know how to add and subtract fractions, although some of you may need just a little review.
5/9 + 2/9 = 7/9
13/12 - 5/12 = 8/12 = 2/3
and that
7/9 – 2/9 = 5/9
13/12 - 5/12 = 8/12 = 2/3
a/c + b/c = (a +b)/c where c does not equal 0
a/c - b/c = (a -b)/c
The properties of addition and subtraction of whole numbers also apply to fractions.
If the denominators are the same— add or subtract the numerators AND use the numerator!!
In order to add two fractions with different denominators, we first find two fractions, with a common denominator, equivalent to the given fractions. Then add these two fractions.
The most convenient denominator to use as a common denominator is the least common denominator of LCD, of the two fractions. That is, the least common multiple of the two denominators.
LCD ( a/b, c/d) = LCM(b, d) where b and d both cannot be equal to 0
For example LCD ( 3/4, 5/6) = LCM(4,6) =12
3/4 = 9/12 and 5/6 = 10/12
Let’s do:
7/15 + 8/9
First find the LCD
LCM(15, 9) Do your factor trees or inverted division – or just by knowing!!
15 = 3• 5
9 = 32
So LCM(15,9) = [every factor to its greatest power] 32•5 = 45
Then find equivalent factions with a LCD of 45, and add
7/15 = 21/45
8/9 = 40/45
21/45 + 40/45 = 61/45 = 1 16/45
5/6- 11/24
Stack them and use the LCD
5/6 = 20/24
-11/24 = -11/24
9/24 = 3/8
7/12 + 4/9 + 3/4
several strategies ca be used. You can find the LCD for all three you can use the C+ and the A+
and change it to
(7/12 + 3/4) + 4/9
then add the first two factions
7/12 + 3/4 becomes 7/12 + 9/12 = 16/12 = 4/3
then add 4/3 + 4/9
change 4/3 to 12/9
12/9 + 4/9 = 16/9 = 1 7/9
What about 17/10 - ( 3/5 + 5/6)
You must do the parenthesis first
so 3/5 + 5/6
3/5 = 18/30
5/6 = 25/30
43/30
Now you have
17/10 - 43/30
stack those
17/10 = 51/30
51/30
-43/30
8/30 = 4/15
n + 1/2 = 5/6
you need to isolate the variable so add 1/2 to BOTH sides of the equation
n + 1/2 = 5/6
-1/2 = -1/2 Change 1/2 to 3/6 and subtract carefully
n = 2/6
n = 1/3
make sure to box your answer
Most of you already know how to add and subtract fractions, although some of you may need just a little review.
5/9 + 2/9 = 7/9
13/12 - 5/12 = 8/12 = 2/3
and that
7/9 – 2/9 = 5/9
13/12 - 5/12 = 8/12 = 2/3
a/c + b/c = (a +b)/c where c does not equal 0
a/c - b/c = (a -b)/c
The properties of addition and subtraction of whole numbers also apply to fractions.
If the denominators are the same— add or subtract the numerators AND use the numerator!!
In order to add two fractions with different denominators, we first find two fractions, with a common denominator, equivalent to the given fractions. Then add these two fractions.
The most convenient denominator to use as a common denominator is the least common denominator of LCD, of the two fractions. That is, the least common multiple of the two denominators.
LCD ( a/b, c/d) = LCM(b, d) where b and d both cannot be equal to 0
For example LCD ( 3/4, 5/6) = LCM(4,6) =12
3/4 = 9/12 and 5/6 = 10/12
Let’s do:
7/15 + 8/9
First find the LCD
LCM(15, 9) Do your factor trees or inverted division – or just by knowing!!
15 = 3• 5
9 = 32
So LCM(15,9) = [every factor to its greatest power] 32•5 = 45
Then find equivalent factions with a LCD of 45, and add
7/15 = 21/45
8/9 = 40/45
21/45 + 40/45 = 61/45 = 1 16/45
5/6- 11/24
Stack them and use the LCD
5/6 = 20/24
-11/24 = -11/24
9/24 = 3/8
7/12 + 4/9 + 3/4
several strategies ca be used. You can find the LCD for all three you can use the C+ and the A+
and change it to
(7/12 + 3/4) + 4/9
then add the first two factions
7/12 + 3/4 becomes 7/12 + 9/12 = 16/12 = 4/3
then add 4/3 + 4/9
change 4/3 to 12/9
12/9 + 4/9 = 16/9 = 1 7/9
What about 17/10 - ( 3/5 + 5/6)
You must do the parenthesis first
so 3/5 + 5/6
3/5 = 18/30
5/6 = 25/30
43/30
Now you have
17/10 - 43/30
stack those
17/10 = 51/30
51/30
-43/30
8/30 = 4/15
n + 1/2 = 5/6
you need to isolate the variable so add 1/2 to BOTH sides of the equation
n + 1/2 = 5/6
-1/2 = -1/2 Change 1/2 to 3/6 and subtract carefully
n = 2/6
n = 1/3
make sure to box your answer
Monday, January 23, 2012
Algebra Honors (Period 6 & 7)
Multiplication of Binomials Containing Radicals 11-9
Chapter 5 taught us how to multiply binomials-- we can use those methods when multiplying binomials that contain square root radicals.
(6 + √11)(6 - √11)
The pattern is
(a +b)(a -b) = a2 - b2
so using that we get
62 - (√11)2
36 - 11 = 25
Simplify (3 + √5)2
The pattern here is
(a + b)2 = a2 + 2ab + b2
so ( 3 + √5)2 =
32 + 2[(3)(√5)] + (√5)2 =
9 + 6√5 + 5 =
14 + 6√5
Simplify (2 √3 - 5√7)2
The pattern here is (a - b)2 = a2 - 2ab + b2
(2 √3 - 5√7)2 =
(2 √3)2 -2[(2)(5)(√3)(√7)] +(5√7)2 =
4(3) -20√21 +25(7) =
12 -20√21+ 175 =
187 -20√21
If both b and d are nonnegative, then the binomials
a√b + c√d AND a√b - c√d are called conjugates of one another. COnjugates differ ONLY in the sign of one term
if a, b, c, and d are all integers then the product (a√b + c√d)(a√b - c√d) will be an integer... see the first example!!
Conjugates can be used to rationalize binomial denominators that contain radicals.. getting rid of the radicals in the denominator
Rationalize
3/(5- 2√7)
3/(5- 2√7) = [ 3/(5- 2√7)] × [((5+ 2√7)/(5+2√7)]
This doesn't show well here hopefully you can remember what was done in class...
= 3(5 +2√7)/25-(2√7)2 =
(15+6√7)/25-28 =
(15+6√7)/-3 =
15/-3 +6√7/-3 =
-5 -2√7
√√
Chapter 5 taught us how to multiply binomials-- we can use those methods when multiplying binomials that contain square root radicals.
(6 + √11)(6 - √11)
The pattern is
(a +b)(a -b) = a2 - b2
so using that we get
62 - (√11)2
36 - 11 = 25
Simplify (3 + √5)2
The pattern here is
(a + b)2 = a2 + 2ab + b2
so ( 3 + √5)2 =
32 + 2[(3)(√5)] + (√5)2 =
9 + 6√5 + 5 =
14 + 6√5
Simplify (2 √3 - 5√7)2
The pattern here is (a - b)2 = a2 - 2ab + b2
(2 √3 - 5√7)2 =
(2 √3)2 -2[(2)(5)(√3)(√7)] +(5√7)2 =
4(3) -20√21 +25(7) =
12 -20√21+ 175 =
187 -20√21
If both b and d are nonnegative, then the binomials
a√b + c√d AND a√b - c√d are called conjugates of one another. COnjugates differ ONLY in the sign of one term
if a, b, c, and d are all integers then the product (a√b + c√d)(a√b - c√d) will be an integer... see the first example!!
Conjugates can be used to rationalize binomial denominators that contain radicals.. getting rid of the radicals in the denominator
Rationalize
3/(5- 2√7)
3/(5- 2√7) = [ 3/(5- 2√7)] × [((5+ 2√7)/(5+2√7)]
This doesn't show well here hopefully you can remember what was done in class...
= 3(5 +2√7)/25-(2√7)2 =
(15+6√7)/25-28 =
(15+6√7)/-3 =
15/-3 +6√7/-3 =
-5 -2√7
√√
Tuesday, January 17, 2012
Algebra Honors (Period 6 & 7)
Adding and Subtracting Radicals 11-8
You can use the Distributive Property to simplify the sums of radicals like
4√7 + 5√7 = (4 +5)√7 = 9√7
UNLIKE radicands CANNOT be combined.
3√6-2√13+5√6
combine the LIKE radicands
8√6 -2√13
Express each radical in simplest form and you can sometimes combine terms in sums and differences of radicals
7√3- 4√6 + 2√48 -6√54 =
7√3 - 4√6 + 2√(16⋅3) -6√(9⋅6) =
7√3 - 4√6 + 8√3 - 18√6 =
15√3 -22√6
To simplify sums or differences of square root radicals
Express each radical in simplest form
Use the DP to add or subtract radicals with LIKE RADICANDS
You can use the Distributive Property to simplify the sums of radicals like
4√7 + 5√7 = (4 +5)√7 = 9√7
UNLIKE radicands CANNOT be combined.
3√6-2√13+5√6
combine the LIKE radicands
8√6 -2√13
Express each radical in simplest form and you can sometimes combine terms in sums and differences of radicals
7√3- 4√6 + 2√48 -6√54 =
7√3 - 4√6 + 2√(16⋅3) -6√(9⋅6) =
7√3 - 4√6 + 8√3 - 18√6 =
15√3 -22√6
To simplify sums or differences of square root radicals
Express each radical in simplest form
Use the DP to add or subtract radicals with LIKE RADICANDS
Tuesday, January 10, 2012
Math 6 Honors ( Periods 1, 2, & 3)
Changing a Fraction to a Decimal 6-5
There are two methods that can be used to change a fraction to a decimal.
1) find an equivalent fraction whose denominator is a power of 10. ( this method does not always work but when it does it becomes really easy to change to a decimal)
13/25 multiply the numerator and the denominator by 4 to get 52/100 and then just close your eyes and see Chapter 3... and 0.52
2) divide the numerator by the denominator. It's a great way to determine your score out of 100 and then figure out your percent.
If you got 67/75 on the last test
divide 67 by 75 carefully 0.8933333... you earned a B+
take 3/8 and divide 3 by 8 8 goes into 3 0.375 times
so 3/8 = 0.375
If the numerator is smaller than the denominator we know our number must be between 0 and 1--> it must be a decimal.
When the remainder is 0 as in the case of dividing 3 by 8, it is called a terminating decimal.
By examining a fraction in lowest terms, we can determine whether the fraction can be expressed as a terminating decimal.
If the denominator has no prime factors other than 2 or 5, the decimal representation will terminate.
7/40
looking at 40 we notice the prime factorization ( oh no, it's Chapter 5)
40 = 23·5 Since the only prime factors are 2 and 5
7/40 must terminate.
What about 5/12 ?
12 = 22· 3 Since 3 is a prime factor of the denominator, the fraction cannot be expressed as a terminating decimal.
What about 9/12 ? At first it looks the same as the one above, but look carefully and realize 9/ 12 = 3/4
Since 4 = 22 , 4 has no other prime factors except 2, this can be expressed as a terminating decimal.
Let's look at 15/22
Since 22 has the prime factor of 11 we know that this will not terminate. In fact when you divide 15 by 22 you end up with 0.681818181...
We write this as ( Please check page 196) Notice that the bar is only over the 81 and represents a block numbers that continues to repeat indefinitely and is called a repeating decimal.
EVERY FRACTION CAN BE EXPRESSED AS EITHER A TERMINATING DECIMAL OR A REPEATING DECIMAL.
Let's look at 4/9 = 0.4444444....
5/9 =
7/9 =
31/99 = 0.3131313131...
8/ 11 = 72/99 = .72727272...
There are two methods that can be used to change a fraction to a decimal.
1) find an equivalent fraction whose denominator is a power of 10. ( this method does not always work but when it does it becomes really easy to change to a decimal)
13/25 multiply the numerator and the denominator by 4 to get 52/100 and then just close your eyes and see Chapter 3... and 0.52
2) divide the numerator by the denominator. It's a great way to determine your score out of 100 and then figure out your percent.
If you got 67/75 on the last test
divide 67 by 75 carefully 0.8933333... you earned a B+
take 3/8 and divide 3 by 8 8 goes into 3 0.375 times
so 3/8 = 0.375
If the numerator is smaller than the denominator we know our number must be between 0 and 1--> it must be a decimal.
When the remainder is 0 as in the case of dividing 3 by 8, it is called a terminating decimal.
By examining a fraction in lowest terms, we can determine whether the fraction can be expressed as a terminating decimal.
If the denominator has no prime factors other than 2 or 5, the decimal representation will terminate.
7/40
looking at 40 we notice the prime factorization ( oh no, it's Chapter 5)
40 = 23·5 Since the only prime factors are 2 and 5
7/40 must terminate.
What about 5/12 ?
12 = 22· 3 Since 3 is a prime factor of the denominator, the fraction cannot be expressed as a terminating decimal.
What about 9/12 ? At first it looks the same as the one above, but look carefully and realize 9/ 12 = 3/4
Since 4 = 22 , 4 has no other prime factors except 2, this can be expressed as a terminating decimal.
Let's look at 15/22
Since 22 has the prime factor of 11 we know that this will not terminate. In fact when you divide 15 by 22 you end up with 0.681818181...
We write this as ( Please check page 196) Notice that the bar is only over the 81 and represents a block numbers that continues to repeat indefinitely and is called a repeating decimal.
EVERY FRACTION CAN BE EXPRESSED AS EITHER A TERMINATING DECIMAL OR A REPEATING DECIMAL.
Let's look at 4/9 = 0.4444444....
5/9 =
7/9 =
31/99 = 0.3131313131...
8/ 11 = 72/99 = .72727272...
Monday, January 9, 2012
Math 6 Honors ( Periods 1, 2, & 3)
Comparing Fractions 6-4
When two fractions have equal denominators it is easy to tell which of the fractions is greater.
We simply compare their numerators.
3/11 < 5/11 since 3< 5 If the fractions have different denominators, there are a variety of methods to consider. We could find a common denominator, which we will need to do when we add or subtract fractions... but when comparing let's try other methods... Take 2/3 and 4/5 Comparing Fractions
Or compare 5/6 and 7/9
again this time you would multiply
5(9) = 45 and 7(6) = 42
so 5/6 > 7/9
Also if the numerator is the same
2/3, 2/7, 2/9, 2/11, 2/21, 2/35
The larger the denominator the smaller the fractions so to list in order from least to greatest start with the largest number in the denominator!!
and if you have fractions with the numerator just one away from the denominator
such as 3/4, 5/6, 7/8, 9/10, 23/24, 45/46
the smallest fraction will be the one with the smallest numbers
3/4 is the smallest fraction and that list is in order from least to greatest!!
What if you need to name a fraction between two fraction 1/6 and 3/8
you could find the LCD
1/6 = 4/24 and
3/8 = 9/24
so you could state
5/24, 6/24 ( but that is really 1/4), 7/24, or 8/24 ( but that is really 1/3.
There are actually an infinite number of fractions... these are only 4 of them
What if you need to find a fraction between 3/7 and 4/7
sometimes you need to change the denominators just to realize that there really are other fractions between
for instance, 3/7 = 6/14 and 4/7 = 8/ 14 so doesn't 7/14 ( or actually 1/2) work!!
... and that's just one of the fractions!!
There were 10 sunny days in February ( 28 total days)
12 sunny days in November (30 total days)
Which month has higher fraction of sunny days?
10/28 or 12/30
First simplify each fraction... then compare
10/28 = 5/14 and 12/30 = 2/5
Now use one oc the methods to compare
Cross products works really well and you discover
NOVEMBER is the month with the higher fraction of sunny days
The Bears won 11 out of 16 games.
The Eagles won 17 out of 24 games.
Which team won a great fraction of their games?
11/16 or 17/24
Again, I would use cross multiplication and we discovered that THE EAGLES won a greater fraction of their games.
If n > 0
Then
if a < b a/n < b/n Think about this one!! Plug in some numbers and see what happens and if a < b, then n/a > n/b
Again, plug in some numbers and see what happens!!
If a/b and c/d are fractions and if ad > bc, which fraction is greater,
a/b or c/d ?
Post your answer below in the comments for extra credit. Make sure to give your reasoning for your answer.
Ordering or comparing fractions:
Different ways:
I. Benchmarks - 0, 1/4, 1/2, 3/4, and 1 (using your gut feeling)
How do you figure out which benchmark to use?
When the numerator is close to the denominator, the fraction is approaching 1
(Ex: 9/11)
When you double the numerator and it's close to the denominator, the fraction is close to 1/2 (Ex: 4/9)
When the numerator is very far from the denominator, the fraction is approaching zero (Ex: 1/8)
Also, if one number is improper or mixed number and other is a proper fraction, then obviously the number greater than 1 will be bigger!
II. LCD - give them all the same denominator using the LCM as the LCD
III. Use cross multiplication when comparing two... do it several times when comparing a list of fractions
IV. Change them to decimals ( works well if you are great at decimals-- but I want you to become GREAT at fractions!!)
When two fractions have equal denominators it is easy to tell which of the fractions is greater.
We simply compare their numerators.
3/11 < 5/11 since 3< 5 If the fractions have different denominators, there are a variety of methods to consider. We could find a common denominator, which we will need to do when we add or subtract fractions... but when comparing let's try other methods... Take 2/3 and 4/5 Comparing Fractions
Or compare 5/6 and 7/9
again this time you would multiply
5(9) = 45 and 7(6) = 42
so 5/6 > 7/9
Also if the numerator is the same
2/3, 2/7, 2/9, 2/11, 2/21, 2/35
The larger the denominator the smaller the fractions so to list in order from least to greatest start with the largest number in the denominator!!
and if you have fractions with the numerator just one away from the denominator
such as 3/4, 5/6, 7/8, 9/10, 23/24, 45/46
the smallest fraction will be the one with the smallest numbers
3/4 is the smallest fraction and that list is in order from least to greatest!!
What if you need to name a fraction between two fraction 1/6 and 3/8
you could find the LCD
1/6 = 4/24 and
3/8 = 9/24
so you could state
5/24, 6/24 ( but that is really 1/4), 7/24, or 8/24 ( but that is really 1/3.
There are actually an infinite number of fractions... these are only 4 of them
What if you need to find a fraction between 3/7 and 4/7
sometimes you need to change the denominators just to realize that there really are other fractions between
for instance, 3/7 = 6/14 and 4/7 = 8/ 14 so doesn't 7/14 ( or actually 1/2) work!!
... and that's just one of the fractions!!
There were 10 sunny days in February ( 28 total days)
12 sunny days in November (30 total days)
Which month has higher fraction of sunny days?
10/28 or 12/30
First simplify each fraction... then compare
10/28 = 5/14 and 12/30 = 2/5
Now use one oc the methods to compare
Cross products works really well and you discover
NOVEMBER is the month with the higher fraction of sunny days
The Bears won 11 out of 16 games.
The Eagles won 17 out of 24 games.
Which team won a great fraction of their games?
11/16 or 17/24
Again, I would use cross multiplication and we discovered that THE EAGLES won a greater fraction of their games.
If n > 0
Then
if a < b a/n < b/n Think about this one!! Plug in some numbers and see what happens and if a < b, then n/a > n/b
Again, plug in some numbers and see what happens!!
If a/b and c/d are fractions and if ad > bc, which fraction is greater,
a/b or c/d ?
Post your answer below in the comments for extra credit. Make sure to give your reasoning for your answer.
Ordering or comparing fractions:
Different ways:
I. Benchmarks - 0, 1/4, 1/2, 3/4, and 1 (using your gut feeling)
How do you figure out which benchmark to use?
When the numerator is close to the denominator, the fraction is approaching 1
(Ex: 9/11)
When you double the numerator and it's close to the denominator, the fraction is close to 1/2 (Ex: 4/9)
When the numerator is very far from the denominator, the fraction is approaching zero (Ex: 1/8)
Also, if one number is improper or mixed number and other is a proper fraction, then obviously the number greater than 1 will be bigger!
II. LCD - give them all the same denominator using the LCM as the LCD
III. Use cross multiplication when comparing two... do it several times when comparing a list of fractions
IV. Change them to decimals ( works well if you are great at decimals-- but I want you to become GREAT at fractions!!)
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