Clearing An Equation of Fractions or Decimals 3-6
You can do equations with fractions or decimals as they are, but many
students find it easier to "get rid of" them!
How? Just use the multiplicative property of equality! (your old friend!)
DECIMALS:
16.3 - 7.2y = -8.18
Multiply by the power of 10 needed to clear ALL decimals!
In the problem above, you would need to multiply by 100 to make -8.18 an integer
100 (16.3 - 7.2y) = 100 (-8.18)
USE THE DISTRIBUTIVE PROPERTY ON THE LEFT SIDE OF THE EQUATION
1630 - 720y = -818
SUBTRACT 1630 FROM EACH SIDE (or add -1630)
1630 - 1630 -720y = -818 - 1630
-720y = -2448
DIVIDE EACH SIDE BY -720
-720y/-720 = -2448/-720
y = 3.4
ALWAYS CHECK YOUR ANSWER!!!!!
Why do students get this wrong?
1)
1) They forget to multiply the integers by the power of 10 because it doesn't have any decimal places to "get rid of".
Remember: to stay in balance, you must distribute to EVERY term.
2) They distribute only the power of 10 that each term needs.
For example, if the problem has .3y = 2.85, you will need to multiply by 100 EQUALLY on both sides to stay in balance, but students will end up with 3y = 285.
THIS IS WRONG!!!
YOU HAVE MULTIPLIED BY 10 ON LEFT SIDE AND 100 ON RIGHT SIDE!
YOU ARE OUT OF BALANCE!!!
FRACTIONS:
Are you "Fraction Phobic"????
Then you're going to love this!!! :)
2/3y + 1/2y = 5/6 + 2y
Instead of finding a common denominator and using fractions, we're going to...
MULTIPLY BY THE LCM OF ALL THE DENOMINATORS
In the problem above, the LCM of 3, 2, and 6 is 6
6 (2/3y + 1/2y) = 6 (5/6 + 2y)
USE THE DISTRIBUTIVE PROPERTY ON BOTH SIDES OF THE EQUATION
6(2/3y) + 6(1/2y) = 6(5/6) + 6(2y)
4y + 3y = 5 + 12y
COMBINE LIKE TERMS ON THE LEFT SIDE OF THE EQUATION
7y = 5 + 12y
SUBTRACT 12y FROM BOTH SIDES TO GET ALL VARIABLES ON ONE SIDE
7y - 12y = 5 + 12y - 12y
-5y = 5
DIVIDE BOTH SIDES BY -5
-5y/-5 = 5/-5
y = -1
ALWAYS CHECK YOUR ANSWER!!!!!
Why do students get this wrong?
1)They forget to multiply the integers by the LCM because it doesn't have any denominator to "get rid of".
Remember: to stay in balance, you must distribute to EVERY term.
2) They distribute only the number that each denominator needs. Again, you're out of balance.
3) They multiply by the wrong LCM...you'll end up still having a denominator.
You have not achieved your objective under this method! Your objective is to get rid of every denominator!
Couldn't I just solve these problems keeping the decimals and fractions?
Yes, you could, but I want you to learn this method because you might like it!
Also, you will to know how to get rid of the denominators in Chapter 10 when you have very complicated denominators with variables.
Friday, September 25, 2009
Math 6H Period 3, 6 & 7
Writing Inequalities 2-3
2 < 7 and 7 > 2 are two inequalities that state the relationship between the numbers 2 and 7
2 < 7 reads 2 is the less than 7
7 > 2 reads 7 is greater than 2
The symbols < and > are called inequality symbols.
Notice the mathematical sentence ( inequality)
Two is less than seven or 2 <7
is different from the mathematical phrase ( expression)
Two less than seven. 7-2
The point of the number line that is paired with a number is called the graph of that number.
Check out the graph on page 39 of our textbook. When you graph numbers on the number line, make sure to place a dot DIRECTLY ON the number line at that particular number's location. Again, check out our textbook for examples!!
Looking at the graph of numbers, we see that the larger number will be to the right of the smaller number.
A number n is between 6 and 12 would be 6 < n < 12 or 12 > n > 6
Notice the subtle differences
in the sentence
Six is greater than a number t
and the phrase
six greater than a number t
Six is greater than a number t becomes 6 > t
while
six greater than a number t becomes t + 6
What about the following inequality:
A number p is greater than a number q
is p > q
The value in cents of d dimes is less than the value in cents of n nickles.
If you need to-- set up your T-charts ( refer to your class notes) one for dimes and the other for nickles.
10d represents the number of dimes and 5n represents the number of nickles
so 10d < 5n
2 < 7 and 7 > 2 are two inequalities that state the relationship between the numbers 2 and 7
2 < 7 reads 2 is the less than 7
7 > 2 reads 7 is greater than 2
The symbols < and > are called inequality symbols.
Notice the mathematical sentence ( inequality)
Two is less than seven or 2 <7
is different from the mathematical phrase ( expression)
Two less than seven. 7-2
The point of the number line that is paired with a number is called the graph of that number.
Check out the graph on page 39 of our textbook. When you graph numbers on the number line, make sure to place a dot DIRECTLY ON the number line at that particular number's location. Again, check out our textbook for examples!!
Looking at the graph of numbers, we see that the larger number will be to the right of the smaller number.
A number n is between 6 and 12 would be 6 < n < 12 or 12 > n > 6
Notice the subtle differences
in the sentence
Six is greater than a number t
and the phrase
six greater than a number t
Six is greater than a number t becomes 6 > t
while
six greater than a number t becomes t + 6
What about the following inequality:
A number p is greater than a number q
is p > q
The value in cents of d dimes is less than the value in cents of n nickles.
If you need to-- set up your T-charts ( refer to your class notes) one for dimes and the other for nickles.
10d represents the number of dimes and 5n represents the number of nickles
so 10d < 5n
Wednesday, September 23, 2009
Algebra Period 4
EQUATIONS 3-3 continued TWO STEPS WITH DISTRIBUTIVE PROPERTY
Usually, you want to do DISTRIBUTE FIRST! UNLESS THE FACTOR OUTSIDE THE ( ) CAN BE DIVIDED OUT OF BOTH SIDES PERFECTLY!!!!
EXAMPLE: 5y - 2(2y + 8) = 16 5y - 4y - 16 = 16 [distribute] y - 16 = 16 [collect like terms] y = 32 [solve by adding 16 to both sides] EXAMPLE: -3(4 + 3x) = -9
[Don't distribute! Divide by -3. 4 + 3x = 3
The -3 goes into both sides perfectly!) 3x = -1 [Subtract 4 from both sides] x = -1/3 [Divide both sides by 3] Chapter 3-5:
TWO STEPS WITH VARIABLES ON BOTH SIDES OF EQUATIONS
simplify each side of the equation first
Then use the ADDITIVE INVERSE PROPERTY to move variable to the other side
Usually, we try to move the smaller coefficient to the larger because sometimes that avoids negative coefficients
But that is not always the case, and you may move whatever side you choose. EXAMPLE: 3y - 10 - y = -10y + 12 2y - 10 = -10y + 12 +10y +10y 12y - 10 = 12 + 10 +10 12y = 22 12 12 y = 11/6
Usually, you want to do DISTRIBUTE FIRST! UNLESS THE FACTOR OUTSIDE THE ( ) CAN BE DIVIDED OUT OF BOTH SIDES PERFECTLY!!!!
EXAMPLE: 5y - 2(2y + 8) = 16 5y - 4y - 16 = 16 [distribute] y - 16 = 16 [collect like terms] y = 32 [solve by adding 16 to both sides] EXAMPLE: -3(4 + 3x) = -9
[Don't distribute! Divide by -3. 4 + 3x = 3
The -3 goes into both sides perfectly!) 3x = -1 [Subtract 4 from both sides] x = -1/3 [Divide both sides by 3] Chapter 3-5:
TWO STEPS WITH VARIABLES ON BOTH SIDES OF EQUATIONS
simplify each side of the equation first
Then use the ADDITIVE INVERSE PROPERTY to move variable to the other side
Usually, we try to move the smaller coefficient to the larger because sometimes that avoids negative coefficients
But that is not always the case, and you may move whatever side you choose. EXAMPLE: 3y - 10 - y = -10y + 12 2y - 10 = -10y + 12 +10y +10y 12y - 10 = 12 + 10 +10 12y = 22 12 12 y = 11/6
Tuesday, September 22, 2009
Math 6H Period 3, 6 & 7
Writing Mathematical Equations 2-2
The process of writing equations really is just writing two equal expressions and joining them by an equals sign. The words "is" "equals" or "equal to" all indicate that two phrases NAME the same number.
The equals sign is the VERB in a mathematical sentence-- without it you have a mathematical expression!!
Eight increased by a number x is equal to thirty-seven.
In translating this mathematical sentence, I always start but placing the equals sign directly under the words "is equal"
so my first step would be
Eight increased by a number x is equal to thirty-seven.
---------------------------------> = <------------------
Then I would translate each mathematical phrase separately.
Yes, thirty-seven is a mathematical phrase!!
8 + x = 37
Ten is two less than a number n
10 = n - 2
Twice a number w equals the sum of the number and four
2w = n + 4
Notice that when you are indicating multiplication the number (coefficient) ALWAYS is placed in front of the variable.
So three times a number b would be 3b
The only time you see the letter first-- is when you are looking for our ROOM-- which is P8... a mathematician did not label the room numbers!! :)
Sometimes we need to write an equation for a word sentence that involves measurements. MAKE SURE that each side of the equation uses the SAME UNIT of Measurement!!
For example. Write an equation for: The value of d dimes is $27.50
WE know that the value of d dimes is 10d cents( from our previous lesson) .. but $27.50 is in terms of dollars so we need to change it to cents . $27.50 is 2750 cents ... So our equation becomes 10d = 2750.
The process of writing equations really is just writing two equal expressions and joining them by an equals sign. The words "is" "equals" or "equal to" all indicate that two phrases NAME the same number.
The equals sign is the VERB in a mathematical sentence-- without it you have a mathematical expression!!
Eight increased by a number x is equal to thirty-seven.
In translating this mathematical sentence, I always start but placing the equals sign directly under the words "is equal"
so my first step would be
Eight increased by a number x is equal to thirty-seven.
---------------------------------> = <------------------
Then I would translate each mathematical phrase separately.
Yes, thirty-seven is a mathematical phrase!!
8 + x = 37
Ten is two less than a number n
10 = n - 2
Twice a number w equals the sum of the number and four
2w = n + 4
Notice that when you are indicating multiplication the number (coefficient) ALWAYS is placed in front of the variable.
So three times a number b would be 3b
The only time you see the letter first-- is when you are looking for our ROOM-- which is P8... a mathematician did not label the room numbers!! :)
Sometimes we need to write an equation for a word sentence that involves measurements. MAKE SURE that each side of the equation uses the SAME UNIT of Measurement!!
For example. Write an equation for: The value of d dimes is $27.50
WE know that the value of d dimes is 10d cents( from our previous lesson) .. but $27.50 is in terms of dollars so we need to change it to cents . $27.50 is 2750 cents ... So our equation becomes 10d = 2750.
Monday, September 21, 2009
Pre Algebra Period 1
Multiplying & Dividing Integers 1-9
They have the SAME rules!!!!
Math Book Rules:
If you have 2 signs that are the SAME -
answer is POSITIVE
If you have 2 signs that are DIFFERENT -
answer is NEGATIVE
Good Guy/Bad Guy Rules:
GOOD thing happens to GOOD person= GOOD = POSITIVE
BAD thing happens to BAD person = GOOD = POSITIVE
(they got what they deserved!)
GOOD thing happens to a BAD person = BAD = NEGATIVE
(we hate when good things happen to people who don't deserve it!)
BAD thing happens to a GOOD person = BAD = NEGATIVE
(we hate when that happens because it's so UNFAIR!)
Finger Rules:
If your index finger represents the negative sign, then if you have 2 negatives, you have the index fingers of both your left hand and your right hand and they make a plus sign!
If you just have one negative, it just stays negative because you don't have another finger to cross it!
If you have more than 2 negatives, you just keep using your index fingers to determine the sign.
We did this in class. It's fun! But once you get it, you won't need to keep doing it! (unless you want to keep having fun!!!)
WHAT IF THERE IS MORE THAN 2 SIGNS?
Use Aunt Sally Rules and go left to right
or
BE A SIGN COUNTER:
an ODD number of NEGATIVES = NEGATIVE
an EVEN number of NEGATIVES = POSITIVE
EXAMPLE: (-2) (-5) (-3) = -30 (odd number of negatives)
(2) (-5) (-3) = +30 (even number of negatives)
(2) (5) (-3) = -30 (odd number of negatives)
Averages ( or the Mean): Adding up all the numbers and dividing by the number of numbers
They have the SAME rules!!!!
Math Book Rules:
If you have 2 signs that are the SAME -
answer is POSITIVE
If you have 2 signs that are DIFFERENT -
answer is NEGATIVE
Good Guy/Bad Guy Rules:
GOOD thing happens to GOOD person= GOOD = POSITIVE
BAD thing happens to BAD person = GOOD = POSITIVE
(they got what they deserved!)
GOOD thing happens to a BAD person = BAD = NEGATIVE
(we hate when good things happen to people who don't deserve it!)
BAD thing happens to a GOOD person = BAD = NEGATIVE
(we hate when that happens because it's so UNFAIR!)
Finger Rules:
If your index finger represents the negative sign, then if you have 2 negatives, you have the index fingers of both your left hand and your right hand and they make a plus sign!
If you just have one negative, it just stays negative because you don't have another finger to cross it!
If you have more than 2 negatives, you just keep using your index fingers to determine the sign.
We did this in class. It's fun! But once you get it, you won't need to keep doing it! (unless you want to keep having fun!!!)
WHAT IF THERE IS MORE THAN 2 SIGNS?
Use Aunt Sally Rules and go left to right
or
BE A SIGN COUNTER:
an ODD number of NEGATIVES = NEGATIVE
an EVEN number of NEGATIVES = POSITIVE
EXAMPLE: (-2) (-5) (-3) = -30 (odd number of negatives)
(2) (-5) (-3) = +30 (even number of negatives)
(2) (5) (-3) = -30 (odd number of negatives)
Averages ( or the Mean): Adding up all the numbers and dividing by the number of numbers
Algebra Period 4
EQUATIONS Sections 3-1 to 3-3
We'll be using our EQUATION BALANCING PROPERTIES OF EQUALITY:
There are 4 of these.
Whatever YOU DO TO BALANCE an equation,
that operation is the property of equality that was used.
So if you have x + 3 = 10, you used the SUBTRACTION PROPERTY OF EQUALITY because you need to SUBTRACT 3 from each side equally. "- prop ="
If you have x - 3 = 10, you used the ADDITION PROPERTY OF EQUALITY because you need to ADD 3 from each side equally. " + prop ="
If you have 3x = 10, you used the DIVISION PROPERTY OF EQUALITY because you need to DIVIDE each side equally by 3. " / prop ="
If you have x/3 = 10, you used the MULTIPLICATION PROPERTY OF EQUALITY because you need to MULTIPLY each side equally by 3. "x prop ="
SOMETIMES, WE SAY THERE ARE ONLY 2 BALANCING PROPERTIES OF EQUALITY
CAN YOU GUESS WHICH 2 ARE "DROPPED OUT"?
Since we say we never subtract and we really never divide, it's those 2.
GOING BACK TO OUR PREVIOUS EXAMPLES:
If you have x + 3 = 10, you could say that we ADDED -3 to each side equally; therefore, we used the ADDITION (not subtraction) PROPERTY.
If you have 3x = 10, you could say that we MULTIPLIED each side equally by 1/3; therefore, we used the MULTIPLICATION (not division) PROPERTY.
(We always multiply by the MULTIPLICATIVE INVERSE).
REVIEW OF SIMPLE EQUATIONS: One and Two Step.
What's the GOAL? Determine the value of the variable
HOW? Isolate the variable (get it alone on one side of equation)
WHAT to do? Use inverse (opposite) operations to "get rid" of everything on the side with the variable
WHAT SHOULD MY FOCUS BE WHEN EQUATIONS GET COMPLICATED? Always focus on the variable(s) first!!!!!!!
TWO STEP EQUATIONS:
1. Use the ADDITION/SUBTRACTION PROPERTIES OF EQUALITY first (get rid of addition or subtraction)
2. Use the MULTIPLICATION/DIVISION PROPERTIES OF EQUALITY second (get rid of mult/division)
IDENTITY PROPERTIES AND INVERSE PROPERTIES are also used to justify solving equations!
When you have a one-step equation such as x + 5 = 12, you ADD -5 (or just subtract 5) from each side equally. The reason you chose -5 is that it was the ADDITIVE INVERSE of 5. The reason the +5 then "disappears" is due to the IDENTITY PROPERTY OF ADDITION. Since +5 + (-5) = 0, it's not necessary to bring down the 0 in the equation.
JUSTIFYING A SIMPLE ONE STEP:
x + 5 = 12 GIVEN
- 5 -5 Subtraction Prop =
X + 0 = 7 Additive Inverse Prop
x = 7 Identity of Addition
COLLECTING TERMS FIRST:
Sometimes, you will have to COLLECT LIKE TERMS ON THE SAME SIDE OF THE EQUATION before balancing
FORMAL CHECK: Follow these three steps---
1. Rewrite original equation
2. Substitute your solution and question mark over the equal sign
3. Do the math and check it!
2- Steps with Distributive Property
USUALLY you want to distribute first!!
(Unless the factor outside the ( ) can be divided out of BOTH SIDES PERFECTLY!!
5y -2(2y +8) = 16
5y -4y -16 = 16 {distribute accurately}
y - 16 = 16 { collect like terms}
y = 32 { solve by adding 16 to both sides!!}
-3(4 +3x) = -9
Wait you could distribute but if you divide both sides by -3
{the -3 goes into BOTH SIDES PERFECTLY}
you have 4 + 3x = 3
3x = -1 { subtract 4 from both sides}
x = -1/3 { Divide both sides by 3}
What if you have variables on BOTH SIDES of the equation?
Simplify each side of the equation first
THEN use the additive inverse property to move the variables to the other side, Usually we try to move the smaller coefficient to the larger because some times that avoids negative coefficients. But it is not always the case.
3y - 10 - y = -10y + 12
combine like terms
2y - 10 = -10y +12
In this case I would add 10y to both sides to get rid of a negative coefficient
12y - 10 = 12
Now add 10 to both sides
12y = 22
Now divide both sides by 12
y = 22/12 or y = 11/6
We'll be using our EQUATION BALANCING PROPERTIES OF EQUALITY:
There are 4 of these.
Whatever YOU DO TO BALANCE an equation,
that operation is the property of equality that was used.
So if you have x + 3 = 10, you used the SUBTRACTION PROPERTY OF EQUALITY because you need to SUBTRACT 3 from each side equally. "- prop ="
If you have x - 3 = 10, you used the ADDITION PROPERTY OF EQUALITY because you need to ADD 3 from each side equally. " + prop ="
If you have 3x = 10, you used the DIVISION PROPERTY OF EQUALITY because you need to DIVIDE each side equally by 3. " / prop ="
If you have x/3 = 10, you used the MULTIPLICATION PROPERTY OF EQUALITY because you need to MULTIPLY each side equally by 3. "x prop ="
SOMETIMES, WE SAY THERE ARE ONLY 2 BALANCING PROPERTIES OF EQUALITY
CAN YOU GUESS WHICH 2 ARE "DROPPED OUT"?
Since we say we never subtract and we really never divide, it's those 2.
GOING BACK TO OUR PREVIOUS EXAMPLES:
If you have x + 3 = 10, you could say that we ADDED -3 to each side equally; therefore, we used the ADDITION (not subtraction) PROPERTY.
If you have 3x = 10, you could say that we MULTIPLIED each side equally by 1/3; therefore, we used the MULTIPLICATION (not division) PROPERTY.
(We always multiply by the MULTIPLICATIVE INVERSE).
REVIEW OF SIMPLE EQUATIONS: One and Two Step.
What's the GOAL? Determine the value of the variable
HOW? Isolate the variable (get it alone on one side of equation)
WHAT to do? Use inverse (opposite) operations to "get rid" of everything on the side with the variable
WHAT SHOULD MY FOCUS BE WHEN EQUATIONS GET COMPLICATED? Always focus on the variable(s) first!!!!!!!
TWO STEP EQUATIONS:
1. Use the ADDITION/SUBTRACTION PROPERTIES OF EQUALITY first (get rid of addition or subtraction)
2. Use the MULTIPLICATION/DIVISION PROPERTIES OF EQUALITY second (get rid of mult/division)
IDENTITY PROPERTIES AND INVERSE PROPERTIES are also used to justify solving equations!
When you have a one-step equation such as x + 5 = 12, you ADD -5 (or just subtract 5) from each side equally. The reason you chose -5 is that it was the ADDITIVE INVERSE of 5. The reason the +5 then "disappears" is due to the IDENTITY PROPERTY OF ADDITION. Since +5 + (-5) = 0, it's not necessary to bring down the 0 in the equation.
JUSTIFYING A SIMPLE ONE STEP:
x + 5 = 12 GIVEN
- 5 -5 Subtraction Prop =
X + 0 = 7 Additive Inverse Prop
x = 7 Identity of Addition
COLLECTING TERMS FIRST:
Sometimes, you will have to COLLECT LIKE TERMS ON THE SAME SIDE OF THE EQUATION before balancing
FORMAL CHECK: Follow these three steps---
1. Rewrite original equation
2. Substitute your solution and question mark over the equal sign
3. Do the math and check it!
2- Steps with Distributive Property
USUALLY you want to distribute first!!
(Unless the factor outside the ( ) can be divided out of BOTH SIDES PERFECTLY!!
5y -2(2y +8) = 16
5y -4y -16 = 16 {distribute accurately}
y - 16 = 16 { collect like terms}
y = 32 { solve by adding 16 to both sides!!}
-3(4 +3x) = -9
Wait you could distribute but if you divide both sides by -3
{the -3 goes into BOTH SIDES PERFECTLY}
you have 4 + 3x = 3
3x = -1 { subtract 4 from both sides}
x = -1/3 { Divide both sides by 3}
What if you have variables on BOTH SIDES of the equation?
Simplify each side of the equation first
THEN use the additive inverse property to move the variables to the other side, Usually we try to move the smaller coefficient to the larger because some times that avoids negative coefficients. But it is not always the case.
3y - 10 - y = -10y + 12
combine like terms
2y - 10 = -10y +12
In this case I would add 10y to both sides to get rid of a negative coefficient
12y - 10 = 12
Now add 10 to both sides
12y = 22
Now divide both sides by 12
y = 22/12 or y = 11/6
Math 6H Period 3, 6 & 7
Writing Mathematical Expressions 2-1
Make sure to glue the 'pink 1/2 sheet' of math word phrases that we associate with each of the four basic operations -- into your spiral notebook (SN)
We can use the same mathematical expression to translate many different word phrases
Five less than a number n
The number n decreased by five
The difference when five is subtracted from a number n
All three of those phrases can be translated into the variable expression
n-5
The quotient of a number y divided by ten becomes y/10. It may look like only a fraction to you-- but if you read y/10 as always " y divided by 10" you have used the proper math language.
Twelve more than three times a number m
Wait-- where are you starting from... in this case you are adding 12 to 3m so you must write
3m + 12
Not all word phrases translate directly into mathematical expressions. Sometimes we need to interpret a situation.. we might need to use relationships between to help create our word phrase.
In writing a variable expression for the number of hours in w workdays, if each workday consists of 8 hours...
First set up a T chart- as discussed in class
put the unknown on the left side of the T chart... The unknown is always the one that reads like " w workdays"
so in this case
w workdays on the left side and under it you put
1
2
3
On the right side put the other variable-- in this case hours
under hours put the corresponding facts you know-- the relationship between workdays and hours as given in this case
hours
8
16
24
all of those would be on the right side of the T chart.
Now look at the relationships and ask yourself--
What do you do to the left side to get the right side?
and in this case
What do you do to 1 to get 8?
What do you do to 2 to get 16?
What do you do to 3 to get 24?
Do you see the pattern?
For each of those the answer is "Multiply by 8" so
what do you do to w-- The answer is Multiply b 8
so the mathematical expression in this case is "8w."
What about writing an expression for
The number of feet in i inches
i inches is the unknown... so that goes on the left side of the T chart... with feet on the right
i inches ___feet
12...............1
24...............2
36...............3
I filled in three known relationships between inches and feet Now, ask your self those questions again...
What do you do to the left side to get the right side?
and in this case
What do you do to 12 to get 1?
What do you do to 24 to get 2?
What do you do to 36 to get 3?
In each of these, the answer is divide by 12
so What do you do to i? the answer is divide by 12
i inches ___feet
12...............1
24...............2
36...............3
i................i/12
and it is written i/12
Some everyday words we use to so relationships with numbers:
consecutive whole numbers are whole numbers that increase by 1 for example 4, 5, 6
So the next consecutive whole number after w is w + 1.
A preceding whole number is the whole number that is 1 less and the next whole number is the whole number that is 1 greater.
So the number which precedes x would be x - 1.
The next number after n is n + 1
Make sure to glue the 'pink 1/2 sheet' of math word phrases that we associate with each of the four basic operations -- into your spiral notebook (SN)
We can use the same mathematical expression to translate many different word phrases
Five less than a number n
The number n decreased by five
The difference when five is subtracted from a number n
All three of those phrases can be translated into the variable expression
n-5
The quotient of a number y divided by ten becomes y/10. It may look like only a fraction to you-- but if you read y/10 as always " y divided by 10" you have used the proper math language.
Twelve more than three times a number m
Wait-- where are you starting from... in this case you are adding 12 to 3m so you must write
3m + 12
Not all word phrases translate directly into mathematical expressions. Sometimes we need to interpret a situation.. we might need to use relationships between to help create our word phrase.
In writing a variable expression for the number of hours in w workdays, if each workday consists of 8 hours...
First set up a T chart- as discussed in class
put the unknown on the left side of the T chart... The unknown is always the one that reads like " w workdays"
so in this case
w workdays on the left side and under it you put
1
2
3
On the right side put the other variable-- in this case hours
under hours put the corresponding facts you know-- the relationship between workdays and hours as given in this case
hours
8
16
24
all of those would be on the right side of the T chart.
Now look at the relationships and ask yourself--
What do you do to the left side to get the right side?
and in this case
What do you do to 1 to get 8?
What do you do to 2 to get 16?
What do you do to 3 to get 24?
Do you see the pattern?
For each of those the answer is "Multiply by 8" so
what do you do to w-- The answer is Multiply b 8
so the mathematical expression in this case is "8w."
What about writing an expression for
The number of feet in i inches
i inches is the unknown... so that goes on the left side of the T chart... with feet on the right
i inches ___feet
12...............1
24...............2
36...............3
I filled in three known relationships between inches and feet Now, ask your self those questions again...
What do you do to the left side to get the right side?
and in this case
What do you do to 12 to get 1?
What do you do to 24 to get 2?
What do you do to 36 to get 3?
In each of these, the answer is divide by 12
so What do you do to i? the answer is divide by 12
i inches ___feet
12...............1
24...............2
36...............3
i................i/12
and it is written i/12
Some everyday words we use to so relationships with numbers:
consecutive whole numbers are whole numbers that increase by 1 for example 4, 5, 6
So the next consecutive whole number after w is w + 1.
A preceding whole number is the whole number that is 1 less and the next whole number is the whole number that is 1 greater.
So the number which precedes x would be x - 1.
The next number after n is n + 1
Wednesday, September 16, 2009
The Number Systems Section 2-2
Natural (counting): 1, 2, 3, ....
Whole: 0, 1, 2, 3, ...
Integers: ...-3, -2, -1, 0, 1, 2, 3, ...
Rational: All numbers that can be written as a fraction or RATIO (decimals-terminating and repeating, fractions, mixed numbers, integers)
Irrational: Decimals that never terminate or repeat - like pi and square root of 2
Real numbers: All the number systems together
Complex or Imaginary Numbers: we don't study these until Algebra II, but they are stated as " a + bi" where i is equal to the square root of -1
DENSITY PROPERTY: Between any 2 rational numbers is another rational number
Rational number = any number that can be expressed as the ratio (fraction) of two integers a/b, where a and b are both integers and b cannot be zero .
"b" cannot be zero because you cannot divide by zero.....IT'S UNDEFINED!
There are positive and negative rational numbers.
One way to put them in order from least to greatest is to simply place them on the REAL number line.
Numbers will be least to greatest if read from the left to the right.
If you have both fractions and decimals, it's often easier to change the fractions into decimals by simply dividing the numerator by the denominator.
REMEMBER: When numbers are NEGATIVE, the closer they are to zero, the bigger they are. Example: -1/2 is greater than -3/4 -2.3 is greater than -2.5
Natural (counting): 1, 2, 3, ....
Whole: 0, 1, 2, 3, ...
Integers: ...-3, -2, -1, 0, 1, 2, 3, ...
Rational: All numbers that can be written as a fraction or RATIO (decimals-terminating and repeating, fractions, mixed numbers, integers)
Irrational: Decimals that never terminate or repeat - like pi and square root of 2
Real numbers: All the number systems together
Complex or Imaginary Numbers: we don't study these until Algebra II, but they are stated as " a + bi" where i is equal to the square root of -1
DENSITY PROPERTY: Between any 2 rational numbers is another rational number
Rational number = any number that can be expressed as the ratio (fraction) of two integers a/b, where a and b are both integers and b cannot be zero .
"b" cannot be zero because you cannot divide by zero.....IT'S UNDEFINED!
There are positive and negative rational numbers.
One way to put them in order from least to greatest is to simply place them on the REAL number line.
Numbers will be least to greatest if read from the left to the right.
If you have both fractions and decimals, it's often easier to change the fractions into decimals by simply dividing the numerator by the denominator.
REMEMBER: When numbers are NEGATIVE, the closer they are to zero, the bigger they are. Example: -1/2 is greater than -3/4 -2.3 is greater than -2.5
Tuesday, September 15, 2009
Algebra Period 4
Using the Distributive Property 2-7
Hanging out with an old friend:
THE DISTRIBUTIVE PROPERTY WITH NEGATIVES
The distributive property works the same when there is subtraction in the ( )
a(b - c) = ab - ac
Inverse of a Sum 2-8
Property of -1:
For any rational number a,
(-1) a = -a
In words: MULTIPLYING BY -1 changes a term to its OPPOSITE SIGN
INVERSE OF A SUM PROPERTY:
DISTRIBUTING THE NEGATIVE SIGN
incognito, it's simply distributing -1
TO EACH ADDEND INSIDE THE PARENTHESES
EXAMPLE: -(3 + x) = -1(3 + x) = (-1)(3) + (-1)(x) = -3 + -x or -3 - x
ALL THAT HAPPENED WAS THAT EACH SIGN CHANGED TO ITS OPPOSITE!
-(a + b) = -a - b
Of course they get MUCH HARDER (but the principle is the same!)
[5(x + 2) - 3y] - [3(y + 2) - 7(x - 3)]
Distribute and simplify inside each [ ] first
[5x + 10 - 3y] - [3y + 6 - 7x + 21]
Now, the subtraction sign between them is really a -1 being distributed!
"Double check" to see this (change the subtraction to adding a negative):
[5x + 10 - 3y] + - 1[3y + 6 - 7x + 21]
Distribute the -1 to all the terms in the 2nd [ ]
[5x + 10 - 3y] + -3y + -6 + 7x + - 21
Simplify by combining like terms:
12x - 6y -17
Hanging out with an old friend:
THE DISTRIBUTIVE PROPERTY WITH NEGATIVES
The distributive property works the same when there is subtraction in the ( )
a(b - c) = ab - ac
Inverse of a Sum 2-8
Property of -1:
For any rational number a,
(-1) a = -a
In words: MULTIPLYING BY -1 changes a term to its OPPOSITE SIGN
INVERSE OF A SUM PROPERTY:
DISTRIBUTING THE NEGATIVE SIGN
incognito, it's simply distributing -1
TO EACH ADDEND INSIDE THE PARENTHESES
EXAMPLE: -(3 + x) = -1(3 + x) = (-1)(3) + (-1)(x) = -3 + -x or -3 - x
ALL THAT HAPPENED WAS THAT EACH SIGN CHANGED TO ITS OPPOSITE!
-(a + b) = -a - b
Of course they get MUCH HARDER (but the principle is the same!)
[5(x + 2) - 3y] - [3(y + 2) - 7(x - 3)]
Distribute and simplify inside each [ ] first
[5x + 10 - 3y] - [3y + 6 - 7x + 21]
Now, the subtraction sign between them is really a -1 being distributed!
"Double check" to see this (change the subtraction to adding a negative):
[5x + 10 - 3y] + - 1[3y + 6 - 7x + 21]
Distribute the -1 to all the terms in the 2nd [ ]
[5x + 10 - 3y] + -3y + -6 + 7x + - 21
Simplify by combining like terms:
12x - 6y -17
Wednesday, September 9, 2009
Algebra Period 4
REVIEW OF THE PROPERTIES AND SOME NEW ONES
Using the Distributive Property 2-7
Hanging out with an old friend:
THE DISTRIBUTIVE PROPERTY WITH NEGATIVES
The distributive property works the same when there is subtraction in the ( )
a(b - c) = ab - ac
Inverse of a Sum 2-8
Property of -1:
For any rational number a,
(-1) a = -a
In words: MULTIPLYING BY -1 changes a term to its OPPOSITE SIGN
INVERSE OF A SUM PROPERTY:
DISTRIBUTING THE NEGATIVE SIGN
incognito, it's simply distributing -1
TO EACH ADDEND INSIDE THE PARENTHESES
EXAMPLE: -(3 + x) = -1(3 + x) = (-1)(3) + (-1)(x) = -3 + -x or -3 - x
ALL THAT HAPPENED WAS THAT EACH SIGN CHANGED TO ITS OPPOSITE!
-(a + b) = -a - b
Of course they get MUCH HARDER (but the principle is the same!)
[5(x + 2) - 3y] - [3(y + 2) - 7(x - 3)]
Distribute and simplify inside each [ ] first
[5x + 10 - 3y] - [3y + 6 - 7x + 21]
Now, the subtraction sign between them is really a -1 being distributed!
"Double check" to see this (change the subtraction to adding a negative):
[5x + 10 - 3y] + - 1[3y + 6 - 7x + 21]
Distribute the -1 to all the terms in the 2nd [ ]
[5x + 10 - 3y] + -3y + -6 + 7x + - 21
Simplify by combining like terms:
12x - 6y -17
Number properties and Proofs 2-10
MORE NEW FRIENDS! (PROPERTIES)
There are 2 types of Properties: Axioms and Theorems
Axioms = properties we accept as obvious and so we don't need to prove them
Theorems = properties that need to be proved USING THE AXIOMS WE ACCEPT AS FACT!
EXAMPLES OF AXIOMS:
Commutative, Associative, Identity, Distributive, Additive Inverse, Multiplicative Inverse
EXAMPLE OF A THEOREM:
Distributive Property in REVERSE (a + b)c = ac + bc
PROPERTIES OF EQUALITY
(these are AXIOMS)
” Prop = “
REFLEXIVE PROP =:
a = a
3 = 3
In words: It looks exactly the same on both sides! (like reflecting in a mirror)
This seems ridiculous, but in Geometry it's used all the time.
SYMMETRIC PROP =:
a = b then b = a
3 + 5 = 8 then 8 = 3 + 5
In words: You can switch the sides of an equation.
We use this all the time to switch the sides if the variable ends up on the right side:
12 = 5y -3
The Symmetric property allows us to switch sides:
5y - 3 = 12
TRANSITIVE PROP =:
a = b and b = c then a = c
3 + 5 = 8, and 2 + 6 = 8 then 3 + 5 = 2 + 6
In words: If 2 things both equal a third thing, then we can just say that the first 2 things are equal.
I've got a pattern that will help you recognize the difference between these 3 properties specifically.
The Reflexive Property only has ONE equation
The Symmetric Property only has TWO equations
The Transitive Property only has THREE equations
Using the Distributive Property 2-7
Hanging out with an old friend:
THE DISTRIBUTIVE PROPERTY WITH NEGATIVES
The distributive property works the same when there is subtraction in the ( )
a(b - c) = ab - ac
Inverse of a Sum 2-8
Property of -1:
For any rational number a,
(-1) a = -a
In words: MULTIPLYING BY -1 changes a term to its OPPOSITE SIGN
INVERSE OF A SUM PROPERTY:
DISTRIBUTING THE NEGATIVE SIGN
incognito, it's simply distributing -1
TO EACH ADDEND INSIDE THE PARENTHESES
EXAMPLE: -(3 + x) = -1(3 + x) = (-1)(3) + (-1)(x) = -3 + -x or -3 - x
ALL THAT HAPPENED WAS THAT EACH SIGN CHANGED TO ITS OPPOSITE!
-(a + b) = -a - b
Of course they get MUCH HARDER (but the principle is the same!)
[5(x + 2) - 3y] - [3(y + 2) - 7(x - 3)]
Distribute and simplify inside each [ ] first
[5x + 10 - 3y] - [3y + 6 - 7x + 21]
Now, the subtraction sign between them is really a -1 being distributed!
"Double check" to see this (change the subtraction to adding a negative):
[5x + 10 - 3y] + - 1[3y + 6 - 7x + 21]
Distribute the -1 to all the terms in the 2nd [ ]
[5x + 10 - 3y] + -3y + -6 + 7x + - 21
Simplify by combining like terms:
12x - 6y -17
Number properties and Proofs 2-10
MORE NEW FRIENDS! (PROPERTIES)
There are 2 types of Properties: Axioms and Theorems
Axioms = properties we accept as obvious and so we don't need to prove them
Theorems = properties that need to be proved USING THE AXIOMS WE ACCEPT AS FACT!
EXAMPLES OF AXIOMS:
Commutative, Associative, Identity, Distributive, Additive Inverse, Multiplicative Inverse
EXAMPLE OF A THEOREM:
Distributive Property in REVERSE (a + b)c = ac + bc
PROPERTIES OF EQUALITY
(these are AXIOMS)
” Prop = “
REFLEXIVE PROP =:
a = a
3 = 3
In words: It looks exactly the same on both sides! (like reflecting in a mirror)
This seems ridiculous, but in Geometry it's used all the time.
SYMMETRIC PROP =:
a = b then b = a
3 + 5 = 8 then 8 = 3 + 5
In words: You can switch the sides of an equation.
We use this all the time to switch the sides if the variable ends up on the right side:
12 = 5y -3
The Symmetric property allows us to switch sides:
5y - 3 = 12
TRANSITIVE PROP =:
a = b and b = c then a = c
3 + 5 = 8, and 2 + 6 = 8 then 3 + 5 = 2 + 6
In words: If 2 things both equal a third thing, then we can just say that the first 2 things are equal.
I've got a pattern that will help you recognize the difference between these 3 properties specifically.
The Reflexive Property only has ONE equation
The Symmetric Property only has TWO equations
The Transitive Property only has THREE equations
Math 6H Period 3, 6 & 7
Distributive Property 1-4
I like to eat...eat...eat..eat Apples and Bananas... now what did that have to do with the distributive property???
The Distributive Property of Multiplication with Respect to Addition
For any whole numbers , a,b, and c,
a(b + c) = ab + ac
and
(b + c)a = ba + ca
so the reason we want to use the Distributive Property is to make our lives easier
for example
13 X 15 could be easily multiply using the distributive property 13(10 + 5) =
13(10) + 13(5) = 130 + 65 = 195
and if you were given (11 X4) + (11 X 6) you could do the reverse and say 11(4 +6) or 11(10) = 110
Similarly, we have the Distributive Property of Multiplication with Resepect to Subtraction
For any whole numbers a, b, and c
a(b - c) = ab - ac
and
(b-c)a = ba - ca
Since multiplication is distributive with respect to both addition and subtraction, we refer to both properties as the DISTRIBUTIVE PROPERTY
99(43) yes, you could multiply it out but if you used the distributive property and thought of two numbers whose difference was 99 -- you could easily do this in your head..
That is,
(100 - 1)43 = 100(43) - 1(43) = 4300 -43 = 4257
51(27) could become (50 + 1)27 = 50(27) + 1(27) = 1350 + 27 = 1377
The distributive property is especially helpful with mental math!!
I like to eat...eat...eat..eat Apples and Bananas... now what did that have to do with the distributive property???
The Distributive Property of Multiplication with Respect to Addition
For any whole numbers , a,b, and c,
a(b + c) = ab + ac
and
(b + c)a = ba + ca
so the reason we want to use the Distributive Property is to make our lives easier
for example
13 X 15 could be easily multiply using the distributive property 13(10 + 5) =
13(10) + 13(5) = 130 + 65 = 195
and if you were given (11 X4) + (11 X 6) you could do the reverse and say 11(4 +6) or 11(10) = 110
Similarly, we have the Distributive Property of Multiplication with Resepect to Subtraction
For any whole numbers a, b, and c
a(b - c) = ab - ac
and
(b-c)a = ba - ca
Since multiplication is distributive with respect to both addition and subtraction, we refer to both properties as the DISTRIBUTIVE PROPERTY
99(43) yes, you could multiply it out but if you used the distributive property and thought of two numbers whose difference was 99 -- you could easily do this in your head..
That is,
(100 - 1)43 = 100(43) - 1(43) = 4300 -43 = 4257
51(27) could become (50 + 1)27 = 50(27) + 1(27) = 1350 + 27 = 1377
The distributive property is especially helpful with mental math!!
Wednesday, September 2, 2009
Algebra Period 4
Solving Equations 1-7
Equations use an = sign to show that two expressions are equivalent
Equations can be true, false, or open
If an equation is NUMERICAL (numbers), it must be either TRUE or FALSE:
3 + 5 = 8 is TRUE
3 + 5 = 9 is FALSE
If an equation is ALGEBRAIC (variables), it is OPEN for discussion
(it all depends on what the variable is!)
y + 4 = 10 is OPEN
SOLUTION = answer
REPLACEMENT SET: The set of answers that you have to choose from for the answer.
SOLUTION SET: All the answers that make a statement true.
{ } set symbol
EXAMPLE: Solve y + 4 = 10 for the replacement set {2, 6, 10}
Substitute each element of the replacement set into the equation to see which one(s) work
Only 6 works, so only 6 is a true solution
The solution set is {6}
Note that if the replacement set was {2, 7, 10} for y + 4 = 10, then the solution set would be the NULL SET which means the solution does not exist in the replacement set.
The symbol for the null set is a zero with a line through it.
Equations must always be in balance.
GOLDEN RULE: WHATEVER YOU DO TO ONE SIDE OF AN EQUATION, YOU MUST DO TO THE OTHER SIDE or you'll be out of balance.
One way they might ask this question on the STAR Test is to ask:
What was done to the first equation to get the second?
EXAMPLE: 3x - 5 = 12 and 9x - 15 = 36
The first equation was multiplied by 3 to get to the second equation so they are equivalent.
Equations use an = sign to show that two expressions are equivalent
Equations can be true, false, or open
If an equation is NUMERICAL (numbers), it must be either TRUE or FALSE:
3 + 5 = 8 is TRUE
3 + 5 = 9 is FALSE
If an equation is ALGEBRAIC (variables), it is OPEN for discussion
(it all depends on what the variable is!)
y + 4 = 10 is OPEN
SOLUTION = answer
REPLACEMENT SET: The set of answers that you have to choose from for the answer.
SOLUTION SET: All the answers that make a statement true.
{ } set symbol
EXAMPLE: Solve y + 4 = 10 for the replacement set {2, 6, 10}
Substitute each element of the replacement set into the equation to see which one(s) work
Only 6 works, so only 6 is a true solution
The solution set is {6}
Note that if the replacement set was {2, 7, 10} for y + 4 = 10, then the solution set would be the NULL SET which means the solution does not exist in the replacement set.
The symbol for the null set is a zero with a line through it.
Equations must always be in balance.
GOLDEN RULE: WHATEVER YOU DO TO ONE SIDE OF AN EQUATION, YOU MUST DO TO THE OTHER SIDE or you'll be out of balance.
One way they might ask this question on the STAR Test is to ask:
What was done to the first equation to get the second?
EXAMPLE: 3x - 5 = 12 and 9x - 15 = 36
The first equation was multiplied by 3 to get to the second equation so they are equivalent.
Tuesday, September 1, 2009
Math 6H Period 3, 6 & 7
Mathematical Expressions 1-1
A variable is a symbol used to represent one or more numbers. The numbers are called the values of the variable.
An expression, such as 3 x n, that involves a variable is called a variable expression.
Expressions, such as 3 x 2, that name a certain number are called numerical expressions
When we write a product that involves a variable, we usually omit the multiplication symbol (whether that be written as x or as ∙ or even with parentheses). Thus, 3 x n is written as 3n
and 2 x a x b is written as 2ab
In numerical expressions for products a multiplication symbol must be used to avoid confusion.
9 x 7 may be written as 9 ∙ 7 or even 9(7)
When a mathematical sentence uses an equal sign, it is called an equation. An equation tells us that two expressions name the same number. The expression to the left of the equals sign is called the left side of the equation and the expression to the right of the equals sign is called the right side.
When a number is substituted for a variable in the variable expression and the indicated operation is carried out, we say that the variable expression has been evaluated. For example, if n has the value 6 in the variable expression 3 x n, then 3 x n has the value 3 x 6, or 18
Example: Evaluate the expression 6a when the variable has the following values:
6a; 2, 4, 6, 8
You would substitute in each vvalue for the variable a
6(2) = 12
6(4) = 24
6(6) = 36
6(8) = 48
A variable is a symbol used to represent one or more numbers. The numbers are called the values of the variable.
An expression, such as 3 x n, that involves a variable is called a variable expression.
Expressions, such as 3 x 2, that name a certain number are called numerical expressions
When we write a product that involves a variable, we usually omit the multiplication symbol (whether that be written as x or as ∙ or even with parentheses). Thus, 3 x n is written as 3n
and 2 x a x b is written as 2ab
In numerical expressions for products a multiplication symbol must be used to avoid confusion.
9 x 7 may be written as 9 ∙ 7 or even 9(7)
When a mathematical sentence uses an equal sign, it is called an equation. An equation tells us that two expressions name the same number. The expression to the left of the equals sign is called the left side of the equation and the expression to the right of the equals sign is called the right side.
When a number is substituted for a variable in the variable expression and the indicated operation is carried out, we say that the variable expression has been evaluated. For example, if n has the value 6 in the variable expression 3 x n, then 3 x n has the value 3 x 6, or 18
Example: Evaluate the expression 6a when the variable has the following values:
6a; 2, 4, 6, 8
You would substitute in each vvalue for the variable a
6(2) = 12
6(4) = 24
6(6) = 36
6(8) = 48
Pre Algebra Period 1
The Order Of Operations 1-2
O3 = Order Of Operations
It’s Aunt Sally’s Rule—Please Excuse My Dear Aunt Sally
Please (XX)
Excuse E2
My Dear (or even Dear My) × ÷ or ÷ ×
Aunt Sally (Sally Aunt) + - or - +
PEMDAS
The most common mistake of O3
when you see multiplication and division—you do whichever comes first in order!!
So if you see division then multiplication, you do division first then multiplication
When you see addition and subtraction, you do whichever comes first in order!
So if you see subtraction then addition, do the subtraction first!
When you see more than one set of parentheses, do the innermost ones first, then work outwards.
They're called nested parentheses because a bird builds his nest from the inside out.
Posted by Jani Nelson at 6:17 AM 0 comments
O3 = Order Of Operations
It’s Aunt Sally’s Rule—Please Excuse My Dear Aunt Sally
Please (XX)
Excuse E2
My Dear (or even Dear My) × ÷ or ÷ ×
Aunt Sally (Sally Aunt) + - or - +
PEMDAS
The most common mistake of O3
when you see multiplication and division—you do whichever comes first in order!!
So if you see division then multiplication, you do division first then multiplication
When you see addition and subtraction, you do whichever comes first in order!
So if you see subtraction then addition, do the subtraction first!
When you see more than one set of parentheses, do the innermost ones first, then work outwards.
They're called nested parentheses because a bird builds his nest from the inside out.
Posted by Jani Nelson at 6:17 AM 0 comments
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