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.
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.
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
Thursday, September 25, 2008
Pre Algebra Period 2 (Thursday)
Simplifying Variable Expressions 2-3
Review of Algebraic terminology:
In the expression, 3y + 5
3 is the coefficient (number attached to variable –
remember, "co" means to go along with)
y is the variable
5 is the constant (number not attached to variable)
terms are separated by ADDITION ONLY!
COMBINING LIKE TERMS:
1) Same variable (or no variable)
2) Same power
You can combine by addition or subtraction LIKE TERMS.
You cannot combine UNLIKE TERMS.
EX: 3a + 4a = 7a
but
3a + 4b = 3a + 4b
3a + 4a2 = 3a + 4a2
YOU SHOULD ALWAYS COMBINE LIKE TERMS BEFORE YOU EVALUATE!
IT'S MUCH SIMPLER!
-25a + 5a - (-10a) when a = -14
First combine like terms: -10a
Then plug in for a = -14: -10(-14) = 140
Review of Algebraic terminology:
In the expression, 3y + 5
3 is the coefficient (number attached to variable –
remember, "co" means to go along with)
y is the variable
5 is the constant (number not attached to variable)
terms are separated by ADDITION ONLY!
COMBINING LIKE TERMS:
1) Same variable (or no variable)
2) Same power
You can combine by addition or subtraction LIKE TERMS.
You cannot combine UNLIKE TERMS.
EX: 3a + 4a = 7a
but
3a + 4b = 3a + 4b
3a + 4a2 = 3a + 4a2
YOU SHOULD ALWAYS COMBINE LIKE TERMS BEFORE YOU EVALUATE!
IT'S MUCH SIMPLER!
-25a + 5a - (-10a) when a = -14
First combine like terms: -10a
Then plug in for a = -14: -10(-14) = 140
Wednesday, September 24, 2008
Algebra Period 3 (Wednesday)
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:
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:
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:
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
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:
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:
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:
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
Tuesday, September 23, 2008
Algebra Period 3 (Monday)
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
Pre Algebra Period 2 (Monday)
Properties of Numbers 2-1
WHAT ARE PROPERTIES? (Why are they your friends?)
You can count on properties. They always work. There are no COUNTEREXAMPLES!
COUNTEREXAMPLE = an example that shows that something does not work
(counters what you have said)
An example from Math: You can't switch the order of subtraction because it's not the same value.
10 - 8 DOES NOT EQUAL 8 - 10
That's a COUNTEREXAMPLE to saying that you can switch subtraction
(We'll say that it's a COUNTEREXAMPLE to the existence of a
COMMUTATIVE PROPERTY OF SUBTRACTION
so that property does not exist!)
An example from Math: You switch the order of addition to make the adding easier.
20 + 547 + 80 = 20 + 80 + 547
(both equal 647, but the right side is much easier!)
What allowed you to switch the order?
A property called the Commutative Property of Addition says you can!
You'll always get the same value!
Now Aunt Sally doesn't like some of the properties because they allow us to do things that are exception to the Order of Operations!
Commutative Property
You can switch the order of all addition or all multiplication
a + b = b + a
ab = ba
3 + 5 = 5 + 3
3 (5) = 5 (3)
(you can HEAR the change in order!)
Aunt Sally says that you always need to go left to right, but Commutative says not necessary if
you have all multiplication or all addition.
Associative Property
You can group all addition or all multiplication any way you want
a + b + c = a + (b + c)
abc = a(bc)
(3 + 2) + 8 = 3 + (2 + 8)
(Why would you want to? Sometimes it's easier!)
[57 x 5] (2) = (57) [ 5 (2) ]
(you can't hear this property! but you can SEE it!)
Aunt Sally says you must always do parentheses first, but Associative says that you can actually take the parentheses away, put parentheses in, or change where the parentheses are if
you have all multiplication or all addition.
These properties give you a choice when it's all multiplication OR all addition
There are no counterexamples for these two operations.
BUT THEY DO NOT WORK FOR SUBTRACTION OR DIVISION
(lots of counterexamples! 10 - 2 does not equal 2 - 10
15 ÷ 5 does not equal 5 ÷ 15)
SO WHY SHOULD YOU CARE????
Because it makes the math easier sometimes!
Which would you rather multiply:
(2)(543)(5) OR (2)(5)(543) ???
Commutative allows you to choose!
ANOTHER EXAMPLE: [(543)(5)](2)
Aunt Sally would say you must do the 543 by the 5 first since it's in [ ]
But our friend the Associative Property allows us to simply move the [ ]
[(543)(5)](2) = (543)[(5)(2)] which is so much easier to multiply in your head!!!
TWO MORE FRIENDS:
THE IDENTITY PROPERTIES
OF ADDITION AND MULTIPLICATION
For addition, we know that adding zero to anything will not change the IDENTITY of what you started with: a + 0 = a (what you started with)
0 is known as the ADDITIVE IDENTITY.
For multiplication, we know that multiplying 1 by anything will not change the IDENTITY of what you started with: (1)(a) = a (what you started with)
1 is known as the MULTIPLICATIVE IDENTITY.
Sometimes 1 is "incognito" (disguised!)
We use this concept all the time to get EQUIVALENT FRACTIONS.
Say we have 3/4 but we want the denominator to be 12
We multiply both the numerator and the denominator by 3 and get 9/12
We actually used the MULTIPLICATIVE IDENTITY of 1, but it was disguised as 3/3
ANYTHING OVER ITSELF = 1 (except zero because dividing by zero is UNDEFINED!)
a + b - c/a + b - c = 1
We also use this property to SIMPLIFY fractions.
We "simplify" all the parts on the top (the numerator) and the bottom (the denominator) that equal 1
(your parents would say that we are reducing the fraction)
6abc/10a =3bc/5 since both the numerator and denominator can be divided by
2a/2a
WE LOVE PROPERTIES BECAUSE THEY MAKE OUR LIFE EASIER!
AUNT SALLY HATES THEM BECAUSE THEY ALLOW US TO BREAK HER RULES!!!
WHAT ARE PROPERTIES? (Why are they your friends?)
You can count on properties. They always work. There are no COUNTEREXAMPLES!
COUNTEREXAMPLE = an example that shows that something does not work
(counters what you have said)
An example from Math: You can't switch the order of subtraction because it's not the same value.
10 - 8 DOES NOT EQUAL 8 - 10
That's a COUNTEREXAMPLE to saying that you can switch subtraction
(We'll say that it's a COUNTEREXAMPLE to the existence of a
COMMUTATIVE PROPERTY OF SUBTRACTION
so that property does not exist!)
An example from Math: You switch the order of addition to make the adding easier.
20 + 547 + 80 = 20 + 80 + 547
(both equal 647, but the right side is much easier!)
What allowed you to switch the order?
A property called the Commutative Property of Addition says you can!
You'll always get the same value!
Now Aunt Sally doesn't like some of the properties because they allow us to do things that are exception to the Order of Operations!
Commutative Property
You can switch the order of all addition or all multiplication
a + b = b + a
ab = ba
3 + 5 = 5 + 3
3 (5) = 5 (3)
(you can HEAR the change in order!)
Aunt Sally says that you always need to go left to right, but Commutative says not necessary if
you have all multiplication or all addition.
Associative Property
You can group all addition or all multiplication any way you want
a + b + c = a + (b + c)
abc = a(bc)
(3 + 2) + 8 = 3 + (2 + 8)
(Why would you want to? Sometimes it's easier!)
[57 x 5] (2) = (57) [ 5 (2) ]
(you can't hear this property! but you can SEE it!)
Aunt Sally says you must always do parentheses first, but Associative says that you can actually take the parentheses away, put parentheses in, or change where the parentheses are if
you have all multiplication or all addition.
These properties give you a choice when it's all multiplication OR all addition
There are no counterexamples for these two operations.
BUT THEY DO NOT WORK FOR SUBTRACTION OR DIVISION
(lots of counterexamples! 10 - 2 does not equal 2 - 10
15 ÷ 5 does not equal 5 ÷ 15)
SO WHY SHOULD YOU CARE????
Because it makes the math easier sometimes!
Which would you rather multiply:
(2)(543)(5) OR (2)(5)(543) ???
Commutative allows you to choose!
ANOTHER EXAMPLE: [(543)(5)](2)
Aunt Sally would say you must do the 543 by the 5 first since it's in [ ]
But our friend the Associative Property allows us to simply move the [ ]
[(543)(5)](2) = (543)[(5)(2)] which is so much easier to multiply in your head!!!
TWO MORE FRIENDS:
THE IDENTITY PROPERTIES
OF ADDITION AND MULTIPLICATION
For addition, we know that adding zero to anything will not change the IDENTITY of what you started with: a + 0 = a (what you started with)
0 is known as the ADDITIVE IDENTITY.
For multiplication, we know that multiplying 1 by anything will not change the IDENTITY of what you started with: (1)(a) = a (what you started with)
1 is known as the MULTIPLICATIVE IDENTITY.
Sometimes 1 is "incognito" (disguised!)
We use this concept all the time to get EQUIVALENT FRACTIONS.
Say we have 3/4 but we want the denominator to be 12
We multiply both the numerator and the denominator by 3 and get 9/12
We actually used the MULTIPLICATIVE IDENTITY of 1, but it was disguised as 3/3
ANYTHING OVER ITSELF = 1 (except zero because dividing by zero is UNDEFINED!)
a + b - c/a + b - c = 1
We also use this property to SIMPLIFY fractions.
We "simplify" all the parts on the top (the numerator) and the bottom (the denominator) that equal 1
(your parents would say that we are reducing the fraction)
6abc/10a =3bc/5 since both the numerator and denominator can be divided by
2a/2a
WE LOVE PROPERTIES BECAUSE THEY MAKE OUR LIFE EASIER!
AUNT SALLY HATES THEM BECAUSE THEY ALLOW US TO BREAK HER RULES!!!
Math 6 H Periods 1, 6 & 7 (Monday)
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
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.
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
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.
Math 6 Honors Periods 1, 6 & 7
Order of Operations 1-5
PEMDAS in my math class... PEMDAS in my math class... just follow the song and make sure you use the rules
Always perform the operation enclosed in the inner pair of group symbols ( such as parentheses and bracket [ ]) FIRST.
Then do all the exponents
Do all multiplication and division in order from left to right
Do all addition and subtractions in order from left to right.
A Problem Solving Model 1-6
make sure to glue in the "Plan for Solving Word Problems" into your spiral notebook
Read the problem carefully. Make sure that you understand what it says. You may need to read it more than once... That's okay!! It is great to re read things!!
Use questions like these in planning the solution:
What is asked for?
What facts are given?
Are there enough facts? Are there some unnecessary facts?
Determine which operation or operations can be used to solve the problem
Carry out the operations CAREFULLY!!
Check your results with the facts given in the problem.
PEMDAS in my math class... PEMDAS in my math class... just follow the song and make sure you use the rules
Always perform the operation enclosed in the inner pair of group symbols ( such as parentheses and bracket [ ]) FIRST.
Then do all the exponents
Do all multiplication and division in order from left to right
Do all addition and subtractions in order from left to right.
A Problem Solving Model 1-6
make sure to glue in the "Plan for Solving Word Problems" into your spiral notebook
Read the problem carefully. Make sure that you understand what it says. You may need to read it more than once... That's okay!! It is great to re read things!!
Use questions like these in planning the solution:
What is asked for?
What facts are given?
Are there enough facts? Are there some unnecessary facts?
Determine which operation or operations can be used to solve the problem
Carry out the operations CAREFULLY!!
Check your results with the facts given in the problem.
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