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
VARIABLES AND EQUATIONS: Section 2-4
Is a given number a solution to an equation?
Substitute and evaluate to see.
Use the set signs { } and a ? over the equal sign as we did in class
9k = 10 - k for k = 1
9(1) = 10 - (1)
9 = 9 checks
Numeric Equations are either True or False
Equations with variables are OPEN (don't know if they are True or False)
Thursday, October 14, 2010
Tuesday, October 12, 2010
Algebra (Period 1)
Inequalities & Their Graphs 4-1
Introduction to INEQUALITIES and graphing them:
Writing an inequality
Graphing an inequality - open dot is < or >
Closed dot mean less than or EQUAL or greater than or EQUAL (think of the = sign as a crayon that you can use to COLOR IN THE DOT!)
Different from equations: Inequalities have many solutions (most of the time an infinite number!)
Example: n > 3 means that every real number greater than 3 is a solution! (but NOT 3)
n ≥ 3 means still means that every real number greater than 3 is a solution, but now 3 is also a solution.
The endpoint ( in this case 3) is called the boundary point.
GRAPHING INEQUALITIES:
First, graphing an equation's solution is easy
1) Let's say you solved an inequality and you discovered that y = 5, you would just put a dot on 5 on the number line
2) But now you have the y ≥ 5
You still put the dot but now also darken in an arrow going to the right
showing all those numbers are also solutions
3) Finally, you find in another example that y > 5
You still have the arrow pointing right, but now you OPEN THE DOT on the 5 to show that 5 IS NOT A SOLUTION!
TRANSLATING WORDS:
Some key words to know
AT LEAST means greater than or equal
AT MOST means less than or equal
I need at least $20 to go to the mall means I must have $20, but I'd like to have even more!
I want at most 15 minutes of homework means that I can have 15 minutes, but I'm hoping for even less!
The Addition Property of Inequalities 4-2
Solving Inequalities with adding or subtracting
Simply use the Additive Inverse Property as if you were balancing an equation!
The only difference is that now you have more than one possible answer.
Example: 5y + 4 > 29
You would -4 from each side, then divide by 5 on each side and get: y > 5
Your answer is infinite! Any real number bigger than 5 will work!
It is IMPOSSIBLE to check every value in an infinite solution set. However, once you have determined the endpoint-- or boundary point, you can verify by checking a representative or sample number from the supposed solution set to verify that your inequality has been solved correctly.
Introduction to INEQUALITIES and graphing them:
Writing an inequality
Graphing an inequality - open dot is < or >
Closed dot mean less than or EQUAL or greater than or EQUAL (think of the = sign as a crayon that you can use to COLOR IN THE DOT!)
Different from equations: Inequalities have many solutions (most of the time an infinite number!)
Example: n > 3 means that every real number greater than 3 is a solution! (but NOT 3)
n ≥ 3 means still means that every real number greater than 3 is a solution, but now 3 is also a solution.
The endpoint ( in this case 3) is called the boundary point.
GRAPHING INEQUALITIES:
First, graphing an equation's solution is easy
1) Let's say you solved an inequality and you discovered that y = 5, you would just put a dot on 5 on the number line
2) But now you have the y ≥ 5
You still put the dot but now also darken in an arrow going to the right
showing all those numbers are also solutions
3) Finally, you find in another example that y > 5
You still have the arrow pointing right, but now you OPEN THE DOT on the 5 to show that 5 IS NOT A SOLUTION!
TRANSLATING WORDS:
Some key words to know
AT LEAST means greater than or equal
AT MOST means less than or equal
I need at least $20 to go to the mall means I must have $20, but I'd like to have even more!
I want at most 15 minutes of homework means that I can have 15 minutes, but I'm hoping for even less!
The Addition Property of Inequalities 4-2
Solving Inequalities with adding or subtracting
Simply use the Additive Inverse Property as if you were balancing an equation!
The only difference is that now you have more than one possible answer.
Example: 5y + 4 > 29
You would -4 from each side, then divide by 5 on each side and get: y > 5
Your answer is infinite! Any real number bigger than 5 will work!
It is IMPOSSIBLE to check every value in an infinite solution set. However, once you have determined the endpoint-- or boundary point, you can verify by checking a representative or sample number from the supposed solution set to verify that your inequality has been solved correctly.
Math 6H (Period 6 & 7)
Problem Solving: Using Mathematical Expression 2-6
Inequalities Continued
We know about > greater and as well as < less than
so now we look at
≥ which means " greater than or equal to" and
≤ which means " less than or equal to"
This time the boundary point ( or endpoint) is included in the solution set.
The good news is that we still solve these inequalities the same way in which we solved equations-- using the properties of equality.
w/4 ≥ 3
we multiply both sides by 4/1
(4/1)(w/4) ≥ 3(4/1) by the X prop =
1w ≥ 12
w ≥ 12 by the ID(x)
Which means that any number greater than 12 is part of the solution AND 12 is also part of that solution
Take the following:
b - 3 ≤ 150 ( we need to add 3 to both sides of the equation)
+3 = +3 using the + prop =
b + 0 ≤ 153
or b ≤ 153 using the ID (+)
Inequalities Continued
We know about > greater and as well as < less than
so now we look at
≥ which means " greater than or equal to" and
≤ which means " less than or equal to"
This time the boundary point ( or endpoint) is included in the solution set.
The good news is that we still solve these inequalities the same way in which we solved equations-- using the properties of equality.
w/4 ≥ 3
we multiply both sides by 4/1
(4/1)(w/4) ≥ 3(4/1) by the X prop =
1w ≥ 12
w ≥ 12 by the ID(x)
Which means that any number greater than 12 is part of the solution AND 12 is also part of that solution
Take the following:
b - 3 ≤ 150 ( we need to add 3 to both sides of the equation)
+3 = +3 using the + prop =
b + 0 ≤ 153
or b ≤ 153 using the ID (+)
Monday, October 11, 2010
Pre Algebra (Period 2 & 4)
EQUATIONS & INEQUALITIES: Properties 2-1
WHAT ARE PROPERTIES? (Why are they your friends?)
You can count on properties. They always work.
There are 0 COUNTEREXAMPLES!
COUNTEREXAMPLE = an example that shows that something does not work (counters what you have said)
Because you can count on them, you can use them to JUSTIFY what you do.
JUSTIFY = a reason for doing what you did
PROPERTIES ARE EXCEPTIONS TO AUNT SALLY:
Commutative (order) Property
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 (groupings) Property
(a + b) + c = a + (b + c)
(ab)c = 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 F
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 = 1
a + b - c
We also use this property to SIMPLIFY fractions.
We simplify all the parts on the top and the bottom that equal 1 (your parents would say that we are reducing the fraction)
6abc/2a = 3bc
2a/2a = 1 and that's why we can divide the fraction by it.
10a/5 = 2a
AGAIN, WE LOVE THESE 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 0 COUNTEREXAMPLES!
COUNTEREXAMPLE = an example that shows that something does not work (counters what you have said)
Because you can count on them, you can use them to JUSTIFY what you do.
JUSTIFY = a reason for doing what you did
PROPERTIES ARE EXCEPTIONS TO AUNT SALLY:
Commutative (order) Property
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 (groupings) Property
(a + b) + c = a + (b + c)
(ab)c = 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 F
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 = 1
a + b - c
We also use this property to SIMPLIFY fractions.
We simplify all the parts on the top and the bottom that equal 1 (your parents would say that we are reducing the fraction)
6abc/2a = 3bc
2a/2a = 1 and that's why we can divide the fraction by it.
10a/5 = 2a
AGAIN, WE LOVE THESE PROPERTIES BECAUSE THEY MAKE OUR LIFE EASIER! AUNT SALLY HATES THEM BECAUSE THEY ALLOW US TO BREAK HER RULES!!!
Math 6H (Period 6 & 7)
Problem Solving: Using Mathematical Expression 2-6
Seventeen less than a number is fifty six
I suggest lining up and placing the "equal sign" right under the word is
then complete the right side = 56
after that take your time translating the left
Start with a "let statement."
A "let statement" tells your reader what variable you are going to use to represent the number in your equation.
So in this case Let b = the number
it becomes
b- 17 = 56
Now solve as we have been practicing for a couple of weeks.
b - 17 = 56
+17 = +17 using the +prop=
b + 0 = 73
b = 73 by the ID(+)
How could we check?
A FORMAL CHECK involves three steps:
1) Re write the equation ( from the original source)
2) substitute your solution or... "plug it in, plug it in...."
3) DO the MATH!! actually do the math to check!!
so to check the above
b - 17 = 56
substitute 73 and put a "?" above the equal sign...
73 - 17 ?=? 56
Now really do the math!! Use a side bar to DO the MATH!!
That is, what is 73- 17? it is 56
so 56 = 56
Practice some of the class exercised on Page 50, Just practice setting up the equations from the verbal sentences.
Seventeen less than a number is fifty six
I suggest lining up and placing the "equal sign" right under the word is
then complete the right side = 56
after that take your time translating the left
Start with a "let statement."
A "let statement" tells your reader what variable you are going to use to represent the number in your equation.
So in this case Let b = the number
it becomes
b- 17 = 56
Now solve as we have been practicing for a couple of weeks.
b - 17 = 56
+17 = +17 using the +prop=
b + 0 = 73
b = 73 by the ID(+)
How could we check?
A FORMAL CHECK involves three steps:
1) Re write the equation ( from the original source)
2) substitute your solution or... "plug it in, plug it in...."
3) DO the MATH!! actually do the math to check!!
so to check the above
b - 17 = 56
substitute 73 and put a "?" above the equal sign...
73 - 17 ?=? 56
Now really do the math!! Use a side bar to DO the MATH!!
That is, what is 73- 17? it is 56
so 56 = 56
Practice some of the class exercised on Page 50, Just practice setting up the equations from the verbal sentences.
Subscribe to:
Posts (Atom)
