Evaluating Expressions 1-3 (continued)
STRATEGY #1: MAKE A T-CHART
To translate known relationships to algebra, it often helps to make a T-Chart.
You always put the unknown variable on the LEFT side and what you know on the right.
Fill in the chart with 3 lines of numbers and look for the relationship between the 2 columns.
Then, you use that mathematical relationship with a variable.
EXAMPLE: The number of hours in d days
Your unknown is d days so that goes on the left side:
d days ---> hours
1 --->24
2 ---> 48
3 --->72
Now look at the relationship between the left column and the right column.
You must MULTIPLY the left column BY 24 to get to the right column
The last line of the chart will then use your variable d
d days ---> hours
1 --->24
2 --->48
3 --->72
d --->24d
EXAMPLE: The number of days in h hours (The flip of the first example)
Your unknown is h hours so that goes on the left side:
h hours -> days
24 --->1
48 --->2
72 --->3
(Why did I start with 24 and not 1 hour this time?)
Now look at the relationship between the left column and the right column.
You must DIVIDE the left column BY 24 to get to the right column
The last line of the chart will then use your variable h
h hours -> days
24 --->1
48 ---> 2
72 ---> 3
h --->h/24
The number of inches in y yards
You know that there are 36 inches in 1 yard.
y yards is the unknown so that must go on the left
y yards -> inches
1 ---> 36
2 --->72
3 ---> 108
Ask yourself
"What did I do to 1 to get 36?"
"What did I do to 2 to get 72?"
"What did I do to 3 to get 108?"
Answer for all three multiplied by 36
so what do you do to y ?
multiply by 36
y yards -> inches
1 ---> 36
2 --->72
3 ---> 108
y ---> 36y
36y is the expression you are looking for!!
How about the number of yards in i inches.
This time i inches is the unknown
i inches -> y
36 ---> 1
72 --->2
108 ---> 3
This time when you ask the question it is
"What did I do to 36 to get 1?"
What did I do to 72 to get 2?"
"What did I do to 108 to get 3?"
With each of these your answer is you divided by 36.
So what must you do to i... divide by 36
i inches -> y
36 ---> 1
72 --->2
108 ---> 3
i ---> i/36
so i/36 is your expression
We drew a grocery bag with the following info:
Food -> Calories
Bread ---> 55
Apple ---> 70
Banana --->100
Egg ---> 110
3 apples + 2 bananas
3(70) + 2 (100)
210 + 200 = 410 calories
a apples + b bananas
becomes 70a + 100b
e eggs + 1 slice of bread?
110e + 55
Tuesday, September 14, 2010
Algebra (Period 1)
Chapter 1 Review ... continued...
CHAPTER 1-2: COMMUTATIVE PROPERTY
PROPERTIES ARE OUR FRIENDS! (mathematically speaking)
YOU CAN ALWAYS DEPEND ON THEM --- THEY HAVE NO COUNTEREXAMPLES!
COMMUTATIVE PROPERTY (works for all multiplication or all addition)
You can SWITCH THE ORDER and still get the same sum or product.
This is the property YOU CAN HEAR because you've switched the order.
a + b = b + a [you can abbreviate as C+]
OR
ab = ba [ you can abbreviate as Cx]
Therefore, we say that both sides of the equations have EQUIVALENT (=) EXPRESSIONS
SO WHY SHOULD YOU CARE????
Because it makes the math easier sometimes!
Which would you rather multiply:
(2)(543)(5) OR
(2)(5)(543) ???
TWO MORE FRIENDS:
THE IDENTITY PROPERTY OF ADDITION AND
THE IDENTITY PROPERTY OF 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 = 1
a + b - c
We also use this property to SIMPLIFY fractions.
We "cross cancel" 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 since both the numerator and denominator can be divided by 2a.
CHAPTER 1-4: ASSOCIATIVE PROPERTY
ANOTHER FRIEND!
This friend allows you to GROUP all multiplication or all addition ANYWAY YOU CHOOSE!
a + (b + c) = (a + b) + c [ you can abbreviate as A+]
a(bc) = (ab)c [you can abbreviate as Ax]
Why? TO MAKE THE MATH EASIER OF COURSE!
This is the property that YOU CAN SEE instead of hearing because you use ( ) but DON'T CHANGE THE ORDER AS IT IS GIVEN.
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!!!
CHAPTER 1-5: DISTRIBUTIVE PROPERTY
A new friend! It's a property. It has NO counterexamples.
a( b + c) = ab + ac
or
(b + c)a = ba + ca (What other property justifies this corollary?)
You don't need to show the "middle step" once you understand this property.
EXAMPLE: -5(3y - 4) = (-5)(3y) + (-5)(-4) = - 15y + 20
You can't combine the y term with the 20 because they are UNLIKE TERMS.
LIKE TERMS:
1. Same variable (s)
2. Same exponent(s)
Constants are like terms because they all have no variables
You can only combine (add or subtract) like terms.
BUT YOU CAN MULTIPLY UNLIKE TERMS!
3a(7y) = 21ay
BUT
3a + 7y cannot be simplified
CHAPTER 1-2: COMMUTATIVE PROPERTY
PROPERTIES ARE OUR FRIENDS! (mathematically speaking)
YOU CAN ALWAYS DEPEND ON THEM --- THEY HAVE NO COUNTEREXAMPLES!
COMMUTATIVE PROPERTY (works for all multiplication or all addition)
You can SWITCH THE ORDER and still get the same sum or product.
This is the property YOU CAN HEAR because you've switched the order.
a + b = b + a [you can abbreviate as C+]
OR
ab = ba [ you can abbreviate as Cx]
Therefore, we say that both sides of the equations have EQUIVALENT (=) EXPRESSIONS
SO WHY SHOULD YOU CARE????
Because it makes the math easier sometimes!
Which would you rather multiply:
(2)(543)(5) OR
(2)(5)(543) ???
TWO MORE FRIENDS:
THE IDENTITY PROPERTY OF ADDITION AND
THE IDENTITY PROPERTY OF 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 = 1
a + b - c
We also use this property to SIMPLIFY fractions.
We "cross cancel" 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 since both the numerator and denominator can be divided by 2a.
CHAPTER 1-4: ASSOCIATIVE PROPERTY
ANOTHER FRIEND!
This friend allows you to GROUP all multiplication or all addition ANYWAY YOU CHOOSE!
a + (b + c) = (a + b) + c [ you can abbreviate as A+]
a(bc) = (ab)c [you can abbreviate as Ax]
Why? TO MAKE THE MATH EASIER OF COURSE!
This is the property that YOU CAN SEE instead of hearing because you use ( ) but DON'T CHANGE THE ORDER AS IT IS GIVEN.
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!!!
CHAPTER 1-5: DISTRIBUTIVE PROPERTY
A new friend! It's a property. It has NO counterexamples.
a( b + c) = ab + ac
or
(b + c)a = ba + ca (What other property justifies this corollary?)
You don't need to show the "middle step" once you understand this property.
EXAMPLE: -5(3y - 4) = (-5)(3y) + (-5)(-4) = - 15y + 20
You can't combine the y term with the 20 because they are UNLIKE TERMS.
LIKE TERMS:
1. Same variable (s)
2. Same exponent(s)
Constants are like terms because they all have no variables
You can only combine (add or subtract) like terms.
BUT YOU CAN MULTIPLY UNLIKE TERMS!
3a(7y) = 21ay
BUT
3a + 7y cannot be simplified
Math 6H (Period 6 & 7)
Inverse Operations 1-3
Inverse Operations undo each other
Addition & subtraction are inverse operations
Multiplication & division are inverse operations
We can use these relationships between inverse operations to simplify some numerical and variable expressions
17- 8 + 8 instead of doing 17 - 8 and then adding 8 again simply cancel the + and the -
That is, 17 - 8 + 8 = 17
9 - x + x = 9 because adding x and subtracting x are inverse operations and they undo each other...
108 ÷ 6 × 6 ... the long way would be to divide 108 by 6 first. That's 18 and then multiply 18 by 6 and get 108.. but why do that? Using the inverse operations you realize that multiplication and division undo each other so 108 ÷ 6 × 6 is simply 108
3n ÷ 3 = 3
Subtraction Property of Zero
For any whole number a
a-0 = a and a-a = 0
Division Property of One
For any whole number a, except 0
a ÷ 1 = a and a ÷ a = 1
Division Property of Zero
For any whole number a, except 0
0 ÷ a = 0
Remember we discussed that we can never divide any number by 0
We said that any number
a ÷ 0 was said to be 'undefined.'
Inverse Operations undo each other
Addition & subtraction are inverse operations
Multiplication & division are inverse operations
We can use these relationships between inverse operations to simplify some numerical and variable expressions
17- 8 + 8 instead of doing 17 - 8 and then adding 8 again simply cancel the + and the -
That is, 17 - 8 + 8 = 17
9 - x + x = 9 because adding x and subtracting x are inverse operations and they undo each other...
108 ÷ 6 × 6 ... the long way would be to divide 108 by 6 first. That's 18 and then multiply 18 by 6 and get 108.. but why do that? Using the inverse operations you realize that multiplication and division undo each other so 108 ÷ 6 × 6 is simply 108
3n ÷ 3 = 3
Subtraction Property of Zero
For any whole number a
a-0 = a and a-a = 0
Division Property of One
For any whole number a, except 0
a ÷ 1 = a and a ÷ a = 1
Division Property of Zero
For any whole number a, except 0
0 ÷ a = 0
Remember we discussed that we can never divide any number by 0
We said that any number
a ÷ 0 was said to be 'undefined.'
Monday, September 13, 2010
Algebra (Period 1)
Chapter 1 Review
Exponential Notation
exponent ---> little number that is raised called superscript on your computer.
43
the 4 is the base and the 3 is the exponent.
the power is how many times the repeated multiplication will occur... in this case the power is 3!!
Some terms to know:
exponentiation notation : written with an exponent (8k)4
expanded notation: 8k⋅ 8k⋅ 8k⋅ 8k
(10c)3 10⋅10⋅10⋅c⋅c⋅c
standard notation: the number value
33 = 27
(3x)3 = 27x3
but 3x3 is just 3⋅x⋅x⋅x or just the same as you started with!! 3x3
Watch your HUGS.. they are very important in life.. and in math!!
(3x)3 where x = 2
is [(3)(2)]3 or 6 3 = 6⋅6⋅6 = 216
but 3x3 is 3 (2)3 or 3⋅2⋅2⋅2⋅ = 24
Powers of negative numbers and the odd/even rule
If there is a negative inside the (hugs)
odd number of negative signs or an odd power ---> its negative
even # of negative signs or an even power ---> positive
(-2) 5 really means (-2)(-2)(-2)(-2)(-2) = -32
(-2) 4 really means (-2)(-2)(-2)(-2) = 16
If there is a negative but NO (hugs)... it is ALWAYS negative
-25 = -32 and
-2 4 = -16
because these really read
"Take the opposite of 2 to the 5th power" and
"Take the opposite of 2 to the 4th power"
Equation Vocab
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
{ } is set symbol
For example
solve y + 4 = 10 if the replacement set is {2, 6, 10}
you discover that
6 + 4 = 10 makes the statement true so 6 is the solution set and you write it
{6}
But what if you have y + 4 = 10
but the replacement set is { 2, 7, 10}
when you plug in each of the values none of them are true
2 + 4 ≠ 10
7 + 4 ≠ 10
10 + 4 ≠ 10
so the solution set is the NULL SET. which you write
{ } or ∅
Equations can be true, false or open!!
If it is numerical they are either true or false...
If equation is Algebraic ( variables) it is always OPEN
Exponential Notation
exponent ---> little number that is raised called superscript on your computer.
43
the 4 is the base and the 3 is the exponent.
the power is how many times the repeated multiplication will occur... in this case the power is 3!!
Some terms to know:
exponentiation notation : written with an exponent (8k)4
expanded notation: 8k⋅ 8k⋅ 8k⋅ 8k
(10c)3 10⋅10⋅10⋅c⋅c⋅c
standard notation: the number value
33 = 27
(3x)3 = 27x3
but 3x3 is just 3⋅x⋅x⋅x or just the same as you started with!! 3x3
Watch your HUGS.. they are very important in life.. and in math!!
(3x)3 where x = 2
is [(3)(2)]3 or 6 3 = 6⋅6⋅6 = 216
but 3x3 is 3 (2)3 or 3⋅2⋅2⋅2⋅ = 24
Powers of negative numbers and the odd/even rule
If there is a negative inside the (hugs)
odd number of negative signs or an odd power ---> its negative
even # of negative signs or an even power ---> positive
(-2) 5 really means (-2)(-2)(-2)(-2)(-2) = -32
(-2) 4 really means (-2)(-2)(-2)(-2) = 16
If there is a negative but NO (hugs)... it is ALWAYS negative
-25 = -32 and
-2 4 = -16
because these really read
"Take the opposite of 2 to the 5th power" and
"Take the opposite of 2 to the 4th power"
Equation Vocab
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
{ } is set symbol
For example
solve y + 4 = 10 if the replacement set is {2, 6, 10}
you discover that
6 + 4 = 10 makes the statement true so 6 is the solution set and you write it
{6}
But what if you have y + 4 = 10
but the replacement set is { 2, 7, 10}
when you plug in each of the values none of them are true
2 + 4 ≠ 10
7 + 4 ≠ 10
10 + 4 ≠ 10
so the solution set is the NULL SET. which you write
{ } or ∅
Equations can be true, false or open!!
If it is numerical they are either true or false...
If equation is Algebraic ( variables) it is always OPEN
Math 6H (Periods 6 & 7)
Properties of Addition & Multiplication 1-2
1, 2, 3, 4,5 .... are counting numbers or natural numbers.
If we add 0... we have 0,1,2,3,4,... and we have the set of whole numbers
Some important properties are found using whole numbers:
Commutative Property of Addition (C+)
for any whole numbers a and b
a + b = b + a
Commutative Property of Multiplication (Cx)
for any whole numbers a and b
ab = ba
You HEAR the differences.. remember we made the big C into an ear... or at least we attempted to!!
However look at the following
(6 +5) + 7 = 11 + 7 = 18
but so does
6 + (5 + 7) = 6 + 12 = 18
Associative Property of Addition ( A+)
For any whole numbers a, b, and c
(a + b) + c = a + ( b + c)
Associative Property of Multiplication ( Ax)
For any whole numbers a, b, and c
(ab)c = a(bc)
You SEE the difference with the hugs ( ).. hugs are really important in life.. and in math!!
Addition Property of Zero from the book
also known as the Identity Property of Addition ( Id+)
for any whole number a
a + 0 = a and 0 + a = a
Multiplication Property of One
also know as the Identity Property of Multiplication (Idx)
For any whole number a
a(1) = a and 1(a) = a
Multiplication Property of Zero
for any whole number a
0(a) = 0 and 0(a) = 0
So how do we justify?
16 + 19 + 34 ---> given
16 + 34 + 19 ---> C+
(16 + 34) + 19---> A+
50 + 19---> simplify
69---> simplify
2 (14)(15)---> given
2 (15) (14) ---> Cx
[2(15)] (14) ---> Ax
30 (14)---> simplify
420---> simplify
how about the following. What would make it easier to simplify???
8⋅ 14⋅ 25 ⋅5 ---> given
14 ⋅ 8 ⋅ 5⋅ 25 ---> Cx
14 ⋅ (8 ⋅ 5)⋅ 25 ---> Ax
14 ⋅ 40⋅ 25--->simplify
14 ⋅ (40⋅ 25) ---> Ax
14 ⋅ 1000 ---> simplify
14000 ---> simplify
Make sure you justify each step of tonight's homework!!
1, 2, 3, 4,5 .... are counting numbers or natural numbers.
If we add 0... we have 0,1,2,3,4,... and we have the set of whole numbers
Some important properties are found using whole numbers:
Commutative Property of Addition (C+)
for any whole numbers a and b
a + b = b + a
Commutative Property of Multiplication (Cx)
for any whole numbers a and b
ab = ba
You HEAR the differences.. remember we made the big C into an ear... or at least we attempted to!!
However look at the following
(6 +5) + 7 = 11 + 7 = 18
but so does
6 + (5 + 7) = 6 + 12 = 18
Associative Property of Addition ( A+)
For any whole numbers a, b, and c
(a + b) + c = a + ( b + c)
Associative Property of Multiplication ( Ax)
For any whole numbers a, b, and c
(ab)c = a(bc)
You SEE the difference with the hugs ( ).. hugs are really important in life.. and in math!!
Addition Property of Zero from the book
also known as the Identity Property of Addition ( Id+)
for any whole number a
a + 0 = a and 0 + a = a
Multiplication Property of One
also know as the Identity Property of Multiplication (Idx)
For any whole number a
a(1) = a and 1(a) = a
Multiplication Property of Zero
for any whole number a
0(a) = 0 and 0(a) = 0
So how do we justify?
16 + 19 + 34 ---> given
16 + 34 + 19 ---> C+
(16 + 34) + 19---> A+
50 + 19---> simplify
69---> simplify
2 (14)(15)---> given
2 (15) (14) ---> Cx
[2(15)] (14) ---> Ax
30 (14)---> simplify
420---> simplify
how about the following. What would make it easier to simplify???
8⋅ 14⋅ 25 ⋅5 ---> given
14 ⋅ 8 ⋅ 5⋅ 25 ---> Cx
14 ⋅ (8 ⋅ 5)⋅ 25 ---> Ax
14 ⋅ 40⋅ 25--->simplify
14 ⋅ (40⋅ 25) ---> Ax
14 ⋅ 1000 ---> simplify
14000 ---> simplify
Make sure you justify each step of tonight's homework!!
Sunday, September 12, 2010
Algebra (Period 1)
Our Class Blog
Welcome to our class blog... where you can earn extra credit by adding your own relevant comments about our class notes for the day.. or where you can find answers from others in your class. Check here often, especially if you have been absent. You might just find out the math strategy that works for you!!
Email me if you are interested in adding notes and/or comments to this blog-- for extra credit!!
Welcome to our class blog... where you can earn extra credit by adding your own relevant comments about our class notes for the day.. or where you can find answers from others in your class. Check here often, especially if you have been absent. You might just find out the math strategy that works for you!!
Email me if you are interested in adding notes and/or comments to this blog-- for extra credit!!
Pre Algebra (Periods 2 & 4)
Our Class Blog
Welcome to our class blog... where you can earn extra credit by adding your own relevant comments about our class notes for the day.. or where you can find answers from others in your class. Check here often, especially if you have been absent. You might just find out the math strategy that works for you!!
Email me if you are interested in adding notes and/or comments to this blog-- for extra credit!!
Expressions 1-1
Expression are NOT equations. An Expression does NOT have an equal sign (=).
Words you need to know:
Addition: sum, increased by, more than
Subtraction: difference, less THAN, subtracted from
Be careful with this one... "3 less than a number n" is n-3. It isn't in the order that you read. Think about "3 less than 10" What would you write ? 10-3...
Multiplication: product, times
Division: quotient or Per... When we say " miles per hour" it is really the MILES DIVIDED by the HOURS!!
The Order Of Operations 1-2
O3 = Order Of Operations
It’s Aunt Sally’s Rule—
Please Excuse My Dear Aunt Sally
Please ( ) parentheses
Excuse E2 exponents
My Dear (or even Dear My) × ÷ or ÷ ×
Aunt Sally (Sally Aunt) + - or - +
also called ... PEMDAS
The most common mistake of O3
Student forget that 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.
Evaluating Expressions 1-3
First substitute in for the variable..
Follow AUNT SALLY's RULE or PEMDAS
4y - 15 where y = 9
4(9) - 15
36 - 15
21
3ab + c/2 where a = 2 b = 5 and c = 10
3(2)(5) + 10/2
30 + 5
35
Be careful when substituting in negatives...
Welcome to our class blog... where you can earn extra credit by adding your own relevant comments about our class notes for the day.. or where you can find answers from others in your class. Check here often, especially if you have been absent. You might just find out the math strategy that works for you!!
Email me if you are interested in adding notes and/or comments to this blog-- for extra credit!!
Expressions 1-1
Expression are NOT equations. An Expression does NOT have an equal sign (=).
Words you need to know:
Addition: sum, increased by, more than
Subtraction: difference, less THAN, subtracted from
Be careful with this one... "3 less than a number n" is n-3. It isn't in the order that you read. Think about "3 less than 10" What would you write ? 10-3...
Multiplication: product, times
Division: quotient or Per... When we say " miles per hour" it is really the MILES DIVIDED by the HOURS!!
The Order Of Operations 1-2
O3 = Order Of Operations
It’s Aunt Sally’s Rule—
Please Excuse My Dear Aunt Sally
Please ( ) parentheses
Excuse E2 exponents
My Dear (or even Dear My) × ÷ or ÷ ×
Aunt Sally (Sally Aunt) + - or - +
also called ... PEMDAS
The most common mistake of O3
Student forget that 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.
Evaluating Expressions 1-3
First substitute in for the variable..
Follow AUNT SALLY's RULE or PEMDAS
4y - 15 where y = 9
4(9) - 15
36 - 15
21
3ab + c/2 where a = 2 b = 5 and c = 10
3(2)(5) + 10/2
30 + 5
35
Be careful when substituting in negatives...
Tuesday, September 7, 2010
Math 6 Honors (Period 6 and 7)
Our Class Blog
Welcome to our class blog... where you can earn extra credit by adding your own relevant comments about our class notes for the day.. or where you can find answers from others in your class. Check here often, especially if you have been absent. You might just find out the math strategy that works for you!!
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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.
expression = expression
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 value for the variable a
6(2) = 12
6(4) = 24
6(6) = 36
6(8) = 48
148 ÷ 4 =
148/4
37
if m = 3 and n = 18
n ÷ m
substitute in
n/m or 18/3 = 6
If y = 18 and x = 8
4y ÷ 3x immediately set this up as
4y/3x
Now substitute in your values
4(18) / 3(8)
72/24 = 3
Welcome to our class blog... where you can earn extra credit by adding your own relevant comments about our class notes for the day.. or where you can find answers from others in your class. Check here often, especially if you have been absent. You might just find out the math strategy that works for you!!
Email me if you are interested in adding notes and/or comments to this blog-- for extra credit!!
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.
expression = expression
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 value for the variable a
6(2) = 12
6(4) = 24
6(6) = 36
6(8) = 48
148 ÷ 4 =
148/4
37
if m = 3 and n = 18
n ÷ m
substitute in
n/m or 18/3 = 6
If y = 18 and x = 8
4y ÷ 3x immediately set this up as
4y/3x
Now substitute in your values
4(18) / 3(8)
72/24 = 3
Friday, June 4, 2010
Congruent Figures 4-7
Two figures are congruent if they have the same size and same shape. If we could lift on of the figures and place it directly on top of the other... all three vertices would match up with the others. In the example in our book ( page 132) we have two congruent triangles ∆ABC and ∆XYZ A would fall on X, B would fall on Y and C would fall on Z. These matching vertices are called corresponding vertices. Angles at corresponding vertices are corresponding angles and the sides joining corresponding vertices are corresponding sides.
The book states Corresponding angles of congruent figures are congruent.
and
Corresponding sides of congruent figures are congruent.
In class we discussed how to abbreviate the above -- when dealing with triangles.
CPCTC
Corresponding PARTS of congruent triangles are congruent!!
When we name two congruent figures we list corresponding vertices in the same order.
∆ABC ≅ ∆XYZ or ∆CAB ≅ ∆ZXY or ∆BCA ≅ ∆YZX
we know that
∠A ≅ ∠X and ∠B ≅ ∠Y and ∠C ≅ ∠Z
and the segments ( which are denoted with a line (but w/o arrows)above each of the two letters
AB ≅ XY and BC ≅ YZ and CA ≅ ZX
If two figures are congruent, we can make the coincide -- occupy the same place-- by using one or more of the following basic rigid motions:
Translation or slide
Rotation
Reflection or flip or mirror
Check the book on page 133 for good examples of these three rigid motions... I like to think of Tetris moves!!
Two figures are congruent if they have the same size and same shape. If we could lift on of the figures and place it directly on top of the other... all three vertices would match up with the others. In the example in our book ( page 132) we have two congruent triangles ∆ABC and ∆XYZ A would fall on X, B would fall on Y and C would fall on Z. These matching vertices are called corresponding vertices. Angles at corresponding vertices are corresponding angles and the sides joining corresponding vertices are corresponding sides.
The book states Corresponding angles of congruent figures are congruent.
and
Corresponding sides of congruent figures are congruent.
In class we discussed how to abbreviate the above -- when dealing with triangles.
CPCTC
Corresponding PARTS of congruent triangles are congruent!!
When we name two congruent figures we list corresponding vertices in the same order.
∆ABC ≅ ∆XYZ or ∆CAB ≅ ∆ZXY or ∆BCA ≅ ∆YZX
we know that
∠A ≅ ∠X and ∠B ≅ ∠Y and ∠C ≅ ∠Z
and the segments ( which are denoted with a line (but w/o arrows)above each of the two letters
AB ≅ XY and BC ≅ YZ and CA ≅ ZX
If two figures are congruent, we can make the coincide -- occupy the same place-- by using one or more of the following basic rigid motions:
Translation or slide
Rotation
Reflection or flip or mirror
Check the book on page 133 for good examples of these three rigid motions... I like to think of Tetris moves!!
Thursday, June 3, 2010
Math 6H ( Periods 3, 6, & 7)
Circle 4-5
A circle is the set of all points in a plane at a given distance from a given point O ( called the center)
A segment joining the center to a point on the circle is called a radius (plural: radii) of the circle ALL radii of a given circle have the same length and this length is called the radius of the circle
A segment joining two points on the circle is called a chord. However a chord passing through the center is called a diameter of the circle. The ends of the diameter divide the circle into two semicircles. The length of the diameter is called the diameter of the circle.
The perimeter of a circle is called the circumference
The quotient
circumference ÷ diameter
is the same for all circles regardless of their size. This quotient is denoted by the Greek letter ∏ ( pronounced 'pie') No decimal gives ∏ exactly.-- but a good approximation is 3.14
If we denote the circumference by C and the diameter by d
we have
C ÷ d = ∏.
We can manipulate this formula to several useful forms:
C = ∏d
or d = C ÷ ∏ .
In addition since we know that d = 2r we can also have the following:
C = 2 ∏ r
or r = C ÷(2∏)
When using the approximate 3.14 for ∏ , give your answer in three digits-- as the book explains.
In addition, use the approximately equal to symbol ≈ when you have replaced ∏ .
A polygon is inscribed in a circle if all of its vertices are on the circle. See the diagram on Page 129 for a good example of an inscribed triangle.
It can be shown that three points NOT ON A LINE determine a circle.
A circle is the set of all points in a plane at a given distance from a given point O ( called the center)
A segment joining the center to a point on the circle is called a radius (plural: radii) of the circle ALL radii of a given circle have the same length and this length is called the radius of the circle
A segment joining two points on the circle is called a chord. However a chord passing through the center is called a diameter of the circle. The ends of the diameter divide the circle into two semicircles. The length of the diameter is called the diameter of the circle.
The perimeter of a circle is called the circumference
The quotient
circumference ÷ diameter
is the same for all circles regardless of their size. This quotient is denoted by the Greek letter ∏ ( pronounced 'pie') No decimal gives ∏ exactly.-- but a good approximation is 3.14
If we denote the circumference by C and the diameter by d
we have
C ÷ d = ∏.
We can manipulate this formula to several useful forms:
C = ∏d
or d = C ÷ ∏ .
In addition since we know that d = 2r we can also have the following:
C = 2 ∏ r
or r = C ÷(2∏)
When using the approximate 3.14 for ∏ , give your answer in three digits-- as the book explains.
In addition, use the approximately equal to symbol ≈ when you have replaced ∏ .
A polygon is inscribed in a circle if all of its vertices are on the circle. See the diagram on Page 129 for a good example of an inscribed triangle.
It can be shown that three points NOT ON A LINE determine a circle.
Wednesday, June 2, 2010
Math 6H ( Periods 3, 6, & 7)
Polygons 4-5
A polygon is a closed figure formed by joining segments—the sides of the polygons at that endpoints—the vertices of the polygon. Polygons are names according to the number of sides they have.
Triangle 3 sides
Quadrilateral 4 sides
Pentagon 5 sides
Hexagon 6 sides
Octagon 8 sides
Decagon 10 sides
A polygon is REGULAR if all its sides and all its angles are congruent.
A regular triangle is the equilateral triangle
A regular quadrilateral is the square.
To name a polygon we name its consecutive vertices in order.
A diagonal of a polygon is a segment joining two non consecutive vertices.
To find the perimeter of a polygon add all the lengths of its sides. The perimeter is the distance around the figure. Finding the perimeter of a parallelogram can be done by computing the sum of the lengths or by using the distributive property to obtain
For instance a parallelogram with sides 9 cm and 6 cm
has a perimeter of 9 + 6 + 9 + 6 = 30 cm
but you could calculate that by 2(9) + 2(6) = 18 + 12 = 30 cm or using the distributive property, even 2(9+6) = 2(15) = 30 cm
If you have a regular polygon you can simple multiple the side by the number of sides in the polygon
For example,
a quadrilateral with side 16.5 m has a perimeter of 4(16.5) = 66 m
The sum of the measures of the angles of any pentagon is 540 degrees. If it is a regular pentagon, what must be the measure of each angle of the regular pentagon? 540/5 = 108 degrees.
The sum of the measures of the angles of any pentagon is 540 degrees. How can you prove that? Draw your pentagon and then draw all the diagonals from ONE of the vertices. Count the number of triangles created. Three. How many degrees does a triangle have? 180. Multiply the number of triangles created by 180… 540 is your answer. IT works every time. So How could you create a general rule or formula for the sum of the measures of the angles of any polygon with n sides?
Practice drawing various polygons—now practice drawing all the diagonals for each of them. Can you determine a general rule for the number of diagonals that can be drawn for any polygon?
A polygon is a closed figure formed by joining segments—the sides of the polygons at that endpoints—the vertices of the polygon. Polygons are names according to the number of sides they have.
Triangle 3 sides
Quadrilateral 4 sides
Pentagon 5 sides
Hexagon 6 sides
Octagon 8 sides
Decagon 10 sides
A polygon is REGULAR if all its sides and all its angles are congruent.
A regular triangle is the equilateral triangle
A regular quadrilateral is the square.
To name a polygon we name its consecutive vertices in order.
A diagonal of a polygon is a segment joining two non consecutive vertices.
To find the perimeter of a polygon add all the lengths of its sides. The perimeter is the distance around the figure. Finding the perimeter of a parallelogram can be done by computing the sum of the lengths or by using the distributive property to obtain
For instance a parallelogram with sides 9 cm and 6 cm
has a perimeter of 9 + 6 + 9 + 6 = 30 cm
but you could calculate that by 2(9) + 2(6) = 18 + 12 = 30 cm or using the distributive property, even 2(9+6) = 2(15) = 30 cm
If you have a regular polygon you can simple multiple the side by the number of sides in the polygon
For example,
a quadrilateral with side 16.5 m has a perimeter of 4(16.5) = 66 m
The sum of the measures of the angles of any pentagon is 540 degrees. If it is a regular pentagon, what must be the measure of each angle of the regular pentagon? 540/5 = 108 degrees.
The sum of the measures of the angles of any pentagon is 540 degrees. How can you prove that? Draw your pentagon and then draw all the diagonals from ONE of the vertices. Count the number of triangles created. Three. How many degrees does a triangle have? 180. Multiply the number of triangles created by 180… 540 is your answer. IT works every time. So How could you create a general rule or formula for the sum of the measures of the angles of any polygon with n sides?
Practice drawing various polygons—now practice drawing all the diagonals for each of them. Can you determine a general rule for the number of diagonals that can be drawn for any polygon?
Tuesday, May 25, 2010
Math 6H ( Periods 3, 6, & 7)
Angles and Angle Measure 4-3
An angle is a figure formed by two rays with the same endpoints. The common endpoint is called the vertex. The rays are called the sides.
We may name an angle by giving its vertex letter if this is the only angle with that vertex, or my listing letters for points on the two sides with the vertex letter in the middle. We use the symbol from the textbook.
To measure segments we use a rule to mark off unit lengths. To measure angles, we use a protractor that is marked off in units of angle measure called degrees.
To use a protractor, place its center point at the vertex of the angle to be measured and one of its zero points on the side.
We often label angels with their measures. When angles have equal measures we can write m angle A = m angle B
We say that angle A and angle B are congruent angles
If two lines intersect so that the angles they form are all congruent, the lines are perpendicular. We use the symbol that looks like an upside down capital T to mean “is perpendicular to.”
Angles formed by perpendicular lines each have measure of 90° . A 90° angle is called a right angle. A small square is often used to indicate a right angle in a diagram
An acute angle is an angle with measure less than 90°. An obtuse angle has measure between 90° and 180°
Two angles are complementary if the sum of the measures is 90°
Two angles are supplementary if the sum of their measures is 180°
Triangles 4-4
A triangle is the figure formed when three points, not on a line are jointed by segments.
Triangle ABC ΔABC
Each of the Points A, B, C is called a vertex
(plural: Vertices) of ΔABC
Each of the angles angle A. angle B. and angle C is called an angle of ΔABC
In any triangle-
The sum of the lengths of any two sides is greater than the length of the third side
The sum of the measures of the angles is 180
There are several ways to name triangles. One way is by angles
Acute Triangle
3 acute angles
Right Triangle
1 right angle
Obtuse Triangle
1 obtuse angle
Triangles can be classified by their sides
Scalene Triangle
no 2 sides congruent
Isosceles Triangle
at least 2 sides congruent
Equilateral Triangle
all 3 sides congruent
The longest side of a triangle is opposite the largest angle and the shortest side is opposite the smallest angle. Two angles are congruent if and only if the sides opposite them are congruent.
An angle is a figure formed by two rays with the same endpoints. The common endpoint is called the vertex. The rays are called the sides.
We may name an angle by giving its vertex letter if this is the only angle with that vertex, or my listing letters for points on the two sides with the vertex letter in the middle. We use the symbol from the textbook.
To measure segments we use a rule to mark off unit lengths. To measure angles, we use a protractor that is marked off in units of angle measure called degrees.
To use a protractor, place its center point at the vertex of the angle to be measured and one of its zero points on the side.
We often label angels with their measures. When angles have equal measures we can write m angle A = m angle B
We say that angle A and angle B are congruent angles
If two lines intersect so that the angles they form are all congruent, the lines are perpendicular. We use the symbol that looks like an upside down capital T to mean “is perpendicular to.”
Angles formed by perpendicular lines each have measure of 90° . A 90° angle is called a right angle. A small square is often used to indicate a right angle in a diagram
An acute angle is an angle with measure less than 90°. An obtuse angle has measure between 90° and 180°
Two angles are complementary if the sum of the measures is 90°
Two angles are supplementary if the sum of their measures is 180°
Triangles 4-4
A triangle is the figure formed when three points, not on a line are jointed by segments.
Triangle ABC ΔABC
Each of the Points A, B, C is called a vertex
(plural: Vertices) of ΔABC
Each of the angles angle A. angle B. and angle C is called an angle of ΔABC
In any triangle-
The sum of the lengths of any two sides is greater than the length of the third side
The sum of the measures of the angles is 180
There are several ways to name triangles. One way is by angles
Acute Triangle
3 acute angles
Right Triangle
1 right angle
Obtuse Triangle
1 obtuse angle
Triangles can be classified by their sides
Scalene Triangle
no 2 sides congruent
Isosceles Triangle
at least 2 sides congruent
Equilateral Triangle
all 3 sides congruent
The longest side of a triangle is opposite the largest angle and the shortest side is opposite the smallest angle. Two angles are congruent if and only if the sides opposite them are congruent.
Monday, May 10, 2010
Math 6H ( Periods 3, 6, & 7)
Simple Interest 9-7
When you borrow money you pay the lender INTEREST for the use of the money. The amount of interest you pay is usually a percent of the amount borrowed figured on a yearly basis. This percent is called the annual rate.
When interest is computed year by year we call it
SIMPLE INTEREST
The formula is I= Prt
Let I = simple interest charges
P = principal ( amount borrowed)
r= annual rate
t = time in years
I = Prt
simple interest is calculated just on the principal.
Let's work through a few examples
$150 borrowed at 12% annual rate for 1 year
I = Prt
I = (150)(.12)(1)
I = 18
so you would owe $18 in interest after 1 year.
The total due would be $150 + 18 = $168
What if instead you borrowed the same amount but for 2 years... nothing was due until the end of two years
I = Prt
I = 150(.12)(2) = 36
You would owe $36 in interest .. so the total due was 150 + 36 = $ 186.
What if you borrowed the same amount for 3 years...
I = 150(.12)(3) = 54 or $54 in interest.
You would owe 150 + 54 = $ 204 after three years...
However, let's say you could only borrow that amount for 6 months...
I = Prt
I = (150)(.12)(.5)
Why 0.5? that is 1/2 a year.
Now you can always multiply by 1/2 as well.. in fact, sometimes that is easier
I = 150(.12)(1/2) = 9 or $ 9.00
After 6 months you would owe $159.
Dylan paid $375 in interest on a loan of $1500 principal at 12.5% interest.
What was the length of time?
Look at what it is asking and see which of the variables you have...
I= Prt
We have the interest paid, the principal and the annual rate so
375= (1500)(.125)(t)
375 = 187.5t
solve this one step equation by dividing both sides by 187.5
375 = 187.5t
187.5 187.5
t = 2
so 2 years
divide carefully...
Alexis paid $ 585 simple interest on a $6500 loan for 6 months.
what was the annual rate?
What do we know?
I = 585
P = 6500
t= 6 months ( which is 0.5 or 1/2)
I = Prt
585 = 6500 (r)(.5)
585 = 3250r
divide both sides by 3250
585 = 3250r
3250 3250
r = 0.18
which means 18%
annual--> once a year
6 months --> 1/2 or 0.5
4 months--> 1/3
3 months --> 1/4 or 0.25
8 month --> 2/3
Compound Interest 9-8
Compound interest is ALWAYS more than simple interest.
interest is compounded on the interest!!
$100 savings earning $10 interest/ annual.. [this only happens NOW if your dad is the one paying you... :)]
I = Prt
at the end of the first year
I = 100(.10)(1) = 10 or $10
add that to the 100
$110.
Now for the 2nd year,
$110 is your principal
so
I = Prt
I = 110(.10)(1) = 11 or $11
so at the end of 2 years you have $110 + 11 or $121
Now for the 3rd year
I = Prt
I = 121(10)(1) = $12.10
So at the end of three years you have $121 + 12.10 = $133.10
What if you had $500 at 8% compounded quarterly for one year.
quarterly means 1/4 or .25
I = Prt
I = 500(.08) (1/4)
calculate the 08(1/4) because that will be the constant you will multiply your principal by each time
(.08)(1/4) = .02
so I = 500(.02) = 10
after the first quarter it is 510
I = Prt for the 2nd quarter
I = 510 (.02) = 10.20
so after the 2nd quarter $510 + 10.20 = $520.20
I = Prt for the third quarter
I = 520.20 (0.02) = about $10.40 ( round to the nearest penny)
so after the third quarter
$520.20 + 10.40 = $530.60
I = Prt
I = 530.60(.02) = about $10.61
So at the end of 4 quarters -- or one year
530.60 + 10.61 = $541.21
compounding terms:
annually--> once a year
semiannually --> twice a year
quarterly--> four times a year
monthly--> 12 times a year
daily--> 365 times a year
When you borrow money you pay the lender INTEREST for the use of the money. The amount of interest you pay is usually a percent of the amount borrowed figured on a yearly basis. This percent is called the annual rate.
When interest is computed year by year we call it
SIMPLE INTEREST
The formula is I= Prt
Let I = simple interest charges
P = principal ( amount borrowed)
r= annual rate
t = time in years
I = Prt
simple interest is calculated just on the principal.
Let's work through a few examples
$150 borrowed at 12% annual rate for 1 year
I = Prt
I = (150)(.12)(1)
I = 18
so you would owe $18 in interest after 1 year.
The total due would be $150 + 18 = $168
What if instead you borrowed the same amount but for 2 years... nothing was due until the end of two years
I = Prt
I = 150(.12)(2) = 36
You would owe $36 in interest .. so the total due was 150 + 36 = $ 186.
What if you borrowed the same amount for 3 years...
I = 150(.12)(3) = 54 or $54 in interest.
You would owe 150 + 54 = $ 204 after three years...
However, let's say you could only borrow that amount for 6 months...
I = Prt
I = (150)(.12)(.5)
Why 0.5? that is 1/2 a year.
Now you can always multiply by 1/2 as well.. in fact, sometimes that is easier
I = 150(.12)(1/2) = 9 or $ 9.00
After 6 months you would owe $159.
Dylan paid $375 in interest on a loan of $1500 principal at 12.5% interest.
What was the length of time?
Look at what it is asking and see which of the variables you have...
I= Prt
We have the interest paid, the principal and the annual rate so
375= (1500)(.125)(t)
375 = 187.5t
solve this one step equation by dividing both sides by 187.5
375 = 187.5t
187.5 187.5
t = 2
so 2 years
divide carefully...
Alexis paid $ 585 simple interest on a $6500 loan for 6 months.
what was the annual rate?
What do we know?
I = 585
P = 6500
t= 6 months ( which is 0.5 or 1/2)
I = Prt
585 = 6500 (r)(.5)
585 = 3250r
divide both sides by 3250
585 = 3250r
3250 3250
r = 0.18
which means 18%
annual--> once a year
6 months --> 1/2 or 0.5
4 months--> 1/3
3 months --> 1/4 or 0.25
8 month --> 2/3
Compound Interest 9-8
Compound interest is ALWAYS more than simple interest.
interest is compounded on the interest!!
$100 savings earning $10 interest/ annual.. [this only happens NOW if your dad is the one paying you... :)]
I = Prt
at the end of the first year
I = 100(.10)(1) = 10 or $10
add that to the 100
$110.
Now for the 2nd year,
$110 is your principal
so
I = Prt
I = 110(.10)(1) = 11 or $11
so at the end of 2 years you have $110 + 11 or $121
Now for the 3rd year
I = Prt
I = 121(10)(1) = $12.10
So at the end of three years you have $121 + 12.10 = $133.10
What if you had $500 at 8% compounded quarterly for one year.
quarterly means 1/4 or .25
I = Prt
I = 500(.08) (1/4)
calculate the 08(1/4) because that will be the constant you will multiply your principal by each time
(.08)(1/4) = .02
so I = 500(.02) = 10
after the first quarter it is 510
I = Prt for the 2nd quarter
I = 510 (.02) = 10.20
so after the 2nd quarter $510 + 10.20 = $520.20
I = Prt for the third quarter
I = 520.20 (0.02) = about $10.40 ( round to the nearest penny)
so after the third quarter
$520.20 + 10.40 = $530.60
I = Prt
I = 530.60(.02) = about $10.61
So at the end of 4 quarters -- or one year
530.60 + 10.61 = $541.21
compounding terms:
annually--> once a year
semiannually --> twice a year
quarterly--> four times a year
monthly--> 12 times a year
daily--> 365 times a year
Monday, May 3, 2010
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