Decimals 3-3
Although decimals ( termed decimal fractions) had been used for centuries, Simon Stevin in the 16th century began using them on a daily basis and he helped establish their use in the fields of sciences and engineering.
Note that
1/10 = 1/101
1/100 = 1/102
1/1000 = 1/103
We also know that
1/10= 0/1
1/100 = 0.01
1/1000 = 0.001
1/10000 = 0.0001
and so on... these strings of digits are called decimals.
Remember how we proved that any number to he zero power was equal to 1
or a0 = 1
Refer back to your notes or to the blog a few days ago...
we also showed how
What happens when you multiply the same bases?
34 ⋅ 32 = 3⋅3⋅3⋅3⋅3⋅3
or 34+2 = 3 6
We just add the exponents if the bases are the same!!
When we divide by the same base we just subtract
34 /32 = 34-2 =32
What would happen if we had
32 / 34 ?
Let's look at what we would actually have
3⋅3
3⋅3⋅3⋅3
Which would be
1
32
or 1/32
but you can write that as 3-2
We just subtract-- using the same rule.
Now let's get back to our decimal lesson and apply that to decimals -- and the Powers of TEN
SO 1/10 = 1/101= 0.01 and it is equal to 10-1
Notice that 10-1 is NOT a negative number-- it is a small number
and 10-21 is not a negative number it is a VERY TINY number
AS with whole numbers, decimals use place values. These place values are to the RIGHT of the decimal point.
We need to be able to write decimals in words as well as expanded notation.
In class we used 0.6394 as our example
zero and six thousand three hundred ninety-four ten-thousandths.
Notice how this number when written in words begins...with "ZERO AND"
Why do we need to do that?
Also notice that there is a hyphen between ten and thousandths in ten-thousandths. It is critical to understand when you must place a hyphen.
We read the entire number to the right of the decimal point as if it represented a whole number, and then we give the place value of the digit farthest to the right.
So, although 0.400 is equivalent to 0.4
we must read 0.400 as "zero and four hundred thousandths."
Now look at the following words
"zero and four hundred-thousandths." What is the subtle difference between those two phrases above?
There is a hyphen in the last phrase-- which means that the hundred and the thousandths are attached and represent a place value so
zero and four hundred-thousandths is 0.00004 while
zero and four hundred thousandths is 0.400
Carefully see the distinction!!
Getting back to our 0.6394
to write it in decimals sums and then in exponents:
0 + 0.6 + 0.03 + 0.009 + 0.0004
0 + 6(0.1) + 3(0.01) +9(0.001) + 4(0.0001)
0(100) + 6(10-1)+ 3(10-2)+ 9(10-3)+ 4(10-4)
14.35 is read as fourteen AND thirty-five hundredths.
When reading numbers, only use the AND to indicate the decimal point
Wednesday, October 20, 2010
Math 6H (Period 6 & 7)
The Decimal System 3-2
Our system of numbers uses the following ten digits:
0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9
Whole numbers greater than 9 can actually be represented as sums. For example
386 = 300 + 80 + 6
or
3(100) + 8(10) + 6(1)
Notice that each place value is ten times the value of the place value to its RIGHT!!
The number 10 is called the BASE of this system of writing numbers.
The system itself is called the DECIMAL SYSTEM from the Latin word decem-- which means ten
Think December- but why is that month the 12th month? hmmm.. Did anyone know from class?
Look at the chart given to you in class and notice the place names for the first several numbers.
To make numbers with MORE THAN four digits easier to read, commas are used to separate the digits into groups of three-- starting from the RIGHT
Think of the comma as a 'signal' for you to mention a place name category!!
In words the number 420,346 is written as
"four hundred twenty thousand, three hundred forty-six."
The expanded notation for 420,346 is given by
4(100,000) + 2(10,00,000) + 0(1000) + 3(100) + 4(10) + 6(1)
Using exponents the expanded notation may be given as
4(105) + 2(104)+0(103)+3(102)+4(101)+6(100)
What hmmm.. how is (100) = 1
any number to the zero power is 1.
From yesterday we noticed that
102 ⋅ 103 = 105
and
106 ⋅ 103 =109
so we wrote a rule for any exponent values a and b
10a ⋅ 10b = 10a+b
Then I asked what would be a rule for
108
105
and
106
104
Students decided that since we added the exponents when multiplying we could subtract the exponents when dividing!!
so could we write a rule for any exponent values a and b?
YES
10a
10b
= 10a-b
but then
105
105
would be
10 5-5 which is 100
BUT WAIT... isn't any number divided by itself equal to ONE. That is,
105
105
= 1
So
100 = 1
279,043
is read as two hundred seventy-nine thousand, forty-three.
and in expanded notation becomes:
2(100,000) + 7(10,000) + 9(1000) + 0(100) + 4(10) + 3(1)
or with exponents:
2(105) + 7(104) + 9(103) + 0(102) + 4(101) + 3(100)
Writing a variable expression to represent two or three digit numbers requires you to think of the value of each place.
For example,
The ten's digit is t and the ones' digit is 2
you can't just put t + 2 WHY???
Let's say you are thinking of the number 12 when we said the expression
"The ten's digit is t and the ones' digit is 2"
If you said t + 2 you would get 1+2
and that = 3
It isn't the two digit number we wanted--- 12.
so what is the place value of the 1?
It is really in the ten's place or written as 1(10)
To write the variable expression we must in include the value
10t + 2 becomes the correct expression
What about the ten's digit is 5: the ones' digit is x? 5t + x. Do I need to put a 1 in front of the x for the ones' digit? No it is... invisible!!
Some of you asked that I put the Magic Number Trick that we did online... try this one on your parents...
Have your 'victim' follow these directions:
Choose any four-digit number
Write the thousand's digit
Write the thousands' and hundreds' digits
Write the thousands' hundred's and tens' digits.
Add these numbers
Multiply by 9
Find the sum of the digits of the original number
add this sum to the previous results.
Your 'victim's' answers should always be the same as their original number chosen.
Example
I used in class 1492
Write the thousand's digit-------> 1
Write the thousands' and hundreds' digits ---> 14
Write the thousands' hundred's and tens' digits--->149
Add these numbers 1 + 14 + 149 = 164
Multiply by 9 or 164(9)= 1476
Find the sum of the digits of the original number 1 + 4 + 9 + 2 = 16
add this sum to the previous results. 1476 + 16 = 1492
Voila!! It worked!!
Email me your parent's comments or post them here!!
Our system of numbers uses the following ten digits:
0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9
Whole numbers greater than 9 can actually be represented as sums. For example
386 = 300 + 80 + 6
or
3(100) + 8(10) + 6(1)
Notice that each place value is ten times the value of the place value to its RIGHT!!
The number 10 is called the BASE of this system of writing numbers.
The system itself is called the DECIMAL SYSTEM from the Latin word decem-- which means ten
Think December- but why is that month the 12th month? hmmm.. Did anyone know from class?
Look at the chart given to you in class and notice the place names for the first several numbers.
To make numbers with MORE THAN four digits easier to read, commas are used to separate the digits into groups of three-- starting from the RIGHT
Think of the comma as a 'signal' for you to mention a place name category!!
In words the number 420,346 is written as
"four hundred twenty thousand, three hundred forty-six."
The expanded notation for 420,346 is given by
4(100,000) + 2(10,00,000) + 0(1000) + 3(100) + 4(10) + 6(1)
Using exponents the expanded notation may be given as
4(105) + 2(104)+0(103)+3(102)+4(101)+6(100)
What hmmm.. how is (100) = 1
any number to the zero power is 1.
From yesterday we noticed that
102 ⋅ 103 = 105
and
106 ⋅ 103 =109
so we wrote a rule for any exponent values a and b
10a ⋅ 10b = 10a+b
Then I asked what would be a rule for
108
105
and
106
104
Students decided that since we added the exponents when multiplying we could subtract the exponents when dividing!!
so could we write a rule for any exponent values a and b?
YES
10a
10b
= 10a-b
but then
105
105
would be
10 5-5 which is 100
BUT WAIT... isn't any number divided by itself equal to ONE. That is,
105
105
= 1
So
100 = 1
279,043
is read as two hundred seventy-nine thousand, forty-three.
and in expanded notation becomes:
2(100,000) + 7(10,000) + 9(1000) + 0(100) + 4(10) + 3(1)
or with exponents:
2(105) + 7(104) + 9(103) + 0(102) + 4(101) + 3(100)
Writing a variable expression to represent two or three digit numbers requires you to think of the value of each place.
For example,
The ten's digit is t and the ones' digit is 2
you can't just put t + 2 WHY???
Let's say you are thinking of the number 12 when we said the expression
"The ten's digit is t and the ones' digit is 2"
If you said t + 2 you would get 1+2
and that = 3
It isn't the two digit number we wanted--- 12.
so what is the place value of the 1?
It is really in the ten's place or written as 1(10)
To write the variable expression we must in include the value
10t + 2 becomes the correct expression
What about the ten's digit is 5: the ones' digit is x? 5t + x. Do I need to put a 1 in front of the x for the ones' digit? No it is... invisible!!
Some of you asked that I put the Magic Number Trick that we did online... try this one on your parents...
Have your 'victim' follow these directions:
Choose any four-digit number
Write the thousand's digit
Write the thousands' and hundreds' digits
Write the thousands' hundred's and tens' digits.
Add these numbers
Multiply by 9
Find the sum of the digits of the original number
add this sum to the previous results.
Your 'victim's' answers should always be the same as their original number chosen.
Example
I used in class 1492
Write the thousand's digit-------> 1
Write the thousands' and hundreds' digits ---> 14
Write the thousands' hundred's and tens' digits--->149
Add these numbers 1 + 14 + 149 = 164
Multiply by 9 or 164(9)= 1476
Find the sum of the digits of the original number 1 + 4 + 9 + 2 = 16
add this sum to the previous results. 1476 + 16 = 1492
Voila!! It worked!!
Email me your parent's comments or post them here!!
Pre Algebra (Period 2 & 4)
Solving Multi-Step Equations 7-2
2 STEP EQUATIONS WITH SOME MORE STEPS BEFORE BALANCING
1) Sometimes equations have LIKE TERMS on the SAME SIDE.
The easiest first step is to COMBINE these before starting to balance:
9x -2x = -42
combine like terms ( those terms on the same side of the = sign)
7x = -42
now divide by 7
7x/7 = -42/7
x= -6
4a + 1 - a = 19
ADD THE 4a to the -1a FIRST
3a + 1 = 19
NOW SOLVE AS A 2 STEP EQUATION
3a + 1 = 19
-1 = -1
3a = 18
then
3a/3 = 18/3
a = 6
2) Sometimes equations have DISTRIBUTIVE PROPERTY on one side:
Usually, you want to do DISTRIBUTE FIRST!
UNLESS THE FACTOR OUTSIDE THE ( ) CAN BE DIVIDED OUT OF BOTH SIDES PERFECTLY!!!!
(meaning it's compatible!)
In the following example you must distribute -2(2y + 8)
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]
But, with the following example you can divide both sides by 3 BEFORE you distribute
EXAMPLE: 3(4 + x) = 9
4 + x = 3 [Don't distribute! Divide both sides by 3.
The 3 goes into both sides perfectly!
x = -1 [Subtract 4 from both sides]
38 = -3(4y +2) + y
38 = -12y + -6 + y
38 = -11y - 6
+6 +6
44 = -11y
so then
44/-11 = -11y/-11
-4 = y
Useful Steps for Solving A Multi-Step Equation
Step 1: Use DP if necessary
Step 2: Combine like terms
Step 3: Undo Addition or subtraction using +prop= or -props=
Step 4: Undo Multiplication or division using xprop+ or ÷prop=
2 STEP EQUATIONS WITH SOME MORE STEPS BEFORE BALANCING
1) Sometimes equations have LIKE TERMS on the SAME SIDE.
The easiest first step is to COMBINE these before starting to balance:
9x -2x = -42
combine like terms ( those terms on the same side of the = sign)
7x = -42
now divide by 7
7x/7 = -42/7
x= -6
4a + 1 - a = 19
ADD THE 4a to the -1a FIRST
3a + 1 = 19
NOW SOLVE AS A 2 STEP EQUATION
3a + 1 = 19
-1 = -1
3a = 18
then
3a/3 = 18/3
a = 6
2) Sometimes equations have DISTRIBUTIVE PROPERTY on one side:
Usually, you want to do DISTRIBUTE FIRST!
UNLESS THE FACTOR OUTSIDE THE ( ) CAN BE DIVIDED OUT OF BOTH SIDES PERFECTLY!!!!
(meaning it's compatible!)
In the following example you must distribute -2(2y + 8)
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]
But, with the following example you can divide both sides by 3 BEFORE you distribute
EXAMPLE: 3(4 + x) = 9
4 + x = 3 [Don't distribute! Divide both sides by 3.
The 3 goes into both sides perfectly!
x = -1 [Subtract 4 from both sides]
38 = -3(4y +2) + y
38 = -12y + -6 + y
38 = -11y - 6
+6 +6
44 = -11y
so then
44/-11 = -11y/-11
-4 = y
Useful Steps for Solving A Multi-Step Equation
Step 1: Use DP if necessary
Step 2: Combine like terms
Step 3: Undo Addition or subtraction using +prop= or -props=
Step 4: Undo Multiplication or division using xprop+ or ÷prop=
Tuesday, October 19, 2010
Algebra (Period 1)
WORD PROBLEMS--REVIEW Continued
Generally, you see these type of word problems for geometry, age problems, and finding two integers that have some sort of relationship to each other.
GEOMETRY:
The length of a rectangle is twice its width. The perimeter is 48 inches. What is the length and width? Let w = width and l = length of the rectangle Using P = 2l + 2w
2l + 2w = 48 and What else do we know? l = 2w
We can't solve either of these 2 equations because they each have 2 different variables. But...we can substitute in for one of the variables and "get rid of it"! :)
2l + 2w = 48 Substitute 2w in for length: ( because we know l = 2w)
2(2w) + 2w = 48
6w = 48
w = 8 inches
l = 2w so l = 2(8) = 16 inches
AGE PROBLEMS:
Mary and Sam's ages sum to 50
Mary's age is 10 less than twice Sam's age. Find their ages
Let M = Mary's age and S = Sam's age
M = 2S - 10
M + S = 50
Substitute (2S - 10) for Mary's age so you'll only have 1 variable:
(2S - 10) + S = 50
3S - 10 = 50
3S = 60
Sam (S) = 20 years old
Mary = 2S - 10 = 2(20) - 10 = 30 years old.
Finding 2 integers:
The sum of 2 integers is 26.
One integer is 10 more than 3 times the other. Find the 2 integers.
x = one integer and y = other integer x + y = 26
x = 3y + 10 Substitute in for x:
(3y + 10) + y = 26 4y + 10 = 26
4y = 16 y = 4
x = 3(4) + 10 = 22
Algebraic Inequalities:
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 $200 to go to the mall means I must have $200, 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!
Because the answers in an inequality are infinite, the word problems usually are worded to either ask for the least or the greatest answers in the solution set.
For example if the problem asks for 2 consecutive odd integers that sum to at least 50, it will say: "give the smallest" in the solution set.
n + (n + 2) > 50
n > 24
The smallest odd integers in the set are 25 and 27
For example if the problem asks for 2 consecutive odd integers that sum to at most 50, it will say: "give the greatest" in the solution set.
n + (n + 2) < 50
n < 24
The greatest odd integer in the set is 23, so the integers are 23 and 25 (n + 2)
Generally, you see these type of word problems for geometry, age problems, and finding two integers that have some sort of relationship to each other.
GEOMETRY:
The length of a rectangle is twice its width. The perimeter is 48 inches. What is the length and width? Let w = width and l = length of the rectangle Using P = 2l + 2w
2l + 2w = 48 and What else do we know? l = 2w
We can't solve either of these 2 equations because they each have 2 different variables. But...we can substitute in for one of the variables and "get rid of it"! :)
2l + 2w = 48 Substitute 2w in for length: ( because we know l = 2w)
2(2w) + 2w = 48
6w = 48
w = 8 inches
l = 2w so l = 2(8) = 16 inches
AGE PROBLEMS:
Mary and Sam's ages sum to 50
Mary's age is 10 less than twice Sam's age. Find their ages
Let M = Mary's age and S = Sam's age
M = 2S - 10
M + S = 50
Substitute (2S - 10) for Mary's age so you'll only have 1 variable:
(2S - 10) + S = 50
3S - 10 = 50
3S = 60
Sam (S) = 20 years old
Mary = 2S - 10 = 2(20) - 10 = 30 years old.
Finding 2 integers:
The sum of 2 integers is 26.
One integer is 10 more than 3 times the other. Find the 2 integers.
x = one integer and y = other integer x + y = 26
x = 3y + 10 Substitute in for x:
(3y + 10) + y = 26 4y + 10 = 26
4y = 16 y = 4
x = 3(4) + 10 = 22
Algebraic Inequalities:
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 $200 to go to the mall means I must have $200, 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!
Because the answers in an inequality are infinite, the word problems usually are worded to either ask for the least or the greatest answers in the solution set.
For example if the problem asks for 2 consecutive odd integers that sum to at least 50, it will say: "give the smallest" in the solution set.
n + (n + 2) > 50
n > 24
The smallest odd integers in the set are 25 and 27
For example if the problem asks for 2 consecutive odd integers that sum to at most 50, it will say: "give the greatest" in the solution set.
n + (n + 2) < 50
n < 24
The greatest odd integer in the set is 23, so the integers are 23 and 25 (n + 2)
Monday, October 18, 2010
Pre Algebra (Period 2 & 4)
Solving Two-Step Equations 7-1
TWO STEP EQUATIONS:
1. Use the ADDITION PROPERTY OF EQUALITY (first)
(get rid of addition or subtraction)
Addition property of equality abbreviated as +prop=
Subtraction property of equality abbreviated as -prop=
2. Use the MULTIPLICATION PROPERTY OF EQUALITY (second)
(get rid of multiplication/division)
Multiplication property of equality abbreviated as ×prop=
Division property of equality abbreviated as ÷ prop=
DID YOU NOTICE THAT THIS IS THE OPPOSITE OF AUNT SALLY?
That's because solving equations is actually doing order of operations backwards!
EXAMPLE:
3x + 7 = 28
-7 -7
SUBTRACT 7 FROM BOTH SIDES
3x = 21
NOW DIVIDE BOTH SIDES BY 3
3x = 21
3 3
x = 7
CHECKING A 2 STEP EQUATION:
You'll need 2 steps to check, too!
(1) Re write the equation
3x + 7 = 28
(2) SUBSTITUTE YOUR ANSWER AND PUT A ? OVER THE =
3(7) + 7 = 28
USE ORDER OF OPERATIONS - MULTIPLY FIRST
(3) Do the Math!! Really do the Math!!
21 + 7 = 28
28 = 28
-d/7 + 21 = 0
(okay addition or subtraction first so... since + 21 O will need to subtract 21_
-d/7 + 21 = 0
-21 -21
-d/7 + 0 = -21
-d/7 = -21
Now since it is minus d divided by 7 I need to multiply both sides by 7
but if I only use 7 I will still have a negative d and I want to end up with just d... so what if we multiply both sides by -7
(-7)(-d/7) = -21(-7)
d = 147
9 - 3p = -27
most students think that the - sign is attached to the 0 but IT IS NOT!!
this is 9 minus 3p so
9- 3p = -27
you need to subtract 0 from both sides to clear the +9 in the original equation
9 -3p = -27
-9 -9
-3p = -36
Wait how did I get -36... well we have -27 - 9 on the right side and using the integer rules... same sign rule... just add them and use their sign...
now
-3p = -36
If I divide both sides by 3 I still would have a negative on the left with the variable so I must divide both sides by -3
-3p = -36
-3 -3
p = 12
TWO STEP EQUATIONS:
1. Use the ADDITION PROPERTY OF EQUALITY (first)
(get rid of addition or subtraction)
Addition property of equality abbreviated as +prop=
Subtraction property of equality abbreviated as -prop=
2. Use the MULTIPLICATION PROPERTY OF EQUALITY (second)
(get rid of multiplication/division)
Multiplication property of equality abbreviated as ×prop=
Division property of equality abbreviated as ÷ prop=
DID YOU NOTICE THAT THIS IS THE OPPOSITE OF AUNT SALLY?
That's because solving equations is actually doing order of operations backwards!
EXAMPLE:
3x + 7 = 28
-7 -7
SUBTRACT 7 FROM BOTH SIDES
3x = 21
NOW DIVIDE BOTH SIDES BY 3
3x = 21
3 3
x = 7
CHECKING A 2 STEP EQUATION:
You'll need 2 steps to check, too!
(1) Re write the equation
3x + 7 = 28
(2) SUBSTITUTE YOUR ANSWER AND PUT A ? OVER THE =
3(7) + 7 = 28
USE ORDER OF OPERATIONS - MULTIPLY FIRST
(3) Do the Math!! Really do the Math!!
21 + 7 = 28
28 = 28
-d/7 + 21 = 0
(okay addition or subtraction first so... since + 21 O will need to subtract 21_
-d/7 + 21 = 0
-21 -21
-d/7 + 0 = -21
-d/7 = -21
Now since it is minus d divided by 7 I need to multiply both sides by 7
but if I only use 7 I will still have a negative d and I want to end up with just d... so what if we multiply both sides by -7
(-7)(-d/7) = -21(-7)
d = 147
9 - 3p = -27
most students think that the - sign is attached to the 0 but IT IS NOT!!
this is 9 minus 3p so
9- 3p = -27
you need to subtract 0 from both sides to clear the +9 in the original equation
9 -3p = -27
-9 -9
-3p = -36
Wait how did I get -36... well we have -27 - 9 on the right side and using the integer rules... same sign rule... just add them and use their sign...
now
-3p = -36
If I divide both sides by 3 I still would have a negative on the left with the variable so I must divide both sides by -3
-3p = -36
-3 -3
p = 12
Algebra (Period 1)
WORD PROBLEMS--REVIEW
Writing algebraic expressions will NOT have an equal sign and you will NOT be able to solve them!
CHAPTER 1-6: WRITING ALGEBRAIC EXPRESSIONS
STRATEGY #1: TRANSLATE WORD BY WORD
Many times you can translate words into Algebra word by word just like you translate English to Spanish or French.
5 more than a number
n + 5
the product of 5 and a number
5n
the quotient of 5 and a number
5/n
the difference of a number and 5
n - 5
NOTE: Because multiplication & addition are both commutative, when solving for a solution the order will not matter for the solution BUT I require that you translate accurately-- similarly to when you speak another language you are required to learn the proper order of words... AND, FOR SUBTRACTION AND DIVISION, YOU MUST BE CAREFUL ABOUT THE ORDER....GENERALLY, THE ORDER FOLLOWS THE ORDER OF THE WORDS EXCEPT (counterexample!)...
5 less THAN a number
or
5 subtracted FROM a number
Both of these are: n - 5
The order SWITCHES form the words because the words state that you have a number that is more than you want it to be so you need to take away 5 from it. If you aren’t sure about the order with these, I suggest that you try plugging in an actually number and see what you would do with the phrase.
For example, if the phrase was
“5 subtracted from 12”
you would immediately know to write
12-5
For word problems like someone's age or the amount of money you have, you also should always check your algebraic expression by substituting actual numbers to see if your expression makes sense.
EXAMPLE: Tom is 3 years older than 5 times the age of Julie
Translating: 3 + 5J
Does that make sense? Is Tom a lot older than Julie or is Julie older?
Try any age for Julie. Say she is 4 years old.
3 + 5(4) = 23
In your check, Tom is 23.
Is Tom 3 years older than 5 times Julie's age?
YES!
You're algebra is correct!
STRATEGY #2: DRAWING A PICTURE
I have 5 times the number of quarters as I have dimes.
Let’s say I first translate to: 5Q = D I
check: If I assume that I have 20 quarters, then 5(20) = 100 dimes
Does this make sense? That would mean I have a lot more dimes than quarters.
The original problem says I have a lot more quarters!
My algebra is WRONG!
I need to switch the variables.
5D = Q
I check: If I assume that I have 20 quarters, then 5D = 20 D = 4
Does this make sense? YES!
I have 20 quarters and only 4 dimes.
Sometimes it helps to make a quick picture.
Imagine 2 piles of coins.
The pile of quarters is 5 times as high as the pile of dimes.
You can clearly see that you would need to multiply the number of dimes to make that pile the same height as the number of quarters!
STRATEGY #3: MAKE A T-CHART -- this is my favorite!!
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 at least 3 lines of numbers and look for the relationship between the 2 columns.
Ask yourself what do you do to the left side to get to the right?
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 l number of hours
1 ------l------24
2 ------l------48
3 ------l------72
Now look at the relationship between the left column and the right column. What do I do to 1 to get 24? What do I do to 2 to get 48? What do I do to 3 to get 72? For each 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 number of hours
d days l number of hours
1 ------l------24
2 ------l------48
3 ------l------72
d ------l-----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 l number of days
24------l------- 1
48------l------- 2
72 ------l-------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. or ask yourself "What do I do to 24 to get 1? What do I do to 48 to get 2? What do I do to 72 to get 3?" 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 number of days
h hours l number of days
24------l------- 1
48------l------- 2
72 ------l-------3
h -------l-------h/24
Some interesting translations used all the time in Algebra:
The next consecutive number after n: n + 1
Does it work? Try it with any number: if you have 5, then 5 + 1 will give you 6
The next EVEN consecutive number after n: n + 2
Does it work? Try it with any EVEN number: if you have 12, then 12 + 2 will give you 14
The next ODD consecutive number after n: n + 2
Does it work? Try it with any number: if you have 9, then 9 + 2 will give you 11
Writing algebraic expressions will NOT have an equal sign and you will NOT be able to solve them!
CHAPTER 1-6: WRITING ALGEBRAIC EXPRESSIONS
STRATEGY #1: TRANSLATE WORD BY WORD
Many times you can translate words into Algebra word by word just like you translate English to Spanish or French.
5 more than a number
n + 5
the product of 5 and a number
5n
the quotient of 5 and a number
5/n
the difference of a number and 5
n - 5
NOTE: Because multiplication & addition are both commutative, when solving for a solution the order will not matter for the solution BUT I require that you translate accurately-- similarly to when you speak another language you are required to learn the proper order of words... AND, FOR SUBTRACTION AND DIVISION, YOU MUST BE CAREFUL ABOUT THE ORDER....GENERALLY, THE ORDER FOLLOWS THE ORDER OF THE WORDS EXCEPT (counterexample!)...
5 less THAN a number
or
5 subtracted FROM a number
Both of these are: n - 5
The order SWITCHES form the words because the words state that you have a number that is more than you want it to be so you need to take away 5 from it. If you aren’t sure about the order with these, I suggest that you try plugging in an actually number and see what you would do with the phrase.
For example, if the phrase was
“5 subtracted from 12”
you would immediately know to write
12-5
For word problems like someone's age or the amount of money you have, you also should always check your algebraic expression by substituting actual numbers to see if your expression makes sense.
EXAMPLE: Tom is 3 years older than 5 times the age of Julie
Translating: 3 + 5J
Does that make sense? Is Tom a lot older than Julie or is Julie older?
Try any age for Julie. Say she is 4 years old.
3 + 5(4) = 23
In your check, Tom is 23.
Is Tom 3 years older than 5 times Julie's age?
YES!
You're algebra is correct!
STRATEGY #2: DRAWING A PICTURE
I have 5 times the number of quarters as I have dimes.
Let’s say I first translate to: 5Q = D I
check: If I assume that I have 20 quarters, then 5(20) = 100 dimes
Does this make sense? That would mean I have a lot more dimes than quarters.
The original problem says I have a lot more quarters!
My algebra is WRONG!
I need to switch the variables.
5D = Q
I check: If I assume that I have 20 quarters, then 5D = 20 D = 4
Does this make sense? YES!
I have 20 quarters and only 4 dimes.
Sometimes it helps to make a quick picture.
Imagine 2 piles of coins.
The pile of quarters is 5 times as high as the pile of dimes.
You can clearly see that you would need to multiply the number of dimes to make that pile the same height as the number of quarters!
STRATEGY #3: MAKE A T-CHART -- this is my favorite!!
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 at least 3 lines of numbers and look for the relationship between the 2 columns.
Ask yourself what do you do to the left side to get to the right?
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 l number of hours
1 ------l------24
2 ------l------48
3 ------l------72
Now look at the relationship between the left column and the right column. What do I do to 1 to get 24? What do I do to 2 to get 48? What do I do to 3 to get 72? For each 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 number of hours
d days l number of hours
1 ------l------24
2 ------l------48
3 ------l------72
d ------l-----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 l number of days
24------l------- 1
48------l------- 2
72 ------l-------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. or ask yourself "What do I do to 24 to get 1? What do I do to 48 to get 2? What do I do to 72 to get 3?" 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 number of days
h hours l number of days
24------l------- 1
48------l------- 2
72 ------l-------3
h -------l-------h/24
Some interesting translations used all the time in Algebra:
The next consecutive number after n: n + 1
Does it work? Try it with any number: if you have 5, then 5 + 1 will give you 6
The next EVEN consecutive number after n: n + 2
Does it work? Try it with any EVEN number: if you have 12, then 12 + 2 will give you 14
The next ODD consecutive number after n: n + 2
Does it work? Try it with any number: if you have 9, then 9 + 2 will give you 11
Math 6H (Period 6 & 7)
Exponents and Powers of Ten 3-1
When two or more numbers are multiplied together--each of the numbers is called a factor of the product.
A product in which each factor is the SAME is called a power of that factor.
2 X 2 X 2 X 2 = 16. 16 is called the fourth power of 2 and we can write this as
24 = 16
The small numeral (in this case the 4) is called the exponent and represents the number of times 2 is a factor of 16.
The number two, in this case, is called the base.
When you are asked to evaluate... simplify... solve... find the answer
That is,
Evaluate
43 = 4 X 4 X 4 = 16 X 4 = 64
The second and third powers of a numeral have special names.
The second power is called the square of the number and the third power is called the cube.
We read 122 as "twelve squared" and to evaluate it
122 = 12 X 12 = 144
Powers of TEN are important in our number system.
Make sure to check out the blue sheet and glue it into your spiral notebook
First Power: 101 but the exponent is invisible = 10
Second Power: 102 = 10 X 10 = 100
Third Power 103 = 10 X 10 X 10 = 1000
Fourth Power 104 =10 X 10 X 10 X 10 = 10,000
Fifth Power 105 = 10 X 10 X 10 X 10 X 10 = 100,000
Take a look at this list carefully and you will probably see a pattern that we can turn into a general rule:
The exponent in a POWER of TEN is the same as the number of ZEROS when the number is written out.
The number of ZEROS in the product of POWERS OF TEN is the sum of the numbers of ZEROS in the factors.
For example Multiply.
100 X 1000
Since there are 2 Zeros in 100 and 3 zeros in 1000,
the product will have 2 + 3 , or 5 zeroes.
100 X 1000 = 100,000
When you need to multiply other bases:
first multiply each
For example
34 X 2 3 would be
(3 X 3 X 3X 3) X ( 2 X 2 X 2)
= 81 X 8 = 648
What happens when you multiply the same bases?
34 ⋅ 32 = 3⋅3⋅3⋅3⋅3⋅3 or 3 6
We just add the exponents if the bases are the same!!
Well then, what about (34)2 ?
Wait.. look carefully isn't that saying 34 Squared?
That would be (34)(34), right?
.. and looking at the rule above all we have to do here is then add those bases or 4 + 4 = 8 so the answer would be 38.
OR
we could have made each (34) = (3⋅3⋅3⋅3)
so (34)2 would be 3⋅3⋅3⋅3⋅3⋅3⋅3⋅3 or still 38
But wait... isn't that multiplying the two powers? So when raising a power to a power-- you multiply!!
(34)2 = 38
1 to any more is still just 1
15 = 1
0 to any power is still 0!!
Evaluate if a = 3 and b = 5
Just substitute in... but use hugs () we all love our hugs!!
a3 + b2
would be (3)3 + (5) 2
= 27 + 25 = 52
We add the following bit of notes on the following day:
looking at the powers of 10 we noticed that
102 ⋅ 103 = 105
and
106 ⋅103 =109
so could we write a rule for any exponent values a and b?
YES, we decided:
10a ⋅10b = 10a+b
Check out this great Video on the Powers of Ten
POWERS OF TEN
When two or more numbers are multiplied together--each of the numbers is called a factor of the product.
A product in which each factor is the SAME is called a power of that factor.
2 X 2 X 2 X 2 = 16. 16 is called the fourth power of 2 and we can write this as
24 = 16
The small numeral (in this case the 4) is called the exponent and represents the number of times 2 is a factor of 16.
The number two, in this case, is called the base.
When you are asked to evaluate... simplify... solve... find the answer
That is,
Evaluate
43 = 4 X 4 X 4 = 16 X 4 = 64
The second and third powers of a numeral have special names.
The second power is called the square of the number and the third power is called the cube.
We read 122 as "twelve squared" and to evaluate it
122 = 12 X 12 = 144
Powers of TEN are important in our number system.
Make sure to check out the blue sheet and glue it into your spiral notebook
First Power: 101 but the exponent is invisible = 10
Second Power: 102 = 10 X 10 = 100
Third Power 103 = 10 X 10 X 10 = 1000
Fourth Power 104 =10 X 10 X 10 X 10 = 10,000
Fifth Power 105 = 10 X 10 X 10 X 10 X 10 = 100,000
Take a look at this list carefully and you will probably see a pattern that we can turn into a general rule:
The exponent in a POWER of TEN is the same as the number of ZEROS when the number is written out.
The number of ZEROS in the product of POWERS OF TEN is the sum of the numbers of ZEROS in the factors.
For example Multiply.
100 X 1000
Since there are 2 Zeros in 100 and 3 zeros in 1000,
the product will have 2 + 3 , or 5 zeroes.
100 X 1000 = 100,000
When you need to multiply other bases:
first multiply each
For example
34 X 2 3 would be
(3 X 3 X 3X 3) X ( 2 X 2 X 2)
= 81 X 8 = 648
What happens when you multiply the same bases?
34 ⋅ 32 = 3⋅3⋅3⋅3⋅3⋅3 or 3 6
We just add the exponents if the bases are the same!!
Well then, what about (34)2 ?
Wait.. look carefully isn't that saying 34 Squared?
That would be (34)(34), right?
.. and looking at the rule above all we have to do here is then add those bases or 4 + 4 = 8 so the answer would be 38.
OR
we could have made each (34) = (3⋅3⋅3⋅3)
so (34)2 would be 3⋅3⋅3⋅3⋅3⋅3⋅3⋅3 or still 38
But wait... isn't that multiplying the two powers? So when raising a power to a power-- you multiply!!
(34)2 = 38
1 to any more is still just 1
15 = 1
0 to any power is still 0!!
Evaluate if a = 3 and b = 5
Just substitute in... but use hugs () we all love our hugs!!
a3 + b2
would be (3)3 + (5) 2
= 27 + 25 = 52
We add the following bit of notes on the following day:
looking at the powers of 10 we noticed that
102 ⋅ 103 = 105
and
106 ⋅103 =109
so could we write a rule for any exponent values a and b?
YES, we decided:
10a ⋅10b = 10a+b
Check out this great Video on the Powers of Ten
POWERS OF TEN
Thursday, October 14, 2010
Pre Algebra (Period 2 & 4)
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)
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)
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.
Wednesday, October 6, 2010
Algebra (Period 1)
More on Solving Equations
2 unique types of equations:
Identity equation: You solve it and you get the same thing on both sides...if you solve until you cannot do anything more, you get 0 = 0.
What this means is that you can pick any number and the equation will work!
The Distributive Property is the simplest example of an Identity Equation:
3(x + 7) = 3x + 21
Distribute on the left side and you'll get:
3x + 21 = 3x + 21
At this point, you should already know this is an Identity!
If you keep solving, you would subtract 3x from each side and you'll get:
21 = 21
and you know again that this is an Identity.
If you now subtract 21 from each side:
0 = 0
BUT I WOULDN'T GO THIS FAR! AS SOON AS YOU HAVE THE SAME THING ON BOTH SIDES, YOU CAN STOP AND SAY IT'S AN IDENTITY EQUATION!!!
Null set equation:
You solve it and you get an impossible answer:
3(x + 7) = 3x + 10
3x + 21 = 3x + 10
21 = 10
WHEN WILL THAT HAPPEN??? NEVER!!! SO THERE IS NO POSSIBLE SOLUTION TO THIS! The answer is the null set.
2 unique types of equations:
Identity equation: You solve it and you get the same thing on both sides...if you solve until you cannot do anything more, you get 0 = 0.
What this means is that you can pick any number and the equation will work!
The Distributive Property is the simplest example of an Identity Equation:
3(x + 7) = 3x + 21
Distribute on the left side and you'll get:
3x + 21 = 3x + 21
At this point, you should already know this is an Identity!
If you keep solving, you would subtract 3x from each side and you'll get:
21 = 21
and you know again that this is an Identity.
If you now subtract 21 from each side:
0 = 0
BUT I WOULDN'T GO THIS FAR! AS SOON AS YOU HAVE THE SAME THING ON BOTH SIDES, YOU CAN STOP AND SAY IT'S AN IDENTITY EQUATION!!!
Null set equation:
You solve it and you get an impossible answer:
3(x + 7) = 3x + 10
3x + 21 = 3x + 10
21 = 10
WHEN WILL THAT HAPPEN??? NEVER!!! SO THERE IS NO POSSIBLE SOLUTION TO THIS! The answer is the null set.
Math 6H (Period 6 & 7)
Solving Equations & Inequalities 2-4 & 2-5
GOAL: You use the INVERSE operation to ISOLATE the variable on one side of the equation
Here are the steps and justifications (reasons)
1. focus on the side where the variable is and focus specifically on what is in the way of the variable being by itself ( isolated)
2. What is the operation the variable is doing with that number in its way?
3. Get rid of that number by using the opposite (inverse) operation
*Use + if there is a subtraction problem
*Use - if there is an addition problem
*Use x if there is a division problem
*Use ÷ if there is a multiplication problem
GOLDEN RULE OF EQUATIONS; DO UNTO ONE SIDE OF THE EQUATION WHATEVER YOU DO TO THE OTHER!!
4. Justification: You have just used one of the PROPERTIES OF EQUALITY
which one?
that's easy-- Whatever operation YOU USED to balance both sides that's the property of equality
We used:
" +prop= " to represent Addition Property of Equality
" -prop= " to represent Subtraction Property of Equality
" xprop= " to represent Multiplication Property of Equality
" ÷prop= " to represent Division Property of Equality
5. You should now have the variable all alone (isolated) on one side of the equal sign.
6. Justification: Why is the variable alone?
For + and - equations you used the Identity Property of Addition (ID+) which simply means that you don't bring down the ZERO because you add zero to anything-- it doesn't change anything... [Note: there is no ID of subtraction]
For x and ÷ equations, you used the Identity Property of Multiplication (IDx) which simply means that you don't bring down the ONE because when you multiply by one it doesn't change anything [NOTE: there is no ID of division]
7. Put answer in the final form of x = ____and box this in.
GOAL: You use the INVERSE operation to ISOLATE the variable on one side of the equation
Here are the steps and justifications (reasons)
1. focus on the side where the variable is and focus specifically on what is in the way of the variable being by itself ( isolated)
2. What is the operation the variable is doing with that number in its way?
3. Get rid of that number by using the opposite (inverse) operation
*Use + if there is a subtraction problem
*Use - if there is an addition problem
*Use x if there is a division problem
*Use ÷ if there is a multiplication problem
GOLDEN RULE OF EQUATIONS; DO UNTO ONE SIDE OF THE EQUATION WHATEVER YOU DO TO THE OTHER!!
4. Justification: You have just used one of the PROPERTIES OF EQUALITY
which one?
that's easy-- Whatever operation YOU USED to balance both sides that's the property of equality
We used:
" +prop= " to represent Addition Property of Equality
" -prop= " to represent Subtraction Property of Equality
" xprop= " to represent Multiplication Property of Equality
" ÷prop= " to represent Division Property of Equality
5. You should now have the variable all alone (isolated) on one side of the equal sign.
6. Justification: Why is the variable alone?
For + and - equations you used the Identity Property of Addition (ID+) which simply means that you don't bring down the ZERO because you add zero to anything-- it doesn't change anything... [Note: there is no ID of subtraction]
For x and ÷ equations, you used the Identity Property of Multiplication (IDx) which simply means that you don't bring down the ONE because when you multiply by one it doesn't change anything [NOTE: there is no ID of division]
7. Put answer in the final form of x = ____and box this in.
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