Dividing Monomials 5-2
There are 3 basic rules used to simplify fractions made up of monomials.
Property of Quotients
if a, b, c, d are real numbers with b≠0 and d ≠0
ac/bd = a/b ⋅c/d
Our example was 15/21 = (3⋅5)/(3⋅7) = 5/7
The rule for simplifying fractions follows ( when a = b)
(bc)/(bd) = c/d
This rule lets you divide both the numerator and the denominator by the same NON ZERO number.
35/42 = 5/6
-4xy/10x = -2y/5 which can also be written (-2/5)x as well as with out the (((HUGS)))
c7/c4 = c4c3/c4 = c3
another way we proved this was to write out all the c's
c⋅c⋅c⋅c⋅c⋅c⋅c⋅/c⋅c⋅c⋅c = and we realized we were left with
c⋅c⋅c = c3
In addition, we noticed that
c7/c4 = = c7-4 = c3
THen we considered
c4/c7 =
c⋅c⋅c⋅c/c⋅c⋅c⋅c⋅c⋅c⋅c = 1/c⋅c⋅c = 1/c3 = c-3
Since we all agreed that any number divided by itself was = 1
(our example was b5/b5 ), we proved the following
1 = b5/b5 = b5-5 = b0
We finally arrived at the Rule of Exponents for Division
if m > n
am/an = a m-n
If n > m
am/an = 1/a n-m
and if m = n
am/an = 1
A quotient of monomials is simplified when
1)each base appears only once in the fraction,
2) there are NO POWERS of POWERS and
3)when the numerator and denominator are relatively prime, that is, they have no common factor other than 1.
35x3yz6/ 56x5yz
5z5/8x2
Finding the missing factor when you are given the following
48x3y2z4 = (3xy2z)⋅ (______)
we find that
48x3y2z4 = (3xy2z)⋅ (16x2z3)
Thursday, October 13, 2011
Wednesday, October 12, 2011
Algebra Honors (Period 6 & 7)
Factoring Integers 5-1
When we write 56= 8⋅7 or 56 = 4⋅14 we have factored 56
to factor a number over a given set, you write it as a product of integers in that set ( the factor set).
When integers are factored over the set of integers, the factors are called integral factors.
We used the T- charts ( students learned in 6th grade) to first find the positive integer factors
56 = 1, 2, 4, 7, 8, 14, 28, 56
A prime number is an integer greater than 1 that has no positive integral factors other than itself and 1.
The first ten prime numbers are
2, 3, 5, 7, 11, 13, 17, 19, 23, 29
To find prime factorization of a positive integer, you express it as a product of primes. We used inverted division (again taught in 6th grade)
504
Try to find the primes in order as divisors.
Divide each prime as many times as possible before going on to the next prime
we found 504 - 2⋅2⋅2⋅3⋅3⋅7
which we write as 23⋅32⋅7
Exponents are generally used for prime factors
The prime factorization is unique--> and the order should be from the smallest prime to the largest.
A factor of two or more integers is called a common factor of the integers.
The greatest common factor (GCF) of two or more integers is the greatest integer that is a factor of all the given integers.
Find the GCF(882, 945)
First find the prime factorization of each integer Then form product of the smaller powers of each common prime factor.
The GCF is only the primes (and the powers) that they SHARE!!
882 = 2⋅32⋅72
945 = 33⋅5⋅7
The common factors are 3 and 7
The smaller powers of 3 and 7 are 32 and 7
You combine these as a PRODUCT and get
the GCF(882, 945) = 32⋅7 = 63
We also talked about listing ALL pairs of factors--> thus including negative integers
For example:
List all the pairs of factors of 20
(1)(20) but also (-1)(-20)
(2)(10) and (-2)(-10)
(4)(5) and (-4)(-5)
Listing all the factors of -20, we discovered
(1)(-20) but also (-1)(20)
(2)(-10) and (-2)(10)
(4)(-5) and (-4)(5)
When we write 56= 8⋅7 or 56 = 4⋅14 we have factored 56
to factor a number over a given set, you write it as a product of integers in that set ( the factor set).
When integers are factored over the set of integers, the factors are called integral factors.
We used the T- charts ( students learned in 6th grade) to first find the positive integer factors
56 = 1, 2, 4, 7, 8, 14, 28, 56
A prime number is an integer greater than 1 that has no positive integral factors other than itself and 1.
The first ten prime numbers are
2, 3, 5, 7, 11, 13, 17, 19, 23, 29
To find prime factorization of a positive integer, you express it as a product of primes. We used inverted division (again taught in 6th grade)
504
Try to find the primes in order as divisors.
Divide each prime as many times as possible before going on to the next prime
we found 504 - 2⋅2⋅2⋅3⋅3⋅7
which we write as 23⋅32⋅7
Exponents are generally used for prime factors
The prime factorization is unique--> and the order should be from the smallest prime to the largest.
A factor of two or more integers is called a common factor of the integers.
The greatest common factor (GCF) of two or more integers is the greatest integer that is a factor of all the given integers.
Find the GCF(882, 945)
First find the prime factorization of each integer Then form product of the smaller powers of each common prime factor.
The GCF is only the primes (and the powers) that they SHARE!!
882 = 2⋅32⋅72
945 = 33⋅5⋅7
The common factors are 3 and 7
The smaller powers of 3 and 7 are 32 and 7
You combine these as a PRODUCT and get
the GCF(882, 945) = 32⋅7 = 63
We also talked about listing ALL pairs of factors--> thus including negative integers
For example:
List all the pairs of factors of 20
(1)(20) but also (-1)(-20)
(2)(10) and (-2)(-10)
(4)(5) and (-4)(-5)
Listing all the factors of -20, we discovered
(1)(-20) but also (-1)(20)
(2)(-10) and (-2)(10)
(4)(-5) and (-4)(5)
Tuesday, October 11, 2011
Math 6 Honors ( Periods 1, 2, & 3)
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
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
Check out this great Video on the Powers of Ten
POWERS OF TEN
Monday, October 10, 2011
Algebra Honors (Period 6 & 7)
Problems Without Solutions 4-10
Not all word problems have solutions. We listed three of the reasons for this:
1) Not Enough Information ( NEI)
2) Unrealistic Results
3) Facts are contradictory
We used the following examples:
Aurenne drove at her normal speed for the first 2 hours of the trip--- but the road repairs slowed her down 10 mph slower than her normal speed. She made the trip in 3 hours. Find her normal speed.
Wait... just looking at this you realize you just dont have enough information.
We even made a chart with the information we had.. and it just was not enough
There is NO SOLUTION Not enough information or NEI
Liam has a beautiful lawn that is 8 m longer than it is wide... and it is surrounded by a wonderful flower bed which he and his brother maintain for his mother The flower bed is 5m wide all around. Find the dimensions of the lawn if the area of the flower bed is 140m2
When you try to solve this problem by letting the dimensions of the lawn be w and w + 8 you find the following equation
(w + 10)(w + 18) - (w)(w +8) = 140
however that leads us to
w = -2
Since the width of the lawn cannot be negative,
There is No SOLUTION and the given facts are unrealistic.
Drehan says he has equal number of dimes and quarters but that he has 3 times as many nickels as he has dimes. He also tells us that the value of his nickels and dimes is 50 cents more than the value of his quarters. How many of each kind of coin does he have?
Let d = the number of dimes
well if he has the same number of quarters as he has dimes
then d also can equal the number of quarters
and with the other information
3d = the number of nickels.
Now looking at the information he gave us
10d + 5(3d) = 25d + 50
But that simplifies to
25d = 25d + 50
which is impossible.
There is NO SOLUTION
The given facts are contradictory
Not all word problems have solutions. We listed three of the reasons for this:
1) Not Enough Information ( NEI)
2) Unrealistic Results
3) Facts are contradictory
We used the following examples:
Aurenne drove at her normal speed for the first 2 hours of the trip--- but the road repairs slowed her down 10 mph slower than her normal speed. She made the trip in 3 hours. Find her normal speed.
Wait... just looking at this you realize you just dont have enough information.
We even made a chart with the information we had.. and it just was not enough
There is NO SOLUTION Not enough information or NEI
Liam has a beautiful lawn that is 8 m longer than it is wide... and it is surrounded by a wonderful flower bed which he and his brother maintain for his mother The flower bed is 5m wide all around. Find the dimensions of the lawn if the area of the flower bed is 140m2
When you try to solve this problem by letting the dimensions of the lawn be w and w + 8 you find the following equation
(w + 10)(w + 18) - (w)(w +8) = 140
however that leads us to
w = -2
Since the width of the lawn cannot be negative,
There is No SOLUTION and the given facts are unrealistic.
Drehan says he has equal number of dimes and quarters but that he has 3 times as many nickels as he has dimes. He also tells us that the value of his nickels and dimes is 50 cents more than the value of his quarters. How many of each kind of coin does he have?
Let d = the number of dimes
well if he has the same number of quarters as he has dimes
then d also can equal the number of quarters
and with the other information
3d = the number of nickels.
Now looking at the information he gave us
10d + 5(3d) = 25d + 50
But that simplifies to
25d = 25d + 50
which is impossible.
There is NO SOLUTION
The given facts are contradictory
Wednesday, October 5, 2011
Algebra Honors (Period 6 & 7)
Rate-Time- Distance Problems 4-8
D = rt
Uniform Motion
Three types of problems:
Motion in opposite direction
Motion in same direction
Round Trip
Motion in opposite direction
For this we used different students bicycling ... Josh K and Josh P in 6th period and Maddie & Jamie from 7th period.
They start at noon -->60 km apart riding toward each other. They meet at 1:30 PM. If Josh K speed is 4 km/h faster than Josh p ( Maddie is greater by the same from Jamie's rate) What are their speeds?
We set up a chart
Motion in Same Direction
Next we had a fictitious story about ANdrew's Helicopter and David's plane ( or Lucas' helicopter and Kitt's Smiling Plane) taking off from Camarillo Airport flying north. The helicopter flies at a speed of 180 mi/hr. 20 minutes later the plane takes off in the same direction going 330 mi/hr. How long will it take David (or Kitt) to over take Andrew's ( or Lucas') helicopter?
Let t = plane's flying time
Make sure to convert the 20 minutes ---> 1/3 hours.
We set up a chart
When the plane over takes the helicopter they have traveled the exact same distance so set them equal
180(t + 1/3) = 330t
180 t + 60 = 330t
60 = 150t
t = 2/5
which means 2/5 hour. or 24 minutes.
Round Trip
A ski life carries Sara ( or Ryan) up the slope at 6 km/h Sare or Ryan snowboard down 34 km/h. The round trip takes 30 minutes.
Did you see the picture?
Let t = time down
then set up a chart
6(.5 -t) = 34 t
3 - 6t = 34 t
3/40 = t
Now, what's that?
0.75 hr or 4.5 minutes
How far did they snowboard... plug it in
34(0.075) = 2.55 km
D = rt
Uniform Motion
Three types of problems:
Motion in opposite direction
Motion in same direction
Round Trip
Motion in opposite direction
For this we used different students bicycling ... Josh K and Josh P in 6th period and Maddie & Jamie from 7th period.
They start at noon -->60 km apart riding toward each other. They meet at 1:30 PM. If Josh K speed is 4 km/h faster than Josh p ( Maddie is greater by the same from Jamie's rate) What are their speeds?
We set up a chart
Motion in Same Direction
Next we had a fictitious story about ANdrew's Helicopter and David's plane ( or Lucas' helicopter and Kitt's Smiling Plane) taking off from Camarillo Airport flying north. The helicopter flies at a speed of 180 mi/hr. 20 minutes later the plane takes off in the same direction going 330 mi/hr. How long will it take David (or Kitt) to over take Andrew's ( or Lucas') helicopter?
Let t = plane's flying time
Make sure to convert the 20 minutes ---> 1/3 hours.
We set up a chart
When the plane over takes the helicopter they have traveled the exact same distance so set them equal
180(t + 1/3) = 330t
180 t + 60 = 330t
60 = 150t
t = 2/5
which means 2/5 hour. or 24 minutes.
Round Trip
A ski life carries Sara ( or Ryan) up the slope at 6 km/h Sare or Ryan snowboard down 34 km/h. The round trip takes 30 minutes.
Did you see the picture?
Let t = time down
then set up a chart
6(.5 -t) = 34 t
3 - 6t = 34 t
3/40 = t
Now, what's that?
0.75 hr or 4.5 minutes
How far did they snowboard... plug it in
34(0.075) = 2.55 km
Math 6 Honors ( Periods 1, 2, & 3)
Problem Solving: Using Mathematical Expressions 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 exercises on Page 50, Just practice setting up the equations from the verbal sentences.
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 (+)
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 exercises on Page 50, Just practice setting up the equations from the verbal sentences.
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 (+)
Tuesday, October 4, 2011
Algebra Honors (Period 6 & 7)
Transforming Formulas 4-7
Formulas are used throughout real life applications-- the book gives an example of the formula for the total piston displacement of an auto engine... we discussed a number of formulas that students recalled such as the following:
A =lw
d = rt
I = Prt
A = ∏r2
A = P(1 + rt)
C = 5/9(F - 32)
y = mx + b
A = bh
C = ∏d
F = Ma
A = ½(b1b2h
E= mc2
A2 + B2 = C2
and even the quadratic formula-- which we will study later this year..
b = ax ; x
just divide both sides by a
b/a = x
Solve P = 2L + 2W for the width, w
P-2L = w
2
C = 5/9(F - 32) Solve for F
We need to multiply both sides by the reciprocal of 5/9
(9/5)C = F - 32
now add 32 to both sides
(9/5)C + 32 = F
Next we tackled
We also discussed the restrictions and found that the denominator could not equal zero.
S = v/r ( solving for r) became one of the homework problems that caused some discussion-- until students realized that they had actually found the reciprocal of r or 1/r instead of solving for r!!
We discussed how to solve that dilemma.
Formulas are used throughout real life applications-- the book gives an example of the formula for the total piston displacement of an auto engine... we discussed a number of formulas that students recalled such as the following:
A =lw
d = rt
I = Prt
A = ∏r2
A = P(1 + rt)
C = 5/9(F - 32)
y = mx + b
A = bh
C = ∏d
F = Ma
A = ½(b1b2h
E= mc2
A2 + B2 = C2
and even the quadratic formula-- which we will study later this year..
b = ax ; x
just divide both sides by a
b/a = x
Solve P = 2L + 2W for the width, w
P-2L = w
2
C = 5/9(F - 32) Solve for F
We need to multiply both sides by the reciprocal of 5/9
(9/5)C = F - 32
now add 32 to both sides
(9/5)C + 32 = F
Next we tackled
We also discussed the restrictions and found that the denominator could not equal zero.
S = v/r ( solving for r) became one of the homework problems that caused some discussion-- until students realized that they had actually found the reciprocal of r or 1/r instead of solving for r!!
We discussed how to solve that dilemma.
Math 6 Honors ( Periods 1, 2, & 3)
Solving Other Equations & Inequalities 2-5 cont'd
Parent: "What did you do today in Math?"
Student: "Today Mrs. Nelson taught us how to wrap and unwrap a present!!!"
Parent: "Huh..."
Student: "Well, it all has to do with PEMDAS... and undoing equations to solve them!!
Someone was paying attention today... :)
In order to solve two step or multi-step equations you must UNDO in the reverse order of PEMDAS...
UNDO by doing the reverse of PEMDAS
4x + 172 = 248
CHECK:
Step 1: Rewrite the problem
Step 2: Substitute the solutions for the variable
Step 3: DO THE MATH
4x + 172 = 248
We found that x = 19 above-- > in the first problem
so for the check
4x + 172 = 248
4(19) + 172 ?=? 248
76 + 172 ?=? 248
248 = 248 CHECK!!!
For the above, you actually pace a “?” above the equal sign….rather than to each side…
Parent: "What did you do today in Math?"
Student: "Today Mrs. Nelson taught us how to wrap and unwrap a present!!!"
Parent: "Huh..."
Student: "Well, it all has to do with PEMDAS... and undoing equations to solve them!!
Someone was paying attention today... :)
In order to solve two step or multi-step equations you must UNDO in the reverse order of PEMDAS...
UNDO by doing the reverse of PEMDAS
4x + 172 = 248
CHECK:
Step 1: Rewrite the problem
Step 2: Substitute the solutions for the variable
Step 3: DO THE MATH
4x + 172 = 248
We found that x = 19 above-- > in the first problem
so for the check
4x + 172 = 248
4(19) + 172 ?=? 248
76 + 172 ?=? 248
248 = 248 CHECK!!!
For the above, you actually pace a “?” above the equal sign….rather than to each side…
Tuesday, September 27, 2011
Math 6 Honors ( Periods 1, 2, & 3)
Solving Other Equations & Inequalities 2-5
Before we began today's lesson, we reviewed some of the difficult mathematical expressions and equations found on Page 448. We discussed the importance of a well placed comma... in MATH as well as in Language Arts!!
Our math example was
the sum of three and a number b, times even
(3 + b)7
and
the sum of three and a number b times seven
3 + 7b
Our Language Arts example... is one of my favorites.
Where does the comma belong in the following:
A woman without her man is nothing.
I insist it is...
A woman, without her, man is nothing.
However, I acknowledge that some men would differ and insist it is
A woman, without her man, is nothing.
So you see, the comma makes all the difference... in math expressions and equations as well as in language arts!!
We solve by "undoing" the operations in each equation.
We use inverse operation & undo Aunt Sally
It's the reverse of PEMDAS!!
GOAL: You use the INVERSE operation to ISOLATE the variable on one side of the equation
GOLDEN RULE OF MATHEMATICS
What you do to one side of the equation you MUST do to the other side!!
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.
Before we began today's lesson, we reviewed some of the difficult mathematical expressions and equations found on Page 448. We discussed the importance of a well placed comma... in MATH as well as in Language Arts!!
Our math example was
the sum of three and a number b, times even
(3 + b)7
and
the sum of three and a number b times seven
3 + 7b
Our Language Arts example... is one of my favorites.
Where does the comma belong in the following:
A woman without her man is nothing.
I insist it is...
A woman, without her, man is nothing.
However, I acknowledge that some men would differ and insist it is
A woman, without her man, is nothing.
So you see, the comma makes all the difference... in math expressions and equations as well as in language arts!!
We solve by "undoing" the operations in each equation.
We use inverse operation & undo Aunt Sally
It's the reverse of PEMDAS!!
GOAL: You use the INVERSE operation to ISOLATE the variable on one side of the equation
GOLDEN RULE OF MATHEMATICS
What you do to one side of the equation you MUST do to the other side!!
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.
Monday, September 26, 2011
Algebra Honors (Period 6 & 7)
Multiplying Polynomials by Monomials 4-5
This is just the distributive property
x(x + 3) = x2 + 3x
-2x(4x2 - 3x + 5)
-8x3 + 6 x2 -10x
The book shows you how to multiply using a vertical method but I think using the original method taught with the distributive property works just as well-- if not better.
n(2-5n) + 5(n2 -2 ) = 0
2n - 5n2 + 5n2 - 10 = 0
2n - 10 = 0
2n = 10
n = 5
and in set notation {5}
1/2(6xc + 4) -2(c + 5/2) = 2/3 (9-3c)
3c + 2 - 2c - 5 = 6 - 2c
3c -3 = 6
3c = 9
c = 3
and in set notation {3}
This is just the distributive property
x(x + 3) = x2 + 3x
-2x(4x2 - 3x + 5)
-8x3 + 6 x2 -10x
The book shows you how to multiply using a vertical method but I think using the original method taught with the distributive property works just as well-- if not better.
n(2-5n) + 5(n2 -2 ) = 0
2n - 5n2 + 5n2 - 10 = 0
2n - 10 = 0
2n = 10
n = 5
and in set notation {5}
1/2(6xc + 4) -2(c + 5/2) = 2/3 (9-3c)
3c + 2 - 2c - 5 = 6 - 2c
3c -3 = 6
3c = 9
c = 3
and in set notation {3}
Math 6 Honors ( Periods 1, 2, & 3)
Solving Equations & Inequalities 2-4
To solve equations you need to ISOLATE the variable on one side of the mathematical sentence.
isolate--> means to get the variable alone on one side of the equal sign or the inequality sign.
Properties of Equality:
Property of Equality allows us to add or subtract the same number from BOTH sides of the equation.
Addition Property of Equality abbreviated as +prop=
Subtraction Property of Equality ( written as -prop= )
Identity Property of Addition ( our textbook calls it the Addition Property of Zero)
We abbreviated it as ID(+) a + 0 = a
Remember to JUSTIFY with the properties
We completed a yellow form-- see tonight's homework assignment if you need to print it out. That sheet gets glued into our spiral notebook
w + 18 = 64 we must undo addition using the inverse of + (that is, subtraction)
- 18 -18
w + 0 = 46
and then we write
w = 46
What property allows us to subtract 18 from both sides of the equation?
The SUBTRACTION property of equality which we abbreviate with
-prop=
What property allows us to write w instead of w + 0
w + 0 = w
That is the Identity Property of Addition.
How about
p - 84 = 102 we must undo subtraction using the inverse of - (which is ADDITION)
p - 84 = 102
+84 +84
p + 0 = 186
and then we write
p = 186
What property allows us to add 84 to both sides of the equation?
the addition property of equality, which we abbreviate as
+prop=
What property allows us to write p instead of p + 0
The Identity property of Addition.
Why is it called the Identity Property of Addition?
The number never changes its identity
a + 0 = a for all numbers!!
What happens if we have
b + 7 > 8
This is an inequality but we solve this as we would an equation
b + 7 > 8
- 7 -7
b + 0 > 1
and then we write b > 1
h - 7 > 7
+ 7 +7
h + 0 > 14
and then we write
h > 14
Our textbook gives the answer as "greater than 14" but I want you to put it in math symbols. That is, please answer with
h > 14.
To solve equations you need to ISOLATE the variable on one side of the mathematical sentence.
isolate--> means to get the variable alone on one side of the equal sign or the inequality sign.
Properties of Equality:
Property of Equality allows us to add or subtract the same number from BOTH sides of the equation.
Addition Property of Equality abbreviated as +prop=
Subtraction Property of Equality ( written as -prop= )
Identity Property of Addition ( our textbook calls it the Addition Property of Zero)
We abbreviated it as ID(+) a + 0 = a
Remember to JUSTIFY with the properties
We completed a yellow form-- see tonight's homework assignment if you need to print it out. That sheet gets glued into our spiral notebook
w + 18 = 64 we must undo addition using the inverse of + (that is, subtraction)
- 18 -18
w + 0 = 46
and then we write
w = 46
What property allows us to subtract 18 from both sides of the equation?
The SUBTRACTION property of equality which we abbreviate with
-prop=
What property allows us to write w instead of w + 0
w + 0 = w
That is the Identity Property of Addition.
How about
p - 84 = 102 we must undo subtraction using the inverse of - (which is ADDITION)
p - 84 = 102
+84 +84
p + 0 = 186
and then we write
p = 186
What property allows us to add 84 to both sides of the equation?
the addition property of equality, which we abbreviate as
+prop=
What property allows us to write p instead of p + 0
The Identity property of Addition.
Why is it called the Identity Property of Addition?
The number never changes its identity
a + 0 = a for all numbers!!
What happens if we have
b + 7 > 8
This is an inequality but we solve this as we would an equation
b + 7 > 8
- 7 -7
b + 0 > 1
and then we write b > 1
h - 7 > 7
+ 7 +7
h + 0 > 14
and then we write
h > 14
Our textbook gives the answer as "greater than 14" but I want you to put it in math symbols. That is, please answer with
h > 14.
Friday, September 23, 2011
Algebra Honors (Period 6 & 7)
Powers of Monomials 4-4
POWER TO ANOTHER POWER
MULTIPLY the POWERS (m5)3 = m15
To check, EXPAND it out: (m5)(m5)(m5) = m15
PRODUCT TO A POWER
DISTRIBUTE the power to EACH FACTOR (m5n4)3 = m15n12
(am)n = a mn
(u4)5 = u20
(2x)3 = (2x)(2x)(2x) = 8x3
(ab)m = (ab)(ab)(ab).... -->m factors<--- = (a⋅a⋅a⋅a⋅a ...)(b⋅b⋅b⋅b⋅b ...) where a is multiplied m number of times and b is multiplied m number of times.... (ab)m = ambm
To find the power of a product, you find the power of each factor and then multiply
Simplify
(-2k)5
= (-2)5 k5 = -32k5
Evaluate if t = 2
a) 3t3
b) (3t)3
c) 33t3
d) -(3t)3
Simplify
(-3x2y5)3
(-3)3(x2)3(y5)3
-27x6y15
RAISING A QUOTIENT TO A POWER:
DISTRIBUTE THE POWER to the numerator and the denominator (m2/n6)3 = m6/n18
1/n6 = n-6
so what does
1/m-7 =?
Let's read it in math terms
it is 1 divided by 1/m7 .. and what do you do when you need to divide by a fraction? You multiply by its reciprocal so
1 divided by 1/m7 = 1 ÷ 1/m7 = 1× m7/1 = m7
POWER TO ANOTHER POWER
MULTIPLY the POWERS (m5)3 = m15
To check, EXPAND it out: (m5)(m5)(m5) = m15
PRODUCT TO A POWER
DISTRIBUTE the power to EACH FACTOR (m5n4)3 = m15n12
(am)n = a mn
(u4)5 = u20
(2x)3 = (2x)(2x)(2x) = 8x3
(ab)m = (ab)(ab)(ab).... -->m factors<--- = (a⋅a⋅a⋅a⋅a ...)(b⋅b⋅b⋅b⋅b ...) where a is multiplied m number of times and b is multiplied m number of times.... (ab)m = ambm
To find the power of a product, you find the power of each factor and then multiply
Simplify
(-2k)5
= (-2)5 k5 = -32k5
Evaluate if t = 2
a) 3t3
b) (3t)3
c) 33t3
d) -(3t)3
Simplify
(-3x2y5)3
(-3)3(x2)3(y5)3
-27x6y15
RAISING A QUOTIENT TO A POWER:
DISTRIBUTE THE POWER to the numerator and the denominator (m2/n6)3 = m6/n18
1/n6 = n-6
so what does
1/m-7 =?
Let's read it in math terms
it is 1 divided by 1/m7 .. and what do you do when you need to divide by a fraction? You multiply by its reciprocal so
1 divided by 1/m7 = 1 ÷ 1/m7 = 1× m7/1 = m7
Algebra Honors (Period 6 & 7)
Multiplying Monomials 4-3
POWER RULES:
MULTIPLYING Powers with LIKE BASES:
Simply ADD THE POWERS
m5m3 = m8
You can check this by EXPANDING: (mmmmm)(mmm) = m8
DIVIDING Powers with LIKE BASES:
Simply SUBTRACT the POWERS
m8/m5 = m3
Again, you can check this by EXPANDING:
mmmmmmmm/mmmmm = mmm
ZERO POWERS:
Anything to the zero power = 1
(except zero to the zero power is undefined)
Proof of this was given in class:
1 = mmmmmmmm/mmmmmmmm
= m8/m8
= m0 (by power rules for division)
By the transitive property of equality : 1 = m0
Review the odd/even rule
IF THERE IS A NEGATIVE INSIDE PARENTHESES:
Odd number of negative signs or odd power = negative
Even number of negative signs or even power = positive
EXAMPLES: (-2)5 = -32
(-2)4 = +16
IF THERE IS A NEGATIVE BUT NO PARENTHESES:
ALWAYS NEGATIVE!!!!
-25 = -32
-24 = -16
JUST REMEMBER
NEGATIVE POWERS MEANS THE NUMBERS ARE FRACTIONS
They're in the wrong place in the fraction
m3/m5 = m-2
m3/m5 = mmm/ mmmmm
= 1/mm
Again, by transitive property of equality:
m3/m5 = m-2 = 1/m2
Remember the rule of powers with ( )
When there is a product inside the ( ), then everything inside is to the power!
If there are no ( ), then only the variable/number right next to the power is raised to that power.
3x-2 does not equal (3x)-2
The first is 3/x2 and the second is 1/9x2
RESTATE A FRACTION INTO A NEGATIVE POWER:
1) Restate the denominator into a power
2) Move to the numerator by turning the power negative
EXAMPLE: 1/32 = 1/(2)5 = (2)-5
POWER RULES:
MULTIPLYING Powers with LIKE BASES:
Simply ADD THE POWERS
m5m3 = m8
You can check this by EXPANDING: (mmmmm)(mmm) = m8
DIVIDING Powers with LIKE BASES:
Simply SUBTRACT the POWERS
m8/m5 = m3
Again, you can check this by EXPANDING:
mmmmmmmm/mmmmm = mmm
ZERO POWERS:
Anything to the zero power = 1
(except zero to the zero power is undefined)
Proof of this was given in class:
1 = mmmmmmmm/mmmmmmmm
= m8/m8
= m0 (by power rules for division)
By the transitive property of equality : 1 = m0
Review the odd/even rule
IF THERE IS A NEGATIVE INSIDE PARENTHESES:
Odd number of negative signs or odd power = negative
Even number of negative signs or even power = positive
EXAMPLES: (-2)5 = -32
(-2)4 = +16
IF THERE IS A NEGATIVE BUT NO PARENTHESES:
ALWAYS NEGATIVE!!!!
-25 = -32
-24 = -16
JUST REMEMBER
NEGATIVE POWERS MEANS THE NUMBERS ARE FRACTIONS
They're in the wrong place in the fraction
m3/m5 = m-2
m3/m5 = mmm/ mmmmm
= 1/mm
Again, by transitive property of equality:
m3/m5 = m-2 = 1/m2
Remember the rule of powers with ( )
When there is a product inside the ( ), then everything inside is to the power!
If there are no ( ), then only the variable/number right next to the power is raised to that power.
3x-2 does not equal (3x)-2
The first is 3/x2 and the second is 1/9x2
RESTATE A FRACTION INTO A NEGATIVE POWER:
1) Restate the denominator into a power
2) Move to the numerator by turning the power negative
EXAMPLE: 1/32 = 1/(2)5 = (2)-5
Subscribe to:
Posts (Atom)
