Algebra Honors ( Periods 5 & 6)

Monomial Factors of Polynomials 5-3

In the first chapters we reflected that if a, b, and c, were real numbers and c was not equal to 0,
then
(a + b)/c = a/c + b/c
It also applies to monomials
(5m + 35)/ 5 = 5m/5 + 35/5... which simplifies to  m + 7

To divide a polynomial by a monomial, divide each term of the polynomial by the monomial and then add the results.
From this point on, our textbook has us assume that not divisor equals 0

Divide the following
26uv - 39v
    13v

You can separate each part so that you have
(26uv)/13v  - 39v/13v
which simplifies to
2n -3

Divide

3x⁴ - 9x³y + 6x²y² 
-3x²

This time you notice that -3x² is a factor of all three terms of this polynomial
and you can separate the polynomial into three separate terms or... as taught in class you can easily do each ( carefully)
so that you simplify the fraction to
-x² + 3xy -2y²

Divide:
x³y - 4y + 6x
     xy

Here you might want to show the three terms to see what is happening to each individual term...

x³y 
  xy

-4y
xy

+6x
xy

which simplifies to

x² -4/y + 6/x

You definitely could simplify by crossing out the common factors  each term shares with the divisor.

One polynomial is evenly divisible or just divisible by another polynomial if the quotient is also a polynomial.
So the first two examples show divisibility but this last one does NOT.

You factor a polynomial by expressing it as a product of other polynomials. The factor set for a polynomial have integral coefficients.
You can use division to test for factors!

The greatest monomial factor of a polynomial is the GCF of its terms!

Factor
5x² + 10x
The greatest monomial factor is 5x
You don't want to change the value of your polynomial-- you just want to factor it!
Pull out the 5x and divide by 5x as well.. because you are NOT CHANGING the VALUE
Its simply using the Distributive Property

5x(5x² + 10x)
     5x

5x(x + 2)
To check-- just multiply out using your knowledge of the distributive property!!

Factor
4x - 6x³ + 14x

The greatest monomial factor is 2x

2x(4x - 6x³ + 14x)
   2x
2x(2x - 3x² + 7)

Factor
8a²bc² - 12ab²c²
The greatest monomial factor is 4abc²


4abc²(8a²bc² - 12ab²c²)
  4abc²

4abc²(2a-3b)

Practice these and you will be able too do the division steps mentally.
Check your factorization by multiplying the resulting factors!
Make sure you end up with where you started-- when you check!!

Math 6H ( Period 3)

Comparing & Ordering Fractions 2.6 

When you decide tat one fraction is less than another fraction, you are comparing fractions. WHen you list fractions from least to greatest you are ordering fractions. You use the symbols > and < to replace words when you compare fractions
> means greater than
< means less than
Compare 2/5 and 3/5
Two fractions with the same denominator are easy to compare. The fraction with the lesser numerator is lesser
so 2/5 < 3/5 because 2<3 p="p">
What about the same numerator ?
 Write 2/5, 2/3, and 2/4 in order
The larger number in the denominator is the smaller sized fraction.
so 2/5 < 2/4< 2/3

What about
8/32  8/24 and 8/16
Using the same process above
8/32 < 8/24 < 8/16
and if you simplify each of the fractions ( from our lesson in 2.4
8/32 = 1/4
8/24 = 1/3
8/16 = 1/2
so we could use the same rule and still get the same results since
1/4 < 1/3< 1/2

What if you were comparing 3/4 and 5/6?
You might need to change to a common denominator
One of the common denominators is always the product of the two denominators so 4 times 6 = 24
3/4 ( 6/6) = 18/24
5/6( 4/4) = 20/24
since 18/24 < 20/24
3/4 < 5/6

OR you could use the Least Common Denominator (LCD) which is really the Least Common Multiple of the Denominators!
LCM(4, 6) = 12 so
change 3/4 into  9/12
and 5/6 = 10/12
since 9/12 , 10/12
3/4 < 5/6

One additional method
When comparing two fractions,

Algebra Honors (Periods 5 & 6)

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 = = c^7-4 = 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 = b^5-5 = b⁰

We finally arrived at the Rule of Exponents for Division

if m > n
a^m/aⁿ = a ^m-n

If n > m
a^m/aⁿ = 1/a ^n-m
and if m = n
a^m/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)

Algebra Honors (Periods 5 & 6)

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)

October 10th: Powers of Ten Day & Binary Day

Check out this great Video on the Powers of Ten
POWERS OF TEN

Algebra Honors (Periods 5 & 6)

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

Math 6H (Period 3)

 Least Common Multiple  2. 5 
A multiple of a number is the product of the number and any NON ZERO whole number. For example the multiples of 5 are 5, 10, 15, 20, 25,  and so on...
A common multiple for two or more numbers is a multiple shared byt he numbers. For example
20 is a common multiple of 2 and 5 because 20 = 2 times 10 and 20 = 5 times 4.

Multiples of 2: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24
Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30
Common multiples of 2 and 3  ( that are < 25) are 6, 12, 18, and 24
and the smalles is the Least Common Multiple and that is 6
LCM(2, 3) = 6
What about the LCM (4, 6) = 12

You can use Prime Factorization to find the LCM
24 and 60
You can use Factor Trees or Inverted Division


LCM is EVERY FACTOR TO ITS GREATEST POWER.

You could list all the multiples but why not use that BOX we used when finding the GCF... it works for  finding the LCM of two numbers.



LCM ( 7, 11) =77
If the two numbers are relatively prime, then the LCM is their product.
Remember: the GCF of two relatively prime numbers is 1
LCM ( 8, 9) = 72   because we know the GCF(8,9) = 1

Math 6A (Periods 2 & 4)

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.


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 (+)

Math 6A (Periods 2 & 4)

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

I will review the Formal Check on Monday but if you want to see-- read on...
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…

Math 6A ( Periods 2 & 4)

Solving Equations & Inequalities 2-4 continued

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!!



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!!





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.

Algebra Honors (Periods 5 & 6)

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 ... Jeff K and Jeff B in 5th period and Rebecca and Cayla  from 6th period.

Last year's examples follow:
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 Stan's Helicopter and Brett's plane ( or David's helicopter and Noah'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 Brett (or Noah) to over take  Stan's ( or David's) helicopter?

Let t = plane's flying time

Make sure to convert the 20 minutes ---> 1/3 hours.
We set up a chart .. Here is the chart with last year's names:

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 lift carries Sammy ( or Jessica) up the slope at 6 km/h Sammy or Jessica 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 6High ( Period 3)

 Greatest Common Factor 2-2
A whole number that is a factor of two or more non zero whole numbers is a common factor of the two numbers.
Find all the common factors of 14 and 42
List all the factors of each number-- making a T chart of the factors as shown in class.
Factors of 14: 1, 2, 7, 14
Factors of 42: 1, 2, 3, 6, 7, 14, 21, 42
You notice that the common factors as 1, 2, 7, and 14
The Greatest of these is 14
GCF(14, 42) = 14

Any two numbers have 1 as a common factor. the largest number that is a factor of two or more nonzero whole numbers is their greatest common factor ( GCF) or greatest common divisor (GCD)
Find the GCF of 28 and 36
Factors of 28: 1, 2, 4, 7, 14, 28
Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
GCF( 28, 36) = 4

Find the GCF of 16 and 37
Factors of 16: 1, 2, 4, 8, 16
Factors of 37: 1, 37  (
GCF( 16, 37) = 1
If the GCF of 2 numbers is 1 the two numbers are said to be relatively prime. Notice the numbers do NOT need to be prime themselves. In this case, 37 is prime BUT 16 is not.

the GCF(8, 9)= 1
because
Factors of 8 : 1, 2, 4, 8
Factors of 9: 1, 3, 9
The only common factor they share is 1. 8 and 9 are said to be relatively prime-- even thought BOTH numbers are composite... neither is actually prime.

GCF ( 72, 84)
Making the T chart for these two numbers you discover that
Factors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72
Factors of 84: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84

The greatest common factor is 12
We write
GCF(72, 84) = 12

What about GCF(66, 96)
Factors of 66: 1, 2, 3, 6, 11, 22, 33, 66
Factors of 96: 1, 2, 3, 4, 6, 8, 12, 18, 24, 32, 48, 96

WE discover that GCF(66, 96) = 6
The BOX method was shown in class. It is another way to discover what the GCF is for two numbers.
If you aren't sure, come in and I will review the BOX method for finding the GCF. You will want to know how to use it because we will use it to find the LCM as well-- later in this chapter.

Math 6A (Periods2 &4)

Solving Equations & Inequalities 2-4

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 seven
(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!!


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. or  ID +

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. ID+
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

Algebra Honors (Periods 5 & 6)

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= mc<sup>2
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</sup> 

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.