Algebra Honors (Period 6 & 7)

Dividing Real Numbers 2-9
Quotient a divided by b or a ÷b should be written as a fraction
a/b
Definition of division --> Which we abbreviated as Def (÷)
a÷ b = a ⋅(1/b)
to divide by a non zero number, multiply by its reciprocal

a+b
c
= a/c + b/c
where c ≠ 0
and

a-b
c
= a/c - b/c

Justifying leads us to the following:
(a+b)/c
= (a+b)(1/c) --> def (÷)
=(a ⋅ 1/c) + (b ⋅ 1/c) --> DP
=a/c + b/c --> def (÷)

Algebra Honors (Period 6 & 7)

Problem Solving Consecutive Integers 2-7


Consecutive- list in natural order from least to greatest
-2, -1, 0, 1 ..

an integer n
the next three numbers are n + 1, n+ 2, and n+ 3
the number that receded n is n-1

Let's create the following equation: The sum of four consecutive integers is 66
Let n be the first integer
n + (n + 1) + (n + 2) + (n + 3) = 66
4n + 6 = 66
n = 15
Now, if the question asked what was the first integer the solution would be {15}
but if the question asks you to state all the integers the solution would be
{15, 16, 17, 18}

If x = 15, how would you write 14 and 16 in terms of x?
14 --> x -1
16 --> x + 1

if m is odd, m + 1 is even

How do you write 4 consecutive even integers starting with x ?
x, x + 2, x + 4, x + 6

How do you write 4 consecutive odd integers, starting with x ?
x , x + 2, x + 4, x + 6


In the above cases, you defined what x was-- in the first case x was even... so you need to add multiples of 2 to continue getting an even number... AND in the 2nd case you defined that this time x was odd.. and again you need to add multiples of 2 to continue getting an odd number.


The sum of 2 consecutive integers is 43. Let x = 1st integer
x + ( x + 1) = 43

The sum of three consecutive odd integers is 40 more than the smallest.
What are the integers?

(Hint: Reread the word problem carefully!!)

Let x represent the 1st odd integer
x + (x + 2) + (x + 4) = x + 40
3x + 6 = x + 40
2x = 34
x = 17
so
{17, 19, 21} is the solution.. and if you substitute in to check, you will see you have solved the question!!

However, in our book, it asks this question a little differently, It says with the domain for the smallest given as {13, 17, 25}.
In this case you could work backwards and substitute each of the three integers to see if it works.

The Reciprocal of Real Numbers 2-8



Two numbers whose product is 1 are called reciprocals or multiplicative inverses
5 and 1/5 are reciprocals because 5⋅1/5 = 1
4/5 and 5/4 are reciprocals 4/5⋅5/4 = 1
and so are (-1.25) and (-0.8) because (-1.25) ⋅ (-0.8) = 1
look closely at (-1.25) and (-0.8) and you realize that they are really -5/4 ⋅ -4/5 = 1

If a is any real number, its reciprocal is 1/a

Property of Reciprocals
for any non zero real number a,
a⋅1/a = 1 and 1/a⋅ a = 1
-a and -1/a are also reciprocals
Notice, we discussed in class that
1/-a = -1/a = -(1/a) those are equivalent

(ab)(1/a⋅1/b) = a (1/a)⋅b(1/b) = a/a⋅b/b= 1⋅1 = 1


so try the following
1/3(42m -3v)
1/3(42m) - (1/3)(3v)
14m - v

Make sure to distribute the negative to each term

How about
-1/20( 5z -4w) - 6(-1/30w - 1/24z)

Again make sure to distribute the negative to each term!!
(-1/4)z + (1/5)w + (1/5)w + (1/4)z
or
-z/4 + w/5 + w/5 + z/4
= 2w/5 also written as (2/5)w

Math 6 Honors ( Periods 1, 2, & 3)

Inverse Operations 1-3

Inverse Operations undo each other
Addition & subtraction are inverse operations
Multiplication & division are inverse operations

We can use these relationships between inverse operations to simplify some numerical and variable expressions

17- 8 + 8 instead of doing 17 - 8 and then adding 8 again simply cancel the + and the -
That is, 17 - 8 + 8 = 17

9 - x + x = 9 because adding x and subtracting x are inverse operations and they undo each other...

108 ÷ 6 × 6 ... the long way would be to divide 108 by 6 first. That's 18 and then multiply 18 by 6 and get 108.. but why do that? Using the inverse operations you realize that multiplication and division undo each other so 108 ÷ 6 × 6 is simply 108

3n ÷ 3 = 3

Subtraction Property of Zero
For any whole number a
a-0 = a and a-a = 0

Division Property of One
For any whole number a, except 0
a ÷ 1 = a and a ÷ a = 1

Division Property of Zero
For any whole number a, except 0
0 ÷ a = 0

Remember we discussed that we can never divide any number by 0
We said that any number
a ÷ 0 was said to be 'undefined.'

4! four factorial
4⋅3⋅2⋅1 = 24

5! = 5⋅4⋅3⋅2⋅1 = 120

Algebra Honors (Period 6 & 7)

Chapter 2 Highlights Continued

Substitution Principle--> an expression may be replaced by another expression that has the same value.

JUSTIFY THE FOLLOWING:
b + (a -b)
= b + [a +(-b)] Definition of subtraction ( we used def-)
= b + [-b + a] C+
=(b + -b) + a A+
= 0 + a Property of Opposites ( How about Prop of Op for this one!!)
= a Id+

Check out the 5 step plan on Page 27... use the index in your textbook to find many concepts and explanations...

Multiplication Property of One a(1) = a and 1(a) = a IDx 1 is the identity element for multiplication

Multiplication Property of Zero A90) = 0 and 0(a) = 0 Ox

Multiplicative Prop of -1
a(-1) = -a and (-1)a = -a

Justify the following
a(-1) + a
= a(-1) + a(1) ID x
= a(-1 +1) DP
=a(0) Prop of Op
=0 Ox

Property of Opposites in Products or the Prop of Opposites in Products or just POP
(-a)b= -ab
a(-b) = -ab
(-a)(-b) = ab

JUSTIFY
a(b-c)
=a[b + (-c)] Def-
=ab + a(-c) DP
ab + (-ac) POP
ab -ac Def -

-3(7c + d) - 2(10d-c)
-21c -3d -20d + 2c
-19c -23d

Math 6 Honors ( Periods 1, 2, & 3)

Properties of Addition & Multiplication 1-2

The following was handed out in class.. if you were absent make sure to get your copy and GLUE it into your Spiral Notebook.
1, 2, 3, 4,5 .... are counting numbers or natural numbers.
If we add 0... we have 0,1,2,3,4,... and we have the set of whole numbers

Some important properties are found using whole numbers:

Commutative Property of Addition (C+)
for any whole numbers a and b
a + b = b + a

Commutative Property of Multiplication (Cx)
for any whole numbers a and b
ab = ba

You HEAR the differences.. remember we made the big C into an ear... or at least we attempted to!!

However look at the following
(6 +5) + 7 = 11 + 7 = 18
but so does
6 + (5 + 7) = 6 + 12 = 18

Associative Property of Addition ( A+)
For any whole numbers a, b, and c
(a + b) + c = a + ( b + c)

Associative Property of Multiplication ( Ax)
For any whole numbers a, b, and c
(ab)c = a(bc)

You SEE the difference with the hugs ( ).. hugs are really important in life.. and in math!!

Addition Property of Zero from the book
also known as the Identity Property of Addition ( Id+)
for any whole number a
a + 0 = a and 0 + a = a

Multiplication Property of One
also know as the Identity Property of Multiplication (Idx)
For any whole number a
a(1) = a and 1(a) = a

Multiplication Property of Zero
for any whole number a
0(a) = 0 and 0(a) = 0

So how do we justify?

16 + 19 + 34 ---> given
16 + 34 + 19 ---> C+
(16 + 34) + 19---> A+
50 + 19---> simplify
69---> simplify


2 (14)(15)---> given
2 (15) (14) ---> Cx
[2(15)] (14) ---> Ax
30 (14)---> simplify
420---> simplify

how about the following. What would make it easier to simplify???
8⋅ 14⋅ 25 ⋅5 ---> given
14 ⋅ 8 ⋅ 5⋅ 25 ---> Cx
14 ⋅ (8 ⋅ 5)⋅ 25 ---> Ax
14 ⋅ 40⋅ 25--->simplify
14 ⋅ (40⋅ 25) ---> Ax
14 ⋅ 1000 ---> simplify
14000 ---> simplify

Make sure you justify each step of tonight's homework!!

Math 6 Honors ( Periods 1, 2, & 3)

Mathematical Expressions 1-1

A variable is a symbol used to represent one or more numbers. The numbers are called the values of the variable.


An expression, such as 3 x n, that involves a variable is called a variable expression.


Expressions, such as 3 x 2, that name a certain number are called numerical expressions


When we write a product that involves a variable, we usually omit the multiplication symbol (whether that be written as x or as ∙ or even with parentheses). Thus, 3 x n is written as 3n
and 2 x a x b is written as 2ab

In numerical expressions for products a multiplication symbol must be used to avoid confusion.

9 x 7 may be written as 9 ∙ 7 or even 9(7)

When a mathematical sentence uses an equal sign, it is called an equation. An equation tells us that two expressions name the same number. The expression to the left of the equals sign is called the left side of the equation and the expression to the right of the equals sign is called the right side.
expression = expression

When a number is substituted for a variable in the variable expression and the indicated operation is carried out, we say that the variable expression has been evaluated. For example, if n has the value 6 in the variable expression 3 x n, then 3 x n has the value 3 x 6, or 18
Example: Evaluate the expression 6a when the variable has the following values:
6a; 2, 4, 6, 8
You would substitute in each value for the variable a
6(2) = 12
6(4) = 24
6(6) = 36
6(8) = 48

148 ÷ 4 =
148/4
37


if m = 3 and n = 18
n ÷ m
substitute in
n/m or 18/3 = 6


If y = 18 and x = 8
4y ÷ 3x immediately set this up as
4y/3x

Now substitute in your values
4(18) / 3(8)
72/24 = 3

Algebra Honors (Period 6 & 7)

Chapter 1 Highlights
Variable→ symbol used to represent one or more numbers
Variable expression contains a variable
Numerical expression names a particular number
Simplifying the expression→Replacing a numerical expression by the simplest name for its value
Grouping symbols→ parentheses, brackets, fraction bar, etc used to enclose an expression that should be simplified first.
Equation→ two numerical or variable expressions that are equal. Represented by an equals sign placed between the two sides of the equation.
Open Sentences→ contain variables, such as 5x – 1 =9
or y + 2 = 2 + y

The given set of numbers that a variable represents is called the domain of the variable.
Use brackets { } to show a set of numbers. A short way to write “the set whose members are 1, 2, and 3” is {1, 2, 3}
Any value of a variable that turns an open sentence into a true sentence is a solution or root of the sentence.
The set of all solutions of an open sentence is called the solution set of the sentence.
Some equations have only one solution, and some equation have no solutions. The sentence y + 2 = 2 + y has an infinite number. The solution set is the set of all numbers. If, however, you are asked to solve of the domain {0,1,2,3}, you state that the solution set is the domain itself {0, 1, 2, 3}.

Another way to express this is

Read the above as
“y belongs to the set whose members are 0, 1, 2, 3”

Real Numbers
Natural Numbers or Counting Numbers→ 1, 2, 3, 4, 5….
Whole Numbers (Natural Numbers + 0) → 0, 1,2, 3, 4, 5 …
Integers (Natural Numbers , their opposites, + 0) → …-4, -3, -2, -1, 0, 1, 2, 3, 4, …
Rational Numbers (quotient of two integers)
So any number that is either positive, negative, or zero is called a real number.

Each number is a pair, such as 4 and -4 is called the opposite of the other number. The opposite of a is written –a.
The numerals -4(lowered minus sign) and ^-4 (raised minus sign) name the same number.
Caution –a read “the opposite of a” is NOT necessarily a negative number.
For example, if a= -2 then –a would be –(-2) = 2

Absolute value is a distance concept. It my be thought of as the distance between the graph of a number and the origin on a number line. The graphs of -4 and 4 are both 4 units from the origin.

Math 6 Honors ( Periods 1, 2, & 3)

Our Class Blog
Welcome to our class blog... where you can earn extra credit by adding your own relevant comments about our class notes for the day.. or where you can find answers from others in your class. Check here often, especially if you have been absent. You might just find out the math strategy that works for you!!

Email me if you are interested in adding notes and/or comments to this blog-- for extra credit!!

Algebra Honors (Period 6 & 7)

Our Class Blog
Welcome to our class blog... where you can earn extra credit by adding your own relevant comments about our class notes for the day.. or where you can find answers from others in your class. Check here often, especially if you have been absent. You might just find out the math strategy that works for you!!

Email me if you are interested in adding notes and/or comments to this blog-- for extra credit!!

Math 6 Honors (Period 6 and 7)

Commission and Profit 9-6

Some sales jobs pay an amount based on how much you sell. This amount is called a commission.

Like a discount, the commission can be expressed as a percent or as an amount of money.

amount of commission = percent of commission X total sales.

Using the examples from our textbook,

Maria sold $42,000 word of insurance in January. If her commission is 3% of the total sales, what was the amount of her commission in January?

amount of commission = percent X total sales
0.03 X 42,000 = 1260
Her commission was $1,260.

Profit is the difference between total income and total operating costs.

profit = total income – total costs

The percent of profit is the percent of total income that is profit

percent of profit = profit/total income
A shoe store had an income of $8600 and operating costs of $7310. What percent of the store's income was profit?

profit= income- total costs = 8600 -7310 = 1290

percent of profit = profit/total income = 1290/8600 = 0.15
So the percent of profit was 15%.

Practice finding 10%-- its easy--- just move the decimal over one place.
We practiced finding 20%. Just double what you got for 10%.



MATH AT WORK:
Caterer
A caterer provides food for parties, weddings, bar/bat mitzvahs, and other events. Caterers plan the menu, buy the ingredients, and cook the food. Often they provide seating and music as well. For each event, a caterer determines the cost per guest. The catering business requires a thorough knowledge of rations, proportions, and percents.

Algebra (Period 1)

Word Problems from Khan Academy
Check some of these out


Trains..

Math 6 Honors (Period 6 and 7)

Discount and Markup 9-5

A discount is a decrease in the price of an item. A markup is an increase in the price of an item. Both of these changes can be expressed as an amount of money or as a percent of the original price of the item. A store may announce a discount of $3 off the original price of $30 basketball, or a discount of 10%

A warm-up suit that sold for $42.50 is on sale at a 12% discount. What is the sale price?

Method 1: Use the formula

amount of change = percent of change X original amount
= 12% X $42.50

therefore the discount is 0.12 X 42.50 or 5.10
The amount of discount is $5.10
The sale price is 42.50 – 5.10 = $37.40

Method 2: Since the discount is 12%, the sale price is 100% - 12% = 88%.
The sale price is 0.88 X 42.50 = $ 37.40

When you know the amount of discount you subtract to find the new price. When dealing with a markup you add to find the new price.

The price of a new car model was marked up 6% over the previous year’s model. If the previous year’s model sold for $7800, what is the cost of the new car? {and what kind of a car could that be?}
Method 1: Use the formula
amount of change = percent of change X original amount

= 6% X 7800

Therefore the markup is 0.06 X7800= $468
The new price is 7800 + 468 = $8268

Method 2: Since the markup is 6% the new price is 100% + 6% or 106% of the original price. so the new price is 1.06 X7800 = $8268

This year a pair of ice skates sells for $46 after a 15% mark up over last year’s price. What was last year’s price?

This year’s price is 100 + 15 or 115% of last year’s price. Let n present last year’s price
46 = (115/100)n

46 = 1.15n
46/.15 = 1.15n/1.115
40 = n

So last year’s price was $40.

A department store advertised eclectic shavers at a sale price of $36.
If this is a 20% discount, what was the original price?
The sale price is 100 - 20 or 80% of the original price. Let n represent the original price.
36 = (80/100)n
36 = .8n
36/.8 = .8n/.8

45 = n
The original price was $45.
Check to see that your answers are logical and reasonable.

Try these: A service station (that’s gas station, now—they no longer provide service!!) give cash customers a 5% discount on the price of gasoline. If gasoline regularly sells for $3.00 a gallon, what is the discounted price?


A store marks up the price of a $5 item to $12. What is the percent of markup?

Algebra (Period 1)

Motion Word Problems 8-5 & 10-7
rt=d problems
(rate)(time)=distance
Back in Pre-Algebra, these were fairly simple word problems:
1) If you go 60 mph for 3 hours, how far have you gone? (180 miles)
2) You've gone 100 miles in 2 hours. What was your average speed? 
(100/2 = 50 mph)
 
3)You've driven 1000 miles at an average speed of 25 mph. How long did it take you? (1000/25 = 40 hours)
 
Now the problems become more difficult. Usually they involve 2 cars, trains, planes, etc. One car is the "slow" car and the other is the "fast" car.
 
Just like the mixture problems, it helps to make a matrix and also draw a picture to help you understand the words.
 
SLOW CAR/FAST CAR LEAVE FROM SAME PLACE, IN THE SAME DIRECTION, AT DIFFERENT TIMES, WHEN WILL THE 2ND CAR CATCH UP WITH THE FIRST CAR? 
 
2 cars leave 3 hours apart. One travels 72 mph. The other travels 120 mph. The slower car leaves first. When with the faster car catch up with the slower car?
 
Use the set up forms from class... you can find more blank forms online!!

SLOW     72           t              72t  
FAST     120        t - 3        120(t - 3) 
         (fast car left 3 hours later so driving 3 less hours or t - 3)
 
At the point when the fast car catches and overtakes the slow car, what is true of the distances of the 2 cars at that exact moment???
They are equal!
WHAT IS THE EQUATION?
Set the 2 cars' distances equal:
72t = 120(t - 3)
 
72t = 120t - 360
-48t = -360
t = 7.5 hours (slow car's time on the road)
t - 3 = 7.5 - 3 = 4.5 hours (fast car's time on the road)
 
CHECK:
The 2 cars should have traveled the same distance because one car catches up with the other car:
slow: (72)(7.5) = 540
fast: (120)(4.5) = 540
 
 
SLOW CAR/FAST CAR LEAVE FROM SAME PLACE, GOING IN DIFFERENT DIRECTIONS, LEAVING AT THE SAME TIME, WHEN WILL THEY BE A CERTAIN DISTANCE APART?
 
2 cars leave going in different directions. One travels 60 mph. The other travels 50 mph. In how many hours will the cars be 605 miles apart?
 
                      CAR     (RATE)    (TIME)   =   DISTANCE
                     SLOW     50            t                  50t 
                     FAST      60            t                  60t
                      (They travel the same amount of time)
 
What is true of the distances the 2 cars travel?
Together they travel 605 miles because they are 605 miles apart.
WHAT IS THE EQUATION?
Set the 2 cars' distances as a SUM to 605.
50t + 60t = 605
110t = 605
t = 5.5 hours
So in 5 1/2 hours the 2 cars will be 605 miles apart.
 
CHECK:
If you plug in 5.5 hours for each car to find each cars distances, they should add to 605 miles.
slow: (50)(5.5) = 275 miles traveled
fast: (60)(5.5) = 330 miles traveled
275 + 330 = 605 miles
 
SLOW CAR/FAST CAR LEAVE FROM SAME PLACE, GOING IN SAME DIRECTION, LEAVING AT THE SAME TIME, WHEN WILL THEY BE A CERTAIN DISTANCE APART?
 
2 cars leave going in the same direction. One travels 45 mph. The other travels 35 mph. In how many hours will the cars be 15 miles apart?
 
                      CAR     (RATE)    (TIME)   =   DISTANCE
                SLOW     35            t             35t 
                FAST      40            t             40t
                      (They travel the same amount of time)
 
What is true of the distances the 2 cars travel?
They are getting further and further apart as the minutes go by.
The DIFFERENCE of the 2 cars is 15 miles after a certain amount of time.
WHAT IS THE EQUATION?
40t - 35t = 15
5t = 15
t = 3 hours
So in 3 hours the 2 cars will be 15 miles apart.
 
CHECK:
If you plug in 3 hours for each car to find each cars distances, their distances should subtract to 15 miles.
slow: (35)(3) = 105 miles traveled
fast: (40)(3) = 120 miles traveled
120 - 105 = 15 miles
 
SLOW CAR/FAST CAR LEAVE FROM SAME PLACE, GOING IN DIFFERENT OR SAME DIRECTION, IN THE SAME AMOUNT OF TIME, EACH TRAVELS A  DIFFERENT DISTANCE, WHAT IS THEIR SPEED? 
One car travels 20 mph faster than the other car. One car travels 240 miles while the other travels 180 miles. Find their average speeds.
 
                      CAR     (RATE)       (TIME)   =   DISTANCE
               SLOW        r           180/r             180 
               FAST      r + 20    240/(r + 20)        240
This time you have the distance and know that the faster car is 20 mph faster than the slower car. To find the time for each car, use the fact that d/r = t so divided each car's distance by their rates.
 
WHAT IS THE EQUATION?
The times of these 2 cars is equal (left at same time and stopped at same time) so set their times equal:
180/r = 240/(r + 20)
Multiply both sides equally by the LCM:
(r)(r + 20)(180/r) = (r)(r + 20)240/(r + 20)
Cross cancel:
180(r + 20) = 240(r)
180r + 3600 = 240r
3600 = 60r
r = 60 mph (the slower car)
r + 20 = 60 + 20 = 80 mph (the faster car)
 
CHECK:
If you plug in the speeds, you should find that both cars traveled the same amount of time:
slow: (60)t = 180   t = 3 hours
fast: (80)t = 240   t = 3 hours
 
A BOAT/PLANE TRAVELS WITH THE CURRENT ON THE DEPARTING LEG OF THE JOURNEY AND TRAVELS AGAINST THE CURRENT ON THE RETURN LEG. WHAT IS THE SPEED OF THE BOAT/PLANE IN STILL WATER/AIR? 
 
A boat travels with a current of 6 mph for 3 hours and then returns home against the same current. The trip home takes 5 hours. What is the speed of the boat in still water?
 
           BOAT/PLANE        (RATE)       (TIME)   =   DISTANCE
      WITH CURRENT        r + 6           3            3(r + 6)  
      AGAINST CURRENT    r - 6           5            5(r - 6) 
r is the speed in still water and 6 is the speed of the current. The boat with need less time to go the same distance with the current than against it. 
 
WHAT IS THE EQUATION?
The distance to the boat's location and the distance home must be equal.
Set the distances equal:
3(r + 6) = 5(r - 6) 
3r + 18 = 5r - 30
48 = 2r
r = 24 mph (speed in still water)
r + 6 = 24 + 6 = 30 mph (speed with the current)
r - 6 = 24 - 6 = 18 mph (speed against the current)
 
CHECK:
If you plug in the speed with and against the current with the hours traveled, you get the distances to and home. Those distances should be equal:
3(30) = 5(18)
90 = 90

Pre Algebra (Period 2 & 4)

Volume of Prisms & Cylinders 10-7



Remember when I said that the basic formula for area is A = bh

Well, the basic formula for volume of a prism or cylinder is:

V = BH

Where capital B = the area of the base of the prism/cylinder

capital H = the height of the prism/cylinder

THE LABEL IS ALWAYS CUBED!!!


Volume of a rectangular, square, or parallelogram prism:

V = BH

Volume = area of the base times the height of the prism

V = (bh)(H)

The lower case b and h are the base and height of the base (this is plane geometry!)


Volume of a triangular prism:

V = BH

Volume = area of the base times the height of the prism

V = (1/2 bh)(H)

The lower case b and h are the base and height of the base (this is plane geometry!)


Volume of a trapezoidal prism:

V = BH

Volume = area of the base times the height of the prism

V = [(average of the 2 bases)(h)](H)

The lower case b and h are the base and height of the base (this is plane geometry!)



Volume of a cylinder:

V = BH

Volume = area of the base times the height of the prism

V = (π r²)(H)

The lower case b and h are the base and height of the base (this is plane geometry!)

Volume of Pyramids, Cones & Spheres 10-9

To find the volume of a pyramid or cone is just as easy!!!

It's just V= 1/3 BH

That means just find the volume as if it was a prism or cone then just divide it by 3!!!


VOLUME OF A SPHERE (ball) (this one is different)
V = 4/3 π r³