Math 6 High (Period 3)

Powers & Exponents 1-4


The power a^m is a product of the form

a^m = a·a·a·a·a·a·a....·a
m factors
The exponent m indicates the number of times the factor a (called the base)  is repeated.
For example, 10³  means 10 ×10×10 where the base  10 is repeated 3 times because the exponent is 3.

4² is 4 ×4 = 16
2³ is 2 ×2×2 = 8
5⁴ = 5⋅5⋅5⋅5 = 625

Evaluating an Expression

Evaluate x⁴ for the given value of x.
a. where x = 3
substitute  3 in for x
so
x⁴ becomes   3⁴
= 3⋅3⋅3⋅3 = 81

b. where x = 17
x⁴ becomes   17⁴
= 17⋅17⋅17⋅17
I would use a calculator for this one ( wouldn't you!)
= 83, 521


c. where x = 1
1⁴ = 1⋅1⋅1⋅1= 1
in fact
1¹²¹ = 1

Try this one:
2^x - x   when x = 5
substitute in 5 for each x
you get
2⁵ - 5
or
2⋅2⋅2⋅2⋅2 - 5
= 32 - 5
= 27

Algebra Honors (Periods 5 & 6)

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 (÷)

Math 6A (Periods 2 & 4)

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.'

Math 6H (Period 3)

Expressions and Variables  1-2

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 (Periods 5 & 6)

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 6A (Periods 2 & 4)

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!!

Algebra Honors (Periods 5 & 6)

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 6A (Periods 2 & 4)

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 (Periods 5 & 6)

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.

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

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