Math 6A ( Periods 1 & 2)

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

Math 6A ( Periods 2 & 4)

Graphs of Equations 11-9

An equation in two variables y = x + 2      produces an infinite number of ordered pairs
If we give x the value of 1, a corresponding value of y is determined
y = (1) + 2 = 3
The ordered pair is (1, 3)

If we let x = 4
y = (4) + 2 = 6
and we get the ordered pair (4, 6)

What happens if x = 0
y = (0) + 2 = 2 ( 0, 2)
or x = -1
y = (-1) + 2 = 1 ( -1, 1)

I like to remember ordered pairs---> ( ordered, pairs)

We graphed the line on a mini graph stickie.
The line is graphed using a ruler and connecting all the points we plotted. Put arrows at each end (since a line continues with out end) and write the line's equation right above the line.


For each value of x there is EXACTLY 1 value of y.
set of ordered pairs in which no two ordered pairs have the same x is called a FUNCTION


y = x + 2
in the future you will see it written as
f(x) = x + 2
so if x = 2
f(2) = (2) + 2 = 4
if x = 5
f(5) = (5) +4 = 9



We used a three column chart to compute our ordered pairs.
Please refer to the work sheet glued into your spiral notebook for the examples we completed from the class exercises found on Page 393 -- if you were absent, please come in one morning and I will review that chart with you.


The following equations create curves that are called PARABOLAS!! Notice the difference in these equations from our previous equations
y = x² +1
when we create your three column table using integers from -2 to 2
we notice
y = (-2)² +1 = 4 + 1 = 5 ordered pair (-2, 5)
y = (-1)² +1 = 1 + 1 = 2 ordered pair (-1, 2)
y = (0)² +1 = 0 + 1 = 1 ordered pair (0, 1)
y = (1)² +1 = 1 + 1 = 2 ordered pair (1, 2)
y = (2)² +1 = 4 + 1 = 5 ordered pair (-2, 5)

When you graph this... you get a "U" shaped graph.

Remember linear equations LINEar equations are lines!
and look like y = x + 2

PARABOLAS have the form y = x² or y = -x²

Let's try
y = 2 - x²
With our 3 column table
for values of x from -2 to 2
we find
y = 2 -(-2)² = 2 -(4) = -2 and the ordered pair is (-2,-2)
y = 2 -(-1)² = 2 - (1) = 1 and the ordered pair is ( -1, 1)
y = 2 -(0)² = 2 - 0 = 2 and the ordered pair is (0, 2)
y = 2 -(1)² = 2 -1 = 1 and the ordered pair is (1, 1)
y = 2 -(2)² = 2 - (4) = -2 and the ordered pair is (2, -2)

When you graph these ordered points you find you have an upside down U
hmmm... y = -x² results in a sad face parabola
and y = x² results in a happy face parabola!!

Math 6A ( Periods 2 & 4)

Graphs of Ordered Pairs 11-8

A PAIR of numbers whose ORDER is important is called an
ordered pair!!
(ordered, pair)

(2,3) is not the same as (3,2)

The two perpendicular lines are called axes.
The x-axis deals with the 1st number of the ordered pair and the y-axis deals with the 2nd number of the ordered pair.
The AXES meet at a point called the Origin (0,0)
The plane is called the coordinate plane

There are 4 quadrants, Use Roman Numerals to name them!!
Quadrant I ---> both the x and y coordinates are positive
(x,y) (+,+)

Quadrant II --> the x coordinate is negative but the y is positive
(-x,y) (-,+)

Quadrant III -->. both the x and y coordinates are negative
(-x,-y) (-,-)

Quadrant IV --> the x coordinate is positive but the y coordinate is negative
(x,-y) (+,-)

Math 6A (Periods 2 & 4)

Equations: All 4 Op's Sections 8.1- 8.5 Summary 

You may be able to solve some of the equations in the homework without pencil and paper. Nevertheless, it is important to show all the steps in your work and to make sure you can tell which transformation you are using in each step.

If the replacement set for an equation is the set of whole numbers, it is not practical to use substitution to solve the equation. Instead we transform or change the given equation into a simpler, equivalent equation. When we transform the given equation, our goal is to arrive at an equivalent equation of the form
variable = number

Transformation by addition: add the same number to both sides
Transformation by subtraction: subtract the same number from both sides

solve x – 2 = 8
our goal is to find an equivalent equation of the form
x = a number

The left side of the given equation is x – 2. Recall that addition and subtraction are inverse operations. If we add 2 to both sides the left sides simplifies to x
x-2 = 8
x – 2 + 2 = 8 + 2 (We usually show the +2 right below each side of the equation)
x = 10


Solve x + 6 = 17
Subtract 6 from both sides of the equation to get an equivalent equation of the form
“ x = a number”

x + 6 = 17
x + 6 – 6 = 17 – 6 (Again, we usually show the -6 right below each side of the equation)
x = 11
the solution is 11

In equations involving a number of steps, it is a good idea to check your answer. This can be done easily by substituting the answer in the original equation.
What about the following
34 – x = 27
add x to both sides
34 – x + x = 27 + x
34 = 27 + x
subtract 27 from both sides
34 – 27 = 27 – 27 + x
7 = x


If an equation involves multiplication or division, the following transformations are used to solve the equation:

Transformation by multiplication: Multiply both sides of the equation by the same nonzero number.
Transformations by division: Divide both sides of the equation by the same nonzero number.

Remember: Do undo on one side what you would do undo the other!!

Our goal is to get the variable alone and to find an equivalent equation of the form
“n = a number”
Our goal is to arrive at the “world’s easiest equation”

Solve 3n = 24
Use the fact that multiplication and division are inverse operations
3n/3 = 24/3
n = 8
Or you could have use the reciprocal of 3--> which is 1/3 and multiplied both sides by 1/3
(1/3)(3n) = 24(1/3)
n = 8 and still arrived at the SAME solution

Solve 5x = 53

5x/5 = 53/5
x = 10 3/5

Solve n/4 = 7
(4)(n/4) = 7(4)
n = 28
How could you know for sure your answer is correct?
Substitute your solution into the ORIGINAL equation
Try:
1. 3r = 57

2. 714 = 7t

3. Solve A = bh for h

4. Solve P = 4s for s

5. Solve C = 2πr for r



Remember when we stated the properties as well-- back in our 1st quarter!!  

Equations with Decimals or Fractions 
You can use transformations to solve equations which involve decimals or fractions
Solve 0.42 x = 1.05
Divide both sides by 0.42
.42x/.42 = 1.05/.42
now, do side bar and actually divide carefully and you will arrive at
x = 2.5

Solve: n/.15 = 92
multiply both sides by .15 to undo the division
(n/.15)(.15) = 92 (.15)
Again, do a sidebar for your calculations and you will arrive at
n = 13.80

How would we solve the following: (2/3)x = 6?

Let’s look at 2x = 6. What do we do?
We divide both sides by 2—or multiply both sides by the reciprocal of 2—which is ½
Remember the product of a number and its reciprocal is 1

Reminder: the ultimate objective is applying transformations to an equation is to obtain an equivalent equation in the form x = c
(where c is a constant.)

Also remember that the understood (invivisble) coefficient of x in the equation x = c is 1.

[Can you picture the poster in the front of the room?]

So to solve (2/3)x = 6 you would divide both sides by 2/3 but that is the same as multiplying by the reciprocal of 2/3, which is 3/2.

If an equation has the form

(a/b)(x) = c,
where both a and c are nonzero,
multiply both sides by b/a, the reciprocal of a/b
Solve (1/3)y = 18

(3/1)(1/3)y = 18(3/1)

y =18(3)
y = 54

Solve the equation: (6/7)n = 8
(7/6)(6/7)n = 8(7/6)

n = 8(7/6)

simplify first , then multiply
n = 28/3
n = 9 1/3
Let’s check

(6/7)n = 8
well, we said that n = 9 1/3 so substitute back, but change to 28/3 first
(6/7)(28/3) ?=? 8
[read ?=? as ‘does that equal?’]

Now really do a side bar with the left side of the equation to see what
(6/7)(28/3) really equals. Simplify before you multiply
2(4) = 8 so
8 = 8


Try:
1. (1/7)a = 13

2. b/8 = 16

3. 3.6d = 0.9

4. (3/8)f = 129


Combined Operations 8-5 

In order to solve an equation of the form
ax + b = c or ax –b = c or b – ax = c
where a, b, c are given numbers and x is the variable, we must use more than one transformation

Solve the equation 3n - 5 = 10 + 6
Simplify the numerical expression

3n - 5 = 10 + 6
3n – 5 = 16

add 5 to both sides

3n – 5 + 5 = 16 + 5
or
3n – 5 = 16
+ 5 = +5

3n = 21

divide both sides by 3 (or multiply each side by the reciprocal of 3)
3n/3 = 21/3

n = 7

General procedures for solving equations

Simplify each side of the equation
If there are still indicated additions or subtractions, use the inverse operation to undo them
If there are indicated multiplications or division involving the variable, use the inverse operations to undo them

The books says you must always perform the same operation on both sides of the equation. I say, “do to one side what you have done to the other side.”

Solve the equation
(3/2)n + 7 = 22
subtract 7 from both sides
(3/2)n + 7 - 7 = 22 - 7
(3/2)n =15

multiply both sides by 2/3, the reciprocal of 3/2

(2/3)(3/2)n = 15(2/3)
n = 10

Solve the equation
40 – (5/3)n = 15
add (5/3)n to both sides

40 – (5/3)n + (5/3)n = 15 + (5/3)n

40 = 15 + (5/3)n
subtract 15 from both sides

40 – 15 = 15-15 + (5/3)n

25 = (5/3)n multiply both sides by 3/5

(3/5)(25) = (5/3)n (3/5)

15 = n

Math 6A (Periods 2 & 4)

Negative Numbers 11-1

On a horizontal number line we use negative numbers for the coordinates of points to the left of zero. We denote the number called ‘negative four’ by the symbol -4. The symbol -4 is normally read ‘ negative 4’ but we can also say ‘ the opposite of 4.’

The graphs of 4 and -4 are the same distance from 0—>but in opposite directions. Thus they are opposites. -4 is the opposite of 4.

The opposite of 0 is 0

Absolute Value is a distance concept. Absolute value is the graph of the distance of a number from 0 on a number line. The absolute value of a number can NEVER be negative!!

Counting (also known as Natural) numbers: 1, 2, 3, 4, ….
Whole numbers 0, 1, 2, 3, 4….
Integers are natural numbers and their opposites AND zero
…-4, -3, -2, -1, 0, 1, 2, 3, 4….

The opposite of 0 is 0.

The integer 0 is neither positive nor negative.

The farther we go to the right on a number line--- the bigger the number. We can compare two integers by looking at their position on a number line.

if x < 0 what do we know? x is negative number if x > 0, what do we know? x is a positive number

We have been practicing representing integers by their graphs, that is, by points on a number line.

Make sure that your number line includes arrows at both ends and a line indicating where zero falls on your number line.
The graph of a number MUST have a closed dot right on the number line at that specific number.
Please see our textbook page 366 for an accurate example.