Math 7 ( Period 4)

Chapter 1.5 relating to Chapter 2

There are 2 formulas in Geometry that are used with equations a great deal:

Perimeter and area

PERIMETER – the sum of all the sides of a polygon

AREA- ( we are only focusing on squares, rectangles and triangles)

Area of a square =  s² where s represents the length of the square

Area of a rectangle = bh or lw

Area of a triangle = bh/2  or ½ bh

(We discussed how we were able to arrive at the formula for a triangle…)

Compound Shapes

Sometimes a shape will be composed of several shapes. Right now we are covering shapes that are composed of squares rectangles and triangles. You break the shape into the parts you know. IF there is a side missing a measurement, you may have to SUBTRACT or ADD using the known measurements of other sides.

We looked at Page 24…

In Chapter 2 you will use the formula to find a missing side using equation solving!

If you know the area of a rectangle is 20 feet squared and one of the sides is 2 feet, what is the other side?

Since A = bh   you can solve for the missing side by plugging in the area and the known side

20 = 2h

h = 10

10 feet

An example of a triangle

The area of a triangle is 20 feet squared. The base of the triangle is 10 feet. How long is the height?

A = b/2

20 = 10h/2

20 = 5h

h = 4

The height must be 4 feet

Always remember the formula of a triangle is half the formula for a rectangle with the same base and height!

This one works nicely because the base was even and we could divide by 2 easily. You can also solve triangles with a base or height that is odd—but we will focus on that later with 2-step equations and equations with fractional coefficients.  

Algebra Honors (Periods 6 & 7)

Powers of Monomials 4-4
To find a power of a monomial that is already a power, you can use the definition of a power and the rule of exponents for products of powers.

(x⁵)³ = x⁵∙ x⁵∙ x⁵ = x⁵⁺⁵⁺⁵= x¹⁵

Notice that  (x⁵)³ = x¹⁵ or x^5∙3

In general

(a^m)ⁿ = a^mn

Rule of Exponents for a Power of a Power

For all positive integers m and n

(a^m)ⁿ = a^mn

To find a power of a power, you multiply the exponents

(u⁴)⁵ = u²⁰

[(-a)²]³ = (a²)³= a⁶

To find the power of a product, you can use the definition of a power AND the commutative and associative properties of multiplication.

(2x)3 = (2x)(2x)(2x) = (2∙2∙2)∙ (x∙ x∙ x)

= 2³∙x³

=8x³

Notice BOTH the 2 and the x are cubed

So (ab)^m = a^mb^m

Rule for Exponents for a Power of a Product

For every positive integer m

(ab)^m = a^mb^m

To find a power of a product, you find the power of each factor and then multiply

(-2k)⁵=(-2)⁵k⁵=-32k⁵

Simplify

(-3x²y⁵)³ = (-3)³(x²)³(y⁵)³

= -27x⁶y¹⁵

Negative Exponents 
If a is a nonzero real number and n is a positive integer

a^-n = 1/aⁿ

so
 10^-3 = 1/10³ = 1/1000
( remember this from Math 6 A)  

5^-4= 1/5⁴ = 1/625

16^-1 = 1/16

Let's look at the rule of exponents for division

Rule of Exponents for Division

If a is a non zero real number and m and n are positive integers, then:

If m>n

a^m/aⁿ = a^m-n 

If n > m

a^m/aⁿ = a^m-n  

but that means it would be 1/a^n-m

If m = n

a^m/aⁿ = a^m-n  = a⁰ = 1

Let's look at the 2nd case using an example

x²/x⁷ =x^2-7 = x^-5 but we write that without negative exponents as 1/x⁵

Looking at the rule above if  n > m

And in this case 7> 2

It says  that the simplified form would be 1/x^7-2 which is 1/x⁵

 This should  help you understand why
a^-n = 1/aⁿ
recall that for m > n a^m/aⁿ= a^m-n
More examples
a⁷/a³ = a^7-3= a⁴
you can also apply this rule when m < n that is when m - n becomes a negative number. For example a³/a⁷ = a^3-7 = a^-4
since
a⁷/a³ and a³/a⁷ are reciprocals then
a⁴ and a^-4 must also be reciprocals.
Thus
a^-4= 1/a⁴
a⁵/ a⁵ = a^5-5 = a⁰
But you already know that a⁵/a⁵ = 1
SO, definition of a⁰
a⁰ = 1
However, the expression 0⁰ has no meaning

All the rules for positive exponents also hold for zero and negative exponents.

Summary of Rules for Exponents 

Let m and n be any integers
Let a and b be any non zero integers
We will be reviewing these throughout the year.. but I want you to begin to understand this concept! We will be incorporating it into an upcoming Project!!

Products of Powers 

b^mbⁿ = b^m+n

Example with negative exponents
2³⋅2^-5 = 2^3+(-5) = 2^-2 = 1/2² = 1/4

Quotients of Powers 

b^m ÷ bⁿ = b^m-n

Example with negative exponents
6³÷6⁷= 6^3-7= 6^-4= 1/6⁴= 1/1296

Power of Powers

(b^m)ⁿ = b^mn

Example with negative exponents
(2³)^-2 = 2^-6 = 1/2⁶ = 1/64

Power of  a Product

(ab)^m= a^mb^m

Example with negative exponents
(3x)^-2 = 3^-2 ⋅x^-2 = 1/3²⋅1/x² = 1/9x²

Power of  a Quotient

(a/b)^m= a^m/b^m

Example with negative exponents
(3/5)^-2= 3^-2/5^-2= (1/3²)/ (1/5²)= 1/3² ÷ 1/5² which means
1/3² ⋅5²/1= 5²/3²= 25/9

Math 6A ( Periods 1 & 2)

Writing Inequalities 2-3

2 < 7 and 7 > 2 are two inequalities that state the relationship between the numbers 2 and 7

2 < 7 reads 2 is the less than 7 and
7 > 2 reads 7 is greater than 2
The symbols < and > are called inequality symbols.

Notice the mathematical sentence (inequality)
Two is less than ten or 2 < 10 is different from the mathematical phrase (expression)
Two less than ten. 10 - 2
A number 2 + x is greater than a number t 2 + x > t

The point of the number line that is paired with a number is called the graph of that number.

Check out the graph in the middle of page 39 of our textbook. When you graph numbers on the number line, make sure to place a dot DIRECTLY ON the number line at that particular number's location.
Again, check out our textbook for examples!!

Looking at the graph of numbers, we see that the larger number will be to the right of the smaller number.

A number n is between 6 and 12 so 6 < n < 12 or 12 > n > 6

Thursday's Lesson: continuing on ....

Notice the subtle differences in the sentence
Six is greater than a number t
and the phrase
six greater than a number t

Six is greater than a number t becomes 6 > t
while
six greater than a number t becomes t + 6

What about the following inequality:
A number p is greater than a number q
is p > q

The value in cents of d dimes is less than the value in cents of n nickels.

If you need to-- set up your T-charts (refer to your class notes) one for dimes and the other for nickels.

10d represents the number of dimes and 5n represents the number of nickels

so 10d < 5n

Algebra Honors ( Periods 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

Math 7 ( Period 4)

Solving Equations Using All for Ops 2.5 and 2.6

What’s the GOAL?  Determine the value of the variable?

How?  Isolate the variable—get it alone on one side of the equation

What do I do? Use inverse (opposite operations to “get rid” of everything on the side with the variable

What should my focus be?  When equations get really complicated…. ALWAYS focus on the variable FIRST!

One Step Equations with all 4 Operations

 –We will be meeting 4 new BFF’s

Equation Balancing Properties of Equality – there are 4 of them

Whatever YOU DO to BALANCE an equation – that operation is the property of equality that was used.

If you have…

X + 3 = 10  …you used the subtraction property  of Equality because you need to SUBTRACT 3 from both sides equally.

x – 3 = 10 …you used the Addition Property of Equality because you need to ADD 3 to each side equally

3x = 9 … you used the Division Property of Equality because you need to DIVIDE  each side by 3 equally

x/3 = 9…  you used the Multiplication Property of Equality because you need to MULTIPLY both sides by 3 equally.

TWO more Properties

Additive Inverse
using the opposite sign of the term given in the equation results in the term dropping out because it simplifies to 0

Multiplicative Inverse
using the reciprocal of the term given in the equation results in the term dropping out because it simplifies to 1

Here are the steps and justifications

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 that the variable is doing with that number in its way?

3. Get rid of that number by using the opposite (inverse) operation

Golden Rule of Equations:

Whatever you do to one side do unto the other side

Doing the same thing on both sides is actually the new set of properties (your new BFF’s)

When you multiply equally, it’s the multiplication property of equality

When you divide equally, it’s the division property of equality

When you add equally, it’s the addition property of equality

When you subtract equally, it’s the subtraction property of equality

4. JUSTIFICATION  You have just used one of the properties of equality… Which one?

Whatever operation you used to balance both sides (not the operation of the original equation)

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 -, you used the Identify Property of Addition ( ID+) which simply means that you don’t bring down the ZERO because when you add ZERO it does not change anything! ( NOTE:  There is no ID of subtraction)

For ×   and  ÷  you used the Identity Property of Multiplication ( ID x) 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!)

FORMAL CHECK:

There are three steps to a formal check:

1. Rewrite the original equation FROM THE ORIGIANL SOUSE – this is just o case that ou find out you copied the problem wrong

2. Substitute your answer where the variable is and QUESTIO you answer by placing a “?” over the equal sign

3. REALLY  Do the math and finally set both sides equal. Place a check mark, a happy face… to indicate that you really did check this!