Classifying Lines and Angles

Point – is represented by a dot

Line segment – has two points and every point in between those two points

Ray – consists of a point and all points going in the direction of another point

Line – names a line which passes through multiple points. The line goes on forever in both directions.

Plane – a flat surface that extends infinitely in all directions.

There are different types of angles, depending on their measurements:

Acute – less than 90°                 Right – 90°

Obtuse – more than 90°, less than 180°

Straight – 180°                          Reflex – more than 180°

Angles – are formed by two rays which share a common endpoint. The angle that you notate will always be the middle letter.

ex: ABC is 45°

Parallel Lines – are lines that will never intersect

Intersecting Lines – will cross at exactly ONE point

Perpendicular Lines – are lines that intersect to form a right angle

Exam 2

No lesson

Sales Tax, Discounts, Percent Increase/Decrease

Sales & Discounts

If you go shopping or have shopped before, you will notice that there are always items on sale. Let’s say that you went shopping for a new pair of sneakers, but the pair you want is a little out of your budget – $100. If you choose you wait a couple of months, the pair of sneaks will be on SALE – meaning that it will be at a reduced price. The reason why things go on sale is because stores and manufacturers have not sold enough of the certain item. Also, as time passes, the more outdated things become.

Shopping Vocabulary:

Sales tax – percentage of your purchase that will be added on to your purchase. This amount is given to the government.

Promotion – something that gets customers to shop at the store. Example: Buy one get one free, 60% off entire store, etc.

Discount – a certain amount that is taken off your purchase. It usually is a percentage of the total price.

Percent Increase/Decrease

To find the percent increase or decrease:

1. Subtract the bigger number from the smaller number

2. Divide by the original number

3. Convert the quotient into a percent

Ex: A Pikachu trading card originally cost $10. Three years later, the card costs $15. Find the percent increase.

1. $15 – $10 = $5

2. $5 /$10 = 0.5

3. 0.5 = 50%

Now, let’s change up the problem to find a percent decrease.

A Pikachu trading card originally cost $10. Three years later, the card costs $8. Find the percent decrease.

1. $10 – $8 = $2

2. $2 /$10 = 0.2

3. 0.2 = 20%

Compound Interest

Compound Interest is much like simple interest, but this situation is when interest is added to the principal, and from that moment on, the interest that is added on also earns interest.

After you find the simple interest for 1 year, you must add that interest back to the original principal and then find the interest again, this time using the total amount from the previous calculation as your NEW PRINCIPAL.

For example:

I = P x r x t    P = $400   r = 6%   t = 3 years

1. I = $400 x .06 x 1 = $24

$24 + $400 = $424 – NEW PRINCIPAL

2. $424 x .06 x 1 = $25.44

$25.44 + $424 = $449.44 – NEW PRINCIPAL

3. $449.44 x .06 x 1 = 26.9664 or $26.97

Total amount – $449.44 + $26.97 = $476.41

Also, you will notice that you will be told how often the interest is “compounded” – semi-annually (twice a year – use 1/2 as your time), quarterly (four times a year – use 1/4 as your time), and monthly (12 times a year – use 1/12 as your time).

Lets say your time given was 1.5 years. If the interest is compounded semi-annually, then you must find the compound interest a total of 3 times. Use the fraction 1/2 as your TIME each instance you are calculating the compound interest.

Simple Interest For “Between Years”

***REMINDER: BRING A CALCULATOR FOR YOUR NEXT CLASS***

There are situations when you can borrow/deposit money for more than one year, less than one year, or something in-between. Nothing really changes here, as you’ll need to convert the fractions into decimals.

However, when we express the Time in days, we will write the time as a fraction over 360 (this number will vary depending on the worksheets we use).

ex: Principal = $200  Rate = 12%  Time = 30 days

I = P x r x t

I = $200 x .12 x 30/360

I = $2

*For your homework today, you will see problems in which you will have to solve for a variable in the interest formula.

ex: Dimitri borrowed money for 4 years at an annual rate of 15%. If he paid $4,800 in interest, how much did he borrow?

I = P x r x t

$4800 = P x 0.15 x 4

$4800 = P x 0.6 or 0.6P (Solve for P; divide each side by 0.6)

$8000 = P

 

Simple Interest

To find Interest, we use a general formula. Interest = Principal x Rate x Time

Interest – the money paid for the borrowing/depositing money

Principal – The amount of money borrowed or deposited

Rate – usually given as a percent

Time – expressed in terms of years

ex: If Steve borrows $300 from Bill at an interest rate of 5% for 1 year, what is the interest?

I = P x r x t

I = $300 x .05 x 1

I = $15

Percent of a Number

All percent problems will have a general structure of:
n = m% of o

PERCENT OF A NUMBER

Remember, we are translating the sentence, “What number is m% of o?” Solve for n.

Ex: What number is 50% of 80?

n = .50 x 80   (translate)

n = 40   (simplify)

40 is 50% of 80

WHAT PERCENT ONE NUMBER IS OF ANOTHER

Remember, we are translating the sentence, “This number is m% of o.” Solve for m.

Ex: 25 is what percent of 125?

25 = m x 125   (translate)

25 = 125m   (simplify)

1/5 = m    (divide each side by 125)

m = 20%

25 is 20% of 125

*Even though your answer is a fraction, remember that “m” is a percent! Convert your answer to a percent.

FINDING A NUMBER WHEN A PERCENT OF IT IS KNOWN

Remember, we are translating the sentence, “This number is m% of o?” Solve for o.
Ex: 105 is  35% of what number?

105 = .35 x o  (translate)

105 = .35o  (simplify)

300 = o   (divide each side by .35)

105 is 35% of 300

Writing Equations, Formulas, Equations Involving Fractions

Writing Equations

Whenever you are working on a word problem and have to set up an algebraic equation, you must first DEFINE YOUR VARIABLES.

ex: Max has two boards that have a combined length of 16 feet. One board is 1 foot longer than twice the length of the other. What is the length of each board?

Since we have two boards, we must assign expressions to the shorter board and the longer board.

Let:

2x + 1 = longer board

x = shorter board

*Whenever you are comparing two variables, the second variable in the problem is usually just the variable.

Now that we know that the two boards have a combined length of 16 feet, lets translate.

2x + 1 + x = 16    (combine like terms)

3x + 1 = 16         (subtract 1 from both sides)

3x = 15        (divide each side by 3)

x = 5

Next, plug in 5 for x to find out the length of each board.

Longer board = 2x + 1 = 2(5) + 1 = 11 ft

Shorter board = x = (5) = 5 ft

Formulas

Formula – A mathematical relationship or rule expressed in symbols.

We have seen formulas in the past – for example, area of shapes                               (length x width, π x r²)

Today, we saw the formula for distance and balancing a lever.

distance = rate x time         d = r x t

weight x distance = weight x distance      w x d = W x D

*For problems involving formulas, you will usually have at least two of the variables defined for you – you will have to solve for the last variable using algebra.

Equations involving Fractions

For equations involving fractions, you follow the same steps in solving for regular algebraic problems – just don’t forget how to operate with fractions!

Combining Like Terms, Equations Involving Negative Integers

As many of you have noticed in the past class sessions, instead of having just a variable in an equation, there are instances in which there is a number in front of the variable.

Whenever you see a number in front of a variable, it is called a coefficient.

A coefficient lets us know how many groups of the variable there are.

For example, lets say “x” represents apples. If we have 5 groups of apples and 7 groups of apples, we can say the total amount of groups of apples is 12.

In an equation, we will have to divide each side by the coefficient at the end.

ex: 5x + 7x + 12 = 144

12x + 12 = 144  (simplify by combining like terms 5x + 7x)

12x = 132   (subtract 12 from each side)

x = 11    (divide each side by 12)

Finally, when you deal with equations involving negative integers, follow the steps to solving for the variable. To review our work with integers, relearn the lessons from earlier in the semester.

Simplifying Both Sides of the Equation, Two-Step Equations, Writing Equations

Today, we practiced simplifying both sides of the equation while solving for the variable (step 1).

ex: 9 x 3 = p + 5

(simplify 9 x 3)           27 = p + 5 

(subtract 5 from both sides)              22 = p

We also practiced solving two-step equations.

*Get rid of the terms that are not associated with the variable first!

ex: 3x + 5 = 26

(Subtract 6 from both sides)     3x = 21

(Divide each side by 3 to isolate the variable)    x = 7

*When writing equations, we must read the information from questions carefully and then use our knowledge of translating English into Algebra.

ex: The temperature has fallen 12 degrees since noon. The present temperature is 57 degrees. What was the noon temperature?

x is the temperature at noon

x – 12 is your present temperature

Equation: x – 12 = 57      x = 69

The noon temperature was 69 degrees.