Algebra Period 3

Compound Sentences 9-2

Conjunction = and - the graph is an intersection ("yo") and inequality looks like our domains and ranges on our projects EXAMPLE: 5 < x < 10
open dots; between 5 and 10 is colored in

Disjunction = or - graph will go opposite ways ("dorky" dancer) and inequality looks like this:
x < -2 OR x > 4
open dots; one arrow goes right at 4 and the other arrow goes left at -2

Equations and Absolute Value - 1 VARIABLE 9-3
Solve the equation twice - Once with the solution positive and once with it negative in this setting [ ] represents absolute value.
[2x - 4] = 10
Solve it twice:
2x - 4 = 10 or 2x - 4 = -10
x = 7 or x = -3

If there is a term on the same side of the equation as the absolute value, move that to the other side of the equation first (just like we did with radical equations!) Then solve twice.

REMEMBER THAT THE SOLUTION GIVEN CANNOT BE NEGATIVE (the null set)

Inequalities and Absolute Value - 1 VARIABLE 9-4
There are 2 possible types of inequalities - less than and greater than
For less thAND:
These are conjunctions and so you solve it twice and the solution ends us between them
[3x] < 15 is equal to -15<3x<15
so x is greater than -5 and less than 5

For greatOR than:
These are disjunctions and are solved twice with the solution infinitely in different directions
[3x] >15 is equal to 3x < -15 and 3x > 15

INEQUALITIES IN 2 VARIABLES 9-5
You will shade an x y graph to find the side of a linear equation that fits the solution
EXAMPLE: x + y > 5
You graph the line with DOTTED line because it cannot be equal to 5
You pick an easy point on one side of the line and substitute to see if that side is a solution.
If that does not work, pick an easy point on the other side to see if that side checks.
I USUALLY USE (0,0) as my first point!
If there is more than 1 Inequality (a system of inequalities), follow the same procedure and where the 2 shadings overlap each other is called the solution of the system of inequalities.

SYSTEMS OF INEQUALITIES IN TWO VARIABLES 9-6
If there is more than 1 Inequality (a system of inequalities),
1. Follow the same procedure as above for one equation
but you will need to do it for each equation.
2. Use a different type of shading for each so you won't get confused
(Ex: Use slanted lines one way and then slanted lines the other way. Use different colors if possible. Make one set of lines wavy and the other set straight)
3. Where the 2 shadings overlap each other is called the solution of the system of inequalities.
(any point in the overlap should work in BOTH inequalities - make sure you check!!!!!)

Pre Algebra Periods 1, 2, & 4

Graphing Review - make sure to look at your book as you review these notes. Look at the examples

Box and Whisker Plots

Suppose you have a large set of data and you want a display that gives a general idea of how the data clusters together. A box-and-whisker plot displays the median, the quartiles, and outliers of a set of data but does not display any other specific values.

To make a box-and-whisker plot:

Write the data in order from least to greatest

Draw a number lime that can show the data in equal intervals – make sure to have intervals that include the least and the greatest

Find the median – Mark it with a dot below ( or above the number line)

Find the upper quartile ( the median of the numbers above the actual median) Mark it with a dot below ( or above the number line)

Find the lower quartile ( the median of the numbers below the actual median) Mark it with a dot below ( or above the number line)

Mark with a dot the upper extreme – the greatest number

Mark with a dot the lower extreme – the lowest number

Draw a box between the lower quartile and upper quartile. Split the box by drawing a vertical line through the median.

Draw two ‘whiskers’ from the quartiles to the extremes.

50% of all the data will be within the box. 25% will be below and 25% will be above.

Frequency Table

A frequency table is a way to show how often an item, a number or a range of numbers occurs.

number	1	2	3	4
frequency	4	0	5	8

The range is the difference between the highest number and the lowest. In this case 4-1 = 3

Line Plots help show the spread of the data. When you look at a line plot you can easily see the range, the mode, and any outliers in the data,

Draw a horizontal line segment on grid paper

Make a scale of numbers below the line. The numbers should include the greatest value and the least value of the set of data.

For each piece of data, draw an X above the corresponding number.

Stem-And-Leaf Plots

Stem and Leaf Plots allow you to easily see the greatest, least, and median values in a set of data.

For example—from class

As of 1997 the following are the ages, in chronological order, at which US Presidents were inaugurated

57,61, 57, 57, 58, 57, 61, 54, 68, 51, 49, 64, 50, 48, 65, 52, 56, 46, 54, 49, 50, 47, 55, 55, 54, 42, 51, 56, 55, 51, 54, 51, 60, 62, 43, 55, 56, 61, 52, 69, 64, 46

To make a stem-and-leaf plot:

Write the data in order from least to greatest

Find the least and greatest values

Choose stem values that will include the extreme values. For this graph, it makes sense to use tens.

Write the tens vertically from least to greatest. Draw a vertical line to the right of the stem values.

Separate each number into stems (tens – in this case) and leaves (ones- in this case). Write each leaf to the right of its stem in order from least to greatest.

Write a key that explains how to read the steams and leaves.

Scatter Plots

Suppose you want to analyze two sets of data to see how closely they are related. On a scatter plot you plot corresponding numbers from two sets of data as order pairs (x,y). You then decide if they are related by determining how close they come to forming a straight line.

For example, here is a frequency table of the number of hours studied and grades of a student

study hours	1.5	1	3	2.5	1.5	4	3.5
grade on test	75	71	88	86	80	97	92

To make the scatter plot

Decide which set of numbers you will plot on each axis and label the axis. In this case, The study time ( in hours) would be the X-axis and the Grade would be the Y-axis.

Choose a scale for each axis.

Plot corresponding numbers as ordered pairs. For example (1.5, 75) and (1, 71) are the first two from the table above.

The dots on the scatter plot are close to forming a straight line ( going up) so this is a strong positive correlation.

If the line formed was going down—it would be a negative correlation

and if you could not determine any line- it would be no correlation.

Math 6 Honors Periods 6 & 7

Percent and Fractions 9-1

The word “percent” is derived from the Latin “per centum” meaning “per hundred” or “out of one hundred” so 28% means 28 out of 100

A percent is a ratio that compares a number to 100. Therefore you can write a percent as a fraction with a denominator of 100, so 28% is also 28/100

Our book’s example is as follows;

During basketball season, Alice made 17 out of 25 free throws, while Nina made 7 out of 10. To see who did better, we compare the fractions representing each girl’s successful free throws.

17/27 or 7/10 or

In comparing fractions it is often convenient to use the common denominator 100, even if 100 is not the LCD of the fractions.

17. 4 = 68 thithithiett7.10=70
24^t4th100thiethithei10t10t100

Since Alice makes 68 free throws per 100 and Nina makes 70 per hundred, Nina is the better free throw shooter.

the ratio of a number to 100 is called a percent. We write percents by using the symbol %

so

17/25=68% and 7/10= 70%

Rule

To express the fraction a/b as a percent, solve the equation

n/100 = a/b

for the variable n and write n%

Just set up as a ratio and compare your fraction to 100. You know three out of the four numbers. Express 17/40 as a percent

Use the proportion method or just divide 17 by 40 multiply both sides by 100

n= 42½

Therefore, 17/40 = 42.5%

Rule

To express n% as a fraction, write the fraction

n/100 in lowest terms

Express 7 1/2 % as a fraction in lowest terms

7 ½ % = 7.5% = 7.5/100 How do we get rid of the decimal? multiply numerator and denominator by ten

7.5 . 10 = 75 simplify
100 t10 t1000

=3/40

Since a percent is the ratio of a number to 100, we can have percents that are greater than or equal to 100%

100 = 100%
100

165 =165%
100

Write 250% as a mixed number in simple form

250% =

250=
100

2 50/100 = 2 1/2

Percents and Decimals 9-2

By looking at the following examples, you will be able to see a general relationship between decimals and percents

51% = 47/100 = 0.57 113% = 113/100 = 1.13

0.79 = 79/100 = 79% .06 = 6/100 = 6%

Rules

To express a percent as a decimal, move the decimal point two places to the left and remove the percent sign

57% = 0.57 thith 113% = 1.13

To express a decimal as a percent, move the decimal point two places to the right and add a percent sign

0.79 = 79% thithith0.06 = 6%

Express each percent as a decimal ( remember- the decimal is so sad to see the % leave that it runs away – two places to the left—from where it was)

1. 83.5% = 0.835

2. 450% = 4.5

3. .25% = 0.0025

Express each decimal as a percent [careful] ( remember- the decimal is so happy to see the % that it runs ( 2 place to the right) towards the %)

1. 10.5 = 1050.%

2. 0.0062 = 0.62%

3. 0.574= 57.4%

In 9-1 you learned one method of changing a fraction into a percent. Here is an alternative method

Rule

To express a fraction as a percent, first express the fraction as a decimal

and then as a percent

Express 7/8 as a percent

Divide 7 by 8

7/8

= 0.875 = 87.5%

Express 1/3 as a percent to the nearest tenth of a percent

divide 1 by 3

1/3

since this becomes a repeating decimal

.33333333….

or .3 with a vinculum, we write 1/3 as

33 1/3%

Pre Algebra Periods 1, 2, & 4

SQUARE ROOTS & PYTHAGOREAN THEOREM 11-1 & 11-2

Square root undoes squaring!
So you're looking for the number/variable that was squared to get the radicand
1) RADICAL sign: The root sign, which looks like a check mark.
If there is no little number on the radical, you assume it's the square root
But many times there will be a number there and then you are finding the root that the number says.
For example, if there is a 3 in the "check mark," you are finding the cubed root.
One more example: The square root of 64 is 8. The cubed root of 64 is 4. The 6th root of 64 is 2.
2)RADICAND : Whatever is under the RADICAL sign
In the example above, 64 was the radicand in every case.
3) ROOT (the answer): the number/variable that was squared (cubed, raised to a power)
to get the RADICAND (whatever is under the radical sign)
4) SQUARE ROOTS: The number that is squared to get to the radicand. Every POSITIVE number has 2 square roots - one positive and one negative.
Example: The square root of 25 means what number squared = 25
Answer: Either positive 5 squared OR negative 5 squared

You can estimate nonperfect square roots by guess and check
Find the 2 numbers that it is between
Example: Square root of 52
It's between the 2 perfect squares: 49 and 64
So the square root is between 7 and 8
Since 52 is only 3 away from 49 and 12 away from 64, the square root will be closer to 7
Guess: 7.2 Square this: (7.2)(7.2) = 51.84 (adjust your estimate as necessary)

REAL NUMBER SYSTEM
2 PARTS: RATIONAL AND IRRATIONAL (both real)

Review LEAP FROG number systems
I: RATIONAL NUMBERS (definitions of different number systems):
Natural = counting = 1, 2, 3, . . .
Whole = natural + 0 = 0, 1, 2, 3, . . .
Integers = whole + opposites = -3, -2, -1, 0, 1, 2, 3, . . .
Rational = integers and all the fractions/decimals in between - terminating and repeating decimals

II: IRRATIONAL - numbers like pi and square root of 3 - never repeat or terminate - round!

PYTHAGOREAN THEOREM
FOR RIGHT TRIANGLES ONLY!
2 legs - make the right angle - called a and b
(doesn't matter which is which because you will add them and adding is COMMUTATIVE!)
hypotenuse - longest side across from the right angle - called c
You can find the third side of a right triangle as long as you know the other two sides:
a² + b² = c²
After squaring the two sides that you know, you'll need to find the square root of that number to find the length of the missing side (that's why it's in this chapter!)
EASIEST - FIND THE HYPOTENUSE (c)
Example #1 from p. 510
8² + 15² = c²
64 + 225 = c²
289 = c²
c = 17

A LITTLE HARDER - FIND A MISSING LEG (Either a or b)
Example #5 from p. 510
5² + b² = 13²
25 + b² = 169
b² = 169 - 25
b² = 144
b = 12

CONVERSE OF PYTHAGOREAN THEOREM
If you add the squares of the legs and that sum EQUALS the square of the longest side, it's a RIGHT TRIANGLE.
If you add the squares of the 2 smallest sides and that sum is GREATER THAN the square of the longest side, you have an ACUTE TRIANGLE.
If you add the squares of the 2 smallest sides and that sum is LESS THAN the square of the longest side, you have an OBTUSE TRIANGLE.

Algebra Period 3 (Monday)

SOLVING SYSTEMS OF EQUATIONS 8-1 TO 8-3:
(2 equations with 2 variables)
You cannot solve an equation with 2 variables - you can find multiple coordinates that work
TO SOLVE MEANS THE
ONE COORDINATE THAT WORKS FOR BOTH EQUATIONS

There are 3 ways to find that point:

1. Graph both equations: Where the 2 lines intersect is the solution
2. Substitution method: Solve one of the equations for either x or y and plug in to the other equation
3. Addition method: Eliminate one of the variables by multiplying the equations by that magical number that will make one of the variables the ADDITIVE INVERSE of the other

Example solved all 3 ways:
Find the solution to the following system:

2x + 3y = 8 and 5x + 2y = -2

1. GRAPH BOTH LINES: Put both in y = mx + b form and graph
Read the intersection point (you should get (-2, 4))

2. SUBSTITUTION:

Isolate whatever variable seems easiest
I will isolate y in the second equation: 2y = -5x - 2
y = -5/2 x - 1
Plug this -5/2 x - 1 where y is in the other equation
2x + 3(-5/2 x - 1) = 8
2x - 15/2 x - 3 = 8
4/2 x - 15/2 x - 3 = 8
-11/2 x - 3 = 8
-11/2 x = 11
-2/11(-11/2 x) = 11(-2/11)
x = -2
Plug into whichever equation is easiest to find y

3. ADDITION:

Multiply each equation so that one variable will "drop out" (additive inverse)
I will eliminate the x, but could eliminate the y if I wanted to
5(2x + 3y) = (8)5
-2(5x + 2y) = (-2)-2

10x + 15y = 40

-10x - 4y = 4

ADD TO ELIMINATE THE x term:
11y = 44
y = 4
Plug into whichever equation is easiest to find x

NOTICE THAT FOR ALL 3 METHODS, THE SOLUTION IS THE SAME!
THEREFORE, USE WHATEVER METHOD SEEMS EASIEST!!!