Intro+to+Algebra

= = =Intro to Algebra =

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Samples and Populations Notes:

Investigation 1:

Measures of Center:
 * Mean - Add up all the data and divide by the number of pieces of data.
 * Median - Put the numbers in order and find the middle. If there are 2 numbers in the middle add them together and divide by 2.
 * Mode - The most frequently occurring number in a set of data.
 * Outlier - A number that is significantly above or below the rest of the data

Measures of variance:
 * Minimum - The smallest number in a set of data
 * Maximum - The largest number in a set of data.
 * Median - The middle number in a set of data.
 * Lower Quartile (LQ) - The middle number between the minimum and the median.
 * Upper Quartile (UQ) - The middle number between the median and the maximum.
 * Range - The maximum minus the minimum
 * Interquartile Range (IQR) - The upper quartile minus the lower quartile

Plots and Graphs
 * Histogram - Label the x-axis in intervals and the y-axis as frequency. Draw a bar in each interval showing how many data points fall within that interval. Border numbers go in the interval on the right.
 * Line Plot - Label the x-axis in intervals and the y-axis as frequency. Draw an X in each interval for every time a number from the interval appears in the data. Border numbers go in the interval on the right.
 * Stem-and-Leaf Plot - A plot used to organize data by putting it in its tens group (the stems).
 * Box-and-Whisker Plot - A plot used to represent a distribution of data showing the five number summary. It is placed above an axis numbered in regular intervals.

Investigation 2 Notes:
Population - The large group we want to gather information about Sample - The small group we question

Sampling Methods:
 * Convenience - You select a sample by choosing those around you or in a convenient location
 * Systematic - You select every person
 * Voluntary Response - You ask for people to volunteer to answer questions
 * Random - you select people from a pool where every person has an equally likely chance of being selected

Best sampling method is the random method!

Ways to choose a random sample:
 * 1) dice
 * 2) spinners
 * 3) names from a hat
 * 4) graphing calculator
 * 5) cards

Ways to predict using a sample:

Percents: Number who answered one way in the sample divided by the number of people in the sample times 100

Number of people: Set up a proportion __number who answered one way in sample__ = __number who would answer that way in the population__ number in the sample number in the population

Investigation 4 Notes:
Scatterplots - used to see if there is a relationship between two variables

To create a scatterplot
 * 1) Put the independent variable on the x-axis and the dependent variable on the y-axis (ex. hours driven is x and distance gone is y since the distance will depend on how many hours we will drive)
 * 2) Number the axes with a scale. The 2 axes can be numbered differently and they do not need to start at 0)
 * 3) plot the data in ordered pairs

Correlation means relationship
 * A positive correlation occurs when as one variable in creases the other will increase as well (dots go up). An example of this would be calories eaten and weight. As I increase the calories I eat my weight will also increase.
 * A negative correlation occurs as one variable increases the other will decrease (dots go down). An example of this is the number of hours a person works out and their weight. As I increase time spent working out weight will decrease.
 * No correlation means that the variables are unrelated (dots spread out all over). An example of this would be age and length of hair. Age has no influence on hair length.

Lines of best fit We can draw in a line of best fit when there is a correlation. We draw it in the middle of the dots, following the trend of the dots. We then can use that line to predict by taking variable we know. Draw a line from the spot on axis where it is to line of best fit and then drawing a line from that spot to the other axes. The place where it hits that other axis is our prediction.

Probability Notes:
Experimental probability is the probability based on the results of an experiment
 * a one means it happened every time
 * a zero means it never happened

Theoretical probability is what we would expect to happen
 * a one means it is guaranteed
 * a zero means it is impossible

Compound probability is the probability that two or more events are going to happen

Independent probability is compound probability where the 1st event has no effect on the 2nd. (ex: dice and then a coin or drawing a card and then replacing it)

Dependent probability is compound probability where the 1st even does have an effect on the 2nd. (ex: drawing a card and not replacing it before drawing again)

To find compound probability you multiply the probabilities of each event happening.

Counting Notes:
__**Fundamental Counting Principle**__ When you are choosing one item from several categories to find out the number of possible outcomes you multiply your choices together. Ex: If I have 5 shirts, 3 pairs of pants, and 2 types of shoes I have 5 x 3 x 2 or 30 possible outfits

Tree Diagram Created by making a new row for each new category in each new row all new choices are listed off of every branch in the previous row.

__**Permutations**__ The number of ways to order or arrange a group of items. We can do this in 3 ways:
 * 1) A List
 * 2) A Tree Diagram
 * 3) Using the Permutation Rule (Multiply from the # of items we are arranging down to zero.) Example: Arranging 4 pictures on a wall is 4 x 3 x 2 x 1 = 24

Factorial (x!) The shortcut way to do the permutation rule uses factorials. 4 x 3 x 2 x 1 is the same as 4!. The factorial button is on your calculator and is used by typing in the number and then hitting that button.

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Notation for permutations is 13 P 5 means you have 13 items and are choosing 5 of them. You would start at the 13 and multiply down until you've multiplied 5 numbers. It would be 13 x 12 x 11 x 10 x 9. The permutation button is also on your calculator and it is used by typing in the first number (the # in the group to choose from) the n P r button and then the second number (the # of items we are choosing).======

__**Combinations**__ The number of ways to make a smaller group from a group of items.

If given n P r where n = 10 and r = 3 we would do 10 x 9 x 8 = 720 With this abc is a different permutation than cba

If given nCr where n = 10 and r = 3 we would do 10 x 9 x 8 divided by 3 x 2 x 1 or 720 divided by 6 which is 120 With this abc is the same combination as cba

Counting Word Problems:

 * Decide if :
 * 1) Fundamental Counting Principal (looking at several groups or categories and choosing one from each)
 * 2) Permutations (order matters; key words include order, arrange, or specific jobs)
 * 3) Combinations (order does not matter; key words include group or committee)


 * Solve problem using method learned previously.

Number Sense Notes:
11/1/10

Classifying Real Numbers:
 * Natural Numbers - counting numbers
 * Whole Numbers - all natural numbers plus zero
 * Integers - all natural numbers, all whole numbers, plus negative counting numbers
 * Rational Numbers - all natural numbers, all whole numbers, all integers, any fraction, and any decimal that ends or repeats
 * Irrational Numbers - any decimal that does not end or repeat

11/2/10

Properties of Real Numbers
 * Commutative Property (add and multiply) - you can switch the order
 * Associative Property (add and multiply) - you can switch the way they are grouped
 * Additive Identity - you can add 0 and not change the identity of a number
 * Multiplicative Identity - you can multiply by 1 and not change the identity of a number
 * Additive Inverse - you add the opposite of a number to get an answer of 0 (the identity)
 * Multiplicative Inverse - you multiply the reciprocal of a number to get an answer of 1 (the identity)
 * Property of Zero - multiplying by 0 will give an answer of 0
 * Property of -1 - Multiplying by negative one will make a number its opposite

Sometimes - give an example of true and one of false Always - no examples needed Never - give an example of false

11/4/10

To compare and order real numbers always convert them to decimals (top of fraction divided by bottom of fraction) Remember the farther right on a number line the bigger it is.

11/9/10

Order of Operations
 * parenthesis (or any grouping symbol)
 * exponents (and square roots)
 * multiplication and division any order moving left to right
 * addition and subtraction any order moving left to right

11/12/10

Distributive Property a(b + c) = ab + ac or a(b - c) = ab - ac

Uses of distributive property:
 * 1) Simplify expressions 3(x + 5) = 3x + 15
 * 2) Combine like terms 4x + 11x = (4 + 11)x = 15x
 * 3) Mental math 6(53) = 6(50) + 6(3) = 300 + 18 = 318

11/16/10 and 11/17/10

Exponent Rules:
 * multiplying - multiply coefficients and add exponents on variables (3x^4)((5x^5) = 15x^9
 * product to a power - distribute the exponent to everything in the parenthesis (4xy)^3 = 4^3 x^3y^3 = 64x^3y^3
 * power to a power - multiply exponents (2x^3)^5 = 2^5 x^15 = 32x^15
 * divide - divide coefficients and subtract exponents on variable 12x^7 / 3x^4 = 4x^3
 * quotient to a power - distribute the exponent to everything in the parenthesis (5x/y)^2 = 25x^2 / y^2
 * exponent of zero - anything raised to the power of zero equals 1 y^0 = 1
 * negative exponents - move to the other side of the fraction making it a positive exponent 5x^-3 = 5/x^3

11/19/10

Scientific Notation

_ . _ _ x 10^ Only one digit to the left of the decimal point The exponent for the 10 is the number of places you move the decimal point (not just the number of zeros) Numbers greater than 1 will have a positive exponent on the 10 and numbers between 0 and 1 will have a negative exponent on the 10

Computing: Use the EE button on the calculator to replace the x10 so 4 x 10^7 would be 4 EE (shows up as single E) 7