Khan Academy Math

Secant

A line that intersects a circle in two points Chord- the segment of a secant line that is contained within the circle

Metric prefixes

Metric prefixes: Giga- 1 billion Mega- 1 million Kilo - 1 thousand Deci- one tenth Centi- one jundreth Milli- 1 thousandth Micro- one millionth

Components of an equation

Terms- things being added and subtracted into the equation 2 x 3 + 4 - 7y Factors: (2x3), 4, and 7y A factor is one part of a product. In the term 8x, the factors are 8 and x.

NUMBERS AND OPERATIONS- Writing numbers in words

*add comma in the words wherever the comma is in the number I.e. 82, 605 eighty-two thousand, six-hundred and five

Prism

A prism is a polyhedron, with two parallel faces called bases. Volume= bh, B is area of base and h is the height Surface Area= 2B + Ph where p is the perimeter of the base

Probability cont.

A probability can be any number from 0 to 1. A probability of 0 or 0% percent means that an event will never happen. A probability of 1/2 or 50% percent means that an event is equally likely to happen or not happen. A probability of 1 or 100% percent means that an event will certainly happen. P(A)= "the probability that event A will happen".

Trapezoids

A quadrilateral with exactly one pair of parallel sides A= 1/2h(b1 + b2) Add base lengths, divide by 2, multiply by h P= a + b1 + c + b2 (the four sides of trapezoid) Isosceles: equal base angles 2 non-parallels have equal length

Formulas for rectangles

quadrilateral with 4 right angles P=2(l+w) A=lw

Using roots to find a quadratic equation

roots x=2 and x=-1 Root x=2 corresponds to equation x-2=0 Root x=-1 = x+1=0 (x-2)(x+1) = 0, x^2 - x-2= 0

ALGEBRA- Linear Functions

the value of the function changes in direct proportion to X constant rate of change, or slope change in y/change in x = ALWAYS constant X Y 1 5 2. 8 3. 16 change is not constant, so it is NOT a linear function Standard form of a linear function: f(x) = mx + d Linear equation- "Luis and Kate have 2 video games they want to play. They plan to spend exactly 45 minutes playing the two games. They want to use equation to express the relationship between the number of minutes they spending each game" x + y = 45 minutes -rearrange into the y= mx + b y = -x + 45

GEOMETRY- triangle types

ANGLES: Acute- 3 angles that each measure than less 90 Right- one 90 degree angle and 2 acute Obtuse- one angle more than 90 and 2 acute SIDES: Equilateral- 3 equal sides Isosceles- at least 2 equal sides Scalene- no equal sides

Reflections

Across X axis: x coordinate stays same, y coordinate becomes opposite (x,y) across x axis (x, -y) Across Y axis- y coordinate remains the same, x becomes opposite (x,y) across y axis (-x, y) Across y=x (x,y) becomes (y,x) Across y=-x (x,y) becomes (-y,-x)

NUMBERS AND OPERATIONS- Properties of Addition

Additive Identity: a + 0 = a Additive Inverse: a + (-a) = 0 Associative of Addition: (a + b) + c = a + (b + c) Commutative of Addition : a + b = b + a Definition of Subtraction : a - b = a + (-b)

Dependent/independent variables

An INDEPENDENT variable is a variable that represents a quantity that is being manipulated in an experiment (usually x) You are doing chores to earn your allowance. For each chore you do, you earn $3. The independent variable is the amount of chores you do, because that is what you have control over. A DEPENDENT variable represents a quantity whose value depends on how the independent variable is manipulated. the dependent variable is the amount of money you make because that DEPENDS on how many chores you do

Cone

V=1/3πr^2h SA= πr² + πrs, s is slant height. s can be calculated using square root of radius squared + height squared In terms of pi, square the radius multiply by the height and divide by 3. A cone with radius 3 and height 4, 3 squared =9 x 4 = 36/3 = 12π

Volume

Volume is the amount of 3-dimensional space an object occupies. Volume is measured in cubic units. Rectangular prism: l x w x h

ALGEBRA- Slope intercept form

-rearrange to slope intercept form y = mx+b EQUATION: y + 2= -2 (x -3) y= -2x + 4 Slope = -2 4x + 2y = -3 Slope intercept: 2y= -4x -3 y= -2x - 3, y intercept = -3, slope = 2 4x = -8 x= -2, X will always =-2 (on a graph, a vertical line down x=-2) 2y = -8 y= -4 (on a graph, a horizontal line on y=-4) HORIZONTAL = negative (7, -1) and (-3, -1) -1 - -1 = 0, the line will be horizontal and have a 0 slope

Experiments

An explanatory variable explains changes in another variable. A response variable measures the result of a study. A treatment is the specific level of the explanatory variable given to individuals in an experiment. An experimental unit is who or what we are assigning to a treatment. I.e. Karina wants to determine if kale consumption has an effect on blood pressure. She recruits 100 households and randomly assigns each household to either a kale-free diet plan or a kale-based diet. At the end of two months, she plans to record the original and final blood pressures for members of each household. explanatory variable- kale consumption response variable- change in blood pressure treatments- kale-based and kale-free diets experimental units- households

Probability

Compound event- event that involves 2 or more independent events (rolling a pair of dice and taking the sum) Independent events- two or more event whose outcomes do not affect one another (two coins tossed at the same time) Dependent events- two or more events whose outcomes do affect one another (drawing 2 cards from the same deck) certain outcome- 100% OR 1 mutually exclusive events- two or more outcomes whose criteria cannot all be satisfied in a single event (a coin coming up heads and tails on the same toss) experimental- based on what has happened theoretical- what should happen

Data representations

Frequency tables- show how frequently each unique value appears in the set Line graphs- lines showing the different values for a set of data -represented in ordered pairs (x,y) -multiple sets of data can be graphed on the same line to give an easy visual comparison Line plots- plotted points not connected by line segments -horizontal axis lists the different possible values for the data, and the vertical axis lists the number of times the individual value occurs Stem and leaf- useful for depicting groups of data that fall into a range of values Bar graph- both axis have categories defined Histograms- a continuous bar for each RANGE of data Scatter plot- useful in determining the type of function represented by the data

ALGEBRA- Writing slope-intercept equations

If Y-intercept is not visible on the graph *plug point on graph in for X and Y and solve for B I.e. "A line has a slope of -3/4 and goes through the point (0,8). What is the equation of this line in slope-intercept form?" Y= -3/4x + b (8) = -3/4(0) + B B= 8 Y= -3/4x + 8

Sets

Intersection- "and" U^M X (9, 4, 3, 1, 8) Y (11, 3, 15) X^Y= (3) would be the overlapping part of a Venn diagram Union- "or" X U Y (9, 4, 3, 1, 8, 11, 15) only write 3 once Complement- the set of all elements that are in the universal set S but not in the set Subtraction (A-B) in A, but not in B A (1, 2, 3) B (3, 5) A-B= (1,2) Mutually exclusive sets- do not have any shared sets, intersection is an empty set

Metric and US

Length: Meter is the base for distance 1 kilometer= 1000 meters (distance from city to city) 1 meter = 100 centimeters (about the size of a person) 1 meter= 1000 millimeters decimeter = 1/10 of a meter centimeter= 1/100 of a meter milimeter= 1/1000 of a meter 1 mile= 5,280 feet 1 yard= 3 feet Volume: 1 liter= 1000 milliliters 1 milliliter= 1 cubic centimeter 1 metric ton= 1,000 kilograms 1 gram= 1,000 milligrams Mass: 1 kilogram is about 2.2 lbs 3 teaspoons = 1 tablespoons 2 tablespoons = 1 fluid oz 8 fluid oz = 1 cup 2 cup = pint 2 pints = quart 4 quarts= a gallon

Representations of data

Line graphs are data displays that show trends over time Scatterplots display data about two quantitative variables as a set of points in the x and y plane.

Outliers

Low outliers: Q1 - 1.5 x IQR High outliers: Q1+ 1.5 x IQR Q1= 20 IQR= 5 Anything below 12.5 is an outlier Q3: 15 Anything more than 22.5 is an outlier

Metric units of distance

Meter- length of a guitar Centimeter- fingernail Milliliter- paper clip Converting larger factors to smaller factors: multiply 1 centimeter= 10 millimeter 1 meter= 100 centimeters 1 meter= 1000 millimeters 1 kilometer= 1000 meters Coverting smaller factors to larger factor: divide

Formula for a square

P=4s A= s^2

GEOMETRY- lines

Perpendicular lines are lines that meet at RIGHT angles. Parallel lines are always the same distance apart from each other — no matter how far they are extended, they will never meet. Intersecting- one point in common Concurrent- multiple lines that intersect at a single point Transversal- a lone that intersects at least 2 other lines, which may or may not be parallel to one another Bisector- divides a line into 2 = lengths Perpendicular- points are equal distance from the endpoints of the segments it is dividing

GEOMETRY- congruent triangles

Side-side-side; SSS : All three pairs of corresponding sides are equal. Side-Angle-side; Two pairs of corresponding sides and the corresponding angles between them are equal. Angle-side-angle; ASA : Two pairs of corresponding angles and the corresponding sides between them are equal. Angle-angle-side; Two pairs of corresponding angles and one pair of corresponding sides (not between the angles) are equal. Hypotenuse-leg; The pair of hypotenuses and another pair of corresponding sides are equal in two right triangles.

NUMBERS AND OPERATIONS- Writing a number in standard/expanded form

Standard: "Write six hundred forty-five million, five hundred eighty four thousand, and four hundred sixty-two in standard form." 645 million 584 thousand 462 hundreds _ _ _ _ _ _ _ _ _ 645 584 462 Expanded: "Write 14, 897 in expanded form" 1 ten thousand 4 thousands 8 hundreds 9 tens 7 ones 1 x 10,000 4 x 1,000 8 x 100 9 x 10 7 x 1

ALGEBRA- Functions

Writing a function from an equation "For a given input value b, the function f outputs a value a to satisfy the following equation. 4a + 7b = -52" Write a function in terms of B. A is the output, when B is the INPUt, so you solve the equation for A. Given a B, what will be the output, A? A= -13 - 7/4B F(B) -13 - 7/4B SLOPE of a function: change in dependent value/change in independent value

Rhombus

a quadrilateral with four congruent sides all rhombuses are parallelograms and kites

GEOMETRY- angles

acute- less than 90 right- 90 obtuse- more than 90, less than 180 straight- exactly 180 reflex- greater than 180, less than 360 full angle- 360 complementary= a + b= 90 supplementary= a + b=180 adjacent angles- share same vertex and share a side

NUMBERS AND OPERATIONS- Regrouping

6 7 5 600 + 70 +5 6 hundreds 7 tens 5 ones could be regrouped to 5 hundreds 17 tens 5 ones or 6 hundreds 6 tens 15 ones

Solving quadratic equations (formula)

set of trinomials of the form y= ax^2 + bx+ c roots: the solutions that satisfy the equation when y = 0, where the graph touches the X axis QUADRATIC FORMULA: (if none of the factors of ac add up to b) -rewrite in standard from ax^2 + bx+ c x = -b _+_ √b^2 - 4ac _______________________ 2a I.e. Find the roots of y= 2x^2 + 8x + 4 -8 + √8^2 - 4(2)(4) __________________________ 2(2) -8 + √32 / 4 -8 + √(16)(2) -8 + 4√2 / 2 x = (-2 + √2) and ) (-2 - √2)

ALGEBRA- Functions

exactly one output value (dependent variable) for each value of the input variable (independent variable) All variables for INPUT variable (x)- domain All variable for OUTPUT function (y)= range from a GRAPH: vertical line test *if the vertical line drawn through the value of x crosses the graph in more than one place, the equation is NOT a function TABLE: can not contain any repeating x-values that correspond with different y-values *can be repeating Y values X = 6 G = 6, still a function X = 3 X=, no longer a function EQUATION: "In the relation x= y^2 + 3, can y be written as a mathematical function of x?" -solve for y Y= square root of x -3 could be positive or negative, so you could get two different results and it is NOT a function WORD: "The value of y is always 3 more than twice x. Is y a function of x?" Y= 3 + 2x It is a function because for every input of X, there is only one Y

ALGEBRA- Slope

increase in vertical/increase in horizontal Change in Y _________________ Change in X given the points (3, 2) and (5, 8) Change in Y: 8-2 = 6 Change in X: 5-3 = 2 6/2 = Slope of 3 Negative: (2,7) and (5,1) Y: 1-7 = -6 X: 5-2= 3 = -2 Decreasing= Negative Slope: y2 - y1/ x2-x1

NUMBERS AND OPERATIONS- Ten/tenths place

1234.56789 1= thousands 2= hundreds 3= tens 4= ones 5= tenths 6= hundredths 7= thousandths 8= ten thousandths 9= hundred thousandths I.e. : nine hundred one thousandths = 0.901 13. 113 = thirteen and one hundred thirteen thousandths

Formula for parallelogram

2 pairs of parallel, congruent sides -opposite interior angles are congruent -consecutive interior angles are supplementary A=bh P=s1​+s2​+s3​ +s4

Solving quadratic equations (Factoring)

-rewrite in standard form x^2 + bx + c = 0 2 factors of c that add up to b x^2 + 6x -16 = 0 Factors 0f -16: -4 and 4, -8 and 2, -2 and 8, -1 and 16, 1 and -16 Factors that add to 6: 8 and -2 -write as product of 2 binomials (x+8) (x-2)= 0 -set each = to 0 x + 8= 0 x - 2 = 0 Roots are 2 and -8

Customary unit of volumes

1 cup= a juice box 1 pint= 1 water bottle 1 quart= a shampoo bottle 1 gallon= milk jug

Customary unit of weights

1 oz= slice of bread 1 lb= soccer ball 16 oz in a pound 30 lbs = 30 x 16= 480 oz 112 oz= 112/16= 7lbs

Mean Absolute Deviation (MAD)

1. Find the mean 6, 8, 3, 2, 6= 5 2. Find distance from each point to mean 6-5 = 1 8-5= 3 5-3= 2 5-2= 3 6-5= 1 3. Calculate the mean of the new data points= 1+3+2+3+1= 10/5 = 2

Interquartile range

1. Put numbers in order 1, 2, 5, 7, 9, 12, 15 2. Find the median 7 3. Place parenthesis around the numbers before and after median (1, 2, 5) 7 (9, 12, 15) 4. Find Q1 and Q3 (median of upper and lower half of data) Q1= 2 and Q3= 12 5. Subtract Q1 from Q3 12-2 = Interquartile range of 3

Standard deviation of a sample

1. find the mean 2. Subtract the mean from each data plot point (can be negative) 3. Square each deviation 4. Find the mean of each deviation, using ONE LESS than the number of data points 4, 5, 6, 8, 12 Mean: 7 4-7= -3 = 9 5-7= -2 = 4 6-7= -1 = 1 8-7= 1= 1 12-7= 5 = 25 25+1+1+4+9= 40/5-1 = 10 Take the square root of 10

NUMBERS AND OPERATIONS- Naming and ordering numbers

10^0 = 1 10^1= 10 10 ^2= 100 10^3 = 1000 10^4 = 10,000 ten thousand 10^5= 100,000 one hundred thousand 10^6 = 1,000,000 one million 10^9 = 1,000,000,000 one billion 10^0 = ones. 10^-1 = tenths 10^-2 = hundreths 10^-3 thousandths 10^-4 - ten thousandths

GEOMETRY- conditions and counterexamples

A mathematical statement is composed of two parts: a condition and a conclusion. Consider the statement "All rectangles are squares": Condition: is a rectangle Conclusion: is a square counterexample: a shape can be a rectangle, but it is not a square.

ALGEBRA- Quadratic equations

A quadratic equation is an equation with at least one variable to the second power as its highest power term and one or more constants. in the quadratic equation 3x^2 = 48 X-variable 2 - power/exponent 3- coefficient 48- constant *The square of a number is equal to the number multiplied by itself. The square root of a positive number x, is the answer to the question "what positive number squared equals x ?" For quadratic equations with coefficients and constants, we need to rearrange the equation until it's the form x2=c then take the square root of both sides of the equation.

Transformation

A translation takes each point in a figure and slides it the same distance in the same direction.

Formula for triangle

A= 1/2 b+h P=s1​+s2​+s3​

Kite

A= 1/2 w x h

Sphere

A= 4πr^2 V= 4/3πr^3 Cube the radius, multiply by 4/3 (about 1.333) then multiply by 3.4

Circles

A=πr² C= 2πr π×diameter the distance around the outside of a circle (its perimeter!). multiply radius by 2, in terms of pi Arc- a portion of a circle, set of points between and including 2 points on a circle (minor arc less than 180, major arc at least 180) every minor arches a corresponding major arc that can be found by subtracting the measure of the minor arc from 360

GEOMETRY- similar triangles

AA : Two pairs of corresponding angles are equal. SSS : Three pairs of corresponding sides are proportional. SAS : Two pairs of corresponding sides are proportional and the corresponding angles between them are equal. If triangles ABC and XYZ are similar, then their corresponding side lengths have the same ratio:

GEOMETRY- interior angles of polygons

As the number of sides of a polygon increases, the sum of its interior angle measures increases as well. For a polygon with n sides, the sum of its interior angle measures is equal to: (n−2)×180∘ For example, for a four-sided polygon such as a square or a rectangle, the sum of interior angles is: (4−2)×180∘=360∘

Sampling methods

Bad: Convenience sample: The researcher chooses a sample that is readily available in some non-random way. Example—A researcher polls people as they walk by on the street. Voluntary response sample: The researcher puts out a request for members of a population to join the sample, and people decide whether or not to be in the sample . Example—A TV show host asks his viewers to visit his website and respond to an online poll. GOOD: Simple random sample: Every member and set of members has an equal chance of being included in the sample. Example—A teachers puts students' names in a hat and chooses without looking to get a sample of students. Stratified random sample: The population is first split into groups. The overall sample consists of some members from every group. The members from each group are chosen randomly. Example—A student council surveys 100 students by getting random samples of 25 freshmen, 25 sophomores, 25 juniors, and 25 seniors. Cluster random sample: The population is first split into groups. The overall sample consists of every member from some of the groups. The groups are selected at random. Example—An airline company wants to survey its customers one day, so they randomly select 5 flights that day and survey every passenger on those flights. Systematic random sample: Members of the population are put in some order. A starting point is selected at random, and every nth member is selected to be in the sample. Example—A principal takes an alphabetized list of student names and picks a random starting point. Every 20th student is selected to take a survey.

GEOMETRY- circles

CIRCUMFERENCE: C=πd or C=2πr AREA: A=πr2

Center and spread

Center describes a typical value of a data point. Two measures of center are mean and median. Spread describes the variation of the data. Two measures of spread are range and standard deviation. Mean= sum of values/ # of values Median= numbers least to greatest odd?- middle number even?- average of two middle values Range= maximum-minimum Standard deviation is the average distance between the mean and a data point.

Central and Inscribed angles

Central angle- vertex on CENTER of the circle, creating an arch with the legs. *measure of central angle is equal to the measure of the MINOR arc it intercepts Inscribed angle- An angle whose vertex lies on a circle and whose legs contain chords of that circle the intercepted arc created by the inscribed angle is TWICE the measure of the inscribed angle therefore any angle inscribed in a semi circle is a right angle (angle of the arc of a semicircle is 180, half of that is 90)

Equivalent expressions

Equivalent expressions are expressions that work the same even though they look different. If 1/2 (6x + 8) = kx + 4, what is the value of K? K must be equal to the coefficient of X on the left side of the equation, since it is the coefficient of X on the right side 3x + 4 = kx +4 K=3 2. If p=mv, what equation correctly shows v in terms of p and m? P/M. = V

Types of experiements

In an observational study, we measure or survey members of a sample without trying to affect them. An observational study cannot indicate a causal relationship between parameters. In a controlled experiment, we assign people or things to groups and apply some treatment to one of the groups, while the other group does not receive the treatment. Causal relationships can be established when subjects are randomly assigned to treatment groups in a controlled randomized experiment.

Metric unit of mass

Mass is used to measure the weight of an object. 1 gram = 1 paperclip 1 kilogram= wooden baseball bat (about 2.2 lbs) Converting kilograms to grams 1 kilogram= 1000 grams 8 kilograms= 8000 grams Converting grams to kilograms 1 gram= 1/1000 kilogram 9 grams= 9/1000= 0.009 Converting grams to miligrams 1 gram= 1000 miligrams 5 gram = 5000 miligrams 3 miligrams = 0.003 grams

NUMBERS AND OPERATIONS- Properties of Multiplication

Multiplicative Identity a * 1 = a Multiplicative Inverse a * (1/a) = 1 (a 0) (Multiplication times 0) a * 0 = 0 Associative of Multiplication (a * b) * c = a * (b * c) Commutative of Multiplication a * b = b * a Distributive Law a(b + c) = ab + ac

GEOMETRY- classification of shapes

Polygons- a closed figure with at least 3 straight sides. *# of sides is ALWAYS = to #of vertices Quadrilaterals are polygons with four sides and four interior angles. Parallelograms are quadrilaterals with two pairs of parallel sides and two pairs of angles with the same measure. The opposite sides have the same length, and adjacent angles are supplementary. Rectangles are parallelograms with four 90, degrees angles. The adjacent sides are perpendicular. While all rectangles are parallelograms, NOT all parallelograms are rectangles. Squares are parallelograms with four sides of equal length and four 90 degrees angles. While all squares are both rectangles and parallelograms, not all parallelograms are squares and not all rectangles are squares. Rhombus- 4 sides, all same length A trapezoid is a quadrilateral with exactly 1 pair of parallel sides

Qualitative/Quantitative data

Qualitative- Mordor University surveys 600 incoming students about which world language they want to study. Here, the qualitative variable is the language that the students want to study *table, bar chart, pictograph, and circle graph Quantitative- Data displays for quantitative data are typically oriented along two numerical axes and relate two quantitative variables. Ms. Buehler asks her homeroom students how long it typically takes them to get to school (in minutes) and records their responses in the following list. Here, one quantitative variable is students' typical travel time to school. dotplots. histograms

NUMBERS AND OPERATIONS- Rational/Irrational numbers

RATIOnal- RATIO of 2 numbers I.e. integers- 1 can be represented as 1/1, 2/2/... Decimals (Repeat = Ratio)- 3.75 can be written as 375/100, 0.3333 = 1/3 Fractions- RATIO 3/5, 2/3 Irrational: # of decimal places is infinite and no recurring pattern of digits, pi, square root of 2 Real: rational and irrational #s

Range and midrange

Range= difference of highest and lowest 95-65= 30 Midrange= average of highest and lowest 95+65/2 = 80

Volume formulas

Rectangular solid- V=lwh

Rotations

Rotations- positive angle measures describe counterclockwise rotations. negative angle measures describe clockwise rotations The angle of rotation is the angle that is formed between the line segments that connect each pair of corresponding pre-image and image points to the center of rotation. Point Q′ is the image of Q (3,−4) under a reflection across the x-axis. What are the coordinates of Q'? Q' (3, 4)

Is a question statistical?

Statistical questions are answered by collecting data with variability. What is the typical height of dog kennels at Keita's Kennels? NOT statistical-How many hairs are there in Gretchen the Gorilla's right armpit? (you are not collecting more than 1 piece of data)

Writing expressions and equations

Sum of A and B= A+B B is added too.. +b the difference between a and b... a-b Product of A and B... a x b b of... b x quotient of a and b... a/b the result is... = If the product of a certain number and 4 is 64, which of the following equations, when solved, would give us the value of the number? 4x = 64 When the sum of a certain number and 1 is multiplied by 3, the result is 6. 3 (1 + x) = 6 At a craft fair, Tony bought 3 items, Isabella bought 1 item, and Tanisha bought 5 times as many items as Tony and Isabella did combined. Which of the following represents the number of items Tanisha bought? 5 (3 + 1)

Formulas for polygons

Sum of interior angles: (n-2) x 180 Apothem- a line segment from the center of a polygon that is perpendicular to a side of a polygon A= 1/2ap, where A is area, a is apothem, and p is perimiter Radius- a line segment from the center of a polygon to a vertex Diagonals- a line segment joining 2 non-adjacent vertices # of diagonals= n(n-3) / 2

Surface Area

Surface area is the amount of space covering the outside of a three-dimensional shape. To calculate surface area, we add the areas of all the faces of the three-dimensional figure.

Quadrants of coordinate plane

The coordinate plane is divided into four quadrants (counterclockwise) Quadrant one (QI) is the top right fourth of the coordinate plane, where there are only positive coordinates. Quadrant two (QII) is the top left fourth of the coordinate plane (negative x, positive y) Quadrant three (QIII) is the bottom left fourth (negative x and y) Quadrant four (QIV) is the bottom right fourth (positive x, negative y)

Triangle inequality rule

The triangle inequality rule states that the longest side of a triangle MUST be SHORTER than the combined lengths of the two other sides. In other words, for a triangle with side lengths a, b, and c: a+b>c For a triangle with only two known side lengths, a and b, the unknown side length, x, must meet one of the following conditions: Shorter than the sum of the two known side lengths Longer than the positive difference of the two known side lengths a−b<x<a+b A: 10 B: 5 X: ? X must be greater than 5 but less than 15 A triangle has side lengths of 10, 20, and x. Which of the following could be the value of x ? 10+20 = 30 20-10 = 10 greater than 10, less than 30. could be 20

ALGEBRA- X and Y intercepts

The x-intercept is the point where a line crosses the x-axis, and the y-intercept is the point where a line crosses the y-axis. EQUATION: 5x + 6y = 30 -Set X for 0 to determine Y axis 5 (0) +6y =30 6y= 30 y=5 Y-intercept (0,5) -Repeat for X 5x + 6y = 30 5x =30 x= 6 X-intercept (6,0) GRAPH: look and identity the points on the graph where the line intersects the x and y axis TABLE: x y 1. -9 3 -6 5 -3 From the graph we can determine that when X increases by 2, Y increases by 3. The slope is 3/2. So when X increases by 1, Y increases by 1.5 6 -.15 7 0 X intercept = (7, 0) 0 -10.5. Y intercept = (0, -10.5)

Dilations

To find the center of the dilation, we can consider the lines that pass through corresponding vertices of quadrilaterals L and L′ prime. The place where these lines meet is the center of dilation.

Reversing a procedure

To reverse the procedure, we need to perform the inverse operations in reverse order: A positive integer is multiplied by 3, and then 1 is added to the result. forward operations: 1st multiply by 3 2nd add one 1st subtract one 2nd divide by 3

GEOMETRY- triangles

Triangles- polygons with 3 sides and 3 interior angles Isosceles triangles have two sides with the same length. The two angles opposite these two sides have the same measure. Equilateral triangles have three sides with the same length. Each interior angle of an equilateral triangle measures 60 degrees. A triangle has three interior angles . The measures of the three interior angles in a triangle add up to 180∘, degrees: x∘+y∘+z∘=180

Bias in surveys

Undercoverage is when the researcher systematically excludes members of the population from being in the sample. Nonresponse is when people chosen for the sample cannot be reached or refuse to participate. Response bias is when people are systematically dishonest when answering a question. Biased wording of survey questions can cause people to favor certain responses over others

Unit reasoning

Unit reasoning helps us make sense of measurements by converting between measurement units and making reasonable estimates.

GEOMETRY- Angles

Usually seen on the same side of an intersection of two lines, the measures of SUPPLEMENTARY angles add up to 180 degrees The angles on the opposite sides of an intersection of two lines are VERTICAL angles. They have the SAME measure. A transversal of two parallel lines creates two sets angles with IDENTICAL angle measures at the intersections

Pyramid

V= 1/3Bh, where B is the area of the base Rectangular: 1/3 lxw x h

Rectangular prisms

V= lwh SA= 2lw + 2hl + 2wh

Cubes

V= s^3, where s is the length of the side. SA= 6s^3

Cylinder

V= πr²h SA= 2πr² + 2πrh In terms of π, square radius and then multiply it by h. Cylinder with a radius of 2 and a height of 8 2 squared= 4 x 8 =32= 32π

Metric units of volume

Volume measures capacity. 1 milliliter= single raindrop 1 liter= 1 liter bottle of soda Converting liters to milliliters 1 liter= 1000 milliliters 8 liters = 8000 milliliters Converting milliliters to liters 1 milliliter= 1/1000 of a liter to convert milliliters to liters, divide number of milliliters by one thousand 45 ml= 45/100= 0.045

Triangular prism

Volume= 1/2 x b x h x l

NUMBERS AND OPERATIONS- Classification of numbers

WHOLE- do not need to be represented with a fraction or. decimal *counting numbers and 0 *can not be negative INTEGERS- whole numbers and their opposites -CAN be negative RATIONAL #s- expressed as a RATIO 44, 0.12, -18/5 IRRATIONAL- can not be expressed as a ratio *whole #s and integers are all rational

Standard deviation

standard deviation cannot be negative. A standard deviation close to 000 indicates that the data points tend to be close to the mean Step 1: Find the mean. Step 2: For each data point, find the square of its distance to the mean. Step 3: Sum the values from Step 2. Step 4: Divide by the number of data points. Step 5: Take the square root. I.e. 6, 2, 3, 1 Mean: 3 Squares of the distance of each point to mean= 6-3 = 3 = 9 3-2 = 1 = 1 3-3= 0 = 0 3-1= 2 = 4 Find new mean: 9+1+0+4= 14/4 = 3.5 Take square root of 3.5= 1.87