Praxis II (5001) Flashcard Study System - Mathematics

Discuss conformability for matrix operations.

All four of the basic operations can be used with operations between matrices (although division is usually discarded in favor of multiplication by the inverse), but there are restrictions on the situations in which they can be used. Matrices that meet all the qualifications for a given operation are called conformable matrices. However, conformability is specific to the operation; two matrices that are conformable for addition are not necessarily conformable for multiplication.

in a bank the banker to customer ratio is 1 to 2. if seven bankers are on duty how many customers are currently in the bank

use proportional reasoning or set up a proportion to solve. Because there are twice as many customers as bankers there must be 14 customers when seven bankers are on duty. Setting up and solving a proportion gives the same result: number of bankers. 1. 7 ————————————— = — = ———————— number of customers. 2. # ofcustomers represent the unknown number of patients as the variable x. 1/2 = 7/x to solve for X, cross multiply: 1 • X = 7×2, so x= 14

Describe number lines and their use.

A number line is a graph to see the distance between numbers. Basically, this graph shows the relationship between numbers. So, a number line may have a point for zero and may show negative numbers on the left side of the line. Also, any positive numbers are placed on the right side of the line.

Discuss absolute value functions

An absolute value function is in the format ƒ(x)= |ax+b|. Like other functions, the domain is the set of all real numbers. However, because absolute value indicates positive numbers, the range is limited to positive real numbers. To find the zero of an absolute value function, set the portion inside the absolute value sign equal to zero and solve for x. An absolute value function is also known as a piecewise function because it must be solved in pieces-one for if the value inside the absolute value sign is positive, and one for if the value is negative. The function can be expressed as: ƒ(x)=+{ax+b if ax+b ≥0 {-(ax+b) if ax+b<0 This will allow for an accurate statement of the range.

A woman's age is thirteen more than half of 60. How old is the woman?

"More than" indicates addition, and "of" indicates multiplication. The expression can be written as 1/2(60)+13. So, the woman's age is equal to 1/2(60)+13 =30+13=43. The patient is 43 years old.

Describe the organization of a stem and leaf plot.

A stem and leaf plot is useful for depicting groups of data that fall into a range of values. Each piece of data is separated into two parts: the first, or left, part is called the stem; the second, or right, part is called the leaf. Each stem is listed in a column from smallest to largest. Each leaf that has the common stem is listed in that stem's row from smallest to largest.

Discuss equivalent units for converting between U.S. standard and metric equivalents of fluids.

Fluid Measurements

Secondary data

Information that has been collected, sorted, and processed by the researcher

Explain the probability of an event.

Probabilities of events are expressed as real numbers from zero to one. They give a numerical value to the chance that a particular event will occur. The probability of an event occurring is the sum of the probabilities of the individual playing elements of that event.

Data

The collective name for pieces of information (singular is datum)

Measurement Conversion Practice Problems a. How many grams are in 13.2 pounds? b. How many pints are in 9 gallons?

a. 13.2 pounds X 1 kilogram/2.2 pounds x 1000 grams/1 kilogram = 6000 grams b. 9 gallons x 4 quarts/1 gallon x 2 pints/1 quarts = 72 pints

A runner's heart beats 422 times over the course of six minutes. About how many times did the runner's heart beat during each minute? Visit mometrix.com/academy for a related video. Enter video code: 126243

"About how many" indicates that you need to estimate the solution. In this case, look at the numbers you are given. 422 can be rounded down to 420, which is easily divisible by 6. A good estimate is 420/6=70 beats per minute. More accurately, the runner's heart rate was just over 70 beats per minute since his heart actually beat a little more than 420 times in six minutes.

Write each decimal in words: 0.06, 0.6, 6.0, 0.009, 0.113, and 0.901

0.06: six hundredths 0.6: six tenths 6.0: six 0.009: nine thousandths 0.113: one hundred thirteen thousandths 0.901: nine hundred one-thousandths

Write the place value of each digit in the following number: 14,059.826. Visit mometrix.com/academy for a related video. Enter video code: 205433

1: ten-thousands 4: thousands 0: hundreds 5: tens 9: ones 8: tenths 2: hundredths 6: thousandths

Describe a bar graph.

A Bar Graph is one of the few graphs that can be drawn correctly in two different configurations -both horizontally and vertically. A bar graph is similar to a line plot in the way the data is organized on the graph. Both axes must have their categories defined for the graph to be useful. Rather than placing a single dot to mark the point of the data's value, a bar, or thick line, is drawn from zero to the exact value of the data, whether it is a number, percentage, or other numerical value. Longer bar lengths correspond to greater data values. To read a bar graph, read the labels for the axes to find the units being reported. Then look where the bars end in relation to the scale given on the corresponding axis and determine the associated value.

Explain survey studies.

A Survey Study is a method of gathering information froma small group in an attempt to gain enough information to make accurate general assumptions about the population. Once a survey study is completed, the results are then put into a summary report. Survey studies are generally in the format of surveys, interviews, or questionnaires as part of an effort to find opinions of a particular group or to find facts about a group. It is important to note that the findings from a survey study are only as accurate as the sample chosen from the population.

Explain the multiplication- counting principle and its application in determining the probablity of two independent events.

A faster way to find the sample space without listing each individual the two events can occur in a ways bways. In the previous example, and a second event in b ways, outcome employs the multiplication counting principle. If one event there are 4 possible can occur in ax number of combinations is 4 x 3 x 2, or 24 shirts, 3 possible pairs of pants, and 2 possible hats, so the possible A similar principle is employed to determine the probability of two independent events. P(A and B) P(A) x P(B), where A is the first of B does not depend on the outcome marbles. The probability event and B is the second such that the outcome choose a marble from a bag marble is found by multiplying the that you would choose a red marble, replace of 2 red marbles, 7 blue marbles, and 4 green of A. For instance, suppose you it, and then choose a green probabilities of each independent event: 2/13 X 4/13=8/169, or 0.047, or 4.7% This method can also be used when finding the probability of more 2 independent events.

Discuss improper fractions and mixed numbers

A fraction whose denominator is greater than its numerator is known as a proper fraction, while a fraction whose numerator is greater than its denominator is known as an improper fraction. Proper fractions have values less than one and improper fractions have values greater than one, A mixed number is a number that contains both an integer and a fraction. Any improper fraction can be rewritten as a mixed number. Example: 8/3=6/3+2/3=2+2/3=2&2/3. Similarly, any mixed number can be rewritten as an improper fraction. Example: 1& 3/5 = 1+3/5= 5/5+3/5=8/5

Explain the lay-out of a line plot or dot plot. Visit mometrix.com/academy for a related video. Enter video code: 754610

A line plot, also known as a dot plot, has plotted points that are NOT connected by line segments. In this graph the horizontal axis lists the different possible values for the data, and the vertical axis lists the number of times the individual value occurs. A single dot is graphed for each value to show the number of times it occurs. This graph is more closely related to a bar graph than a line graph. Do not connect the dots in a line plot or it will misrepresent the data.

Define matrix and the associated terminology.

A matrix (plural: matrices) is a rectangular array of numbers or variables, often called elements, which are arranged in columns and rows. A matrix is generally represented by a capital letter, with its elements represented by the corresponding lowercase letter with two subscripts indicating the row and column of the element. For example, ηₙₙ represents the element in a row n column n of matrix N. N = [n₁₁ n₁₂ n₁₃] [n₂₁ n₂₂ n₂₃] A matrix can be described in terms of the number of rows and columns it contains in the format a x b, where a is the number of rows and b is the number of columns. The matrix shown above is a 2 x 3 matrix. Any a x b matrix where a = b is a square matrix. A vector is a matrix that has exactly one column (column factor) or exactly one row (row factor).

Discuss the characteristics of polynomial functions.

A polynomial function is a function with multiple terms and multiple powers of x, such as: ƒ(x) =aₙxⁿ¹+aₙ-₁xⁿ⁻¹+ aₙ-₂xⁿ⁻²+...a₁x+a₀ where n is a non-negative integer that is the highest exponent in the polynomial, and aₙ≠0. The domain of a polynomial function is the set of all real numbers. If the greatest exponent in the polynomial is even, the polynomial is said to be of even degree and the range is the set of all real numbers that satisfy the function. If the greatest exponent in the polynomial is odd, the polynomial is said to be odd and the range, like the domain, is the set of all real numbers.

Discuss rational functions

A rational function is a function that can be constructed as a ratio of two polynomial expressions: ƒ(x) = p(x)/q(x), where p(x) and q(x) are both polynomial expressions and q(x) ≠ 0. The domain is the set of all real numbers, except any values for which q(x)=0. The range is the set of real numbers that satisfies the function when the domain is applied. When you graph a rational function you will have vertical asymptotes wherever q(x) = 0. If the polynomial in the numerator is of lesser degree than the polynomial in the denominator, the x-axis will also be a horizontal asymptote. If the numerator and denominator have equal degrees, there will be a horizontal asymptote not on the x-axis. If the degree of the numerator is exactly one greater than the degree of denominator, the graph will have an oblique, or diagonal, asymptote. The asymptote will be along the line y=pₙ/qₙx + pₙ-1/qₙ-1, where pₙ and qₙ-1 are the coefficients of the highest degree terms in their respective polynomials.

Discuss samples, including random samples, representative samples, and sample size.

A sample is a piece of the entire population that is selected for a particular study in an effort to gain knowledge or information about the entire population. For most studies, a Random Sample is necessary to produce valid results. Random samples should not have any particular influence to cause sampled subjects to behave one way or another. The goal is for the random sample to be a Representative Sample, or a sample whose characteristics give an accurate picture of the characteristics of the entire population. To accomplish this, you must make sure you have a proper Sample Size, or an appropriate number of elements in the sample.

Define sample and statistic.

A sample is a portion of the entire population. Where as a parameter helped describe the population, a statistic is a numerical value that gives information about the sample, such as mean, median, mode, or standard deviation.

Give the geometric description of a vertical hyperbola. Include the equation of a hyperbola, vertices, foci, center, and asymptotes.

A vertical hyperbola is formed when a plane makes a vertical cut through two cones that are stacked vertex-to-vertex. a2 b2 hyperbola is -k (x-h)2 = 1, The standard equation of a vertical where a, b, k, and h are real numbers. The center is the point (h, k), the vertices are the points (h, k + a) and (h, k -a), and the foci are the points that every point on one of the parabolic curves is equidistant from and are found using the formulas (h, k + c) and (h, k - c), where c2 = a2 +b2. The asymptotes are two lines the graph of the hyperbola approaches but never reach, and are given by the equations y =(a/b) (x-h) + k and y =-(a/b) (x-h) +k

Discuss the Fundamental Theroem of Algebra , the Remainder Theroem, and the Factor Theroem as they apply to functions.

According to the Fundamental Theroem of Algebra, every non-constant, single variable polynomial has exactly as many roots as the polynomials highest exponent. For example, if X^4 is the largest exponent of a term, the polynomial will have exactly 4 roots. However, some of these roots may have multiplicity or be non-real numbers. For instance, in the polynomial function of f(x) = x^4 - 4x + 3, the only real roots are 1 and -1. The root 1 has multiplicity of 2 and there is one non-real root (-1 -√2i)

Describe the differences between algebraic functions and transcendental functions.

Algebraic functions are those that exclusively use polynomials and roots. These would include polynomial functions, rational functions, square root functions, and all combinations of these functions, such as polynomials as the radicand. These combinations may be joined by addition,s subtraction, multiplication, or division, but may not include variables as exponents. Transcendental functions are all functions that are non-algebraic. Any function that includes logarithms, trigonometric functions, variables as exponents, or any combination that includes any of these not algebraic in nature, even if the function includes polynomials or roots.

Describe the following types of triangles: acute, right, and obtuse. Tell the sum of the angles of a triangle.

An acute triangle is a triangle whose three angles are all less than 90°. If two of the angles are equal, the acute triangle is also an isosceles triangle. If the three angles are all equal, the acute triangle is also an equilateral triangle. A right triangle is a triangle with exactly one angle equal to 90°. All right triangles follow the Pythagorean Theorem. A right triangle can never be acute or obtuse. An obtuse triangle is a triangle with exactly one angle greater than 90° The other two angles may or may not be equal. If the two remaining angles are equal, the obtuse triangle is also an isosceles triangle. The sum of the measures of the interior angles of a triangle is always 180°. Therefore, a triangle can never have more than one angle greater than or equal to 90°.

Describe the following types of triangles: equilateral, isosceles, scalene.

An equilateral triangle is a triangle with three congruent sides. An equilateral triangle will also have three congruent angles. isosceles triangle will also have two congruent angles opposite the two An isosceles triangle is a triangle with two congruent sides. An congruent sides. A scalene triangle is a triangle with no congruent sides. A scalene triangle will also have three angles of different measures. The angle with the largest measure is opposite the longest side, and the angle with the smallest measure is opposite the shortest side.

Define event.

An event, represented by the variable E, is a portion of a sample space. It may be one outcome or a group of outcomes from the same sample space. If an event occurs, then the test or experiment will generate an outcome that satisfies the requirement of that event. For example, given a standard deck of 52 playing cards as the sample space, and defining the event as the collection of face cards, then the event will occur if the card drawn is a J, Q, or K. If any other card is drawn, the event is said to have not occurred.

Define expected value.

Expected value is a method of determining expected outcome in a random situation. It is really a sum of the weighted probabilities of the possible outcomes. Multiply the probability of an event occurring by the weight assigned to that probability ( such as the amount of money won or lost). A practical application of the expected value is to determine whether a game of chance is really fair. If the sum of the weighted probabilities is greater than or equal to 0, the game is generally considered fair because the player has a fair chance to win, or at least to break even. If the expected value is less than 1, then players lose more than they win.

Explain exponential functions and logarithmic functions and the relationship between the two.

Exponential functions are equations that have the format y=mx+b^x, where base b>0 and b≠1. The exponential function can also be written ƒ(x)=b^x. Logarithmic functions are equations that have the format y=logₙx or ƒ(x)=logₙx. The base b may be any number except one; however, the most common bases for logarithms are base 10 and base e. The log base e is known the natural logarithm, or ln, expressed the function ƒ(x)= ln x. Any logarithm that does not have an assigned value of b is assumed to be base 10: log x= log₁₀x. Exponential functions and logarithmic functions are related in that one is the inverse of the other. If ƒ(x)= b^x, then ƒ⁻¹(x) = logₙx. This can perhaps be expressed more clearly by the two equations: y=b^x and x= logₙy. The following properties apply to logarithmic expressions: logₙ 1= 0 logₙ b=1 logₙ bⁿ = p logₙ MN = logₙM-logₙN logₙ M/N = logₙM-logₙN logₙM^p = p logₙM

Explain roots in quadratic equations.

Find the roots of y = x² + 6x - 16 and explain why these values are important. The roots of a quadratic equation or the solutions when ax² + bx + c = 0. To find these roots of a quadratic equation, first replace y with 0. If 0=x² +6x-16, then to find the values x, you can factor the equation if possible. When factoring a quadratic equation where a = 1, find the factors of c that add up to b. That is the factors of -16 that add up to 6. The factors of -16 include, -4 and 4, -8 and 2, and -2 and 8. The factors that add up to equal 6 are -2 and 8. Write these factors as the product of two binomials, 0=(x-2)(x+8). You can verify that these are the correct factors by using FOIL to multiply them together. Finally, since these binomials multiply together to equal zero, set them each equal to 0 and solve for X. This results in x-2=0, which simplifies to X = 2 and x+8=0, which simplifies to x=-8. Therefore, the roots of the equation are 2 and -8. These values are important because they tell you where the graph of the equation crosses the X-axis. The points of the intersection are (2,0) and (-8,0).

How do you solve for x in the proportion 0.50/2=1.50/x?

First, cross multiply then solve for x. .50/2=1.50/x .50(x)=1.50(2) .50x=3 .50x/.50=3/.50 x=6 Or notice that .50 x 3/2 x 3= 1.50/6 so x=6

Solve 45%÷12% = 15%÷ X for X

First, cross multiply; then, solve for x: 45% / 12% = 15% / X 0.45/0.12 = 0.15/x 0.45(x) = 0.12(0.15) 0.45x = 0.0180 0.45x / 0.45 = 0.0180 / 0.45 x= 0.04 = 4% Alternatively, notice that 45% ÷ 3 / 12% ÷ 3 = 4% = 15%. So x= 4%

Demonstrate how to subtract 189 from 525 using regrouping.

First, set up the subtraction problem: 525 -189 ----- Notice that the numbers in the ones and tens columns of 525 are smaller than the numbers in the ones and tens columns of 189. This means you will need to use regrouping to preform subtraction. 5 2 5 -1 8 9 -------- To subtract 9 from the 5 in the ones column you will need to borrow from the 2 in the tens column. 5 1 15 -1 8 9 --------- 6 Next, to subtract 8 from the 1 in the tens column you will need to borrow from the 5 in the hundreds column. 4 11 15 -1 8 9 ---------- 3 6 Last, subtract the 1 from the remaining 4 in the hundreds column. 4 11 15 -1 8 9 ---------- 3 3 6

Explain the process for matrix addition and subtraction.

For two matrices to be conformable for addition or subtraction, they must be of the same dimension; otherwise, the operation is not defined. If matrix M is a 3 x 2 matrix and matrix N is a 2 x 3 matrix, the operations M + N and M - N or meaningless. If matrices M and N are the same size, the operation is as simple as adding or subtracting all of the corresponding elements: [m₁₁ m₁₂]+[n₁₁ n₁₂] [m₁₁+n₁₁ m₁₂+n₁₂] [m₂₁ m₂₂]+[n₂₁ n₂₂] = [m₂₁+n₂₁ m₂₂+n₂₂] [m₁₁ m₁₂]-[n₁₁ n₁₂] [m₁₁-n₁₁ m₁₂-n₁₂] [m₂₁ m₂₂]-[n₂₁ n₂₂] = [m₂₁-n₂₁ m₂₂-n₂₂] The result of addition or subtraction is a matrix of the same dimension as the two original matrices involved in the operation.

Contrast unimodal and bimodal distributions of data.

If a distribution has a single peak, it would be considered unimodal. If it has two discernible peaks it would be considered bimodal. Bimodal distributions may be an indication that the set of data being considered is actually the combination of two sets of data with significant differences.

Define the following terms: line of best fit, regression coefficients, residuals, and least-squares regression line.

In a scatter plot, the Line of Best Fit is the line that best shows the trends of the data. The line of best fit is given by the equation ý = ax + b, where 'a' and 'b' are the regression coefficients. The regression coefficient 'a' is also the slope of the line of best fit, and 'b' is also the y- coordinate of the point at which the line of best fit crosses the x-axis. Not every point on the scatter plot will be on the line of best fit. The differences between the y-values of the points in the scatter plot and the corresponding y-values according to the equation of the line of best fit are the residuals. The line of best fit is also called the least-squares regression line because it is also the line that has the lowest sum of the squares of the residuals.

Define odds in favor.

In probability, the odds in favor of an event are the number of times the event will occur compared to the number of times the event will not occur. To calculate the odds in favor of an event, use the formula P(A)/1-P(A), where P(A) is the probability that the event will occur. Many times, odds in the form a/b or a:b, where a is the probability in favor is given as a ratio b probability of the event not occurring. If the odds in favor are given as of the event occurring and b is the complement of the event, the 2:5, that means that you can expect the event to occur two times for every 5 times that it does not occur. In other words, the probability that 2 the event will occur is 2/2+5 =2/7

Explain simple regression.

In statistics, Simple Regression is using an equation to represent a relation between an independent and dependent variables. The independent variable is also referred to as the explanatory variable or the predictor, and is generally represented by the variable x in the equation. The dependent variable, usually represented by the variable y, is also referred to as the response variable. The equation may be any type of function - linear, quadratic, exponential, etc. The best way to handle this task is to use the regression feature of your graphing calculator. This will easily give you the curve of best fit and provide you with the coefficients and other information you need to derive an equation.

Define population and parameter.

In statistics, the Population is the entire collection of people, plants, etc., that data can be collected from. For example, a study to determine how well students in the area schools perform on a standardized test would have a population of all the students enrolled in those schools, although study may include just a small sample of students from each school. A Parameter is a numerical value that gives information about the population, such as the mean, median, mode, or standard deviation. Remember that the symbol for the mean of a population is u and the symbol for the standard deviation of a population is o.

A teacher tells her students that she has five pieces of fruit in her refrigerator. Two are apples, and the rest are oranges. She asks the students how many oranges are in her refrigerator. What type of problem is the teacher attempting to model?

In this type of problem, there is an unknown addend. When dealing with unknown addends and numbers up to 20 (typically in the elementary classroom), this type of problem is known as a put- together/take apart problem. Ideally, the teacher's goal is for the students to visualize this the scenario as 2+? = 5. Depending on the students' approach (drawing a picture, subtracting, number facts, modeling, etc.), they may choose to put together or take apart the problem in a variety of ways. To analyze the students' process, the teacher should check that each student has developed a strategy that will result in correctly identifying the missing addend in a way that could be applied to other problems similar in nature, continually resulting in the correct answer.

Define inferential statistics and sampling distribution.

Inferential Statistics is the branch of statistics that uses samples to make predictions about an entire population. This type of statistics is often seen in political polls, where a sample of the population is questioned about a particular topic or politician to gain an understanding about the attitudes of the entire population of the country. Often, exit polls are conducted on election days using this method. Inferential statistics can have a large margin of error if you do not have a valid sample. Statistical values calculated from various samples of the same size make up the sampling distribution. For example, if several samples of identical size are randomly selected from a large population and then the mean of each sample is calculated, the distribution of values of the means would be a Sampling Distribution.

Discuss the importance of providing students with the opportunity to explore problem structures with unknowns in all positions.

It is important to provide students with the opportunity to explore and find solutions for problem structures to develop their higher-level thinking skills and problem-solving strategies. By exposing students to problems with unknowns in all positions, students are forced to not only memorize procedures but also to analyze the framework of a problem, make connections for relationships within the problem, and develop strategies for solving problems that may not follow specific rules or procedures. Problems that include put-together/take-apart scenarios are excellent problem structures for teachers to use in an effort to develop higher-level thinking skills. In addition, providing students the opportunity to use arrays to model their solutions provides teachers with a glimpse into the thought processes of each student's solution.

Explain observational studies.

Observational studies are the opposite of experimental studies. In observational studies, the tester cannot change or in any way control all the variable in the test. For example, a study to determine which gender does better in math classes in school is strictly observational. You cannot change a person's gender, and you cannot change the subject bing studied. The big downfall of observational studies is that you have no way of proving a cause-and-effect relationship because you cannot control outside influences. Events outside of school can influence a student's performance in school, and observational studies cannot take that into consideration.

Explain probability and probability measure.

Probability is a branch of statistics One classic example heads or tails. The likelihood, or probability, is a coin toss. There are that deals with the likelihood of only two possible results: something taking place. that the coin will land as heads is 1 out of 2 (1/2, 0.5, 50%). Tails has the same probability. probability of any given number coming up is 1 out of 6. Another common example is a 6-sided die roll. The is a function OCcur. The probability event must have a real multiple mutually exclusive events must equal space must equal one, and the probability less than or equal to one. Also, the probability measure of the sample For a probability number probability measure that is greater measure to be accurate, every that assigns a real number probability, from zero measure, also called the distribution, to one, to each event. For every sample space, each possible event has a probability that it will than or equal to zero and measure of the union of the sum of the individual probability measures.

Describe how to manipulate equations to find missing values.

Sometimes you will have variables missing in equations. So, you need to find the missing variable. To do this, you need to remember one important thing: whatever you do to one side of an equation, you need to do to the other side. If you subtract 100 from one side of the equation you need to subtract 100 from the other side of the equation. This will allow you to change the form of the equation to find missing values.

explain 5-number summary and how it relates to a box-and-whiskers plot.

The 5-number summary of a set of data gives a very informative picture of the set. The five number in the summary include the minimum value, maximum value, and the three quartiles. This information gives the reader the range and median of the set, as well as an indication, of how the data is spread about the median. A box-and-whiskers plot is a graphical representation of the 5-number summary. To draw a box-and-whiskers plot, plot the points of the 5-number summary on a number line. Draw a box whose ends are through the points for the first and third quartiles. Draw a vertical line in the box through the median to divide the box in half. Draw a line segment from the first quartile point to the minimum value, and from the third quartile point to the maximum value.

Define variance.

The Variance of a population, or just variance, is the square of the standard deviation of that population. While the mean of a set of data gives the average of the set and gives information about where a specific data value lies in relation to the average, the variance of the population gives information about the degree to which the data values are spread out and tell you how close an individual value is to the average compared to the other values. The units associated with variance are the same as the units of the data values squared.

Explain the Addition Rule for probability.

The addition rule for formula P(A or B) P(A) +P(B) - probability is used for finding the probability of a compound event. Use the P(A and B), where P(A and B) is the probability of both events occurring to find the probability of a compound event. The probability of both events occurring at the same time must be subtracted to eliminate any overlap in the first two probabilities.

A patient was given blood pressure medicine at a dosage of 2 grams. The patient's dosage was then decreased to 0.45 grams. By how much was the dosage decreased?

The decrease is represented by the difference between the two amounts: 2 grams-0.45 grams=1.55 grams. Remember to line up the decimal point before subtracting. 2.00 -0.45 ------- 1.55

A car is traveling at a speed of 40 miles per hour for 2.5 hours. How far, in miles, has the car traveled? Describe the steps for arriving at your answer.

The distance formula is d = rt, where d represents the distance, r represents the rate of speed, and t represents the time elapsed. The distance formula tells us that by multiplying the rate of speed by the time elapsed we can determine the distance traveled. Here, when we substitute our rate, 40, for r and 2.5 for our time, t, we have the expression d = 40(2.5). The product of 40 and 2.5 is 100, which simplifies our expression to d = 100. We can now conclude that the car has traveled a distance of 100 miles in 2.5 hours ata rate of 40 miles per hour.

Explain the Multiplication Rule for probability.

The multiplication rule can be used to find the probability of two independent events occurring using the formula P(A and B) = P(A) X P(B), where P(A and B) is the probability of two independent events occurring, P(A) is the probability of the first event occurring, and P(B) is the probability of the second event occurring. The multiplication rule can also be used to find the probability of two dependent events occurring using the formula P(A and B) = P(A) x P(BIA), where P(A and B) is the probability of two dependent events occurring and P(B|A) is the probability of the second event oCcurring after the first event has already occurred. Before using the multiplication rule, you MUST first determine whether the two events are dependent or independent

Describe the characteristics of the sampling distribution of the mean.

The sampling distribution of the mean is the distribution sample mean, x, derived from random samples of a given size. It has three important characteristics. First, the mean of the sampling distribution of the mean is equal to the mean of the population that was sampled. Second, assuming the standard deviation is non-zero, the standard deviation of the sampling distribution of the mean equals the standard deviation of the sampled population divided by the square root of the sample size. This is sometimes called the standard error. Finally, as the sample size gets larger, the sampling distribution of the mean gets closer to a normal distribution via the Central Limit Theorem.

Define Median as it applies to statistics.

The statistical Median is the value in the middle of the set of data. To find the median, list all data values in order from smallest to largest or from largest to smallest. Any value that is repeated in the set must be listed the number of times it appears. If there are an odd number of data values, the median is the value in the middle of the list. If there is an even number of data values, the median is the arithmetic mean of the two middle values.

What is 30% of 120?

The word "of: indicates multiplication, so 30% of 120 is found by multiplying 30% by 120. First, change 30% to a fraction or decimal. Recall that "percent" means per hundred, so 30%=30/100=.30. 120(.3) is 36.

Determine whether (-2,4) is a solution of the inequality y ≤ -2x + 3.

To determine whether a coordinate is a solution of an inequality, you can either use the inequality or its graph. Using (-2,4) as (x,y), substitute the values into the inequality to see if it makes a true statement. This results in 4 ≤ -2(-2) + 3. Using the integer rules, simplify the right side of the inequality by multiplying and then adding. The result is 4≤7, which is a false statement. Therefore, the coordinate is not a solution to the inequality. You can also use the graph of an inequality to see if a coordinate is a part of the solution. The graph an an inequality is shaded over the section of the coordinate grid that is included in the solution. The graph of y ≥ -2x + 3 includes the solid line y= -2x + 3 and is shaded to the right of the line, representing all of the points greater than and including the points on the line. This excludes the point (-2,4), so it is not a solution to the inequality.

The area of a closet is 25 1/4 square feet. New carpet (including labor and materials) costs $8.00 per square foot to install. What will it cost to re-carpet the entire closet?

To find the total cost to re-carpet the closet, we must multiply the area of the closet, 25 1/4, by the cost of the carpet, $8.00 per square foot. To multiply these values, we must convert both of them to improper fractions, creating the expression 101/4 x $8/1. We can use cross-cancellation to simplify our expression by dividing both 4 and 8 by 4, which changes our expression to 101/1 x $2/1. When we multiply across, we arrive at our product, $202.00. Without cross-cancellation, our product would be 808/4, which still reduces to $202.00. It will cost $202.00 to re-carpet the closet.

Tell how to find the probability that at least one of something will occur.

Use a combination of the multiplication rule and the rule of complements to find the probability that at least outcome of the elements will occur. This is given the general formula P(at least one event occurring)= 1- P(no outcome occurring). You can always used a tree diagram or make a chart to list the possible outcomes when the sample space is small, such as dice-rolling.

Explain the difference between variables that vary directly and those that very inversely.

Variables that vary directly are those that either both increase at the same rate or both decrease at the same rate. For example, in the functions ƒ(x)=kx or ƒ(x)=kxⁿ, where k and n are positive and the value of ƒ(x) increases as the value of x increases and decreases as the value of x decreases. Variables that vary inversely are those where one increases while the other decreases. For example, in the functions ƒ(x) = k/x or ƒ(x) = k/xⁿ where k is a positive constant, the value of y increases as the value of x decreases, and the value of y decreases as the value of x increases. In both cases, k is the constant of variation.

Discuss the importance of analyzing the use of appropriate mathematical concepts, procedures, and vocabulary when evaluating student solutions.

When evaluating student solutions, it is important to analyze the use of appropriate mathematical concepts, procedures, and vocabulary for a variety of reasons. First and foremost, we must be sure that we have provided adequate practice and instruction of important concepts before assessing them. Once we have established that instruction is sufficient, we must ensure that students are following the appropriate procedures when faced with various tasks and that those procedures are executed correctly. Finally, we must hold students accountable for using high-level vocabulary to ensure that students are able to read, understand, and communicate their mathematics thoughts at age- and grade-appropriate levels.

Discuss using permutation and combination to calculate the number of outcomes.

When trying to calculate the probability of an event using the desired outcomes/ total outcomes formula, you may frequently find that there are too many outcomes to individually count them. Permutation and combination formulas offer a shortcut to counting outcomes. A permutation is an arrangement of a specific number of a set of objects in a specific order. the number of permutations of r items given a set of n items can be calculated as nPr= n!/ (n-r)!. Combinations are similar to permutations, except there are no restrictions regarding the order of the elements. While ABC is considered a different permutation than BCA, ABC and BCA are considered the same combination. The number of combinations of r items given a set of n items can be calculated as nCr= n!/r!(n-r)! or nCr=nPr/r!.

Explain the possible disadvantages of using the mean as the only measure of central tendency. Visit mometrix.com/academy for related video Enter video code: 286207

While the mean is relatively easy to calculate and averages are understood by most people, the mean can be very misleading if used as the sole measure of central tendency. If the data set has outliers (data values that are unusually high or unusually low compared to the rest of the data values), the mean can be very distorted, especially if the data set has a small number of values. If unusually high values are countered with unusually low values, the mean is not affected as much. Whenever the mean is skewed by outliers, it is always a good idea to include the median as an alternate measure of central tendency.

Measurement conversion practice problems a. Convert 1.4 meters to centimeters. b. Convert 218 centimeters to meters.

Write ratios with the conversion factor 100 cm/1 m. Use proportions to convert the given units. a. 100 cm/1 m = x cm/1.4 m. Cross multiply to get x = 140. So, 1.4 m is the same as 140 cm. b. 100 cm/1 m = 218 cm/x m. Cross multiply to get 100x = 218, or x = 2.18. So, 218 cm is the same as 2.18 m.

Measurement conversion practice problems A. Convert 42 inches to feet. B. Convert 15 feet to yards.

Write ratios with the conversion factors 12 in/1 foot and 3 foot/1 yard. Use proportions to convert the given units. A. 12 in/1 ft = 42 in/x ft. Cross multiply to get 12 X = 42, or X = 3.5. So, 42 inches is the same as 3.5 feet. B. 3 ft/1 yd = 15 ft/x yd. Cross multiply to get 3X = 15, or X = 5. So, 15 feet is the same as 5 yards.

Round each number: 1. To the nearest ten: 11, 47, 118 2. To the nearest hundred: 78, 980, 248 3. To the nearest thousand: 302, 1274, 3756

1. Remember, when rounding to the nearest ten, anything ending in 5 or greater rounds up. So, 11 rounds to 10, 47 rounds to 50, and 118 rounds to 120 2. Remember, when rounding to the nearest hundred, anything ending in 50 or greater rounds up. So, 78 rounds to 100, 980 rounds to 1000, and 248 rounds to 200 3. Remember, when rounding to the nearest thousand, anything ending in 500 or greater rounds up. So, 302 rounds to 0, 1274 rounds to 1000, and 3756 rounds to 4000. When you are asked for the solution a problem, you may need to provide only an approximate figure or estimation for your answer. In this situation, you can round the numbers that will be calculated to a non- zero number. This means that the first digit in the number is not zero and the following numbers are zeros.

What is 150% of 20?

150% of 20 is found by multiplying 150% by 20. First, change 150% to a fraction or decimal. Recall that "percent" means per hundred, so 150%=150/100=1.50. So, (1.50)(20)=30. Notice that 30 is greater than the original number of 20. This makes sense because you are finding a number that is more than 100% of the original number.

Write each number in words: 29, 478, 9, 542, 302, and 876.

29: twenty-nine 478: four hundred seventy-eight 9,435: Nine thousand four hundred thirty-five 98,542: ninety-eight thousand five hundred forty-two 302,876: three hundred two thousand eight hundred seventy-six

Explain fractions, numerators and denominators

A fraction is a number that is expressed as one integer written above another integer, with a dividing line between them.(x/y) It represents the quotient of the two numbers "x divided by y". It can also be thought of as "x out of y equal parts" The top number of a fraction is called the numerator, and it represents the number of parts under consideration. The 1 in 1/4 means that the 1 part out of the whole is being considered in the calculation. The bottom number of the fraction is called the denominator, and it represents the total number of equal parts. The 4 in 1/4 means that the whole consists of 4 equal parts. A fraction cannot have a denominator of 0, this is referred to as undefined.

Explain the measure of central tendency.

A measure of central tendency is a statistical value that gives a general tendency for the center of a group of data. there are several different ways of describing the measure of central tendency. Each one has a unique way it is calculated, and each way gives a slightly different perspective on the data set. Whenever you give a measure of central tendency, always make sure the units are the same. If the data has different units, such as hours, minutes, seconds, convert all the data to the same unit, and use the same unit in the measure of central tendency. If no units are given in the data, do not give units for the measure of central tendency.

Explain the measure of dispersion.

A measure of dispersion is a single value that helps to "interpret" the measure of central tendency by providing more information about how the data values in the set are distributed about the measure of central tendency. The measure of dispersion helps to eliminate or reduce the disadvantages of using the mean, median, or mode as a single measure of central tendency, and give a more accurate picture of the data set as a whole. To have a measure of dispersion, you must know or calculate the range, standard deviation, or variance of the data set.

Discuss percentage problems and the process to be used for solving them.

A percentage problem can be presented three main ways: (1) Find what percentage of some numbers another number is. Example: What percentage of 40 is 8? (2) Find what number is some percentage of a given number.Example: What number is 20% of 40? (3) Find what number another number is a given percentage of. Example: What number is 8 20% of? The three components in all of these cases is the same: a whole (W), a part (P) and a percentage (%). These are related by the equation: P=W x %. This is the form of the equation you would use to solve the problems of type (2). To solve for types (1) & (3), you would use these two forms: %=P/W and W=P/%. The thing that frequently makes percentage problems problems difficult is that they are often also word problems, so a large part of solving them is finding out which quantities are what. Example: In a school cafeteria, 7 students choose pizza, 9 choose hamburgers and 4 choose tacos. Find all the parts: 7+9+4=20. The percentage can then be found by dividing the part by the whole (%=P/W): 4/20-20/100=20%.

Describe a pictograph and tell how to read one.

A pictograph is a graph, generally in the horizontal orientation, that uses pictures or symbols to represent the data. Each pictograph must have a key that defines the picture or symbol and give the quantity each picture or symbol represents. Pictures and symbols on a pictograph are not always shown as whole elements. In this case, the fraction of the picture or symbol shown represents that same fraction of the quantity a whole picture or symbol stands for.

Explain the terms ray, angle, and vertex.

A ray is a portion of a line extending from a point in one direction. It has a definite beginning, but no ending. An angle is formed when two rays meet at a common point. It may be a common starting point, or it may be the intersection of rays, lines, and/or line segments. A vertex is the point at which two segments or rays meets to form an angle. If the angle is formed by intersecting rays, lines, and/or line segments, the vertex is the point at which for angles are formed.

Discuss square root functions.

A square root function is a function that contains a radical and is in the format (x)= √ax + b. The domain is the set of all real numbers that yields a positive radicand or a radicand equal to zero. Because square root values are assumed to be positive unless otherwise identified, the range is all real numbers from zero to infinity. To find the zero of a square root function, set the radicand equal to zero to solve for x. The graph of a square root function is always to the right of the zero and always above the x-axis.

Define z-scores.

A z-score is an indication of how many standard deviations a given value falls from the mean. To calculate a z-score, use the formula = x-µ/σ, where x is the data value, µ is the mean of the data set, and σ is the standard deviation of the population. If the z-score is positive, the data value lies above the mean. If the z-score is negative, the data value falls below the mean. These scores are useful in interpreting data such as standardized test scores, where every piece of data in the set has been counted rather than just a small random sample. In cases where standard deviations are calculated from a random sample of the set, the z-scores will not be as accurate.

Discuss the central limit theorem

According to the central limit theorem, regardless of what the original distribution of a sample is, the distribution of the means tends to get closer and closer to a normal distribution as the sample size gets larger and larger (this is necessary because the sample is becoming more all-encompassing of the elements of the population). As the sample size gets larger, the distribution of the sample mean will approach a normal distribution with a mean of the population mean and variance of the population variance divided by the sample size.

Discuss the rational root theorem as it applies to functions.

According to the rational root theorem, any rational root of a polynomial function ƒ(x)=aₙxⁿ+aₙ-₁xⁿ⁻¹+...+a₁x+a₀ with integer coefficients will, when reduced to its lowest terms, be a positive or negative fraction such that the numerator is a factor of a₀ and the denominator is a factor of aₙ. For instance , if the polynomial function ƒ(x)=x³+3x²-4 has any rational roots, the numerators of those roots can only be factors of 4(1,2,4), and the denominators can only be factors of 1(1). The function in this example has roots of 1(or1/1) and -2(or-2/1).

Describe exponents and parentheses.

An exponent is a superscript number placed next to another number at the top right. It indicates how many times the base number is to be multiplied by itself. Exponents provide a shorthand way to write what would be a longer mathematical expression. Example: a²=axa; 2⁴=2x2x2x2. A number with an exponent of 2 is said to be "squared," while a number with an exponent of 3 is said to be "cubed." The value of a number raised to an exponent is called its power. So, 8⁴ is read "8 to the 4th power," or "8 raised to the power of 4." A negative exponent is the same as the reciprocal of a positive exponent. Example: a-² = 1/a². Parentheses are used to designate which operations should be done first when there are multiple operations. Example: 4 - (2+1) =1; the parentheses tell us that we must add 2 and 1, and then subtract the sum from 4, rather than subtracting 2 from 4 and then adding 1 (this would give us an answer of 3).

List some guidelines for using manipulative materials in the mathematics classroom.

As with all classroom supplies, the student must understand that there are rules for their use, including how to store the materials when they are not in use. In addition: The teacher should discuss with the students the purpose of the manipulatives and how they will help the students to learn, The student should understand that the manipulatives or intended for use with specific problems in activities; however, time for free exploration should be made available so students are less tempted to play when assigned specific tasks, A chart posted in the classroom of the manipulatives with their names will help the students to gain familiarity with them and develop mathematical literacy skills, and Loans of manipulatives for home use with the letter of explanation to the parents about the purpose and value of the manipulatives will encourage similar strategies with homework.

Describe a histogram and compare it to a bar graph and a stem and leaf plot.

At first and histogram looks like a vertical bar graph. The differenced is that a bar graph has a separate bar for each piece of data and a histogram has one continuous bar for each range of data. While a bar graph has numerical values on one axis, a histogram has numerical values on both axes. Each range is of equal size, and they are ordered from left to right from lowest to highest. The height of each column in a histogram represents the number of data values within that range. Like a stem and leaf plot, a histogram makes it easy to glance at the graph and quickly determine which range has the greatest quantity of values.

Describe the organization and purpose of scatter plots in terms of bivariate data.

Bivariate data is simply data from 2 different variables (The prefix bi- means 2). In a scatter plot, each value in the set of data is plotted on a grid similar to a Cartesian plane, where each axis represents one of the two variables. By looking at the pattern formed by the points on the grid you can often determine whether or not there is a relationship between the two variables, and what that relationship is, if it exists. The variables may be directly proportionate, inversely proportionate, or show no proportion at all. It may also be possible to determine if the data is linear, and if so, to find an equation to relate the two variables.

Discuss equivalent units for converting between U.S. standard and metric equivalents of capacity.

Capacity Measurements:

Describe circle graphs (pie charts) and tell how to interpret data.

Circle graphs, also known as pie charts, provide a visual depiction of the relationship of each type of data compared to the whole set of data. The circle graph is divided into sections by drawing radii to create central angles whose percentage of the circle is equal to the individual data's percentage of the whole set. Each 1% of data is equal to 3.6⁰ of the circle graph. Therefore, data represented by a 90⁰ section of the circle graph makes up 25% of the whole. When complete the circle graph often looks like a pie cut into uneven wedges.

Explain correlational studies.

Correlation studies seek to determine how much one variable is affected by changes in a second variable. For example, correlation studies may look for a relationship between the amount of a time student spends studying for a test and the grade that student earned on the test or between student scores on college admissions tests and student grades in college. It is important to note that correlational studies cannot show a cause and effect, but rather can show only that two variables are or are not potentially correlated.

Discuss ways in which a teacher can create structured of students experiences for small and large groups according to the cognitive complexity of the task

Depending on the complexity of the task, teachers should modify the delivery of instruction. For less complex tasks, the teacher may opt for whole-group instruction, in which the entire class is introduced to a new concept together. Whole-group instruction usually includes a connection to prior knowledge, direct instruction, and some form of media. For more complex tasks, the teacher may choose small-group instruction, where students are grouped based on ability levels. The more accelerated learners are given a quicker, more direct form of instruction, whereas the struggling learners work independently at a math center or station activity. After a rotation, the struggling learners are given a modified form of instruction by the teacher, which has been differentiated and allows for more gradual release based on individual needs and abilities.

Explain empirical probability.

Empirical probability is based on conducting numerous repeated experiments and observations rather than by applying pre-defined formulas to determine the probability of an event occurring. To find the empirical probability of an event, conduct repeated trials (repetitions of the same experiment) and record your results. The empirical probability of an event occurring is the number of times the event occurred in the experiment divided by the total number of trials you conducted to get the number of events. Notice that the total number of trials is used, not the number of unsuccessful trials. A practical application of empirical probability is the insurance industry. There are no set functions that define life span, health, or safety. Insurance companies look at factors from hundreds of thousands of individuals to find patterns that they then use to set the formulas for insurance premiums.

Explain experimental studies.

Experimental studies take correlational studies one step farther, in that they attempt to prove or disprove a cause-and-effect relationship. These studies are performed by conducting a series of experiments to test the hypothesis. For a study to scientifically accurate, it must have both an experimental group that receives the specified treatment and a control group that does not get the treatment. This is the type of study pharmaceutical companies do as part of drug trials for new medications. Experimental studies are only valid when proper scientific method has been followed. In other words, the experiment must be well-planned and executed without bias in the testing process, all subjects must be selected at random, and the process of determining which subject is in which of the two groups must also be completely random.

Define extraneous variables.

Extraneous variables are, as the name implies, outside influences that can affect the outcome of a study. They are not always avoidable, but could trigger bias in the result.

Define the term factor and explain common and prime factors with examples. Visit mometrix.com/academy for a related video. Enter video code: 920086

Factors are numbers that are multiplied together to obtain a product. For example, in the equation 2 x 3 = 6, the numbers 2 and 3 are factors. A prime number has only two factors (1 and itself), but other numbers can have many factors. A common factor is a number that divides exactly into two or more other numbers. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12, while the factors of 15 are 1, 3, 5, and 15. The common factors of 12 and 15 are 1 and 3 A prime factor is also a prime number. Therefore, the prime factors of 12 are 2 and 3. For 15, the prime factors are 3 and 5.

Simplify: 0.22 + 0.5 - (5.5 + 3.3/3)

First, evaluate the terms in the parentheses (5.5+3.3/3) using order of operations. 3.3/3=1.1 and 5.5+1.1=6.6. Next, rewrite the problem: 0.22+0.5-6.6 Finally, add and subtract from left to right: 0.22+0.5=0.72; 0.72-6.6=-5.88. Answer: -5.88

At a hotel, 3/4 of the 100 rooms were occupied today. Yesterday, 4/5 of the 100 rooms were occupied. On which day were more of the rooms occupied and by how much more?

First, find the number of rooms occupied each day. To do so, multiply the fraction of rooms occupied by the number of rooms available: Number occupied=fraction occupied x total number. Today: Number of rooms occupied = 3/4 x 100 =75 Today, 75 rooms are occupied. Yesterday: Number of rooms occupied = 4/5 x 100 = 80 Yesterday, 80 rooms were occupied. The difference in the number of rooms occupied is 80-75=5 rooms. Therefore, five more rooms were occupied yesterday than today.

Demonstrate how to subtract 477 from 620 using regrouping.

First, set up the subtraction problem. 620 - 477 Notice that the numbers in the ones and tens columns of 620 are smaller than the numbers in the ones and tens columns of 477. This means you will need to use regrouping to preform subtraction. 6 2 0 -4 7 7 To subtract 7 from 0 in the ones column you will need to borrow from the 2 in the tens column. 6 1 10 -4 7 7 ⁻⁻⁻⁻⁻⁻⁻⁻⁻⁻⁻⁻ 3 Next, to subtract 7 from the 1 that's still in the tens column you will need to borrow from the 6 in the hundreds column. 5 11 10 -4 7 7 ⁻⁻⁻⁻⁻⁻⁻⁻⁻⁻⁻ 4 3 Lastly, subtract 4 from the remaining 5 in the hundreds column to get: 5 11 10 -4 7 7 ⁻⁻⁻⁻⁻⁻⁻⁻⁻⁻⁻⁻ 1 4 3

Two weeks ago, 2/3 of the 60 customers at a skate shop were male.Last week, 3/6 of the 80 customers were male. During which week were there more male customers?

First, you need to find the number of male customers that were in the skate shop each week. You are given this amount in the terms of fractions. To find the actual number of male customers, multiply the number of male customers by the number of customers in the store. Actual number of male customers=fraction of male customers X total number customers. Two weeks ago: actual number of male customers= 2/3 X 60 2/3 X 60/1 = 2X60/3x1=120/3=40 Two weeks ago 40 of the customers were male. Last week: actual number of male customers= 3/6 X 80 3/6 X 60/1= 3 X 80/6 X 1= 240/6=40 Last week, 40 of the customers were male. The number of male customers was the same both weeks.

Describe the process for adding, subtracting, multiplying, and dividing fractions.

If to fractions have a common denominator, they can be added or subtracted simply by adding or subtracting the two new writers and retaining the same denominator. Example: 1/2+14 equals 2/4+14 equals 3/4. If the two fractions do not already have the same denominator, one or both of them must be manipulated to achieve a common denominator before they can be added or subtracted. New two fractions can be multiplied by multiplying the two numerators to find the new numerator and The two denominators to find the new denominator. Example: 1/3×2/3 = 1x2/3x3=2/9. Two fractions can be divided by flipping the numerator and denominator of the second fraction and then proceeding as though it were multiplication. Example: 2/3÷3/4 = 2/3×4/3 = 8/9.

Discuss linear functions.

In linear functions, the value of the function changes in direct proportion to x. The rate of change, represented by the slope on its graph, is constant throughout. The standard form of a linear equation is ax+by=c, where a,b and c are real numbers. As a function, the equation is commonly written as y=mx+b or ƒ(x)=mx+b. This is known as slope intercept form, because the coefficients give the slope of the graphed function (m) and its y-intercept(b). Solve the equation mx+b=0 for x to get x=-b/m, which is the only zero of the function. The domain and range are both the set of all real numbers.

A teacher is assessing a students ability to multiply two-digit numbers. A student in the class arrives at the correct answer without using the traditional algorithm. What should the teacher look for when assessing the students learning?

In mathematics, there is often more than one way to arrive at the correct answer. The focus of the teacher should be to analyze the validity of the student' mathematical process to determine if it is a process that could be applied to other multiplication problems, which would result in the correct answer. If the student has modeled his or her thought process, or can argue that the inventive strategy is applicable across all multiplication problems, then that student should be considered to have mastered the skill of multiplication. However, if the student's argument or model is not applicable to other multiplication problems, that student should be provided with more instruction and opportunity for improvement.

Graph the inequality 10 > -2x + 4

In order to graph the inequality 10 > -2x + 4, you must first solve for x. The opposite of addition is subtraction, so subtract 4 from both sides. This results in 6 > -2x. Next the opposite of multiplication is division, so divide both sides by -2. Don't forget to flip the inequality symbol since you are dividing by a negative number. This results in -3 < x. You can rewrite this as x < -3. To graph an inequality, you create a number line and put a circle around the value that is being compared to x. If you are graphing a greater than or less than inequality, as the one shown, the circle remains open. This represents all of the values excluding -3. If the inequality happens to be greater than or equal to, or less than or equal to, you draw a closed circle around the value. This would represent all of the values including the number. Finally, take a look at the values the solution represents and shade the number line in the appropriate direction. You are graphing all of the values greater than -3 and since this is all of the numbers to the right of -3, shade this region on the number line.

Define odds against.

In probability, the odds against an event are the number of times the event will not occur compared to the number of times the event will occur. To calculate the odds against an event, use the formula 1-P(A)/ P(A), where P(A) is the probability that the event will occur. Many times, odds against is given as a ratio in the form b/a or b:a, where b is the probability that the event will not occur and a is the probability the event will occur.

Discuss bias and extraneous variables as they relate to samples and studies.

In statistical studies, biases must be avoided. Bias is an error that causes the study to favor one set of results over another. Extraneous variables are, as the name implies, outside influences that can effect the outcome of a study. They are not always avoidable, but could trigger bias in the result.

Qualitative data

Information (such as colors, scents, tastes, and shapes) that cannot be measured using numbers

Continuous data

Information (such as time and temperature) that can be expressed by any value within a given range

Discrete data

Information that can be expressed only by specific value, such as whole or half numbers. For example, since people can be counted only in whole numbers, a population count would be discrete data.

Ordinal data

Information that can be placed in numerical order, such as age or weight

Nominal data

Information that cannot be placed in numerical order, such as names or places

Primary data

Information that has been collected directly from a survey, investigation, or experiment, such as a questionnaire or the recording of daily temperatures. Primary data that has not yet been organized or analyzed is called raw data.

Describe scatter plots.

Scatter plots are useful in determining the type of function represented by the data and finding the simple regression. Linear scatter plots may be positive or negative. Nonlinear scatter plots are generally exponential or quadratic.

Explain the organization and usefulness of a line graph. Visit mometrix.com/academy for a related video. Enter video code: 480147

Line graphs have one or more line of varying styles (solid or broken) to show the different values for a set of data. The individual data are represented as ordered pairs, much like on a Cartesian plane. In this case, the x and y axes are defined in terms of their units, such as dollars or time. The individual plotted points are joined by line segments to show whether the value of the data is increasing (line sloping upward), decreasing (line slopping downward) or staying the same (horizontal line). Multiple sets of data can be graphed on the same line graph to give an easy visual comparison. An example of this would be graphing achievement test scores for different groups of students over the same time period to see which group had the greatest increase or decrease in performance from year-to-year.

Quantitative data

Measurements (such as length, mass, and speed) that provide information about quantities in numbers

List standard units for measurement in metric.

Metric conversions 1000 mcg (microgram) 1 mg 1000 mg (milligram). 1 g 1000 g. (grams) 1 kg 1000 kg (kilogram) 1 metric ton 1000 mL (milliliter). 1 L 1000 um (micrometer). 1 mm 1000 mm (millimeter). 1 m 100 cm (centimeter). 1 m 1000 m (meter). 1 km

Define the following common arithmetic terms specific to numbers: integers, prime, composite, even, and odd.

Numbers are the basic building blocks of mathematics. Specific features of numbers are identified by the following terms: Integers - The set of positive and negative numbers, including zero. Integers do not include fractions (1/3), decimals (0.56), or mixed numbers (7 3/4). Prime number - a whole number greater than one that has only two factors, itself and one; that is, a number that can be divided evenly only by one and itself. Composite number - a whole number greater than one that has more than two different factors; in other words, any whole number that is not a prime number. For example: the composite number 8 has the factors of 1, 2, 4, and 8. Even number - any integer that can be divided by two without leaving a remainder. For example: 2, 4, 6, 8, and so on. Odd number - any integer that cannot be divided evenly by two. For example: 3, 5, 7, 9, and so on.

Compare and contrast objective probability and subjective probability.

Objective probability is based on mathematical formulas and documented evidence. Examples of objective probability include raffles or lottery drawings where there is a pre-determined number of possible outcomes and a predetermined number of outcomes that correspond to an event. Other cases of objective probability include probabilities of rolling dice, flipping coins, or drawing cards. Most gambling is based on objective probability. Subjective probability is based on personal or professional feelings and judgments. Often, there is a lot of guesswork following extensive research. Areas where subjective probability is applicable include sale trends and business expenses. Attractions set admission prices based on subjective probabilities of attendance based on varying admission rates in an effort to maximize their profit.

Describe solving quadratic equations with graphing. Explain the quadratic formula.

One way to find a solution or solutions of a quadratic equation is to use its graph. The solution(s) of a quadratic equation or the values of x when y = 0. On the graph, y = 0 is where the parabola crosses the x-axis, or the X-intercepts. This is also referred to as the roots, or zeros of a function. Given a graph, you can locate the X-intercepts to find the solutions. If there are no x-intercepts, the function has no solution. If the parabola crosses the x-axis at one point, there is one solution and if it crosses at two points, there are two solutions. Since the solutions exist where y = 0, you can also solve the equation by substituting zero in for y. Then, try factoring the equation by finding the factors of ac that add up to equal b. You can use the guess and check method, the box method, or grouping. Once you find a pair that works, write them as the product of two binomials and set them equal to zero. Finally, solve for x to find the solutions. The last way to solve a quadratic equation is to use the quadratic formula. The quadratic formula is: ______ x=-b±√(b²-4a) -------------- 2a Substitute the values of a, b, and c into the formula and solve for X. Remember that blank refers to two different solutions. Always check your solutions with the original equation to make sure they are valid.

Solve for x in the equation 40/8 = x/24.

One way to solve for x is to first cross multiply. 40/8 = x/24 40(24) = 8(x) 960=8x 960/8=8x/8 x=120 Or notice that: 40 x 3/ 8 x 3 = 120/24, so x=120

Discuss the use of parentheses in operations.

Parentheses are used to designate which operations should be done first when there are multiple operations. Example: 4-(2+1)=1; the parentheses tell us we must add 2 and 1, and then subtract the sum from 4, rather than subtracting 2 from 4 and then adding 1(this would give us the answer of 3)

Explain the relationships between percentages, fractions and decimals.

Percentages can be thought of as fractions that are based on a whole of 100; that is, one whole is equal to 100%. The word percent means "per hundred". Fractions can be expressed as percentages by finding equivalent fractions with the denomination of 100. Example: 7/10=70/100=70%; 1/4=25/100=25%. To express a percentage as a fraction, divide the percentage number by 100 and reduce the fraction to its simplest possible terms. Example: 60%=60/100=3/5; 96%=96/100=24/25. Converting decimals to percentages and percentages to decimals is as simple as moving the decimal point. To convert from a decimal to a percentage, move the decimal point two places to the right. To convert from a percentage to a decimal, move it two places to the left. Example: 0.23=23%; 5.34=534%; .007=.7%; 700%=7.00; 86%=.86; .15%=.0015. It may be helpful to remember that the percentage number will always be larger than the equivalent decimal number.

Discuss percentiles and quartiles.

Percentiles and quartiles are other methods of describing data within a set. Percentiles tell what percentage of the data in the set fall below a specific point. For example, achievement test scores are often given in percentiles. A score at the 80th percentile is one which is equal to or higher than 80 percent of the scores in the set. In other words, 80 percent of the scores were lower than that score. Quartiles are percentile groups that make up quarter sections of the data set. The first quartile is the 25th percentile. The second quartile is the 50th percentile; this is also the median of the data set. The third quartile is the 75th percentile.

Describe perpendicular lines and perpendicular bisectors.

Perpendicular lines are lines that intersect at right angles. The are represented by the symbol ⊥. The shortest distance from a line to a point not on the line is a perpendicular segment from the point to the line. In a plane, the perpendicular bisector of a line segment is a line comprised of the set of al points that are equidistant from the endpoints of the segment. This line always forms a right angle with the segment in the exact middle of the segment. Note that you can only find perpendicular bisectors of segments.

Explain rational expressions. Review the operations of rational expressions.

Rational expressions are fractions with polynomials in both the numerator and the denominator; the value of the polynomial in the denominator cannot be equal to zero. To add or subtract rational expressions, first find a common denominator, then rewrite each fraction as an equivalent fraction with the common denominator. Finally, add or subtract the numerators to get the numerator of the answer, and keep the common denominator as the denominator of the answer. When multiplying rational expressions factor each polynomial and cancel like factors (a factor which appears in both the numerator and the denominator). Then, multiply all remaining factors in the numerator to get the numerator of the product, and multiply the remaining factors in the denominator to get the denominator of the product. Remember - cancel entire factors, not individual terms. To divide rational expressions, take the reciprocal of the divisor (the rational expression you are dividing by) and multiply by the dividend.

Explain the importance of instruction connecting math and science to real life for gifted students, and provide some examples that teachers can use.

Since math and science are so often heavy in calculations and facts, students are usually not taught how the subjects relate to the real world. All students, especially gifted students, need to find meaning in an academic subject and understand how it applies in life. We may not appreciate enough that math and science or not like people whose opinions we can disagree with; they provide hard, objective, unchangeable facts. Teachers can show students the consequences of ignoring facts with examples like these: mathematicians and engineers advised not launching the Challenger space shuttle, but management overruled them, therefore leading to the deadly explosion. Pop singer Aaliyah died in a plane crash after pilot and crew ignored the mathematics indicating airplane overload and flew regardless. A mathematician prove racial bias in jury selection by calculating that the mathematical probability of their selection was approximately 1 in 1,000000,000,000,000.

Discuss skewness. including positively skewed and negatively skewed.

Skewness is a way to describe the symmetry or asymmetry of the distribution of values in a data set. If the distribution of values is symmetrical, there is no skew. In general the closer the mean of a data set is to the median of the data set, the less skew there is. Generally, if the mean is to the right of the median, the data set is positively skewed, or right-skewed, and if the mean is to the left of the median the data set is negatively skewed, or left-skewed. However, this rule of thumb is not infallible. When the data is graphed on a curve, set with no skew will be a perfect bell curve.

Explain how to solve systems of two linear equations by illumination. Solve using elimination: x+6y=15 3x-12y=18

Solve a system of equations using elimination, begin by rewriting both equations in standard form blank. Check to see if the coefficient of one pair of like variables add to zero. If not, multiply one or both of the equations by non-zero number to make one set of like variables add to zero. Add the two equations to solve for one of the variables. Substitute back into either original equation to solve for the other variable. Check your work by substituting into the other equation. Example: solve the system using elimination: x+6y=15 3x-12y=18 If we multiply the first equation by 2, we can eliminate the y terms: 2x+12y=30 3x-12y=18 Add the equations together and solve for x: 5x=48 x=48/5=9.6 Plug value for x back into either of the original equations and solve for y: 9.6+6y=15 y=15-9.6/6=0.9

Explain the complement of an event.

Sometimes it may be easier to calculate the possibility of something not happening, or the complement of an event. Represented by the symbol A (with hyphen over it), the complement of A is the probability that event A does not happen. When you know the probability of event A occurring, you can use the formula P(A) (with hyphen)= 1-P(A), where P(A) is the probability of event A not occurring, and P(A) is the is the probability of event occurring.

Standard Deviation

Standard deviation is a measure of dispersion that compares all the data values in the set to the mean of the set to give a more accurate picture. Use this formula: where σ is the standard deviation of a population, x represents the individual values of the data set, M is the mean of the data values, and n is the number of data values in the set. The higher the value of the standard deviation is, the greater the variance of the data values from the mean. The units associated with the standard deviation are the same as the units of the data values.

Describe symmetry and skew in data sets.

Symmetry is a characteristic of the shape of the plotted data. Specifically, it refers to how well the data on one side of the median mirrors the data on the other side. A skewed data set is one that has a distinctly longer or fatter tail on one side of the peak or the other. A data set that is skewed left has more of its values to the left of the peak, while a set that is skewed right has more of its values to the right of the peak. When actually looking at the graph, these names may seem counterintuitive since, in a left-skewed data set, the bulk of the values seem to be on the right side of the graph, and vice versa. However, if the graph is viewed strictly in relation to the peak, the direction of skewness makes more sense.

describe systems of equations.

System of equations: a set of simultaneous equations that all use the same variables. A solution to a system of equations must be true for each equation in the system. consistent system: a system of equations that has at least one solution. inconsistent system: a system of equations that has no solution. Systems of equations may be solved using one of four methods: substitution, elimination, transformation of the augmented matrix and using the trace feature on a graphing calculator.

Discuss the 68-95-99.7 rule.

The 68-95-99.7 rule describes how a normal distribution of data should appear when compared to the mean. This is also a description of a normal bell curve. According to this rule, 68% of the data values in a normally distributed set should fall within one standard deviation of the mean (34% above and 34% below the mean), 95% of the data values should fall within two standard deviations of the means (47.5% above and 47.5% below the mean), and 99.7% of the data values should fall within three standard deviations of the mean, again, equally distributed in either side of the mean. This means that only 0.3% of all data values should fall more than three standard deviations from the mean.

The McDonald's are taking a family road trip, driving 300 miles to their cabin. It took them two hours to drive the first 120 miles. They will drive at the same speed all the way to their cabin. Find the speed at which the McDonald's are driving and how much longer it will take them to get to their cabin.

The McDonald's are driving 60 mph. This can be found by setting up a proportion to find the unit rate, the number of miles they drive one hour: 120 over two equals X over one. Cross multiplying yields 2X equals 120 and division by two shows the X equals 60. since the McDonald's will drive the same speed, it will take them another three hours to get to their cabin. this can be found by first finding how many miles the McDonald's have left to drive which is 300-120 = 180. The McDonald's are driving at 60 mph, so a proportion can be set up to determine how many hours it will take them to drive 180 miles: 180/x = 60/1. cross multiplying yields 60 X equals 180, and division by 60 shows that X equals three. This can also be found by using the formula D=r•t, where 180 equals 60 times T, and division by 60 shows that t equals three

Explain correlation coefficient.

The correlation coefficient is the numerical value that indicates how strong the relationship is between two variables of a linear regression equation. A correlation coefficient of ⁻1 is a perfect negative correlation. A correlation coefficient of ⁺1 is a perfect positive correlation. Correlation coefficients close to ⁻1 or ⁺1 are very strong correlations. A correlation coefficient equal to zero indicates there is no correlation between the two variables. This test is a good indicator for the line of best fit is accurate. The formula for the correlation coefficient is:

A patient was given pain medicine at a dosage of 0.22 grams. The patient's dosage was then increased to 0.80 grams. By how much was the patient's dosage increased?

The first step is to determine what operation(addition, subtraction, multiplication, or division) the problem requires. Notice the keywords and phrases " by how much" and "increased". "Increased" means that you go from a smaller amount to a larger amount. This change can be found by subtracting the smaller number from the larger number: 0.80 grams-0.22 grams=0.58 grams. Remember to line up the decimal when subtracting. 0.80 -0.22 ------ 0.58

Describe the five general shapes of frquenct curves.

The five general shapes of frequency curves are symmetrical, u-shaped, skewed, j-shaped, and multimodal. Symmetrical curves are also known as bell curves or normal curves. Values equidistant from the median have equal frequencies. U-shaped curves have two maxima- one at each end. Skewed curves have the maximum on the right side of the graph so there is longer tail and lower slope on the left side. The opposite is true for curves that are positive skewed, or right skewed. J-shaped curves have a maximum at one end and a minimum at the other end. Multimodal curves have multiple maxima.

Define main diagonal as it applies to a matrix.

The main diagonal of a matrix is the set of elements on the diagonal from the top left to the bottom right of a matrix. Because of the way it is defined, only square matrices will have a main diagonal. For the matrix shown below the main diagonal consist of the elements n₁₁, n₂₂, n₃₃, n₄₄. [n₁₁ n₁₂ n₁₃ n₁₄] [n₂₁ n₂₂ n₂₃ n₂₄] [n₃₁ n₃₂ n₃₃ n₃₄] [n₄₁ n₄₂ n₄₃ n₄₄] A 3 x 4 matrix such as the one shown below would not have a main diagonal because there is no straight line of elements between the top left corner in the bottom right corner that joins the elements. [n₁₁ n₁₂ n₁₃ n₁₄] [n₂₁ n₂₂ n₂₃ n₂₄] [n₃₁ n₃₂ n₃₃ n₃₄]

Explain the possible disadvantages of using the median or mode as an only measure of central tendency.

The main disadvantage of using the median as a measure of central tendency is that it relies solely on a value's relative size as compared to the other values in the set. When the individual values in a set of data are evenly dispersed, the median can an accurate tool. However, if there is a group of rather large values or a group of rather small values that are not offset by a different group of values, the information that can be inferred from the median may not be accurate because the distribution of values is skewed. The main disadvantage of the mode is that the values of the other data in the set have no bearing on the mode. The mode may be the largest value, the smallest value, or a value anywhere in between the set. The mode only tells which value or values, if any, occurred the most number of times. It does not give any suggestions about the remaining values in the set.

Convert 3 2/5 to decimal and into a percentage.

The mixed number three and two fifths has a whole number in a fractional part the fractional part to this can be written as a decimal by dividing five into two which gives 0.4. adding the whole to the part gives 3.4 alternatively note that three and 2 fifths equals 3 4/10 equals 3.4 to change your decimal to a percentage multiply it by 100. 3.4 equals 340%.notices this percentage is greater than 100% this makes sense because the original mixed number three and two fifths is greater than one

Define range as it applies to statistics.

The range of a set of data is the difference between the greatest and lowest values of the data in the set. To calculate the range identify the highest and lowest values. Use the formula: Range= highest value-lowest value

Define mean as it is applied to statistics.

The statistical mean of a group of data is the same as the arithmetic average of that group. To find the mean add all the values together and count the total number of data values. If a value appears more than once, count it more than once. Divide the sum of the values by the total number of values and apply the units, if any. Use the formula: Mean= sum of the data values/quantity of the data values

Define mode as it applies to statistics.

The statistical mode is the data value that occurs the most number of times in the data set. It is possible to have exactly one mode, more than one mode, or no mode. To find the mode of a set of data, arrange the data like you do to find the median (all values in order, listing multiples of data values). Count the number of times each value appears in the data set. If all values appear an equal number of times, there is no mode. If on value appears more than any other value, that value is the mode. If two or more values appear the same number of times, but there are other values that appear fewer times and no values that appear more times, all of those values are the mode.

A student is asked to find the area of a rectangle measuring 8' x 4'. The student is able to break the rectangle into 32 squares but states that the area is 24 ft.². Identify the student's error and offer an exclamation to help him or her arrive at the correct answer.

The student calculated the perimeter of the rectangle rather than the area. The perimeter of a rectangle is the sum of its sides Here, the rectangle's sides measures 8 feet, 8 feet, 4 feet, and 4 feet, which total to 24 feet. To calculate area, the student must multiply the length by the width (8x4), which equals 32 ft.². To help the student correct his or her error, the teacher can explain that the student was on the right track when he or she broke the rectangle into 32 squares. Because area is the amount of unit squares that can be contained in a two-dimensional figure, the student could have also opt to simply count the unit squares he or she created, which would have also led him on her to the correct answer of 32 ft.².

Describe the difference between theoretical and experimental probability.

Theoretical probability is used to predict the likelihood of an event. Experimental probability expresses the ratio of the number of times a event actually occurs to the number of trials performed in an experiment.

List the four basic mathematical operations and give examples of each.

There are four basic mathematical operations: Addition increases the value of one quantity by the value of another quantity. Example: 2+4=6; 8+9=17. The result is called the sum. With addition, the order does not matter. 4+2=2+4. Subtraction is the opposite operation to addition; it decreases the value of one quantity by the value of another quantity. Example: 6-4=2; 17-8=9. The result is called the difference. Note that with subtraction, the order does matter. 6-4≠4-6. Multiplication can be thought of as repeated addition. One number tells how many times to add the other number to itself. Example: 3x2 (three times two) =2+2+2=6. With multiplication, the order does not matter. 2x3 (or 3+3) =3x2 (2+2+2). Division is the opposite operation to multiplication; one number tells us how many parts to divide the other number into. Example: 20 ÷ 4 = 5; if 20 is split into 4 equal parts, each part is 5. With division, the order of the numbers does matter. 20÷4≠4÷20.

A fish tank measures 5 feet long, 3 feet wide, and 3 feet tall. Each cubic foot of the tank holds 7.48 gallons of water. The fish tank is filled with water, leaving 6 inches of empty space at the top of the tank. How much water, in cubic feet, is in the fish tank?

To calculate the volume of a rectangular prism we would typically multiply the length, width, and height of the prism. However, the prompt tells us that 6 inches (or half the fluid) of the tank is left empty. Given this information, we must use a measurement of 2.5 in place of 3 feet for the height of our tank. To calculate the volume of the portion of the tank filled with water we must multiply 5×3 x2.5, which equals 37.5 ft.³. If each cubic foot holds 7.48 gallons of water, we can multiply 37.5 by 7.48 to determine the number of gallons of water contained in the tank. The product of 37.5 and 7.48 is 280.5. There are 280.5 gallons of water in the fish tank.

Review division of polynomials.

To divide polynomials, begin by arranging the terms of each polynomial in order of one variable. You may arrange in ascending or descending order, but be consistent with both polynomials. To get the first term of the quotient, divide the first term of the dividend by the first term of the divisor. Multiply the first term of the quotient by the entire divisor and subtract that product from the dividend. Repeat for the second and successive terms until you either get a remainder of zero or a remainder whose degree is less than the degree of the divisor. If the quotient has a remainder, write the answer as a mixed expression in this form: Quotient + remainder/divisor.

Jane ate lunch at a local restaurant. She ordered a $4.99 appetizer, a $12.50 entree, and a $1.25 soda. If she wants to tip her server 20%, how much money will she spend in all?

To find the total amount, first find the sum of the items she ordered from the menu and then add 20% of this sum to the total. In other words: $4.99+$12.50+$1.25=$18.74 Then 20% of $18.74 is (20%)($18.74)=(0.20)($18.74)=$3.75. So, the total she spends is the cost of the meal plus the tip or $18.74+$3.75=$22.49. Another way to find this sum is to multiply 120% by the cost of the meal: $18.74(120%)=$18.74(1.20)=$22.49

Explain how to solve a quadratic equation by factoring.

To solve a quadratic equation by factoring, begin by rewriting the equation in standard form, if necessary. Factor the side with the variable then set each of the factors equal to zero and solve the resulting linear equations. Check your answers by substituting the roots you found into the original equation. If, when writing the equation in standard form, you have an equation in the form x²+c=0 or x²-c=0, set x²=-c or x=c and take the square root of c. If c = 0, the only real route is zero. If c is positive, there are two real roots: the positive and negative square root values. If c is negative, there are no real roots because you cannot take the square root of a negative number.

Explain how to solve a quadratic equation by completing the square.

To solve the quadratic equation by completing the square, rewrite the equation so that all terms containing the variable are on the left side of the equal sign, and all the concerts are on the right side of the equal sign. Make sure the coefficient of the square term is 1. If there is a coefficient with the square term, divide each term on both sides of the equal sign by that number. Next, work with the coefficient of the single-variable term. Square half of this coefficient, and add that value to both sides. Now you can factor the left side (the side containing the variable) as the square of a binomial. x²+2ax+a²=c→(x+a²)²=c, where x is the variable, and a and c are constants. Take the square root of both sides and solve for the variable. Substitute the value of the variable in the original problem to check your work.

Name each point on the number line below: A B C D <—I—I—I—I—I—I—I—I—I—I—I—I—> 0 1 2 3

Use the dashed lines on the number line to identify each point. Each dashed line between two whole numbers is 1/4. The line halfway between two numbers is 1/2. 1/4 1&1/2 2 2&3/4 <—I—•—I—I—I—I—•—I—•—I—•—I—> 0 1 2 3

Discuss the importance of evaluating the validity of a mathematical model or argument when analyzing a solution.

When assessing a students proficiency, simply arriving at the correct answer does not validate true mastery of a mathematical concepts. To determine that a student has truly mastered the skill or concept, that student must be able to explain or defend his or her process as well as the solution. By requiring a model or compelling students to defend their answers, teachers can truly assess the validity of each solution and determine whether or not each student has a true grasp on the concept being assessed. A teacher can be confident that a student has truly mastered a concept when that student can describe his or her process, explain the meaning behind his or her solution, and defend why that solution is logical and appropriate.

Discuss converting from smaller units to larger units and from larger units to smaller units of measurement.

When going from a larger unit to a smaller unit, multiply the number of the known amount by the equivalent amount. When going from a smaller unit to a larger unit, divide the number of the known amount by the equivalent amount. Also, you can set up conversion fractions. In these fractions, one fraction is the conversion factor. The other fraction has the unknown amount in the numerator. So, the known value is placed in the denominator. Sometimes the second fraction has the known value from the problem in the numerator, and the unknown in the denominator. Multiply the two fractions to get the converted measurement.

Measurement Conversion Practice Problems a. How many kilometers are in 2 miles? b. How many centimeters are in 5 feet?

a. 2 miles x 1.609 kilometers/1 mile = 3.218 kilometers b. 5 feet x 12 inches/1 foot x 2.54 centimeters/1 inch = 152.4 centimeters

explain how to factor a polynomial.

first, check for a common Manal meal factor. When the greatest comment but no meal factor has been factored out, look for patterns of special products: differences of two squares, the sum or difference of two cubes for binomial factors, or perfect Trinomial squares for trinomial factors. If the factor is a trinomial but not a perfect trinomial square, look for a factor will form such as Xsquared + (a+b)x + ab = (x+a)(x+b) or (ac)Xsquared + (ad+by)x + bd = (ax+b)(cx+d) for factors with four terms, look for groups to factor. Once you have found the factors, write the original polynomial as the product of all the factors. Make sure all of the polynomial factors are prime monomial factors maybe prime or composite. Check your work by multiplying the factors to make sure you get the original polynomial.

order the following rational numbers from greatest to least 0.3, 27%, the square root of 100, 72/9, 1/9, 4.5.

recall that the term rational simply means that the number can be expressed as a ratio or fraction. the set of rational numbers includes integers and decimals. notice that each of the numbers in the problem can be written as a decimal or integer: 27% equals 0.27 The square root of 100 equals 10 72/9 equals eight 1/9 is approximately equal to 0.11 so the answer is the square root of 100, 72/9, 4.5, 0.3, 27%, and 1/9.

explain graphing and the Cartesian coordinate plane

when algebraic functions and equations are shown graphically, they are usually shown on a Cartesian coordinate plane. The Cartesian coordinate plane consists of two number lines placed perpendicular to each other, and intersecting at the zero point, also known as the origin. The horizontal number line is known as the X axis, with positive values to the right of the origin, and negative values to the left of the origin. The vertical number line is shown as the y-axis with positive values about the origin, and negative values below the origin. II. I III. IV any point on the plane can be identified by an ordered pair in the form of (X, Y), called coordinates. The X value of the coordinate is called the abscissa, And the Y value of the coordinate is called the ordinate. The two number lines divide the plane into four quadrants: I, II, III, and IV.

discuss the use of x and Y in functional relationships.

when expressing functional relationships, the variables X and y are typically used. These values are often written as the coordinates (X,y). The X value is the independent variable and the y value is the dependent variable. A relation is a set of data in which there is not a unique y value for each x value in the data set. This means that there can be two of the same x values assigned to different y values. A relation is simply a relationship between the x and Y values in each coordinate but does not apply to the relationship between the values of X and Y in the data set. A function is a relation where one quantity depends on the other. For example, the amount of money that you make depends on the number of hours that you work. In a function, each X value in the data set has one unique Y value because the Y value depends on the X value.