COMMON CORE ACHIEVE Mastering Essential Test Readiness Skills TASC Test Student Supplement

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1 COMMON CORE ACHIEVE Mastering Essential Test Readiness Skills TASC Test Student Supplement MATHEMATICS Bothell, WA Chicago, IL Columbus, OH New York, NY

2 TASC Test Assessing Secondar Completion is a trademark of McGraw-Hill School Education Holdings LLC McGraw-Hill Education is not affiliated with The After-School Corporation, which is known as TASC The After-School Corporation has no affiliation with the Test Assessing Secondar Completion ( TASC test ) offered b McGraw-Hill Education, and has not authorized, sponsored or otherwise approved of an of McGraw-Hill Education s products and services, including TASC test MHEonlinecom Copright McGraw-Hill Education All rights reserved The contents, or parts thereof, ma be reproduced in print form for non-profit educational use with Common Core Achieve, provided such reproductions bear copright notice, but ma not be reproduced in an form for an other purpose without the prior written consent of McGraw-Hill Education, including, but not limited to, network storage or transmission, or broadcast for distance learning Send all inquiries to: McGraw-Hill Education Orion Place Columbus, OH ISBN: ---- MHID: --- Printed in the United States of America ONL

3 Table of Contents To the Student v Number Concepts Imaginar and Comple Numbers Review Probabilit and Statistics Conditional Probabilit Addition Rule of Probabilit Standard Deviation Gathering Data Data Correlation Two-Wa Frequenc Tables Densit Review Functions, Equations, and Inequalities Logarithms Inverse Functions Sequence Functions Linear and Eponential Models Parallel and Perpendicular Lines Linear Inequalities Transformations Even and Odd Functions Review Congruence Points, Lines, and Angles Parallelograms Triangles Triangle Congruence Transformations Review Triangles and Right-Triangle Trigonometr The Pthagorean Theorem The Distance Formula The Midpoint Formula Similarit Trigonometric Ratios Trigonometric Identities Review iii Table of Contents

4 Circles Circle Basics Circle Constructions Radian Measures Arcs and Sectors The Equation of a Circle The Equation of a Parabola Periodic Phenomena Review Posttest Posttest Answer Ke and Check Your Understanding Answer Ke TASC Test Mathematics Reference Sheet iv Table of Contents

5 To the Student Congratulations! If ou are using this book, it means that ou are taking a ke step toward achieving an important new goal for ourself You are preparing to take the TASC Test Assessing Secondar Completion, one of the most important steps in the pathwa toward career, educational, and lifelong well-being and success Common Core Achieve: Mastering Essential Test Readiness Skills is designed to help ou learn or strengthen the skills ou will need when ou take the TASC test The TASC Test Math Student Supplement provides ou with additional instruction and practice of the ke concepts, core skills, and core practices required for success on test da and beond How to Use This Supplement This supplement is designed to support or enhance the lessons in the Core Student Module and Eercise Book Each topic provides instruction and practice on concepts that ma be tested on the TASC test but go beond the foundational instruction in the Core Student Module Understanding these concepts will help ou better prepare for the TASC test and for college-level math courses At the back of this supplement, ou will find the answer ke The answer to each question in the supplement is provided along with a rationale for wh the answer is correct If ou get an answer incorrect, please return to the appropriate page to review the specific concept again The TASC Test Mathematics Reference Sheet is provided after the answer ke and lists all of the formulas that are available during the test About the TASC Test for Mathematics The TASC test for Mathematics assesses across five content categories (approimate percentage of test questions in each categor is shown in parenthesis): Number and Quantit (%), Geometr (%), Statistics and Probabilit (%), Algebra (%), and Functions (%) The test is divided into two sections: Section is minutes and Section is minutes Students are permitted to use a calculator for Section but not for Section Questions are primaril multiple choice; however, there are gridded response items for a small number of questions As ou work through the supplement, ou will practice using the gridded response item format v To the Student

6 Number Concepts Use with Lesson OBJECTIVES Identif and write imaginar and comple numbers Simplif epressions containing imaginar and comple numbers Use the imaginar unit i and properties of real numbers to add, subtract, and multipl comple numbers Solve quadratic equations that have comple solutions CORE SKILLS & PRACTICES Use Math Tools Appropriatel Solve Quadratic Equations Ke Terms comple number the sum of a real number and an imaginar number imaginar number a value whose square is negative real number an number that can be plotted on a number line Vocabular simplif to rewrite in a different form that is less complicated solve to find the values that make an equation true Ke Concept All numbers can be written as comple numbers in the form a + bi, where a and b are real numbers and i is an imaginar number equal to the square root of Operations can be performed on imaginar and comple numbers, and the can be used to epress solutions to quadratic equations Imaginar and Comple Numbers In the beginning, people onl used numbers to count So the onl needed natural numbers Then someone invented zero, and that led to the question, How do we show values that are less than zero? In response, people invented negative integers As new problems and ideas developed, so did new kinds of numbers fractions, decimals, irrational numbers Finall, people needed a wa to solve problems that involved square roots of negative numbers There was onl one thing to do create imaginar and comple numbers! Imaginar Numbers An number that is not a real number is an imaginar number The main wa to know if a number is real or imaginar is to square it If squaring gives ou a negative number, ou are dealing with an imaginar number Real Numbers: an number whose square is positive Eamples: π Imaginar Numbers: an number whose square is negative Eamples: = i For eample, - is imaginar because it means that some number times itself equals - The square root of - is the basic imaginar unit, i i = = i You can use the basic imaginar unit to simplif, or rewrite in a different form, epressions with negative square roots This means ou can use i to rewrite the epression so that it does not have a negative number inside the radical To simplif, appl the Multiplication Rule: a b = ab Eample : Simplif Epressions with i Simplif Step Appl the Multiplication Rule to factor out Step Simplif the positive square root Step Replace with i Solution: = i = = = = i TASC Test Math Student Supplement Number Concepts

7 CORE PRACTICE Use Math Tools Appropriatel Real, imaginar, and comple numbers are three sets of numbers A Venn diagram is a good tool for showing and analzing the relationships between sets This Venn diagram shows the relationships between eight sets of numbers Make a larger cop of the diagram and write at least one eample of each kind of number inside its oval or rectangle Then describe at least three relationships that our completed diagram shows Comple Numbers Imaginar Numbers Real Numbers Irrational Numbers Rational Numbers Integers Whole Numbers Natural Numbers Powers of i You can also use the imaginar unit and rules for eponents to simplif epressions with powers of i Look at the chart Notice that the values of the power of i repeat after i The repeat in this pattern: i,, i, The pattern shows that ever power of i that is a multiple of, such as i, i, and i, is equal to Eample : Simplif Powers of i Simplif i Step Divide the eponent b = R Step Write an equivalent power of i i = i i = (i ) i = i = i Step Simplif the new epression i = Solution: i = Comple Numbers When ou add an imaginar number to a real number, ou get a comple number An comple number can be written in the form a + bi, where a and b are real numbers, and i is the imaginar unit equal to a real part + bi imaginar part Eample : Eamples of Comple Numbers An number can be written as a comple number + i - πi - i i + i i = i i = i i = = i = i i = i = i i = i i = ( ) = i = i i = i = i i = i i = ( ) = i = i i = ( i) = i i = i i = = + + i Look at the two eamples in the last column The show that either the real or the imaginar part of a comple number can equal You can write i as + ( i), and ou can write as + i Even can be written as a comple number: = + i Operations with Comple Numbers You can add, subtract, and multipl comple numbers The trick is to treat i like a variable But alwas remember that i is not a variable Its value is alwas TASC Test Math Student Supplement Number Concepts

8 Eample : Add Comple Numbers Add ( + i) + ( - i) Step Group the real values and imaginar values ( + ) + (i + ( i)) Step Add the real values + = Step Add the imaginar values i + ( i) = ( + ( ))i = i Step Combine the two sums Solution: ( + i) + ( - i) = - i Subtracting comple numbers is just like adding them The onl difference is that ou first need to distribute the minus sign to the subtracted comple number Eample : Subtract Comple Numbers Subtract ( - i) - ( - i) Step Distribute the minus sign ( -i) - ( - i) = - i - + i Step Group the real and imaginar parts - i - + i = ( - ) + ( i + i) Step Add the real and imaginar parts separatel and simplif ( - ) + ( i + i) = + ( i) = - i Solution: ( - i) - ( - i) = - i Comple numbers are epressions with two terms So ou can use FOIL to multipl them Recall that FOIL is a wa to appl the Distributive Propert to multipl epressions with two terms The letters in FOIL stand for the order to multipl pairs of terms: First, Outer, Inner, and Last FOIL: (a + b)(c + d) = ac + ad + bc + bd Eample : Multipl Comple Numbers Multipl ( + i)( - i) Step Use FOIL to multipl the epressions ( + i)( - i) = ( ) + ( i) + (i ) + (i i) = + i + i + ( i ) = + i + ( )( i ) Step Substitute for i and simplif + i + ( )( i ) = + i + ( )( ) = + i + = + i Solution: ( + i)( - i) = + i Comple Solutions of Quadratic Equations When ou solve a quadratic equation in the form a + b + c =, ou find the value(s) of that make the equation equal Sometimes ou get a solution with a negative square root When that is the case, the equation has no real solutions It has comple solutions TASC Test Math Student Supplement Number Concepts

9 CORE SKILL Solve Quadratic Equations When ou solve a quadratic equation, one wa to check our solution is to graph the equation and identif the -intercepts The -coordinates of those intercepts are the solutions But this graphing method does not work when the solutions are comple numbers Wh is that so? Graph quadratic equations w + w + = and - + = Use the graphs to eplain wh ou cannot use the graphing method to check our solutions to such equations What other solving method(s) could ou use to check? Eample : Use the Quadratic Formula Solve - + = Step Identif the coefficients and constants for a + b + c = a = b = c = Step Substitute the values into the quadratic formula and simplif = b ± b - ac a ( ) ± ( ) - ()() = () = ± = ± = ± i = ± i Solutions: = + i or = - i You can also use the completing the square method to find comple solutions Eample : Solve b Completing the Square Solve w + w + = Step Isolate the terms with variables b subtracting from both sides Step Divide the coefficient of the -term b and square the result Step Add the result to both sides to complete the square Step Write the left side of the equation as a square Step Take the square root of both sides Step Solve for w and simplif with i Solutions: w = i or w = + i w + w = = ; = w + w + = + (w + ) = w + = ± w = ± i Think about Math Directions: Answer the following questions What is the product of i( - i)? A + i B i C i D + i What is the solution to the equation in simplest form? - + = A ± i B ± i C ± D ± TASC Test Math Student Supplement Number Concepts

10 Vocabular Review Directions: Write the missing term in the blank comple number imaginar number real number When ou square a(n), the result is a positive number A(n) is the sum of a real and an imaginar number When ou square a(n), the result is a negative number Skill Review Directions: Read each problem and complete the task Which of these shows - written as a comple number in the form a + bi? A + B + i C + i D + i Simplif: A i B i C D i What is the product of - + i and - i? A - i B - i C + i D + i Consider this equation - + = What values of are solutions to the equation? A = + i and = - i B = + i and = - i C = + i and = - i D = + i and = - i Skill Practice Directions: Read each problem and complete the task What is the value of + i written as an integer? Please enter our response in the grid / / / Consider this epression ( + i) + ( - i) What is the sum? A + i B + i C i D + i Consider this equation - + = What values of are solutions to the equation? A = + i and = - i B = + i and = - i C = + i and = - i D = + i and = - i Consider this epression ( - i) - ( + i) What is the difference? A - i B + i C + i D - i TASC Test Math Student Supplement Number Concepts

11 Probabilit and Statistics Use with Lessons,,, and OBJECTIVES Find the conditional probabilit of events Appl the addition rule to calculate the probabilit of events Calculate and analze the standard deviation of a data set Compare methods and best practices for gathering data Distinguish between correlation and causation Construct and interpret two-wa frequenc tables Appl concepts of densit CORE SKILLS & PRACTICES Analze Events and Ideas Attend to Precision Appl Number Sense Concepts Ke Terms conditional probabilit the chances that one event will happen given that another event has happened correlation a measure of how much sets of data values are related densit the measure of how crowded, full, or solid something is mutuall eclusive when two or more events cannot occur at the same time relative frequenc how often something happens as compared to all the possible outcomes standard deviation a measure of variation that tells how spread out the data values in a set are from the mean Vocabular analze to eamine carefull in order to identif essential features, causes, and relationships among parts Ke Concept Probabilit can be used to predict the likelihood of events happening Correctl gathering, displaing, and analzing data can solve problems and lead to valid conclusions Conditional Probabilit You know how to find the probabilit of a simple event, such as picking a face card from a group of four cards You also know how to find the probabilit of a compound event, such as picking two face cards at the same time But what if ou want to know the probabilit of picking a face card after ou alread picked a face card? That s when ou need to find a conditional probabilit Conditional Probabilit Formula Use the notation P(B A) to show conditional probabilit It means the probabilit that event B will happen, given the knowledge that event A has alread happened You can use a formula to calculate a conditional probabilit The formula shows that the probabilit of B given A is a fraction of A s outcomes probabilit of A and B together P(B A) = probabilit of A P(A and B) P(B A) = P(A) Eample : Use the Formula You randoml draw a card from the four cards shown Without replacing that card, ou randoml draw another card What is the probabilit that our second card is a face card, given that the first card was a face card? Step Define events A and B Event A: picking a face card on the first draw Event B: picking a face card on the second draw Step Find P(A) ( P(A) ( ) = face cards = cards = Step Find P(A( and B) P(A( and B) = P(A) ( ) P(B) ( ) = = = Step Substitute the probabilities into the conditional probabilit formula P(B( A) = P(A ( and B) P(A) ( ) P(B( A) = = = = TASC Test Math Student Supplement Probabilit and Statistics

12 CORE SKILL Analze Events and Ideas Here are the rules to find the probabilit of compound events: If A and B are independent, P(A and B) = P(A) P(B) If A and B are dependent, P(A and B) = P(A) P(B A) You can use conditional probabilit to determine whether two events are independent If P(A B) = P(A), then A is independent of B If P(B A) = P(B), then B is independent of A Eample: In a surve, it is found that % of respondents drink coffee, % own a laptop, and % drink coffee and own a laptop Event A: Respondent drinks coffee Event B: Respondent owns a laptop P(A and B) P(A B) = P(B) = = = P(A) P(A and B) P(B A) = P(A) = = = P(B) Conclusion: Events A and B are statisticall independent Use conditional probabilit to show if the following pairs of events are independent You toss a fair coin twice Event A: First toss is heads Event B: Second toss is heads You toss a fair coin times Event A: First toss is heads Event B: Eactl two of the tosses are heads Conditional Probabilit with Percents In some conditional probabilit problems, the probabilities of A and B occurring together and the probabilit of A or B happening alone are given in percents In those cases, ou just need to substitute the percents into the correct parts of the formula Eample : Use Given Percents At ABC Foods Compan, % of all emploees are men and % of all the emploees are men who have a college degree If ou randoml select a male emploee at ABC Foods, what is the probabilit that he has a college degree? Step Define events A and B Event A: The emploee is a man Event B: The emploee has a college degree Step Substitute the given probabilities into the conditional probabilit formula P(B( A) = P(A ( and B) = P(man and college degree) = P(A) ( ) P(man) % % Step Write the percents as decimals, divide, and then write the result as a percent P(B( A) = % % = = = % Think about Math Directions: Answer the following questions A number cube with faces labeled - is rolled twice What is the probabilit that it lands on an odd number for the second roll, given that the first roll lands on a number less than? A B C D Addition Rule of Probabilit In Jefferson College, % of the students take Spanish and French, while % of the students take Spanish If ou randoml select a student in the school who takes Spanish, what is the approimate probabilit that the student also takes French? A % B % C % D % You know how to find the compound probabilit of two events happening together, P(A and B) You can also find the compound probabilit of either event happening, P(A or B) To do so, ou can use a formula called the general addition rule General Addition Rule P(A or B) P(A) P(B) P(A and B) A and B ma be independent or dependent events TASC Test Math Student Supplement Probabilit and Statistics

13 This rule works whether or not the events are mutuall eclusive Two events are mutuall eclusive when it is impossible for them to happen together Eample : Events that Are Not Mutuall Eclusive A number cube with faces labeled is rolled What is the probabilit that a multiple of or an even number is rolled? Step Define events A and B Event A: a multiple of is rolled Event B: an even number is rolled A roll of is both a multiple of and an even number In other words, events A and B can happen together So events A and B are not mutuall eclusive Step Find the probabilit that each event happens separatel, and the probabilit that both happen together P(A) = P(B) = P(A and B) = Step Substitute the probabilities into the general addition rule formula and simplif P(A or B) = P(A) + P(B) - P(A and B) = + - = = Eample : Mutuall Eclusive Events Sheila is sick She estimates that there is a % chance she will go to work all da tomorrow, and a % chance that she will sleep all da tomorrow What is the probabilit that Sheila will go to work or sleep all da tomorrow? Step Define events A and B Event A: Sheila goes to work all da tomorrow Event B: Sheila sleeps all da tomorrow Step Find the probabilit that each event happens separatel, and the probabilit that both happen together P(A) = P(B) = P(A and B) = Both events can t happen at once she can t work and sleep all da So the probabilit of A and B together is Step Substitute the probabilities into the general addition rule formula and simplif P(A or B) = P(A) + P(B) - P(A and B) = + - = The third term in the general addition rule is alwas equal to for mutuall eclusive events So, ou can drop the third term in the formula to write the simplified addition rule for mutuall eclusive events TASC Test Math Student Supplement Probabilit and Statistics

14 CALCULATOR SKILL You can use the TI-XS Multiview calculator to compute the standard deviation of a data set Step Press data and enter all the data values in L NOTE: To input a data value, tpe the value and press enter Use the scroll arrow to move between data values Step Press nd data Select : -Var Stats For the Data option, select L For the FRQ option, select one since this data set does not include a frequenc list Select Calc for calculate Scroll down to : σ This is the standard deviation! Use a TI-XS Multiview calculator to find the standard deviation of this data set:,,,,,,, Think about Math Directions: Answer the following questions Emil selects a card from a standard deck of plaing cards What is the probabilit that the card is a heart or a face card (king, queen, or jack)? A B C D Standard Deviation A bag contains black marbles, red marbles, blue marbles, and white marbles You randoml pick a marble from the bag What is the probabilit that it is black or white? A B C D People often determine probabilities based on statistics For eample, sports journalists can find the probabilit of a team winning a game based on data collected from previous games Or teachers might analze probabilities of getting certain test scores In that case, the teacher might want to find the standard deviation of all the test scores This will show how close the scores are to the average score Calculate Standard Deviation Standard deviation is a measure of variation It describes how much the data are spread awa from their mean The standard deviation is calculated b finding the square root of the averages of the squared differences of the data values from the mean The smbol for standard deviation is σ standard deviation average of squared differences from the mean Eample : Find Standard Deviation The following scores were recorded for a test:,,, and What is the standard deviation of the data set? Step Find the mean of the scores mean = = = Step Find the difference of each σ = ( - ) + ( - ) + ( - ) + ( - ) score from the mean Step Square each σ = ( ) + ( ) + ( ) + ( ) difference Step Find the mean σ = of the resulting squares Step Take the square root of this mean σ = σ = Solution: The standard deviation of the scores is approimatel TASC Test Math Student Supplement Probabilit and Statistics

15 CORE SKILL Interpret Data Displas Comparing the mean and standard deviation of two or more data sets is a good wa to compare what is tpical about the sets and how much the var But ou should also consider the shape of the data distribution in our comparisons These dot plots show data collected for two classrooms showing how man pets each student has Pet Ownership Class A Pet Ownership Distributions In a normal distribution, data values are spread out evenl around the mean For this reason, a normal distribution is sometimes called smmetric % mean % % % % % Standard Deviations % of the data values are within standard deviation of the mean % of the data values are within standard deviations of the mean % of the data values are within standard deviations of the mean Eample : Analze Distributions Recall that the test scores are,,, and, and the mean score is Do the test scores have a normal distribution? Step Add and subtract standard deviation to and from the mean mean + σ = + = mean - σ = - = Step Compare this range to % of the data values If the test scores had a normal distribution, % of the data values would be between and, but the are not Onl % of the scores are in that range So the test scores do not have a normal distribution Class B The mean for Class A is and the standard deviation is The mean for Class B is and the standard deviation is Use these data measures and the shape of the two dot plots to compare the two data sets Remember to consider the effects of outliers Of course, this eample analzes a ver small data set onl four values Standard deviation is best used to analze larger data sets, such as all the test scores for an entire grade or ear Think about Math Directions: Use the data set below to answer the following questions Miles Jogged Each Da This Week:,,,,,, What is the standard deviation of the data set? A B C D Which fact would best show that the data set has a normal distribution? A % of the das were between and miles B % of the das were between and miles C % of the das were between and miles D % of the das were between and miles TASC Test Math Student Supplement Probabilit and Statistics

16 Gathering Data Man decisions that affect our life are based on the analsis of the mean, standard deviation, and other measures of data sets For eample, the number of teachers in a school, the price of movie tickets, and even the products available in stores are all results of such data analsis For this reason, the wa that data are collected is etremel important Three Methods This chart shows the three most common methods of collecting data All three methods use a sample of the studied population Surve Method Eperiment Observational Stud Description Ask the same question(s) of each person in the sample Conduct trials Change a sample and compare to the original Create groups from the sample, change all but one group, and compare Gather information without changing or influencing the sample Eample : Compare Collection Methods You want to stud how taking an SAT prep class affects scores How could ou collect our sample data? In a surve, ou might ask a sample of students in an SAT prep class what their scores were before and after taking the class In an eperiment, ou might have onl half of the sample take an SAT prep class and compare their before and after scores to the half that did not take the SAT prep class In an observational stud, ou might just collect scores for a sample of students taking an SAT prep class Best Practices Regardless of which data collection method ou use, ou need to follow certain best practices These practices ensure that the data ou collect is valid and can be used to draw valid conclusions You must make ever effort to avoid bias This means avoiding an sstematic favoritism, such as leading surve questions or choosing our sample from a limited part of our population In general, randomness helps prevent bias TASC Test Math Student Supplement Probabilit and Statistics

17 Eample : Use Best Practices The chart below lists some was ou can appl best practices when ou gather data using an of the three methods Best Practices for Data Collection Randomness Ever member of population is equall likel to be in sample Members of sample are equall likel to be assigned to subgroups Size Use the largest sample possible Conduct the largest number of trials possible Bias Avoid leading questions or answer choices Avoid allowing subgroups to know in which group the are placed As an observer, be aware of our own biases Simulations Randomness is also important when ou perform a simulation A simulation is a wa to model random events It is a common method used to collect probabilit data and to decide if a model is consistent with results Eample : Perform a Simulation A board game for two plaers comes with a coin to decide who goes first One plaer flips the coin If it lands on heads, that plaer goes first If it lands on tails, the other plaer goes first For the first five games plaed, the coin landed on tails Is the coin a fair coin? Step Describe all possible outcomes The random event is tossing a coin There are two possible outcomes for each toss: heads or tails Assumption: The coin is fair, so each outcome is equall likel to occur Step Assign random digits to each outcome = tails = heads Step Use a calculator, spreadsheet, or other random number generator to get a group of random digits to model the coin tosses Each group of random digits is a trial Note the simulated outcome The first trial did not get digits, so it did not simulate tails in a row Trial Outcome Result Unsuccessful Step Repeat step man times Number of successful trials = Number of unsuccessful trials = TASC Test Math Student Supplement Probabilit and Statistics

18 CORE PRACTICE Attend to Precision In most cases, it is impossible to gather data for an entire population So a random sample is used to draw inferences about the population This is not a perfect process A certain amount of error is bound to occur simpl because a sample is used For this reason, surve results often include a description of the margin of error the maimum amount b which the sample results are epected to differ from those of the actual population In most cases, the results will also include a confidence level This tells ou how sure ou can be about the margin of error Most researchers use a % confidence level Eample: According to a recent surve, % of voters plan to vote for Candidate X, with a margin of error of % calculated for a confidence level of % This means that ou can be % confident that between % ( - ) and % ( + ) of all voters (population) will vote for Candidate X Eplain wh ou think it is important to include a margin of error in surve results Step Calculate the simulated probabilit = = % This suggests that there is a % chance of getting tails in a row with a fair coin Step Interpret the results The simulated data shows that it is ver unlikel to get tails in a row with a fair coin So ou can conclude that the board game s coin is not fair The theoretical probabilit of getting tails in a row with a fair coin is = = % At %, the simulated probabilit was slightl lower, but still close If ou had generated more than groups of digits (conducted more simulated trials), the two probabilities would likel be much closer Think about Math Directions: Use the information below to answer the following questions Out of students in her school, Maria randoml selected during lunch period and asked them this question: What is our opinion of our school principal? o Prett good o Great o Awesome Which data collection method did Maria use? A eperiment B observational stud C simulation D surve What is the biggest error in Maria s collection methods? A She did not ask a single question B She did not use a large enough sample C The question was biased to a positive response D Onl asking students during lunch period biased the sample Data Correlation People can make mistakes when the gather data The can also make mistakes when the analze data One common eample is confusing correlation with causation There ma seem to be a relationship between two variables, but this does not prove that changes in one variable cause changes in the other variable TASC Test Math Student Supplement Probabilit and Statistics

19 CALCULATOR SKILL You can use a number called a correlation coefficient to determine how strong a correlation is between two variables The correlation coefficient ranges between and = perfect negative correlation = no correlation = perfect positive correlation Follow these steps to use the TI-XS Multiview calculator to compute the correlation coefficient of the two-variable data set below Height (in) Weight (lb) Step Press data Enter all the -values in L Enter all the -values in L Press nd quit mode to eit screen Step Press nd data Select : -Var Stats Select L for -data and L for -data Select Calc for calculate Scroll down to F: r = This is the correlation coefficient! Eplain what this correlation coefficient tells ou about the relationship between height and weight Eample : Identif Correlation Recall that a scatterplot is a graph that shows the relationship between two data variables The relationship is called correlation The closer the data points cluster together in a pattern, the stronger the correlation is between the two variables Positive Correlation As increases, increases Negative Correlation As increases, decreases No Correlation A change in does not seem to affect Eample : Correlation vs Causation This scatterplot shows the relationship between the grades students received on a test and the number of hours the spent plaing video games the week before the test Video Games vs Grades Time Spent Plaing Video Games (hrs/wk) Grade (%) Based on the scatterplot, the teacher concludes that plaing more video games causes students to get lower grades on the test But this is not necessaril true Man other factors could be involved It might be that students did not stud enough Or mabe the teacher did not eplain the material well Or the test room could have been too hot or too cold Correct Conclusions: There is a strong negative correlation between hours spent plaing video games and scores on the test As video gameplaing hours increase, test scores tend to decrease So, plaing more hours seems to have a negative effect on test scores TASC Test Math Student Supplement Probabilit and Statistics

20 Think about Math Directions: Use the scatterplot to answer the following questions Dail Cold Medicine Sales ($) Ice Cream Sales vs Cold Medicine Sales Dail Ice Cream Sales ($) Which of the following best describes the trend in the data? A positive correlation B negative correlation C no correlation D weak correlation Based on the scatterplot, which is the best conclusion? A Buing more ice cream causes people to bu less cold medicine B Buing less cold medicine causes people to bu ice cream C As ice cream sales increase, cold medicine sales tend to decrease D As cold medicine sales increase, ice cream sales tend to increase Two-Wa Frequenc Tables Another wa to displa relationships between data sets is a frequenc table Such tables are useful for analzing data that can be divided into categories For eample, a pizzeria owner might analze the number of pizzas she sold in two categories tpe and size nd Categor st Categor Small Medium Large Totals Cheese Pepperoni Pepper and Onion Totals The rows of the table indicate how man of each tpe of pizza cheese, pepperoni, or pepper and onions were sold The columns show how man pizzas of each size small, medium, or large were sold You can add the rows and add the columns to find the totals for each categor For eample, the owner sold pepperoni pizzas and large pizzas The total number of pizzas sold can be found b adding the row totals across or adding the column totals down The sums will be the same, You can also find the total b adding all the numbers inside the table: = TASC Test Math Student Supplement Probabilit and Statistics

21 Relative Frequencies A two-wa frequenc table can show relative frequencies A relative frequenc is the ratio of an event occurring compared to the total number of events You can convert a two-wa frequenc table to a two-wa relative frequenc table b writing each value as a ratio compared to the total number of pizzas sold, Then write each ratio as a percent Small Medium Large Totals Cheese % % % % Pepperoni % % % % Pepper and Onion % % % % Totals % % % % You can use the relative frequenc table to solve problems For eample, ou can use the table to find marginal frequencies The percents in the totals row and totals column show marginal frequencies The represent data for onl one categor Eample : Find Marginal Frequencies What percent of all the pizzas sold were large? % Small Medium Large Totals Cheese % % % % Pepperoni % % % % Pepper and Onion % % % % Totals % % % % The percents in the middle of the table show joint frequencies The represent data for two categories together Eample : Find Joint Frequencies What percent of all the pizzas sold were large and pepperoni? % Small Medium Large Totals Cheese % % % % Pepperoni % % % % Pepper and Onion % % % % Totals % % % % You can also use the table to calculate conditional frequencies A conditional frequenc is the ratio of a joint relative frequenc compared to a related marginal relative frequenc In other words, conditional frequenc can be seen as the frequenc, or percentage, of a subset of a larger categor TASC Test Math Student Supplement Probabilit and Statistics

22 Eample : Find Conditional Frequencies What percent of all the large pizzas sold were pepperoni? Small Medium Large Totals Cheese % % % % Pepperoni % % % % Pepper and Onion % % % % Totals % % % % large and pepperoni pepperoni, given large = large = % % = % Think about Math Directions: Use the pizza relative frequenc table above to answer the following questions What percent of all the pizzas sold were medium and cheese? A % B % C % D % What percent of all the small pizzas sold were pepper and onion? A % B % C % D % Densit When ou stud data, ou often want to compare one measure to another A common eample of such comparisons is densit Densit measures how solid, full, or crowded something or someplace is It compares a measure of what is inside something or someplace to the size of that thing or place You can appl densit to area the amount of space inside a two-dimensional figure or surface For eample, population densit tells the average number of people or other living things in an area of a certain size Eample : Appl Densit to Area The table shows data collected about three states What is the population densit of California? State Area (square miles) Population, Alaska,, California,,, Teas,,, Step Find the data ou need California population:,, people California area:, square miles TASC Test Math Student Supplement Probabilit and Statistics

23 CORE SKILL Appl Number Sense Concepts Densit is a ratio a comparison of two values So it is measured in derived units, such as people per square mile and grams per cubic centimeter These diagrams show the number of plants in two identical gardens and the number of balls in two identical boes Find and compare their densities Can ou compare the densit of either garden to either bo? Eplain Based on these diagrams, what would be an inappropriate densit measurement for the garden or bo? m m m m m m Step Write the data values into the population densit formula and simplif population count population densit = area measure,, people =, square miles people/m i This densit does not mean that people live on ever square mile of California But if ou took all the people in the state and spread them out equall, that is about how man would be in each square mile You can also appl densit to volume the amount of space inside a threedimensional figure Eample : Appl Densit to Volume A silver bar is in the shape of a rectangular prism that is centimeters long, centimeters wide, and centimeters tall It has a mass of grams What is the densit of the silver bar? Step Find the volume of the bar volume = length width height = = cubic centimeters Step Write the values in the densit formula and simplif densit = mass volume = g = g/c m c m m m m m Think about Math Directions: Use the data table in Eample to answer question What is the population densit of Alaska? A people per square mile B people per square mile C people per square mile D people per square mile An ice cube measuring centimeters on each side has a densit of grams per milliliter What is the mass of the ice cube? ( c m = ml) A grams B grams C grams D grams TASC Test Math Student Supplement Probabilit and Statistics

24 Vocabular Review Directions: Match each term to its definition Terms correlation densit mutuall eclusive relative frequenc standard deviation conditional probabilit Definitions A a measure of how much sets of data values are related B the chances that one event will happen given that another event has happened C how often something happens as compared to all the possible outcomes D a measure of variation that tells how spread out the data values in a set are from the mean E the measure of how crowded or full a certain area or volume is F when two or more events cannot occur at the same time Skill Review Directions: Read each problem and complete the task Two number cubes are each numbered You roll one cube and then the other What is the probabilit of getting a on the second roll given that ou get an even number on the first roll? A B C D In which two data gathering methods do ou conduct trials? A surves and eperiments B eperiments and observational studies C observational studies and simulations D simulations and eperiments When a data set has a normal distribution, what percent of the data values in the set should be within standard deviation of the mean? A % B % C % D % A block of iron shaped like a rectangular prism is centimeters long, centimeters wide, and centimeters tall and has a mass of grams What is the densit of the block of iron, in grams per cubic centimeter? Round our answer to the nearest tenth Please enter our response in the grid / / / TASC Test Math Student Supplement Probabilit and Statistics

25 Which conclusion is best supported b the scatterplot below? Running Speed (mph) Running Speed of Children Age (ears) A Getting older causes children to run faster B As children get older, the tend to run faster C There is no relationship between a child s age and running speed D All children aged run slower than miles per hour Skill Practice Directions: Read each problem and complete the task Consider the data set,,,,,,,,, What is the standard deviation? A B C D There are four classrooms, each in the shape of a rectangle Each classroom will have students and teacher Which classroom will have the greatest densit? A Room A is feet long and feet wide B Room B is feet long and feet wide C Room C is feet long and feet wide D Room D is feet long and feet wide Directions: Use the two-wa frequenc table to answer questions and Movie Theater Audience Males Females Total Adults Children Total Approimatel what percent of the female audience members were children? A % B % C % D % If ou randoml selected an audience member, approimatel what is the probabilit that the person would be male and adult? A % B % C % D % There are blue, ellow, green, and red marbles in a bag Marbles are selected at random with replacement What is the probabilit of selecting a red marble or a green marble? A % B % C % D % Which of the following would most likel bias data collected in an eperiment? A Using a simulation in the trials B Randoml selecting test subjects C Conducting a large number of trials D Allowing test subjects to know in which group the are placed TASC Test Math Student Supplement Probabilit and Statistics

26 Directions: The two-wa frequenc table below shows the responses to a surve of several high schools Each respondent identified how he or she prefers to read books, in paper formats or digital formats Use the two-wa frequenc table to answer questions and High School Reading Preferences If ou randoml selected a surve respondent who preferred digital formats, what is the probabilit that the person would be a teacher? A About % B About % C About % D About % Students Teachers Total Digital Paper Total Which statement is an accurate interpretation of the data? A About % of the surveed students prefer paper formats B About % of those surveed were students who prefer digital formats C About % of the surveed teachers prefer digital formats D About % of the surveed teachers prefer paper formats TASC Test Math Student Supplement Probabilit and Statistics

27 Functions, Equations, and Inequalities Use with Lessons,, and OBJECTIVES Epress eponential models as logarithms, and use to solve problems Find inverse functions Construct linear and eponential functions, including arithmetic and geometric sequences Distinguish between situations that can be modeled with linear functions and with eponential functions Use slope to identif parallel and perpendicular lines and solve geometric problems Graph the solutions to inequalities and sstems of linear inequalities Identif the effect on the graph of replacing f() b f() + k, k f(), f(k), and f( + k) for specific values of k Identif even and odd functions from their graphs and equations CORE SKILLS & PRACTICES Interpret Graphs and Functions Appl Number Sense Concepts Model with Mathematics Make Sense of Problems Ke Terms inverse function a function that reverses another function logarithm an epression that shows the result of multipling a number b itself sequence an ordered set of numbers transformation a change to the equation of a function that changes its graph Vocabular graph to create a visual model of a function on a coordinate plane Ke Concept Functions define the relationship between two values in the real world Equations and inequalities model that relationship, and graphs model the equations and inequalities Changing the equation or inequalit changes the graph and vice versa Logarithms Most banks offer savings accounts that earn compound interest If ou know the principal (the amount deposited) and the interest rate, ou can use eponents to find the account balance at an time If ou know the balance and want to know the interest rate, ou can use a logarithm Logarithmic Notation An eponent tells how man times to multipl a number b itself A logarithm does the opposite It tells the result of multipling a number b itself Eponents and logarithms are inverses of each other As a result, an equation written with eponential notation can also be written with logarithmic notation, as long as the bases are positive and not equal to If no base is written, the base is Eponential Notation = Question: What is the product when ou multipl b itself times? Answer: Logarithmic Notation log = Question: How man times do ou need to multipl b itself to get a product of? Answer: The equation = is equivalent to the equation log = Recall that ou can read the eponential equation as base raised to the rd power is You can read the logarithmic equation as log base of is When writing an equation in logarithmic notation, use the base of the power as the base of the logarithmic epression, and use the eponent as the logarithm = log = You can use eponential notation to find the value of a logarithmic epression Eample : Find the Value of a Logarithmic Epression What is the value of if log =? Step Rewrite the equation in eponential form = Step Solve the equation Ask ourself: raised to what power equals? How man times do I need to multipl b itself to get? Solution: Because =, log = TASC Test Math Student Supplement Functions, Equations, and Inequalities

28 CALCULATOR SKILL The log button on the TI-XS Multiview calculator onl evaluates logarithms with base In order to evaluate logarithms of other bases, ou have to appl the change-ofbase formula log a = log log a Eample: Evaluate log log ) Logarithmic Graphs When ou graph an equation, ou use a set of points on a coordinate plane to displa the values that make the equation true To graph = log b, make a table for and values, and then plot each pair as a point on a coordinate plane The equation = log looks like this Here are some facts about the graphs of all logarithmic functions: The graph alwas crosses the ais at (, ) The domain is all positive real numbers The range is all real numbers The graph gets ver close to the -ais, but it never touches or crosses it O log ) enter log log = = log Use our calculator to find log and eplain what our result means Eample : Graph a Logarithmic Function Graph the equation = log Step Make a table of values for the function log = because = log = because = log = because = log = because = You can also use our calculator to find the -values Step Plot the points (, ) and connect them with a smooth, continuous curve Recall that the graph gets ver close to the -ais, but it never touches or crosses it O Think about Math Directions: Answer the following questions Which eponential equation is equivalent to log? A = B = C = D = Which logarithmic equation does this graph model? O A = log B = log C = log D = log TASC Test Math Student Supplement Functions, Equations, and Inequalities

29 CORE SKILL Interpret Graphs and Functions Comparing the graphs of eponential and logarithmic functions helps ou understand wh the are inverse functions Look at the graphs of the functions = and = log = Inverse Functions You just saw that eponents and logarithms are inverses the undo each other The same is true for eponential and logarithmic functions The output of an eponential function becomes the input of its inverse, a logarithmic function This is true for all inverse functions An inverse function is a function that undoes what another function does Ever one-to-one function f() has an inverse function f () Read f () as f inverse of The inverse does the opposite of the function It undoes whatever the original function f() did to the input For eample, if f() adds something to the input, f () will subtract it To find the inverse of a function, switch the domain and the range O Eample : Find the Inverse Function What is the inverse of f() =? Step Replace f() with the variable Step Switch and in the resulting equation = = O = log Step Solve the new equation for Step Replace with f () Solution: The inverse of f() = is f () = = = f () The two graphs are a reflection of each other over the line = Each point (, ) for = becomes (, ) for log The inputs and outputs are switched in the two functions, so the are inverse functions Because the inputs and outputs of a function are switched in its inverse, the graph of a function and the graph of its inverse are reflections over the line = = f() = f () = O O Choose one of the Think about Math questions on the net page Use graphs to show that the two equations model inverse functions Think about Math Directions: Answer the following questions What is the equation for the For which function is f () = - inverse of f() = -? the inverse? A f () = + A f() = + B f () = + B f() = + C f () = + C f() = + D f () = Sequence Functions D f() = + Some functions model patterns Numerical patterns in mathematics are called sequences Specificall, a sequence is an ordered list of numbers with each term (number) defined b its position on the list One of the most famous sequences is the Fibonacci sequence This sequence is often modeled beautifull in nature through the structures of plants and other living things TASC Test Math Student Supplement Functions, Equations, and Inequalities