Regents Examination in Algebra II (Common Core) Sample Question Spring 2015

Transcription

1 Regen nts Examination in Algebra II (Common Ce) Sample Questions Spring 015 i May 015

2 THE STATE EDUCATION DEPARTMENT / THE UNIVERSITY OF THE STATE OF NEW YORK / ALBANY, NY 134 New Yk State Common Ce Sample Questions: Regents Examination in Algebra II (Common Ce) With the adoption of the New Yk P-1 Common Ce Learning Standards (CCLS) in ELA/Literacy and Mathematics, the Board of Regents signaled a shift in both instruction and assessment. Educats around the state have already begun instituting Common Ce instruction in their classrooms. To aid in this transition, we are providing sample Regents Examination in Algebra II (Common Ce) questions to help students, parents, and educats better understand the instructional shifts demanded by the Common Ce and the rig required to ensure that all students are on track to college and career readiness. These Questions Are Teaching Tools The sample questions emphasize the instructional shifts demanded by the Common Ce. F Algebra II (Common Ce) we have provided fourteen questions. These questions include multiplechoice and constructed response. The sample questions are teaching tools f educats and can be shared freely with students and parents. They are designed to help clarify the way the Common Ce should drive instruction and how students will be assessed on the Regents Examination in Algebra II measuring CCLS beginning in June 016. NYSED is eager f feedback on these sample questions. Your input will guide us as we develop future exams. These Questions Are NOT Test Samplers While educats from around the state have helped craft these sample questions, they have not undergone the same extensive review, vetting, and field testing that occurs with actual questions used on the State exams. The sample questions were designed to help educats think about content, NOT to show how operational exams look exactly to provide infmation about how teachers should administer the test. How to Use the Sample Questions Interpret how the standards are conceptualized in each question. Note the multiple ways the standards are assessed throughout the sample questions. Look f opptunities f mathematical modeling, i.e., connecting mathematics with the real wld by conceptualizing, analyzing, interpreting, and validating conclusions in der to make decisions about situations in everyday life, society, the wkplace. Consider the instructional changes that will need to occur in your classroom. ii May 015

3 Notice the application of mathematical ways of thinking to real-wld issues and challenges. Pay attention to the strong distracts in each multiple-choice question. Don t consider these questions to be the only way the standards will be assessed. Don t assume that the sample questions represent a mini-version of future State exams. Understanding Math Sample Questions Multiple-Choice Questions Sample multiple-choice math questions are designed to assess CCLS math standards. Math multiple-choice questions assess procedural fluency and conceptual understanding. Unlike questions on past math exams, many require the use of multiple skills and concepts. Within the sample questions, distracts will be based on plausible missteps. Constructed Response Questions Math constructed response questions are similar to past questions, asking students to show their wk in completing one me tasks solving me extensive problems. Constructed response questions allow students to show their understanding of math procedures, conceptual understanding, and application. Fmat of the Math Sample Questions Document The Math Sample Questions document is fmatted so that headings follow each question to provide infmation f teacher use to help interpret the question, understand measurement with the CCLS, and infm instruction. A list of the headings with a brief description of the associated infmation is shown below. Key: This is the crect response, in the case of multiple-choice questions, the crect option. Measures CCLS: This question measures the knowledge, skills, and proficiencies characterized by the standards within the identified cluster. Mathematical Practices: If applicable, this is a list of mathematical practices associated with the question. Commentary: This is an explanation of how the question measures the knowledge, skills, and proficiencies characterized by the identified cluster(s). Rationale: F multiple-choice questions, this section provides the crect option and demonstrates one method f arriving at that response. F constructed response questions, one me possible approaches to solving the question are shown, followed by the scing rubric that is specific to the question. Note that there are often multiple approaches to solving each problem. The rationale section provides only one example. The scing rubrics should be used to evaluate the efficacy of different methods of arriving at a solution. iii May 015

4 1 If a, b, and c are all positive real numbers, which graph could represent the sketch of the graph of px axbx cxc =? 1

5 Key: 1 Measures CCLS Cluster: A-APR.B Mathematical Practice: 4 Commentary: This question measures A-APR.B because students demonstrate understanding of the relationship between the facts of the polynomial and the zeros, and apply this understanding to constructing a rough graph of the function. Rationale: Option 1 is crect. The zeros of the polynomial are at b, and c. The sketch of a polynomial of degree 3 with a negative leading coefficient should have end behavi showing as x goes to negative infinity, f ( x ) goes to positive infinity. The multiplicities of the roots are crectly represented in the graph.

6 Which equation represents a parabola with a focus of (0,4) and a directrix of y =? (1) y x 3 () y x 1 (3) y x 3 (4) x y 3 4 3

7 Key: 4 Measures CCLS Cluster: G-GPE.A Mathematical Practice:, 7 Commentary: This question measures G-GPE.A because students need to determine the equation of a parabola given its focus and directrix. Rationale: Option 4 is crect. A parabola with a focus of (0,4) and a directrix of y = is sketched as follows: 4

8 By inspection, it is determined that the vertex of the parabola is (0,3). It is also evident that the distance, p, between the vertex and the focus is 1. It is possible to use the fmula x h 4py k to derive the equation of the parabola as follows: x 0 41y 3 x x x 4 4y y +3= y A point ( x, y ) on the parabola must be the same distance from the focus as it is from the directrix. F any such point ( x, y ), the distance to the focus is directrix is y. Setting this equal leads to: ( x0) ( y 4) and the distance to the ( x0) ( y4) y x y 8y16 y 4y4 x x y4 3 y 5

9 3 If the terminal side of angle θ, in standard position, passes through point ( 4,3), what is the numerical value of sin θ? (1) () (3) (4)

10 Key: 1 Measures CCLS Cluster: F-TF.A Mathematical Practice: 5 Commentary: This question measures F-TF.A because students use the unit circle to find the numerical value of a trigonometric function. Rationale: Option 1 is crect. A reference triangle can be sketched using the codinates ( 4,3) in the second quadrant to find the value of sin θ. 7

11 4 A study of the annual population of the red-winged blackbird in Ft. Mill, South Carolina, shows the population, B t, can be represented by the function number of years since the study began. t Bt , where the t represents the In terms of the monthly rate of growth, the population of red-winged blackbirds can be best approximated by the function t (1) Bt t () B t (3) B t t (4) B t t 1 8

12 Key: Measures CCLS Cluster: A-SSE.B Mathematical Practice: 4, 6 Commentary: This question measures A-SSE.B because students use properties of exponents to transfm an exponential function. Rationale: B t B t B t 1 1t 1t 1t 1 1t Bt ( ) is used, as opposed to annual rate that is not compounded monthly Bt () t, because the growth is an 9

13 5 Use the properties of rational exponents to determine the value of y f the equation: 3 8 x x y x, x 1. 10

14 Key: 4 3 Measures CCLS Cluster: N-RN.A Mathematical Practice: Commentary: This question measures N-RN.A because students extend their knowledge of integer exponents to rewriting and wking with radicals in terms of rational exponents. Rationale: 8 x 3 4 x 3 4 x 3 x x 4 y 3 y y Rubric: [] 4 3 an equivalent value is written, and crect wk using rational exponents is shown. [1] Appropriate wk is shown, but one computational err is made. [1] Appropriate wk is shown, but one conceptual err is made. [1] [1] 8 x3 4 x 3 is written, but no further crect wk is shown. 4, but no wk is shown. 3 [0] A zero response is completely increct, irrelevant, incoherent is a crect response that was obtained by an obviously increct procedure. 11

15 6 Write 5 yi4 3i 5 yi4 3i in a bifm, where y is a real number. 1

16 Key: 1y + 16yi Measures CCLS Cluster: N-CN.A Mathematical Practice: 1, 7 Commentary: This question measures N-CN.A because students add, subtract, and multiply complex expressions, and apply the concept that i 1. The question rewards seeing structure and can be rewritten efficiently by applying the distributive property. The expression 1 y 16yi is crectly written in a bi fm. However, if a student writes 16yi 1 y, while not in a bi fm, no credit should be deducted. Rationale: 43i5 yi 5yi 4 3i4yi 16yi 1 yi 1 y 16yi Rubric: [] 1y 16 yi, and crect wk is shown. [1] Appropriate wk is shown, but one computational, facting, simplification err is made. [1] Appropriate wk is shown, but one conceptual err is made. [1] 1y 16 yi, but no wk is shown. [0] A zero response is completely increct, irrelevant incoherent is a crect response that was obtained by an obviously increct procedure. 13

17 7 Use an appropriate procedure to show that x 4 is a fact of the function f x x x x Explain your answer. 14

18 Key: See rationale below. Measures CCLS Cluster: A-APR.B Mathematical Practice: 3 Commentary: The question measures A-APR.B because an appropriate procedure is used to show 4 is the positive zero f f x. Rationale: f f x 3x 1 3 x 4x 5x 11x 4 3 x 8x 3x 11x 3x 1x x 4 x 4 0 Rubric: Any method that demonstrates 4 is a zero of f ( x ) confirms that x 4 is a fact, as suggested by the Remainder Theem. [] Crect wk is shown confirming x 4 is a fact, and a crect explanation is written. [1] Appropriate wk is shown, but one computational err is made. [1] Appropriate wk is shown, but one conceptual err is made. [1] Crect wk is shown, but no explanation an increct explanation is written. [0] A zero response is completely increct, irrelevant, incoherent is a crect response that was obtained by an obviously increct procedure. 15

19 8 Solve algebraically f all values of x: x 5 x 7 16

20 Key: 6 Measures CCLS Cluster: A-REI.A Mathematical Practice: 3, 6 Commentary: This question measures A-REI.A because the problem requires students to solve a radical equation and identify extraneous solutions. Rationale: x x 5 7 x x 5 7 x x x x 54914x x 15x x 60 x 6 x 90 x Accept Reject 17

21 Rubric: [] 6 and crect algebraic wk is shown. [1] Appropriate wk is shown, but one computational simplification err is made. [1] Appropriate wk is shown, but one conceptual err is made. [1] 6, but a method other than algebraic is used. [1] Appropriate wk is shown, but 9 is not rejected. [0] 6, but no wk is shown. [0] A zero response is completely increct, irrelevant, incoherent is a crect response that was obtained by an obviously increct procedure. 18

22 9 Monthly mtgage payments can be found using the fmula below: M r r P n r n M = monthly payment P = amount browed r = annual interest rate n = number of monthly payments The Banks family would like to brow $10,000 to purchase a home. They qualified f an annual interest rate of 4.8%. Algebraically determine the fewest number of whole years the Banks family would need to include in the mtgage agreement in der to have a monthly payment of no me than $70. 19

23 Key: 3 Measures CCLS Cluster: A-SSE.B Mathematical Practice: 1, 6 Commentary: This question measures A-SSE.B because students wk with the sum of a finite geometric series through a related mtgage fmula. Rationale: A crect equation inequality solved algebraically should receive full credit , n n n n n log log 3 log log n n n years Rubric: [4] 3, and crect algebraic wk is shown. [3] Appropriate wk is shown, but one computational rounding err is made. [3] Appropriate wk is shown to find n, but no further crect wk is shown. [] Appropriate wk is shown, but two me computational rounding errs are made. [] Appropriate wk is shown, but one conceptual err is made. [] Appropriate wk is shown to find n 3, but no further crect wk is shown. [] 3, but a method other than algebraic is used. 0

24 [1] Appropriate wk is shown, but one conceptual err and one computational rounding err are made. [1] A crect substitution is made into the monthly mtgage fmula but no further crect wd is shown. [1] 3, but no wk is shown. [0] A zero response is completely increct, irrelevant, incoherent is a crect response that was obtained by an obviously increct procedure. 1

25 10 Solve the following system of equations algebraically f all values of x, y, and z: x 3y 5z 45 6x 3y z 10 x 3y 8z 7

26 Key: x, y 4, and z 7 Measures CCLS Cluster: A-REI.C Mathematical Practice: 1, 6 Commentary: This question measures A-REI.C because students are required to solve a 3 3 system of equations. Rationale: Crect algebraic wk is shown below. x 3y 5z 45 6x 3y z 10 x 3y 8z 7 (1) () (3) 7x7z 35 xz 5 4x 10z 6 4( xz 5) 4x10z 6 (1) + () () + (3) 6z z 7 x 75 x 3y 5(7) 45 3y 10 3y 1 (1) y 4 3

27 Rubric: [4] x, y 4, and z 7, and crect algebraic wk is shown. [3] Appropriate wk is shown, but one computational err is made. [3] Appropriate wk is shown to find two of the solutions, but no further crect wk is shown. [] Appropriate wk is shown, but two me computational errs are made. [] Appropriate wk is shown, but one conceptual err is made. [] Appropriate wk is shown to find one of the solutions, but no further crect wk is shown. [] x, y 4, and z 7, but a method other than algebraic is used. [1] Appropriate wk is shown, but one conceptual err and one computational err are made. [1] Appropriate wk is shown to eliminate y to create a system of two equations, but no further crect wk is shown. [1] x, y 4, and z 7, but no wk is shown. [0] A zero response is completely increct, irrelevant, incoherent is a crect response that was obtained by an obviously increct procedure. 4

28 11 Write an explicit fmula f a n, the n th term of the recursively defined sequence below. a a 1 n x 1 x an 1 F what values of x would a 0whenn 1? n 5

29 Key: n 1 n n1 n n a x x 1 a x x, and x0 and x 1 Measures CCLS Cluster: F-BF.A Mathematical Practice:, 8 Commentary: This question measures F-BF.A because students are required to translate a sequence from its recursive fm to an explicit fm. Rationale: a a 1 x 1 x x 1 a x x 3 n1 1 a x x 1 n a 1 x 1 a x x 3 a x x 3 n n a x x n 1 x n 1 0 ( x 1) 0 x 0 x 1 Note: Students are not required to show wk to solve 0. It is expected that x raised to a positive value set equal to 0 yields the solution x = 0. n x 1 Rubric: n [4] a x 1 x1 n equivalent and x 0andx 1, and crect wk is shown. [3] Appropriate wk is shown, but one computational simplification err is made. [] Appropriate wk is shown, but two me computational simplification errs are made. [] Appropriate wk is shown, but one conceptual err is made. [] Appropriate wk is shown to find n1 a x x 1 x 0and x 1. n 6

30 [1] Appropriate wk is shown, but one conceptual err and one computational simplification err are made. n1 [1] an x x 1and x 0andx1, but no wk is shown. [0] A zero response is completely increct, irrelevant, incoherent is a crect response that was obtained by an obviously increct procedure. 7

31 1 Stephen s Beverage Company is considering whether to produce a new brand of cola. The company will launch the product if at least 5% of cola drinkers will buy the product. Fifty cola drinkers are randomly selected to take a blind taste-test of products A, B, and the new product. Nine out of fifty participants preferred Stephen s new cola to products A and B. The company then devised a simulation based on the requirement that 5% of cola drinkers will buy the product. Each dot in the graph shown below represents the proption of people who preferred Stephen s new product, each of sample size 50, simulated 100 times. Assume the set of data is approximately nmal and the company wants to be 95% confident of its results. Does the sample proption obtained from the blind taste-test, nine out of fifty, fall within the margin of err developed from the simulation? Justify your answer. The company decides to continue developing the product even though only nine out of fifty participants preferred its brand of cola in the taste-test. Describe how the simulation data could be used to suppt this decision. 8

32 Key: See rationale below. Measures CCLS Cluster: S-IC.B Mathematical Practice:, 4 Commentary: This question measures S-IC.B since students estimate the margin of err based on a simulation model and use the data from the simulation to evaluate a company s decision. Rationale: Yes. The margin of err from this simulation indicates that 95% of the observations fall within ± 0.1 of the simulated proption, 0.5. The margin of err can be estimated by multiplying the standard deviation, shown to be 0.06 in the dotplot, by, applying the p(1 p) (0.5)(0.75) estimated standard err fmula, and multiplying by. n 50 The interval 0.5 ± 0.1 includes plausible values f the true proption of people who prefer Stephen s new product. The company has evidence that the population proption could be at least 5%. As seen in the dotplot, it can be expected to obtain a sample proption of 0.18 (9 out of 50) less several times, even when the population proption is 0.5, due to sampling variability. Given this infmation, the results of the survey do not provide enough evidence to suggest that the true proption is not at least 0.5, so the development of the product should continue at this time. 9

33 [4] Yes, and a crect justification is given, and a crect description is given. [3] Appropriate wk is shown, but one computational err is made. [] Appropriate wk is shown, but two me computational errs are made. [] Appropriate wk is shown, but one conceptual err is made. [] Yes, and a crect justification is given, but no further crect wk is shown. [] A crect description is given, but no further crect wk is shown. [1] Appropriate wk is shown, but one conceptual err and one computational err are made. [1] A crect margin of err is stated and yes, but no further crect wk is shown. [0] A zero response is completely increct, irrelevant, incoherent is a crect response that was obtained by an obviously increct procedure. 30

34 13 In contract negotiations between a local government agency and its wkers, it is estimated that there is a 50% chance that an agreement will be reached on the salaries of the wkers. It is estimated that there is a 70% chance that there will be an agreement on the insurance benefits. There is a 0% chance that no agreement will be reached on either issue. Find the probability that an agreement will be reached on both issues. Based on this answer, determine whether the agreement on salaries and the agreement on insurance are independent events. Justify your answer. 31

35 Key: 0.4 an equivalent answer, and no with a crect justification. Measures CCLS Cluster: S-CP.A Mathematical Practice:, 4 Commentary: This question measures S-CP.A because the student must understand events as subsets of the sample space, including the concepts of union, intersection, and complement. Additionally, the student must reason using properties of probability about whether not events are independent. Rationale: This scenario can be modeled with a Venn Diagram: Since PSI PS I c 0., 0.8 Then, P S I P S P I P S I If S and I are independent, then the Product Rule must be satisfied. However, Therefe, salary and insurance have not been treated independently. If S and I are independent, the conditional probability of PS I PS Therefe, salary and insurance have not been treated independently. /. However,

36 Rubric: [4] 0.4 an equivalent answer and no, and crect wk is shown, and a crect justification is given. [3] Appropriate wk is shown, but one computational, simplification, rounding err is made. [3] Appropriate wk is shown to find 0.4 and no, but no justification is given. [] Appropriate wk is shown, but two me computational, simplification, rounding errs are made. [] Appropriate wk is shown, but one conceptual err is made. [] 0.4 an equivalent answer, and crect wk is shown, but no further crect wk is shown. [1] Appropriate wk is shown, but one conceptual err and one computational, simplification, rounding err are made. [1] 0.4 and no, but no wk is shown. [0] A zero response is completely increct, irrelevant, incoherent is a crect response that was obtained by an obviously increct procedure. 33

37 14 The ocean tides near Carter Beach follow a repeating pattern over time, with the amount of time between each low and high tide remaining relatively constant. On a certain day, low tide occurred at 8:30 a.m. and high tide occurred at 3:00 p.m. At high tide, the water level was 1 inches above the average local sea level; at low tide it was 1 inches below the average local sea level. Assume that high tide and low tide are the maximum and minimum water levels each day, respectively. Write a cosine function of the fm f () t Acos( Bt), where A and B are real numbers, that models the water level, f () t, in inches above below the average Carter Beach sea level, as a function of the time measured in t hours since 8:30 a.m. On the grid below, graph one cycle of this function. People who fish in Carter Beach know that a certain species of fish is most plentiful when the water level is increasing. Explain whether you would recommend fishing f this species at 7:30 p.m. 10:30 p.m. using evidence from the given context. 34

38 Key: f () t 1cos t equivalent function, a crect graph is drawn, and 10:30 pm with 13 appropriate evidence from the context. Measures CCLS Cluster: F-IF.B Mathematical Practice:,4 Commentary: The question measures F-IF.B because the student must interpret a key feature of the graph of a function, and intervals on which the function is increasing. This question also measures F-TF.B and F-IF.C because the student must choose and graph a trigonometric function to model a periodic phenomenon, the level of water near Carter Beach over time. Rationale: The amplitude, 1, can be interpreted from the situation, since the water level has a minimum of 1 and a maximum of 1. The value of A is 1 since at 8:30 it is low tide. The period of the function is 13 hours, and is expressed in the function through the parameter B. By experimentation with technology using the relation P (where P B is the period), it is determined that B

39 Rubric: In der to answer the question about when to fish, the student must interpret the function and determine which choice, 7:30 pm 10:30 pm, is on an increasing interval. Since the function is increasing from t = 13 to t = 19.5 (which cresponds to 9:30 pm to 4:00 am), 10:30 is the appropriate choice. [6] f () t 1cos t equivalent function, a crect graph is drawn, and 10:30 pm with 13 appropriate evidence from the context. [5] Appropriate wk is shown, but one computational graphing err is made. [5] Appropriate wk is shown, but the explanation is incomplete. [5] Appropriate wk is shown, but an err is made in determining A B in the function. 36

40 [4] Appropriate wk is shown, but two computational graphing errs are made. [4] Appropriate wk is shown, but one conceptual err is made. [4] f () t 1cos t and a crect graph is drawn, but no further crect wk is shown. 13 [3] Appropriate wk is shown, but three me computational errs are made. [3] Appropriate wk is shown, but one conceptual and one computational err are made. [] Appropriate wk is shown, but two conceptual errs are made. [] Appropriate wk is shown, but one conceptual and two me computational errs are made. [] 10:30 pm and a crect explanation with evidence from the context is written, but no further crect wk is shown. [1] Appropriate wk is shown, but two conceptual and one computational errs are made. [1] 1cos t but no further crect wk is shown. 13 [0] A zero response is completely increct, irrelevant, incoherent is a crect response that was obtained by and obviously increct procedure. 37

41 Regen nts Examination in Algebra II (Common Ce) Sample Questions Fall 015 i 015

42 THE STATE EDUCATION DEPARTMENT / THE UNIVERSITY OF THE STATE OF NEW YORK / ALBANY, NY 134 New Yk State Common Ce Sample Questions: Regents Examination in Algebra II (Common Ce) With the adoption of the New Yk P-1 Common Ce Learning Standards (CCLS) in ELA/Literacy and Mathematics, the Board of Regents signaled a shift in both instruction and assessment. Educats around the state have already begun instituting Common Ce instruction in their classrooms. To aid in this transition, we are providing sample Regents Examination in Algebra II (Common Ce) questions to help students, parents, and educats better understand the instructional shifts demanded by the Common Ce and the rig required to ensure that all students are on track to college and career readiness. These Questions Are Teaching Tools The sample questions emphasize the instructional shifts demanded by the Common Ce. F Algebra II (Common Ce) we have provided seventeen questions. These questions include multiple-choice and constructed response. The sample questions are teaching tools f educats and can be shared freely with students and parents. They are designed to help clarify the way the Common Ce should drive instruction and how students will be assessed on the Regents Examination in Algebra II measuring CCLS beginning in June 016. NYSED is eager f feedback on these sample questions. Your input will guide us as we develop future exams. These Questions Are NOT Test Samplers While educats from around the state have helped craft these sample questions, they have not undergone the same extensive review, vetting, and field testing that occurs with actual questions used on the State exams. The sample questions were designed to help educats think about content, NOT to show how operational exams look exactly to provide infmation about how teachers should administer the test. How to Use the Sample Questions Interpret how the standards are conceptualized in each question. Note the multiple ways the standards are assessed throughout the sample questions. Look f opptunities f mathematical modeling, i.e., connecting mathematics with the real wld by conceptualizing, analyzing, interpreting, and validating conclusions in der to make decisions about situations in everyday life, society, the wkplace. Consider the instructional changes that will need to occur in your classroom. Notice the application of mathematical ways of thinking to real-wld issues and challenges. ii 015

43 Pay attention to the strong distracts in each multiple-choice question. Don t consider these questions to be the only way the standards will be assessed. Don t assume that the sample questions represent a mini-version of future State exams. Understanding Math Sample Questions Multiple-Choice Questions Sample multiple-choice math questions are designed to assess CCLS math standards. Math multiple-choice questions assess procedural fluency and conceptual understanding. Unlike questions on past math exams, many require the use of multiple skills and concepts. Within the sample questions, distracts will be based on plausible missteps. Constructed Response Questions Math constructed response questions are similar to past questions, asking students to show their wk in completing one me tasks solving me extensive problems. Constructed response questions allow students to show their understanding of math procedures, conceptual understanding, and application. Fmat of the Math Sample Questions Document The Math Sample Questions document is fmatted so that headings follow each question to provide infmation f teacher use to help interpret the question, understand measurement with the CCLS, and infm instruction. A list of the headings with a brief description of the associated infmation is shown below. Key: This is the crect response, in the case of multiple-choice questions, the crect option. Measures CCLS: This question measures the knowledge, skills, and proficiencies characterized by the standards within the identified cluster. Mathematical Practices: If applicable, this is a list of mathematical practices associated with the question. Commentary: This is an explanation of how the question measures the knowledge, skills, and proficiencies characterized by the identified cluster(s). Rationale: F multiple-choice questions, this section provides the crect option and demonstrates one method f arriving at that response. F constructed response questions, one me possible approaches to solving the question are shown, followed by the scing rubric that is specific to the question. Note that there are often multiple approaches to solving each problem. The rationale section provides only one example. The scing rubrics should be used to evaluate the efficacy of different methods of arriving at a solution. iii 015

44 1 What is the solution set of the equation 3 x 5 5 3? x 7 x (1) () (3) (4) 3,7 7, 3 3,7 7, 3 1

45 Key: 4 Measures CCLS Cluster: A-REI.A Mathematical Practice:, 7 Commentary: This question measures A-REI.A because students must solve a rational equation. Rationale: Option 4 is crect. 3 x x x ; x 7, x 0 x 7 x x x x x x 3x 5x5x 35x 3x1 x 13x10 x x x x x 3 0 x 3

46 Functions f, g, and h are given below. sin f x x g x f x 1 Which statement is true about functions f, g, and h? (1) f x and gx are odd, () f x and gx are even, (3) f x is odd, gx is neither, (4) f x is even, gx is neither, hx is even. hx is odd. hx is even. hx is odd. 3

47 Key: 3 Measures CCLS Cluster: F-BF.B Mathematical Practice:, 5 Commentary: This question measures F-BF.B because students must be able to recognize even and odd functions. Rationale: Option 3 is crect. f x f x f is symmetric about the igin f x odd = even h x h x h x h is symmetric about the y-axis F example, consider x 1 g 1 = g 1 =.0907 gx gx not even gxgx not odd gx neither 4

48 3 The expression 3 6x 17x 10x x 3 equals (1) () (3) (4) 5 3x 4x 1 x 3 5 6x 8x x x x 13 x x 13x x 3 5

49 Key: 1 Measures CCLS Cluster: A-APR.D Mathematical Practice: 8 Commentary: This question measures A-APR.D because students must rewrite a simple rational expression in quotient-remainder fm. Rationale: Option 1 is crect. 3x 4x 1 3 x3 6x 17x 10x 3 6x 9x 8x 10x 8x 1x x x x 4x1 x 3 6

50 4 The solutions to the equation (1) 6 i () 6 19 (3) 6 i (4) x 6x0 are 7

51 Key: 3 Measures CCLS Cluster: A-REI.B Mathematical Practice: 7 Commentary: This question measures A-REI.B because students must solve a quadratic equation with complex solutions. Rationale: Option 3 is crect. Method 1: Method : 1 x 6x 0 1 x 6x 0 1 x 6x x x 1 x 6 i x x x 1x 40 1x40 0 1x x 6 4 x 6 i x 6 i 8

52 4 3 5 What is the completely facted fm of k 4k 8k 3k 1k 48? (1) k k k 3k 4 () k k k 6k (3) k k k 3k 4 (4) k k k 6k 9

53 Key: 4 Measures CCLS Cluster: A-SSE.A Mathematical Practice: 5, 7 Commentary: This question measures A-SSE.A because students use the structure of an expression to identify ways to rewrite it. Rationale: Option 4 is crect. 4 3 k 4k 8k 3k 1k 48 k 4 4 k 8 k 3 3 k 1 k k k k k k k 4 k 8 k 1 kkk6k 10

54 3cos 4 7? 3 6 Which statement is increct f the graph of the function y x (1) The period is 6. () The amplitude is 3. (3) The range is [4,10]. (4) The midline is y 4. 11

55 Key: 4 Measures CCLS Cluster: F.IF.C Mathematical Practice: 5, 7 Commentary: This question measures F-IF.C because students must determine key features of the graph of a given trigonometric function. Rationale: Option 4 states an increct midline. The midline is y 7 since 7 is the average of the endpoints of the range. 1

56 7 Algebraically determine the values of x that satisfy the system of equations below. y x 1 y x 3x 1 13

57 Key: 0, 5 Measures CCLS Cluster: A-REI.C Mathematical Practice:, 7 Commentary: This question measures A-REI.C because students must be able to solve a linearquadratic system in two variables. Rationale: x 1 x 3x1 x 5x 0 x x5 0 x 0 x x Rubric: [] 5 0, and crect algebraic wk is shown. [1] Appropriate wk is shown, but one computational facting err is made. [1] Appropriate wk is shown, but one conceptual err is made. 5 [1] 0, but a method other than algebraic is used. 5 [1] 0, but no wk is shown. [0] A zero response is completely increct, irrelevant, incoherent is a crect response that was obtained by an obviously increct procedure. 14

58 8 The results of a poll of 00 students are shown in the table below: Preferred Music Style Techno Rap Country Female Male F this group of students, do these data suggest that gender and preferred music styles are independent of each other? Justify your answer. 15

59 Key: See rationale below Measures CCLS Cluster: S-CP.A Mathematical Practice: 1,, 3 Commentary: This question measures S-CP.A because students must demonstrate an understanding of conditional probability and interpret independence of events. Rationale: Based on these data, the two events do not appear to be independent. The probability that a student is female given that she prefers techno music is while the probability that 90 a student is female is These probabilities are not the same. This suggests that 00 the events are not independent. Other music styles can be used such as P 5 7 Female Rap 0.385; Female Country P Rubric: [] The events are not independent and crect wk is shown. [1] Appropriate wk is shown, but one computational err is made. [1] Appropriate wk is shown, but one conceptual err is made. [0] Not independent, but no wk is shown. [0] A zero response is completely increct, irrelevant, incoherent is a crect response that was obtained by an obviously increct procedure. 16

60 3 1 9 F the function f xx f x 3 1,find. 17

61 Key: x Measures CCLS Cluster: F-BF.B Mathematical Practice: 7 Commentary: This question measures F-BF.B because students must write the inverse of a given function. Rationale: x y x1 y3 x13 y3 1 1 x1 3 y3 1 x1 3 3 y f x x1 3 3 Rubric: [] x an equivalent expression and crect wk is shown. [1] Appropriate wk is shown, but one computational err is made. [1] Appropriate wk is shown, but one conceptual err is made. [1] x y [1] x is written, but no further crect wk is shown , but no wk is shown. [0] A zero response is completely increct, irrelevant, incoherent is a crect response that was obtained by an obviously increct procedure. 18

62 10 Given: h x x x x x 5 k x State the solutions to the equation hx kx, rounded to the nearest hundredth. 19

63 Key: 5.17, 1.13, and Measures CCLS Cluster: A-REI.D Mathematical Practice: 5, 6 Commentary: This question measures A-REI.D because students are required to find the approximate h x k x solutions to. Rationale: Using technology and y hx y kx 1 and, the intersect function is used to determine all values of x f which y y 1. On their calculat screens, students should see an image similar to the one below. 0

64 Rubric: [] 5.17, 1.13, and [1] One computational rounding err is made. [1] One conceptual err is made. [1] Only two crect values are found. [1] ( 5.17, 1.38), ( 1.13, 4.1), and (1.75, 3.77) are written. [0] A zero response is completely increct, irrelevant, incoherent is a crect response that was obtained by an obviously increct procedure. 1

65 11 Algebraically prove that the difference of the squares of any two consecutive integers is an odd integer.

66 Key: See rationale below. Measures CCLS Cluster: A-APR.C Mathematical Practice: 1, 8 Commentary: This question measures A-APR.C because students must prove a polynomial identity. Rationale: Let x the first integer x 1 the next integer The difference of their squares is x 1 x x x1x x1 x is an even integer, therefe x 1 is an odd integer. x x1 x x x1 x1 x is an even integer, therefe x 1 is an odd integer. Rubric: [] A crect algebraic proof is written. [1] Appropriate wk is shown, but one computational err is made. [1] Appropriate wk is shown, but one conceptual err is made. [1] Appropriate wk is shown to find x 1 x 1, but no concluding statement is written. [1] An incomplete algebraic proof is written. [0] A zero response is completely increct, irrelevant, incoherent is a crect response that was obtained by an obviously increct procedure. 3

67 1 Rewrite the expression x x x x as a product of four linear facts. 4

68 Key: See rationale below. Measures CCLS Cluster: A-SSE.A Mathematical Practice: 1,, 7 Commentary: This question measures A-SSE.A because students produce an equivalent fm of an expression. Rationale: The problem is of the fm y 5y 6, which facts to y y 6 1. Therefe: x x x x 4x 5x 64x 5x x 3x 4x 1x 1 Rubric: [] 4x 3x 4x 1x 1 and crect wk is shown. [1] Appropriate wk is shown, but one computational facting err is made. [1] Appropriate wk is shown, but one conceptual err is made. [1] 4x 5x64x 5x1 is written, but no further crect wk is shown. 4x 3 x 4x 1 x 1, but no wk is shown. [1] [0] A zero response is completely increct, irrelevant, incoherent is a crect response that was obtained by an obviously increct procedure. 5

69 13 After sitting out of the refrigerat f a while, a turkey at room temperature (68 F) is placed into an oven at 8 a.m., when the oven temperature is 35 F. Newton s Law of Heating explains that the temperature of the turkey will increase proptionally to the difference between the temperature of the turkey and the temperature of the oven, as given by the fmula below: T T T T e a o a kt T a the temperature surrounding the object T o the initial temperature of the object t the time in hours T the temperature of the object after t hours k decay constant The turkey reaches the temperature of approximately 100 F after hours. Find the value of k, to the nearest thousandth, and write an equation to determine the temperature of the turkey after t hours. Determine the Fahrenheit temperature of the turkey, to the nearest degree, at 3 p.m. 6

70 Key: 0. k 0.066, T e 066t, 163 Measures CCLS Cluster: A-CED.A Mathematical Practice: 1, 4 Commentary: This question measures A-CED.A because students must create an exponential equation and use it to solve problems. Rationale: e 5 57e 5 ln 57 k k k T e 0.066t k At 3 pm, t 7. T T 35 57e

71 Rubric: [4] 0. k 0.066, T e 066t, 163, and crect wk is shown. [3] Appropriate wk is shown, but one computational rounding err is made. 0. [3] Appropriate wk is shown, T 35 57e 066t is written, but no further crect wk is shown. [3] Appropriate wk is shown, but the equation is written without T t. [] Appropriate wk is shown, but two me computational rounding errs are made. [] Appropriate wk is shown, but one conceptual err is made. [] Appropriate wk is shown to find k = 0.066, but no further crect wk is shown. [] The expression 35 57e t is written, but no further crect wk is shown. [1] Appropriate wk is shown, but one conceptual and one computational rounding err is made. [1] 0.066, but no wk is shown. [0] A zero response is completely increct, irrelevant, incoherent is a crect response that was obtained by an obviously increct procedure. 8

72 14 Seventy-two students are randomly divided into two equally-sized study groups. Each member of the first group (group 1) is to meet with a tut after school twice each week f one hour. The second group (group ), is given an online subscription to a tutial account that they can access f a maximum of two hours each week. Students in both groups are given the same tests during the year. A summary of the two groups final grades is shown below: Group 1 Group x S x Calculate the mean difference in the final grades (group 1 group ) and explain its meaning in the context of the problem. A simulation was conducted in which the students final grades were rerandomized 500 times. The results are shown below Use the simulation to determine if there is a significant difference in the final grades. Explain your answer. 9

73 Key: See rationale below. Measures CCLS Cluster: S-IC.B Mathematical Practice: 1, 3, 6 Commentary: This question measures S-IC.B because students use a simulation to determine if a difference in sample means in an experiment is significant. Rationale: The mean difference between the students final grades in group 1 and group is This value indicates that students who met with a tut had a mean final grade of 3.64 points less than students who used an on-line subscription. One can infer whether this difference is due to the differences in intervention due to which students were assigned to each group by using a simulation to rerandomize the students final grades many (500) times. If the observed difference 3.64 is the result of the assignment of students to groups alone, then a difference of 3.64 less should be observed fairly regularly in the simulation output. However, a difference of 3 less occurs in only about % of the rerandomizations. Therefe, it is quite unlikely that the assignment to groups alone accounts f the difference; rather, it is likely that the difference between the interventions themselves accounts f the difference between the two groups mean final grades. The rerandomization process always involves the following steps: 1. Shuffle all observations.. Divide the observations into equal groups. 3. Find the mean difference between the groups. 4. Repeat steps 1 through 3 many times. 30

74 Rubric: [4] 3.64 and a crect explanation, and yes and a crect explanation is given. [3] Appropriate wk is shown, but one computational err is made. [3] Appropriate wk is shown, but the mean difference is calculated increctly. [] Appropriate wk is shown, but one conceptual err is made. [] Appropriate wk is shown, but two me computational errs are made. [] 3.64 and a crect explanation, but no further crect wk is shown. [1] 3.64, but no wk is shown. [0] Yes, but no explanation is given. [0] A zero response is completely increct, irrelevant, incoherent is a crect response that was obtained by an obviously increct procedure. 31

75 3 15 Given zx x bx x z z zeros of zx , 35, and 5 0, algebraically determine all the 3

76 Key: 3 1, and 5 3 Measures CCLS Cluster: A-APR.B Mathematical Practice: 1, 7 Commentary: This question measures A-APR.B because students must apply the Remainder Theem and then identify the zeros of a polynomial when a suitable factization is available. Rationale: Find b: 3 b b b 19 b 3 b b 475 5b 19 b z x 6x 19x 5x15 z 3 5 0, by the Remainder Theem; x 11x30 6x 9xx30 3xx31x30 x x z x 30 3 x 3x 1 0 z x x

77 Rubric: [4] 3 1,,and 5, and crect algebraic wk is shown. 3 [3] Appropriate wk is shown to find 3 and 1 3, only. [3] Appropriate wk is shown, but one computational err is made. [] Appropriate wk is shown, but two me computational errs are made. [] Appropriate wk is shown, but one conceptual err is made. [] Appropriate wk is shown, but a method other than algebraic is used. [1] Appropriate wk is shown, but one conceptual and one computational err are made. [1] Appropriate wk is shown to find b = 19, but no further crect wk is shown. 3 1 [1],,and 5 but no wk is shown. 3 [0] A zero response is completely increct, irrelevant, incoherent is a crect response that was obtained by an obviously increct procedure. 34

78 16 Two versions of a standardized test are given, an April version and a May version. The statistics f the April version show a mean sce of 480 and a standard deviation of 4. The statistics f the May version show a mean sce of 510 and a standard deviation of 0. Assume the sces are nmally distributed. Joanne took the April version and sced in the interval What is the probability, to the nearest ten thousandth, that a test paper selected at random from the April version sced in the same interval? Maria took the May version. In what interval must Maria sce to claim she sced as well as Joanne? 35

79 Key: See rationale below. Measures CCLS Cluster: S-ID.A Mathematical Practice: 1, 3, 5 Commentary: This question measures S-ID.A because students must be able to use their calculats to estimate the area under the curve. Rationale: The probability of a sce being between 510 and 540 on the April exam can be found using the nmal probability cumulative density function, nmcdf(510, 540, 480, 4) = Use z-sces to compare the two sets of data. Joanne s sces crespond to z 1.5 to z Calculating equivalent sces, 1.5 x x x 535 x 560 Maria must sce in the interval

80 Rubric: [4] and [535,560] and crect wk is shown. [3] Appropriate wk is shown, but one computational rounding err is made. [] Appropriate wk is shown, but two me computational rounding errs are made. [] Appropriate wk is shown, but one conceptual err is made. [] Appropriate wk is shown to find , but no further crect wk is shown. [] Appropriate wk is shown to find [535,560], but no further crect wk is shown. [] and [535,560], but no wk is shown. [1] Appropriate wk is shown, but one conceptual and one computational rounding err are made. [1] [535,560], but no wk is shown. [0] A zero response if completely increct, irrelevant, incoherent is a crect response that was obtained by an obviously increct procedure. 37

81 17 Titanium-44 is a radioactive isotope such that every 63 years, its mass decreases by half. F a sample of titanium-44 with an initial mass of 100 grams, write a function that will give the mass of the sample remaining after any amount of time. Define all variables. Scientists sometimes use the average yearly decrease in mass f estimation purposes. Use the average yearly decrease in mass of the sample between year 0 and year 10 to predict the amount of the sample remaining after 40 years. Round your answer to the nearest tenth. Is the actual mass of the sample the estimated mass greater after 40 years? Justify your answer. 38

82 Key: See rationale below. Measures CCLS Cluster: F-BF.A Mathematical Practice:, 4 Commentary: This question measures F-BF.A because students must write a function that describes a relationship between two quantities. This question also measures F-IF.B because students must calculate and interpret the average rate of change of a function. Rationale: Method 1: t At , where t = time in years and A t amount of titanium-44 remaining after t years. A A At t 40, the estimated mass is g The actual mass is A The estimation is less than the actual. 39

83 Method : y e kt e 1 e 63k 1 ln 63k = k 63k e t y 100, t = time in years. where y amount of titanium-44 remaining after t years and y y At t 40, the estimated mass is ( ) 58.3 grams The actual mass is y 100e The estimation is less than the actual. 40

84 Rubric: [6] A crect function with defined variables is written, 58.3, actual, and a crect justification is given. [5] Appropriate wk is shown, but one computational err is made. [5] Appropriate wk is shown, but the function s variables are not defined. [4] Appropriate wk is shown, but two computational errs are made. [4] Appropriate wk is shown, but one conceptual err is made. [4] Appropriate wk is shown to find 58.3, actual, and a crect explanation are stated, but no further crect wk is shown. [3] Appropriate wk is shown, but three me computational errs are made. [3] Appropriate wk is shown, but one conceptual and one computational err is made. [3] A crect function and 58.3 are stated, but no further crect wk is shown. [] Appropriate wk is shown, but two conceptual errs are made. [] Appropriate wk is shown, but one conceptual and two me computational errs are made. [] A crect function is written with defined variables, but no further crect wk is shown. [1] Appropriate wk is shown, but two conceptual and one computational errs are made. [1] 58.3, but no wk is shown. [0] Actual, but no wk is shown. [0] A zero response is completely increct, irrelevant, incoherent is a crect response that was obtained by an obviously increct procedure. 41