Tips for Using the QualityCore Mathematics Benchmark Assessments

Transcription

1 Tips for Using the QualityCore Mathematics Benchmark Assessments Each QualityCore course has its own set of Benchmark Assessments based on the QualityCore Formative Item Pool. Algebra I has four Benchmark Assessments and Algebra II, Geometry, and Precalculus each have five Benchmark Assessments. Each assessment consists of 15 to 25 multiple-choice items and one constructed-response item. The assessments are presented as a PDF file to maintain the visual consistency of graphics, special characters, and symbols. Each assessment is bookmarked for easy navigation through the PDF file. The PDF file also contains the corresponding QualityCore Reference Sheet. Each Benchmark Assessment is introduced by a cover sheet displaying the item Identification Number (ID), the correct answer (Key), the cognitive level, and the alphanumeric code for each ACT Course Standard covered by that item. (See the applicable ACT Course Standards document.) The scoring criteria and a scoring rubric follow the constructed-response item by ACT, Inc. Permission granted to reproduce this page for QualityCore educational purposes only.

2 QualityCore Reference Sheet Precalculus Triangles Law of Sines a = b = c sin A sin B sin C Law of Cosines a 2 = b 2 + c 2 2bc cos A Area of a Triangle Area = 1_ 2 bc sin A Area = s(s a)(s b)(s c) Conic Sections Circle (x h) 2 + (y k) 2 = r 2 Parabola, opening vertically Parabola, opening horizontally Ellipse, major axis horizontal y = a(x h) 2 + k x = a(y k) 2 + h (x h) 2 a 2 (y + k) 2 = 1, a > b b 2 C b A a c s = semi-perimeter = (h,k) = center, r = radius (a + b + c) 2 axis of symmetry x = h focus h, k + 1 4a, directrix y = k axis of symmetry y = k focus h + 1 4a, k, directrix x = h foci (h ± c, k) where c 2 = a 2 b 2 B 1 4a 1 4a Ellipse, major axis vertical Area of an ellipse Hyperbola, transverse axis horizontal Hyperbola, transverse axis vertical (y k) 2 a 2 A = πab (x h) 2 a 2 (y k) 2 a 2 + (x h) 2 = 1, a > b b 2 (y k) 2 = 1 b 2 (x h) 2 = 1 b 2 foci (h, k ± c) where c 2 = a 2 b 2 foci (h ± c, k) where c 2 = a 2 + b 2 foci (h, k ± c) where c 2 = a 2 + b 2 Sequences and Series Arithmetic Sequence a n = a 1 + (n 1)d Arithmetic Series s n = n_ (a 1 + a n ) 2 Geometric Sequence a n = a 1 r n 1 Finite Geometric Series s n = a 1 a 1 r n 1 r where r 1 Infinite Geometric Series a s = 1 where r <1 1 r a n = nth term a 1 = first term n = number of the term d = common difference r = common ratio s n = sum of the first n terms s = sum of all the terms Exponential Functions Discretely Compounded Interest A = p 1 + r_ n nt A = amount of money after t years p = starting principal r = interest rate Continuously Compounded Interest A = pe rt n = compound periods per year t = number of years e Discrete, Continuous Exponential Growth N t = N 0 (1 + r) t, N t = N 0 e rt N t = value after t time periods r = rate of growth t = time periods continued

3 Polar Coordinates and Vectors De Moivre s Theorem [r(cos θ + i sin θ)] n = r n (cos nθ + i sin nθ) Conversion: Polar to x = r cos θ Rectangular Coordinates y = r sin θ r = radius, distance from origin θ = angle in standard position n = exponent Conversion: Rectangular to Polar Coordinates Product of Complex Numbers in Polar Form Inner Product of Vectors r = x 2 + y 2, θ = arctan y_, when x > 0 x θ = arctan y_ + π, when x < 0 x r 1 (cos θ 1 + i sin θ 1 ) r 2 (cos θ 2 + i sin θ 2 ) = r 1 r 2 [cos(θ 1 + θ 2 ) + i sin(θ 1 + θ 2 )] a b = a 1 b 1 + a 2 b 2 + a 3 b 3 a = a 1,a 2 vector in the plane a = a 1,a 2,a 3 vector in space Matrices Determinant of a 2 2 Matrix a b det c = ad bc d Determinant of a 3 3 Matrix det = e a det b det d + c det d e d e Inverse of a 2 2 Matrix M 1 = 1 d c where M = b b a a g b h c fj det M h f j g a c f j d g h Trigonometry Sum and Difference Identities sin(α ± β) = sin α cos β ± cos α sin β cos(α ± β) = cos α cos β tan α ± tan β tan(α ± β) = 1 ± tan α tan β ± ± sin α sin β α, β, θ = angles, from positive x-axis Double-Angle Identities sin 2θ = 2 sin θ cos θ cos 2θ = cos 2 θ sin 2 θ 2 tan θ tan 2θ = 1 tan 2 θ Half-Angle Identities sin α =± 2 cos α =± 2 1 cos α cos α 2 tan α =± 1 cos α, where cos α cos α 2010 by ACT, Inc. All rights reserved *0190D1080* Rev 2

4 QualityCore Benchmark Assessment Precalculus Benchmark 1 Polynomial Expressions, Equations, and Functions; Sequences and Series The following pages contain one of the Benchmark Assessments for this course. The table below gives the ID number for each item, the correct answer (Key), the cognitive level, and the alphanumeric code for each ACT Course Standard measured by the item. (The language associated with each code appears in the ACT Course Standards document for this course.) The items in this PDF file appear in the order presented in the table. Multiple-choice (MC) directions follow the table and are followed by a name sheet and the MC items. Following the MC items, you will find a constructed-response (CR) item followed by its scoring criteria and/or scoring rubric. DO NOT DISTRIBUTE SCORING CRITERIA TO STUDENTS. The scoring rubric can be included or excluded at your discretion. ID Key Cognitive Level Standard D L1 E.1.b D L1 E.2.a C L1 E.2.c D L2 E.1.a C L2 E.1.b B L2 E.2.a C L2 E.2.b D L2 E.2.e C L2 E.2.f B L2 E.2.g B L2 G.2.a A L2 G.2.b A L3 E.1.a D L3 E.2.c A L3 E.2.d D L3 E.2.e A L3 E.2.e C L3 G.2.a D L3 G.2.b A L3 G.2.c L3 B.1.c C.1.b E.1.a E.1.b 2008 by ACT, Inc. Permission granted to reproduce this page for QualityCore educational purposes only.

5 Directions: Solve each problem, choose the best answer, and then circle the corresponding letter. Do not linger over problems that take too much time. Solve as many as you can; then return to the others in the time you have left for this test. You are permitted to use a calculator on this test. You may use your calculator for any problems you choose, but some of the problems may best be solved without using a calculator. Note: Unless otherwise stated, all of the following assumptions apply to these problems. 1. Illustrative figures are NOT necessarily drawn to scale. 2. Geometric figures lie in a plane. 3. The word line indicates a straight line. 4. The word average indicates the arithmetic mean by ACT, Inc. Permission granted to reproduce this page for QualityCore educational purposes only.

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13 Scoring Criteria: 21) A 4-point response may include, but is not limited to, the following points: A. Correct sketch: Correct polynomial equation: 32 = πx 3 + 5πx 2 or πx 3 + 5πx 2 32 = 0 Appropriate work needed to find the equation: V = πr 2 h 32 = πx 2 (x + 5) B. Solutions of the equation: x = 4.50, 1.79, and 1.27 Explanation of how the answers were found: First, I rewrote the equation in the form πx 3 + 5πx 2 32 = 0 and then set the expression on the left side equal to y. This yielded y = πx 3 + 5πx Then, I graphed y = πx 3 + 5πx 2 32 on my calculator. This produced a graph that crossed the x-axis 3 times. Finally, I used the CALC zero function of the calculator to find each zero. Note: Depending on the type of calculator the students use, the solutions and explanation will vary. C. Radius and height of the can: Radius = 1.27 inches, height = 6.27 inches Explanation of how the answers were found: The solutions 4.5 and 1.79 must be excluded because they are negative lengths for the radius, which is not possible. Since x = 1.27, this produces a can with a radius of 1.27 inches. I know the height is 5 inches more than the radius, so I added 5 to x to get the height.

14 Rubric: 4 A response at this level provides evidence of thorough knowledge and understanding of the subject matter. The response addresses all parts of the question or problem correctly. The response demonstrates efficient and accurate use of appropriate procedures. The explanation of strategies used in the response shows evidence of a good understanding of mathematical concepts and principles, and it does not contain any misconceptions. The explanation in the response is clear and coherent. 3 A response at this level provides evidence of competent knowledge and understanding of the subject matter. The response addresses most parts of the question or problem correctly. The response includes some minor errors but generally uses appropriate procedures accurately. The explanation of strategies used in the response shows some evidence of a good understanding of mathematical concepts and principles, and it contains few, if any, misconceptions. The explanation in the response is mostly clear and coherent. 2 A response at this level provides evidence of a basic knowledge and understanding of the subject matter. The response addresses some parts of the question or problem correctly. The response includes a number of errors but demonstrates some use of appropriate procedures. The explanation of strategies used in the response shows a little evidence of understanding of mathematical concepts and principles, but it may contain some evidence of misconceptions. The explanation in the response is partially clear, but some parts may be difficult to understand. 1 A response at this level provides evidence of minimal knowledge and understanding of the subject matter. The response addresses a few parts of the problem correctly, but the response is mostly incorrect. The response includes inappropriate procedures or simple manipulations that show little or no understanding of correct procedures. The explanation of strategies used in the response shows little or no evidence of understanding of mathematical concepts and principles, and it may contain evidence of significant misconceptions. Many parts of the explanation are difficult to understand. 0 A response at this level is not scorable. The response is off-topic, blank, hostile, or otherwise not scorable.

15 QualityCore Benchmark Assessment Precalculus Benchmark 2 Conic Sections The following pages contain one of the Benchmark Assessments for this course. The table below gives the ID number for each item, the correct answer (Key), the cognitive level, and the alphanumeric code for each ACT Course Standard measured by the item. (The language associated with each code appears in the ACT Course Standards document for this course.) The items in this PDF file appear in the order presented in the table. Multiple-choice (MC) directions follow the table and are followed by a name sheet and the MC items. Following the MC items, you will find a constructed-response (CR) item followed by its scoring criteria and/or scoring rubric. DO NOT DISTRIBUTE SCORING CRITERIA TO STUDENTS. The scoring rubric can be included or excluded at your discretion. ID Key Cognitive Level Standard D L1 C.1.b C L1 D.1.d C L2 C.1.a D L2 C.1.a D L2 C.1.b B L2 D.1.b B L2 D.1.d D L2 D.1.e B L3 C.1.b C L3 C.1.b C L3 C.1.b A L3 D.1.a A L3 D.1.b A L3 D.1.b B L3 D.1.e L3 B.1.c D.1.e 2008 by ACT, Inc. Permission granted to reproduce this page for QualityCore educational purposes only.

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28 Scoring Criteria: 16) A 4-point response may include, but is not limited to, the following points: A. Correct equation of the ellipse: ( x 2 ) ( y 4 )2 16 = 1 Appropriate work needed to find the answer: Center of ellipse falls at +, 2 2 = (2,4) midpoint of major axis Length of major axis = 12 ft a = 12 2 = 6 a2 = 36 Length of minor axis = 8 ft b = 8 2 = 4 b2 = 16 B. Coordinates of the foci of the ellipse: ( 247,4. ) and ( 647,4. ) Appropriate work needed to find the answer: c 2 = = 20 c = 20 c 4.47 Explanation of how the answer was found: The foci are ( h c, k) ± where c 2 = a 2 b 2 and (h,k) is the center of the ellipse. To solve for c 2, I substituted 36 and 16 for a 2 and b 2 that I found in Part A. Then, I wrote the coordinates of the foci based on the coordinates of the center, which I found in Part A as well. C. Sum of the lengths of the two line segments: 12 ft Appropriate work needed to find the answer: Distance from F 1 to point on major axis (8,4) = 8 ( 2 2 5) = Distance from F 2 to point on major axis (8,4) = 8 ( ) = ( ) + ( )

29 Explanation of how the answer was found: Since, by definition, the sum of the distances from the foci to any point on the ellipse is constant, I used the sum of the distances from the foci to one of the endpoints of the major axis, (8,4) to find the sum of the lengths. I found the lengths from both foci to (8,4), and then added the two lengths together. Note: Students may use any point on the ellipse.

30 Rubric: 4 A response at this level provides evidence of thorough knowledge and understanding of the subject matter. The response addresses all parts of the question or problem correctly. The response demonstrates efficient and accurate use of appropriate procedures. The explanation of strategies used in the response shows evidence of a good understanding of mathematical concepts and principles, and it does not contain any misconceptions. The explanation in the response is clear and coherent. 3 A response at this level provides evidence of competent knowledge and understanding of the subject matter. The response addresses most parts of the question or problem correctly. The response includes some minor errors but generally uses appropriate procedures accurately. The explanation of strategies used in the response shows some evidence of a good understanding of mathematical concepts and principles, and it contains few, if any, misconceptions. The explanation in the response is mostly clear and coherent. 2 A response at this level provides evidence of a basic knowledge and understanding of the subject matter. The response addresses some parts of the question or problem correctly. The response includes a number of errors but demonstrates some use of appropriate procedures. The explanation of strategies used in the response shows a little evidence of understanding of mathematical concepts and principles, but it may contain some evidence of misconceptions. The explanation in the response is partially clear, but some parts may be difficult to understand. 1 A response at this level provides evidence of minimal knowledge and understanding of the subject matter. The response addresses a few parts of the problem correctly, but the response is mostly incorrect. The response includes inappropriate procedures or simple manipulations that show little or no understanding of correct procedures. The explanation of strategies used in the response shows little or no evidence of understanding of mathematical concepts and principles, and it may contain evidence of significant misconceptions. Many parts of the explanation are difficult to understand. 0 A response at this level is not scorable. The response is off-topic, blank, hostile, or otherwise not scorable by ACT, Inc. Permission granted to reproduce this page for QualityCore educational purposes only.

31 QualityCore Benchmark Assessment Precalculus Benchmark 3 Advanced Functions The following pages contain one of the Benchmark Assessments for this course. The table below gives the ID number for each item, the correct answer (Key), the cognitive level, and the alphanumeric code for each ACT Course Standard measured by the item. (The language associated with each code appears in the ACT Course Standards document for this course.) The items in this PDF file appear in the order presented in the table. Multiple-choice (MC) directions follow the table and are followed by a name sheet and the MC items. Following the MC items, you will find a constructed-response (CR) item followed by its scoring criteria and/or scoring rubric. DO NOT DISTRIBUTE SCORING CRITERIA TO STUDENTS. The scoring rubric can be included or excluded at your discretion. ID Key Cognitive Level Standard D L1 F.2.a A L1 F.2.b A L2 F.1.a B L2 F.1.b C L2 F.2.a D L2 F.2.b A L2 F.2.c C L2 F.2.e B L2 F.2.f A L3 F.1.a B L3 F.1.b D L3 F.2.c B L3 F.2.d C L3 F.2.e C L3 F.2.f L3 F.2.f 2008 by ACT, Inc. Permission granted to reproduce this page for QualityCore educational purposes only.

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38 Scoring Criteria: 16) A 4-point response may include, but is not limited to, the following points: A. Number of years she should leave the money in the account: 7.5 yr Appropriate work needed to find the answer: 2500 = 2000( )4t 2500 = ( )4t log ( 5 4 ) log ( 5 4 ) 4t = log 1 + 4t = t = = log( )4t = 4t log( ) 5 log ( ) 4 B. Correct answer: No Explanation needed to find the answer: When Janie first invested the money, her starting principal was only $2,000. It took 7.5 yr to earn $500. If she left the money in the account to earn an additional $500, her starting principal would now be $2,500. As a result, it would take less than 7.5 yr to earn an additional $500. The longer money stays in the account, the more quickly the principal increases. Therefore, more interest is earned each time it is compounded. Note: Students may find the actual time (13.6 yr) it would take to earn $1,000 to show that it is not twice as long. C. Amount she would have to invest: $3, Appropriate work needed to find the answer: P = P ( ) (. ) P = P 1000 =.2513P P =

40 QualityCore Benchmark Assessment Precalculus Benchmark 4 Trigonometric and Periodic Functions The following pages contain one of the Benchmark Assessments for this course. The table below gives the ID number for each item, the correct answer (Key), the cognitive level, and the alphanumeric code for each ACT Course Standard measured by the item. (The language associated with each code appears in the ACT Course Standards document for this course.) The items in this PDF file appear in the order presented in the table. Multiple-choice (MC) directions follow the table and are followed by a name sheet and the MC items. Following the MC items, you will find a constructed-response (CR) item followed by its scoring criteria and/or scoring rubric. DO NOT DISTRIBUTE SCORING CRITERIA TO STUDENTS. The scoring rubric can be included or excluded at your discretion. ID Key Cognitive Level Standard C L1 F.3.e D L1 F.3.l C L2 F.3.a C L2 F.3.b B L2 F.3.d D L2 F.3.f B L2 F.3.g A L2 F.3.h A L2 F.3.j B L2 F.3.k D L2 F.3.l B L3 F.3.a C L3 F.3.c A L3 F.3.c D L3 F.3.d C L3 F.3.e C L3 F.3.f B L3 F.3.h D L3 F.3.i A L3 F.3.j L3 F.3.i F.3.j 2008 by ACT, Inc. Permission granted to reproduce this page for QualityCore educational purposes only.

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51 Scoring Criteria: 21) A 4-point response may include, but is not limited to, the following points: Correct answer: x = 0, π; 3 π, 2π 2 Appropriate work needed to find the answer: sin 2 x + cos 2 x = 1 cos 2 x = 1 sin 2 x sin x cos 2 x + 1 = 0 sin x (1 sin 2 x) + 1 = 0 sin x 1 + sin 2 x + 1 = 0 sin 2 x + sin x = 0 sin x(sin x + 1) = 0 sin x = 0 or sin x = 1 Explanation of how the answer was found: I used the Pythagorean identity sin 2 x + cos 2 x = 1. First, I solved the identity for cos 2 x. Then, I substituted 1 sin 2 x for cos 2 x. This altered the equation so all the trigonometric functions were in terms of sin x. The equation could then be factored and solved. Note: Students can also solve the identity for sin 2 x, substitute sin 2 x for 1 cos 2 x, and continue from there.

53 QualityCore Benchmark Assessment Precalculus Benchmark 5 Polar Coordinates and Vectors The following pages contain one of the Benchmark Assessments for this course. The table below gives the ID number for each item, the correct answer (Key), the cognitive level, and the alphanumeric code for each ACT Course Standard measured by the item. (The language associated with each code appears in the ACT Course Standards document for this course.) The items in this PDF file appear in the order presented in the table. Multiple-choice (MC) directions follow the table and are followed by a name sheet and the MC items. Following the MC items, you will find a constructed-response (CR) item followed by its scoring criteria and/or scoring rubric. DO NOT DISTRIBUTE SCORING CRITERIA TO STUDENTS. The scoring rubric can be included or excluded at your discretion. ID Key Cognitive Level Standard C L1 I.1.a A L2 I.1.a D L2 I.1.b C L2 I.1.c C L2 I.1.e C L2 I.1.g A L2 I.1.h D L2 I.1.i D L2 I.1.j B L3 I.1.b B L3 I.1.d A L3 I.1.f B L3 I.1.g C L3 I.1.h B L3 I.1.i L3 I.1.f 2008 by ACT, Inc. Permission granted to reproduce this page for QualityCore educational purposes only.

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62 Scoring Criteria: 16) A 4-point response may include, but is not limited to, the following points: A. Correct answers: u = 4,3 and v = 7, 2 Appropriate work needed to find the answer: u = 5 1,5 2 v = 13 6, 1 1 Explanation of how answer was found: To get u and v in component form, I had to find the change in both the x and y directions. Thus, I subtracted the coordinates of the initial points from the terminal points for both u and v. B. Correct answers: u + v = 11,1, u v = 3,5 Appropriate work needed to find the answer: u + v = 4,3 + 7, 2 = 4+ 7,3+ ( 2) u v = 4,3 7, 2 = 4 7,3 ( 2 ) C. Correct graph: