Item and Scoring Sampler

Transcription

1 Algebra I Item and Scoring Sampler 2011

2 KEYSTONE ALGEBRA I SAMPLER Table of Contents I. INFORMAT ON BOUT LGEBRA I INTRODUCT ON... 3 ABOUT THE KEYSTONE X MS... 3 AL GNMENT... 3 KEYSTONE X M FORM T... 4 ENER L ESCR PT ON O COR NG U DEL NES OR ALGEBR I... 5 FORMUL HEET... 6 II. ODULE 1 ULT PLE- HO CE QUEST ONS... 7 ONSTRUCTED- ESPONSE QUEST ONS III. ODULE 2 ULT PLE- HO CE QUEST ONS ONSTRUCTED- ESPONSE QUEST ONS Pennsylvania Keystone Algebra I Item Sampler

3 KEYSTONE ALGEBRA I SAMPLER ALGEBRA I Information About Algebra I INTRODUCT ON he ennsylvania epartment of duca on ( ) provides districts and schools with tools to assist in delivering focused instruc onal programs aligned to the state assessment system. hese tools include assessment anchor documents, assessment handbooks, and content-based item and scoring samplers. his 2011 Algebra I Item and coring ampler is a useful tool for ennsylvania educators in the prepara on of local instruc onal programs and in preparing students for the Keystone xams. his Item and coring ampler contains mul ple-choice ques ons and constructed-response ques ons that are aligned to the Keystone Assessment Anchors and ligible ontent. hey provide examples of the types of ques ons that will appear on the opera onal pring 2011 Keystone xams. ach ques on has been through a rigorous review process to ensure alignment with the Assessment Anchors and ligible ontent. he ques ons in this sampler may be used as examples for crea ng assessment ques ons at the classroom level, and they may also be copied and used as part of a local instruc onal program. 1 lassroom teachers may nd it bene cial to have students respond to the constructed-response ques ons in this sampler. ducators can then use the sampler as a guide to score the responses either independently or together with colleagues within a school or district. BOUT THE KEYSTONE XAMS he Keystone xams are end-of-course assessments designed to assess pro ciency in various subject areas, including Algebra I, Alge bra II, iology, hemistry, ivics and overnment, nglish omposi on, eometry, iterature,.. istory, and World istory. he Keystone xams are just one component of ennsylvania s high school gradua on requirements. tudents must also earn state-speci ed credits, ful ll the state s servicelearning and a endance requirements, and complete any addi onal local school system requirements to receive a ennsylvania high school diploma. For detailed informa on about how the Keystone xams are being integrated into the ennsylvania gradua on require ments, please contact the ennsylvania epartment of duca on or visit the Web site at L GNMENT he Algebra I Keystone xam consists of exam ques ons arranged into two modules: pera ons and inear qua ons & Inequali es and inear Func ons and ata rganiza ons. ach module corresponds to speci c content aligned to statements and speci ca ons included in the course-speci c assessment anchor documents. he Algebra I content included in the Keystone Algebra I mul ple-choice ques ons will align with the Assessment Anchors as de ned by the ligible ontent statements. he process skills, direc ves, and ac on statements will also speci cally align with the Assessment Anchors as de ned by the ligible ontent statements. he content included in Algebra I constructed-response ques ons aligns with content included in the ligible ontent statements. he process skills, direc ves, and ac on statements included in the performance demands of the Algebra I constructed-response ques ons align with speci ca ons included in the Assessment Anchor statements, the Anchor escriptor statements, and/or the ligible ontent statements. In other words, the verbs or ac on statements used in the constructed-response ques ons or stems can come from the ligible ontent, Anchor escriptor, or Assessment Anchor statements. 1 he permission to copy and/or use these materials does not extend to commercial purposes. Pennsylvania Keystone Algebra I Item Sampler

4 KEYSTONE ALGEBRA I SAMPLER Information About Algebra I KEYSTONE XAM ORMAT he Algebra I Keystone xam includes ques ons that require students to select the best answer from four possible answer op ons. tudents read each ques on and record their answers in the space provided. he correct answer for each mul ple-choice ques on is worth one point. he Algebra I Keystone xam also includes ques ons that require students to write responses. tudents read the ques on and write their responses in the spaces provided. ach constructed-response ques on is designed to take about ten minutes to complete. uring an actual exam administra on, students are given addi onal me as necessary to complete the exam. ach constructed-response ques on in Algebra I is scored using an item-speci c scoring guideline based on a 0 4 point scale. In this sampler, each item-speci c scoring guideline is combined with sample student responses represen ng each score point to form a prac cal, itemspeci c scoring guide. he sampler also includes the eneral escrip on of coring uidelines for Algebra I used to develop the item-speci c scoring guidelines. hese general guidelines should be used if any addi onal item-speci c scoring guidelines are created for use within local instruc onal programs. Pennsylvania Keystone Algebra I Item Sampler

5 KEYSTONE ALGEBRA I SAMPLER ALGEBRA I Information About Algebra I ENERAL ESCR PT ON OF COR NG U DEL NES FOR LGEBRA I 4 O NTS he response demonstrates a thorough understanding of the mathema cal concepts and procedures required by the task. he response provides correct answer(s) with clear and complete mathema cal procedures shown and a correct explana on, as required by the task. esponse may contain a minor blemish or omission in work or explana on that does not detract from demonstra ng a thorough understanding. 3 O NTS he response demonstrates a general understanding of the mathema cal concepts and procedures required by the task. he response and explana on (as required by the task) are mostly complete and correct. he response may have minor errors or omissions that do not detract from demonstra ng a general understanding. 2 O NTS he response demonstrates a par al understanding of the mathema cal concepts and procedures required by the task. he response is somewhat correct with par al understanding of the required mathema cal concepts and/or procedures demonstrated and/or explained. he response may contain some work that is incomplete or unclear. 1 O NT he response demonstrates a minimal understanding of the mathema cal concepts and procedures required by the task. 0 O NTS he response has no correct answer and insu cient evidence to demonstrate any understanding of the mathema cal concepts and procedures required by the task for that grade level. Pennsylvania Keystone Algebra I Item Sampler

6 KEYSTONE ALGEBRA I SAMPLER Information About Algebra I ORMULA HEET Formulas that you may need to work questions in this sampler are found below. You may use calculator or the number Arithmetic Properties w A = lw Additive Inverse: a + (ˉa) = 0 l 1 Multiplicative Inverse: a = 1 a h V = lwh Commutative Property: a + b = b + a a b = b a l w Associative Property: (a + b) + c = a + (b + c) (a b) c = a (b c) Linear Equations Identity Property: a + 0 = a a 1 = a y 2 Slope: m = y 1 x 2 x 1 Point-Slope Formula: (y y 1 ) = m(x x 1 ) Distributive Property: a (b + c) = a b + a c Multiplicative Property of Zero: a 0 = 0 Slope-Intercept Formula: y = mx + b Additive Property of Equality: If a = b, then a + c = b + c Standard Equation of a Line: Ax + By = C Multiplicative Property of Equality: If a = b, then a c = b c Pennsylvania Keystone Algebra I Item Sampler

7 ALGEBRA I MODULE 1 I - I Q I An expression is shown below Ï } 51x Which value of x makes the expression equivalent to 10 Ï } 51? A. 5 B. 25 * C. 50 D. 100 A student could determine the correct answer, op on, by factoring 10 Ï } 51 as 2 5 Ï } 51, then moving the 5 inside the radical as 2 Ï } = 2 Ï } A student could arrive at an incorrect answer by either using an incorrect method or by making errors in computa on. For example, a student would arrive at op on A if he/she failed to square 5 when he/she moved it under the radical. Pennsylvania Keystone Algebra I Item Sampler

8 ALGEBRA I MODULE Simplify: 2(2 Ï } 4 ) 2 A. B. 1 8 * 1 4 C D. 32 A student could determine the correct answer, op on A, by recognizing 2(2 Ï } 4 ) 2 2 = 2 Ï } 4 2 Ï } 4 = Ï } 4 Ï } 4 = = 1. 8 A student could arrive at an incorrect answer by failing to follow correct order of opera ons or by not knowing how to use radicals or nega ve exponents. For example, a student would arrive at op on if he/she ignored the nega ve exponent and treated 2(2 Ï } 4 ) 2 as 2(2 Ï } 4 ) 2. Pennsylvania Keystone Algebra I Item Sampler

9 ALGEBRA I MODULE A polynomial expression is shown below. (mx 3 + 3) (2x 2 + 5x + 2) (8x x 4 ) The expression is simplified to 8x 3 + 6x x + 6. What is the value of m? A. 8 B. 4 C. 4 * D A student could determine the correct answer, op on, by using correct order of opera ons and the distribu ve property to expand (mx 3 + 3) (2x 2 + 5x + 2) to 2mx 5 + 5mx 4 + 2mx 3 + 6x x + 6. he student could then combine like terms and realize that 2mx 5 8x 5 = 0x 5, so 2m = 8 and m = 4. A student could arrive at an incorrect answer by failing to follow order of opera ons, making an error with the distribu ve property, or incorrectly combining like terms. For example, a student would arrive at op on if he/she failed to distribute and then set mx 3 = 8x 3, so m = Which is a factor of the trinomial x 2 2x 15? A. (x 13) B. (x 5) * C. (x + 5) D. (x + 13) A student could determine the correct answer, op on, by factoring the trinomial x 2 2x 15 as (x 5)(x + 3) and iden fying (x 5) as a factor. A student could arrive at an incorrect answer by failing to correctly factor the trinomial. For example, a student would arrive at op on if he/she factored x 2 2x 15 as (x + 5)(x 3) and iden ed (x + 5) as a factor. Pennsylvania Keystone Algebra I Item Sampler

10 ALGEBRA I MODULE Simplify: x 2 3 x 10 }} x x + 8 ; x 4, 2 A. 1 } 2 x 5 } 4 B. x 2 1 } 2 x 5 } C. D. x 5 } x + 4 x + 5 } x 4 * A student could determine the correct answer, op on, by factoring both the numerator and denominator, then reducing x2 3x 10 x 2 = (x 5)(x + 2) + 6x + 8 (x + 4) (x + 2) = x 5 x + 4. A student could arrive at an incorrect answer by failing to factor the numerator and denominator or by incorrectly factoring the numerator and denominator. For example, a student would arrive at op on by factoring x2 3x 10 x 2 as (x + 5)(x 2) + 6x + 8 (x 4)(x 2). Pennsylvania Keystone Algebra I Item Sampler

11 ALGEBRA I MODULE Anna burned 15 calories per minute running for x minutes and 10 calories per minute hiking for y minutes. She spent a total of 60 minutes running and hiking and burned 700 calories. The system of equations shown below can be used to determine how much time Anna spent on each exercise x + 10y = 700 x + y = 60 What is the value of x, the minutes Anna spent running? A. 10 B. 20 * C. 30 D. 40 A student could determine the correct answer, op on, by solving the system of equa ons using subs tu on. olving the equa on x + y = 60 for y yields y = 60 x. ubs tu ng 60 x in the place of y in the equa on 15x + 10y = 700 yields 15x + 10(60 x) = 700. sing the distribu ve property yields 15x x = 700. ombining like terms and subtrac ng 600 from both sides yields 5x = 100. ividing both sides by 5 yields x = 20. A student could arrive at an incorrect answer by either using an incorrect method for solving a system of equa ons or by making errors in computa on. For example, a student would arrive at op on by incorrectly solving for y as y = x + 60 and then failing to distribute when subs tu ng, yielding 15x + x + 60 = 700. ombining like terms and subtrac ng 60 from both sides yields 16x = 640. ividing both sides by 16 yields x = 40. Pennsylvania Keystone Algebra I Item Sampler

12 ALGEBRA I MODULE Samantha and Maria purchased flowers. Samantha purchased 5 roses for x dollars each and 4 daisies for y dollars each and spent $32 on the flowers. Maria purchased 1 rose for x dollars and 6 daisies for y dollars each and spent $22. The system of equations shown below represents this situation. Which statement is true? A. A rose costs $1 more than a daisy. * B. Samantha spent $4 on each daisy. 5x + 4y = 32 x + 6y = 22 C. Samantha spent more on daisies than she did on roses. D. Maria spent 6 times as much on daisies as she did on roses. A student could determine the correct answer, op on A, by solving the system of equa ons and correctly interpre ng the solu on x = 4 and y = 3. he x-variable refers to the price of a rose and the y-variable refers to the price of a daisy. 4 3 = 1 A student could arrive at an incorrect answer by either making errors in solving the system of equa ons or by incorrectly interpre ng the solu on set. For example, a student would arrive at op on if he/she interpreted the x-value as the price of a daisy. Pennsylvania Keystone Algebra I Item Sampler

13 ALGEBRA I MODULE Which is a graph of the solution of the inequality ) 2x 1 ) 5? A. B. C. D * A student could determine the correct answer, op on A, by simplifying the absolute value inequality. ) 2x 1 ) 5 is eqivalent to 2x 1 5 and 2x 1 5. olving the rst inequality yields x 3. olving the second inequality yields x 2. A student could arrive at an incorrect answer by failing to split the absolute value inequality into simple inequali es before manipula ng to solve the equa on. For example, a student would arrive at op on if he/she rst added 1 to each side of the absolute value inequality, divided both sides by 2, then split the absolute value inequality into simple inequali es. Pennsylvania Keystone Algebra I Item Sampler

14 ALGEBRA I MODULE A baseball team had $1,000 to spend on supplies. The team spent $185 on a new bat. New baseballs cost $4 each. The inequality b 1,000 can be used to determine the number of new baseballs (b) that the team can purchase. Which statement about the number of new baseballs that can be purchased is true? A. The team can purchase 204 new baseballs. B. The minimum number of new baseballs that can be purchased is 185. C. The maximum number of new baseballs that can be purchased is 185. D. The team can purchase 185 new baseballs, but this number is neither the maximum nor the minimum. * A student could determine the correct answer, op on, by solving the inequality and interpre ng the solu on b he variable b represents the number of baseballs that can be purchased. It is a true statement that A student could arrive at an incorrect answer by either making errors in solving the system of equa ons or by incorrectly interpre ng the solu on set. For example, a student would arrive at op on A if he/she switched the sign of the inequality when dividing by 4. Pennsylvania Keystone Algebra I Item Sampler

15 ALGEBRA I MODULE Tyreke always leaves a tip of between 8% and 20% for the server when he pays for his dinner. This can be represented by the system of inequalities shown below, where y is the amount of tip and x is the cost of dinner y > 0.08x y < 0.2x Which of the following is a true statement? A. When the cost of dinner, x, is $10 the amount of tip, y, must be between $2 and $8. B. When the cost of dinner, x, is $15 the amount of tip, y, must be between $1.20 and $3.00. * C. When the tip, y, is $3, the cost of dinner, x, must be between $11 and $23. D. When the tip, y, is $2.40, the cost of dinner, x, must be between $3 and $6. A student could determine the correct answer, op on, by interpre ng the system of inequali es in the context of the problem situa on. When 15 is subs tuted for the x-variable, y > 0.08(15) or y > 1.2 and y < 0.2(15) or y < 3. A student could arrive at an incorrect answer by either making errors in computa on or in interpreta on of the system of inequali es. For example, a student would arrive at op on A if he/she incorrectly calculated 0.08(10) as 8 and switched the signs of both inequali es. Pennsylvania Keystone Algebra I Item Sampler

16 ALGEBRA I MODULE 1 - Q I Keng creates a painting on a rectangular canvas with a width that is four inches longer than the height, as shown in the diagram below. h h + 4 A. Write a polynomial expression, in simplified form, that represents the area of the canvas. Keng adds a 3-inch-wide frame around all sides of his canvas. B. Write a polynomial expression, in simplified form, that represents the total area of the canvas and the frame. Go to the next page to nish question 11. Pennsylvania Keystone Algebra I Item Sampler

17 ALGEBRA I MODULE Continued. Please refer to the previous page for task explanation. Keng is unhappy with his 3-inch-wide frame, so he decides to put a frame with a different width around his canvas. The total area of the canvas and the new frame is given by the polynomial h 2 + 8h + 12, where h represents the height of the canvas. C. Determine the width of the new frame. Show all your work. Explain why you did each step Pennsylvania Keystone Algebra I Item Sampler

18 ALGEBRA I MODULE 1 I - I I I I I ITEM # 11, ODULE 1 ssessment nchor: his item is reported under pera ons with eal umbers and xpressions peci c ligible ontent addressed by this item: coring uide: A Add, subtract, and/or mul ply polynomial expressions (express answers in simplest form). A Factor algebraic expressions, including di erence of squares and trinomials. core he student demonstrates a thorough understanding of opera ons with real numbers and expressions by correctly solving problems with clear and complete procedures and explana ons when required. he student demonstrates a general understanding of opera ons with real numbers and expressions by solving problems and providing procedures and explana ons with only minor errors or omissions. he student demonstrates a par al understanding of opera ons with real numbers and expressions by providing a por on of the correct problem solving, procedures, and explana ons. he student demonstrates a minimal understanding of opera ons with real numbers and expressions. he student does not demonstrate any understanding of opera ons with real numbers and expressions. Pennsylvania Keystone Algebra I Item Sampler

19 ALGEBRA I MODULE 1 op coring esponse: art : What? Why? h 2 + 4h equivalent (1 score point) 1 point for correct expression art : What? Why? h h + 60 equivalent (1 score point) 1 point for correct expression art : What? Why? 1 inch Sample Explana on o do this problem, I factored h 2 + 8h + 12 into (h + 6)(h + 2) to nd the length and height of the canvas and frame. he new height, h + 2, is 2 more than the height of the canvas, h, so the new frame must add a total of 2 inches, 1 inch on each side. (2 score points) 1 point for correct answer 1 point for correct work and explana on Pennsylvania Keystone Algebra I Item Sampler

20 ALGEBRA I MODULE esponse core: 4 points 11. Keng creates a painting on a rectangular canvas with a width that is four inches longer than the height, as shown in the diagram below. h h + 4 A. Write a polynomial expression, in simplified form, that represents the area of the canvas. tudent has given a correct expression. Keng adds a 3-inch-wide frame around all sides of his canvas. B. Write a polynomial expression, in simplified form, that represents the total area of the canvas and the frame. tudent has given a correct expression. Go to the next page to nish question 11. Pennsylvania Keystone Algebra I Item Sampler

21 ALGEBRA I MODULE Continued. Please refer to the previous page for task explanation. Keng is unhappy with his 3-inch-wide frame, so he decides to put a frame with a different width around his canvas. The total area of the canvas and the new frame is given by the polynomial h 2 + 8h + 12, where h represents the height of the canvas. C. Determine the width of the new frame. Show all your work. Explain why you did each step. tudent has given a correct answer. tudent has shown work. tudent has given an explana on ased on coring uidelines, 4 points is representa ve of a thorough understanding. Pennsylvania Keystone Algebra I Item Sampler

22 ALGEBRA I MODULE esponse core: 3 points 11. Keng creates a painting on a rectangular canvas with a width that is four inches longer than the height, as shown in the diagram below. h h + 4 A. Write a polynomial expression, in simplified form, that represents the area of the canvas. tudent has given a correct expression. Keng adds a 3-inch-wide frame around all sides of his canvas. B. Write a polynomial expression, in simplified form, that represents the total area of the canvas and the frame. tudent has given a correct expression. Go to the next page to nish question 11. Pennsylvania Keystone Algebra I Item Sampler

23 ALGEBRA I MODULE Continued. Please refer to the previous page for task explanation. Keng is unhappy with his 3-inch-wide frame, so he decides to put a frame with a different width around his canvas. The total area of the canvas and the new frame is given by the polynomial h 2 + 8h + 12, where h represents the height of the canvas. C. Determine the width of the new frame. Show all your work. Explain why you did each step. tudent has given a correct answer. tudent has shown no work. tudent has given no explana on ased on coring uidelines, 3 points is representa ve of a general understanding. Pennsylvania Keystone Algebra I Item Sampler

24 ALGEBRA I MODULE esponse core: 3 points 11. Keng creates a painting on a rectangular canvas with a width that is four inches longer than the height, as shown in the diagram below. h h + 4 A. Write a polynomial expression, in simplified form, that represents the area of the canvas. tudent has given a correct expression. Keng adds a 3-inch-wide frame around all sides of his canvas. B. Write a polynomial expression, in simplified form, that represents the total area of the canvas and the frame. tudent has given a correct expression. Go to the next page to nish question 11. Pennsylvania Keystone Algebra I Item Sampler

25 ALGEBRA I MODULE Continued. Please refer to the previous page for task explanation. Keng is unhappy with his 3-inch-wide frame, so he decides to put a frame with a different width around his canvas. The total area of the canvas and the new frame is given by the polynomial h 2 + 8h + 12, where h represents the height of the canvas. C. Determine the width of the new frame. Show all your work. Explain why you did each step. tudent has given no answer, the result of a calcula on error of omission. tudent has correct work (all procedures necessary to solve problem are shown). tudent has correct explana on (picture helps) ased on coring uidelines, 3 points is representa ve of a general understanding. Pennsylvania Keystone Algebra I Item Sampler

26 ALGEBRA I MODULE esponse core: 2 points 11. Keng creates a painting on a rectangular canvas with a width that is four inches longer than the height, as shown in the diagram below. h h + 4 A. Write a polynomial expression, in simplified form, that represents the area of the canvas. tudent has given a correct expression. Keng adds a 3-inch-wide frame around all sides of his canvas. B. Write a polynomial expression, in simplified form, that represents the total area of the canvas and the frame. Go to the next page to nish question 11. tudent has given an incorrect expression. tudent has failed to take into considera on that the frame is on all four sides of the gure and not just on two. Pennsylvania Keystone Algebra I Item Sampler

27 ALGEBRA I MODULE Continued. Please refer to the previous page for task explanation. Keng is unhappy with his 3-inch-wide frame, so he decides to put a frame with a different width around his canvas. The total area of the canvas and the new frame is given by the polynomial h 2 + 8h + 12, where h represents the height of the canvas. C. Determine the width of the new frame. Show all your work. Explain why you did each step. tudent has given a correct answer. tudent has shown no correct work. tudent has given no correct explana on ased on coring uidelines, 2 points is representa ve of a par al understanding. Pennsylvania Keystone Algebra I Item Sampler

28 ALGEBRA I MODULE esponse core: 2 points 11. Keng creates a painting on a rectangular canvas with a width that is four inches longer than the height, as shown in the diagram below. h h + 4 A. Write a polynomial expression, in simplified form, that represents the area of the canvas. tudent has given a correct expression. Keng adds a 3-inch-wide frame around all sides of his canvas. B. Write a polynomial expression, in simplified form, that represents the total area of the canvas and the frame. Go to the next page to nish question 11. tudent has given an incorrect expression. tudent has failed to take into considera on that the frame is on all four sides of the gure and not just on two. Pennsylvania Keystone Algebra I Item Sampler

29 ALGEBRA I MODULE Continued. Please refer to the previous page for task explanation. Keng is unhappy with his 3-inch-wide frame, so he decides to put a frame with a different width around his canvas. The total area of the canvas and the new frame is given by the polynomial h 2 + 8h + 12, where h represents the height of the canvas. C. Determine the width of the new frame. Show all your work. Explain why you did each step. tudent has given an incorrect answer. tudent has shown correct work up to interpre ng answer. tudent has given correct explana on ased on coring uidelines, 2 points is representa ve of a par al understanding. Pennsylvania Keystone Algebra I Item Sampler

30 ALGEBRA I MODULE esponse core: 1 point 11. Keng creates a painting on a rectangular canvas with a width that is four inches longer than the height, as shown in the diagram below. h h + 4 A. Write a polynomial expression, in simplified form, that represents the area of the canvas. tudent has given a correct expression. Keng adds a 3-inch-wide frame around all sides of his canvas. B. Write a polynomial expression, in simplified form, that represents the total area of the canvas and the frame. tudent has given an incorrect expression. Go to the next page to nish question 11. Pennsylvania Keystone Algebra I Item Sampler

31 ALGEBRA I MODULE Continued. Please refer to the previous page for task explanation. Keng is unhappy with his 3-inch-wide frame, so he decides to put a frame with a different width around his canvas. The total area of the canvas and the new frame is given by the polynomial h 2 + 8h + 12, where h represents the height of the canvas. C. Determine the width of the new frame. Show all your work. Explain why you did each step. tudent has given an incorrect answer. tudent has shown no correct work. tudent has given no correct explana on ased on coring uidelines, 1 point is representa ve of a minimal understanding. Pennsylvania Keystone Algebra I Item Sampler

32 ALGEBRA I MODULE esponse core: 1 point 11. Keng creates a painting on a rectangular canvas with a width that is four inches longer than the height, as shown in the diagram below. h h + 4 A. Write a polynomial expression, in simplified form, that represents the area of the canvas. tudent has given a correct expression. Keng adds a 3-inch-wide frame around all sides of his canvas. B. Write a polynomial expression, in simplified form, that represents the total area of the canvas and the frame. tudent has given an incorrect expression. Go to the next page to nish question 11. Pennsylvania Keystone Algebra I Item Sampler

33 ALGEBRA I MODULE Continued. Please refer to the previous page for task explanation. Keng is unhappy with his 3-inch-wide frame, so he decides to put a frame with a different width around his canvas. The total area of the canvas and the new frame is given by the polynomial h 2 + 8h + 12, where h represents the height of the canvas. C. Determine the width of the new frame. Show all your work. Explain why you did each step. tudent has given an incorrect answer. tudent has shown no correct work. tudent has given no correct explana on ased on coring uidelines, 1 point is representa ve of a minimal understanding. Pennsylvania Keystone Algebra I Item Sampler

34 ALGEBRA I MODULE esponse core: Keng creates a painting on a rectangular canvas with a width that is four inches longer than the height, as shown in the diagram below. h h + 4 A. Write a polynomial expression, in simplified form, that represents the area of the canvas. tudent has given an incorrect expression. Keng adds a 3-inch-wide frame around all sides of his canvas. B. Write a polynomial expression, in simplified form, that represents the total area of the canvas and the frame. tudent has given an incorrect expression. Go to the next page to nish question 11. Pennsylvania Keystone Algebra I Item Sampler

35 ALGEBRA I MODULE Continued. Please refer to the previous page for task explanation. Keng is unhappy with his 3-inch-wide frame, so he decides to put a frame with a different width around his canvas. The total area of the canvas and the new frame is given by the polynomial h 2 + 8h + 12, where h represents the height of the canvas. C. Determine the width of the new frame. Show all your work. Explain why you did each step. tudent has given an incorrect answer. tudent has shown no correct work. tudent has given no correct explana on ased on coring uidelines, 0 points is representa ve of no understanding. Pennsylvania Keystone Algebra I Item Sampler

36 ALGEBRA I MODULE The diagram below shows 5 identical bowls stacked one inside the other. Bowls 5 inches 2 inches The height of 1 bowl is 2 inches. The height of a stack of 5 bowls is 5 inches. A. Write an equation using x and y to find the height of a stack of bowls based on any number of bowls. equation: B. Describe what the x and y variables represent. x-variable: y-variable: Go to the next page to nish question 12. Pennsylvania Keystone Algebra I Item Sampler

37 ALGEBRA I MODULE Continued. Please refer to the previous page for task explanation. C. What is the height, in inches, of a stack of 10 bowls? height: inches Pennsylvania Keystone Algebra I Item Sampler

38 ALGEBRA I MODULE 1 I - I I I I I ITEM # 12, ODULE 1 ssessment nchor: his item is reported under inear qua ons peci c ligible ontent addressed by this item: coring uide: A Write, solve, and/or apply a linear equa on (including problem situa ons). A Interpret solu ons to problems in the context of the problem situa on (linear equa ons only). core he student demonstrates a thorough understanding of linear equa ons by correctly solving 4 problems. he student demonstrates a general understanding of linear equa ons by solving problems 3 with only minor errors or omissions. he student demonstrates a par al understanding of linear equa ons by providing a por on 2 of the correct problem solving. 1 he student demonstrates a minimal understanding of linear equa ons. 0 he student does not demonstrate any understanding of linear equa ons. Pennsylvania Keystone Algebra I Item Sampler

39 ALGEBRA I MODULE 1 op coring esponse: art : What? y = 0.75x equivalent Why? (1 score point) 1 point for correct equa on art : What? x-variable: the number of bowls y-variable: the height of the stack of bowls equivalent Why? (2 score points) 1 point for correct descrip on of x-variable 1 point for correct descrip on of y-variable art : What? 8.75 inches equivalent Why? (1 score point) 1 point for correct answer Pennsylvania Keystone Algebra I Item Sampler

40 ALGEBRA I MODULE esponse core: 4 points 12. The diagram below shows 5 identical bowls stacked one inside the other. Bowls 5 inches 2 inches The height of 1 bowl is 2 inches. The height of a stack of 5 bowls is 5 inches. A. Write an equation using x and y to find the height of a stack of bowls based on any number of bowls. equation: tudent has given a correct equa on. B. Describe what the x and y variables represent. x-variable: y-variable: tudent has given two correct descrip ons. Go to the next page to nish question 12. Pennsylvania Keystone Algebra I Item Sampler

41 ALGEBRA I MODULE Continued. Please refer to the previous page for task explanation. C. What is the height, in inches, of a stack of 10 bowls? height: inches tudent has given a correct answer ased on coring uidelines, 4 points is representa ve of a thorough understanding. Pennsylvania Keystone Algebra I Item Sampler

42 ALGEBRA I MODULE esponse core: 3 points 12. The diagram below shows 5 identical bowls stacked one inside the other. Bowls 5 inches 2 inches The height of 1 bowl is 2 inches. The height of a stack of 5 bowls is 5 inches. A. Write an equation using x and y to find the height of a stack of bowls based on any number of bowls. equation: tudent has given an incorrect equa on. B. Describe what the x and y variables represent. x-variable: y-variable: tudent has given two correct descrip ons. Go to the next page to nish question 12. Pennsylvania Keystone Algebra I Item Sampler

43 ALGEBRA I MODULE Continued. Please refer to the previous page for task explanation. C. What is the height, in inches, of a stack of 10 bowls? height: inches tudent has given a correct answer based on an incorrect equa on from art A. ( rror is carried through correctly, so student is not penalized twice.) ased on coring uidelines, 3 points is representa ve of a general understanding. Pennsylvania Keystone Algebra I Item Sampler

44 ALGEBRA I MODULE esponse core: 3 points 12. The diagram below shows 5 identical bowls stacked one inside the other. Bowls 5 inches 2 inches The height of 1 bowl is 2 inches. The height of a stack of 5 bowls is 5 inches. A. Write an equation using x and y to find the height of a stack of bowls based on any number of bowls. equation: tudent has given a correct equa on in frac on form. B. Describe what the x and y variables represent. x-variable: y-variable: tudent has given two correct descrip ons. Go to the next page to nish question 12. Pennsylvania Keystone Algebra I Item Sampler

45 ALGEBRA I MODULE Continued. Please refer to the previous page for task explanation. C. What is the height, in inches, of a stack of 10 bowls? height: inches tudent has given an incorrect answer ased on coring uidelines, 3 points is representa ve of a general understanding. Pennsylvania Keystone Algebra I Item Sampler

46 ALGEBRA I MODULE esponse core: 2 points 12. The diagram below shows 5 identical bowls stacked one inside the other. Bowls 5 inches 2 inches The height of 1 bowl is 2 inches. The height of a stack of 5 bowls is 5 inches. A. Write an equation using x and y to find the height of a stack of bowls based on any number of bowls. equation: tudent has given an incorrect equa on. B. Describe what the x and y variables represent. x-variable: y-variable: tudent has given two correct descrip ons. Go to the next page to nish question 12. Pennsylvania Keystone Algebra I Item Sampler

47 ALGEBRA I MODULE Continued. Please refer to the previous page for task explanation. C. What is the height, in inches, of a stack of 10 bowls? height: inches tudent has given an incorrect answer ased on coring uidelines, 2 points is representa ve of a par al understanding. Pennsylvania Keystone Algebra I Item Sampler

48 ALGEBRA I MODULE esponse core: 2 points 12. The diagram below shows 5 identical bowls stacked one inside the other. Bowls 5 inches 2 inches The height of 1 bowl is 2 inches. The height of a stack of 5 bowls is 5 inches. A. Write an equation using x and y to find the height of a stack of bowls based on any number of bowls. equation: tudent has given an incorrect equa on. B. Describe what the x and y variables represent. x-variable: y-variable: tudent has given one incorrect descrip on followed by one correct descrip on. Go to the next page to nish question 12. Pennsylvania Keystone Algebra I Item Sampler

49 ALGEBRA I MODULE Continued. Please refer to the previous page for task explanation. C. What is the height, in inches, of a stack of 10 bowls? height: inches tudent has given a correct answer based on an incorrect equa on from art A. ( rror is carried through correctly, so student is not penalized twice.) ased on coring uidelines, 2 points is representa ve of a par al understanding. Pennsylvania Keystone Algebra I Item Sampler

50 ALGEBRA I MODULE esponse core: 1 point 12. The diagram below shows 5 identical bowls stacked one inside the other. Bowls 5 inches 2 inches The height of 1 bowl is 2 inches. The height of a stack of 5 bowls is 5 inches. A. Write an equation using x and y to find the height of a stack of bowls based on any number of bowls. equation: tudent has given an incorrect equa on. B. Describe what the x and y variables represent. x-variable: y-variable: tudent has given one correct descrip on followed by one incorrect descrip on. Go to the next page to nish question 12. Pennsylvania Keystone Algebra I Item Sampler

51 ALGEBRA I MODULE Continued. Please refer to the previous page for task explanation. C. What is the height, in inches, of a stack of 10 bowls? height: inches tudent has given an incorrect answer ased on coring uidelines, 1 point is representa ve of a minimal understanding. Pennsylvania Keystone Algebra I Item Sampler

52 ALGEBRA I MODULE esponse core: 1 point 12. The diagram below shows 5 identical bowls stacked one inside the other. Bowls 5 inches 2 inches The height of 1 bowl is 2 inches. The height of a stack of 5 bowls is 5 inches. A. Write an equation using x and y to find the height of a stack of bowls based on any number of bowls. equation: tudent has given an incorrect equa on. B. Describe what the x and y variables represent. x-variable: y-variable: tudent has given two incorrect descrip ons. Go to the next page to nish question 12. Pennsylvania Keystone Algebra I Item Sampler

53 ALGEBRA I MODULE Continued. Please refer to the previous page for task explanation. C. What is the height, in inches, of a stack of 10 bowls? height: inches tudent has given a correct answer ased on coring uidelines, 1 point is representa ve of a minimal understanding. Pennsylvania Keystone Algebra I Item Sampler

54 ALGEBRA I MODULE esponse core: The diagram below shows 5 identical bowls stacked one inside the other. Bowls 5 inches 2 inches The height of 1 bowl is 2 inches. The height of a stack of 5 bowls is 5 inches. A. Write an equation using x and y to find the height of a stack of bowls based on any number of bowls. equation: tudent has given an incorrect equa on. B. Describe what the x and y variables represent. x-variable: y-variable: tudent has given two incorrect descrip ons. Go to the next page to nish question 12. Pennsylvania Keystone Algebra I Item Sampler

55 ALGEBRA I MODULE Continued. Please refer to the previous page for task explanation. C. What is the height, in inches, of a stack of 10 bowls? height: inches tudent has given an incorrect answer ased on coring uidelines, 0 points is representa ve of no understanding. Pennsylvania Keystone Algebra I Item Sampler

56 ALGEBRA I MODULE 2 I - I Q I Which graph shows y as a function of x? A. y B. * y x x C. y D. y x x A student could determine the correct answer, op on, by examining the graphs of each of the rela ons and using the ver cal line test. A student could arrive at an incorrect answer by either not knowing the de ni on of a func on or not knowing how to interpret closed and open circles. For example, a student would arrive at op on if he/she thought func on meant con nuous. Pennsylvania Keystone Algebra I Item Sampler

57 ALGEBRA I MODULE The graph of a function is shown below. f(x) x Which value is not in the range of the function? A. 0 B. 3 * C. 4 D. 5 A student could determine the correct answer, op on, by examining the graph and seeing that it never intersects with the horizontal line y = 3. A student could arrive at an incorrect answer by either not knowing the de ni on of range or not knowing how to interpret closed and open circles. For example, a student would arrive at op on if he/she thought range meant that there was only one value of x for each value of y. Pennsylvania Keystone Algebra I Item Sampler

58 ALGEBRA I MODULE A pizza restaurant charges for pizzas and adds a delivery fee. The cost (c), in dollars, to have any number of pizzas (p) delivered to a home is described by the function c = 8p + 3. Which statement is true? A. The cost of 8 pizzas is $11. B. The cost of 3 pizzas is $14. C. Each pizza costs $8 and the delivery fee is $3. * D. Each pizza costs $3 and the delivery fee is $8. A student could determine the correct answer, op on, by interpre ng the linear equa on c = 8p + 3 in the context of the problem situa on. A student could arrive at an incorrect answer by misinterpre ng the linear equa on c = 8p + 3 in the context of the problem situa on. For example, a student would arrive at op on if he/she interpreted the cost of a pizza to be the y-intercept and the delivery fee to be the slope. Pennsylvania Keystone Algebra I Item Sampler

59 ALGEBRA I MODULE The table below shows values of y as a function of x. x y Which linear equation best describes the relationship between x and y? A. y = 2.5x + 5 B. y = 3.75x * C. y = 4x + 1 D. y = 5x A student could determine the correct answer, op on, by iden fying the linear equa on which will map every value of x in the table to the corresponding value of y. A student could arrive at an incorrect answer by checking only one of the (x, y) coordinate pairs in the table. For example, a student could arrive at op on A if he/she only checked to see that the equa on worked when x = 2 and y = 10. Pennsylvania Keystone Algebra I Item Sampler

60 ALGEBRA I MODULE Jeff s restaurant sells hamburgers. The amount charged for a hamburger, h, is based on the cost for a plain hamburger plus an additional charge for each topping, t, as shown in the equation below h = 0.60t + 5 What does the number 0.60 represent in the equation? A. the number of toppings B. the cost of a plain hamburger C. the additional cost for each topping * D. the cost of a hamburger with 1 topping A student could determine the correct answer, op on, by interpre ng the linear equa on h = 0.60t + 5 in the context of the problem situa on. A student could arrive at an incorrect answer by misinterpre ng the linear equa on h = 0.60t + 5 in the context of the problem situa on. For example, a student would arrive at op on A if he/she interpreted the number of toppings to be the rate of change. Pennsylvania Keystone Algebra I Item Sampler

61 ALGEBRA I MODULE A graph of a linear equation is shown below. y x Which equation describes the graph? A. y = 0.5x 1.5 B. y = 0.5x + 3 C. y = 2x 1.5 D. y = 2x + 3 * A student could determine the correct answer, op on, by examining the graph to obtain the slope and y-intercept. A student could arrive at an incorrect answer by either not knowing how to nd the slope or y-intercept of a graph. For example, a student would arrive at op on if he/she used the x-intercept instead of the y-intercept. Pennsylvania Keystone Algebra I Item Sampler

62 ALGEBRA I MODULE The scatter plot below shows the cost, y, of ground shipping packages from Harrisburg, PA, to Minneapolis, MN, based on the package weight, x. y Ground Shipping Cost (in dollars) x Weight (in pounds) Which equation best describes the line of best fit? A. y = 0.37x B. y = 0.37x C. y = 0.68x D. y = 0.68x * A student could determine the correct answer, op on, by drawing and deriving the equa on of the line of best t. A student could arrive at an incorrect answer by either not knowing how to draw a line of best t or not knowing how to nd the equa on of that line. For example, a student would arrive at op on if he/she drew a line such that all of the data points are at or above the line. Pennsylvania Keystone Algebra I Item Sampler

63 ALGEBRA I MODULE The daily high temperatures in degrees Fahrenheit in Allentown, PA, for a period of 10 days are shown below Which statement correctly describes the data? A. The median value is 98. B. The interquartile range is 16. * C. The lower quartile value is 76. D. The upper quartile value is 96. A student could determine the correct answer, op on, by nding the di erence between the third and rst quar le. Arranging the data from lowest to highest shows that the median value is the average of 89 and 91. he third quar le value is the median of the upper half of the data, 98, and the rst quar le value is the median of the lower half of the data, = 16. A student could arrive at an incorrect answer by not knowing how to use or compute median or interquar le range. For example, a student would arrive at op on A if he/she confused median and mode. Pennsylvania Keystone Algebra I Item Sampler

64 ALGEBRA I MODULE Vy asked 200 students to select their favorite sport and then recorded the results in the bar graph below. Votes Favorite Sport football basketball baseball hockey volleyball track and field Sports Vy will ask another 80 students to select their favorite sport. Based on the information in the bar graph, how many more students of the next 80 asked will select basketball rather than football as their favorite sport? A. 10 * B. 20 C. 25 D. 30 A student could determine the correct answer, op on A, by using the bar graph to obtain probabili es for basketball ( = 0.375) and football ( = 0.25), subtract the di erence in the probabili es ( = 0.125) and mul ply by the new sample ( = 10). A student could arrive at an incorrect answer by using an incorrect method or making a computa onal error. For example, a student would arrive at op on if he/she mul plied the probability di erence by 200 instead of 80 ( = 25). Pennsylvania Keystone Algebra I Item Sampler

65 ALGEBRA I MODULE A number cube with sides labeled 1 6 is rolled two times, and the sum of the numbers that end face up is calculated. What is the probability that the sum of the numbers is 3? A. B. C. D. 1 } 18 * 1 } 12 1 } 9 1 } 2 A student could determine the correct answer, op on A, by realizing that the possible combina ons are 2 and 1 or 1 and 2. here are 2 ways to get a number for the rst number cube out of 6 possible outcomes, 2 6, and only 1 way to get a number for the second number cube, 1 } 6. ul plying the probabili es together } 2 6 } 1 6 = } 2 36 which can be reduced to } A student could arrive at an incorrect answer by using an incorrect method or making a computa onal error. For example, a student would arrive at op on if he/she decided the probability for picking the rst number cube was 2 } 6 and that the second number cube was also 2 6, then 2 } 6 2 } 6 = 4 } 36 which can be reduced to 1 } 9. Pennsylvania Keystone Algebra I Item Sampler

66 ALGEBRA I MODULE 2 - Q I Hector s family is on a car trip. When they are 84 miles from home, Hector begins recording their distance driven each hour in the table below. The pattern continues. Time in Hours Distance by Hour Distance in Miles A. Write an equation to find distance driven in miles (d) after a given number of hours (h). B. Hector also kept track of the remaining gasoline. The equation shown below can be used to find the gallons of gasoline remaining (g) after distance driven (d ). g = 16 1 } 20 d Use the equation to find the missing values for gallons of gasoline remaining. Distance Driven in Miles (d ) Gallons of Gasoline Remaining (g ) Go to the next page to nish question 11. Pennsylvania Keystone Algebra I Item Sampler

67 ALGEBRA I MODULE Continued. Please refer to the previous page for task explanation. C. Draw the graph of the line formed by the points in the table from part B. g d D. Explain why the slope of the line drawn in part C must be negative Pennsylvania Keystone Algebra I Item Sampler

68 ALGEBRA I MODULE 2 I - I I I I I ITEM # 11, ODULE 2 ssessment nchor: his item is reported under unc ons peci c ligible ontent addressed by this item: coring uide: A Analyze a set of data for the existence of a pa ern and represent the pa ern algebraically and/or graphically. A reate, interpret, and/or use the equa on, graph, or table of a linear func on. core he student demonstrates a thorough understanding of func ons by correctly solving 4 problems with clear and complete procedures and explana ons when required. he student demonstrates a general understanding of func ons by solving problems and 3 providing procedures and explana ons with only minor errors or omissions. he student demonstrates a par al understanding of func ons by providing a por on of the 2 correct problem solving, procedures, and explana ons. 1 he student demonstrates a minimal understanding of func ons. 0 he student does not demonstrate any understanding of func ons. Pennsylvania Keystone Algebra I Item Sampler