CEAFE Fall 2012 Item Results and Learning Outcomes Correspondence · 2017. 4. 17. · Fall 2012 Item Results and Learning Outcomes Correspondence Purpose of Exam: Students enrolled

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CUNY Office of Institutional Research and Assessment 2/5/2013

CUNY Elementary Algebra Final Exam (CEAFE)

Fall 2012 Item Results and Learning Outcomes Correspondence

Purpose of Exam:

Students enrolled in any CUNY Elementary Algebra developmental courses, workshops or other interventions

demonstrate readiness for college level courses in mathematics by: 1) passing the University-wide CUNY Elementary

Algebra Final Exam with a score of 60 or higher; and 2) earning an overall average of a 74 or higher in the intervention.

The CUNY Elementary Algebra Final Exam is worth 35% of the final grade.

Description of Exam:

In the Fall 2012 implementation, the Question Pool contained 743 questions divided into 25 Question Groups. Each

Question Group contained 16-80 variations of each question. Each student’s test was randomly generated by selecting

one item from each Question Group, and the items were randomly ordered.

Data presented in the tables below include approximately 10,790 students University-wide who tested 12/1/2012-

12/24/2012.

Table 1 contains summary data for each question, and Table 2 contains more detailed information about each question.

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Description of Columns in Table 1:

Learning Outcomes Headings are from the CUNY Elementary Algebra Proficiency Standards Details (as amended

3/30/2012).

Mean Correct is the quotient of the number of correct responses by the number of nonblank responses. The inclusion of

available data on blank responses would result in maximum variance of .006 from listed values and does not result in any

change in rankings listed below. Table 1: Summary Item Outcomes and Correspondence, sorted by Mean Correct

Question Group

Learning Outcomes Heading Mean Correct

3 Operations with scientific notation 0.278

22 Equations of vertical and horizontal lines 0.419

18 Solve linear inequalities in 1 variable 0.472

2 Operations with radicals (multiplication and division) 0.483

20 Graph a line from an equation 0.489

4 Operations with exponents 0.490

9 Factor polynomials (non-monic trinomials) 0.498

23 Slope and 𝑦-intercept of a line from an equation 0.522

10 Factor polynomials (grouping; multiple variables) 0.540

13 Solve a system of linear equations 0.547

21 Equation of a line from two points 0.564

15 Solve factorable quadratic equations 0.571

8 Factor polynomials (difference of squares; two steps) 0.572

16 Solve quadratic equations with no linear term 0.578

25 Word problem: Percent 0.590

1 Operations with radicals (addition and subtraction) 0.637

17 Radicals and the Pythagorean Theorem 0.651

5 Polynomial operations (addition and subtraction) 0.673

14 Solve literal equations 0.689

7 Polynomial operations (division by a monomial) 0.694

11 Translate words into algebraic relationships 0.696

12 Solve a linear equation in one variable 0.736

6 Polynomial operations (multiplication) 0.744

19 Use function notation 0.784

24 Word problem: Proportions 0.843

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Description of Columns in Table 2:

Learning Outcomes listed are from the CUNY Elementary Algebra Proficiency Standards Details (as amended 3/30/2012).

Sample Questions are from the August 2012 Sample-A exam. An additional sample item is available in the August 2012 Sample-B exam.

Mean Correct is the quotient of the number of correct responses by the number of nonblank responses. The inclusion of available data on blank responses

would result in maximum variance of .006 from listed values.

Table 2: Detailed Item Outcomes and Correspondence, sorted by Question Group

Question Group

Learning Outcomes Sample Question Mean Correct

1 Operations with radicals (addition and subtraction) 1)a. Radicals. Includes only square roots of nonnegative numbers. 1)a.i. Simplify radical terms (no variable in the radicand). (AN2) 1)a.ii. Perform addition, subtraction, multiplication and division using like and unlike radical terms and express the result in simplest form. (AN3*) 1)a.ii.1. Multiplication should involve at most one factor of

the form 𝑎 + 𝑏√𝑑 with 𝑎 ≠ 0. 1)a.ii.2. All divisors and denominators should be of the form

𝑎 + 𝑏√𝑑 with 𝑎 = 0.

Simplify.

𝟕√𝟐𝟒 − 𝟑√𝟔

A) 42√2 − 3√6 B) 25√6 C) 11√6 D) 12√2

0.637

2 Operations with radicals (multiplication and division) 1)a. Radicals. Includes only square roots of nonnegative numbers. 1)a.i. Simplify radical terms (no variable in the radicand). (AN2) 1)a.ii. Perform addition, subtraction, multiplication and division using like and unlike radical terms and express the result in simplest form. (AN3*) 1)a.ii.1. Multiplication should involve at most one factor of

the form 𝑎 + 𝑏√𝑑 with 𝑎 ≠ 0. 1)a.ii.2. All divisors and denominators should be of the form

𝑎 + 𝑏√𝑑 with 𝑎 = 0.

Simplify completely.

√𝟓(√𝟏𝟓 + √𝟓)

A) 5√3 + √5 B) 25√3 C) 3√5 + 5 D) 5√3 + 5

0.483

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Question Group


3 Operations with scientific notation 1)b.i. Convert between standard decimal and scientific notation. 1)b.ii. Understand and use scientific notation to compute products and quotients of numbers. (AN4)

Multiply. Give the answer in scientific notation.

(𝟑 × 𝟏𝟎𝟔)(𝟒 × 𝟏𝟎−𝟐)

A) 12 × 104 B) 1.2 × 104 C) 1.2 × 105 D) 1.2 × 103

0.278

4 Operations with exponents 1)c. Exponents. Multiply and divide monomial expressions with a common base using the properties of exponents. All exponents are integral. (AA12)

Simplify.

𝒙𝟓𝒙𝟕

(𝒙𝟑)𝟐

A) 𝑥2

B) 𝑥6

C) 𝑥7

D) 𝑥29

0.490

5 Polynomial operations (addition and subtraction) 2)b. Add and subtract monomials and polynomials. (AA13*)


(𝟓𝒙𝟐 − 𝟕𝒙 + 𝟗) − (−𝟐𝒙𝟐 − 𝟑𝒙 + 𝟐)

A) 3𝑥2 − 4𝑥 + 7 B) 7𝑥2 − 4𝑥 + 7 C) 7𝑥2 − 10𝑥 + 7 D) 7𝑥2 − 4𝑥 + 11

0.673

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Question Group


6 Polynomial operations (multiplication) 1)c. Exponents. Multiply and divide monomial expressions with a common base using the properties of exponents. All exponents are integral. (AA12) 2)c. Multiplication of a monomial and binomial by any degree polynomial. (AA13*)

Multiply.

(𝟐𝒙 − 𝟓)(𝒙𝟐 + 𝟒𝒙 − 𝟔)

A) 2𝑥3 + 3𝑥2 − 32𝑥 + 30 B) 2𝑥3 + 8𝑥2 − 12𝑥 + 30 C) 2𝑥3 + 3𝑥2 − 12𝑥 + 30 D) 2𝑥3 + 8𝑥2 − 32𝑥 + 30

0.744

7 Polynomial operations (division by a monomial) 1)c. Exponents. Multiply and divide monomial expressions with a common base using the properties of exponents. All exponents are integral. (AA12) 2)d. Divide a polynomial by a monomial, where the quotient has no remainder. (AA14*)


𝟐𝟓𝒙𝟑 − 𝟑𝟓𝒙𝟐 + 𝟓𝒙

−𝟓𝒙

A) −5𝑥2 + 7𝑥 B) 25𝑥3 − 35𝑥2 C) 5𝑥2 − 7𝑥 + 1 D) −5𝑥2 + 7𝑥 − 1

0.694

8 Factor polynomials (difference of squares; two steps) 2)e.i. Identify and factor the greatest common factor from an algebraic expression. 2)e.ii. Identify and factor the difference of two perfect squares. (AA19) 2)e.v. Factor algebraic expressions completely where the factorization requires more than one step (e.g. first remove the GCF and then factor the remaining factor). (AA20*)

Factor completely.

𝟑𝟔𝒙𝟐𝒚 − 𝟏𝟎𝟎𝒚𝟑

A) 4(9𝑥2𝑦 − 25𝑦3) B) 4𝑦(9𝑥2 − 25𝑦2) C) 4𝑦(3𝑥 − 5𝑦)(3𝑥 + 5𝑦) D) 4𝑦(3𝑥 − 5𝑦)2

0.572

9 Factor polynomials (non-monic trinomials) 2)e.iii. Factor all trinomials of a single variable, including a leading coefficient other than 1.

Which of the following is a factor of the polynomial?

𝟐𝒙𝟐 − 𝒙 − 𝟓𝟓

A) 𝑥 + 11 B) 𝑥 − 5 C) 2𝑥 + 11 D) 2𝑥 − 11

0.498

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Question Group


10 Factor polynomials (grouping; multiple variables) 2)e.iv. Factor algebraic expressions by grouping with up to 4 terms, possibly with multiple variables.

Which of the following is a factor of the polynomial? 𝟐𝟏𝒂𝒃 − 𝟏𝟒𝒂𝒙 + 𝟏𝟓𝒃𝒚 − 𝟏𝟎𝒙𝒚

A) 3𝑏 − 2𝑥 B) 3𝑏 + 2𝑥 C) 7𝑎 − 5𝑦 D) 7𝑎 + 2𝑦

0.540

11 Translate words into algebraic relationships 2)a. Translate a quantitative verbal phrase into an algebraic expression. (AA1) 3)a. Translate verbal sentences into mathematical equations. (AA4)

If 𝑛 represents a number, which equation is a correct translation of the sentence?

𝟏𝟓 is 𝟏𝟐 less than 2 times a number.

A) 15 = 12 − 2𝑛 B) 15 = 2(𝑛 − 12) C) 15 = 2𝑛 − 12 D) 15 = 2(12 − 𝑛)

0.696

12 Solve a linear equation in one variable 3)b. Solve all types of linear equations in one variable. (AA22)

Solve for 𝑛.

𝟓(𝟖 − 𝒏) = 𝟑𝒏 − 𝟏𝟔

A) 𝑛 = 3

B) 𝑛 = −3

C) 𝑛 = −7

D) 𝑛 = 7

0.736

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Question Group


13 Solve a system of linear equations 3)c. Systems of Linear Equations (2x2) † 3)c.i. Solve systems of two linear equations in two variables algebraically. (AA10) 3)c.ii. Graph and solve systems of linear equations with rational coefficients in two variables. (AG7*) 3)c.† Note: On a multiple choice exam it is impossible to impose a solution method on students. As a result, we will combine these two objectives into a single test item and assume students may use either method when answering the question.

What is the value of the 𝑦-coordinate of the solution to the system of

equations?

𝒙 + 𝟑𝒚 = 𝟐

−𝟑𝒙 − 𝟖𝒚 = 𝟒

A) 𝑦 = −2

B) 𝑦 = 10

C) 𝑦 = 6

D) 𝑦 = −10

0.547

14 Solve literal equations 3)d. Solve literal equations for a given variable. (AA23) (Area and perimeter formulas should be included as one source of examples.)

Solve for 𝑥.

𝒛 = 𝟓𝒙 + 𝒚

A) 𝑥 =𝑧+𝑦

5

B) 𝑥 =𝑧−𝑦

5

C) 𝑥 = 𝑧 5

− 𝑦

D) 𝑥 = 5(𝑧 − 𝑦)

0.689

15 Solve factorable quadratic equations 2)e.iii. Factor all trinomials of a single variable, including a leading coefficient other than 1. 3)e.i. Understand and apply the multiplication property of zero to solve quadratic equations with integral coefficients. (AA27*)

Find all solutions to the equation.

𝟒𝒃𝟐 + 𝟖𝒃 = 𝟎

A) Only 𝑏 = −2 B) Only 𝑏 = 2 C) 𝑏 = 0 or 𝑏 = 2 D) 𝑏 = 0 or 𝑏 = −2

0.571

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Question Group


16 Solve quadratic equations with no linear term 3)e.ii. Solve quadratic equations with no linear term.

Find all solutions to the equation.

𝟏𝟎𝒙𝟐 = 𝟒𝟗𝟎

A) 𝑥 = 7 or 𝑥 = 49 B) 𝑥 = 0 or 𝑥 = 49 C) 𝑥 = 7 or 𝑥 = −7 D) Only 𝑥 = 7

0.578

17 Radicals and the Pythagorean Theorem 1)a.i. Simplify radical terms (no variable in the radicand). (AN2) 3)e.iii. Determine the measure of a third side of a right triangle using the Pythagorean Theorem, given the lengths of any two sides. (AA45)

What is the value of 𝑥 in the right triangle?

A) 6√2 B) 2√3 C) 6 D) 3√10

0.651

3

9 𝑥

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Question Group


18 Solve linear inequalities in 1 variable 3)f. Linear inequalities in a single variable 3)f.i. Solve linear inequalities in one variable. (AA24) 3)f.ii. Represent solutions to linear inequalities as a single inequality. 3)f.iii. Represent the solution to a linear inequality in one variable on a number line.

Find the graph of the solution to the inequality. 𝟑𝒙 + 𝟓 < 𝟔𝒙 − 𝟏

A)

B)

C)

D)

0.472

19 Use function notation 4) Functions and functional notation. This is an introduction

to basic notational representation and should not include any explicit discussion of functions vs. relations, domain, range and vertical line test, etc.

4)a. Use function notation to compute a single output for simple linear and quadratic relationships.

Evaluate 𝑔(2) for the function 𝑔(𝑥).

𝒈(𝒙) = 𝟑𝒙𝟐 + 𝟓𝒙 − 𝟐

A) 44 B) 24 C) 20 D) 48

0.784

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Question Group


20 Graph a line from an equation 5)a.vi. Write and transform equations of lines in the following forms 5)a.vi.1. Point-Slope form 5)a.vi.2. Slope Intercept form 5)a.vi.3. 𝐴𝑥 + 𝐵𝑦 = 𝐶 form 5)b. Draw and recognize graphs of lines.

Which of the following is the graph of the equation? −𝟑𝒙 + 𝟒𝒚 = 𝟏𝟐

A)

B)

C)

D)

0.489

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Question Group


21 Equation of a line from two points 5)a.i. Determine the slope of a line, given the coordinates of two points on the line. (AA33) 5)a.ii. Write the equation of a line, given its slope and the coordinates of a point on the line. (AA34) 5)a.iii. Write the equation of a line, given the coordinates of two points on the line. (AA35) 5)a.vi. Write and transform equations of lines in the following forms 5)a.vi.1. Point-Slope form 5)a.vi.2. Slope Intercept form 5)a.vi.3. 𝐴𝑥 + 𝐵𝑦 = 𝐶 form

Find the equation of the line passing through the points (−2, 3) and (1, −3). Write the equation in slope-intercept form.

A) 𝑦 = −2𝑥 + 3 B) 𝑦 = 2𝑥 + 7 C) 𝑦 = −6𝑥 − 9 D) 𝑦 = −2𝑥 − 1

0.564

22 Equations of vertical and horizontal lines 5)a.iv. Write the equation of a line parallel to the 𝑥- or 𝑦- axis. (AA36) 5)a.vi. Write and transform equations of lines in the following forms 5)a.vi.1. Point-Slope form 5)a.vi.2. Slope Intercept form 5)a.vi.3. 𝐴𝑥 + 𝐵𝑦 = 𝐶 form

Find the equation of the vertical line passing through the point (−5, −2).

A) 𝑦 = 𝑥 − 2

B) 𝑦 = −2

C) 𝑥 = −5

D) 𝑦 =2

5𝑥 − 2

0.419

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Question Group


23 Slope and 𝒚-intercept of a line from an equation 5)a.v. Determine the slope and 𝑦-intercept of a line, given its equation in any form. (AA37*) 5)a.vi. Write and transform equations of lines in the following forms 5)a.vi.1. Point-Slope form 5)a.vi.2. Slope Intercept form 5)a.vi.3. 𝐴𝑥 + 𝐵𝑦 = 𝐶 form

Find the slope and 𝑦-intercept for the graph of the equation. 𝟑𝒙 + 𝟒𝒚 = 𝟖

A) Slope= − 34

and 𝑦-intercept = (0, 2)

B) Slope=

43


C) Slope=

34


D) Slope= − 4

3 and 𝑦-intercept = (0, 8)

0.522

24 Word problem: Proportions 6)a. Solve simple verbal problem with two quantities that are proportional.

If 6 gallons of gas cost $24, how much does 10 gallons of gas cost?

A) $60 B) $30 C) $96 D) $40

0.843

25 Word problem: Percent 6)b. Solve simple verbal problem involving a single percent and/or a single percent increase/decrease.

During the course of a year, the price of a house increased from $200,000 to $250,000. What was the percent increase in price?

A) 5% B) 20% C) 25% D) 50%

0.590

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CEAFE Fall 2012 Item Results and Learning Outcomes Correspondence · 2017. 4. 17. · Fall 2012 Item Results and Learning Outcomes Correspondence Purpose of Exam: Students enrolled