CUNY Elementary Algebra Final Exam (CEAFE) Fall 2012 Item Results and Learning Outcomes Correspondence

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1 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,

1 1 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 Pool contained 743 questions divided into 25 s. Each contained variations of each question. Each student s test was randomly generated by selecting one item from each, and the items were randomly ordered. Data presented in the tables below include approximately 10,790 students University-wide who tested 12/1/ /24/2012. Table 1 contains summary data for each question, and Table 2 contains more detailed information about each question.

2 Description of Columns in Table 1: Learning Outcomes Headings are from the CUNY Elementary Algebra Proficiency Standards Details (as amended 3/30/2012). Mean 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 Learning Outcomes Heading Mean 3 Operations with scientific notation Equations of vertical and horizontal lines Solve linear inequalities in 1 variable Operations with radicals (multiplication and division) Graph a line from an equation Operations with exponents Factor polynomials (non-monic trinomials) Slope and y-intercept of a line from an equation Factor polynomials (grouping; multiple variables) Solve a system of linear equations Equation of a line from two points Solve factorable quadratic equations Factor polynomials (difference of squares; two steps) Solve quadratic equations with no linear term Word problem: Percent Operations with radicals (addition and subtraction) Radicals and the Pythagorean Theorem Polynomial operations (addition and subtraction) Solve literal equations Polynomial operations (division by a monomial) Translate words into algebraic relationships Solve a linear equation in one variable Polynomial operations (multiplication) Use function notation Word problem: Proportions

3 3 Description of Columns in Table 2: Learning Outcomes listed are from the CUNY Elementary Algebra Proficiency Standards Details (as amended 3/30/2012). Sample s are from the August 2012 Sample-A exam. An additional sample item is available in the August 2012 Sample-B exam. Mean 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 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. 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. A) B) 25 6 C) 11 6 D) 12 2 Simplify completely. A) B) 25 3 C) D) ( )

4 4 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. ( )( ) A) B) C) D) 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. A) x 2 x 5 x 7 (x 3 ) B) x 6 C) x 7 D) x 29 5 Polynomial operations (addition and subtraction) 2)b. Add and subtract monomials and polynomials. (AA13*) Simplify completely. (5x 2 7x + 9) ( 2x 2 3x + 2) A) 3x 2 4x + 7 B) 7x 2 4x + 7 C) 7x 2 10x + 7 D) 7x 2 4x

5 5 6 Polynomial operations (multiplication) 1)c. 2)c. Exponents. Multiply and divide monomial expressions with a common base using the properties of exponents. All exponents are integral. (AA12) Multiplication of a monomial and binomial by any degree polynomial. (AA13*) Multiply. A) 2x 3 + 3x 2 32x + 30 B) 2x 3 + 8x 2 12x + 30 C) 2x 3 + 3x 2 12x + 30 D) 2x 3 + 8x 2 32x + 30 (2x 5)(x 2 + 4x 6) 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*) Simplify completely. A) 5x 2 + 7x B) 25x 3 35x 2 C) 5x 2 7x + 1 D) 5x 2 + 7x 1 25x 3 35x 2 + 5x 5x 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*) 9 Factor polynomials (non-monic trinomials) 2)e.iii. Factor all trinomials of a single variable, including a leading coefficient other than 1. Factor completely. A) 4(9x 2 y 25y 3 ) B) 4y(9x 2 25y 2 ) C) 4y(3x 5y)(3x + 5y) D) 4y(3x 5y) 2 36x 2 y 100y 3 Which of the following is a factor of the polynomial? 2x 2 x 55 A) x + 11 B) x 5 C) 2x + 11 D) 2x

6 6 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 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) Solve a linear equation in one variable 3)b. Solve all types of linear equations in one variable. (AA22) Solve for n. A) n = 3 B) n = 3 C) n = 7 D) n = 7 5(8 n) = 3n

7 7 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 = 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. A) x = z+y 5 z = 5x + y B) x = z y 5 C) x = z 5 y D) x = 5(z y) 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. 4b 2 + 8b = 0 A) Only b = 2 B) Only b = 2 C) b = 0 or b = 2 D) b = 0 or b =

8 8 16 Solve quadratic equations with no linear term 3)e.ii. Solve quadratic equations with no linear term. Find all solutions to the equation. 10x 2 = 490 A) x = 7 or x = 49 B) x = 0 or x = 49 C) x = 7 or x = 7 D) Only x = 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) x

9 9 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) 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) = 3x 2 + 5x 2 A) 44 B) 24 C) 20 D)

10 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)

11 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 Equations of vertical and horizontal lines 5)a.iv. Write the equation of a line parallel to the x- or yaxis. (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 5 x

12 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= 3 4 B) Slope= 4 3 C) Slope= 3 4 and y-intercept = (0, 2) and y-intercept = (0, 8) and y-intercept = (0, 2) D) Slope= 4 3 and y-intercept = (0, 8) 24 Word problem: Proportions 6)a. Solve simple verbal problem with two quantities that are proportional. 25 Word problem: Percent 6)b. Solve simple verbal problem involving a single percent and/or a single percent increase/decrease. If 6 gallons of gas cost $24, how much does 10 gallons of gas cost? A) $60 B) $30 C) $96 D) $40 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%

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