Analysis. Correct Answer : meters (m) The student was epected to know and use the Pythagorean theorem to find the missing side. a + b c 8 + 7 64 + 89 89 64 SKILL: Use the Pythagorean theorem to find the missing side of a right triangle.. Correct Answer : -4 The student was epected to use knowledge of proportions and solving equations to find the value of. The sample solutions show two possible ways to solve this problem: Solution A:. Cross multiply to solve the proportion.. Solve the equation by isolating the variable on one side. Solution B: + 6 6( + ) ( ) 6 + 8 + 8 4. Rewrite the two fractions with common denominators.. Set the numerators equal to one another.. Solve the equation by isolating the variable on one side. + 6 6 + 8 0 0 6 + 8 + 8 4 SKILLS: Use ratios to solve real-world problems. Solve equations with variables on both sides. Math-U-See: PreCalculus Placement Assessment Page
a. Correct Answers : or a b -6 6 b. A fraction can be rewritten as a division problem. ab ( ab ) ( ba) ba. Division by an eponent can be rewritten as multiplication by its opposite. ( ) ( ) ( ) ( ab ba ab b a ). Only the same variables can be multiplied together. ( ) ( ) ( ) ( ) ab b a a a b b 4. When variables are multiplied, they can be combined by adding the eponents. 6 a ( ) ( ) a a b b ab SKILL: Simplify a rational epression. 4. Correct Answer : 8qr r. Variables being multiplied within a term can be combined by adding the eponents. q - rq + qr r qr + qr r. Terms with the eact same variables can be added or subtracted. qr + qr r 8qr r SKILL: Add and subtract to simplify a rational epression. +. Correct Answer :. A fraction can be rewritten as a division problem. + +. Division can be rewritten as multiplication by the opposite. + + or b 6. To multiply fractions, multiply the numerators together and the denominators together. + ( + ) + SKILL: Simplify a comple fraction. Math-U-See: PreCalculus Placement Assessment Page
6. Correct Answer : In order to solve this problem, the student needed to know the following concepts:. Fractions need to have common denominators before they can be combined. ( ) The first fraction needed to be multiplied by, ( ) ( ) and the second fraction needed to be multiplied by. ( ) + + 6 + ( ) ( ) + ( )( ) ( )( ). The Distributive Property needed to be used to create new numerators. ( ) ( ) + ( )( ) ( )( ) 6 + ( )( ) ( )( ). The numerators needed to be combined over the new denominator, and both numerator and denominator needed to be simplified. + 6 ( )( ) + 6 ( )( ) + 6 + SKILL: Add fractions with polynomials in the denominator. Math-U-See: PreCalculus Placement Assessment Page
7. Correct Answer : 78 The student needed to understand how to perform operations with radicals. The sample solutions show two ways in which these concepts could be applied: Solution A: Here the radicals are combined first and then simplified. ( 6) ( ) ( 9 8) 0 7 + 9 8 ( ) + 0 8 + 9 8 0 8 + 9 8 ( ) 9 8 9 78 Solution B: Here the radicals are simplified first and then combined. ( 6) ( ) ( 9 8) ( 6)( 4 ) ( 8 ) 0 8 + 9 8 + + 60 + 8 78 SKILL: Simplify epressions with radicals. 8. Correct Answer :. A negative eponent indicates the reciprocal of the base. 8 7 7 8. A fractional eponent indicates a root. In this case, the eponent means the cube root of the base. 7 7 or 8 8 7. The cube root is a number that it is multiplied by itself three times to obtain the given number. In this case, 7, and 8. 7 7 or 8 8 SKILL: Simplify an epression with a fractional eponent. 8 Math-U-See: PreCalculus Placement Assessment Page 4
9. Correct Answer : 4 SAMPLE SOLUTIONS:.. 8 7 8 7 8 7 96 4 8 4 7 7 49 7 7 8 4 4 7 SKILL: Simplify a division epression with irrational numbers. 0. Correct Answer : ( 4b) ( + 4b) The student should have recognized the epression as the difference of two squares and factored it accordingly. SKILL: Factor the difference of two squares.. Correct Answer : -7 The student needed to know that i is an imaginary number representing the square root of -. He then needed to apply this information to rename square roots of negative numbers and simplify epressions, as follows: ( ) ( )( ) ()( ) ( ) ( i) ( i) ( i) i ( i ) 9 64 9 64 9 64 9 64 (9)( )(8) 7 SKILL: Simplify epressions with imaginary numbers.. Correct Answers : or ( ) The student was epected to know that he needed to multiply the denominator by the conjugate ( ) in order to complete the square and obtain a whole-number denominator. + + ( ) ( ) ( ) (+ )( ) or ( ) SKILL: Rationalize the denominator of a fraction. Math-U-See: PreCalculus Placement Assessment Page
. Correct Answer : ± 7 or + 7, 7 The student was epected to notice that the equation could not be solved by factoring; therefore, he needed to know and use the quadratic formula to find the solution. SKILL: Use the quadratic formula to solve an equation. 4. Correct Answer : E; + y 6 The student was epected to know the general equation for a circle: ( h) +(y k) r The student should also know the general equations for the other answer choices: A. y + 0 Quadratic formula: ± b b 4ac a ± ( ) 4 ± 8 ± 7» This is the general equation of a line: y m + b» This equation can be rewritten as: y + 0 B. y 4 -» This is the general equation of a hyperbola : y c» This equation can be rewritten as: 4 y - C. y 4» This is also the general equation of a hyperbola: y c» This equation can be rewritten as: y 4 D. y -» This is the general equation of a parabola: y SKILL: Identify the shape of a graph from its equation. Math-U-See: PreCalculus Placement Assessment Page 6
. Correct Answer: cm mo m m/yr (or 0.6 m/yr) mo yr 00 cm The glacier moves 0.6 meters in a year. SKILL: Use unit multipliers (dimensional analysis) to convert units. 6. A student who has mastered the prerequisite concepts should be able to complete the written assessment in about 4 minutes. 7. A student who has mastered the prerequisite concepts should feel confident in his or her ability to solve the problems and should not need to ask for assistance. Math-U-See: PreCalculus Placement Assessment Page 7