The Daily Item, Sunbury, PA

December 1, 2012

Keystone algebra: Are you smarter than an 11th-grader?

Pass the test and graduate from high school!


 Are you smarter than an 11th-grader?

Could you pass the newest state test needed in order to graduate from high school? Take a sample test in algebra of the new Keystone Exams, which have replaced the Pennsylvania System of School Assessment tests. Students will need to pass the tests before they can graduate from high school. Answers at bottom:



Pennsylvania Keystone Algebra I Item Sampler 2011




1. An expression is shown below.

2√51X



Which value of x makes the expression equivalent to 10√51

51 ?

A. 5

B. 25

C. 50

D. 100



2. Simplify: 2(2√4 ) −²

A. ⅛

B. ¼

C. 16

D. 32



3.  A polynomial expression is shown below.

(mx³ + 3) (2x³+ 5x + 2) – (8x^5 + 20x^4). The expression is simplified to 8x³ + 6x²+ 15x + 6. What is the value of m?

A. –8

B. –4

C.  4

D.  8



4. Which is a factor of the trinomial x²– 2x – 15?

A. (x – 13)

B. (x – 5)

C. (x + 5)

D. (x + 13)



5. Simplify:

x² – 3 x – 10⁄x² + 6 x + 8

; x ≠ –4, –2

A. – 1⁄ 2x – 5⁄4

B. x² – 1 ⁄2x – 5⁄4

C. x – 5⁄x + 4

D. x + 5⁄ x – 4



6. 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.

15x + 10y = 700

x + y = 60



What is the value of x, the minutes Anna spent running?

A. 10

B. 20

C. 30

D. 40



7. 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.

5x + 4y = 32

x + 6y = 22

Which statement is true?



A. A rose costs $1 more than a daisy.

B. Samantha spent $4 on each daisy.

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.





8. 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 185 + 4b ≤ 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.





ANSWER KEY AND EXPLANATIONS:

1. An expression is shown below.

2√51X



Which value of x makes the expression equivalent to 10√51?

A. 5

B. 25

C. 50

D. 100





A student could determine the correct answer, option B, by factoring 10 √51 as 2 × 5 √51 , then moving the 5 inside the radical as 2√51 × 5 × 5 = 2√51 × 25.

A student could arrive at an incorrect answer by A student could arrive at an incorrect answer by either using an incorrect method or by making errors incomputation. For example, a student would arrive at option A if he/she failed to square 5 when he/she moved it under the radical.



2. Simplify: 2(2√4 ) −²

A. ⅛

B. ¼

C. 16

D. 32



A student could determine the correct answer, option A, by recognizing

2(2√4 ) –2 = ____2__ ___     =    ______2_ _ _____   = __1__ =  1/8.

                       2√4 × 2√4             2 × 2 ×√4 ×√4               2 × 4



A student could arrive at an incorrect answer by failing to follow correct order of operations or by not knowing how to use radicals or negative exponents. For example, a student would arrive at option D if he/she ignored the negative exponent and treated 2(2√4 ) -² as 2(2√4 ) ².



3.  A polynomial expression is shown below.

(mx³ + 3) (2x³+ 5x + 2) – (8x^5 + 20x^4)

The expression is simplified to 8x³ + 6x²+ 15x + 6. What is the value of m?

A. –8

B. –4

C.  4

D.  8

A student could determine the correct answer, option C, by using correct order of operations and the distributive

property to expand (mx³ + 3) (2x² + 5x + 2) to 2mx^5 + 5mx^4 + 2mx³+ 6x²+ 15x + 6. The 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 operations, making an error with the distributive property, or incorrectly combining like terms. For example, a student would arrive at option D if he/she failed to distribute and then set mx³= 8x³, so m = 8.



4. Which is a factor of the trinomial x2 – 2x – 15?

A. (x – 13)

B. (x – 5)

C. (x + 5)

D. (x + 13)



A student could determine the correct answer, option B, by factoring the trinomial x² – 2x – 15 as (x – 5)(x + 3) and identifying (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 option C if he/she factored x² – 2x – 15 as (x + 5)(x – 3) and identified (x + 5) as a factor.



5. Simplify:

x² – 3 x – 10⁄x² + 6 x + 8; x ≠ –4, –2

A. – 1⁄ 2x – 5⁄4

B. x² – 1 ⁄2x – 5⁄4

C. x – 5⁄x + 4

D. x + 5⁄ x – 4



A student could determine the correct answer, option C, by factoring both the numerator and denominator, then reducing  x2–3x –10⁄  x2 + 6x + 8 = (x – 5)(x + 2)/(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 option D by factoring x2–3x –10/x2 + 6x + 8  as (x + 5)(x – 2)/(x –4)(x –2) .



6. 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.

15x + 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, option B, by solving the system of equations using substitution.

Solving the equation x + y = 60 for y yields y = 60 – x. Substituting 60 – x in the place of y in the equation 15x + 10y = 700 yields 15x + 10(60 – x) = 700. Using the distributive property yields 15x + 600 – 10x = 700.

Combining like terms and subtracting 600 from both sides yields 5x = 100. Dividing 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 equations or by making errors in computation. For example, a student would arrive at option D by incorrectly solving for y as y = x + 60 and then failing to distribute when substituting, yielding 15x + x + 60 = 700. Combining like terms and subtracting 60 from both sides yields 16x = 640. Dividing both sides by 16 yields x = 40.



7. 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.

5x + 4y = 32

x + 6y = 22



Which statement is true?

A. A rose costs $1 more than a daisy.

B. Samantha spent $4 on each daisy.

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, option A, by solving the system of equations and correctly interpreting the solution x = 4 and y = 3. The 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 equations or by incorrectly interpreting the solution set. For example, a student would arrive at option B if he/she interpreted the x-value as the price of a daisy.



8. 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 185 + 4b ≤ 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, option D, by solving the inequality and interpreting the solution b ≤ 203.75. The variable b represents the number of baseballs that can be purchased. It is a true statement that 185 ≤ 203.75.

A student could arrive at an incorrect answer by either making errors in solving the system of equations or by incorrectly interpreting the solution set. For example, a student would arrive at option A if he/she switched the sign of the inequality when dividing by 4.