Math
Spring Operational 2015
Algebra 1
PBA Item #17
Complete the Square
VF650016
Prompt
Rubric
Task is worth a total of 4 points.
Score
2
VF650016 Rubric Part A
Description
Student response includes the following 2 elements.
• Reasoning component = 1 point
o Algebraic or written explanation for solving the equation
•
Computation component = 1 point
o Solution of x = 2 or -14
x 2 + 12x − 28 =
0
x 2 + 12x =
28
Sample Student Response: x
2
+ 12x + 36 = 28 + 36
( x + 6)
2
64
=
x +6 =
±8
x =
8 − 6 =2 or -8 − 6 =−14
x 2 + 12x − 28 =
0
x 2 + 12x =
28
x 2 + 12x + 36 = 28 + 36
( x + 6)
2
=
64
x + 6 =±8
x =
8 − 6 =2 or -8 − 6 =−14
1
0
Score
2
Solution 1: 2
Solution 2: –14
Student response includes 1 of the 2 elements.
Student response is incorrect or irrelevant.
VF650016 Rubric Part B
Description
Student response includes the following 2 elements.
• Reasoning component = 1 point
o Valid explanation
•
Computation component = 1 point
o Solution of c = 36
Sample Student Response:
There would be only one solution if the factors of the polynomial
are the same. If the factors are the same, then the identity
( x + a)2 =x 2 + 2ax + a2 can be used. The middle term is 12, so c
would have to be the square of half of that number. Therefore c =
36
1
0
Student response includes 1 of the 2 elements.
Student response is incorrect or irrelevant.
Anchor Set
A1 – A10
A1
Part A: Score Point 2
Part B: Score Point 2
Annotations
Anchor Paper 1
Part A: Score Point 2
This response receives full credit. The student provides both of the required elements:
• the steps to solve the equation by completing the square
(I x2+12x-28=0 . . . VI x= -6 + 8 x= -6-8)
• both correct solutions (x=2,x=-14)
Part B: Score Point 2
This response receives full credit. The student provides both of the required elements.
• an explanation of why the value of c makes the equation have only one solution (in
order to have only one solution, the number under the radical in the quadratic formula
must equal zero, and 36 is the only value that will satisfy this). Use of the quadratic
formula is acceptable for the explanation. The quadratic formula is
x =
−b ± b2 − 4ac
2a
. In
this equation, a = 1 and b = 12. For the equation to have one solution, the value of
2
b
b2-4ac would have to equal 0, so c would have to equal=
4a
•
a correct value of 36 (c will have to be equal to 36).
144
= 36.
4(1)
A2
Part A: Score Point 2
Part B: Score Point 2
Annotations
Anchor Paper 2
Part A: Score Point 2
This response receives full credit. The student provides both of the required elements.
• the steps to solve the equation by completing the square (1. x2+12x-28=0 . . . 6.
x+6 = 8 x+6=-8)
• both correct solutions (x=2,x=-14)
Part B: Score Point 2
This response receives full credit. The student provides both of the required elements.
• an explanation of why the value of c makes the equation have only one solution (the
equation is a perfect square trinomial so both solutions would be -6).
• a correct value of 36 (c equals 36)
A3
Part A: Score Point 1
Part B: Score Point 2
Annotations
Anchor Paper 3
Part A: Score Point 1
This response receives partial credit. The student provides one of the two required elements.
• both correct solutions (2, -14).
Because the student arrives at the correct answer by using the quadratic formula instead of
completing the square as required by the prompt, no credit is given for the steps used to
complete the square.
Part B: Score Point 2
This response receives full credit. The student provides both of the required elements.
• an explanation of why the value of c makes the equation have only one solution
((x+6)(x+6), x=-6). By showing that both factors lead to the same answer, the
student explains how this value for c leads to a single value for x.
• a correct value of 36 (36 makes the equation only have 1 solution).
A4
Part A: Score Point 2
Part B: Score Point 1
Annotations
Anchor Paper 4
Part A: Score Point 2
This response receives full credit. The student provides both of the required elements.
• the steps to solve the equation by completing the square
(x2 + 12x – 28 = 0 . . . x + 6 = ± 8)
• both correct solutions (x = 2, x = -14).
Part B: Score Point 1
This response receives partial credit. The student provides both of the required elements,
but a precision point is deducted.
• The student explains why the value leads to one solution (a perfect square is formed
since the factored form would be (x + 6)2).
• a correct value of 36 (c is 36).
Only the exact value of 36 would give the equation a single solution. Values greater than 36
[indicated by “>” in “c ≥ 36”] would have imaginary [i.e., complex] solutions. One point is
lost for including numbers that are not solutions to this problem.
A5
Part A: Score Point 2
Part B: Score Point 0
Annotations
Anchor Paper 5
Part A: Score Point 2
This response receives full credit. The student provides both of the required elements.
•
•
the steps to solve the equation by completing the square (1 Move the 28 to the right
of the equal sign by adding it . . . Subtract 6 from both sides).
both correct solutions (x =-14 and x = 2).
Part B: Score Point 0
This response receives no credit. Although the response contains (x + 6 = 0) and (answer: 6), it does not provide the c value of 36 nor explain why 36 results in a single answer.
A6
Part A: Score Point 1
Part B: Score Point 1
Annotations
Anchor Paper 6
Part A: Score Point 1
This response receives partial credit. The student provides one of the two required elements.
•
both correct solutions (x=2, x=-14).
The student uses the quadratic equation to find the solutions. Therefore no credit is awarded
for completing the square.
Part B: Score Point 1
This response receives partial credit. The student provides one of the two required elements.
•
a correct value of 36 (c = 36).
The response provides the steps to finding the square, but the student does not explain why
this would result in one solution.
A7
Part A: Score Point 1
Part B: Score Point 0
Annotations
Anchor Paper 7
Part A: Score Point 1
This response receives partial credit. The student provides one of the two required elements.
•
both correct solutions (x=2, x=-14)
The student finds the correct solutions by using the quadratic equation and therefore
receives no credit for completing the square.
Part B: Score Point 0
This response receives no credit. The values given for c (c>36) exclude the value that would
result in a single solution and are therefore incorrect. The explanation provided is for finding
values with no real solutions, rather than one solution (If the part under the square root is
negative, then there are no solutions).
A8
Part A: Score Point 1
Part B: Score Point 0
Annotations
Practice Set 1, Paper 8
Part A: Score Point 1
This response receives partial credit. The student provides one of the two required elements:
•
the steps to solve the equation by completing the square (x2 + 12x – 28 = 0 . . . x +
6 = 8). The response does not include the steps [x + 6 = ±8] and [x + 6 = -8], but
enough of the work is shown to understand how the solutions were derived, so the
response receives the point.
One of the solutions provided is not correct (x = 14 or x = 2), so the second point is not
awarded.
Part B: Score Point 0
This response receives no credit. The explanation and value provided are not correct (If you
make c be a negative number, because then you will only have 1 answer because that one
answer will be an imaginary number). It is still possible to have two solutions when they are
both imaginary.
A9
Part A: Score Point 0
Part B: Score Point 0
Annotations
Anchor Paper 9
Part A: Score Point 0
This response receives no credit. The equation is factored, but the response does not contain
the final solutions. The steps to complete the square are also not given.
Part B: Score Point 0
This response receives no credit. The student evaluates the equation from Part A instead of
the equation given in Part B.
A10
Part A: Score Point 0
Part B: Score Point 0
Annotations
Anchor Paper 10
Score Point
Part A: Score Point 0
This response receives no credit. The steps to completing the square that follow (add 28 to
both sides) are incorrect. The resulting solution is also incorrect.
Part B: Score Point 0
This response receives no credit. The explanation for what value of c is needed is incomplete
(Whatever value (x2+12x) have c will have but as the opposite positive/negative value). The
value of 36 is not given.
Practice Set
P101 - P105
P101
P102
P103
P104
P105a
P105b
Practice Set
Paper
Score
P101
0, 1
P102
1, 2
P103
0, 0
P104
1, 1
P105
2, 2