Sample
questions for Precalculus/CT
Objective 1:
Students will identify, analyze, and evaluate arguments as they occur in
their own or others' work.
Sample problem:
(8 pts)
Consider this calculation:
x2
+ 2x + 2 = 0
(x2
+ 2x + 1) + 1 = 0 (step 1)
(x
+ 1)2 + 1 = 0 (step
2)
(x
+ 1)2 = -1 (step
3)
a) For
each of the three steps above, say whether or not it is correct.
b) Is it
fair to conclude that x2 + 2x + 2 = 0 has no solutions in the set of
real numbers?
(Explain, briefly.)
Sample "Exceeds" response:
a) Yes,
all three steps are correct.
b)
Yes. If there were a value of x
which worked, then a square would be negative.
But this is impossible.
Sample "Meets" response:
a) Yes,
all steps are correct.
b)
Yes. Otherwise it wouldn't work.
Sample "Approaches" response:
a) Yes,
all steps are O.K.
b) No,
not enough information is given.
Sample "Does not meet" response:
a) Step
2 is wrong. (x + 1)2 = x2
+ 12 = x2 + 1.
b) Yes.
Objective 2:
Students will develop well-reasoned arguments.
Sample problem:
(8 pts)
Carefully showing all work, find all real solutions of:
x3
- 4x = 0
Sample "Exceeds" response:
x3
- 4x = 0
x(x2
- 4) = 0
x
= 0
or x2 - 4 = 0
x2
- 4 = 0
x2
= 4
x
= +2
or x = -2
So
x is 0, 2, or -2.
Sample "Meets" response:
x3 - 4x =
0
x(x2
- 4) = 0
x
= 0 x2 - 4 = 0
x2
= 4
x
= 2
or x = -2
Sample "Approaches" response:
x3 - 4x =
0
x3 = 4x
x2
= 4
x
= +2
or x = -2
Sample "Does not meet" response:
x3 - 4x =
0
x3 = 4x
x2
= 4
x
= 2