Sample Test Questions
Viewing the mathematical equations on this page requires certain system requirements.
Current section: Essay > Short Response
Sample Question 1
Emily's monthly income consists of a monthly salary of $2,000, plus a commission if her monthly sales exceed $10,000. The commission is equal to 5 percent of the amount by which her monthly sales exceed $10,000. (For example, if Emily's monthly sales are $15,000, she receives a commission of 5 percent of $5,000.)
(a)
Write an equation that gives Emily's monthly income, y, in terms of her monthly sales, x, if her monthly sales are less than or equal to $10,000 (x ≤ 10,000). Write a second equation that gives Emily's monthly income, y, in terms of her monthly sales, x, if her monthly sales are greater than $10,000 (x > 10,000).
(b)
Using the equations you wrote in part (a) of this question, draw a graph of Emily's monthly income, y, as a function of her monthly sales, x, for monthly sales of $0 to $60,000 (0 ≤ x ≤ 60,000). Graph her monthly sales on the x-axis and her monthly income on the y-axis. Label each axis and show the units and scales used.
(c)
Use the graph you drew in part (b) of this question to estimate the increase in Emily's monthly income if her monthly sales were to increase from $8,000 to $30,000. Show your work.
Sample Response That Received a Score of 3:
(a)
(b)
(c)
The increase in Emily's income would be $1,000. If she made $8,000 in monthly sales, her income would be $2,000 since her sales did not exceed $10,000. If she made $30,000 in monthly sales, her income would be y =
Note: An alternative to this equation (easier for calculation though not as concrete) would be
Commentary on Sample Response that Earned a Score of 3
This response received a score of 3 because it responded appropriately to all parts of the question: the response to part (a) included both equations requested; the graph was correctly drawn in part (b); and the response to part (c) was obtained from the graph, as requested. Two additional strengths of this response are 1) the observation that the algebraic solution to part (c) is consistent with the graphical solution and 2) the footnote giving an alternative form of the second equation in part (a), with the explanation that, although it would be easier to use for computation, it is not as concrete.
Sample Response That Received a Score of 2:
(a)
Monthly income = Monthly salary + .05x any sales over 10,000
y = 2,000 (x ≤ 10,000)
y = 2,000 + (.05(x – 10,000))
(b)
(c)
Increase if sales were to go from $8,000 to $30,000
At $8,000 sales she gets no extra since 8,000 ≤ 10,000. So only gets $2,000 base salary.
At $30,000 she gets basic $2,000 plus .05 times 20,000 since that's the amount over $10,000. So it goes from $2,000 on the graph to $3,500 at 30,000 monthly sales.
Commentary on Sample Response that Earned a Score of 2
This response received a score of 2 because it responds appropriately to most parts of the question and demonstrates a sufficient knowledge of the concepts relevant to the question. In particular, both of the equations requested in part (a) are correct. The graph in part (b) is partially correct. The values plotted for x > 10,000 reflect calculating Emily's income without subtracting $10,000 when evaluating and graphing the second equation. It should not have the discontinuity shown at monthly sales x = $10,000. The response to part (c) is correct based on the graph in part (b). The correct process is described in part (c), but the examinee missed an opportunity to identify the mistake in part (b) by not carrying out this process and then comparing the algebraic solution to the graphical solution.
Sample Response That Received a Score of 1:
(a)
2,000y + x ≤ 10,000
2,000y + .05x > 10,000
(b)
(c)
≈ $2,200
The graph moves up $500 for every 10,000 sales.
At 8,000 sales, the income would be ≈ $1,800
At 30,000 sales, the income would be ≈ $3,000 judging by the dots.
Commentary on Sample Response that Earned a Score of 1
This response received a score of 1 because it demonstrates a weak knowledge of the concepts relevant to the question. In part (a), the equations (actually inequalities as written) are not correct. The response does not provide evidence of being able to translate a verbal representation into algebraic equations. Because part (a) is incorrect, the graph in part (b) is not the graph of a piecewise linear function. However, the discrete values that are graphed in part (b) are correct. In part (c), the general description of the increase in income is correct for values of sales (x) greater than $30,000; however, the estimation of the income associated with sales of $8,000 as different from $2,000 is further evidence of misunderstanding of this piecewise linear function. There was also an arithmetic error in finding the difference between $3,000 and $1,800. The response received some credit because portions of parts (b) and (c) are correct.
Sample Response That Received a Score of 0:
(a)
y = .05x(10,000 –
y = 2,000 + .05(x) ≤ 10,000
(b)
(c)
8,000
Commentary on Sample Response that Earned a Score of 0
This response received a score of 0 because it demonstrates extremely limited understanding of the topic. The equations in part (a) are not correct. The graph in part (b) does not receive any credit because it is inaccurately drawn and does not reflect the piecewise linear characteristics of the situation. Most of the discrete values were graphed incorrectly, even though, based on the table, they appear to have been computed correctly. Part (c) is either incomplete or totally incorrect. The evidence of correct thinking provided by this response does not reach the threshold required to receive a score above 0.
Sample Question 2
In triangle EFG, side EF has length 8 and side FG has length 10.
(a)
Two of the possible lengths of side EG are 3 and 16. Draw triangle EFG with side EG of length 3. Draw a second triangle EFG with side EG of length 16. For each of the triangles, label all of the vertices and show the lengths of all of the sides. [Note: Your drawings are not expected to be exact but should reasonably represent the relative lengths of the sides.]
(b)
The length of side EG could not be 1 or 20. Explain why not. Draw figures to support your explanation.
(c)
If triangle EFG is a right triangle, what are the two possible lengths of side EG? Draw the two right triangles. For each triangle, label all of the vertices and show the lengths of all of the sides. Indicate the right angle.
Sample Response That Received a Score of 3:
(a)
(b)
The sum of the lengths of two sides of a triangle must be greater than the length of the other side. In triangle EFG, if the length of side EG was 1, the sum of the lengths of sides EG and EF would be 9. This value is less than the length of side FG, which is 10. Also if the length of side EG was 20, the sum of the lengths of sides EF and FG would be 18. This value is less than the length of side EG, which is 20.
(c)
If EFG were a right , the length of side EG could be either 6 or approximately 12.81.
Commentary on Sample Response that Earned a Score of 3
This response received a score of 3 because it responds appropriately to all parts of the question and demonstrates strong knowledge of the concepts relevant to the question. Part (a) includes two drawings that show reasonable relative lengths of the sides and that show clearly that angle F is acute when side EG has length 3 and obtuse when side EG has length 16. Part (b) both describes and illustrates the triangle inequality property correctly. Part (c) correctly shows the two possible right triangles with sides of 8 and 10. (The one minor error in the response is that the right angle is not labeled in the second drawing. However, angle E appears to be a right angle and is consistent with the opposite side being the hypotenuse; thus, the error is not considered significant enough to warrant a lower score.)
Sample Response That Received a Score of 2:
(a)
(b)
From Pythagorean theorem a2 + b2 = c2.
FG is hypotenuse C
EF2+EG2=FG2
82+b2=102
b could not be true by being 1 or 20.
(c)
c2 – b2 = a2
100 – 64 = 36
a = 6
c2 = 102 + 82
c2 = 100 + 64
c =
c ≈ 12.8
Commentary on Sample Response that Earned a Score of 2
This response received a score of 2 because it demonstrates a sufficient knowledge of the concepts relevant to the question. Part (a) includes two drawings that show reasonable relative lengths of the sides and that show clearly that angle E is acute when side FG has length 3 and obtuse when side EG has length 16. Part (b) receives no credit because it assumes incorrectly that triangle EFG must be a right triangle. Part (c) correctly shows the two possible right triangles with sides of 8 and 10.
Sample Response That Received a Score of 1:
(a)
(b)
There are no ways to construct triangles with measured lengths of the 3 sides to be 1, 8, 10 due to the fact that the angles have to add up to 180°. Using trigonometry sin, cos, and tan, they will not add up.
There is no way that the angles will add up to 180°.
Once again, it just proves that there is no way the angles are going to add up to 180°.
(c)
One possible solution is the EG can be 6 according to Pyth. theorem.
Commentary on Sample Response that Earned a Score of 1
This response received a score of 1 because it demonstrates a weak knowledge of the concepts relevant to the question. In part (a), both triangles are drawn (incorrectly) as right triangles. The relative lengths of the sides are acceptable in the 3-8-10 triangle, but not in the 8-10-16 triangle. In the 8-10-16 triangle, the side of length 10 is shown longer than the side of length 16 and the largest angle (which should be obtuse) is not opposite the longest side. The response to part (b) is incorrect. Credit is given for the correct possible right triangles shown in part (c).
Sample Response That Received a Score of 0:
(a)
(b)
For EG to have a length of 1 or 20 would not be possible because 1 would create a triangle too small to the relative dimensions of the other lengths while 20 would create a triangle which has one length too big.
(c)
Commentary on Sample Response that Earned a Score of 0
This response received a score of 0 because it demonstrates extremely limited understanding of the topic. Both triangles in part (a) are drawn as right triangles. Neither of the triangles drawn shows reasonable relative lengths of the sides. In the 3-8-10 triangle, the side of length 8 is longer than the side of length 10. In the 8-10-16 triangle, the side of length 8 is shown as the longest side and is opposite the greatest angle. None of the angles in the second triangle is shown as obtuse. No credit is given for part (b) since the two figures are drawn showing triangles with sides of 1, 8, and 10 and 8, 10, and 20, respectively, but the explanation indicates that such triangles are not possible. The response does not sufficiently explain the triangle inequality property. Part (c) contains one of the two possible configurations of right triangle EFG. Although a portion of part (c) is correct, the evidence of knowledge about triangle geometry provided by this response does not reach the threshold required to receive a score above 0.
Sample Question 3
In a certain experiment, a researcher plans to label each sample with an identification code consisting of either a single letter or 2 different letters in alphabetical order. For example, if the researcher uses the 3 letters A, B, and C, then there are 6 possible identification codes that can be formed: A, B, C, AB, BC, and AC. The 2-letter combinations BA, CA, and CB would not be identification codes because the letters are not in alphabetical order.
(a)
If the researcher uses the 4 letters A, B, C, and D, how many identification codes can be formed that consist of a single letter? How many 2-letter identification codes can be formed that begin with the letter A? How many 2-letter identification codes can be formed that begin with the letter B? How many 2-letter identification codes can be formed that begin with the letter C? How many 2-letter identification codes can be formed that begin with the letter D? List all of the identification codes that can be formed using the letters A, B, C, and D.
(b)
Recall the formula: 1 + 2 + 3 + … + n =
Explain how this formula can be applied to answer the following question: If the researcher uses all 26 letters, what is the maximum possible number of identification codes that can be formed?
(c)
How many different letters did the researcher use if a maximum of 45 possible identification codes could have been formed? Show your work.
Sample Response That Received a Score of 3:
(a)
4 identification codes can be used with a single letter.
3 2-letter codes beginning with A.
2 2-letter codes beginning with B.
1 2-letter codes beginning with C.
0 2-letter codes beginning with D.
A, B, C, D, AB, AC, AD, BC, BD, CD
(b)
1 + 2 + 3 + … + n =
This formula can be applied to this problem because to find the number of possible labels when there are n-letters, there are n-possible single letter labels, n – 1 possible labels beginning with the first letter, n – 2 possible ways beginning with the second letter, and so on all the way down to one possible way for the second to last letter and zero ways for the last letter. When adding these numbers to find the total possible labels, we get the equation n + (n – 1) + (n – 2) + (n – 3) + … + 1 + 0. This is the sum of all the numbers from 0 to n which is what the expression
Therefore, there are
(c)
9 letters were used.
Commentary on Sample Response that Earned a Score of 3
This response received a score of 3 because it responds appropriately to all parts of the question and demonstrates strong knowledge of the concepts relevant to the question. Part (a) presents the correct number for each possible identification code and a complete and correct list of all the possible codes that could be formed with the letters A, B, C, and D. The response demonstrates a systematic approach to counting and identifying the possible codes. The response to part (b) correctly shows that 351 codes can be formed using all 26 letters and provides an appropriate explanation of how the formula for finding the sum of the first n integers can be applied to this question. Part (c) provides a correct algebraic solution that shows that a maximum of 45 possible identification codes can be formed if 9 letters are used.
Sample Response That Received a Score of 2:
(a)
Single letter — 4 (A, B, C, D)
2-Letter (Begin with A) — 3 (AB, AC, AD)
2-Letter (B) — 2 (BC, BD)
2-Letter (D) — 0
A, B, C, D
AB, AC, AD
BC, BD 2(4 + 1)=10
CD
(b)
The formula is for combinations and can be used to find how many different combinations are created using 26 letters.
(c)
cannot be negative
Commentary on Sample Response that Earned a Score of 2
This response received a score of 2 because it demonstrates a sufficient knowledge of the concepts relevant to the question. Part (a) shows a systematic identification and counting of the possible codes that can be formed with the letters A, B, C, and D. The response omits the number of 2-letter codes that can be formed beginning with the letter C but correctly identifies the one code that begins with the letter C as part of the complete list of the 10 possible codes that can be formed with these 4 letters. The calculations to the right in the response to part (a) appear to be an application of the formula given in part (b) to predict or confirm the total number of codes in part (a). These calculations are considered irrelevant in evaluating the response to part (a). In part (b), the given formula is used correctly to calculate the number of possible codes that could be formed with 26 letters, but the explanation of how this formula can be applied to this question is totally incorrect. Part (c) provides a correct algebraic solution that shows that a maximum of 45 possible identification codes can be formed if 9 letters are used.
Sample Response That Received a Score of 1:
(a)
Single letter A, B, C, D
2 Letter (A) AB, AC, AD
(B) BC, BD
(C) CD
(D) none
(b)
Each letter will only be paired up once, so the formula shows that for 26 letters, there would be
351 combinations
(c)
45 =
45 =
Commentary on Sample Response that Earned a Score of 1
This response received a score of 1 because it demonstrates a weak knowledge of the concepts relevant to the question. Although the response to part (a) does not explicitly answer the questions about the numbers of each type of identification code, it does present a systematic and correct identification of the 10 possible codes that can be formed using the letters A, B, C, and D. The evaluation of the formula given in part (b) for 26 letters is correct, but the explanation of how this formula can be applied to this question is totally incorrect. The response to part (c) is incomplete and does not provide evidence of the ability to solve equations such as these.
Sample Response That Received a Score of 0:
(a)
4
3
1
1
0
A, B, C, D, AB, BC, CD, AC, AD
(b)
331 codes
(c)
45
Commentary on Sample Response that Earned a Score of 0
This response received a score of 0 because it demonstrates extremely limited understanding of the topic. The response to part (a) shows a partially complete list of the possible codes that can be formed with the letters A, B, C, and D. The response does not demonstrate an understanding of how to identify and count the possible codes systematically, beginning with each of the letters. Part (b) provides an incorrect numerical answer with no accompanying work or explanation of how the formula given can be applied to this question. The response to part (c) is either just recording the information given or is an incorrect response with no work shown. The evidence of correct thinking provided by this response does not reach the threshold required to receive a score above 0.