This Web article, a supplement to a broader critique of the New York State Regents Mathematics A exam, offers a detailed look at individual items on recent instances of the Math A exam. Past instances of this exam are posted on the Regents Examinations Web site, under the link to Mathematics A. My comments concern the exams of August, 2002, and of January and June, 2003.
The format of the test is identical over the three instances. There are 35 questions. Questions 1-20 are four-way multiple choice, and are worth 2 points each. Questions 21-25 are open response questions worth 2 points each, questions 26-30 are open response worth 3 points each, and questions 31-35 are open response worth 4 points each. Partial credit is possible on the open response questions.
Students must be supplied with a calculator and with a straight-edge and compass. The straight-edge and compass is required on only one or two questions each exam, but the calculator will be used extensively.
Aug 2002, Q4: "Juan has three blue shirts, two green shirts, seven red shirts, five pairs of denim pants and two pairs of khaki pants. How many different outfits consisting of one shirt and one pair of pants are possible? Answ: (1) 19; (2) 84; (3) 130; (4) 420"
A flawed question. If this were open ended then some students would decide that within each given category the items are to be viewed as indistinguishable, and they would find that there are 6 possible outfits. It appears that the test authors have decided to disambiguate the question through the choice of answers.
Aug 2002, Q5: "Given the statement: 'If two lines are cut by a transversal so that the corresponding angles are congruent, then the lines are parallel.' What is true about the statement and its converse? (1) The statement and its converse are both true. (2) The statement and its converse are both false. (3) The statement is true but its converse is false. (4) The statement is false but its converse is true."
Requires students to know what is meant by the "converse" of an implication. Many mathematicians would have to guess.
Aug 2002, Q6: "If the area of a square garden is 48 square feet, what is the length, in feet, of one side of the garden? (1) 12*sqrt(2); (2) 4*sqrt(3); (3) 16*sqrt(3); (4) 4*sqrt(6)"
The question looks entirely reasonable except that calculators are allowed on the exam, and so students will just try all the four given answers.
Aug 2002, Q14: "If the lengths of two sides of a triangle are 4 and 10, what could be the length of the third side? Answ: (1) 6; (2) 8; (3) 14; (4) 16"
This question is another instance of many in which the test authors have disambiguated their question by the choice of answers. A mathematician would find the question ambiguous, because it is not stated explicitly that "degenerate" triangles are disallowed. If we admit only proper triangles then the third side must have length in the open interval (6,14), else in the closed interval [6,14].
Aug 2002, Q20: "In the graph of y .le. -x , which quadrant is completely shaded? (1) I; (2) II; (3) III; (4) IV"
It is clear what is meant, but a mathematician would not use those words. Rather: "Which quadrant is completely included in the graph of y .le. -x?" . In fact, I would not have thought to test Math A students on their knowledge of the numbering of the quadrants. This knowledge only becomes routine when one is taking trigonometry or other more advanced mathematical analysis.
Aug 2002, Q22: The question gives what mathematicians call the multiplication table for an abstract group of four elements, and asks for the inverse of a certain element. I'm not going to reproduce it here; just go to the Web site. To me the question looks quite unreasonable for the intended test takers.
Aug 2002, Q27: "Tamika could not remember the scores from five mathematics tests. She did remember that the mean (average) was exactly 80, the median was 81, and the mode was 88. If all her scores were integers with 100 the highest score possible and 0 the lowest score possible, what was the lowest score she could have received on any one test?"
Mathematicians don't like this question because it requires one to know what is the "mode" of a data set - a silly concept if ever there was one. It is a rather difficult question in any case.
Aug 2002, Q34: "Greg is in a car at the top of a roller-coaster ride. The distance, d, of the car from the ground as the car descends is determined by the equation d = 144-16*t^2, where t is the number of seconds it takes the car to travel down to each point on the ride. How many seconds will it take Greg to reach the ground?"
More appropriate to the given formula: "Greg has just jumped off a high bridge..." (I'm assuming distances are in feet.)
Jan 2003, Q1: Students must know what is a "box-and-whisker" plot. Mathematicians and scientists would generally not care for this concept.
Jan 2003, Q3: Students must know what is the "inverse" of an implication. Many mathematicians would have the guess what is meant.
Jan 2003, Q7: "There are 12 people on a basketball team, and the coach needs to choose 5 to put into a game. How many different possible ways can the coach choose a team of 5 if each person has an equal chance of being selected?"
The "equal chance" bit does not belong in this question.
Jan 2003, Q8: "Given the true statement: 'If a person is eligible to vote, then that person is a citizen.' Which statement must also be true? (1) Kayla is not a citizen; therefore she is not able to vote. (2) Juan is a citizen; therefore he is eligible to vote. (3) ... etc."
A bad question, although the students can figure out what is meant and select the intended correct answer. The proper correct answer is that on the basis of what is given, none of the four statements must also be true. We were not given any information about Kayla or Juan or Marie or Morgan.
Jan 2003, Q9: "Line P and line C lie on a coordinate plane and have equal slopes. Neither line crosses the second or the third quadrant. Lines P and C must... (1) form an angle of 45 deg; (2) be perpendicular; (3) be horizontal; (4) be vertical."
It is questionable to refer to the slope of a vertical line. See also my comments under Aug 2002, Q20, about testing Math A students on their knowledge of the numbering of the quadrants.
Jan 2003, Q13: "If the measure of an angle is represented by 2x, which expression represents the measure of its complement? (1) 180-2x; (2) 90-2x; (3) 90+2x; (4) 88x"
Many mathematicians would cheerfully admit that they don't know which of answers (1) and (2) is the complement and which is the supplement. Further, a cleaner phrasing would be: "If the measure of an angle is 2x (degrees), what is the measure of its complement?".
Jan 2003, Q15: Requires students to know what is the "mode" of a data set.
Jan 2003, Q18: "What are the factors of x^2-10*x-24? (1) (x-4)*(x+6); (2) ... etc."
Should have asked: "What is a factorization of ..."
Jan 2003, Q20: Requires students to identify a pair of "alternate interior angles" in a figure of a line crossing a pair of parallel lines. Many mathematicians would cheerfully admit that they don't know the definition of this concept.
Jan 2003, Q34: "Sarah's mathematics grades for one marking period were 85, 72, 97, 81, 77, 93, 100, 75, 86, 70, 96, and 80. Question A: Complete the tally sheet and frequency table below and label a frequency histogram for Sarah's grades using the accompanying grid. (The given intervals are 61-70, 71-80, 81-90, 91-100.) Question B: Which interval contains the 75th percentile (upper quartile)?"
The "clarification" in question B is faulty. Either the question is what is the 75th percentile, or the question is which scores make up the upper quartile. They are different questions. For the given data it just happens to be the case that the 75th percentile (midway between 93 and 96) and the upper quartile scores (96, 97, 100) all lie in the same interval, 91-100.
Jun 2003, Q1: "The number 8.375*10^(-3) is equivalent to: (1) 0.0008375; (2) 0.008375; (3) 0.08375; (4) 8,375"
A fifth-grade question to start the exam. Mathematicians would use "equals" instead of "is equivalent to". (The same applies to Question 12.) Students will probably use their calculators for this question.
Jun 2003, Q3: "Which expression represents an irrational number? (1) sqrt(2); (2) 1/2; (3) 0.1; (4) 0"
Better: "Which of the following numbers is irrational".
Jun 2003, Q4: "Which of the following does not have rotational symmetry: (1) trapezoid; (2) regular pentagon; (3) square (4) circle?"
Mathematicians would not use this formulation, because of ambiguity in the expression "rotational symmetry". The circle has complete rotational symmetry, the square and regular pentagon have symmetry with respect to rotations over multiples of 90 or 72 degrees, and only the trapezoid has, in general, no rotational symmetry at all.
Jun 2003, Q14: "If the expression 3-4^2+6/2 is evaluated, what would be done last?"
Obviously faulty, and indeed the scoring rubrics were quickly corrected to allow both addition and subtraction as a correct answer. (Tough luck, still, for those students that thought to bring all terms to a common denominator of 2 and do the division last.) It would have been unambiguous to ask "If ... is evaluated without any regrouping, ...". The examiners may have felt it would be pointless to ask for the value of 3-4^2+6/2, because students would just key it into their calculators.
Jun 2003, Q16: "The sum of sqrt(18) and sqrt(72) is... (1) sqrt(90); (2) 9*sqrt(2); (3) 3*sqrt(10); (4) 6*sqrt(3)"
Note that students can just try the answers on their calculator.
Jun 2003, Q17: "What is the inverse of the statement 'If Julie works hard, then she succeeds'?"
Many mathematicians would have to guess what is meant by the inverse of an implication.
Jun 2003, Q18: If one factor of 56*x^4*y^3-42*x^2*y^6 is 14*x^2*y^3, what is the other factor?"
Better: What is 56*x^4*y^3-42*x^2*y^6 divided by 14*x^2*y^3?
Jun 2003, Q20: "How many different five-member teams can be made from a group of eight students, if each student has an equal chance of being chosen?"
The "equal chance" bit does not belong in the question.
Jun 2003, Q24. Please see the original. The notation m(angle)a would be unfamiliar to most mathematicians, although one can guess what is meant (the measure of the angle labelled "a"). The question should state that angles are measured in degrees.
Jun 2003, Q26. "Seth has one less than twice the number of compact discs (CDs) that Jason has. Raoul has 53 more CDs than Jason has. If Seth gives Jason 25 CDs, Seth and Jason will have the same number of CDs. How many CDs did each of the three boys have to begin with?"
The question does not state that Seth is giving Jason 25 CDs from his own collection, and the student has to guess that that was intended. An adjustment of the scoring guidelines to account for this would have been in order.
Jun 2003, Q29: "A certain state is considering changing the arrangement of letters and numbers on its license plates. The two options the state is considering are:
Option 1: three letters followed by a four-digit number with repetition of both letters and digits allowed
Option 2: four letters followed by a three-digit number without repetition of either letters or digits
[Zero may be chosen as the first digit of the number in either option.] Which option will enable the state to issue more license plates? How many more different license plates will that option yield?"
A rather unelegant and wordy question. The authors might have considered just license plates consisting of four letters followed by three digits, and could then have asked how many possibilities there are if repetitions are allowed and how many if repetitions are not allowed. Please note that student treatment of this question involves the calculator function "P": students are expected to key in something like 26(P)4 to obtain the number of four-letter sequences without repetition.
Jun 2003, Q31: Deborah built a box by cutting 3-inch squares from the corners of a rectangular sheet of cardboard, as shown on the accompanying diagram, and then folding the sides up. [The diagram shows the rectangular cardboard, with 3in * 3in cut-outs at the four corners, and dotted lines in the obvious places to indicate the folds.] The volume of the box is 150 cubic inches, and the longer side of the box is five inches more than the shorter side. Find the number of inches in the shorter side of the original sheet of cardboard."
A difficult four-point question, which has a quadratic equation buried inside a multi-stage geometry problem. There is a serious issue with the scoring rubric. A student might write: "The box is 3in high, the volume is 150 in^3, therefore the area of the base is 50 in^2. I recognize that 50=5*10, and with these factors indeed the difference between the lengths of the sides comes out to 5in. Therefore the shorter side of the base of the box is 5in, and the shorter side of the original piece of paper is (5+2*3)in, which equals (ANSW:) 11in." The scoring rubrics imply that this correct answer would receive just 1 point out of the 4 maximum. The issue is that, although the scoring rubrics allow the use of trial and error to solve the quadratic equation, to receive full credit one must use at least three trials with appropriate checks. An answer that contains two trials may receive 2 points, and the answer above, with only a single trial, may receive just one point.
June 2003, Q34. "A straw is placed into a rectangular box that is 3 inches by 4 inches by 8 inches, as shown in the accompanying diagram. If the straw fits exactly into the box diagonally from the bottom left front corner to the top right back corner, how long is the straw, to the nearest tenth of an inch?"
For students that know the three-dimensional version of Pythagoras this question is easy. The answer is sqrt(3^2+4^2+8^2) rounded to the nearest tenth. The strange thing is then that this is would be a four-point question. It appears, also from the scoring key, that the examiners do not consider the 3-d version of Pythagoras to be standard knowledge, and they actually expect students first to recognize the diagonal of length 5 in the base and the top of the container, and then to recognize that now the two-dimensional version of Pythagoras applies to the diagonal of a rectangle of sides 5 and 8. With that perspective the question looks a great deal harder - unreasonably hard in my opinion.
June 2003, Q35. "The senior class is sponsoring a dance. The cost of a student disk jockey is $40, and tickets sell for $2 each. Write a linear equation and, on the accompanying grid, graph the equation to represent the relationship between the number of tickets sold and the profit from the dance. Then find how many tickets must be sold to break even."
If students have seen any economic modelling at all then this question is easy. We are given the fixed cost and the unit price, and there is a single commodity. The question can be difficult only if this very elementary situation has not been covered in the curriculum. Apparently a similar question has not previously appeared on the Regents Math A, and many students were not prepared for this one. The scoring key says to subtract two points if a student graphs y=2x (revenue instead of profit) and intersects this with y=40 to obtain the correct break-even point.
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