Last month, June 2022, I participated as a scorer for the New York State Algebra 2 Regents Exam. I was assigned problems 31 through 34, which accounts for one-third of the available points for the open-ended responses. I can safely say that I read and scored hundreds of papers, as a conservative estimate. And over the course of nearly a week, several common errors and misconceptions emerged more than I would have imagined.
What follows are the most common mistakes we encountered in our team for these four questions. First, let me start off by saying that no particular student or school is being singled out. These occurred from school to school, district to district.
Second, I will stick to the avoidable mistakes, the responses where the student knew the proper procedure or at least showed some idea of how to solve the problem but show careless mistake or misunderstanding cost them one or more points. These are the problems with students may have done 95% of the work but only receive 50% (or less) of the credit.
Question 31
Graph y = 2 cos(1/2 x) + 5 on the interval [0,2π], using the axes below.
Among students who knew how to create the graph, most knew that the midline was 5, the maximum was 7 and the minimum was 3. However, there were three common errors, any two of which would result in zero credit.
First, the frequency was 1/2. Many responses had other frequencies, either 2 or some other value.
Second, the interval was stated. If they drew a line that went out of those boundaries without clearly marking off [0,2π], then a point was deducted. Similar to this, and possibly more aggravating, were the graphs that fit into the correct interval, but then included arrows at the endpoints. Arrows indicate that the graph continues when, in this instance, it should not.
The third of the common mistakes was not labeling the x-axis. The scorer could not assume the scale or where, say, 1/4 π was based on the curve above it.
For this question, I'll also point out that a large number of the papers I scored included students who relied on the table of values in their calculators. This was apparent from the fact that the x-axis was labeled 1,2,3,..., etc. In almost every instance, the student lost at least one credit for going beyond 2π. Many of these papers had a second error, making the question worth zero credits.
Question 32
A cup of coffee is left out on a countertop to cool. The table below represents the temparture, F(t), in degrees Fahrenheit, of the coffee after it is left out for t minutes.
[TABLE OMITTED]
Based on these data, write an exponential regression equation, F(t), to model the temperature of the coffee. Round all values to the nearest thousandth.
Of the students who knew the procedure, among the paper I scored, rounding errors were almost non-existant. This made me happy as I expected more of them.
The number one error that caused a one-credit deduction was using x instead of t, as in either of the following:
y = 169.136(0.971)x
or
F(t) = 169.136(0.971)x
This was probably the most careless and costliest error that I encountered. A lot of work is involved putting the data into those tables to get this response, and a single letter reduces the credit by half. It's maddening to me as an educator as well as a scorer.
There were few who did Linear regressions by mistake, which would be worth 1 credit if done "correctly". However, many of those also used x, which cost the other point.
For those who didn't know what to do, many wrote a function or equation for exponential decay.
Question 33
On the set of axes below, graph y = f(x) and y = g(x) for the given functions:
f(x) = x3 - 3x2
g(x) = 2x - 5
State the number of solutions to the equation f(x) = g(x).
It was quite clear from the response that many, MANY students had no idea what that last sentence meant or how to find solutions to f(x) = g(x), despite the graph that they had just created.
For the record, if they had answered "3" without a graph, they would have received one credit. And if they drew an incorrect graph and then responded with the number of intersections that appeared on their graph, they would have received one credit. (Consistent error, they already lost the points on the graph.) The student did not have to explain nor justify nor provide coordinates. They only had to provide a number. Many times that number did not match the graph or was omitted completely.
Additionally, some students tried to solve f(x) = g(x) algebraically, with various degrees of success. Again, this just reinforces my belief that they didn't know what solutions the questions was asking for.
Speaking of solutions, there was a substantial subset that assumed that "solutions" meant the "zeroes" for each of the graphs, which were listed separately.
While looking for graphing errors, most of the scorers I met worked with for this question focused on the domain -2 < x < 3, unless the lines were significantly incorrect beyond those points. It was usually clear when a point was misplaced or omitted (maybe they misread their table) and when a student was just sketching something that looked like what he saw on the screen. For example, many graphs of g(x) not only weren't slope of 2 and/or didn't have a y-intercept of -5.
I found the graphing errors a little disturbing because the functions were calculator-ready -- not manipulation or calculations had to be done before entering them into the calculators.
There were some "forced" errors. That is, a number of students tried to force one line or the other to intersect at integers, particularly (1,-2) or (1,-3).
This questions was worth 4 credits, which were essentially 2 credits for a correct graph of f(x), 1 credit for the correct graph of g(x), and 1 credit for "3" or a number consistent with the graph provided.
Question 34
A Foucault pendulum can be used to deomnstrate that the Earth rotates. The time, t, in seconds, that it takes for one swing or period of the pendumulm can be modeled by the equation t = 2π√(L/g), where L is the length of the pendulum in meters and g is a constant of 9.81 m/s2.
The first Foucault pendulum was constructed in 1851 and has a pendulum length of 67 m. Determine, to the nearest tenth of a second, the time it takes this pendulum to complete one swing.
Another Foucault pendulum at the United Nations building takes 9.6 seconds to complete one swing. Determine, to the nearest tenth of a meter, the length of this pendulum.
The most common mistake, by far, was A CALCULATOR ISSUE, so it only cost one credit instead of two. Many students, somewhere in their calculations, divided by 2π instead of dividing by (2π).
To state this more explicitly, many wrote (9.6/2π)2 correctly on their papers and then entered it exactly that way into their calculators, instead of entering (9.6/(2π))2. As a result, 9.6 was divided by 2 but then MULTIPLIED by π. THis usually resulted in a response of 2230.8, unless another error was made.
The second most common computational error was rounding in the middle of the problem. This meant that they lost accuracy and their responses would be off by as much as a meter.
A concepual error that I encountered quite a few times involved this equation:
9.6 = 2π√(L/9.81)
Some students attempted to square both sides of the equation to get rid of the radical, but they did not square the 2π. A second conceptual error involved dividing by 9.81 instead of multiplying.
Final Comments
Arguing with the format of the Regents exams won't get me anywhere -- particularly in this space and on this blog. (Please, just don't.) However, as long as students are stuck with them, educators need to stress the pitfalls of cutting any corners or simply not checking the work thoroughly. Mistakes happen not just through laziness or carelessness, but also through stress or just misreading or misinterpreting a few words.
The number of papers where students had absolutely no idea what to do tells me as an educator that I need to review a topic. But the ones who knew what they were doing but were tripped up on silly things, I really feel for them. I don't want them to be discouraged by a grade that doesn't affect their ability. It's even more frustrating because they'll never know why they received the score they did. (They might get a breakdown in points, but not any explanations.)
And honestly, I want to make sure that next year's students don't get tripped up by these things.
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