The New York Regents Exams are the standardized final exams for high school students in New York State. For many years, the math Regents were very well made tests that teachers (most that I know) seemed to like. But lately the math Regents have poorly constructed tests. As much as I’d like to blame this on The Common Core, it actually has nothing to do with the Common Core and everything to do with whoever has been tasked to write these tests. In two recent posts I’ve analyzed some questions on the June 2017 Algebra II Regents, and will continue with more examples here.

Modeling real-world scenarios with exponential functions is a very interesting topic in the course which lasts for several weeks. There are so many good ways to assess if students understand what the different numbers in the exponential equation are. This Roman numeral thing with two Roman numerals and the four choices (I), (II), Both, Neither, is very lazy test making. Basically they have turned this into two True/False questions that you get either get full credit for getting both, or no credit even if you know one of the two.

Roman numeral I is unnecessarily confusing. When there is a number like a 110 in front of the *e*, it is generally the ‘starting point.’ So in this case it means that when *t*=0, or 0 years after 2010, or just 2010, the population was 110 million. But look at Roman numeral I. Rather than say the population in 2017 is 110 million, they say ‘The current population’ which can throw people off since ‘the current’ could mean 2010 when it was 2010. It is a poor wording for this question which still wouldn’t be a great question even if this was worded correctly.

For Roman numeral II, I think that the ‘approximately’ can lead to extra confusion as well. Had they made the exponent 0.039t, then they wouldn’t need to put the ‘approximately’ in at all. But if they want to use that, they should tell what it would need to be rounded to otherwise it becomes a matter of opinion if 3.922% is approximately 3.9%.

A better question would be to have the students select from different functions to say which makes the best model after being given the starting population and the growth rate. Or they could make four choices and ask something like “What does the 0.03922 represent?” or “What does the 110 represent?”.

I know that many people reading this are not math teachers, but I can tell you from a math teacher’s point of view that this is an awful way to test this topic, of rational equations. This is the only question on this entire test about rational equations, a topic that must take about two weeks or more when you include the various word problems that go along with it (like the ones where two people paint a house together).

The equation, itself, is fine for this. But then instead of just asking for the solution set to the equation, which requires understanding that one of the two apparent solutions needs to be rejected, they do something extremely odd, which is telling the students what the first step in the solution is. There are actually two ways to solve this problem (you could combine the fractions on the left and ‘cross multiply’), and it shouldn’t matter which way the student chooses since either way will lead to one actual solution and one extraneous solution.

Choice (3) is what’s known as a ‘distractor’ since it is something that might look tempting to someone who has a partial understanding of the topic. They might assume that just because 0 does make one of the terms undefined, it must be an extraneous solution even though it isn’t a solution to the equation that pops up in the process.

I think in all my years of teaching this topic and in making and seeing tests on this topic, I have never seen this tested in just this way. Perhaps if you have a test on just this topic and you have 10 questions and most of them are pretty straight forward and you want to put one like this on so that getting a 100 is tough, then I don’t have a problem with it. But if on a statewide Regents exam, this is your only question about this unit and it is an all-or-nothing multiple choice question, this is bad test making.

There is no reason to give the *f*(9)=-2 unless you are trying to force the students to calculate the *d* value by doing -2 = -8 + d(9-1), but why would anyone do that when it is clear that *d*=0.75 by just looking at the sequence itself.

Using the formula, you would get f(n)=-8.75 + 0.75(n-1) which looks a lot like choice (2) though choice (2) is not correct since there is a minus sign instead of a +. Personally, I would have made choice (3) in the f(n)=-8.75 + 0.75(n-1) form instead of the equivalent form that they have.

For a Regents, and maybe some people will say that this is too straightforward but I think it is the right way to test this topic, what is the 1000th term of the sequence -8, -7.25, -6.5, -5.75, … or if you want to make it more difficult, in the sequence -8, -7.25, -6.5, … what term number will be 315.25, or something like that. Since this is pretty much the only question on this topic which is a four or five day topic, you’ve got to keep it straight forward on the Regents. Being able to answer the two questions that I propose is a true indication that you understand this formula and how to apply it. The way this question is, it is quite possible for a student who would get something like an 80% on a full test on this topic to get zero out of two points on this one, the same as a student who would have gotten a 0% on a full test on this topic.

They should change the function to something like f(x)=x^2+2x, or something of that complexity. The point here is to see if the students know how to calculate the average rate of change, which I think is something they should know how to do. But by making this unnecessarily complicated function that requires putting the calculator into radian mode even (even though I could argue that this is ambiguous, you could have degrees with a pi in them).

This turns into a question about how well the students can manipulate the calculator and a student who understands the concept of average rate of change but presses one button out of the 40 or so buttons that need to be pushed for this, will get the same amount of credit, none, as the student who doesn’t know what average rate of change is.

I think this question, more than any so far, reveals how this test making team just did not ‘get’ the idea of what a good test question is and why.

So these four questions, were actually four consecutive questions on this test. Each could have been improved or edited to be a fine question. If they had someone in charge who is an expert on making good test questions, maybe they wouldn’t have had to curve this test so that 30% got curved up to a 65.

Thanks, Gary (and others) for pointing out the deteriorating quality of the Regents.

Problem 18 (the exponential model) is even a little worse. Yes, the “instantaneous” rate of change is 3.922%, which may or may not be “approximately 3.9%” (depending on context). But instantaneous rate of change is not the same as “the increase per year” mentioned in the problem. That is:

[e^.03922(t+1) -e^.03922t] / e^.03922t

= e^.03922 – 1

which turns out to be .0399992583 … and that’s essentially 4%. Is that approximately 3.9%? At best it’s confusing.

The writer of this question is either not quite sure of the mathematics or very sloppy. My guess is both.

Thank you. I love that someone cares enough about this to spend the time to analyze it.