Algebra 2/Trigonometry Regents for the Algebra 1 Student & Teacher

Today was the New York State Algebra 2/Trigonometry Regents exam. I don't teach this course, so I won't comment personally on how good a test it was for Trig students, other than to say that a couple of colleagues called it a "fair exam". What I can say about this exam is this: Algebra 1 teachers can use many of these multiple choice questions in their own classes with little to no adjustments. If I might so boldly and "arrogantly" claim, the top students in my Algebra 1 class could have solved 8 of the first 9 problems. An above average student would've gotten at least five of those correct.

With this in mind, I'd like to once again go over the Algebra 2 problems which I believe Algebra 1 students could handle, even if only as challenge problems.

Algebra 2/Trigonometry

1. Which survey is least likely to contain bias?
1. surveying a sample of people leaving a movie theater to determine which flavor of ice cream is the most popular
2. surveying the members of a football team to determine the most-watched TV sport
3. surveying a sample of people leaving a library to determine the average number of books a person reads in a year
4. surveying a sample of people leaving a gym to determine the average number of hours a person exercises per week

Not having my students the entire year, I didn't get to "bias" in Common Core Algebra (I believe the previous teacher should have touched on it). I know it was covered in the Integrated Algebra course. The second, third and fourth choices are going to places to ask a question pertaining to the place where the questions are asked; e.g., readers at a library. Only the first one goes to a place where you will find different types of people, not just ice cream lovers. Could there be bias in Choice 1? Of course, it could. Not all people go to movies. But it is still less biased than the other three.

2. The expression (2a)^-4 is equivalent to ...?

If you know that a negative exponent means to (basically) take the reciprocal, then you'll get 1/(16a⁴ as your answer.

Question 3 is a trigonometry question. We'll skip that.

4. Expressed in its simplest form,

is

This could easily be used in Algebra 1 without the negatives under the radicals. It could be used as an extension if there's time. Some of my students knew about imaginary numbers, even if they weren't sure exactly what they were. And they knew they had something to do with square roots.

It's also easy to reason out the answer from the choices. Once you realize that i is involved in both radicals and can be factored out, you've eliminated choices (1) and (2). Realizing that you're subtracting a bigger number from a smaller number indicates that the answer will be negative, eliminating choice (4). (3) is the answer.

5. Theresa is oomparing the graphs of y = 2^x and y = 5^x. Which statement is true?

First of all, both graphs have a y-intercept of (0, 1). Choices (1) and (4) are silly. (Really, "neither graph has a y-intercept"?) Of the two, y = 5^x is steeper. You can check this in your graphing calculator if you weren't sure.

6. The solution set of the equation

is

For Algebra 1 students (and some Trig students), the fastest method is to plug in the choices. Trying -2 doesn't work. Trying 2 does work. Only one solution set contains 2. It also contains 4, which also works.

How are you supposed to solve this? Square both sides and solve the resulting quadratic equation. For multiple choice, plugging in is much faster.

7. The expression is equivalent to

(2)(2) = 4; (-3)(x)^.5 X (-3)(x)^.5 = 9x; (2)(2)(-3)(x)^.5 = -12(x)^.5
The correct choice is (3).

8. Which step can be used when solving x² - 6x - 25 = 0 by completing the square.

Okay, I never did completing the square in Integrated Algebra. It might've been there in the textbook, but it wasn't covered in the curriculum, and it wasn't on the Algebra Regents. That said, it was in the Common Core Algebra this year, and my students picked it up pretty easily. (Well, most of them did.)

To complete the square, you need to halve the -6, getting -3, and then squaring that, getting 9. So +9 is added to each side of the equation and +25 is also added to each side of the equation to get rid of the -25 on the left. The correct choice is (1).

9. Which graph represents a function?

Seriously? This is an Algebra 1 question. If there aren't two y values for the same x-value, then it is a function. Choice (1).

Question 10 is a trigonometry question. We'll skip that.

11. What is the common difference of the arithmetic sequence below?
-7x, -4x, -x, 2x, 5x, . . .

Algebra students should recognize the pattern and deduce that the "common difference" is 3x.

Jumping ahead...

14. What is the product of the roots of the quadratic equation 2x² - 7x = 5?

I should include questions like this. There's no reason not to, and it will get an extra step of them. First solve the quadratic equation, and then multiply the roots. The only problem I have with this -- and maybe it isn't a problem at all -- is that the most common mistake my students make in solving quadratics in flipping the sign. If they flipped both signs and then multiply the answer, then the mistakes will cancel out.

Quick use of the quadratic formula will get you ... two radical conjugates. Okay, so this goes beyond the scope of Integrated Algebra, but a teacher could modify this one a little. But anyway, the product is one-sixteenth of (49 - 89), which is -5/2.

It's actually simpler than this: the rule for product of roots is c/a, which is -5/2. Introducing this right after doing a long problem might be a good way to make them remember the shortcut. It also reinforces the fact that if you can't remember the formulas and shortcuts, it helps to know where they come from, so you can derive them if you have to.

* * *

Continuing the thread...

15. What is the equation of the circle passing through the point (6, 5) and centered at (3, -4)?

This question gets asked on the Geometry Regents at least 3 or 4 times on every test. The only difference here is that the radius is an irrational number, but big deal. Geometry students need to deal with irrational numbers, and the square of the number is needed anyway. (6 - 3)² + (5 - -4)² = 90. So the equation is
(x - 3)² + (y + 4)² = 90.

16. The formula to determine continuously compounded interest is A = Pe^rt, where A is the amount of money in the account, P is the initial investment, r is the interest rate and t is the time, in years. Which equation could be used to determine the value of an account with an $18,000 initial investment, at an interest rate of 1.25% for 24 months?

As complicated as this looks, this is a simple substitution question. It could be given to my freshmen as an extension, just to see if they really can parse a question. You don't have to explain e yet, if you don't want to be, because it could be considered just any variable for the moment. (I realize that it's a constant, but let's not confuse matters at the moment.) The only "trick" to the problem is to remember that 24 months is 2 years. This trips up some students with I=PRT, too.

Question 17 is interesting. Without the "+ 1", it's a simple proportion that leads to a quadratic equation if you don't factor the difference of squares and multiply the fraction on the right by (x + 3)/(x + 3). The "+ 1" makes the addition a little more interesting. Lots of possibilities with this equation for Algebra students.

18. The graph below shows the average price of gasoline, in dollars, for the years 1997 to 2007. [GRAPH NOT SHOWN] What is the approximate range of this graph?

Seriously? Range measures the y values on the graph. The lowest point appears to be about 1.00 or lower, and the highest point is between 2.00 and 2.50. The correct choice would be 0.97 < y < 2.38. Choices 1 and 2 relate to the domain of the graph.

What are your opinions of all this?