Tuesday, July 27, 2021

Algebra Problems of the Day (Integrated Algebra Regents, August 2012)



Now that I'm caught up with the current New York State Regents exams, I'm revisiting some older ones.

More Regents problems.

August 2012

Part I: Each correct answer will receive 2 credits.


16. The rectangular prism shown below has a length of 3.0 cm, a width of 2.2 cm, and a height of 7.5 cm.

What is the surface area, in square centimeters?

1) 45.6
2) 49.5
3) 78.0
4) 91.2

Answer: 4) 91.2


The surface area is the sum of the areas of the six sides of the prism. Since it is a rectangular prism, the opposite sides have the same area.

You have find the L * W, L * H, and W * H, double each of those, and find the sum.

L * W = 3 * 2.2 = 6.6

L * H = 3 * 7.5 = 22.5

W * H = 2.2 * 7.5 = 16.5

2 * (6.6 + 22.5 + 16.5) = 91.2





17. Which set of coordinates is a solution of the equation 2x - y = 11?

1) (-6, -1)
2) (-1, 9)
3) (0, 11)
4) (2, -7)

Answer: 4) (2, -7)


You can try each of the choices to see which makes a true statement, or you can rewrite the equation, solving for y, and put that into your graphing calculator and look at the Table of Values for the correct entry.

2x - y = 11
2x - 11 = y, or y = 2x - 11

Ironically, it wasn't until I typed the above equation just now that I noticed that the first three choices are all solutions to y = 2x + 11. I would've just did the mental math.





18. The graph of a parabola is represented by the equation y = ax2 where a is a positive integer. If a is multiplied by 2, the new parabola will become

1) narrower and open downward
2) narrower and open upward
3) wider and open downward
4) wider and open upward

Answer: 2) narrower and open upward


Since a is positive, it opens upward. Eliminate Choices (1) and (3).

If you double a, then the y values will increase more rapidly. That will cause the parabola to become thinner, or narrower.





19. Which equation represents a line that has a slope of 3/4 and passes through the point (2,1)?

1) 3y = 4x - 5
2) 3y = 4x + 2
3) 4y = 3x - 2
4) 4y = 3x + 5

Answer: 3) 4y = 3x - 2


The choices given are not in any familar form. However, they could easily be put into slope-intercept form by dividing by the coefficient of y to get y = mx + b.

If you divide Choices (1) and (2) by 3, then the value of m will be 4/3 instead of 3/4. Eliminate choices (1) and (2).

Next, substitute x = 2 into Choices (3) and (4).

4y = 3(2) - 2 = 6 - 2 = 4; 4y = 4, y = 1. This is the solution we are looking for.

4y = 3(2) + 5 = 6 + 5 = 11; 4y = 11, y = 11/4. This is NOT the solution we are looking for.





20. What is the value of | ( 4(-6) + 18 ) / 4! | ?
1) 1/4
2) - 1/4
3) 12
4) -12

Answer: 1) 1/4


I hate questions like this. I had a student come up to me after the exam and ask me, "Mr. Burke, what does an exclamation point in a math problem mean?" That's not to say that he shouldn't have known it, but the notation wasn't used as much as the concept it stands for. However, the reason I hate this problem is because it's meaningless and bizarre. There is no reason to put a factorial symbol in an equation like this other than to see if students were paying attention. It's a meaningless Order of Operations question with an added twist to it that only makes it longer, but not more challenging or interesting.

Order of Operations: Evaluate the numerator, multiply and add. Then evaluate the factorial, 4!, which is 4*3*2*1. Then simplify the fraction. Finally, take the absolute value.

Note that the Absolute Value is the final step, so the answer cannot be negative. Eliminate Choices (2) and (4).

| ( 4(-6) + 18 ) / 4! |
= | ( -24 + 18 ) / 4! |
= | -6 / 4! |
= | - 6/24 |
= | -1/4 |
= 1/4

I'm a little curious where they got 12 from. That should come from an "obvious" mistake, but I'm not seeing it.




More to come. Comments and questions welcome.

More Regents problems.

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