What follows is a portion of the Common Core Integrate Algebra exam. Other parts will be posted on other days. Illustrations will be added at a later time when they become available.
Part II is available here.
August 2016 Algebra Regents, Part III
33. The data table below shows the diameter of grains of sand and the slop of the beach for 9 naturally occurring ocean beaches.
Using this equation, predict the slope of a beach, to the nearest tenth of a degree, on a beach with grains of sand having a median diameter of 0.65 mm.
Write the linear regression equation for this set of data, rounding all values to the nearest thousandth.
First, a naturally occurring ocean beach is one that isn't artificial or man-made. The sand comes onto the shore because of nature. This is not important for solving the problem, but don't let it throw you if you don't know what it is.
Second, I cannot stress this enough, if you do not round everything correctly, you will lose 1 of the 3 points for a -- yes, I will go there -- STUPID reason. "Nearest thousandth" means three decimal places.
Put the data into LISTs on the graphing calculator. Put the x values in L1 and the y values in L2. Double check your values, making sure you have all of them entered correctly, and that the x and y values match those on the table. (If you switch two x values without also switching the y values, you will get the wrong answer. It's like you mixed up coordinates!)
Press STAT. Right arrow to CALC. Press 4 for LinReg(ax + b)4. Hit Enter. (Depending upon your calculator's operating system, you may have to hit Enter again.)
You should get the following:
y=ax+b
a=17.15936928
b=-2.475925196
33. Shawn incorrectly graphed the inequality -x - 2y > 8 as shown below.
Explain Shawn's mistake.
Graph the inequality correctly on the set of axes below.
Shawn shaded the graph incorrectly. If you pick (0, 0) as a test point,
-(0) - 2(0) is not greater than or equal to 8, 0 < 8. He should have shaded below the line.
One possible error, but we cannot be sure of this because we don't have his work, is that Shawn forgot to flip the sign when rewriting the equation in slope-intercept form when he divided by -2.
You can graph the equation either by finding the x-intercept and the y-intercept or by rewriting the equation in slope-intercept form, which can then be put in your calculator.
If -x - 2(0) = 8
the -x = 8
and x = -8, so there is a point at (-8, 0)
If -(0) - 2y = 8
the -2y = 8
and y = -4, so there is a point at (0, -4)
Draw a solid line. That includes these points. The slope is down 4, right 8, which is -4/8 or -1/2.
Checking a test point, (0, 0), which is not on the line, we can see (as mentioned above) that (0, 0) is NOT a solution to this inequality, so it cannot be in the shaded region. That means that the other side of the graph, below the line, needs to be shaded. You can check this by picking another point, say (0, -10).
-(0) - 2(-10) > 8?
0 + 20 > 8?
20 > 8. Check.
Your graph should like as follows:
35. A drama club is selling tickets to the spring musical. The auditorium holds 200 people. Tickets cost $12 at the door and $8.50 if purchased in advance. The drama club has a goal of selling at least $1000 worth of tickets to Saturday's show.
Write a system of inequalities that can be used to model this scenario.
If 50 tickets are sold in advance, what is the minimum number of tickets that must be sold at the door so that the club meets its goal? Justify your answer.
Let x be the number of advance-sale tickets, and let y be the number of at-the-door tickets. The total number of tickets must be less than or equal to 200 people.
So x + y < 200.
The amount of money raised by selling x advance tickets is $8.50x. The amount of money raised at the door is $12y. The total of these two amounts needs to be more than $1000.
So 8.50x + 12.00y > 1000
That is your system of inequalities.
If 50 tickets are sold in advance, then the second inequality becomes
425 + 12.00y > 1000
12.00y > 575
y > 47.9166666...
So y has to be 48 or higher. Therefore, the minimum number of tickets sold at the door is 48.
Pointswise: if you had the correct system of inequalities, you scored 2 points, 48 was worth another point, and the work to justify 48 is the final point.
36. Janice is asked to solve 0 = 642 + 16x - 3. She begins the problem by writing the following steps:
Explain the method Janice used to solve the quadratic equation.
Line 1 0 = 642 + 16x - 3
0 = B2 + 2B - 3
0 = (B + 3)(B - 1)
Use Janice's procedure to solve the equation for x.
Janice simplified the problem by substituting the letter B in place of 8x, which is a factor of both 64x2 and 16x.
(8x)2 = 64x2 and 2(8x) = 16x. These substitutions will give you the original equation back.
Following Janice's logic:
B + 3 = 0 or B - 1 = 0
So B = -3 or B = 1
However, B = 8x
So 8x = -3 or 8x = 1
Therefore x = -3/8 or x = 1/8.
If you didn't do this kind of substitution, you likely would have been forced to use the Quadratic Formula to solve this equation, and there would have been BIG numbers to worry about.
If you solved it using another method, you lost a point.
August 2016 Algebra Regents, Part IV
37. For a class picnic, two teachers went to the same store to purchase drinks. One teacher purchased 18 juice boxes and 32 bottles of water, and spent $19.92. The other teacher purchased 14 juice boxes and 26 bottles of water, and spent $15.
76.
Write a system of equations to represent the costs of a juice box,j, and a bottle of water, w.
Kara said that the juice boxes might have cost 52 cents each and that the bottles of water might have cost 33 cents each. Use your system of equations to justify that Kara's prices are not possible.
Solve your system of equations to determine the actual cost, in dollars, of each juice box and each bottle of water.
The first part is a straightforward system of equations, similar to the inequalities we did in the earlier problem.
The equations are
14j + 26w = 15.76
To show that Kara is incorrect, substitute 52 for j and 33 for w in BOTH equations.
9.36 + 10.56 = 19.92 (check)
14(.52) + 26(.33) = 15.76?
7.28 + 8.58 = 15.76?
15.86 =/= 15.76 (does not check)
For the correct solution, you need to find the least common multiple of 14 and 18. (Yes! I hate the numbers in this problem!) The LCM is 126.
You need to multiply the first equation by 7, and the second equation by -9.
-126j - 234w = -141.84
add the equations
-10w = 2.40
w = .24.
A bottle of water = $.24.
14j + 26(.24) = 15.76
14j + 6.24 = 15.76
14j = 9.52
j = .68
A bottle of juice = $.68.
Partial credit would be given. A computational error would cost one point. Two or more computational errors would cost 2 points.
How did you do? Any questions?
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