Monday, November 30, 2020

STAAR (State of Texas Assessments of Academic Readiness) Algebra I, May 2018, cont.

More STAAR problems.

Read each question carefully. For a multiple-choice question, determine the best answer to the question from the four answer choices provided. For a griddable question, determine the best answer to the question.

21. The graph shows the linear relationship between the maximum area in square feet that can be painted and the number of gallons of paint used.

Which of these best represents the rate of change of the maximum area painted with respect to the number of gallons of paint used?

A 200 ft2/gal
B 1/200 ft2/gal
C 400 ft2/gal
D 1/400 ft2/gal

If you go to 1 gallon on the x-axis, that lines up with 400 ft2 on the y-axis. Change in y / Change in x, it is 400 ft2 / gal.

22. What are the solutions to (x + 7)2 = 81?

F -74 and 88
G -2 and 16
H -88 and 74
J -16 and 2

Take the square root of both sides: x + 7 = + 9
Split and solve: x + 7 = 9 or x + 7 = -9
x = 2 or x = -16

You could also have plugged in 2 for x and saw that it was a correct solution, while all of incorrect choices would be way off.

23. What appears to be the domain of the part of the exponential function graphed on the grid?

A -1 < x < 3
B -1 < y < 3
C 5.3 < x < 27
D 5.3 < y < 27

Answer: A -1 < x < 3
This gets repeated many times, but the domain is the x values. Eliminate Choice B and D.

The leftmost point has an x coordinate of -1, so the correct choice is A.

Choices B and C are just nonsense.

24. The tables of ordered pairs represent some points on the graph of lines q and v.

Which system of equations is represented by lines q and v?

F 21x − y = 9 ; 5x + 6y = 40
G 3x − y − = −27 ; x + 2y = 16
H 21x −y = 9 ; 5x + 6y = 20
J 9x − y = −27 ; x + 2y = 8

Answer: G 3x − y = −27 ; x + 2y = 16
Working backward from the answers is the quickest route. You could create the linear equations for each table, but then you would have to rewrite them in Standard form to match the choices.

My preference is to work with 0 when it is available. Looking at the table for line q, we see the point (-9, 0). Therefore x = -9 and y = 0 is a solution to one of those equations. (Note that it doesn't have to be the first of the pair. Nothing says that it is.)

21(9) - 0 =/= 9. 5(-9) + 0 =/= 40.
3(-9) - 0 = -27. (Choice G looks good so far).
21(9) - 0 =/= 0. 5(-9) + 0 =/= 20.
9(-9) - 0 =/= -27. -9 + 0 =/= 8.

Only Choice G is possible. Check the second equation for (0, 8). 0 + 2(8) = 16. Check!

25. Which situation does NOT show causation?

A When the student population at a school increases, the number of teachers at the school increases.
B When the amount of sugar in a quart of apple juice is reduced, there are fewer calories in each serving.
C When there are more workers on a project, the project is completed in less time.
D When there is more protein in an athlete’s diet, the athlete scores more points in a game.

Answer: D When there is more protein in an athlete’s diet, the athlete scores more points in a game.

If you get more students, then you need more teachers. One causes the other.

If you reduce the sugar, you reduce the number of calories.

If you put more workers on a project, it gets completed in less time. (There's a point of diminishing returns, but that's a little beyond the level of this question. Also, there's a joke about the project being a symphony -- a bigger orchestra won't complete it sooner. But this is more about "man-hours" to finish.)

There are many factors in athletic performance. Is increasing protein one of them? It doesn't show causation. Maybe if said what the protein did, or how that affected performance, you might show causation. But this itself does not.

More to come. Comments and questions welcome.

More STAAR problems.

Sunday, November 29, 2020

STAAR (State of Texas Assessments of Academic Readiness) Algebra I, May 2018, cont.

More STAAR problems.

Read each question carefully. For a multiple-choice question, determine the best answer to the question from the four answer choices provided. For a griddable question, determine the best answer to the question.

16. The water level of a river was measured each day during a two-week period. The graph models the linear relationship between the water level of the river in feet and the number of days the water level was measured.

Which statement best describe the y-intercept of the graph?

F The water level increased by 0.25 ft per day.
G The maximum water level was 19.5 ft.
H The initial water level was 16 ft.
J The water level was measured for 14 days.

Answer: H The initial water level was 16 ft.
The y-intercept represents the initial amount in a graph or fuction.

The other statements are all true, but have nothing to do with the y-intercept. In F, the incease in water level per day is the slope. In G, the maximum water level is a part on the graph, but not the initial point. In J, we've established the domain, 0 - 14, which concerns the x values, not the y values.

17. There are 1,024 players in a tennis tournament. In each round, half the players are eliminated. Which function can be used to find the number of players remaining in the tournament at the end of x rounds?

A f(x) = 1,024(1.50)x
B f(x) = 1,024(0.50)x
C f(x) = 1,024(1.05)x
D f(x) = 1,024(0.05)x

If half the teams are eliminated, then the rate of decay is .50, or 50%.

Choice A shows a 50% increase, meaning more players are joining the tournament. Choice C shows a 5% increase. Choice D shows a 95% decrease, with only 5% advancing to the next round.

18. Which statement about g(x) = x2 − 576 is true?

F The zeros, −288 and 288, can be found when 0 = (x + 288)(x − 288).
G The only zero, 288, can be found when 0 = (x − 288)2.
H The zeros, −24 and 24, can be found when 0 = (x + 24)(x − 24).
J The only zero, 24, can be found when 0 = (x − 24)2.

Answer: H The zeros, −24 and 24, can be found when 0 = (x + 24)(x − 24).
The Difference of Squares rule states that x2 - n2 can be factored into (x + n)(x - n). The "inner" and "outer" terms when multiplied form a zero pair and cancel out.

In this problem, 576 represents n2, so n is SQRT(576), which is 24, not 288.

19. Which graph best represents the solution set -4x < 6y - 54?

Two questions to ask yourself: Is it a solid or a broken line? Is (0, 0) part of the solution?

Since we want "less than or equal to" (<) and not just "less than" (<), the graph needs to have a solid line, so Choices B and C are eliminated.
Substitute (0, 0) and we get 0 < 0 - 54, which is not a true statement. Therefore, (0, 0) is not a part of the solution, and that side of the line cannot be shaded. Choice D is eliminated.

The longer method involves rewriting the problem into slope-intercept form.

-4x < 6y - 54
6y - 54 > -4x
6y > -4x + 54
y > -4/6 x + 9

At this point, you can see that the line needs to be shaded above, not before. Also, you can see that choice A have a slope of -4/6 (or -2/3), while D has a slope of -6/4 (or -3/2).

20.Given f(x) = 1/3 (4 - x)2, what is the value of f(16)?

f(16) = 1/3 (4 - 16)2
= 1/3 (-12) 2
= 1/3 (144)
= 48

More to come. Comments and questions welcome.

More STAAR problems.

Friday, November 27, 2020

STAAR (State of Texas Assessments of Academic Readiness) Algebra I, May 2018, cont.

More STAAR problems.

Read each question carefully. For a multiple-choice question, determine the best answer to the question from the four answer choices provided. For a griddable question, determine the best answer to the question.

11. Which expression is equivalent to x2 − 17x − 60?

A (x - 20)(x + 3)
B (x - 5)(x - 12)
C (x - 20)(x - 3)
D (x + 5)(x - 12)

Answer: A (x - 20)(x + 3)
The minus sign in front of the 60 tells you that one factor is positive and one factor is negative. That eliminates Choices B and C.

What are two factors of -60 that add up to -17? -20 + 3 = -17, Choice A is the answer.

12. A zookeeper recorded the feeding schedule for a baby rhinoceros for 20 weeks. The table and scatterplot show the percentage of the baby rhinoceros’s body mass that was used to determine the amount of food given at each feeding as a linear function of its age in weeks.

What is the best prediction of the percentage of the baby rhinoceros’s body mass that should be used to determine the amount of food given at each feeding when it is 25 weeks old?

F 8.5%
G 6%
H 2.5%
J 10%

Put the data into your graphing calculator and do a linear regression. You will get a = -.5448... and b = 19.80...
That makes your line of best fit, rounded to the nearest hundredth, y = -.54x + 19.80.
Substitute 25 for x and you get y = -.54(25) + 19.80 = 6.3, which is approximately 6%, which is Choice G.

13. What are the domain and range of f(x) = −37?

A Domain: All real numbers greater than or equal to −37; Range: All real numbers
B Domain: {−37} ; Range: All real numbers
C Domain: All real numbers ; Range: All real numbers greater than or equal to −37
D Domain: All real numbers ; Range: {−37}

Answer: D Domain: All real numbers ; Range: {−37}
The range is {-37}. No matter what the input is, the output is -37. That is the entirety of the range. That gives you exactly one option, Choice D.

The input is All real numbers. It doesn't matter what x is, and there are no restrictions on it. There are no values that are undefined.

14. The graph of quadratic function p is shown on the grid.

If k(x) = x2 and p(x) = k(x) + n, what is the value of n? Record your answer and fill in the bubbles on your answer document.

The graph of the parent function k(x) = x2 goes through the origin, (0, 0). The illustrated graph of p(x) has its vertex at (0, -6). Therefore, p(x) = x2 - 6, which means that n = -6.

You can check this by substituting x = -3 or x = 3. Both numbers squared give 9 which after subtracting 6 becomes 3.

15. Two customers went to a post office to buy postcards and large envelopes. Each postcard costs the same amount, and each large envelope costs the same amount.

• • The first customer paid \$12 for 14 postcards and 5 large envelopes.
• • The second customer paid \$24.80 for 10 postcards and 15 large envelopes.

What was the cost in dollars of each large envelope?

A \$1.42
B \$0.35
C \$1.15
D \$0.63

The system of equations is as follows:

14x + 5y = 12
10x + 15y = 24.80
multiply the first equation by 3, and then subtract
42x + 15y = 36.00
10x + 15y = 24.80
32x = 11.20
x = 0.35, which is the cost of a postcard
14(.35) + 5y = 12
4.90 + 5y = 12
5y = 7.10
y = 1.42, which is the cost of a large envelope

More to come. Comments and questions welcome.

More STAAR problems.

Thursday, November 26, 2020

Happy Thanksgiving 2020!

(Click on the comic if you can't see the full image.) (C)Copyright 2020, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

From start to finish, there's a strong aromatic sequence from one course to the next.

A lot of food for one small family of four, but how many people can cook smaller Thanksgiving dinners and still call it a Thanksgiving dinner? I mean, what are you going to leave out?

And you know that there will be "food runs" to the relatives who couldn't be there.

Happy Thanksgiving from (x, why?)!

Come back often for more funny math and geeky comics. Wednesday, November 25, 2020

Call Me Mike

(Click on the comic if you can't see the full image.) (C)Copyright 2020, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

It just sounds wrong, right?

Students who keep in touch on social media still call me Mr. B, which is fine. But I've seen a couple socially, and they've tried not calling me that, and it doesn't work for them. Even though people younger than them have no problem.

This, of course, carries over to all of my parents' friends. Keep in mind, I'm older than Mike.

On a comic note: Moira (first time mentioning a name) was a student here when she was younger. She became a teaching student of Michele's shortly after the wedding.

Come back often for more funny math and geeky comics. STAAR (State of Texas Assessments of Academic Readiness) Algebra I, May 2018, cont.

More STAAR problems.

Read each question carefully. For a multiple-choice question, determine the best answer to the question from the four answer choices provided. For a griddable question, determine the best answer to the question.

6. Which expression is equivalent to (144k2r14)(1/2) for all positive values of k and r?

F 12kr7
G 72k2r14
H 144kr7
J 12k2r14

The 1/2 exponent gets applied to each of the factors in the parentheses. When a variable with an exponent is raised to another exponent, you multiply the exponents. So k2r14 becomes k1r7. (We don't need to write the 1.)

The 144 also gets the 1/2 exponent, and 144(1/2) is the square root of 144, which is 12.

2. The graph shows the height in feet of an object above the ground t seconds after it was launched from the ground.

Which function is best represented by the graph of this situation?

A h(t) = -16t2 - 64t
B h(t) = -16t2 + 128t - 256
C h(t) = -16t2 + 64t
D h(t) = -16t2 - 128t - 256

Answer: C h(t) = -16t2 + 64t
Notice that the y-intercept is 0, so Choices B and D can be eliminated.

If you substitute t = 2 into Choice A, you will get -16(2)2 - 64(2), which you can see is a negative number before you put it into your calculator. (It's -192.)

If you substitute t = 2 into Choice C, you will get -16(2)2 + 64(2), which is 64.

8. What value of n makes the equation 4(0.5n − 3) = n − 0.25(12 − 8n) true?

F 3
G -9
H 0
J -15

Solve the equation for n:

4(0.5n − 3) = n − 0.25(12 − 8n)
2n - 12 = n - 3 + 2n
-12 = n - 3
-9 = n

Check:
4(0.5(-9) - 3) = -9 - 0.25(12 - 8(-9))
4(-4.5 - 3) = -9 - 0.25(12 + 72)
4(-7.5) = -9 - 0.25(84)
-30 = -9 - 21
-30 = -30 (check!)

9. In a sequence of numbers, a3 = 0, a4 = 6, a5 = 12, a6 = 18, and a7 = 24. Based on this information, which equation can be used to find the nth term in the sequence, an?

A an = 6n + 18
B an = -18n + 6
C an = 6n - 18
D an = 18n - 6

Answer: C an = 6n - 18
The Common Difference in this arithmetic is 6 - 0 = 6, 12 - 6 = 6, 18 - 12 = 6, etc. So the coefficient of the variable is 6. Eliminate choices B and D.

At this point, we would need to work backward to find the y-intercept (a0), but it isn't necessary. The choices are -18 and +18. However, a6 is 18 and the numbers are getting larger as the sequence continues. So it has to be -18, which is Choice C.

Working backward: a3 = 0, so a2 = 0 - 6 = -6, then a1 = -6 - 6 = -12, and a0 = -12 - 6 = -18.

10. The value of y varies directly with x. If x = 3, then y = 21. What is the value of x when y = 105??

F 3/5
G 1 2/3
H 7
J 15

It the value of y varies directly with x, then the slope of the line is simply the ratio y/x. (The y-intercept of the line is 0.)
This means that a proportion can be set up:

y / x = y / x
21 / 3 = 105 / x
21x = 315
x = 15

Or you might have noticed that 21/3 = 7, so the ratio is 7/1. Divide 105 by 7, and you get 15, which is J.

You might also have divided 105 / 21 = 5 and then multiplied 3 x 5 = 15.

Honestly, I've surprised that 735 was not one of the incorrect choices. Someone was likely to find the ratio 7 and then multiply instead of divide.

More to come. Comments and questions welcome.

More STAAR problems.

Tuesday, November 24, 2020

STAAR (State of Texas Assessments of Academic Readiness) Algebra I, May 2018

More STAAR problems.

Read each question carefully. For a multiple-choice question, determine the best answer to the question from the four answer choices provided. For a griddable question, determine the best answer to the question.

1. At a restaurant jars of tomato sauce are stored in boxes in the pantry. Each box contains 8 jars of tomato sauce. A cook uses 2 jars from 1 of the boxes.
Which function shows the relationship between y, the total number of jars of tomato sauce remaining in the pantry, and x, the number of boxes in the pantry?

A y = 8x + 6
B y = 8x
C y = 8x - 2
D y = 6x

Answer: C y = 8x - 2
We don't know how many jars of sauce are in a single box. However, we know that if one box holds x jars than 8 boxes holds 8x jars.

If 2 jars are taken out of one of the boxes -- not 2 boxes of jars, not 2 jars from every box -- than 2 is subtracted from the 8x, which is 8x - 2.

2. Which expression is equivalent to -28x2 + 35x?

F 7x(4x + 5)
G -7x(4x - 5)
H 7x(4x - 5)
J -7x(4x + 5)

Notice that all of the choices have a positive 4x. If you divide -28x2 by 4x, you will get -7x. Eliminate choice F and H.

Next, if you divide + 35x by -7x, you will get -5.

3. Which graph best represents a system of equations that has no solution?

The solution to a system of equations is where the two lines meet. Choices A and C actually show the points of intersection. Eliminate them.

Next, look at B. The lines don't meet on the portion of the graph that is visible, but the lines are moving closer together. If you extend the lines, they were intersect at point just off to the right.

In graph D, the two lines are parallel. They have the same slope. That means that they will never get any closer to each other and definitely will NOT intersect. There is no solution.

4. Which statment about the graph of y = 8(0.25)x is true?

F The coordinates of the x-intercept are (0.25, 0).
G The coordinates of the y-intercept are (0, 8).
H The equation of the asymptote is x = 0.
J The graph includes the point (2, 1).

Answer: G The coordinates of the y-intercept are (0, 8).
The graph is an exponential function which is getting smaller as x increases, but will never cross the x-axis. This means that there is NO x-intercept. Eliminate Choice F.

When x = 0, 0.250 = 1, and 8 * 1 = 8. Therefore, the y-intercept is (0, 8). Choice G is the correct answer.

As stated above, the asymptote for the graph is the x-axis. The equation for the x-axis is y = 0 NOT x = 0. Eliminate Choice H.

If you substitube x = 2, then y = 8(0.25)2 = 8(.0625) = .5, not 1. Eliminate Choice J.

5. What is the range of y = -x2 - 2x + 3?

A x < 4
B x > -4
C y < 4
D y > -4

The range is the y values so eliminate Choices A and B.

Because the leading coefficient of the quadratic equation is negative, you know that it will have a maximum value, and the range must be less than or equal to that amount. Only Choice C has a y and less than or equal to.
You don't even need to calculate what the maximum value will be to answer the question.

More to come. Comments and questions welcome.

More STAAR problems.

Monday, November 23, 2020

STAAR (State of Texas Assessments of Academic Readiness) Algebra I, May 2019, cont.

More STAAR problems.

Read each question carefully. For a multiple-choice question, determine the best answer to the question from the four answer choices provided. For a griddable question, determine the best answer to the question.

51. A customer at a store paid \$64 for 3 large candles and 4 small candles. At the same store, a second customer paid \$4 more than the first customer for 1 large candle and 8 small candles.
The price of each large candle is the same, and the price of each small candle is the same. Which system of equations can be used to find the price in dollars of each large candle, x, and each small candle, y?

A 4y = 3x + 64 ; 8y = x + 68
B 4y = 3x + 64 ; 8y = x + 60
C 3x + 4y = 64 ; x + 8y = 68
D 3x + 4y = 64 ; x + 8y = 60

Answer: C 3x + 4y = 64 ; x + 8y = 68
The total price of the candles is \$64. That tells you that it will be Choice C or D.
If the second customer paid \$4 more, than that customer paid \$68, not \$60, so Choice D is eliminated.

Choices A and B would work if the equations said 4y = 64 - 3x, with subtraction and not addition.

52. Linear parent function f is graphed on the grid.

Which graph best represents h(x) = −f(x) + 3?

The -f(x) will reflect the graph over the x-axis. All of the negative values will become positive. All of the positive values will become negative. Eliminate choices F and J.
The +3 will add 3 to the y coordinate after it has been reflected. Choice G is correct.

53. Which expression is equivalent to 4(sqrt(147))?

A 196(sqrt(3))
B 12(sqrt(7))
C 3(sqrt(7))
D 28(sqrt(3))

You can check the decimal equivalents of each of the choices, just be careful when entering the expression.

To simplify the radical, look at it's factors.
The factors of 147 are 3 x 49 = 3 x 7 x 7
So we can split SQRT(147) into SQRT(3) * SQRT(72), which is 7 SQRT(3).
Therefore, 4(SQRT(147)) = 4(7*SQRT(3)) = 28(SQRT(3))

54. The total distance in centimeters a toy robot moves varies directly with the time in seconds.
The toy robot moves a total distance of 264 centimeters in 11 seconds.
What is the time in seconds the toy robot moves when the total distance is 408 centimeters?

F 24 s
G 17 s
H 13 s
I 37 s

The robot moved 264 cm in 11 sec, meaning it traveled at a rate of (264/11) (cm/s), or 24 cm/s.
If you divide 408 cm by 24 cm/s, you get 17 s.

End of exam.

More to come. Comments and questions welcome.

More STAAR problems.

Sunday, November 22, 2020

STAAR (State of Texas Assessments of Academic Readiness) Algebra I, May 2019, cont.

More STAAR problems.

Read each question carefully. For a multiple-choice question, determine the best answer to the question from the four answer choices provided. For a griddable question, determine the best answer to the question.

46. The graph of quadratic function k is shown on the grid.

Which statements are best supported by the graph of k? I. The x-intercept is located at (-3 0). II. The coordinates of the y-intercept are (0, 9). III. The axis of symmetry is x = -3.

F I and II only
G I and III only
H II and III only
J I, II, and III

Answer: J I, II, and III
The vertex is the x-intercept, and the axis of symmetry goes through the vertex at x = -3.
The y-intercept is (0, 9), which is a little difficult to see in the image. However, if you notice, the points go from (-3, 0), (-2, 1), (-1, 4), (0, 9). The y-coordinate increases +1, +3, +5, etc.

47. A college student has two different jobs. Her combined work schedules consist of no more than 48 hours in one week.
Which graph best represents the solution set for all possible combinations of x, the number of hours she worked at her first job, and y, the number of hours she worked at her second job, in one week?

The more hours she works at one job, the fewer hours she can work at the other job. That is a declining slope from the y-intercept (Only works the second job) to the x-intercept (only works the first job).
And that line is the most she could work. She could work less, which is the area under the line.

48.Which function is equivalent to q(x) = 9x2 − 24x + 16?

F y = (9x - 4)(x - 4)
G y = (3x + 4)2
H y = (9x + 4)(x + 4)
J y = (3x - 4)2

Answer: J y = (3x - 4)2
You have to find those two middle terms and find which have a sum of -24x.
It should be obvious that G and H should be eliminated because the positive numbers cannot have a negative product.
Choice F has (9x)(-4) + (-4)(x) = -40x, which is incorrect.
Choice J has (3x)(-4) + (3x)(-4) = -24x, which is what we are looking for.

49. Which graph best represents this system of equations and its solution?

8x − 4y = −16
3x + 15y = −6

One option is to find the x-intercept and y-intercept of each line.

8x - 4y = -16
8x = -16 and -4y = -16
x = -2 and y = 4
(-2, 0) and (0, 4)
Only Choice B has these two points.

3x + 15y = -6
3x = -6 and 15y = -6
x = -2 and y = -6/15
(-2, 0) and (0, -6/15)
Choice B has these points. (You can estimate the y-intercept)

50. What are the domain and range of g(x) = -1/4(x - 17)2 + 61?

F Domain: All real numbers; Range g(x) < 61
G Domain: x < 17; Range g(x) < 61
H Domain: All real numbers; Range g(x) < 17
I Domain: x > 17; Range g(x) < 17

Answer: A y = 1.2x - 6
All values of x are valid input for this function. There are no value for which the function would not be defined. Choices G and J are eliminated.
The function is a quadratic. It's a parabola with a vertex at (17, 61). The leading coefficient is negative, so the vertex is a maximum. All values of the range must be below the vertex, so g(x) < 61, which is Choice F.
Also notice that in Choice H, the range is given in terms of x, which makes no sense.

More to come. Comments and questions welcome.

More STAAR problems.

Saturday, November 21, 2020

STAAR (State of Texas Assessments of Academic Readiness) Algebra I, May 2019, cont.

More STAAR problems.

Read each question carefully. For a multiple-choice question, determine the best answer to the question from the four answer choices provided. For a griddable question, determine the best answer to the question.

41. A part of an exponential function is graphed on the grid

Which inequality best represents the domain of the part shown?

A x > -2
B y > 4.5
C x > 4.5
D y > -2

We want the domain, which means the x values. Eliminate Choices B and D.
The x-coordinate of the starting point is -2. Its y-coordinate is 4.5. We want x.

42. What is the solution to -(6m + 8) = 4(17 - m)?

-(6m + 8) = 4(17 - m)
-6m - 8 = 68 - 4m
-2m = 76
m = -38

Don't forget to distribute the minus sign. Basically, watch your signs everywhere.

43. Which function is equivalent to y = 3(x + 2)2 + 7?

A y = 3x2 + 12x + 33
B y = 3x2 + 12x + 19
C y = 3x2 + 19
D y = 3x2 + 33

Answer: B y = 3x2 + 12x + 19
Option 1: Graph them all and look.
Option 2: Multiply 3(x + 2)(x + 2) and get your answer.
Option 3: which of the four choices has a vertex of (-2, 7)?

The equation in the question is in Vertex form. The choices are in standard form.
The vertex is on the Axis of Symmetry, and the formula for the Axis is x = -b/(2a).
Since neither C nor D have a b term, they can be eliminated.
Note that A and B both have 12 as the middle coefficient, so we don't need to check. Just subsitute -2 for x and see which on gives you 7.

y = 3(-2)2 + 12(-2) + 33 = 21
y = 3(-2)2 + 12(-2) + 19 = 7. The answer is Choice B.

If you wanted to multiply, then do the following:

y = 3(x + 2)2 + 7
y = 3(x + 2)(x + 2) + 7
y = 3(x2 + 4x + 4) + 7
y = 3x2 + 12x + 12 + 7
y = 3x + 12x + 19

44. Which table shows y as a function of x?

[SEE IMAGE]

In F, G, and J, there are repeating x values, which is not allowed in a functions. For a relation to be a function, each value in the domain (x) can be mapped to only one value in the range (y).
The y values are allowed to repeat.

45. What is the equation in slope-intercept form of the line that passes through the point (5, 0) and is parallel to the line represented by y = 1.2x + 3.8?

A y = 1.2x - 6
B y = -1.2x + 6
C y = 1.2x + 5
D y = -1.2x - 5

Answer: A y = 1.2x - 6
Parallel lines have the same slope. If the slope of the given line is 1.2, then the slope of the answer is 1.2. Eliminate choices B and D.

Next, if the line goes through the point (5, 0), then when 5 is substituted for x, y must be 0.
y = 1.2(5) - 6 = 6 - 6 = 0. Choice A is correct.
y = 1.2(5) + 5 = 6 + 5 = 11. Choice C is incorrect.

More to come. Comments and questions welcome.

More STAAR problems.

Friday, November 20, 2020

(x, why?) Mini: Inscribed

(Click on the comic if you can't see the full image.) (C)Copyright 2020, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

You know that he has an angle there, right?

It could just be that the pink square's name is Valerie or Valentina or Veronica or something.

Come back often for more funny math and geeky comics. STAAR (State of Texas Assessments of Academic Readiness) Algebra I, May 2019, cont.

More STAAR problems.

Read each question carefully. For a multiple-choice question, determine the best answer to the question from the four answer choices provided. For a griddable question, determine the best answer to the question.

36. Which expression is equivalent to 210d2 - 63d?

F 21d(10d - 3)
G 21d(10d + 3)
H 21(10d - 3)
J 21(10d + 3)

Simple factoring. Or use the Distributive Property on the choices.
21 is a factor of both terms. There is also a factor of d in both of those terms.
There is no reason when factoring to flip the sign from minus to plus.
Also, for either H or J to be correct, there would need to be another factor of d inside the parentheses because 21(10d) is only 210d, not 210d2.

32. What is the value of x in the solution to this system of equations?

3x - 5y = 22
y = -5x + 32

A -6.5
B 0.5
C 6.5
D -0.5

If you do the Substitution, you get

3x - 5(-5x + 32) = 22
3x +25x - 160 = 22
28x - 160 = 22
28x = 182
x = 6.5

If you work from the choices, you have to substitute into the second equation to get y, and then substitute both x and y into the first equation to see if you get 22.

38. Which graph best represents y = 10(0.085)x?

Since 0.85 is less than 1, the graph is shrinking not expanding. So Choices F and H are eliminated.

When x = 0, the initial amount is 10. The fits both G and J.
When x = 1, y = 10(0.85) = 8.5, which is graph G. Graph J is far below this, and can be eliminated.

39. The table shows a linear relationship between x and y.

What is the rate of change of y with respect to x?

A -9/2
B 2/9
C -2/9
D 9/2

Pick any two points. I prefer smaller numbers, easier to simplify.

m = (33 - 15) / (-6 - -2) = 18 / -4 = -9 / 2.

40. Which value of x is a solution to this equation?

5x2 - 36x + 36 = 0

F x = -6
G x = 4
H x = -1.8
J x = 1.2

5x2 - 36x + 36 = 0
Multiply ac: (5)(36) = 180. What are the factors of 180 that add up to -36?
Both must be negative
1 & 180, 2 & 90, 3 & 60, 4 & 45, 5 & 38, 6 & 30 < --- Stop Here

5x2 - 30x - 6x + 36 = 0
(5x2 - 30x) + (-6x + 36) = 0
5x(x - 6) - 6(x - 6) = 0
(5x - 6)(x - 6) = 0
5x - 6 = 0 or x - 6 = 0
5x = 6 or x = 6
x = 1.2 or x = 6

More to come. Comments and questions welcome.

More STAAR problems.

Thursday, November 19, 2020

STAAR (State of Texas Assessments of Academic Readiness) Algebra I, May 2019, cont.

More STAAR problems.

Read each question carefully. For a multiple-choice question, determine the best answer to the question from the four answer choices provided. For a griddable question, determine the best answer to the question.

31. The table represents some points on the graph of an exponential function.

Which function represents the same relationship?

A f(x) = 15(5/6)x
B f(x) = 18(6/5)x
C f(x) = 15(6/5)x
D f(x) = 18(5/6)x

Two observations.
First, the numbers are getting bigger, which means growth, not decay. So the base has to be greater than one (6/5) not less than one (5/6). Eliminate choices A and D.
Second, when x = 0, (6/5)0 = 1. We are left with just the initial amount. Since f(x) = 18, the initial amount in the function must be 18, which is choice B.
Other observations: 15 was just silly. And you could graph them and check the table of values, if you really needed to.

32. The table shows the amount of pet food in cups remaining in an automatic feeder as a function of the number of meals the feeder has dispensed.

Based on the table, which function models this situation?

F f(n) = -3n + 24
G f(n) = -1/3 n + 16
H f(n) = -3n + 64
J f(n) = -1/3 n + 8

Answer: F f(n) = -3n + 24
Find the slope:
(15 - 21) / (3 - 1) = -6 / 2 = -3
Eliminate choices G and J
If the slope is -3 and you have a point of (1, 21), then you can easily see that (0, 24) is also a point on the graph. This gives you choice F.

If you didn't see that, write the following equation:

y - 21 = -3(x - 1)
y - 21 = -3x + 3
y = -3x + 24

33. The graph of f(x) = x2 was transformed to create the graph of g(x) = f(x) - 9. Which statement about the graphs is true?

A The graph of g is a reflection of the graph of f across the x-axis.
B The vertex of the graph of g is 9 units to the right of the vertex of the graph of f.
C The graph of g is a reflection of the graph of f across the y-axis.
D The y-intercept of the graph of g is 9 units below the y-intercept of the graph of f.

Answer: D The y-intercept of the graph of g is 9 units below the y-intercept of the graph of f.
The output of f(x) is the y-coordinate on the graph.
If g(x) subtracts 9 units from f(x), then the y-coordinate will be 9 units lower.
That is, the graph shifts down.
There is no reason why the graph should be a reflection across either axis. To make the graph shift to the right, the -9 would have to be inside the parentheses, where it would affect the input and not the output.

34. The expression (x22)(x7)3 is equivalent to xp. What is the value of p?

The rules for exponents say you have to multiply the 7 and 3 and then add 22 to the result.
22 + 7 * 3 = 43

If you're aren't sure why that is, consider writing out the second term three times.
(x22)(x7)(x7)(x7)
The sum of the exponents is 22 + 7 + 7 + 7 = 43.

35. The graph of linear function k passes through the points (-7, 0) and (1, 8)

Which statement must be true?

A The slope of the graph is -4/3
B The graph of k passes through the point (-1, -8)
C The zero of k is 7.
D The x-intercept of the graph of k is -7.

Answer: D The x-intercept of the graph of k is -7.
The blank graph is given as an aid. You can plot the two points, figure out the slope from the rise and run and then check which of the answers is true.

However, if you read all the choices, that wouldn't be necessary.

Keep in mind that "the zero of k" and "the x-intercept" mean the same thing. Where does the line cross the x-axis. At the x-axis, the value of y is 0.

But we are given the point (-7, 0), so we know that the x-intercept (the zero) is -7. That is choice D.

It is easy to tell from the points that the slope is rising, and therefore positive, so -4/3 is not correct for the slope. (It is, in fact, 1 if you want to check.)
Likewise, if the y values are rising between 0 and 8 then a value of -8 is inconsistent.
Choice C is out because we know it is -7, not +7.

More to come. Comments and questions welcome.

More STAAR problems.

Wednesday, November 18, 2020

Hi-Lo

(Click on the comic if you can't see the full image.) (C)Copyright 2020, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

This is up there with ''Yello!''/''Green!''. Only mathier.

I have no idea when Michele's due date is. One more story line up-ended by this lost year we're living. That said, the AP in my department had a baby today! Congratulations!

Come back often for more funny math and geeky comics. STAAR (State of Texas Assessments of Academic Readiness) Algebra I, May 2019, cont.

More STAAR problems.

Read each question carefully. For a multiple-choice question, determine the best answer to the question from the four answer choices provided. For a griddable question, determine the best answer to the question.

26.The graph of linear function f passes through the point (1, −9) and has a slope of -3.

What is the zero of f?

F 2
G 4
H -6
J -2

I included the blank graph because the visual makes the problem easier to solve. Make a point at (1, -9). If the slope is -3, it's going down 3 and to the right 1. That means it also goes up 3 and 1 to the left. If you move that way, you get to (0, -6). Keep going and you get to the x-axis at (-2, 0).
If you don't want to use the graph, you could solve the equation

y - (-9) = -3(x - 1)
0 - (-9) = -3(x - 1)
9 = -3x + 3
6 = -3x
-2 = x

27. What is the value of the y-intercept of the graph of h(x) = 29(5.2)x?

At the y-intercept, x = 0.
h(0) = 29(5.2)0 = 29(1) = 29

28. The graph of a quadratic function is shown on the grid.

Which function is best represented by this graph?

F h(x) = x2 - 3x - 9
G h(x) = x2 + 3x - 9
H h(x) = x2 - 6x
J h(x) = x2 + 6x

Answer: H h(x) = x2 - 6x
As always, you could put the functions into your graphing calculator and check. However, we can reason these out.
First of all, for a parabola that opens upward, when the middle term, the x term, of a quadratic function is negative, the parabola is shifted to the right. If it is positive, it gets shifted to the left. So we can eliminate G and J.
Second, the constant is the y intercept, which in the graph is the origin, or 0. That eliminates choice F. J is the answer.
If you needed a third reason, the Axis of Symmetry is x = -b / (2a). The graph shows the Axis (the vertical line going through the vertex) at x = 3. This eliminates F and G immediately, and J after you do a quick mental check on the signs.

29. Which expression is equivalent to 24x2 - 22x + 5?

A (12x + 5)(2x + 1)
B (8x - 5)(3x - 1)
C (12x - 5)(2x - 1)
D (8x + 5)(3x + 1)

Answer: C (12x - 5)(2x - 1)
There are choices that you can "FOIL" (box method, distribute, etc) instead of trying to factor this mess.
Look at the signs. You have -22x and + 5. The only way to get that is if there are minus signs in both of the binomials. This eliminates A and D.
A quick check shows that both B and C will produce 24 and +5, so you have to check the two middle terms, the Outer and Inner.
(8)(-1) + (-5)(3) = -23
(12)(-1) + (-5)(2) = -22. Choice C is the answer.

30. A system of equations is graphed on the grid.

Which system of equations does the graph represent?

F y = -x - 4 ; y = 2x - 2
G y = -x + 4 ; y = 2x - 4
H y = -x - 4 ; y = 2x - 2
J y = -x + 4 ; y = 2x - 4

Answer: J y = -x + 4 ; y = 2x - 4
A curious problem, given that the y-intercept are obviously 4 and -4, Choices F and H can be immediately eliminated.
The line that's increasing gets the + 4, and it has to have a positive slope. That makes J the correct answer. Likewise, the decreasing line gets the - 4, which is also J.
You don't have to calculate a thing for this problem.
I would've thought that they would have you calculate at least one slope, just to be sure. However, that isn't necessary at all.

More to come. Comments and questions welcome.

More STAAR problems.

Tuesday, November 17, 2020

STAAR (State of Texas Assessments of Academic Readiness) Algebra I, May 2019, cont.

More STAAR problems.

Read each question carefully. For a multiple-choice question, determine the best answer to the question from the four answer choices provided. For a griddable question, determine the best answer to the question.

21. Which graph best represents y = -x2 + 6x - 1?

[SEE IMAGE]

Since the coeffiicient of x2 is negative, the parabola opens downward. So choices B and C are eliminated.
When x = 0, y = -1. The point (0, -1) is on the line in graph A but not graph D. So choice D is eliminated. This leaves B as the answer.
You can check by trying any other point, such as the vertex. The zeroes aren't a good choice in this example because they are not integers.

22.Which graph best represents the solution set of y > 3x − 4?

F 77
G 137
H 83
J 105

I put all the data into a table in the graphing calculator and did a linear regression to find the line of best fit.
My results were y=5.5392709845975 + 0.051934923490403x (You don't need that many decimals.)
To get a prediction for 1500, substitute 1500 for x.
Then 5.54 + 0.052(1500) = 83.54, which is closest to H. (If you used more decimal places, you would get even closer.)

23. The graph of a linear function is shown in the grid.

Which equation is best represented by this graph?

A y + 2 = 7/5 (x + 7)
B y - 2 = 7/5 (x - 7)
C y + 2 = 5/7 (x + 7)
D y - 2 = 5/7 (x - 7)

Answer: D y - 2 = 5/7 (x - 7)
The choices are given in Point-Slope form, which is written in the form:

y - y1 = m(x - x1)

Let's look at the slope first.
There appears to be a point at (0, -3). It looks like there's a point at (4, 0), which would give a slope of 3/4. However, that isn't a choice, so I'm deducing that (4, 0) isn't really a point on the line, but is so close to the line that it is hard to tell.
There also appears to be a point at (7, 2). The choices are all full of 7s and 2s, so this looks good. The slope of a line from (0, -3) to (7, 2) would be found as follows:
m = (y2 - y1) / (x2 - x1) = (2 - -3) / (7 - 0) = 5 /7

There is a slope of 5/7 in choices C and D.

Next, notice that the form has minus signs in it. For y + 2 and x + 7 to be in the equation, then the point (-7. -2) would have to be on the line. However, it is not.
On the other hand, we know that the point (7, 2) is on the line. So Choice D is the answer.

24. Which expression is equivalent to (xy-6)2 for all values of x and y where the expression is defined?

F xy-36
G xy36
H x2y-12
J x2y12

The rule for exponents is that the exponent goes only with the constant or the variable right next to it on its left. If there is a parenthesis there, then the exponent is a applied to everything within the parentheses.
So the x, which doesn't show an exponent, gets an exponent of 2. Because the y already has an exponent of -6, it gets multiplied by 2, which makes it -12.

Another way to look at it is this:

(xy-6)2 = (xy-6)(xy-6) = x x y-6 y-6 = x2 y-12

25. A college student completed some courses worth 3 credits and some courses worth 4 credits. The student earned a total of 59 credits after completing 18 courses.
How many courses worth 3 credits did the student complete?

A 13
B 5
C 20
D 39

You can set up a system of equations that says

3x + 4y = 59
x + y = 18
Multiply the second equation by -4 so we can eliminate the y terms
4x + 4y = 72
3x + 4y = 59
x = 13

The student took 13 three-credit course (which means the student took 5 four-credit courses).
Choices C and D are actually the number of the 59 credits earned from four- and three-credit courses.

More to come. Comments and questions welcome.

More STAAR problems.

Monday, November 16, 2020

STAAR (State of Texas Assessments of Academic Readiness) Algebra I, May 2019, cont.

More STAAR problems.

Read each question carefully. For a multiple-choice question, determine the best answer to the question from the four answer choices provided. For a griddable question, determine the best answer to the question.

16. Which expression is equivalent to (10 + 7r - r2) + (−6r2 − 18 + 5r)

F -7r2 + 2r + 8
G 7r2 + 12r + 8
H -7r2 + 12r - 8
J 7r2 + 2r - 8

Answer: H -7r2 + 12r - 8
Combine the like terms: r2 goes with r2, r goes with r, and constant goes with constant. Then, arrange them in standard order, which is from highest exponent to lowest.
10 + 7r - r2 + −6r2 − 18 + 5r
-r2 − 6r2 + 7r + 5r + 10 − 18
-7r2 + 12r − 8

17.Which graph best represents the solution set of y > 3x − 4?

[SEE IMAGE]

The inequality have a > sign, not a > sign. Therefore, it will have a broken (dotted or dashed) line to represent the boundary that is NOT part of the solution. And since it is "greater" and not "less than", the points above the line are the ones that have to be shaded.
That's Choice D.

18. A bank account earning annual compound interest was opened, and no additional deposits or withdrawals were made after the initial deposit. The balance in the account after x years can be modeled by b(x) = 850(1.025)x.
Which statement is the best interpretation of one of the values in this function?

F The initial balance of the account decreases at a rate of 97.5% each year.
G The balance in the account increases at a rate of 2.5% each year.
H The initial balance of the account was \$1,025.
J The balance in the account at the end of one year is \$850.

Answer: G The balance in the account increases at a rate of 2.5% each year.
Looking at the numbers in the equation:
850 represents the opening balance before any time has passed.
1.025 is the growth rate, which as a percentage is 102.5%. That means an additional 2.5% is being added to the balance
x is the number of years that pass, the more years, the bigger the balance.

The initial balance is increasing, not decreasing, so F is eliminated
2.5% is the rate of increase, so this is the answer.
The initial balance is \$850. 102.5% represents the growth. Eliminate H.
The initial balance of \$850 is at Year 0 before any time has passed. It is not the balance after one year. Eliminate choice J.

19. A company collected data for the number of text messages sent and received using a text-message application since October 2011. The table shows the number of text messages sent and received in billions over time. The data can be modeled by a quadratic function.

Which function bests models the data?

Which function best models the data?
A n(t) = −0.002t2 + 0.55t + 5.02
B n(t) = 0.072t2 − 0.15t + 2.73
C n(t) = −0.002t2 + 5.02
D n(t) = 0.072t2 + 2.73

Answer: B n(t) = 0.072t2 − 0.15t + 2.73
You have a few options here.
You can plot each of the four graphs in your calculator and do a visual check. You can enter the data into a table and do a quadratic regression. Or you can reason out which choices to eliminate.
For example, in Choices C and D, the middle coefficent, b is 0, so the vertex is 0. The positive x values are rising, so the t2 term can't have a negative coefficient. So C is eliminated.
Calculating -b/2a for A, we get -.55/(2(-0.002)) = 137.5. The values would be rising for x < 137.5, but the rate of change would be decresasing. Choice A can be eliminated.
Calculating -b/2a for B, we get -0.15/(2(0.072)) = 1.041666..., so this is also a possibility.
Now there are only two formulas to check.
0.072(40)2 - 0.15(40) + 2.73 = 111.93
0.072(40)2 + 2.73 = 117.93
Choice B is a better fit.
0.072(10)2 - 0.15(40) + 2.73 = 8.43
0.072(20)2 - 0.15(20) + 2.73 = 28.53
0.072(30)2 - 0.15(30) + 2.73 = 63.03
This look pretty close with the -0.15t middle term.

All of that said, I put all of the numbers into a Desmos quadratic regression calcuator and got the following results:
a = 0.0721429, b = -0.14881, c = 2.73214. Which is Choice B.

20. The graph of a linear function is shown on the grid.

What is the rate of change of y with respect to x for this function?

What is the rate of change? That is, what is the slope of this line?

Change in y / change in x = (y2 - y1) / (x2 - x1) = (2 - 3.6) / (5 - -3)
= - 1.6 / 8 = -16 / 80 = -1 / 5 or -0.2.

More to come. Comments and questions welcome.

More STAAR problems.

(Click on the comic if you can't see the full image.) (C)Copyright 2020, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

As long as we need random letters anyway, why not pick words. Easier to say and remember!

Coming up with five words was the easy part, apparently. I had intended for BAD to be 85o, not 95. Then I looked at the picture was I was done. It was easier to change the number.

Ken could've asked Bibi about minor arc BI and major arc BI, I guess.

The original idea was that BOD would be double the size of BAD, BED, and BID, which are all congruent because they are all inscribed angles that intercept the same arc. Instead, it is double the size of angle BUD. It just doesn't flow as nicely.

Come back often for more funny math and geeky comics. Sunday, November 15, 2020

STAAR (State of Texas Assessments of Academic Readiness) Algebra I, May 2019, cont.

More STAAR problems.

Read each question carefully. For a multiple-choice question, determine the best answer to the question from the four answer choices provided. For a griddable question, determine the best answer to the question.

11. A student worked out at a gym continuously for 50 minutes. The graph shows the remaining percentage of the workout as a linear function of x, the time in minutes.

Which answer choice best describes the domain and range of the function for this situation?

A Domain: All real numbers greater than or equal to 0 and less than or equal to 100
Range: All real numbers greater than or equal to 0 and less than or equal to 50
B Domain: {-2}
Range: {100}
C Domain: All real numbers greater than or equal to 0 and less than or equal to 50
Range: All real numbers greater than or equal to 0 and less than or equal to 100
D Domain: {100}
Range: {-2}

Answer: C Domain: All real numbers greater than or equal to 0 and less than or equal to 50
Range: All real numbers greater than or equal to 0 and less than or equal to 100

The domain is all of the possible x values. The range is all of the possible y values.
The domain goes from 0 to 50. The range goes from 0 to 100.

12.The graph of quadratic parent function f was transformed to create the graph of g(x) = f(x + 2) −5. Which graph best represents g?

[SEE IMAGE]

The parent function would have a vertex at the origin, (0, 0).
The transformation y = a(x - h)2 + k moves the graph h units to the left and k units up. However, in this equation, h is +2 and k is -5. We have to flip the signs. So the graph will move two spaces to the right and five units down.

13. Which expression is equivalent to (45 m-6p2v12) / (15 m-2p8v-4)

A (3 v8) / (m8p6)
B (3 v16) / (m4p6)
C (30 m3) / (p4p3)
D (30 v3) / (m3p4)

Answer: B (3 v16) / (m4p6)
Put whatever confusions or misconceptions you may have about exponents aside for one moment. The first thing you need to do is divide 45 by 15 and get 3. Eliminate Choices C and D immediately. If you subtracted 45 - 15 and got 30, you need to review the basics. (Harsh, maybe, and sorry, but very true.)

There are two rules that have to be used here. First, when you have the same variables with exponents in both the numerator and denominator, you subtract the exponents. Second, a negative exponent on top becomes a postive exponent by moving the variable to the bottom, and vice versa.

Let's subtract the exponents: m, -6 - (-2) = -4; p, 2 - 8 = -6; v, 12 - (-4) = 16
So we have m-4, p-6 and v16
Only the v is staying on top. The m and p go to the bottom.
The correct choice is B.

14.What is the positive solution to this equation?

4x2 + 12x = 135

First step in solving a quadratic equation is setting the equation equal to 0. Then we can either factor or use the quadratic formula.

4x2 + 12x = 135
4x2 + 12x - 135 = 0
Find ac: (4)(135) = -540
Are there factors of -540 that sum to +12?
(-2, 270), (-3, 180), (-4, 135), (-5, 108), (-6, 90),
(-9, 60), (-10, 54), (-12, 45), (-15, 36), (-18, 30) < -- This one.
4x2 - 18x + 30x - 135 = 0

At this point, I would usually draw a table, and do the box method in reverse. Instead, we'll work by grouping.
(4x2 - 18x) + (30x - 135) = 0

Find the Greatest Common Factor of each pair of terms, and factor it out.
2x(2x - 9) + 15(2x - 9) = 0

Factor (2x - 9) from each expression.
(2x - 9)(2x + 15) = 0
2x - 9 = 0 or 2x + 15 = 0
2x = 9 or 2x = -15
x = 4.5 or x = -7.5

The positive solution is 4.5.

NOTE: After you factor, if you aren't sure if you factored correctly, "FOIL" your answer and see if you got your original expression back.
(2x - 9)(2x + 15) = 4x2 - 18x + 30x - 135.

15. A grill at a barbecue restaurant will be used to cook sausage links that are 2 lb each and briskets that are 6 lb each. No more than 120 lb of sausage links and briskets will be cooked on the grill.
Which inequality represents all possible combinations of x, the number of sausage links that will be cooked on the grill, and y, the number of briskets that will also be cooked?

A 6x + 2y < 120
B 2x + 6y < 120
C 6x + 2y > 120
D 2x + 6y > 120

Answer: B 2x + 6y < 120
If x is the number of sausage links and each link weighs 2 lbs, then the weight of all the sausage links is 2x. Choices A and C are eliminated.
The weight needs to be "no more than" 120, which means "less than or equal to" 120. Only one choice has a "less than or equal to" symbol, and that's Choice B.

More to come. Comments and questions welcome.

More STAAR problems.

Saturday, November 14, 2020

STAAR (State of Texas Assessments of Academic Readiness) Algebra I, May 2019, cont.

And now for something completely different.

While I'm waiting for new Regents exams to come along, I looked to other states for standardized tests that have been made public.

It's nice to say how other states live, right?

More STAAR problems.

Read each question carefully. For a multiple-choice question, determine the best answer to the question from the four answer choices provided. For a griddable question, determine the best answer to the question.

6. What is the equation in slope-intercept form of the line that passes through the points(-4, 2)and (12, 6)?

F y = 0.25x + 3
G y = 0.25x - 4.5
H y = 4x + 18
J y = 4x - 42

Answer: F y = 0.25x + 3
First step is to find the slope, which will eliminate two choices immediately.
After that, you have two methods to choose from: you can use slope-intercept form and solve for the y-intercept; or, you can use point-slope form, and rewrite the equation in slope-intercept form.

m = (y2 - y1) / (x2 - x1) = (6 - 2) / (12 - (-4)) = 4 / 16 = 1 / 4 = 0.25. Eliminate choices H and J.
Point-slope form: y - 6 = .25(x - 12) or y = .25(x - 12) + 6 (I prefer the latter because it looks like a transformation.)
y = .25x - 3 + 6
y = .25x + 3, which is choice F.

2. Which graph best represents -5y = -6x + 15?

[SEE IMAGE FOR CHOICES]

Rewrite in slope-intercept form by dividing the equation by -5: y = 6/5 x - 3
Choices A and B have negative slopes and can be eliminated.
Choice C goes through (0, -3), which is what we want. Choice D goes through (0, 3), which is incorrect.

8. A baker determined the annual profit in dollars from selling pies using p(n) = 52n − 0.05n2, where n is the number of pies sold. What is the annual profit if the baker sells 400 pies?

F \$20,780
G \$12,800
H \$28,800
J \$20,760

As I would see on a New York test, this question seems very silly. It would imply that there will come a point where the baker will start to lose money making more pies. Then again, maybe this takes into account that he can only sell so many, and after that the go into the dumpster.

To solve the problem, substitute 400 for n and evaluate the expression.
52(400) - 0.05(400)2 = 12,8000

9.An exponential function is graphed on the grid.

The graph is positive, so the rate can't have a negative sign in front of it: C and D are eliminated.
The graph is showing decay, so A is correct. B would show growth.
Of course, you could just put each of these equations into a calculator and look.

10. What is the equation in slope-intercept form of the line that crosses the x-axis at 36 and is perpendicular to the line represented by y = −4/9 x + 5 ?

F y = 4/9 x + 16
G y = 4/9 x - 16
H y = 9/4x + 81
J y = 9/4 x - 81

Answer: J y = 9/4 x - 81
A slope that it perpendicular to a slope of -4/9 would be its negative multiplicative inverse -- that is, the inverse reciprocal.
Flip the fraction and switch the sign and -4/9 becomes +9/4.
Two slopes are perpendicular if their product is -1, and (-4/9)(9/4) = -1.

We've eliminated F and G.
To finish the problem, check H and J for the point (36, 0)

y = 9/4 (36) + 81 = 9 (9) + 81 = 162 =/= 0
y = 9/4 (36) - 81 = 9 (9) - 81 = 0
Hopefully, it was obvious that it you were adding two positive numbers, you could get a sum of 0 in choice H.

More to come. Comments and questions welcome.

More STAAR problems.

Friday, November 13, 2020

(x, why?) Mini: Transformation

(Click on the comic if you can't see the full image.) (C)Copyright 2020, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

It'll drive you insa-a-a-a-ane!

Come back often for more funny math and geeky comics. Thursday, November 12, 2020

And now for something completely different.

While I'm waiting for new Regents exams to come along, I looked to other states for standardized tests that have been made public.

It's nice to say how other states live, right?

More STAAR problems.

Read each question carefully. For a multiple-choice question, determine the best answer to the question from the four answer choices provided. For a griddable question, determine the best answer to the question.

1. What is the slope of the graph of y = 12x - 19?

A -19
B -12/19
C 19/12
D 12

The exquation is in slope-intercept form, y = mx + b. The slope, m, is the coefficient of the variable x, which in this case is 12.
The 19 represents the y-intercept and isn't relevant to this questions.

2. A golfer hit a golf ball from a tee box that is 6 yards above the ground. The graph shows the height in yards of the golf ball above the ground as a quadratic function of x, the horizontal distance in yards of the golf ball from the tee box.

What is the domain of the function for this situation?
F 0 < x < 230
G 6 < y < 36
H 0 < x < 36
J 6 < y < 230

Answer: F 0 < x < 230
The domain refers to the x values, not the y values, so choices G and J are immediately eliminated. Also, the 36 refers to the height, which is y. The horizontal distance is 230.

3. Which value of x makes the equation 1.25(4x − 10) = 7.5 true?

A 3.5
B -1
C -0.5
D 4

Since it is multiple choice, you could just check each solution.
In fact, with the TI calucluators you could enter

1.25(4*{3.5 -1 -0.5 4} - 10)
and see which value gives you 7.5.\

Or you can solve it

1.25(4x - 10) = 7.5
4x - 10 = 6
4x = 16
x = 4

4.Which function is equivalent to g(x) = x2 + 15x - 54 ?
F g(x) = (x + 9)(x - 6)
G g(x) = (x + 18)(x - 3)
H g(x) = (x - 9)(x + 6)
J g(x) = (x - 18)(x + 3)

Answer: G g(x) = (x + 18)(x - 3)
Which pair of binomials has a product that is equivalent to the given statement? Or, how do you factor the given quadratic function?

The factors of -54 are -6 and +9, +9 and -9, -3 and + 18, and +3 and -18. Of those pairs, which sums to +15? The answer is +18 and -3.

Tricky question because they could have asked about x2 + 15x + 54 and got an entirely different answer that doesn't involve just flipping signs.

5. The table shows the linear relationship between the average height in feet of trees on a tree farm and the number of years since the trees were planted.

What is the rate of change of the average height in feet of the trees on the farm with respect to the number of years since the trees were planted?

A 14 ft/yr
B 3 ft/yr
C 7 ft/yr
D 10 ft/yr

To find the average, find the rate of change from the first and the last point, (1, 10) and (15, 108).
So m = (108 - 10) / (15 - 1) = 98 / 14 = 7.

More to come. Comments and questions welcome.

More STAAR problems.

I also write Fiction!

Check out In A Flash 2020, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides.
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Thank you. Wednesday, November 11, 2020

Happy Veterans Day 2020

(Click on the comic if you can't see the full image.) (C)Copyright 2020, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

Their masks are in their pockets, okay. They'll take them out as they need them.

I would think that if he couldn't go to the American Legion post today, my father would be over in his community garden where he grew tomatoes and peppers. It's actually a nice day outside, except for some light drizzle or mist.

Happy Veterans Day to all verterans, and Thank you.

Come back often for more funny math and geeky comics. 