Wednesday, February 28, 2018

January 2018 Common Core Geometry Regents, Part 2 (open-ended)

The following are some of the multiple questions from the recent January 2018 New York State Geometry Regents exam.
The questions and answers to Part I can be found here.

January 2018 Geometry, Part II

Each correct answer is worth up to 2 credits. Partial credit is available. Work must be shown. Correct answers without work receive only 1 point.

25. Given: Parallelogram ABCD with diagonal AC drawn

Prove: triangle ABC = triangle CDA

Answer: You can give either a paragraph or two-column proof. However, when writing a paragraph, you still need to remember to have all the statements and reasons.
ABCD is a parallelogram; Given.
AB = CD, AD = BC; Opposite sides of a parallelogram are congruent.
AC = AC; Reflexive property
triangle ABC = triangle CDA; SSS

26. The diagram below shows circle 0 with diameter AB. Using a compass and straightedge, construct a square that is inscribed in circle 0. [Leave all construction marks.]

Answer: Strategy: if you draw a perpendicular bisector between A and B, you will get a vertical line through the center of the circle O. Where the line intersects the circle will be the other two vertices of the square. Use your straightedge to draw the square.

27. Given: Right triangle ABC with right angle at C
If sin A increases, does cos B increase or decrease? Explain why

Answer: Cos B will increase because the sin A = cos B. This is only a partially correct response. You need to back it up with a definition or explanation.
Sine is the ratio of opposite over hypotenuse (you can write this as a fraction.)
Cosine is the ratio of adjacent over hypotenuse.
The side that is opposite angle A is the same side that is adjacent to angle B.
Therefore sin A and cos B are the same ratio.

28. In the diagram below, the circle has a radius of 25 inches. The area of the unshaded sector is 500 pi in2.

Determine and state the degree measure of angle Q, the central angle of the shaded sector.

Answer: Find the area of the circle. Subtract 500pi from it. Compare the result to the original as a fraction. Multiply that fraction by 360 degrees.
A = (pi)(r)2
A = (pi)(25)2
A = 625pi
The shaded area is 625pi - 500pi = 125pi
The fraction of the circle that's shaded is (125pi)/625pi) = 125/625 = 1/5
1/5 (360) = 72 degrees.

29. A machinist creates a solid steel part for a wind turbine engine. The part has a volume of 1015 cubic centimeters. Steel can be purchased for $0.29 per kilogram, and has a density of 7.95 g/cm3.
If the machinist makes 500 of these parts, what is the cost of the steel, to the nearest dollar?

Answer: Find the mass of one part, and multiply it by 500 to get the mass of 500 parts in grams. Divide by 1000 to change it into kilograms. Then multiply the mass by $0.29 per kilogram.
D = m/V
7.95 = m/1015
m = 1015(7.95) = 8069.25
8069.25 * 500 = 4,034,625
4,034,625 / 1000 = 4,034.625
4,034.625 * 0.29 = 1,170.04125
$1,170 (to the nearest dollar)
Note: If you do all this work and forget to round to the nearest dollar, you will lose 1 of the 2 available points.
That stinks, but there's nothing you can do about that.

30. In the graph below, triangle ABC has coordinates A(-9,2), B(-6,-6), and C(-3, -2), and triangle RST has coordinates R(-2,9), S(5,6), and T(2,3).

Is triangle ABC congruent to triangle RST? Use the properties of rigid motions to explain your reasoning.

Answer: They are not congruent. If they were congruent then you could map ABC onto RST with a series of rigid motions. If you reflect ABC over the line y = -x, A would map to R and C would map to T, but B would map to (6, 6), not to S(5, 6). So the triangles are not congruent.

You could also use the distance formula to show that the lengths of the sides of the triangles are not the same, but you still need to mention the properties of rigid motions to get full credit.

31. Bob places an 18-foot ladder 6 feet from the base of his house and leans it up against the side of his house. Find, to the nearest degree, the measure of the angle the bottom of the ladder makes with the ground.

Answer: Sketch a little picture, if it helps. The ladder is the hypotenuse, and the distance to the house is the adjacent side.
Use cos x = adj / hyp
cos x = 6 / 18
x = cos-1(6/18) = 70.5287793655... = 71 degrees.

End of Part II

How did you do?
Questions, comments and corrections welcome.

Tuesday, February 27, 2018

January 2018 Common Core Geometry Regents, Part 1 (mult choice)

The following are some of the multiple questions from the recent January 2018 New York State Geometry Regents exam.

January 2018 Geometry, Part I

Each correct answer is worth up to 2 credits. No partial credit. Work need not be shown.

1. In the diagram below, a sequence of rigid motions maps ABCD onto JKLM.

If m<A = 82°, m<B = 104°, and m<L = 121°, the measure of <M is

Answer: (1) 53°.
Because they were rigid motions, the corresponding angles are congruent. A corresponds to J, which is 82 degrees; B corresponds to K, which is 104 degrees. L is given as 121 degrees. The quadrilateral has a total of 360 degrees.
360 - (82 + 104 + 121) = 53

2. Parallelogram HAND is drawn below with diagonals HN and AD intersecting at S.

Which statement is always true?

Answer: (2) AS = 1/2 AD .
The diagonals of a parallelogram bisect each other.

3. The graph below shows two congruent triangles, ABC and A'B'C'.

Which rigid motion would map triangle ABC onto triangle A'B'C'?

Answer: (4) a reflection over the line y = x.
A translation wouldn't change the orientation. If there were a rotation, each point would have more 1 quadrant (for 90 degrees) or 2 quadrants (for 180 degrees).

4. A man was parasailing above a lake at an angle of elevation of 32° from a boat, as modeled in the diagram below.

If 129.5 meters of cable connected the boat to the parasail, approximately how many meters above the lake was the man?

Answer: (1) 68.6.
The height is opposite the 32° angle, and the hypotenuse of 129.5 is given. Use the sine ration: sin = opp/hyp
sin 32 = x / 129.5
x = 129.5 (sin 32) = 68.6245447...
Be sure to have the calculator in DEGREE mode! If the calculator is set to radians, you will get an incorrect answer that might seem to be reasonable (71.4), but isn't one of the choices.
Speaking of "reasonable", because 32 degrees is close to 30 degrees, you could have estimated a multiple choice answer as follows: in a 30-60-90 degree triangle, the side opposite the 30 degree angle will ALWAYS be half of the hypotenuse. In that case the height would be 64.75. Since the angle is 32 degrees, it should be a little bigger than this number. Only 68.6 would be reasonable.

5. A right hexagonal prism is shown below. A two-dimensional cross section that is perpendicular to the base is taken from the prism.

Which figure describes the two-dimensional cross section?

Answer: (2) rectangle.
Parallel to the base would give a hexagon. Perpendicular to the base, whether it's "front to back", "side to side" or any other direction will yield a rectangle.

6. In the diagram below, AC has endpoints with coordinates A(-5,2) and C(4, -10).

If B is a point on AC and AB:BC = 1:2, what are the coordinates of B?

Answer: (1) (-2,-2).
A ratio of 1:2 means that AB is 1/3 the length and BC is 2/3 the length.
If you look at the change in the x-coordinates, the distance from -5 to 4 is 9. (4 - (-5) = 9.)
One third of 9 is 3, so the change in the x-coordinate from A to B is +3.
-5 + 3 = -2, so the only choice is (-2, 2).
Checking the y-coordinate: (-10) - 2 = -12. (1/3)(-12) = -4.
2 + (-4) = -2, which is the y-coordinate of B.

7. An ice cream waffle cone can be modeled by a right circular cone with a base diameter of 6.6 centimeters and a volume of 54.45(pi) cubic centimeters. What is the number of centimeters in the height of the waffle cone?

Answer: (3) 15.
V = (1/3)(pi)r2h
54.45(pi) = (1/3)(pi)(3.3)2h
h = 54.45(pi) / ((1/3)(pi)(3.3)2) = 15
You could solve the equation for h first, or you can plug in the numbers first and then solve. Note that you can divide both sides of the equation by pi to remove it from the equation entirely. Note that you were given a diameter of 6.6, which makes the radius 3.3.

8. The vertices of triangle PQR have coordinates P(2,3), Q(3,8), and R(7,3). Under which transformation of triangle PQR are distance and angle measure preserved?

Answer: (4) (x,y) -> (x + 2, y + 3).
Distance is not preserved if one of the coordinates is multiplied, which eliminates choices (1), (2), and (3). Moreover, if the coordinates are not multiplied by the same scale factor, then angle measure will not be preserved either.
Choice (4) is a translation, which preserves distance and angle measure.

9. In triangle ABC shown below, side AC is extended to point D with m<DAB = (180 - 3x)°, m<B = (6x - 40)°, and m<C = (x + 20)°.

What is m<BAC?

Answer: (3) 60°.
According to the Exterior Angle Theorem, the sum of the exterior angle equals the sum of the two remote angles.

(180 - 3x)= (6x - 40) + (x + 20)
180 + 40 - 20 = 6x + x + 3x
200 = 10x
x = 20

The measure of angle BAC is supplementary to DAB.
Notice that 180 - (180 - 3x) = 3x, and 3(20) = 60 degrees.
The longer way: m<DAB = 180 - 3(20) = 180 - 60 = 120, and then M&BAC = 180 - 120 = 60 degrees

10. Circle O is centered at the origin. In the diagram below, a quarter of circle O is graphed.
Which three-dimensional figure is generated when the quarter circle is continuously rotated about the y-axis?

Answer: (4) hemisphere.
If you reflect the quarter circle, you would get a semicircle in two dimensions. In three dimensions, that would be a hemisphere.

11. Rectangle A'B'C'D' is the image of rectangle ABCD after a dilation centered at point A by a scale factor of 2/3. Which statement is correct?

Answer: (1) Rectangle A'B'C'D' has a perimeter that is 2/3 the perimeter of rectangle ABCD.
The perimeter would shrink. It would only be 2/3 of the original.
The area would be reduced to (2/3)2, or 4/9, of the original area.

12. The equation of a circle is x2 + y2 - 6x + 2y = 6. What are the coordinates of the center and the length of the radius of the circle? (1) center (-3,1) and radius 4 (2) center (3, -1) and radius 4 (3) center (-3,1) and radius 16 (4) center (3, -1) and radius 16

Answer: (2) center (3, -1) and radius 4.
Rewrite the formula into standard form,
(x - h)2 + (y - k)2 = r2, by grouping the variables and then completing the squares:

x2 + y2 - 6x + 2y = 6
x2 - 6x + y2 + 2y = 6
Half of -6 is -3, and (-3)2 = 9. Add 9 to both sides.
x2 - 6x + 9 + y2 + 2y = 6 + 9
Half of 2 is 1, and (1)2 = 1. Add 1 to both sides.
x2 - 6x + 9 + y2 + 2y + 1= 6 + 9 + 1
Factor the polynomials into binomials
(x - 3)2 + (y + 1)2 = 16

The center is (3, -1) and the radius is 4.
Remember to flip the signs to get the coordinates, and take the square root of r2.

13. In the diagram of triangle ABC below, DE is parallel to AB, CD = 15, AD= 9, and AB= 40.

The length of DE is

Answer: (3) 25.
CD / DE = CA / AB
15 / x = (15 + 9) / 40
(15 + 9)x = (15)(40)
24x = 600
x = 25

14. The line whose equation is 3x - 5y = 4 is dilated by a scale factor of 5/3 centered at the origin. Which statement is correct?

Answer: (1) The image of the line has the same slope as the pre-image but a different y-intercept.
When dilating a line, the slope will not change. The new line will either be parallel to the original line, or coincident (i.e., the same line) if the center of the dilation is a point on the line.
If the center of the dilation is a point on the line, then the y-intercept would not change either, but that is not the case in this example. If the origin, (0, 0), were a point on the line, then 3(0) - 5(0) = 4, which is not true.

15. Which transformation would not carry a square onto itself?

Answer: (3) a 180° rotation about one of its vertices.
A 180° rotation about the center of the square would carry it onto itself.

16. In circle M below, diameter AC, chords AB and BC, and radius MB are drawn.

Which statement is not true?

Answer: (4) mAB = (1/2)m<ACB .
The measure of arc AB would be TWICE the measure of angle ACB. In other words, the measure of angle ACB would be half of arc AB.
ABC must be a right triangle, because angle B must be a right angle because it intercepts a semicircle.
ABM must be isosceles because both AM and BM are radii.
The measure of arc BC is equal to the measure of its central angle, angle BMC.

17. In the diagram below, XS and YR intersect at Z. Segments XY and RS are drawn perpendicular to YR to form triangles XYZ and SRZ.

Which statement is always true?

Answer: (4) XY / SR = YZ / RZ.
We have enough information to show that the two triangles are similar, but not congruent. Eliminate choices (2) and (3).
Angles Y and R are right angles because they are perpendicular to YR, so they are congruent.
Angles XZY and RZS are congruent because they are vertical angles.
Therefore, XYZ is similar to RSZ by AA.
That means that the corresponding sides are proportional in length. Choice (4) is a correct proportion

18. As shown in the diagram below, ABC || EFG and BF = EF.

If m<CBF = 42.5°, then m<EBF is

Answer: (2) 68.75°.
Because BF = EF, then BEF is an isosceles triangle.
So angle BEF is congruent to angle FBE because they are the base angles and BFE is the vertex angle.
Angle BFE is congruent to angle CBF because they are alternate interior angles.
Then CBF = BFE = 42.5 degrees.
So x + x + 42.5 = 180
2x = 137.5
x = 68.75°

19. A parallelogram must be a rhombus if its diagonals

Answer: (4) are perpendicular to each other .
Diagonals of a rectangle are also congruent, so choice (1) is incorrect.
Diagonals of a rectangle also bisect each other, so choice (2) is incorrect.
Diagonals of a rhombus do bisect each other, so choice (3) is incorrect.

20. What is an equation of a line which passes through (6,9) and is perpendicular to the line whose equation is 4x - 6y = 15?

Answer: (1) y - 9 = (-3/2)(x - 6).
The original line is in Standard Form. Find the slope of the line, using -A/B, or by rewriting it into point-slope or slope-intercept form.
A = 4, B = -6, so the slope is (-4)/(-6) = 2/3
The slope of a line perpendicular to this is -3/2, the negative reciprocal. This eliminates choices (2) and (4).
Point-slope form is y - y0 = m(x - x0), so the solution is y - 9 = (-3/2)(x - 6)

21. Quadrilateral ABCD is inscribed in circle 0, as shown below.
If m<A = 80°, m<B = 75°, m<C = (y + 30)°, and m<D = (x -10)°, which statement is true?

Answer: (4) x = 115 and y = 70.
The sum of angles A and C is 180 degrees, and the sum of angles B and D is 180 degrees.
Angle C is 180 - 80 = 100 degrees, so y = 70.
Angle D is 180 - 75 = 105 degrees, do x = 115.

22. A regular pyramid has a square base. The perimeter of the base is 36 inches and the height of the pyramid is 15 inches. What is the volume of the pyramid in cubic inches?

Answer: (2) 405.
The perimeter of the square base is 36, so each side of the square is 9 (not 6). The area of the base is 92 = 81.
The volume of the pyramid is (1/3) times the Area of the Base times the height:
V = (1/3)(81)(15) = 405

23. In the diagram below of triangle ABC, <ABC is a right angle, AC = 12, AD = 8, and altitude BD is drawn.
What is the length of BC?

Answer: (2) 4 * SQRT(3).
Triangle BCD is similar to ABC because they both have right angles and they both have angle C (Reflexive Property).
So you can compare leg / hypotenuse = leg / hypotenuse
DC / BC = BC / AC
(12 - 8) / BC = BC / 12
BC2 = (12)(4) = 48
BC = SQRT(48) = SQRT(16 * 3) = 4 * SQRT(3)

24. In the diagram below, two concentric circles with center 0, and radii OC, OD, OGE, and ODF are drawn.

If OC = 4 and OE = 6, which relationship between the length of arc EF and the length of arc CD is always true?

Answer: (3) The length of arc EF is 1.5 times the length of arc CD.
OE is 1.5 times the length of OC because 6/4 = 1.5.
The outer circle is a dilation of the inner circle with a scale of 1.5 centered on O. That means that the lengths of the corresponding arcs will have a scale of 1.5 as well.

End of Part I.

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Monday, February 26, 2018

Parallel Lines

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(C)Copyright 2018, C. Burke.

Yeah, try representing a 3D activity in a 2D comic with an extra dimension for humor, which is often on an imaginary axis ... Maybe it would've looked better if I hadn't insisted that the lines be functions.

Parallel lines always have the same slope and are always the same distance apart.

With curve lines, the distance bewteen them isn't constant -- however, the vertical distance for any x value is always constant. At least, in this example because it's a simple transformation of the function; i.e. B(x) = R(x) + 2.

What about other examples? Well, that's left as an exercise to the reader. Yeah, that's the ticket!

Come back often for more funny math and geeky comics.

Friday, February 23, 2018

Snow Bored

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(C)Copyright 2018, C. Burke.

Got stuck trying to do something with 'curling'.

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Monday, February 19, 2018

Basic Transformations

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(C)Copyright 2018, C. Burke.

Not that Kitty is not a Rigid Motion because Kitties will change size and shape to fill the available space on your lap.

Seriously, three months ago I made a notation in my notepad that read
"(X, why?) explains transformations: reflection, translation, rotations, kitty."
without any notation what "kitty" was supposed to be.

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Friday, February 16, 2018


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(C)Copyright 2018, C. Burke.

Unlike Michele and Ken, I haven't actually set a date.

Nor do I know if it's going to happen "on camera" or not.

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Wednesday, February 14, 2018

The One That I Want

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(C)Copyright 2018, C. Burke.

The One I need? Oh, yes! Indeed!

I was going to call her Olivia Newton-One, but she's clearly Roman, not Australian!

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Monday, February 12, 2018

Olympic Ski Jumping: Normal Hill

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(C)Copyright 2018, C. Burke.

Really ... there's nothing 'normal' about this. Cue 'Goofy voice': Ya-hoo-hoo-hooey!!!

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Wednesday, February 07, 2018

(x, why?) Mini: Happy e Day

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(C)Copyright 2018, C. Burke.

If you'd told me 10 years ago that I'd need to have comics ready for 3/14/15 and 2/7/18, I'd've called you crazy.

But then, everyone calls me crazy, so I'm not even sure what that means any more.

Come back often for more funny math and geeky comics.

Tuesday, February 06, 2018

January 2018 Common Core Algebra I Regents, Part 1 (mult choice)

The following are some of the multiple questions from the recent January 2018 New York State Common Core Algebra I Regents exam.
The answers to Part II can be found here
The answers to Parts III and IV can be found here

January 2018 Algebra I, Part I

Each correct answer is worth up to 2 credits. No partial credit. Work need not be shown.

1. When solving the equation 12x2 - 7x = 6 - 2(x2 - 1), Evan wrote computations. 12x2 - 1x = 6 - 2x2 + 2 as his first step. Which property justifies this step?

Answer: (4) distributive property of multiplication over subtraction.

2. Jill invests $400 in a savings bond. The value of the bond, V(x), in hundreds of dollars after x years is illustrated in the table below.

x V(x)
0 4
1 5.4
2 7.29
3 9.84

Which equation and statement illustrate the approximate value of the bond in hundreds of dollars over time in years?

Answer: (3) V(x) = 4(1.35)x and it grows.
Choices (1) and (2) are no good because the base is less than 1.00, which would cause decay, not growth. Choice (4) is out because it is not decay.

3. Alicia purchased H half-gallons of ice cream for $3.50 each and P packages of ice cream cones for $2.50 each. She purchased 14 items and spent $43. Which system of equations could be used to determine how many of each item Alicia purchased?

Answer:(1) 3.50H + 2.50P = 43, H+P=14
The sum of the prices is $43.00. The total number of items is 14.

4. A relation is graphed on the set of axes below.

Based on this graph, the relation is

Answer: (2) a function because it passes the vertical line test

5. Ian is saving up to buy a new baseball glove. Every month he puts $10 into a jar. Which type of function best models the total amount of money in the jar after a given number of months?

Answer: (1) linear
There is a constant rate of change in the amount he has saved.

6. Which ordered pair would not be a solution to y = x3 - x?

Answer: (4) (-1,-2)
(-1)3 - (-1) = 0, not -2.

7. Last weekend, Emma sold lemonade at a yard sale. The function P(c) = .50c - 9.96 represented the profit, P(c), Emma earned selling c cups of lemonade. Sales were strong, so she raised the price for this weekend by 25 cents per cup. Which function represents her profit for this weekend?

Answer: (4) P(c) = .75c - 9.96
Raising the price per cup would mean increasing the cost per cup from 50 cents (or .50) to 75 cents (or .75).

8. The product of Sqrt(576) and Sqrt(684) is

Answer: (3) irrational because one factor is irrational
Sqrt(576) is 24, which is rational. Sqrt(684) is irrational. Their product is irrational.

9. Which expression is equivalent to y4 - 100?

Answer: (3) (y2 + 10)(y2 - 10)
Difference of Squares rule.

10. The graphs of y = x2 - 3 and y = 3x - 4 intersect at approximately

Answer: (3) (0.38, -2.85) and (2.62, 3.85)
If you graph both functions, you will see that they have two points of intersection, so choices (1) and (2) are out. The second one occurs when x is less than 3, so choice (4) is out.
You can also put the x values into each equation and check the y values.

11. The expression -4.9t2 + 50t + 2 represents the height, in meters, of a toy rocket t seconds after launch. The initial height of the rocket, in meters, is

Answer: (2) 2
When t = 0, the expression is 0 + 0 + 2 = 2.
Note: The -4.9 will show up in gravity questions when meters are the units being used. Sometimes it's rounded to -5.

12. If the domain of the function f(x) = 2x2 - 8 is {-2, 3, 5}, then the range is

Answer: (3) {O, 10, 42}
Simple Order of Operations question.
Calculator tip: type 2x^{-2,3,5} - 8 into your graphing calculator and it will give you the range in one shot.

13. Which polynomial is twice the sum of 4x2 - x + 1 and -6x2 + x - 4?

Answer: (3) -4x2 - 6
Twice the sum means add them first then double it.
(4x2 - x + 1) + (-6x2 + x - 4) = -2x2 - 3, (the middle term cancelled out)
2 (-2x2 - 3) = -4x2 - 6

14. What are the solutions to the equation 3(x - 4)2 = 27?

Answer: (1) 1 and 7

3(x - 4)2 = 27
(x - 4)2 = 9
x - 4 = 3 or x - 4 = -3
x = 7 or x = 1

15. A system of equations is shown below.

Equation A: 5x + 9y = 12
Equation B: 4x - 3y = 8

Which method eliminates one of the variables?

Answer: (2) Multiply equation B by 3 and add the result to equation A.
9y + (-9y) will eliminate the y variable so you can solve for x.

16. The 15 members of the French Club sold candy bars to help fund their trip to Quebec. The table below shows the number of candy bars each member sold.

When referring to the data, which statement is false?

Answer: (1) The mode is the best measure of central tendency for the data.
The median (middle value when data is sorted) is 53. The range is 120 - 0 = 120. There are two outliers: 0 and 120, which are very low and very high compared to the rest of the data. The mode is 68, which comes from 3 of the 5 highest values of the 15 pieces of data. That is not the best measure of central tendency to use for this data.

17. Given the set {x| -2 < x < 2, where x is an integer}, what is the solution of -2(x - 5) < 10?

Answer: (2) 1, 2
You could check 0, 1, -1 and have enough information to find the correct choice. (You wouldn't need -1, in fact, in this case.) Or you can simplify the inequality first.

-2(x - 5) < 10
x - 5 > -5
x > 0

18. If the pattern below continues, which equation(s) is a recursive formula that represents the number of squares in this sequence?

Answer: (3) a1 = 3, an = an-1 + 2
Choices (1) and (2) are not recursive. The first design, a1 has 3 squares in it, so choice (4) is incorrect.

19. If the original function f(x) = 2x2 - 1 is shifted to the left 3 units to make the function g(x), which expression would represent g(x)?

Answer: (2) 2(x + 3)2 - 1
Shifting the vertex 3 units left gives you "(x + 3)". A shift the the right would be "(x - 3)".
Choices (3) and (4) represent shifts up and down, respecitvely.

20. First consider the system of equations y = (-1/2)x+ 1 and y = x- 5.
Then consider the system of inequalities y > (-1/2)x + 1 and y < x- 5.
When comparing the number of solutions in each of these systems, which statement is true?

Answer: (3) The system of inequalities has more solutions.
The system of equations has a single solutions. Two linear equations meet at a single point unless they have the same slope. The first line has a slope of -1/2, the second 1.
This system of inequalities has an infinite number of solutions; e.g., (6,0), (7,0), (8,0), etc.

21. Nora inherited a savings account that was started by her grandmother 25 years ago. This scenario is modeled by the function A(t) = 5000(1.013)t + 25, where A(t) represents the value of the account, in dollars, t years after the inheritance. Which function below is equivalent to A(t)?

Answer: (4) A(t) = 5000(1.013)t (1.013)25
When you multiply two expressions that have the same base, you add the exponents. The reverse is true: if you have a base with an exponent, you can factor it by keeping the base and reducing the exponents to two addends (i.e., to two numbers or expressions that add up to the original expression).
So, for example, x8 = x5 * x3, or
(m + 3)n+5 = (m + 3)n * (m + 3)5

22. The value of x which makes

(2/3)((1/4)x - 2) = (1/5)((4/3)x - 1)
true is

Answer: (4) -11.3... (repeating decimal; i.e., 1/3)

Multiple both side by 3*5, which is 15
(15)(2/3)((1/4)x - 2) = (15)(1/5)((4/3)x - 1)
(10)((1/4)x - 2) = (3)((4/3)x - 1)
2.5x - 20 = 4x - 3
-17 = 1.5x
-11.3... = x

23. Which quadratic function has the largest maximum over the set of real numbers?

Answer: (2) k(x)

Choice (3) is in vertex form. It's maximum point occurs at (5, 5).
Choice (1) can be entered into the graphing calculator, or you can use x = -b/a = -2/-1 = 1
f(1) = -(1)2 + 2(1) + 4 = -1 + 2 + 4 = 5, which gives (1, 5). So neither (1) or (3) can be the answer.
Choice (4) does not show a value higher than 3, but neither of those two points are the vertex. You could do a quadratic regression to find the function and then find the vertex, but look at the rate of change in the table. It goes +6, +4, +2, ... getting smaller with each step. You can deduce that h(1.5) will NOT be greater than 5.
Choice (2), like (4) doesn't show the vertex. However, the highest number it does show is 5. The vertex must be higher than 5. That makes choice (2) the answer.

24. Voting rates in presidential elections from 1996-2012 are modeled below.

Which statement does not correctly interpret voting rates by age based on the given graph?

Answer: (2) From 1996-2012, the average rate of change was positive for only two age groups.
The rate of change was negative for the "45 to 64 years" age group, decreasing from 68.2 to 67.9. It increased for the other three groups from 1996-2012.

End of Part I

How did you do?

Questions, comments and corrections welcome.

Monday, February 05, 2018

January 2018 Common Core Algebra I Regents, Parts 3 and 4

The following are some of the open-ended questions from the recent January 2018 New York State Common Core Algebra I Regents exam, parts 3 and 4
Part II can be found here.

January 2018 Algebra I, Part III

Each correct answer is worth up to 4 credits. Work must be shown.

33. Jim is a furniture salesman. His weekly pay is $300 plus 3.5% of his total sales for the week. Jim sells x dollars' worth of furniture during the week. Write a function, p(x), which can be used to determine his pay for the week.

Use this function to determine Jim's pay to the nearest cent for a week when his sales total is $8250.

Answer: p(x) = 300 + .035x
$300 is the fixed amount, .035 is 3.5% in decimal form, x is the variable.

To find his pay, substitute 8250 for x.
P(8250) = 300 + .035(8250)
P(8250) = $588.75

34. Omar has a piece of rope. He ties a knot in the rope and measures the new length of the rope. He then repeats this process several times. Some of the data collected are listed in the table below.

State, to the nearest tenth, the linear regression equation that approximates the length, y, of the rope after tying x knots.
Explain what the y-intercept means in the context of the problem.
Explain what the slope means in the context of the problem.

Answer: Put all of the values into L1 and L2 on your graphing calculator. Then perform a Linear regression to find the slope (a) and y-intercept (b)
This will give you y = -8.5x + 99.2

The y-intercept means that when there are no knots in the rope, the length of the rope will be 99.2 cm.

The slope of the equation means that each knot made will decrease the length of the rope by 8.5 cm.

35. The drama club is running a lemonade stand to raise money for its new production. A local grocery store donated cans of lemonade and bottles of water. Cans of lemonade sell for $2 each and bottles of water sell for $1.50 each. The club needs to raise at least $500 to cover the cost of renting costumes. The students can accept a maximum of 360 cans and bottles.

Write a system of inequalities that can be used to represent this situation.

The club sells 144 cans of lemonade. What is the least number of bottles of water that must be sold to cover the cost of renting costumes? Justify your answer.

Answer: Let L = the number of cans of lemonade and W = the number of bottles of water.
Note: Use a capital L or a script l -- don't let your L's look like ones!

2L + 1.5W > 500
L + W < 360

2L + 1.5W > 500
2(144) + 1.5W > 500
288 + 1.5W > 500
1.5W > 212
W > 141.33333

Round up to 142 bottles.
Do NOT round down. That won't be enough -- you will have less than $500.

36. A manager wanted to analyze the online shoe sales for his business. He collected data for the number of pairs of shoes sold each hour over a 14-hour time period. He created a graph to model the data, as shown below.

The manager believes the set of integers would be the most appropriate domain for this model. Explain why he is incorrect.

State the entire interval for which the number of pairs of shoes sold is increasing.

Determine the average rate of change between the sixth and fourteenth hours, and explain what it means in the context of the problem.

Answer: The manager is incorrect. (Don't forget to state this with your explanation.)
The most appropriate domain would be whole numbers numbers which are positive and zero. Integers would include negative numbers, but you cannot sell a negative number of shoes.

The number of pairs of shoes sold is increasing during the interval 0 < t < 6.
After 6 hours, fewer pairs are being sold each hour.

To find the average rate of change, use the slope formula for the sixth hour (6, 120) and the fourteenth hour (14, 0)

slope = (y2 - y1) / (x2 - x1) = (0 - 120) / (14 - 6) = (-120) / 8 = -15
In the context of the problem, this means that 15 fewer shoes were being sold each hour between hours 6 and 14.

EDIT: The scoring guidelines were updated after the exam to allow for "0 to 14" as an acceptable answer because of the imprecise wording of the problem.

January 2018 Algebra I, Part IV

This answer is worth up to 6 credits. Work must be shown.

37. At Bea's Pet Shop, the number of dogs, d, is initially five less than twice the number of cats, c. If she decides to add three more of each, the ratio of cats to dogs will be 3/4.
Write an equation or system of equations that can be used to find the number of cats and dogs Bea has in her pet shop.

Could Bea's Pet Shop initially have 15 cats and 20 dogs? Explain your reasoning.

Determine algebraically the number of cats and the number of dogs Bea initially had in her pet shop.

Note: I'd have to check the archives, but I seem to remember a very similar question years ago.

Answer: Let d = the number of dogs and c = the number of cats
The first equation can be translated directly from the first sentence:

d = 2c - 5
Remember that "five less than" means "- 5".
The next sentence gives us two ratios, so the second equation is a proportion:
(c + 3)/(d + 3) = 3/4

Second part: Plug in the values for 15 cats and 20 dogs.
Check (20) = 2(15) - 5?
20 = 30 - 5 = 25, Incorrect.
No, 15 cats and 20 dogs cannot be the initial number of pets because the first equation would not be true.

Third part: You need to solve by substitution and then cross-multiplying the proportion.
Substitute (2c - 5) for d in the proportion.

Bea initially had 9 cats and 13 dogs.

End of exam.

How did you do?

Questions, comments and corrections are welcome.

Friday, February 02, 2018

Squares Are Cool

(Click on the comic if you can't see the full image.)

(C)Copyright 2018, C. Burke.

Just don't call him late for dinner. He likes his three squares!

And if you define trapezoids more inclusively as "at least one pair of parallel sides", then squares, and all parallelograms, will be trapezoids as well.

Come back often for more funny math and geeky comics.

Thursday, February 01, 2018

January 2018 Common Core Algebra I Regents, Part 2

The following are some of the open-ended questions from the recent January 2018 New York State Common Core Algebra I Regents exam.

January 2018 Algebra I, Part II

Each correct answer is worth up to 2 credits. Work must be shown.

25. On the set of axes below, graph f(x) = |x - 3| + 2.

Answer: See below. You could use a graphing calculator to find a table of values, or you could realize that the parent functions has been translated 3 units to the right and 2 up. (In other words, the vertex is at the point (3, 2).) The slope on the left is -1 and the slope on the right is 1, so finding the other points on the line is simple.

26. Determine all the zeros of m(x) = x2 - 4x + 3, algebraically.

Answer: Pretty straight forward. Set the expression equal to zero and factor it. The factors of +3 that add up to -4 are +3 and +1.
If you didn't see that, you can always use the Quadratic Formula. It's longer, but it will give you the answer.

x2 - 4x + 3 = 0
(x - 3)(x - 1) = 0
x - 3 = 0 or x - 1 = 0
x = 3 or x = 1

27. The distance traveled is equal to the rate of speed multiplied by the time traveled. If the distance is measured in feet and the time is measured in minutes, then the rate of speed is expressed in which units? Explain how you arrived at your answer.

Answer: The rate of time is measured in feet per minute.
Because distance = rate * time,
Then rate = distance / time, which gives you feet / min.

28. Determine if the point (0,4) is a solution to the system of inequalities graphed below. Justify your answer.

Answer: (0, 4) is NOT a solution to the system. It is the point on the y-axis where the two boundary lines intersect. However, one of those lines is broken, which means that those points are NOT point of the solution to the system. (0, 4) is on the broken line.

Note that each box in the scale is 2 units. If you put a point on the fourth line, which would be (0, 8), you would probably have lost a point.

29. If the zeros of a quadratic function, F, are -3 and 5, what is the equation of the axis of symmetry of F? Justify your answer.

Answer: If the zeroes are at -3 and 5, then the axis of symmetry has to be right in the middle of them. Average the two numbers:
x = (-3 + 5) / 2 = 2 / 2 = 1. x = 1 (Note: "equation" means you MUST include "x = ")

One longer way would be to find the equation of the function and then the axis of symmetry:
f(x) = (x + 3)(x - 5) = x2 - 2x - 15.
Formula for Axis of Symmetry is x = -b / (2a)
x = -(-2) / ((2)(1)) = 2 / 2 = 1. x = 1.

30. The formula Fg = (GM1M2)/r2 calculates the gravitational force between two objects where G is the gravitational constant, M1 is the mass of one object, M2 is the mass of the other object, and r is the distance between them. Solve for the positive value of r in terms of Fg, G, M1, and M2.


Fg = (GM1M2)/r2
Fg r2 = (GM1M2)
r2 = (GM1M2) / Fg
r = SQRT ( (GM1M2) / Fg )

31. At Mountain Lakes High School, the mathematics and physics scores of nine students were compared as shown in the table below.

State the correlation coefficient, to the nearest hundredth, for the line of best fit for these data.
Explain what the correlation coefficient means with regard to the context of this situation.

Answer: Put all the values into a list and then do a linear regression.
You will see that the correlation coefficient r = .92.
In this example, it means that there is a strong positive correlation between math and physics scores.

32. The graph of the function f(x) = ax2 + bx + c is given below.

Could the factors of f(x) be (x + 2) and (x - 3)? Based on the graph, explain why or why not.

Answer: Yes. Those are the factors because (x + 2) means that there is a zero at -2 and (x - 3) means that there is a zero at +3, which is what the graph shows.

End of Part II

How did you do?
Questions, comments and corrections welcome. Typos happen.