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.

Come back often for more funny math and geeky comics.

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

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.

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

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(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.