Thursday, March 14, 2024

Pi Day 2024

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(C)Copyright 2024, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

The co-sin as well!

I know I usually save these guys for Talk Like A Priate Day, and use Sherlock Pi on Pi Day (3/14) but I decided to shake things up a little.

I've been getting a lot of suggestions for using the Sin of Pie is 0, which is amusing, but it's not my joke, so unless I can work it into my characters, I can't use it. (I could, but I choose not to.)

On the other hand, if you quarter pi, then the sin is radical 2 over 2. And so is the cos.

And your tan will be the one.



I also write Fiction!


You can now order my newest book Burke's Lore, Briefs: A Heavenly Date / My Damned Best Friend, written by Christopher J. Burke, which contains the aforementioned story and a bonus story.
Order the softcover or ebook at Amazon.

Also, check out In A Flash 2020, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides.
Available in softcover or ebook at Amazon.

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Come back often for more funny math and geeky comics.



Tuesday, March 12, 2024

January 2024 Algebra 2 Regents, Part III


This exam was adminstered in January 2024.

More Regents problems.

Algebra 2 January 2024

Part III: Each correct answer will receive 4 credits. Partial credit can be earned. One computational mistake will lose 1 point. A conceptual error will generally lose 2 points (unless the rubric states otherwise). It is sometimes possible to get 1 point for a correct answer with no correct work shown.


33. A researcher wants to determine if nut allergies and milk allergies are related to each other. The researcher surveyed 1500 people and asked them if they are allergic to nuts or milk. The survey results are summarized in the table below.


Determine the probability that a randomly selected survey respondent is allergic to milk
Determine the probability that a randomly selected survey respondent is allergic to milk, given that the person is allergic to nuts.
Based on the survey data, determine whether nut allergies and milk allergies are independent events. Justify your answer.

Answer:


Add up the rows and columns. You will see that that it adds to 1500, as stated in the problem.

There are 45 respondents out of 1500 who are allergic to milk, so the probability is 45/1500. (You don't need to simplify the fraction.)

There are 15 people who are allergic to nuts, and of those, 3 are also allergic to milk, so the probability is 3/15.

If the nut allergies and milk allergies are independent, then the previous two answers would be the same because P(A) would have to be equal to P(A|B). However, 45/1500 = 0.03 and 3/15 = 0.2. So the events are not independent.





34. Algebraically solve for x: 2x = 6 + 2√(x - 1)

Answer:


Isolate the radical. Then square both sides. Finally, solve the quadratic equation.

2x = 6 + 2√(x - 1)
2x - 6 = 2√(x - 1)
x - 3 = √(x - 1)
(x - 3)2 = (√(x - 1))2
x2 - 6x + 9 = x - 1
x2 - 7x + 10 = 0
(x - 5)(x - 2) = 0
x - 5 = 0 or x - 2 = 0
x = 5 or x = 2

Throw out x = 2 as extraneous because 2(2) =/= 6 + 2√(2-1).

The only solution is x = 5.





35. During the summer, Adam saved $4000 and Betty saved $3500. Adam deposited his money in Bank A at an annual rate of 2.4% compounded monthly. Betty deposited her money in Bank B at an annual rate of 4% compounded quarterly. Write two functions that represent the value of each account after t years if no other deposits or withdrawals are made, where Adam’s account value is represented by A(t), and Betty’s by B(t).

Using technology, determine, to the nearest tenth of a year, how long it will take for the two accounts to have the same amount of money in them. Justify your answer.


Answer:


Write the functions A(t) and B(t) using the given initial amounts and rates. Note that that A is compounded monthly, so 0.024 will be divided by 12 and the exponent will be multiplied by 12. Likewise, B is compounded quarterly, so 0.04 will be divided by 4 and the exponent will be multiplied by 4.

A(t) = 4000(1 + 0.024/12)12t
or A(t) = 4000(1.002)12t

B(t) = 3500(1 + 0.04/4)12t
or B(t) = 3500(1.01)4t

"Using technology" means that you can graph the two functions to see when they intersect rather than writing an equation and solving. Put the two equations into your graphing calculator. Graph and trace the functions, or look at the table of values, setting the calculator to show every 0.1 value of x.

At t=8.4, A(8.4) = 4892.40 and B(8.4) = 4889.50, a difference of $3.10, which is the smallest difference to the nearest tenth of year.





36.
On the graph below, draw at least one complete cycle of a sine graph passing through point (0,2) that has an amplitude of 3, a period of p, and a midline at y = 2.
Based on your graph, state an interval in which the graph is increasing.

Answer:


The period of π instead of 2π means that means we need sin(2x) instead of sin(x). The amplitude of 3 is a mulitplier in front of the function and the midline of 2 is an addition after the function.

y = 3 sin(2x) + 2

The graph would look like the one below. The y-intercept is at (0,2). The maximum value is 5 and the minimum is -1.

One interval where it is increasing would be from 3π/4 to 5π/4.




End of Part III

How did you do?








More to come. Comments and questions welcome.

More Regents problems.

I also write Fiction!


You can now order my newest book Burke's Lore, Briefs: A Heavenly Date / My Damned Best Friend, written by Christopher J. Burke, which contains the aforementioned story and a bonus story.
Order the softcover or ebook at Amazon.

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Available in softcover or ebook at Amazon.

If you enjoy it, please consider leaving a rating or review on Amazon or on Good Reads.



Friday, March 08, 2024

Arctan

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(C)Copyright 2024, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

I know, the Ham joke was a little cheesy.

I haven't used Noah and his sons in quite a while. They were one of the early themes, and I thought I'd revisit them as we closed in on comic #2000.

Also, as anyone with fair skin can tell you, you can get burned on cloudy days. UV rays don't care about clouds.

Since I have such fair skin, I tend to stay covered up even in the summer, so I have more of an inverse tan.



I also write Fiction!


You can now order my newest book Burke's Lore, Briefs: A Heavenly Date / My Damned Best Friend, written by Christopher J. Burke, which contains the aforementioned story and a bonus story.
Order the softcover or ebook at Amazon.

Also, check out In A Flash 2020, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides.
Available in softcover or ebook at Amazon.

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Come back often for more funny math and geeky comics.



Wednesday, February 28, 2024

School Life #40: Beagle

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(C)Copyright 2024, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

Everyone know Charlie Brown and Snoopy went to the Moon, not to the Galápagos!

This one has been in the pipeline for a while, waiting to be done. Problem was I forget about it a couple of times, but the good ones always come back.

Sometimes the bad ones do, too, like a bad (knock, knock, knock) Penny.



I also write Fiction!


You can now order my newest book Burke's Lore, Briefs: A Heavenly Date / My Damned Best Friend, written by Christopher J. Burke, which contains the aforementioned story and a bonus story.
Order the softcover or ebook at Amazon.

Also, check out In A Flash 2020, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides.
Available in softcover or ebook at Amazon.

If you enjoy it, please consider leaving a rating or review on Amazon or on Good Reads.





Come back often for more funny math and geeky comics.



Monday, February 19, 2024

January 2024 Algebra 2, Part II


This exam was adminstered in January 2024.

More Regents problems.

Algebra 2 January 2024

Part II: Each correct answer will receive 2 credits. Partial credit can be earned. One mistake (computational or conceptual) will lose 1 point. A second mistake will lose the other point. It is sometimes possible to get 1 point for a correct answer with no correct work shown.


25. Factor the expression x3 + 4x2 - 9x - 36 completely.

Answer:


Factor by grouping, and then factor the quadratic you get after the first step.

There are two ways to group, and either should work in any question of this kind.

x3 + 4x2 - 9x - 36
(x3 + 4x2) - (9x + 36)
x2(x + 4) - 9(x + 4)
(x2 - 9)(x + 4)
(x + 3)(x - 3)(x + 4)

You can also switch the two middle terms around. This is just the way I learned it, so I usually do it, especially if it helps me avoid factoring out a minus sign.

x3 - 9x + 4x2 - 36
(x3 - 9x) + (4x2 - 36)
x(x2 - 9) + 4(x2 - 9)
(x + 4)(x2 - 9)
(x + 4)(x + 3)(x - 3)

Note: This was Very Similar to Question 25 on the Auguest 2023 Regents. Right down to the (x + 3)(x - 3).





26. Determine if x + 4 is a factor of 2x3 + 10x2 + 4x - 16. Explain your answer.

Answer:


If (x + 4) is a factor of the polynomial, then the value of the polynomial must be 0 when x = -4.

2(-4)3 + 10(-4)2 + 4(-4) - 16 = 0

Since the expression is equal to zero when x = -4, then (x + 4) must be a factor.

You could also solve this using polynomial division.

(x + 4) divides evenly, with no remainder, so it is a factor.





27. An initial investment of $1000 reaches a value, V(t), according to the model V(t) = 1000(1.01)4t, where t is the time in years.
Determine the average rate of change, to the nearest dollar per year, of this investment from year 2 to year 7.

Answer:


Calculate V(7) and V(2). Subtract them and divide by 7 - 2, which is 5. You are looking for the rate of change (or slope, if you prefer).

V(7) = 1000(1.01)4(7) = 1321.29

V(2) = 1000(1.01)4(2) = 1082.86

Rate of change = (1321.29 - 1082.86) / 5 = 47.686, which is $48 to the nearest dollar.





28. When ( 1 / ∛(y2) ) y4 is written in the form yn, what is the value of n? Justify your answer.

Answer:


Use the laws of exponents to change the radical into a fraction. The combine the terms.

( 1 / ∛(y2) ) y4
( 1 / (y2/3) y4
(y-2/3) y4
y10/3

n = 10/3.





29. The heights of the members of a ski club are normally distributed. The average height is 64.7 inches with a standard deviation of 4.3 inches. Determine the percentage of club members, to the nearest percent, who are between 67 inches and 72 inches tall.

Answer:


They don't use the chart with the normal distribution and all the standard deviations marked off any more. They just assume that you have and will use a calculator for this.

You need to use the normalcdf function.

Enter the command normalcdf(67,72,64.7,4.3) and you will get .2515... or 25%.

All of the numbers that go into the command are in the question. Lower bound, upper bound, median, standard deviation.





30. The explicit formula an = 6 + 6n represents the number of seats in each row in a movie theater, where n represents the row number. Rewrite this formula in recursive form.

Answer:


A recursive function needs an initial value (a1) and an equation for an is terms of an-1.

The inition value a1 = 12.

Then an = an-1 + 6, because the common difference (rate of change) is 6.





31.Write (2xi3 - 3y)2) in simplest form.

Answer:


Square the binomial, substitute the powers of i, and Combine Like Terms.

(2xi3 - 3y)2)

(2xi3 - 3y)(2xi3 - 3y)

4x2i6 - 6xyi3 - 6xyi3 + 9y2

-4x2 - 12xyi3 + 9y2

-4x2 + 12xyi + 9y2





32. A survey was given to 1250 randomly selected high school students at the end of their junior year. The survey offered four post-graduation options: two-year college, four-year college, military, or work. Of the 1250 responses, 475 chose a four-year college. State one possible conclusion that can be made about the population of high school juniors, based on this survey

Answer:


This seems almost too simple a problem. If you divide 475/1250, you get .38 or 38%.

One conclusion you can draw is that the population of high school juniors that would chose a four-year college would probably be about 38% and 62% would choose a different option.




End of Part II

How did you do?





More to come. Comments and questions welcome.

More Regents problems.

I also write Fiction!


You can now order my newest book Burke's Lore, Briefs: A Heavenly Date / My Damned Best Friend, written by Christopher J. Burke, which contains the aforementioned story and a bonus story.
Order the softcover or ebook at Amazon.

Also, check out In A Flash 2020, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides.
Available in softcover or ebook at Amazon.

If you enjoy it, please consider leaving a rating or review on Amazon or on Good Reads.



Friday, February 16, 2024

January 2024 Geometry Regents Part IV


This exam was adminstered in January 2024.

January 2024 Geometry, Part IV

A correct answer is worth 6 credits. Partial credit can be given for correct statements in the proof.


35. Quadrilateral MATH has vertices with coordinates M(-1,7), A(3,5), T(2,-7), and H(-6,-3).
Prove that quadrilateral MATH is a trapezoid.
[The use of the set of axes on the next page is optional.]

State the coordinates of point Y such that point A is the midpoint of MY.

Prove that quadrilateral MYTH is a rectangle. [The use of the set of axes below is optional.]

Answer:


You don't neeed to use the grid but it may be helpful for visualizing. If you do use it, you still need to answer the questions fully and completely. You can't rely on whatever is in the grid to be sufficient.

To show that MATH is a trapezoid, you have to that there is one pair of parallel sides. You can do this by find the slopes of the four sides. This is easy if you graph it because you can count boxes without worrying about subtracting signed numbers (if that's a problem for you). Only two of the sides will be the same and the other two will be different.

Slope MA (7 - 5) / (-1 - 3) = 2/-4 = -1/2

Slope AT = (-7 - 5) / (2 - 3) = -12/-1 = 12

Slope TH = (-3 - -7) / (-6 - 2) = 4/-8 = -1/2

Slope HM = (7 - -3) / (-1 - -6) = 10/5 = 2

Since there is one pair of parallel sides, the quadrilateral is a trapezoid.

To find point Y, find the change in x-value and y-value fro M to A and add those numbers again to get Y.

M(-1,7) -> A(3,5) is a translation of +4,-2. Y(3+4,5-2) = Y(7,3)
This was worth a point even if you didn't show work.

To show that MYTH is a rectangle, you could show that the opposite sides are parallel (same slope) and that two consecutive sides are parallel (slopes are inverse reciprocals).

Slope MY (7 - 3) / (-1 - 7) = 4/-8 = -1/2

Slope YT = (-7 - 3) / (2 - 7) = -10/-5 = 2

You previously found:
Slope TH = (-3 - -7) / (-6 - 2) = 4/-8 = -1/2
Slope HM = (7 - -3) / (-1 - -6) = 10/5 = 2

Opposite sides are parallel, so it is a parallelogram.

(-1/2)(1) = -1. MY is perpendicular to YT, so angle Y is a right angle. Therefore, MYTH is a rectangle.




End of Exam

How did you do?

Questions, comments and corrections welcome.

I also write Fiction!


You can now order my newest book Burke's Lore, Briefs: A Heavenly Date / My Damned Best Friend, written by Christopher J. Burke, which contains the aforementioned story and a bonus story.
Order the softcover or ebook at Amazon.

Also, check out In A Flash 2020, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides.
Available in softcover or ebook at Amazon.

If you enjoy it, please consider leaving a rating or review on Amazon or on Good Reads.



January 2024 Algebra 1 Regents Part IV



This exam was adminstered in January 2024.

More Regents problems.

January 2024

Part IV: A correct answer will receive 6 credits. Partial credit can be earned.


37. Jim had a bag of coins. The number of nickels, n, and the number of quarters, q, totaled 28 coins. The combined value of the coins was $4.

Write a system of equations that models this situation.

Use your system of equations to algebraically determine both the number of quarters, q, and the number of nickels, n, that Jim had in the bag.

Jim was given an additional $3.00 that was made up of equal numbers of nickels and quarters. How many of each coin was he given? Justify your answer.

Answer:


Write one equation for the total number of the nickels and quarters, and then write a second equation for the total value of those nickels and quarters.
n + q = 28
5n + 25q = 400

Use substition or elimination to solve the equation. For example, you could replace q with 28 - n.

5n + 25(28 - n) = 400

5n - 25n + 700 = 400

-20n = -300

n = 15

15 + q = 28

q = 13

If he was given an equal amount of nickels and quarters, then n = q. Therefore,

5q + 25q = 300
30q = 300
q = 10
n = 10

He received 10 more nickels and 10 more quarters.




End of Exam

How did you do?





More to come. Comments and questions welcome.

More Regents problems.

I also write Fiction!


You can now order my newest book Burke's Lore, Briefs: A Heavenly Date / My Damned Best Friend, written by Christopher J. Burke, which contains the aforementioned story and a bonus story.
Order the softcover or ebook at Amazon.

Also, check out In A Flash 2020, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides.
Available in softcover or ebook at Amazon.

If you enjoy it, please consider leaving a rating or review on Amazon or on Good Reads.



Thursday, February 15, 2024

January 2024 Geometry Regents Part III


This exam was adminstered in January 2024.

January 2024 Geometry, Part III

Each correct answer is worth up to 4 credits. Partial credit can be given. Work must be shown or explained.


32. Trish is a surveyor who was asked to estimate the distance across a pond. She stands at point C, 85 meters from point D, and locates points A and B on either side of the pond such that A, D, and B are collinear.


Trish approximates the measure of angle DCB to be 35° and the measure of angle ACD to be 75°.
Determine and state the distance across the pond, AB, to the nearest meter.

Answer:


Using the 75 degree angle, you can find the length of AD, which we'll call x. Using the 35 degree angle, you can find the length of DB, which we'll call y. Add the two together to find the length of AD.

In both cases, we have the opposite side and we need to find the adjacent side. So we need to use the tangent ratio twice.

tan 75 = x / 85
x = 85 * tan 75 = 317.224...

tan 35 = y / 85
y = 85 * tan 35 = 59.517...

AB = x + y = 317.224 + 59.517 = 376.741 = 377 meters




33. A candle in the shape of a right pyramid is modeled below. Each side of the square base measures 12 centimeters. The slant height of the pyramid measures 16 centimeters.


Determine and state the volume of the candle, to the nearest cubic centimeter
The wax used to make the candle weighs 0.032 ounce per cubic centimeter. Determine and state the weight of the candle, to the nearest ounce.

Answer:


Notice that they gave you the slant height and not the height. You need the height to find the Volume. If you take a vertical slice (cross-section) of the pyramind, you would get an isosceles triangle with a base of 12 and two legs that were 16 cm. If you draw an altitude, you will get two congruent right triangles with a base of 6 and a hypotenuse of 16. Use this information to find the height.

(6)2 + b2 = (16)2
36 + b2 = 256
b2 = 220
b = √(220) = 14.832...

Use this value to find the Volume.

V = (1/3) Area of Base * height = (1/3) * 12 * 12 * 14.832 = 711.936 = 712 cu cm.

The weight is equal to the Volume times the Density: W = (712) * (0.032) = 22.784 = 23




34. In the diagram of quadrilateral ABCD below, AB ≅ CD, and AB || CD.
Segments CE and AF are drawn to diagonal BD such that BE ≅ DF.


Prove: CE ≅ AF

Answer:


TO prove that CE is congruent to AF, you are going to have to show that triangles BEC and DFA are congruent and then use CPCTC. To show that the triangles are congruent, you can use SAS.

Your proof should look like this:

Statement Reasons
Quadrilateral ABCD, AB ≅ CD, AB || CD, and BE ≅ DF. Given
ABCD is a parallelogram A quadrilateral with one pair of sides that are parallel and congruent is a parallelogram
BC ≅ AD Opposite sides of parallelograms are congruent.
BC || AD Opposite sides of parallelograms are parallel.
∠ CBE ≅ ∠ ADF Alternate Interior Angles
△BCE ≅ △DAF SAS Postulate
(AB) / (AE) = (TR) / (TE) Corresponding sides of similar triangles are proportional
CE ≅ AF CPCTC

End of Part III

How did you do?

Questions, comments and corrections welcome.

I also write Fiction!


You can now order my newest book Burke's Lore, Briefs: A Heavenly Date / My Damned Best Friend, written by Christopher J. Burke, which contains the aforementioned story and a bonus story.
Order the softcover or ebook at Amazon.

Also, check out In A Flash 2020, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides.
Available in softcover or ebook at Amazon.

If you enjoy it, please consider leaving a rating or review on Amazon or on Good Reads.



Tuesday, February 13, 2024

January 2024 Algebra 1 Regents Part III


This exam was adminstered in January 2024 .

More Regents problems.

January 2024

Part III: Each correct answer will receive 4 credits. Partial credit can be earned.


33. While playing golf, Laura hit her ball from the ground. The height, in feet, of her golf ball can be modeled by h(t) = - 16t + 48t, where t is the time in seconds.
Graph h(t) on the set of axes below.


What is the maximum height, in feet, that the golf ball reaches on this hit?
How many seconds does it take the golf ball to hit the ground?

Answer:


Look at the graph below. You can put the equation in the graphing calculator. You might want to set it so that it shows every 0.5 increment of x because the high point is going to happen at x = 1.5

The maximum height the ball reaches is 36 feet (which happens at 1.5 seconds).

It takes 3 seconds for the ball to hit the ground.





34. The table below shows the number of SAT prep classes five students attended and the scores they received on the test.

State the linear regression equation for this data set, rounding all values to the nearest hundredth.
State the correlation coefficient, rounded to the nearest hundredth.
State what this correlation coefficient indicates about the linear fit of the data.

Answer:


Enter all the data into two lists in your graphing calculator. You will have to run a linear regression. Make sure you have DIAGNOSTICS ON set on your calculator.

When you run the linear regression, you will get a = 40.48 and b = 363.81, rounded to the nearest hundredth.
So the equation is y = 40.48x + 363.81

The correlation coefficient, r, is 0.84.

There is a strong positive correlation between the number of SAT prep courses attended and the score on the Math SAT.





35. Julia is 4 years older than twice Kelly’s age, x. The product of their ages is 96.
Write an equation that models this situation.
Determine Kelly’s age algebraically.
State the difference between Julia’s and Kelly’s ages, in years.

Answer:


Kelly's age is x. Write an expression for Julia in terms of x. The product of that expression and x will be 96. Solve the quadratic equation that results from it.

J = 2x + 4
x(2x + 4) = 96

2x2 + 4x = 96
2x2 + 4x - 96 = 0
x2 + 2x - 48 = 0
(x + 8)(x - 6) = 0
x + 8 = 0 or x - 6 = 0
x = -8 or x = 6

Throw out the negative answer because age cannot be negative. Therefore, Kelly is 6 years old.

Julia is 2(6) + 4 = 16. The difference between their ages is 10 years.

If you messed up the signs and thought that Kelly was 8 years old, then Julia would 20, and the difference would be 12 years. If you made one mistake, you would have lost only one point if the rest of your answers were consistent with that mistake.





36. On the set of axes below, graph the following system of inequalities:
2x - y > 4
x + 3y > 6

Label the solution set S.

Is (4,2) a solution to this system? Justify your answer.

Answer:


Rewrite the inequalities into slope-intercept form. Remember when you divide an inequality by a negative number, you have to flip the direction of the inequality symbol.

2x - y > 4
- y > -2x + 4
y < 2x - 4

x + 3y > 6
3y > -x + 6
y > -1/3 x + 2

Both inequalites will have broken lines. Shade above the line y > 1/3x + 2, and below y < 2x - 4. Mark the area with the crisscross with a big "S". This is your solution.

(4,2) is a solution to the system because it's in the double-shaded area.




End of Part III

How did you do?








More to come. Comments and questions welcome.

More Regents problems.

I also write Fiction!


You can now order my newest book Burke's Lore, Briefs: A Heavenly Date / My Damned Best Friend, written by Christopher J. Burke, which contains the aforementioned story and a bonus story.
Order the softcover or ebook at Amazon.

Also, check out In A Flash 2020, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides.
Available in softcover or ebook at Amazon.

If you enjoy it, please consider leaving a rating or review on Amazon or on Good Reads.



Monday, February 12, 2024

A Fine Line

(Click on the comic if you can't see the full image.)
(C)Copyright 2024, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

There's a fine line between humor and informational context.

You know there's a great chance that Mike is not only aware that it was a joke but it's something that he would've said.

And everyone likes Mike. Right? And I ask this not because people say he's my avatar in this comic.

I almost had Scott tell it to Ken, but their rivalry is a little different (bordering, perhaps, on animosity).



I also write Fiction!


You can now order my newest book Burke's Lore, Briefs: A Heavenly Date / My Damned Best Friend, written by Christopher J. Burke, which contains the aforementioned story and a bonus story.
Order the softcover or ebook at Amazon.

Also, check out In A Flash 2020, by Christopher J. Burke for 20 great flash fiction stories, perfectly sized for your train rides.
Available in softcover or ebook at Amazon.

If you enjoy it, please consider leaving a rating or review on Amazon or on Good Reads.





Come back often for more funny math and geeky comics.