School Life #24: One-to-One and Onto

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

Bonus Panel

(C)Copyright 2021, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

Same old stories: boys want girls to be into them, not onto them.

Ob Math: Originally the bonus panel was going to be the map of the relation, but then the Kyung idea occurred to me. When you have a relation of two sets of data, if each member of the input (domain) maps to one distinct member of the output (range), then it is a one-to-one relation. If every member of the output has an input mapped to it then it's "onto". If there's an extra element, then it's not onto. The first comic is one-to-one and onto. The bonus panel is one-to-one but not onto.

The one-to-one relation is mapped below:

I also write Fiction! You can now preorder Devilish And Divine, edited by John L. French and Danielle Ackley-McPhail, which contains (among many, many others) three stories by me, Christopher J. Burke about those above us and from down below. Preorder the softcover or ebook at Amazon.

Come back often for more funny math and geeky comics.

Families of Functions

(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's a Not All in the Family moment.

Come back often for more funny math and geeky comics.

(x, why?) Mini: Asymptomatic

(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?).

Some values have no signs. And absolute values have no signs absolutely.

This comic was inspired by #asymptotic comic from a comic of days ago.

Ob-Math: For students who found this comic page, I'm curous -- HOW?. Anyway, the fuction on the right, g(x) is asymptotic, not asymptomatic. (Also, another note for students. There's no such thing as an asymptomatic function. That's the joke.)

Come back often for more funny math and geeky comics.

(x, why?) Mini: Asymptotic

(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?).

Keeping it within the lines.

This comic was inspired by #asymptotic trending a few nights ago along with #asymptomatic.

When is a function #asymptomatic? ... Well, maybe you'll find out on Wednesday! (I was about to waste a punchline.)

Ob-Math: For students who found this comic page, the function on the left is quadratic (or is it?), and the function on the right could be exponential or part of an inverse function. (It's hard to tell.) Both functions have been transformed from their parent functions.

Come back often for more funny math and geeky comics.

(x, why?) Mini: Evaluations

(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?).

These things always lead to an argument.

Okay, not really. That's just an expression.

Come back often for more funny math and geeky comics.

Hearsay

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

(C)Copyright 2019, C. Burke.

Don't confuse this with the rem(n) function!

I did just see the band last week, so, yes, I was thinking about math and comics during a concert.

And the "rem()" function, whatever that is, will have to wait for another day. Not that I'm likely to see them in concert any time soon.

Update: Bonus Comic

Hearsay being in the news right now, and my seeing REO Speedwagon in concert last week, several ideas came to mind, and I had to decide which one to use.
Here is another take on the song, more topical, but less mathematical.




Come back often for more funny math and geeky comics.

Cup of Tea

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

(C)Copyright 2019, C. Burke.

If you're an English teacher, this joke might not by your Cup(tea). But maybe it is.

It seemed more like a Mike joke, but I didn't think it was a good Sharon setup. And, so far, one character is an establish tea drinker.




Come back often for more funny math and geeky comics.

Happy Mothers Day 2018

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

(C)Copyright 2018, C. Burke.

Worry is a function of motherhood.

As for those transformations, if the parent function f(x) = x² is transformed into g(x), then g(x) = -(x + 11)² + 3

If the parent function h(x) = |x| is transformed into j(x), then j(x) = 2|x - 14)² + 2

(I realize that I haven't been doing Mothers Day for a while. I hope you can understand.)




Come back often for more funny math and geeky comics.

Problem of the Day: Quadratic and Absolute Value Functions

During the break, an online friend, who knows I'm a math teacher (the user handle @mrburkemath is generally a tip-off), sent me the following math review problem:

"Given the functions h(x) = |x - 4| + 1 and k(x) = x² + 3, which intervals contain a value of x for which h(x) = k(x)?
This was followed by a list of intervals in the form number < x < number. I've left these out as I have to assume the entire problem is copyrighted from a review book. I'm hoping my excerpt is covered by "fair use".

First thing I said what, in a purely multiple-choice format, just plug the two functions into a calculator (or use an online app, if you're at home without one) and look for the answers. Then select the intervals that include those values.

Otherwise, we can work it out. First thing to realize in that dealing with quadratics and absolute values, there can be, at most, two real number answers. (The problem didn't involve imaginary roots.)

The friend wasn't sure if she was supposed to set each equal to zero and solve, or set them equal to each other. It's in the question: h(x) = k(x).

So here is what we have:

h(x) = k(x)
|x - 4| + 1 = x² + 3

First thing is to isolate the absolute value, by subtracting 1 from each side:

|x - 4| = x² + 2

Many of you can skip the mini-review I'm going to do right now.
When solving an absolute value equation like |x - 4| = 7, you have to split the equation into two possibilities, one positive, one negative, and solve each.

x - 4 = 7

x - 4 = -7

Applying that rule to this equation, you get
When solving an absolute value equation like |x - 4| = 7, you have to split the equation into two possibilities, one positive, one negative, and solve each.

x - 4 = x2 + 2	x - 4 = -(x2 + 2)
x - 4 = x2 + 2	x - 4 = -x2 - 2
0 = x2 - x + 6	x2 + x - 2 = 0
No real roots	Factor (x + 2)(x - 1) = 0 x = -2 or x = 1

If you check the discriminant (b² - 4ac) for the first equation, you will get a number less than 0, meaning that there are no real roots.

Going back to the original problem, they wanted you to select any of the intervals that contained either -2 or 1 or both.

Had they asked for the points of intersection, the values of the functions for those x values, then you have to plug them in:
h(-2) = |-2 - 4| + 1 = |-6| + 1 = 6 + 1 = 7, (-2, 7)
h(1) = |1 - 4| + 1 = |-3| + 1 = 3 + 1 = 4, (1, 4)

Simple, right?

Function Transformation

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

(C)Copyright 2017, C. Burke.

Unlike math, technology can quickly become obsolete with its function becoming nothing.

Mike gets a chuckle every time he says ''question Mark''.




Come back often for more funny math and geeky comics.

Daily Regents: Functions (Movie Rentals)

I'll be reviewing a New York State Regents Exam Question every day from now until the Regents exams begin next month. At least, that is the plan.

June 2014, Questions 35

35. Caitlin has a movie rental card worth $175. After she rents the first movie, the card's value is$172.25. After she rents the second movie, its value is $169.50. After she rents the third movie, the car is worth $166.75. Assuming the pattern continues, write an equation to define A(n), the amount of money on the rental card after n rentals.

Caitlin rents a movie every Friday night. How many weeks in a row can she afford to rent a movie using the rental card only? Explain how you arrived at your answer.

First, "Explain" means in words; equations will NOT be enough information. I am not kidding.

Subtract 175-172.25 and you get $2.75. Check the next and the next to be sure, but you see that each rental is $2.75
That makes the equation A(n) = -2.75n + 175.

To answer the second part, make an inequality greater than 0, and find the largest whole number which makes it true. (Or make an equation equal to 0 and drop everything after the decimal.)

-2.75n + 175 > 0
-2.75n > -175
n < 63.63
Each movie costs $2.75 to rent. If you divide $175 by $2.75, you can get 63 rentals. There is money left over, but not enough for another rental using the card only.

At one rental per week, she can do this for 63 weeks before she runs out of money.

Any questions?

---

If anyone in Brooklyn is looking for an Algebra or Geometry Regents Prep tutor, send me a note. I have a couple of weekly spots available between now and June.

Daily Regents: Functions and Factoring

I'll be reviewing a New York State Regents Exam Question every day from now until the Regents exams begin next month. At least, that is the plan.

June 2014, Questions 30 and 31

30. The function f has a domain of {1, 3, 5, 7} and a range of {2, 4, 6}. Could f be represented by {(1, 2), (3, 4), (5, 6), (7, 2)}? Justify your answer.

I'm still not happy about the wording of this one. I've been told that I'm "splitting hairs", but, in my opinion, the question is asked backward.

The question is asking if the set given could be function f. In other words, is it a function and does it have the correct domain and range?

Let's answer that question.

The set of ordered pairs is a function because no x value is mapped to more than one y value. The domain (the x values) are {1, 3, 5, 7} as in f. The range (the y values) are {2, 4, 6} which matches the range of f. So, yes, it could represent f.

Remember that "yes" or "no" without an explanation is worth ZERO points.

31. Factor the expression x⁴ + 6x² - 7 completely.

Usually, a “factor completely” question has a Greatest Common Factor (GCF) component to it. This one doesn't, but it does have an extra step that isn't obvious at first.

If you are thrown off by the exponents being 4 and 2, instead of 2 and 1, that's okay. They will act pretty much the same way.

x⁴ + 6x² - 7 factors into (x² + 7) (x² - 1). You now have two binomials, each with an x² term. If you look closely, you can see that we aren't finished. We can factor further.

Using the Difference of Squares Rule (x² - 1) can be factored into (x – 1)(x + 1).

This makes the final answer:

(x² + 7) (x – 1)(x + 1)
Note that (x² + 7) has no real roots and cannot be factored further.

Any questions?

---

If anyone in Brooklyn is looking for an Algebra or Geometry Regents Prep tutor, send me a note. I have a couple of weekly spots available between now and June.

Movie Functions Quiz

(Click on the comic if you can't see the full image.)
(C)Copyright 2016, C. Burke.

I wanted to use this as a stumper "__(M + M) = 1939"

I easily had over 50 of these, but they were a bit repetitive after a while, which is why I added the years to the equations. If this is popular, then, as with all good movies and many bad ones, there will be a sequel.

The Answer Key will be up soon -- as soon as I find it. Yes, a couple of them are stumping me now and I wrote the thing!




Come back often for more funny math and geeky comics.

Function

(Click on the comic if you can't see the full image.)
(C)Copyright 2016, C. Burke.

The question you have to ask here is how do you have an "age of -3"? You would think that there was something wrong with the domain.

Also, when will we get Ultron squared, if ever?




Come back often for more funny math and geeky comics.

The Superman Function

(Click on the comic if you can't see the full image.)
(C)Copyright 2015, C. Burke.

You know the hardest part was finding a graph that would reasonably fit on the screen.

Alternatively, it could've been 0.5x⁵ + 1.5x⁴ - 2.5x³ - 7.5x² + 2x + 6.
A toss of the coin, really.

And, LOOK! It's a co-ordinate plane! But watch out when it's Super 3-D because evil things start to appear!

Come back often for more SUPER funny math comics. Okay?

Recursive Function Brewing

(Click on the cartoon to see the full image.)
(C)Copyright 2014, C. Burke.

f(0) is 'No more bottles of beer on the wall', but that's just sad.

You could call this a follow-up to T-test discussion about Guinness or a follow-up to the article about recursive functions. Either way.

And I just want to take a line to remember the 70th anniversary of D-Day. I didn't have anything special planned. It's not that I forgot when D-Day was -- I knew it was coming. I just forgot that today was the sixth of June until I was typing the date on today's comic. And considering that this needed to be short because I haven't finished packing, there wouldn't have been time to work on something worthy of the occasion. (In fact, some of you may be reading this while I'm on a plane to Denver for niece's wedding.)

Enjoy your weekend.

June 2014 Common Core Algebra 1 Regents Exam, Part 3

Update: I now have a Common Core Regents Review books available on Amazon.

This is the second thread dealing with the Common Core Algebra test, and probably the last for a while, as I'll be away from the weekend. The first thread was about Part 2 of the test. This post deals with part 3 and 4.

Algebra 1 (Common Core), Part 3

Questions is Part 3 are worth four points. Again, I won't assume the point distribution for partial credit.

33. Write an equation that defines m(x) as a trinomial where m(x) = (3x - 1)(3 - x) + 4x² + 19. Solve for x when m(x) = 0.

You need to multiply the binonials and then combine all the like terms.
(3x - 1)(3 - x) = 9x - 3x² - 3 + x = -3x² + 10x - 3
To that, add 4x² + 19.
You get m(x) = x² + 10x + 16. Don't forget to write it as an equation!

For the second part of the question, take the equation you found in the first part (they allow for consistent errors) and substitute 0 for x. The answer should be obvious, but I would still show the work -- especially because it says solve

m(0) = (0)² + 10(0) + 16 = 0 + 0 + 16 = 16.

Frankly, I thought this problem was easier (or at least less complicated) than some of the Part 2 problems.

EDIT: As you may have seen in the comments below, in my rush to complete this section before leaving for the weekend, I misread my notes in solving this problem, despite having typed it correctly above. My original instinct was that this problem was too easy, and now I see why. Generally, when I get the feeling that something is too easy, there's usually something I missed, and I go look for it.

You needed to find when m(x) = 0, so x² + 10x + 16 = 0. That's a simple quadratic equation to solve. What two factors of 16 add up to 10? That's 8 and 2.
So (x + 8)(x + 2) = 0
x + 8 = 0 or x + 2 = 0
Then x = -8 or x = -2.

So that would be a Conceptual error for this part of the question, which is more points off than a simple Computational error. It could have been worse. I might have been incredibly wrong and arrogantly displayed a lack of understanding of how functions work, but that would be entirely overstating the mistake made. In any event, I am told that some people like to make damning accusations under the guise of anonymity rather than step out into the light of day. But we don't talk about such things in polite company.

Now to continue.... End of EDITED SECTION

34. A rectangular garden measuring 12 meters by 16 meters is to have a walkway installed around it with a width of x meters, as shown in the diagram below (Note: This will be added later) Together the walkway and the garden have an area of 396 meters.

Write an equation that can be used to find x, the width of the walkway.

Describe how your equation models the situation.

Determine and state the width of the walkway, in meters.

Unlike the previous question, this was a majorly complicated problem, with more binomial multiplication.

The length of the rectangle is (2x + 16). The width of the rectangle is (2x + 12). The area of the rectangle is 396.

So (2x + 16)(2x + 12) = 396
4x² + 24x + 32x + 192 = 396
4x² + 56x - 204 = 0
x² + 14x - 51 = 0
(x + 17)(x - 3) = 0
x + 17 = 0 or x - 3 = 0
x = -17 or x = 3
Discard the negative value and you are left with the width of the walkway being 3 meters wide.

Hopefully, that's enough.

35. Caitlin has a movie rental card worth $175. After she rents the first movie, the card's value is$172.25. After she rents the second movie, its value is $169.50. After she rents the third movie, the car is worth $166.75. Assuming the pattern continues, write an equation to define A(n), the amount of money on the rental card after n rentals.

Caitlin rents a movie every Friday night. How many weeks in a row can she afford to rent a movie using the rental card only? Explain how you arrived at your answer.

First, "Explain" means in words; equations will NOT be enough information. I am not kidding.

Subtract 175-172.25 and you get $2.75. Check the next and the next to be sure, but you see that each rental is $2.75
That makes the equation A(n) = -2.75n + 175.

To answer the second part, make an inequality greater than 0, and find the largest whole number which makes it true. (Or make an equation equal to 0 and drop everything after the decimal.)
-2.75n + 175 > 0
-2.75n > -175
n < 63.63
Each movie costs $2.75 to rent. If you divide $175 by $2.75, you can get 63 rentals. There is money left over, but not enough for another rental using the card only.

36. An animal shelter spends $2.35 per day to care for each cat and $5.50 per day to care for each dog. Pat noticed that the shelter spent $89.50 caring for cats and dogs on Wednesday.

Write an equation to represent to represent the possible numbers of cats and dogs that could have been at the shelter on Wednesday. Pat said that there might have been 8 cats and 14 dogs at the shelter on Wednesday. Are Pat's numbers possible? Use your equation to justify your answer. Later, Pat found a record sowing that there were a total of 22 cats and dogs at the shelter on Wednesday. How many cats were at the shelter on Wednesday?

This is a lot of work for a system of equations. I heard some teachers complaining about it (one of whom has seen the scoring rubric), and I heard students tell me that this was so easy. The students explained to me how they did it, so they might be correct in how "easy" they found it.

We'll use C for the number of cats and D for the number of dogs. That makes 2.35C the amount spent on cats and 5.50D is the amount spent on dogs.

The equation for Wednesday is 2.35C + 5.50D = 89.50.

For the second part, substitute 8 for C and 14 for D, and see if it's a true statement.

2.35(8) + 5.50(14) = 89.50 ?
18.80 + 77.00 = 89.50 ?
95.80 =/= 89.50 X
This is NOT a possible combination.

The final part states that C + D = 22. We now have a system of equations. Multiply this equation by -2.35

-2.35C + -2.35D = -51.70
2.35C + 5.50D = 89.50
combine: 3.15D = 37.80
divide: D = 12
There are 12 Dogs, so there are 22 - 12 = 10 = C, 10 Cats.

* * * * * * * * * * *

Okay, that's enough for me. This is taking a lot out of me.

What are your opinions of all this?

June 2014 Common Core Algebra 1 Regents Exam, Part 2

Update: I now have a Common Core Regents Review books available on Amazon.

Today was the first ever New York State Common Core Algebra 1 Regents. No one knew what to expect. Sure, math is math, and Algebra is Algebra. What questions could they ask, right? Well, it’s not just a matter of knowing the material. Some of this was covered nearly a year ago and not revisited. Not everything in the course scaffolds into new topics; not every new topic spirals back into the old.

And then there’s the question of presentation. You can do practice problems until the kids’ pencils are worn to nubs, but if the test problems are suddenly presented in a different -- particularly in an odd – way, a young teen might freeze up and yield the opportunity to work it out.

A lot of the test came down to vocabulary, and not necessarily math vocabulary, and reading comprehension. If you could figure out what they were asking, you could figure out what the answer might be. Or should I say “is”. It should be “is”, but who can be sure?

Once again, I’ll be reviewing the test. I’m starting with the open-ended. We’ll spiral back to the multiple-choice in the coming days. Part 1 is shorter than the older test and Part 2 makes up for it. Big Time.

Note: I won’t even pretend to guess at how many points you’ll get for writing what, other than to say if it’s perfect, you’ll get full credit. But who can be sure what “perfect” means?

Algebra 1 (Common Core), Part 2

25. Draw the graph of y = SQRT(x) – 1.

If you put this in your calculator, you had to be sure to close the parentheses after the x. Otherwise, the “- 1” would be part of the expression beneath the radical.

The trick to remember here is that the domain is x > 0. You can’t use negative numbers. The y-intercept is (0, -1). You should have, at the least, plotted the points (0, -1), (1, 0), (4, 1) and (9,2) before drawing a curve through them. There should be an arrow on the right side of the curve because it continues to the right. There is no arrow on the left because the line starts with (0, -1).

26. The breakdown of a sample of a chemical compound is represented by the function p(t) = 300(0.5)^t, where p(t) represents the number of milligrams of the substance and t represents the time, in years. In the function p(t), explain what 0.5 and 300 represent.

I don’t know just how specific an answer they are looking for here.

• 0.5 is the rate of decay of the substance. It is the base in the exponential function.
• 300 is the initial amount of the substance. It is the y-intercept of the function and the co-efficient of the base.

27. Given 2x + ax – 7 > -12, determine the largest integer value of a when x = -1.

Confused? Don’t feel bad. Make sure you use x = -1 and not a = -1.
Plug is -1 for x and simplify the inequality before you do anything else.
2(-1) + a(-1) – 7 > -12
-2 – a – 7 > -12
-a – 9 > -12
-a > -3
a < 3
Remember to flip the inequality symbol when you divide by -1. If a < 3, then the largest integer value of a will be 2.

28. The vertex of the parabola represented by f(x) = x² - 4x + 3 has coordinates (2, -1). Find the coordinates of the vertex of the parabola defined by g(x) = f(x -2). Explain how you arrived at your answer.

The notation for this is confusing. And when my students see this, I know that they’ll want to solve something because of the equal sign, but it’s a definition, not an equation.

Others will look at this and think it’s a recursive function because we just reviewed those a few days ago. Sigh.

For every value of x, g(x) will have the same value that the f() function had when x was 2 less than it is now. So the entire parabola will shift two places to the right. That means that the coordinates of the vertex with be (4, -1).

There are more complicated ways of achieving the same result, which, for 2 miserable points, I hope that they aren’t looking for.

29. On the set of axes below, draw the graph of the equation y = (-3/4)x + 3. Is the point (3, 2) a solution to the equation? Explain your answer based on the graph drawn.

This seems to be the easiest, most straightforward question, so far. Okay, so it’s a graph. Do the graph. You have a calculator to help you, if you need it. The y-intercept is (0, 3). The slope is -3/4 – down 3, 4 to the right, make another point, down 3, 4 to the right, make another point, … when you’re at the end of the graph, go back up the other direction. LABEL THE LINE

(3, 2) is not a solution. How do you show this using the graph? Put the point on the graph at (3, 2). Label it (3, 2). Respond: (3, 2) is not on the line so it is not a solution to the equation.

Do NOT plug (3, 2) into the equation to check. That’s not what they asked for, so they won’t give you points for it.

30. The function f has a domain of {1, 3, 5, 7} and a range of {2, 4, 6}. Could f be represented by {(1, 2), (3, 4), (5, 6), (7, 2)}? Justify your answer.

This one led to a bit of a discussion in the Math Department. One side was quite sure of their superiority of knowledge, and the other side still wasn’t satisfied with the explanation. To put it plainly: I think I know what they are asking, but I’m not entirely sure. And I’ve learned in the past, you can’t always go for what you think they want – sometimes, you have to go with what they ask.

The argument boils down to semantics, really. Or maybe it’s syntax. I don’t know. I’m not an English teacher. However, I have a problem with the word “could”. Seriously.

Is this question asking if the relation they gave fits the domain and range of f? If so, the answer is YES. Or is this question asking if the relation is the ONLY POSSIBLE FUNCTION f? If that’s the case, it’s NO. We don’t know how f is defined. There is no mapping function. It could be that this relation represents f, but it might not be. Is that what it’s asking? Literally, yes, that is what it says, word for word. And yet I’m still not sure if that’s what they mean, and I’m not sure that my students will catch that meaning as well. Nuance? I don’t know. Maybe I’m overthinking it.

Another way for me to put it is like this: Could A represent B if A is only a subset of B?

Unfortunately, not all my students are native speakers, so I hope there isn’t a problem.

One thing I know: “Yes” or “No” without a good explanation will be worth nothing.

UPDATE: I spoke with a teacher who has been to training on how to grade these exams, and he had an answer key with sample responses and their point values. Basically, the answer is YES for reasons given above. When I explained my concerns about the wording, he thought I was splitting hairs. To be honest, I agree with that. That said, the Regents has been know to split hairs in the past.

31. Factor the expression x⁴ + 6x² - 7 completely.

They changed it up a bit. Usually, a “factor completely” question has a Greatest Common Factor (GCF) component to it.

x⁴ + 6x² - 7 factors into (x² + 7) (x² - 1). If you think that this seems a little simplistic for “factor completely” instead of “factor into two binomials”, you are not wrong.

That’s because using the Difference of Squares Rule (x² - 1) can be factored into (x – 1)(x + 1), making the final answer:

(x² + 7) (x – 1)(x + 1) Note that (x² + 7) has no real roots and cannot be factored further.

32. Robin collected data on the number of hours she watched television on Sunday through Thursday nights for a period of 3 weeks. The data are shown in the table below. … Using an appropriate scale on the number line below, construct a box plot for the 15 values.

Note: A picture of the table will be added later.

Put the 15 data values in order. The appropriate scale would be start at 1 and increment by .5.

The data are: 1, 1.5, 1.5, 2, 2, 2.5, 2.5, 3, 3, 3, 3.5, 4, 4, 4.5, 5.
Note: if you don’t have 15 values, you left something out. Also, your calculator will do all this for you -- but copy it ALL down on your paper anyway!

Your five-number summary is as follows: Min: 1, Q1: 2 (4th value), Median: 3 (8th value), Q3: 4 (12th value), Max: 5. Number the scale from 1 to 5, counting by .5. Plot these five points. Draw a box using Q1 and Q3, with a vertical line through the median. Draw whiskers from Q1 to min and Q3 to max.

Done.

And that will do it for Part 2, which is much longer than the Integrated Algebra Part 2.

Day 30 of 30: You Want a Piece of This Function?

This is day 30 of the 30-day blogging challenge. It's the checkered flag! Maybe I'll take tomorrow off.

I briefly thought about doing something concerned with the Indy 500, but it's almost 10pm ET as I write this and that race is long over. I'm sure the excitement isn't for those present, but it would be forced now. So, continuing from last night's discussion on functions, let's talk, by which I mean complain, about piecewise functions.

Okay, one Indy-related question:

Q: How many ways can 33 cars be arranged at the start of the race?

A: One. That's what Time Trials are for!

That could've been a comic for today had I had time to make it. Busy weekend. Busier with the cleaning and the grading. But back to functions.

Just explaining to a student what a piecewise function is is not a simple task. Explaining how to read one takes time and patience, along with repetition of the phrase, "when you see the comma, think 'when'". Oddly, I once said, "say 'when'", and it sounded like I was pouring beverages out. I had to switch that up.

Worst of all are the examples that they give. They make no sense whatsoever. They are purely abstract creations that you wonder if they might ever come up -- even just in another math class.

This isn't to say that piecewise functions aren't useful in the real world, or even in mathematics. I could even justify them in Algebra as opposed to waiting for Trigonometry/Algebra 2. But do they have to make them so confusing off the bat. (Hell, the name itself is confusing -- and might I add that it isn't even recognized by my spell checker!)

Examples of reasonable math functions that they could have brought up? First, the absolute value function, which looks like this:

"When" x is less than 0, you want to flip the sign, (i.e., take the negative of x because a negative of a negative number is positive). Otherwise, leave x alone. Note that the last condition has to cover all other possibilities in the domain. You don't want to leave a value out(*), and you definitely don't want to repeat a value, because then it won't be a function. (*) Yeah, there are times you'll leave something out, but not here, not now and not with absolute value!

Another one that they can use which makes for an interesting graph, but doesn't have any variables, is the Sign function, not to be confused with the Sine function:

Negative numbers return -1, positive numbers return positive 1 and zero returns 0. This was useful when I was programming computers, something that I'm glad I've done and something to which I'll refer often. Why not steer the kids in that direction if it's something challenging that might interest them? By the way, the Sign function would be the basis for some kind of trinary system of anything when binary gets boring.

So we have two good examples to start. So what do the books give us? Something like this:

Okay, maybe nothing that nuts, and maybe nothing with e or i in the exponent, but it might as well have been. Non-continuous functions that mean nothing even in the abstract.

On the other hand, finding relevant, relateable uses for piecewise functions was a little crazy. I could've tried my Financial Algebra textbook which is constantly trying to get the students to create some of these (and eventually did). And then there are the old standbys, which don't mean as much any more. I used to use the example of different phone plans when talking about systems of equations. This is easily adaptable into a piecewise function. There's a minimum charge for a certain number of minutes and then you have to pay, say, $.10 per minute after you used your allotment. There are two problems with this example: first, the minimum means that there will be a constant, a line with a slope of zero and no variable. Second, what kid in my class a) pays for their phone, or b) doesn't have unlimited minutes. Minute plans are already a thing of the past.

Pay phones are right behind them, but you can tell them that they're at the mall or the airports and they'll believe that they're there somewhere and just haven't noticed.

My other example suffers similar problems: if you live in an apartment and aren't allowed to have a washing machine, then you have to go a laundromat where you can do it yourself, or you can pay to have them do it for you. It's usually done by weight. I gave an example (and I haven't dropped off laundry in a long time) of a cost of $.60/pound with a $6.00 minimum, and let them figure out that you're paying for at least ten pounds whether or not you bring ten pounds. Same problem with the flat minimum and just as unrelateable.

On the other hand, if I put enough of these problems on the board, they'll see enough commas and say "When".

Like I am now because this is too much.

30 Days. When!

Day 29 of 30: Recursive Functions

This is Day 29 of my 30-day blogging challenge. I can see the checkered flag in the distance.

Common Core Algebra seems more concerned about functions and families of functions than the previous Integrated Algebra in New York State was. It used to be that it was all linear equations and if you saw f(x), you just think "y". They had their uses, but we didn't get to talk much about them -- there was already too much to talk about in that course, and we had a hard time making all of it happen.

Well, Common Core is no different. They just give you a different "all that" to cover, and they really do want you to cover it all. When it comes to functions, they not only want the standard linear, quadratic, exponential and absolute function, but also cubic, square root, piecewise and recursive. Some of them need to be graphed, some only have to be used for evaluating, for example, f(3) or f(-5).

I can get to piecewise another time. My students are getting the hang of them. Well, some of them are. Some of the time. Okay, so maybe not. And I don't blame them. I didn't have to deal with those until later on.

Recursive functions, on the other hand, I don't remember from math class at all. Seriously. I remember them from computer programming. This isn't to say that I hadn't seen them in a math class first before I programmed a recursive function, but I know what left its mark on my memory and where it took.

A recursive function is one that calls itself. It uses itself in its own definition. The two most obvious examples (and Common Core won't use obvious examples are:

Factorial: f(n) = f(n - 1) * n, f(1) = 1

Triangular Numbers: t(n) = t(n - 1) + n, t(1) = 1

To calculate f(n) when n = 5, you do the following: f(5) = f(5-1) * 5 = f(4) * 5. But you need to know what f(4) is.
Well, that's easy. f(4) = f(4-1) * 4 = f(3) * 4. Okay, so now we need to know f(3) ... and then f(2) ... and so on down to f(1), which we are given.
As I told me students: you might as well start with f(2) and work your way up. We'll have to do all that work anyway. Then find f(3), f(4) and so on. If you are programming a computer ... well, I'd probably do it the same way for simple things, but use a recursive function when the professor tells me to do so. Most of the recursive things I've ever coded worked fine with a For...Next or Do...End loop.

Likewise, t(5) would work the same way. The only difference is that I used the letter t for triangular because I already used f for factorial in the same example. Be careful! Students can actually get confused by this. (I'm not kidding.)

Real-World Connection: Forgetting about where factorials and triangular numbers may occur in your everyday existence, recursive functions are part of the real world. Take the function clean(n). Suppose you wish to evaluate this function for the value of n = floor. There's a problem with this because your wife may tell you that you can't clean the floor until you evaluate clean(counters). And you can't do that until you evaluate clean(cabinets).

Now keep in mind that this is just supposed to be an illustration of a recursive function. However, it isn't exactly a true parallel example. For one thing, one you can't calculate 5! (5 factorial) without first calculating 4! (4 factorial). On the other hand, you can clean the floor without cleaning the counters or the cabinets. However, in both cases, according to my wife, at least, you'd be incorrect.