If Exponent Rules Are Aspirin, Then How Do You Create The Headache?

This Week’s Skill

Exponent Rules.

Rules like these are too quickly abstracted, memorized, confused, and forgotten.

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We can attach to them meaning and purpose by asking ourselves, why did we come up with these shortcuts? If these shortcuts are aspirin, then how do we create the headache?

What a Theory of Need Recommends

Again, with Harel’s “need for computation,” students need to experience the “longcut” before they learn the shortcut. Otherwise it’s just another trick in the endless series of tricks students call “math class.”

Several people suggested the same in last week’s thread, of course. Ask students to calculate expressions like these the long way before discussing shortcuts the students may have noticed (or that you may have noticed as a member of the class also).

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Chris Hulitt was one of my workshop participants in Norristown, PA, and his group suggested an important addition to this idea. Ask students to calculate this expression instead

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Looks the same as the last, right? But whereas the last expression resolves to 16, this expression resolves to 1. That headache is a little bit sharper. How did this gangly mess of numbers result in such a simple answer? Could I have realized that in advance?

Again, this isn’t real world, or relevant, per our usual definitions of the term. And yet this approach may still endow exponent rules with a purpose they often lack.

Next Week’s Skill

Determining if a relationship is a function or not.

This is another skill that can become quickly instrumental (run a vertical line over the graph, etc.) and obscure why it is aspirin for a particular kind of headache.

Let us know your ideas for motivating the definition of a function in the comments.

What You Recommended

Tom Hall:

I think of my 6th grade students writing down the prime factorizations of whole numbers (Why learn that? Oh yeah, to improve number sense and seeing structure behind numbers, among other reasons). When you work with something simple like 24, writing out 2*2*2*3 is not so bad. When you up the stakes to something like 256, writing out 2*2*2*2*2*2*2*2 becomes annoying. It’s not so much a headache as a tedious process that any normal person would like to make quicker and easier. At this point, I discuss exponents as a means of communicating all of that multiplication without having to write it all.

About 
I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. More here.

29 Comments

  1. It’s a very CS approach, but I like to start by presenting functions as a way of getting at properties of objects. I usually use a group of students (sometimes just the ones in my class, sometimes the whole student body, whatever works) as the objects, and introduce functions like “age(___)”, “height(___)”, “numsiblings(___)” (however you choose to define “sibling”). Also throw in some more fun ones like “birdpoop(___)” (the number of times in the past year the student has been pooped on by a bird) and “driverslicensenumber(___)”, which is only defined on a subset of the “domain”.

    This definitely has its shortcomings, but I think framing it this way makes it clear why an input cannot correspond to two different outputs.

  2. This year, I reviewed the definition of function with family trees, and I really liked it. Given a family tree diagram, I asked students, “Who is the mother of X?” and “Who is the sister of X?” The former was simple, the latter created a headache … or maybe it would have if this was new material instead of review.

  3. I think the scientific notation dovetail is the best.

    That is, compare trying to do 5000000 * 2000000000
    with
    5 * 10^6 * 2 * 10^9
    or 5000000 / 2000000000
    with
    (5 * 10^6) / (2 * 10^9)

    You can even have it then pop up in bona fide real-life problems, if that’s your thing. I tend to use astronomy (let’s say your flying your spaceship to the edge of the milky way, how long will it take) but there’s lots of strong bio applications too.

  4. Hola! I’ve been reading your blog for a long time now and finally got the bravery to go ahead and give you a shout out from New Caney Texas! Just wanted to tell you keep up the fantastic work!

  5. Mike Lawler

    July 1, 2015 - 9:56 am -

    Interesting coincidence with this topic – here’s my interaction with exponents just in the last two days (plus two older fun projects).

    (1) My older son and I had a 3 hour drive yesterday and we were talking a little math for part of it. Specifically, we were talking through some problems from the 2005 AMC 10a (picked at random). I like talking through these tests because you never know what topics will come up, and we had a great conversation about exponents starting from this problem:

    http://artofproblemsolving.com/wiki/index.php/2005_AMC_10A_Problems#Problem_11

    “A wooden cube n units on a side is painted red on all six faces and then cut into n^3 unit cubes. Exactly one-fourth of the total number of faces of the unit cubes are red. What is n?”

    From this easy to state problem, you get lots of different exponential conversations and also get to see the rules you list above in action with specific integers as well as with the abstract “n”

    (2) A couple of days ago lots of people on Twitter linked to this SMBC comic:

    http://www.smbc-comics.com/index.php?id=3777

    The last frame is what people found intriguing: “The harmonic series grows to infinity unless you through out the numbers with a 9 in the denominator”

    Although a detailed discussion of why either of those two series behave the way they do probably isn’t that interesting to most kids, the idea that most numbers have a 9 as one of their digits probably a little surprising. Talking through that idea involves exponents (at least the way we talked through it) and made for a fun project yesterday morning:

    https://mikesmathpage.wordpress.com/2015/06/30/counting-and-a-fun-harmonic-series-fact/

    (3) Calculations with exponents come up all the time in my work. Sorry to get finance wonky for a second, but . . . .

    Just an hour ago I was looking at the present value of a series of cash flows. The problem was essentially that I knew the value was $1,000 three months ago and I needed to know what it was today. Also, about $30 of the cash flows happened in the last three months and they weren’t around any more.

    Interest rates have gone up in the last three months so I was expecting the value today to be less than it was three months ago. My little model did indeed produce a lower number – call it $980. But something seemed wrong.

    When you spend time doing these sorts of calculations for work you pick up little tricks – tricks that involve approximating exponential functions – that allow you to do quick approximations. The little mental arithmetic I’d done ahead of time told me the value should be more like $950. That little bit of extra familiarity with exponential functions helped me notice and then find the mistake in my model and avoid a little “whoops” moment at work. Again, this was just an hour ago.

    (4) Away from the last couple of days we’ve done a couple of really fun project that have involved exponents. The first was a conversation about Graham’s number. This is a great way to explore exponents because the number you are studying is so mind bogglingly big:

    Just talking about Graham’s number:

    https://mikesmathpage.wordpress.com/2014/04/12/an-attempt-to-explain-grahams-number-to-kids/

    Calculating the last 4 digits of Graham’s number:

    https://mikesmathpage.wordpress.com/2014/11/28/the-last-4-digits-of-grahams-number/

    Finally, Evan Schwartz’s book “Really Big Numbers” was the source for one of the best conversations that we’ve had about exponents:

    https://mikesmathpage.wordpress.com/2015/05/02/a-few-projects-for-kids-from-richard-evan-schwartzs-really-big-numbers/

    So, I guess that I agree that introducing and then memorizing a bunch of exponential rules isn’t really that interesting. But there are neat properties of numbers and neat problems involving exponents that can make fun projects which allow for a little sneaky review / practice of exponent rules. I think kids will find these type of project to be much more interesting than memorizing.

  6. Functions! Ok, that’s probably the most literal time I make headaches. I’m not sure I can pull this off without pictures, but I’ll try my best.

    First part:

    I show that one graph you threw up once showing water use in Canada during the Olympic hockey game, and how everyone in the country was flushing their toilets at the same time.

    I then show the same graph turned sideways. It really, really, hurts.

    People naturally want the y axis to reflect a single answer. Showing multiple graphs with the x and y axis flipped can make one psychologically quesy.

    The next part:

    I have this lesson where students pick a particular make and model and get prices from newspapers and plot “years old” versus price. It is kind of interesting because this is NOT a function — the same 2004 car might be $4000 or $8000 depending on condition — but the end result is something hard to read, and in practical terms people want a simple input -> output — if the car I want is this old, what average should I expect to spend?

    In other words, almost without prompting, students want to turn a graph which has lots of data points for each x value into one with a single data point for each x value. They want to turn it into a function.

    Now I’m ready to introduce functions formally.

  7. Hi Dan,

    I don’t know if you have been to PCMI or not, but I think Bowen and Doug do a great job of introducing problems which get tedious quickly but are based on some minimal context.

    For example, a couple of weeks ago we were looking at a card shuffling problem and within a couple of days, everyone I was working with had a great need to be able to calculate 2^n mod p in a time-efficient way.

    So then this leads me to wonder; is there some context which involves multiplying powers of numbers together and can this task get tedious enough quickly enough that problem solvers want to look for a short-cut. I think combinatorics would be the right area to think about increasing powers of different bases multiplied together, but exactly what, I’m not sure.

    I’m going to think about it and maybe get back to you.

    David

  8. Going back to quadratics… if it’s a headache to randomly guess the zeros of a function (which I agree it is), why is learning to factor much better? It still involves a lot of guessing for most students and it doesn’t work for all quadratics. Guessing factors is a headache. Why don’t we always use the quadratic formula? It’s fail-proof- you can even use it for non-integer factors and imaginary numbers. The main downside I see is that the quadratic formula may be a little heavy handed in cases where the factors are obvious (to math teachers) like x^2+3x+2. I don’t think factoring is worth the pain it causes students.

  9. The headache for exponents is Calculus. Once kids hit calculus they need to be able to manipulate their exponents as easily as they add, subtract, multiply and divide.

    The problem is that they learn exponent rules as a chapter. Learn the rules.Move on to the next chapter. Forget the rules. Sure, they see them in quadratic equations, but nothing like the constant stream of them that occurs in calculus.

    Here’s what I noticed with my children: when they were introduced to exponents they still weren’t very comfortable with the idea of generalizing. In other words, when the rules were explained like a^m *a^n = a^(m+n) it didn’t really click. It seems like an opportunity is lost here as it is really so easy to figure out the laws of exponents by examining examples that use numbers. So if a student sees examples that look like this 2^5 * 2^7 = 2 ^12 they will figure out the generalization for themselves if given the opportunity.

    The rules of exponents are easy for teenagers to forget. What seemed to work for my son was having a complete set of samples of exponent operations using numbers, which was on the same “foldable” note paper that reminded him of the power rule, chain rule, quotient rule etc.

  10. Lisa Soltani

    July 2, 2015 - 4:28 am -

    It seems to me that the harder question to answer isn’t what’s the headache for the exponent rules, but what’s the headache for expressions like: 3^5*2^4*3^2/3^4*2^2*3^3.

    I like for students to think of these problems as puzzles. I think they can’t be inherently fun, but then again I am a math teacher. Otherwise, I find myself saying things like, “you’ll need this later” when …

  11. I commented on Zack’s post (linked above) yesterday regarding the definitions.
    here it is:
    “If you want to minimise the problems with 2^0 and negative powers, consider defining 2^3 as 1*2*2*2 and so on. This is quite logical, and better than 2^3 is 2 times itself 3 times.
    1*2*2*2 reads 1 multiplied by 2 three times
    Then 2^0 is 1 multiplied by 2 no times
    and negative powers are then so obvious that it’s painful.”

  12. Seems that people still want to talk about exponent rules.
    Okay, so one of my headaches is that these aren’t rules, at least not for integer exponents. I know it isn’t stated that the exponents are integers, but the use of variables n and m is suggestive, by tradition.

    Sticking with integer exponents for the moment, these are actually properties and I think properties can get discovered, but sometimes we need a reason to go looking.

    As a general class of headache, this is: can we extend something we already know into a new domain? A classic mathematician habit, but also one that is really delightful when done well by either a student or a cutting edge researcher.

    So, my proposed headache for the students is: assume a>0, does a^q have a meaning, when q is a rational number? If it does have a meaning, what is the meaning? That probably leads to some backtracking:
    – what does a^n (n a positive integer) mean?
    – what properties do we notice about exponents for positive integers? these are the rules in your table.
    – what about a^0?
    – if we want to be consistent with those rules/properties, what would a^n (for n a negative integer) mean?
    – Now, can we use those properties to extend to rational numbers?
    – what ideas do they have about extending this to all real numbers?

    Now, this might seem very theoretical/axiomatic, but I’ve done it successfully with 10th grade students by first working through an analogous issue: multiplying by negative numbers. The progression:
    – when m is positive multiplying n*m can be considered repeated addition of n (I said “can” btw)
    – when all are positive, we have a distributive property:
    (n+m)*j = n*j + m*j
    – What about 0? think (n+0)*j
    – What about -n (n still positive)? Think (n – n) *j
    This tends to go over really well because it gives them a familiar result (something they probably memorized) and now makes them feel that they’ve actually understood why it should be that way.

    With that feeling in place, they are ready for the exponent extension discussion.

  13. Jan McNulty

    July 2, 2015 - 4:55 am -

    What are your “feelings” about the need to simplify radicals these days. Is this math that should become obsolete. I certainly use a lot of properties of exponents to teach this but have been struggling of late with the necessity. I love giving the students a history lesson in life without calculators (which unfortunately I am old enough to understand).

  14. @Jan

    This is now almost entirely omitted in our curriculum although I make a point of covering at least the basics. There isn’t much need at all in my opinion unless someone happens to give you a problem with radicals in it, but I hardly call that realistic or meaningful. I typically take a while with kids ‘unlearning’ radical expressions when we get to calculus anyway, so I feel the same was as you regarding necessity.

    @Dan

    Have to disagree regarding the Norristown question. You can argue whether the answer is vaguely trick-like(it is) but the more important aspect is that you’re telling kids to take one of those fictional 4-function-calculators which none of them own. Why do this? It’s an artificial problem limitation that would be better served by simply asking them to do it sans calculator at all, rather than deliberately not using at tool we’d have already been encouraging them to make the most out of all along.

  15. Is a relation a function?

    Well, why do we care if something is a function anyway? For me the key is when I am making a definition and want to know that it is “well defined,” that there is a meaning to my definition and it is unambiguous.

    Continuing from the exponent example above, say that I have fixed a>0 and want to define f(q) = a^q when q is a rational to be (a^n)/(a^m) for n and m integers with q = n/m. Is f() a function?

    A priori, we should probably guess it isn’t. We have made a big choice of n and m from all the pairs that could be used to represent q, so why should our result be independent of that choice? Of course, we’re fine in this case.

    As another example, we can explore function inverses geometrically, as a reflection of the x- and y-axes across the x=y line. Given a function, then, I can verbally/visually define the inverse without having to be explicit.

    However, do I always end up with something that is well defined? Of course not and it depends on the function you are considering as well as the domain over which you are viewing it. The x^2 and sqrt(x) pair is a core example. So tempting to be lazy and say sqrt(x) = y is defined to mean that y^2 = x.

    Ok, so that’s a mathematically mature take. Here’s a slightly different direction you probably weren’t expecting:

    What’s the area where everyone has a badly defined relation that they are treating like a well-defined function? In racial/gender/etc biases. It is only a slight caricature so say that we form our definitions by going through this process:

    (1) identify a distinct collection of people, say group X?
    (2) Sample a couple of individuals from group X
    (3) What do we perceive to be their common characteristic? Call it Y.
    (4) Then, make the following function definition: f(X) = Y. Or, for all z in X, belief(z) = Y.
    Repeat for other groups of people (other Xs).

    Isn’t that a terrible (and terribly common) non-function worth calling out?

  16. My favorite application* for the tool** x^m / x^n = x^(m-n) is 3^1003 / 3^1000.

    * “headache” is so unpleasant. Is that really a good association for our subject? I am a programmer, too, and my day is filled with puzzles, many of my own sloppy creation, and with many in my field I cannot believe I get paid to do this. Each day is a day of play, hardly headaches.

    ** more accurate I think than “trick” or “shortcut”. Tool creation is a hallmark of intelligence. When we discover a pattern with simpler ratios and can express it as the formalism x^(m-n), we have a tool that can save us when PEMDAS lets us down (forcing us to compute 3^1003 before dividing). And that is not all.

    Once we conceive a possible new tool, as good little mathematicians we must validate it before adding it to our toolkit. Does it work for any value of X? Is there an edge case out there we need to add as a constraint on the tool’s use? Shucks, in some cases we may decide the tool simply does not hold — a regularity we observed was just over a small sample where it happened to hold.

    While validating our tool, a good detective will explore the case where m=n. Hang on, what is X^0?! Well, our tool says it is 1. What do mathematicians say? Hang on. Is 0^0 = 1? Which edge case prevails?

    Another challenge comes from n>m. 3^1 / 3^3 = 3^-2?! Let’s expand each power…OK, 3 / 27 = 1 /9…hey, that’s 1/3^2! Are we saying x^(-n) = 1 / x^n? Wow, we have another tool to validate.

  17. I have very little comment on your comments. Learning a lot. Thanks for the links, especially.

    Joshua Greene:

    Well, why do we care if something is a function anyway? For me the key is when I am making a definition and want to know that it is “well defined,” that there is a meaning to my definition and it is unambiguous.

    So far most people who have tackled the function question have identified its lack of ambiguity as its key advantage, its primary capacity for pain-relief (if you will).

    If that’s true, then the teacher should try to put students, at least momentarily, in a place of ambiguity.

    How do we do that? I’m enjoying thinking about several of your suggestions.

  18. Shawn Thomas-Royster

    July 4, 2015 - 6:49 am -

    I have always used the idea of mechanically functioning machines help students understand functions. The reason we use functions is so we can use its patterns to predict outcomes. In other words, you use a function because you want to be certain that if you put in a particular input you will be able to confidently determine the outcome.

    So let’s take a soda machine. If you put your money in and the press the button labeled coke, you expect a Coke to come out. -What if a Coke does come out? Do you know that the machine is functioning properly?
    -Now imagine that you press that same button again (with a new dollar of course), and out comes a Sprite. Is the machine functional or not?
    -Now imagine that the first time you press the Coke button you get a Sprite, but every subsequent time you press the Coke button you always get a Sprite. Is the machine functioning properly?
    -What if both the button labeled Coke and the button labeled Sprite both consistently give you a Sprite? Is the machine functioning?

    I generally let the students discuss which situations show the machine is functioning properly (in a mechanical sense).

    The students then come up with the definition that it is a function if a certain input always gets you the same output. They also realize that it is okay if two or more inputs happen to yield the same output, as long as they each do so all the time. Parenthetically, they might also point out that human error can make a function look like it’s not a function. :)

    This helps when we figure out what the vertical line tests is helping you to look for. It also connects later when we look at scatter plots and lines of fit, as it is our attempt to force a non-function into the closest function possible so that we can make more confident predictions about outcomes.

    p.s. – I sometimes use the buttons on a cellphone, in addition to the button on a soda machine, to further the discussion.

  19. Adam Poetzel

    July 4, 2015 - 12:40 pm -

    One fun way to create a small “headache” around the concept of a function involves explorations with a motion detector. Students begin my matching distance time graphs that are all continuous functions (although we don’t use that vocabulary yet). Through that activity, they begin to deepen their understanding of reading two axes of the graph and articulating their relationship. After some success, give them a graph to match like a loop. Many of them will try and reproduce the graph by spinning around. AFter some engaging unsuccessful attempts, students realize it is impossible. When pressed to explain, they usually start saying things like “you can’t be in more than one place at one time”. Bingo. That connects perfectly to the idea that graphs that can be made by a motion detector are special graphs called functions. Functions are special relationships in which each input (time) has exactly one output (Distance from the motion detector).

  20. I quite like the idea of linking functions to distance/time graphs as pupils can generate fairly simple and personal connections to a formal mathematical concept, however I see two problems with this approach:

    1) The teacher is still essentially telling the rules – ‘impossible motion graphs are not functions, all others are’.

    2) Creativity is discouraged – I love distance time graphs, I think a good cue that pupils really understand what is going on is when they can link elaborate stories to even the most bizarre looking graphs. When defining functions you need to limit some potential interesting interpretations, i.e. ‘no time travel allowed’ (one-to-many mapping), whilst permitting others, i.e. teleportation can be ok…as long as it’s continuously near-instant (step functions).

    In this sort of situation when dealing with the formalities of mathematical definition I’m happy for that to be it’s own headache. My approach usually is along the lines of:

    -A simple table, headed ‘is a function’ and ‘is not a function’
    – A couple graph diagram examples in each to start
    – A host of more graphs to try to predict and sort (make sure to include lots of nice subtle differences, e.g. step function vs step function with little gap, x^2 vs sqrt(x), sin/cos/tan etc.)

    From there you can tease out what sort of rules pupils were trying to use, what did they notice and how can they describe it (you might get things along the lines of ‘there can’t be gaps’, or ‘it can’t go back on itself’). Once the final sorting has been decided get pupils to see if they can put into more formal definitions, little cues like writing ‘input’ and ‘output’ on the axes to start with and then later replacing with ‘x’ and ‘y’ might help. Get pupils to draw their own after (and maybe even try matching to a story!).

    If you really want to get fancy then from here you might want to repeat similarly introducing different domains (e.g. only select integers, coordinate pairs, complex numbers), headings for ‘injective’, ‘surjective’, ‘bijective’ , matrix or 3D functions etc.

  21. I think it may be ill-advised to search for these “headaches” without a bit of a look at the history of mathematics. If you want to know why someone today would care to learn about a mathematical idea, maybe figure out why the first person started to care about it.

    Historically, as I recall, the notion of “function” was developed for purely foundational reasons, centuries after differential calculus was practically all fleshed out without it. My suspicion is that the main reason the “function” idea is developed so carefully in algebra classes nowadays is simply because “they’ll need it for calculus.” (Which wouldn’t even be true, if we started calculus with Leibniz and Newton’s beautiful original vision, instead of the hyperformalized modern stuff.)

    But cynicism aside, I think some basic calculus-like questions could be good headaches for the aspirin of functions.

  22. How about a classical headache? If I have a function, there aren’t more objects in the range than in the domain. We can use functions to explore cardinality, how many objects are in a set.

    It’s easy to know how many numbers are in the set {1, 2, 3, . . . n} so if I make a function (and make sure the inverse is a function) from any set to the counting numbers, the size of the set will be the destination of the last element. How many 5-digit multiples of 3 are there? Well, what function maps 10002 to 1 and 10005 to 2? One that divides by 3 then subtracts 3333. Then 99999 -> 30000, so there are 30000 of them. I can use functions to count all sorts of sets.

    If I can define a function with an inverse from the integers to the counting numbers, I’d know those sets have the same number of elements, that is, the same cardinality, and if not, there must be ‘more integers’ than counting numbers. How about the rationals or the reals then? Is “infinity times infinity” from the AT&T commercial really mind-blowingly larger than infinity?

  23. Mary Anne Hardy

    July 9, 2015 - 6:00 am -

    I too struggle with the best way to introduce functions in my algebra one classroom without being too abstract in my reasoning. Unfortunately I found this year my 9th graders “learned” about functions in 8th grade and memorized cute little rules such as functions are “one to one” or “many to one” but that “one to many” is not a function. And they learned to say it’s a function because it “passes the vertical line test”. But they can’t tell me what any of those phrases mean and how it relates to an overall understanding of “what is a function?”. I also like the comment that Dan appreciated:
    .

    Joshua Greene:

    Well, why do we care if something is a function anyway? For me the key is when I am making a definition and want to know that it is “well defined,” that there is a meaning to my definition and it is unambiguous.

    As I work on my new blog sharing my experiences using “interactive student notebooks” for the first time in my classes I am pondering how to introduce functions with my 9th grade Algebra One that gives them more understanding than they had developed in 8th grade. I like the Khan Academy introduction https://www.khanacademy.org/math/algebra/algebra-functions/evaluating-functions/v/what-is-a-function to look at it mathematically (and circles are unambiguous and well defined so they kind of conflict with the above justification of functions…). Then from there I bring it down a notch and do my “Ice Cream Shoppe” view of functions. Which I’m currently trying to illustrate and will share here later. It’s cute and helps makes sense of what it means to have an input produce only one output.

  24. Mary Anne:

    Then from there I bring it down a notch and do my “Ice Cream Shoppe” view of functions. Which I’m currently trying to illustrate and will share here later. It’s cute and helps makes sense of what it means to have an input produce only one output.

    Love to see it, though there seems to be something categorically different between “creating a headache” and “creating a metaphor.” Another example of a metaphor.

  25. There is a neat little rational exponents on the Shell Centre website, under that “Improving Learning in Mathematics” module that Dan recommended years ago and which I still use (the British precursor to the Math Assessments Project I think).

    The students usually already know the exponent rules by then, but the lesson starts by having them extend them to rationals by asking stuff like:

    Rewrite 2*4 = 8 only twos as bases and whatever exponents are needed, so (2^1) * (2^2) = (2^3)

    And then more complicated stuff like

    16*(1/4) = 4

    And then finally

    2^? * 2^? *2^? = 2

  26. Sally Cosgrove

    August 1, 2015 - 11:19 am -

    http://imgur.com/gallery/CBRUP

    Have you seen this meme and argument?
    360 million dollars is 1 million dollars to 317 million people and have 43 million left? Hilarious and scary.
    It would be a good “headache” to show the students and have them prove for/against using exponent rules. Perhaps a fun Socratic seminar.

  27. Very good hook. Favorite student comment, “I want my 30 minutes back.” One class made up a problem for me. That worked as an extra hook.