Teaching for Tricks or Sensemaking

Here are two approaches to teaching zero exponents that are worth comparing and contrasting:

First, a Virtual Nerd video:

Once you have your non-zero number or variable picked out, put it to the zero power. Now no matter what number or variable you picked, once you put it to the zero power, I know what the answer is. One. How do I know this? Well in math, if we put any non-zero number or variable to the zero power it always equals one. No matter what.

Second, Cathy Yenca’s activity, which has students completing the following table:

160921_1

One approach will lead students to understand that math is a fragile set of rules that have to be transmitted and validated by adults. The other will help students realize that these rules are strong and flexible, and exist to make math internally coherent, with or without any adults around.

BTW. This is as good a time as any to re-mention Nix the Tricks, the MTBOS’s collection of meaningless math tricks and great strategies for teaching those concepts with meaning instead.

BTW. Check out Cathy Yenca’s own post on the comparison.

BTW. Check out the comments on that YouTube video. Interesting, right? What do we do with that?

Featured Comment

Tracy Zager:

I wrote a story about this exact moment in my book. Cliffs notes version: 8th grade. Mr. Davis told us the rule a^0=1. I questioned why. “Because that’s the rule.” I said, “But why?” Stern voice now. “Because that’s what’s I just told you.” The first boy I’d ever kissed said, “Give it up T! It’s in the book, that’s why!” Everybody laughed. Me too, on the outside. Not on the inside. On the inside I was angry and frustrated and humiliated. If I write my math autobiography, this moment goes in it. It’s one of the moments math lost me.

By the way, my favorite way to see the pattern Cathy shows here is with Cuisinaire rods. You literally go from cubes to squares to rods to single unit squares. It’s this amazing moment to see one as the fundamental unit.

But I didn’t get to see that in 8th grade. I didn’t see that until my late 30s.

The saddest part? Most kids don’t try again after they’re burned. They never come back. The not-quite-saddest-part-but-still-sad-part? I doubt Mr. Davis ever learned why either.

That’s the biggest legacy of that story to me, as a teacher. How do we handle the moment when it becomes clear, in front of the class, that we don’t understand some math we thought we understood? Do we handle it like Mr. Davis did, and view kids’ questions as challenges to our expertise and authority? Or do we say, “You know, your question is making me realize I don’t understand this as deeply as I thought I did. That’s awesome, because now I get to learn something. Let’s figure it out together.”

Kent Haines:

Something that has been effective for me (that I mention in the video above) is to really emphasize that 1 is the origin and invisible starting point of all multiplication problems.

Scott Farrand offers another helpful way for students to make sense of zero exponents:

There’s a truly great old lesson on exponentiation that I believe comes from Project SEED, that has been used with amazing success in hundreds of elementary classrooms.

education realist brings up a Ben Orlin post that includes a) a beautiful technique for teaching lots of rules of exponentiation and b) this beautiful paragraph:

Math’s saving grace, though, is that it can make us feel smart for another reason: because we’ve mastered an ancient, powerful craft. Because we’ve laid down rails of logic, and guided a train of thought smoothly to its destination. Because we’re masters—not over our peers, but over the deep patterns of the universe itself.

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.

50 Comments

  1. I recently went over this with my 13-yr old, and we went over division of a factor with different exponents.

    So 2^3 / 2^2 = 2^1, etc. So 2^2 / 2^2 should be 2^0 = 1. And that’s why you can’t do 0^0 — because you’re dividing by zero!

    I like Cathy’s activity

  2. I wrote a story about this exact moment in my book. Cliffs notes version: 8th grade. Mr. Davis told us the rule a^0=1. I questioned why. “Because that’s the rule.” I said, “But why?” Stern voice now. “Because that’s what’s I just told you.” The first boy I’d ever kissed said, “Give it up T! It’s in the book, that’s why!” Everybody laughed. Me too, on the outside. Not on the inside. On the inside I was angry and frustrated and humiliated. If I write my math autobiography, this moment goes in it. It’s one of the moments math lost me.

    By the way, my favorite way to see the pattern Cathy shows here is with Cuisinaire rods. You literally go from cubes to squares to rods to single unit squares. It’s this amazing moment to see one as the fundamental unit.

    But I didn’t get to see that in 8th grade. I didn’t see that until my late 30s.

    The saddest part? Most kids don’t try again after they’re burned. They never come back. The not-quite-saddest-part-but-still-sad-part? I doubt Mr. Davis ever learned why either.

    That’s the biggest legacy of that story to me, as a teacher. How do we handle the moment when it becomes clear, in front of the class, that we don’t understand some math we thought we understood? Do we handle it like Mr. Davis did, and view kids’ questions as challenges to our expertise and authority? Or do we say, “You know, your question is making me realize I don’t understand this as deeply as I thought I did. That’s awesome, because now I get to learn something. Let’s figure it out together.”

  3. I’ve always (4 yrs) taught the topic this way and unfortunately haven’t found it to be any more effective at helping kids use zero power exponents several months down the road. Perhaps the method itself is the best we’ve got, but I need to continue working to create an environment where kids are used to thinking through things on their own in the future so they can recreate the work Cathy has done here on their own in order to arrive at the correct interpretation.

  4. There is another way:
    2^3 = 8 so 2^2 = 4, divide by 2 and get 8/2 = 4
    2^2 = 4 so 2^1 = 2, divide by 2 and get 4 /2 = 2
    2^1 = 2, so divide by 2 and get 1
    and it is simple to go further and get 2^-1 = 1/2

  5. Something that has been effective for me (that I mention in the video above) is to really emphasize that 1 is the origin and invisible starting point of all multiplication problems.

    What I mean is, with addition and subtraction we start with 0 and then move up or down the number line based on the problem. So every problem such as 6 + 5 can be thought of as 0 + 6 + 5. It is not a coincidence that adding 0 doesn’t change the value of a number. Zero is the additive identity and the starting point of all addition.

    But multiplication is a different operation than adding. It is about scaling up and scaling down from an original size. Multiply a number by 6 and you scale up to 6 times its original size. Multiply by 1/3 and you scale down to 1/3 of the original size. But if you multiply by 1, nothing happens. It is the multiplicative identity.

    So what is the invisible starting point of all multiplication? What should be this “original size?” It can’t be 0 because anything multiplied by 0 is 0. So why not the multiplicative identity? Indeed, it works. Any multiplication problem like 5*6 can be thought of as 1*5*6 without changing any answers.

    Then with exponents, a problem like 3^4 becomes 1*3*3*3*3 because you multiply the starting point by four 3s. Similarly, 3^0 becomes 1, because you multiply the starting point by zero 3s.

    So I don’t say in class, over and over, “anything to the power of zero is one.” I say, over and over, “One is the starting point of all multiplication.”

    This is a better mantra for my students because it helps them remember/understand both zero exponents and negative exponents.

  6. I use a clone of Kathy’s activity as an intro when I do the intro for exponent rules. Then, about a half hour later, right as it’s kind of slipped their mind, I give them an exponent multiplication problem with a zero exponent.

    The look crosseyed for a bit, and then decide that the rules mean they should just add the two exponents, and get the same non zero exponent.

    At this point, I ask them why they got the same value that they were multiplying by, and get a good solid minute of “oooooohs” from around the room as the almost forgotten info reconnects with what they see in front of them.

  7. If students are familiar with expanded form and the shortcuts for exponent operations, you can also show them that x^0=1 is consistent with everything else they’ve learned: x^5/x^5=xxxxx/xxxxx=1=x^(5-5)=x^0

  8. I do an area model approach with rectangles. I start with a 2×1 rectangle and we double the area for a few terms, drawing each new rectangle to the left of the previous rectangle, similar in fashion to a number line, (graph paper works well for this) and establish the idea that the areas are being multiplied by two. Students write the area of each rectangle as a string of multiplication by 2.
    Ex. rectangle of area 8 = 2x(2×2). This leads to writing the area as an exponent.

    Then the fun part! I ask the students, what if we went the opposite direction? Instead of doubling the area, what if we halved the area? This leads students to draw a square with the area of 1 (following the /2 pattern). Then they draw rectangles with areas of 1/2, 1/4, 1/8, etc. Write those as a string of multiplications, then exponents .

    They observe the patterns starting at the far left; 2^3…2^2…2^1…2^0… Students then extend the pattern to their negative exponents.

    It’s not “magic”, but it is a visual way of introducing non-positive exponents.

  9. In addition to Cathy’s method (which is more intuitive) I show the rule for zero by using the subtraction rule for powers (which is more proofy):

    (1) a^m ÷ a^m = a^(m-m) = a^0

    (2) a^m ÷ a^m = 1

    (3) So a^0 must also equal 1 since we started with the same expression in each case.

  10. The biggest takeaway is to compare the approaches to teaching.

    Exponent rules are among the most difficult to teach because students almost never have experience using them. They are always the abstract concept of (x^3*y^2)^2/(x^2*y^(-2))…. I think a lot of people do some type of patterning, but our charge (with tools like Desmos) is to create rich activities that really get students to understand that raising a power to the 0 is 1.

    Even with the pattern, what concrete models do we create for students for zero and negative exponents? I teach 8th grade and community college calculus and the lack of understanding what 2^(-1) really means is the same at both levels. Did they all have poor teaching? Probably not. My guess is that even when we teach by a “pattern”, we rush to the rule.

    Unfortunately, time is always working against us. Desmos provides an opportunity possibly to create a “series” of slides that maybe could create more “stickiness” with this topic.

    By the way, I just learned about labs and slides from Cathy’s Activity! Wow, that opens up a whole new set of questions!! Excited to start working with that!

  11. Scott Immel says “My guess is that even when we teach by a “pattern”, we rush to the rule.
    Unfortunately, time is always working against us”

    I definitely struggle with this. I see it over and over. I give enough examples for a few students to discover the rule, not most/all. Then for most students, it’s just a trick. It’s so hard to slow down, and what do you do with the ones who see it and want to share it? Many students have been taught with tricks for so long that they are eager to hear it, not realizing it will be soon forgotten, and stunt their long-term understanding.

  12. We just did this in my Algebra I class of 8th grade students this week! I showed them two different reasons why (as I could reason) a^0 = 1. First we based it on our work with division of numbers in exponential form 3^2/3^2 = 3^0, by subtracting the exponents, which was a “shortcut” we discovered in previous lessons; and 3^2/3^2 = 1 because anything divided by itself is 1. We had a good discussion about equality and agreed that if two values are equal to the same thing, then these values are equal to each other.

    Then we looked at the pattern. They “got it,”, but several students were upset because when discussing the pattern, they claimed that 2^0 did not fit the definition they had been taught about exponents…that an exponent tells you how many times you multiply the base by itself. They were right, and I had to admit it. However, they also had to concede that just because there was a “hole” there, it did not negate the pattern or the equality discussion.

    That night I went home and found Sal Khan’s explanation of exponents (and one that has been shared here already) that exponents are how many times we multiply the base by 1. Brilliant! Hole filled! I brought it back to my students, and I feel confident that we created a solid understanding that math works…all the time!

  13. This comparison is a bit of a straw man argument. As an entire lesson on zeroth powers, it fails, but as a refresher I think it works.

    Fluency comes partly from internalizing simple rules. As long as a student can make sense of the result as well, then this trick is valuable.

    • As long as a student can make sense of the result as well, then this trick is valuable.

      That’s the problem. The video sets the student up to trust the answer (because the teacher on the video said it was true) but not make sense of it.

  14. I find this concept so interesting because so many people don’t truly understand why this property hold, in fact I would say that I didn’t understand it too well until freshman year of college when I had a professor show me that as an explanation. I thought it was really cool and makes me think of a similar proof for the property 0.999…=1, I think it is simple proofs like these that often show the tricks better than the trick themselves. I am really interested in that book you mentioned, I downloaded from the website so I could read it at some point, and I am excited to see what wonderful knowledge it may have in store for me. I hope that as a math teacher in the future, that I can make sure to emphasize conceptual understanding rather than just computational understanding, because both are important and conceptual understanding leads to a more thorough understanding of computations.

  15. There is another view of all this.
    a = a
    a^2 = a x a (definition)
    a^3 = a x a x a (definition)

    a^1 = a (consequence of definition) (a times itself once is a bizarre concoction)
    a^0 is undefined

    so we define it as equal to 1
    because it is sensible AND consistent

    Now I am wondering why (-1) x (-1) is equal to 1

  16. Those lines, “How do I know this?” “Well in math …” really yank my chain, because it not only fails to provide a way to understand this, it also pretends that it does provide an explanation. By misidentifying what it means to know something “in math,” it sends a message that not understanding why is what to expect if you ask questions about why things are true.

    Math is the subject where everything can make sense, and where student expectations that it make sense are part and parcel of their learning of the subject.

    That activity from Cathy Yenca that produces the result about 0th powers provides some opportunities to make sense of the result, especially if there is conversation to accompany the activity, for example to highlight why dividing by 4 as you move down the chart isn’t just a coincidence – it connects with what is previously known about exponentiation, in this case with the definition.

    Without that connection to the meaning of exponentiation, it could be little more than noticing a pattern, and extending that pattern, which doesn’t inherently assist understanding. More than this, without justification for the pattern, it isn’t a valid argument — patterns don’t necessarily extend beyond the cases for which they have been verified, not without some additional reasoning.

    Several comments have provided a valid argument. My personal preference is to hang it all on the additive law of exponents: (a^s) x(a^t)=a^(s+t), which follows directly from the definition of exponentiation, when s and t are positive integers. All of the standard “rules” for exponentiation follow from the additive law of exponents, as long as the meaning of exponentiation for other exponents (for zero, for negative integers, for fractions, …) is assumed to also obey the additive law.

    There’s a truly great old lesson on exponentiation that I believe comes from Project SEED, that has been used with amazing success in hundreds of elementary classrooms. Students deduce the laws of exponents by first finding the additive law of exponents, and then deduce the meaning of other powers, starting with the zero power:

    Students guess what 2^0 will equal. Typically, they guess 0.
    Teacher writes (2^0)x(2^3) on the board and asks what it must equal, according to the additive law, and so we have (2^0)x(2^3)=2^(0+3)
    Now students work with this equation to identify the numerical value of any pieces they can, and get:
    (2^0)x8=8
    Teacher goes back to the class guess, that 2^0=0 to see how that works in this equation, and the students then argue that 2^0 must be 1 in order for this equation to be true.
    Next students create the similar argument to determine the value of 3^0.

    • That’s dynamite. Added to the post.

      What all these methods have in common – whichever angle they take on the concept – is that math must make sense. Your angle in is the additive property. Others are working with the subtractive property. Cathy is working with the multiplicative property.

      None of them is working with the “Just take my word for it” property.

  17. Reading the thoughts and comments here is refreshing. While I have always offered a proof for this problem to my students, I have never thought about it in so many wonderful ways. I wifi definitely use the Desmos activity this year. With that said, was the first video real? Do people actually teach it that way? I’m baffled.

    • While I have always offered a proof for this problem to my students, I have never thought about it in so many wonderful ways.

      Agreed. I’ve learned a lot here.

      With that said, was the first video real? Do people actually teach it that way? I’m baffled.

      Numberwang is fake. Sadly, Virtual Nerd is real.

    • People do actually teach it that way – I’ve seen it. In fact, if you google any particular math topic, you’ll find multiple videos embracing the “do these steps because I said so” approach. Finding good activities to develop rich understanding is a lot harder.

  18. Showed Cathy’s version on the second day of school during a lesson on products of exponents when asked about the zeroth power. Grade 9 . Bright student who must have asked other teachers, but never gotten a satisfactory response. He did the whole class a favour by asking.

  19. The zero exponent rule used to drive me crazy when I taught exponent rules. I like the exponential pattern idea and use it often to provide even more structure for students to conceptualize zero and negative exponents.

    I like it best when I can explain what math notation is really trying to say. Obviously this is easier said then done though!

    I find it easiest to explain and talk about exponents as an expression of factors. If you think of 7^2 as two factors of 7, 7^1 then is one factor of 7, and 7^0 then is no factors of 7 (no multiplies by 7).

    The only last thing to clear up for students is that any number (term) can be written with a factor of 1….because there’s always 1 group of that. So just like an apple is 1 apple, and x is 1x, then 7 should obviously be 1(7). Therefore, 7^0 is 1(7^0) is just 1 and no factors of 7….ahhh just 1!

    It can easily continue from there as 7^(-1) is the opposite of a factor of 7….a divide by 7, specifically 1(7^-1) so why it’s 1/7.

    7^(1/2) then is half a factor of 7….or halfway to 7 by multiplying (just like 3.5 would be hallway to 7 by adding). It makes sense now why you square (2) root!

    8^(2/3) is then two thirds of a factor of 8…or two factors, of a third of a factor of 8.

  20. Surprised no one has mentioned Ben Orlin’s sublime https://mathwithbaddrawings.com/2015/09/09/the-exponential-bait-and-switch , “The Exponential Bait and Switch”, which completely altered the way I teach exponents. I show this blog to my pre-calc students, and teach this method to my algebra 2 kids. Pretty much the same as “hanging it all on the additive”, but instead of deriving the rules, I give them problems and tell them to use the additive rule.

    Hopefully this will spur me to write it up.

    • Oh that’s a winner. I’ll add it to the post. The technique and also this bit of exposition:

      Math’s saving grace, though, is that it can make us feel smart for another reason: because we’ve mastered an ancient, powerful craft. Because we’ve laid down rails of logic, and guided a train of thought smoothly to its destination. Because we’re masters—not over our peers, but over the deep patterns of the universe itself.

  21. coming out of hibernation to respond to this. if we just teach tricks, then we negate what we *claim* the purpose of math education is: to get kids to critically and logically think and problem solve. because let’s face it, 99% of the kids we teach will never need to think about what anything to the 0 power is after graduation, so we better be doing something more important than imparting the factoid that it equals 1.

    blog post about this topic here: https://inpursuitofnerdiness.wordpress.com/2016/09/23/what-is-the-purpose-of-math-education/

  22. christophe meudec

    September 23, 2016 - 5:00 am -

    This does no explain anything.
    ‘Why is it always equal to 1?’ is the question that should answered by the video, not just the mere fact.

  23. The lesson by Cathy Yenca is awesome. I can imagine that both approaches can take about the same amount of time to get the lesson done, but it goes to show that putting in a little more thought and effort into your lesson planning can go a long way for your students. It’s crazy because from a high-stakes testing point of view both are teaching “correctly” because they are teaching the necessary skills to pass. But the lesson from Cathy also teaches students that math is not just a set of arbitrary rules that work. They get to develop skills that helps them to explore the why aspects as well.

  24. This video by James Tanton changed really expanded my view of exponent rules. He makes the case the the fundamental law x^a * x^b = x^(a+b) drives our decisions about how to treat zero exponents, negative exponents, and fractional exponents. It’s a unifying principle!!!

  25. Michael Paul Goldenberg

    September 23, 2016 - 2:55 pm -

    Lots of lovely replies here, and a James Tanton video as the “cherry on top.” :) (I use some of the videos from his Quadratics unit in my intermediate algebra course to great effect).

    I’ve generally argued that if the multiplication and division laws for exponents make sense, we’re forced to decide that for any real number m not 0, m^0 has to equal 1. And that follows from looking at cases like (m^5) / (m^2) = (m*m*m*m*m) / (m*m) = ((m*m*m)*(m*m)) /(m*m) = (m*m*m) = m^3, which agrees with the division law for exponents: (m^5) / (m^2) = m^(5-2) = m^3.

    After students are satisfied that this makes sense, we look at a case (m^3) / (m^3) for non-zero m. Everyone agrees that this should be 1. Then we look at it from the point of view of exponent laws and we get m^0. So the notion that m^0 should be 1 seems sensible.

    Finally, we discuss what happens if m itself is 0. And this gets us to our favorite “dividing a real number by 0 is undefined” situation, something we’ve already endeavored to make sense of from a couple of perspectives (none of which is “that’s the rule”).

  26. Michael Paul Goldenberg

    September 23, 2016 - 3:16 pm -

    Now having watched (or rewatched) that Tanton video, there are a few important things to note. Whenever he does this sort of “What should this mathematical expression equal?” video (see the one on 0! for another example), he argues along these lines: If you accept this previous law, then it seems to follow logically that THIS definition of said expression should be true.” And then he adds an interesting point: you can choose to accept this result or not.

    Some places he expands that point. I think it’s useful to do that in class. What happens if you choose not to accept that x/0 is undefined for all real numbers x (not equal to 0)? What do we do with 0/0? How about x^0 for non-zero x if you don’t want to accept that it equals 1?

    I think I first raise these sorts of things in my class when we discuss order of operations and why it was important to have universally accepted, yet ultimately arbitrary rules about such things. What happens if the kids across the hall have a different set of rules for order of operations? Big deal or not?

    I’ve been inclined over the last several years to define mathematics as “The Science of What If?” What if we change this definition? What if we restrict ourselves to a smaller domain? What if we look at an expanded domain? What if not all infinities are the same size?

    What we’re looking at here are examples of perfect places for such discussions. The thought of allowing students to watch and accept the take in VirtualNerd video makes me ill. Such a nice young person, so friendly, and so NOT talking about mathematics.

  27. Interesting post. This is something I passionately believe in and have done for my tutoring life (16 years). I noticed that students would memorise the rules of indices without being taught why things were. I use a similar approach to explain why 7^0=1 as above. It is one of the first abstract concepts – as teachers we have to admit that. Another one like this is the question of why a minus times a minus is a plus. This is NEVER explained, but I explain it as well so that the student knows that there must be nothing that is not understood.

    • How do you explain that multiplying two negatives gives a positive? I do it by showing the pattern that you get by multiplying a negative by 3, the 2, then 1, then 0, then -1, -2 etc. I’ve also told students that -7*-5 is asking you for the opposite of 7*-5. I’d love to hear additional explanations.

  28. Another aproach is for maintaining the rule 2^n * 2^m = 2^{n+m}
    And another aproach is to see this video [http://somenxavier.github.io/operes-en-tres-actes/Pay-it-forward.html] and notice that in Step 0 there is just Trevor. So 3^0 = 1.

  29. What happens to 0^0?

    Based on the first teacher, 0^0 = 1.

    Based on the second teacher 0^0 = 0/0 = indeterminent.