Teaching Exponents ConceptuallyPart 1

Exponents offer a great opportunity to develop more of a knack for teaching conceptually, and offers students a chance to change how they approach math.

The transition is difficult, for all parties, but well worth it.  In this episode we discuss ways to help you and your students make the switch.  I’d encourage you to download the exponents packet.  To do so, click this link.  It is 100% free if you use the promo code “exponents4free” at check out.

Please visit the pages here on this site that cover exponents.  There are videos, assignments, all kinds of wonderful resources.  Start with the basic introduction by clicking here.

What’s wrong with teaching students exponents by:

Step 1:  Rewrite any base to the power of zero as one.

Step 2:  Look for a power raised to a power … multiply those exponents

Step 3:  Look for negative exponents, flip the base, change the sign of the exponent

Step 4:  If you have like bases that are multiplying, add the exponents.

Step 5:  If you have like bases that are divided, subtract the exponents, top minus bottom.

Step 6:  Clean up any remaining coefficients.

Before getting into that, which I paraphrased from information found doing a google search, let me suggest that exponents are a gate-way topic.  Students, and teachers, can really start to approach mathematics conceptually, engaging their literacy to make sense of information when it comes to exponents.

But it’s a tricky change to make, on your end especially.  You’ll likely have ZERO guidance at your school, and will receive push back.  Without the experience of long-term success with the approach, you might lack conviction to stick to your plan.

I can tell you that what I’m going to share does work, and works over the long run.  However, it can be a little messy early on.  So a best practice would be to explain to the students the change in approach.  Ask them to be willing and patient participants.  Also explain that you understand this might be a difficult task for them initially, but the pay-off will be worth it, for them.  Also, make them a deal … if they struggle early on, but show an appropriate level of mastery over time, weigh the latter grades much more heavily.

It will often feel, to them, as though you’re intentionally tricking them, that they’re not being supported, and that the questions aren’t what was shown in class.  Explain to them that this is okay.  The reason it feels that way is you will be asking them to apply what they know in creative ways, and that you also must ask questions that will uncover misconceptions.  There’s no way to provide information so flawlessly that misconceptions aren’t created, and further, learning really occurs when confusion is confronted.  Your job is to give them the information as clearly and easily consumable as possible, but then also to ask questions that will uncover confusion.  You’re testing your own delivery of content as much as their absorption.

If students only see problems that they’ve witnessed before, they’re mimicking, it’s not authentic learning.  It would be like writing a short-story by copying Old Man and the Sea.

It might be a good idea, depending on the nature and density of helicopter parents, to get them involved, as well as other team member and administration.  Just run them through the idea, the overall plan, how remediation will work, and the long-term pay off you expect.

The payoff is there.  I have lots of data, going back almost a decade, but this is a podcast … so we’ll use an anecdote.  I teach a two-year course called Cambridge IGCSE.  It’s a fantastic curriculum.  Like always, I taught the exponents early in the first year, around week 3 and 4.  The group I had was one of my lowest achieving groups in a few years.  Fast forward this group to the spring of their sophomore year.  They’d seen a few exponents as they naturally occur in mathematics, but no review or revisiting.  The students were practicing a past Cambridge exam and they came to a problem that involved a tricky combination of coefficient, variable, and rational exponent, like (27x27)2/3.  Even the low students got it right.  So I was convinced some Tomfoolery was at play.  I asked one of the suspect children how he knew his answer was right and explained with perfect clarity, and with a touch of strained patience in his voice, exactly what he understood.  He knew the relationship between the exponents and bases and understood that the power of 2/3 meant, “what number cubed, then squared?”  Since 27 was a cube, he rewrote it as 33, then just multiplied the exponent of 3 with 2/3, and ended up with three squared, or nine.  The exponent of 27 multiplied with 2/3, giving 18.

I asked other students and they all had various, but conceptually similar explanations.  This retention and clarity has typically been the case, but it surprised me out of these students.

After going over all of this, I see why it is so much easier to open the book to page 92 and just start there…and you’ll end up with the six steps we discussed at the beginning.  Those steps could be followed to get right answers, after all.  So what’s wrong with them?

Those steps completely fail in a few key areas.  First, exponents share information, not at all unlike how multiplication shares information.  Both of these compress numbers.  5 to the power of 11 is a HUGE number, one that will likely be difficult to understand.  But, when written as an exponent, it is easy to write, read, reference, and think about.

The “rules,” or “laws,” are consequences of multiplication, or patterns. Six steps, each with at least one rule to remember, and they don’t even get into rational exponents.  Now, rational exponents, if I remember correctly, aren’t covered until grade 11, by CCSS, though the basics we covered in our six steps are done in grade 8.

What will be shared here can be used at any time, and will include rational exponents.  Once students are reading and understanding exponents correctly, rational exponents are super easy to understand … plus, the students will be rewriting numbers in response to what they’ve read and understood.  It’s great.

But, you can use this at any point.  This is great the first time students see exponents, but is also fully appropriate in an Algebra 2 or an Algebra 1 setting.

One last thing, before we really jump in.  This is a difficult type of topic for a podcast.  I’m new at this, and learning, so I’ll do my best to describe carefully and concisely the approaches I’ve used.  I’ll encourage you to visit the show notes at https://thebeardedmathman.com/podcast/e13.  There you’ll find lessons, sample problems, homework, the works.  You can download the packet which I sell in my store for free.  Use the code exponents4free to download the packet.  Most of the resources in the packet can be found the exponent pages linked in the show notes, but this organizes everything and provides you access in word document form so you can edit and print, whatever your heart desires.

Let’s get to it.  Let’s discuss opening question, what is wrong with these six steps?  Mathematically, nothing, at least not that jumps out at me.  But what’s wrong with it pedagogically, is a lot.  In the defense of the authors, I don’t imagine they wrote that to help students to learn, but instead have published it as a reference.  There’s a big difference.

Students need to understand what exponents are before they can really master the concepts at play, right?  It would be hard to learn to bake cookies if you didn’t know what cookies are.  If everything went perfectly as planned, no problems.  But, what if an ingredient was out of stock, or slightly different?  What if you needed to make a new type of cookie?  That’s similar to our situation here.

I believe it is important that students can read and understand expression with exponents.  It allows them different paths to solutions, unexpected connections, and much greater retention over time. Here’s how I start.

3 × 5 = 3 + 3 + 3 + 3 + 3.  While it is true that 5 + 5 + 5 is the same sum, it is fundamentally different.  And the order with exponents is of substantial importance.

Idea #1:  The take away is this:  Multiplication is a short hand way of writing repeated addition (of the same number).  No need to write out five threes all adding to one another.  We use multiplication for that.  Students may or may not understand this.  Don’t labor the point, but do go through the effort of having students explain and demonstrate how they understand this.

Idea #2:  35, with three as the base, and 5 as the exponent, is a short hand way of writing repeated multiplication.  3 × 3 × 3 × 3 × 3.  Instead of writing out all five threes, we use the notation instead.

So exponents are NOT something to be solved, or even a new topic in math, not really.  Everybody knows how to multiply, or should, at this level. What this really is about is writing values in different ways!

Idea #3:  The first two rules … if the bases are the same and they’re multiplying, the exponents are added.  Have students explain why 35 × 33 = 38 by applying idea #2.   Have them also explain why (35)3 = 315, use idea #2.  This is an imperative understanding.  Before moving on, be prepared to ask some tricky questions here.  I’ll do my best to describe them here, but you can find them in the show notes.

Question set #1:  Simplify the following, writing your answer with a single base and a positive exponent.

1. 45 + 45 + 45 + 45 =
2. x2 + x2 + x2 =
3. x2 × x2 × x2 =
4. a5a =
5. Solve for m: (3x3)m = 27x9

The idea here is not to just confuse kids and leave them helpless.  The idea is to uncover common misconceptions.  These are just a few ideas of problems that will do that.  It is incredibly important NOT to just tell kids what the answers are, and equally important that they really TRY these, with focus and without distraction.

When kids are stuck, have them revert to the first ideas we’ve covered and either find resolution (if stuck) or confirmation of an answer, through application of those ideas.

Idea #4:  If we’re dealing with like bases, then division is really reducing.  Either demonstrate, or guide students, in writing out examples like these that follow.

Example A:   56 ÷ 52

Example B:  52 ÷ 56

At some point, either you or the students need to articulate that you’re reducing and that the exponents describe how many of each bases exists in the numerator and the denominator.  So, in example A, there are four more 5’s, all multiplying together, in the number than in the denominator.  So it is just 54.  In example B, there are four more 5’s, all multiplying, in the denominator, so the reduced form is “one over five to the fourth power.”

There’s no need to confuse things with reciprocals, negative exponents, top minus the bottom.  This is 100% accessible, and procedurally proficient, with this approach.  Also, because it was explored and the pattern discovered, students will be able to retrace this discovery process, and do so quickly, should they forget in the future.  (That’s a great tip for students, too.)

Idea #5:  Anything to the power of zero is one.  Show this through application of idea #4.  It will be revisited after we introduce idea #6.  But for now, it should be apparent that 5k divided by 5k = 1, because they all reduce, and k – k = 0.

Idea #6:  This is where the wheels can fall off!  Negative exponents take a little more time.  It is possible, with a sharp group, to effectively get through all 5 of the first ideas in a single lesson, and an average group can get through all very well in three lessons.  But this takes a little time and patience, at least a day on its own.

Let’s break this down so it is accessible.  It is important that students really understand, and revert back to, these ideas as they struggle to make sense of problems involving negative exponents.  It will keep them solidly grounded in concept.

First, negative, in the context of math, means opposite.

Second, exponents are repeated multiplication.

Third, the opposite of multiplication is division.

So, a negative exponent is a way of writing repeated division.

To that point, the following are exactly the same meaning:

÷ 2, × ½, 2-1

At this level, we rarely use the division symbol as it is awkward to manipulate.  Instead, we multiply by the reciprocal, which usually means we write it as a fraction. This isn’t too difficult for students as long as the base is already in the numerator, like 3x-2.  That exponent of -2 means dividing by x twice.  We write that as 3/x2.  Why does the sign of the exponent change?  Well, for the same reason the division symbol changes to multiplication when we rewrite division as multiplication by the reciprocal…we’ve addressed it and are translating that information with a different symbol.

Where it’s tricky is if you have a base in the denominator and it has a negative exponent.  Try this… ask students, without prompt, what is 8 divided by ½.  Or, what is ½ divided by 8.  The vast majority of the time the students will say 4!  The issue that makes negative exponents of bases in the denominator tricky is not isolated to exponents.  The issue is division and multiplication by the reciprocal.

Explore these ideas with students until they can articulate why it is common that students are taught about negative exponents “flip the base and change the sign of the exponent.”

What students really need is practice, practice and more practice.  But not practice of the kutasoft variety.  This isn’t rote procedure.  They need practice reading, breaking down the information, and applying the ideas here.  That brings us to the next idea.

Idea #7:  When dealing with an expression that has multiple bases, like numeric coefficients, and multiple variables like x, y, and z, break the problem into the product of separate expressions…break the problem apart.  Simplify each portion, then pull it back together.  If students do this a few times, they’ll get the idea and be able to see and think more clearly, without writing it out.

As with our first few examples, always look for ways to expose misconception and confusion, this will push forward the learning.  Coach students on what it is you’re doing and why.  This will help them to be more willing participants and academic risk takers.  You can find examples of these in the show notes on my webpage.

Give students lots of encouragement, and don’t let these ideas get too far out of mind.  Use them frequently for bellwork, homework, and on quizzes, even with other topics.  The reason is that doing so will help those that are almost “there” get “there,” and will help those that get it to keep in touch with it.

I firmly believe that rational exponents should be taught in conjunction with what’s been covered so far.  But, that’s what we’ll talk about in the next episode.  I typically only devote a few days to the topic, and that is sufficient.  But, at this point we have already got a lot of information for you to chew on.