Teaching Direct Variation Conceptually (as a foundation for slope)

OK, the following may rate as one of my best test questions ever.  It might take you a second to see the point:


If you don’t get it yet, compare it to this alternate version:


By now you’ve probably seen what I’m getting at: the answer to #1 is “Not enough information”, because there’s a jar there, so the weight per candy is not 7 grams.

Let me share the struggles that led to this idea, and the way this approach will underpin my teaching of linear functions next year:

First of all, I want to continue to make my math real-world (even if nobody at Dan Meyer’s blog can articulate why), but too often that means students have to wade through lots of WORDS to understand a question or task.  This is especially daunting for English Language Learners, and I’m working on meeting their needs better.  So I need to find ways to capture the essence of concepts and story problems with mostly pictures and a minimum of text.

Secondly, my students often have a hard time seeing why rates are often calculated as (y2 – y1)/(x2 – x1) instead of y/x. With a good context like Domino Effect by Mathalicious, students will discover the idea for themselves, and they’ll be able to memorize it and apply it in real-life situations that obviously give them 2 points.  But I’ve always taught direct variation at the end of linear functions, as a special case after the main concept…and after that lesson, some students revert to calculating slope as y/x.  They can’t distinguish cases when you need to subtract first from cases when you can just divide.

So instead of teaching direct variation last, next year I’ll teach it first, as the introduction to linear functions.  Since they won’t know (y2 – y1)/(x2 – x1) yet, we’ll just start by sorting situations into 2 categories: ones in which you can find the rate (not “slope”–that word won’t exist for us yet) by dividing y/x, and situations in which that makes no sense.  We’ll make predictions for new situations in the first category by writing proportions, writing equations of the form y=kx, drawing graphs that pass through the origin, etc.  And mixed in with all of that practice will be situations in which direct variation and proportionality don’t work:


(The bar itself weighs something here).

Once we’ve really hammered the idea that there are 2 categories of linear scenarios, ones in which the rate is just y/x and ones in which it’s not, we can finally do Domino Effect and figure out how to deal with the second category.  As an aside, I do need to find more 3-Acts that are simple linear scenarios and are still enticing for students.  If anybody knows of some, please tell me in the comments.  I tried Dan Meyer’s Pencil Sharpener task, which he seems to have taken down from 101q’s, but it gave me a really poor prediction.  That’s no good, because early in the year I want my students to feel impressed by their new ability to predict things.

As a final note, I think even before starting the direct variation lesson, we might use Simpsons Sunblock to get students thinking about proportions.  In the past, they’ve tended to make their predictions by writing and solving a proportion (or at least informally by doubling the x- and y-values from a particular point in their data table to predict the needed values).  Then when we formally introduce the concept of proportionality/direct variation, we can return to Simpson’s Sunblock and talk about how the method they already used there was an example of it.  And we can translate the proportions they used into direct variation equations.

Yay!  Why is it that March is always when I realize what I should have been doing in September/October?

Cryptography and Horizontal Function Transformations

This lesson has gone well enough for me that I think it might be worth putting out there for others to use.  I’ve been troubled for years by the lame explanations I had to give for why horizontal transformations are backwards.  Why does y=(x – 3)2 get shifted to the right?  Every explanation I’d give was met with the polite “Oh, okay” that’s student for “I have no idea what you just said.”

So here’s a taste of what how I taught the topic using the context of cryptography.  I gave students the following function and asked them something to set a low barrier to entry: if someone sends you the message “8-9”, then what are they saying to you?


(The answer is “Hi”, of course).  But that code is too obvious, so then I introduced the idea of transforming the code to something more secret.  For example, you and a co-conspirator could agree that to read a message, you’d first add 3 to each number, and then look up the corresponding letter on the graph.  So if I send you the number 1, what letter did I send?  (The answer is that you first do 1 + 3 to get 4, so the correct letter is D).  That’s simple enough, so we practiced it and built up to something more complicated:

Easy Transformations

The answer to question e) is 55.  This is not easy to see without doing some calculations, and those calculations already get you thinking in the backwards logic of horizontal transformations.  Since “S” is the number 19, you need (x+2)/3 to come out to 19.  So you first multiply 19 by 3, and then subtract 2.  Students should show work like this: f[(55+2)/3)] = f(57/3) = f(19) = S.

After that discussion, it’s time to let the students have some fun by sending a secret message to a classmate with the code f(2x-3).

By then, we’ve noticed that it’s annoying to have to recalculate each letter you want to send.  Wouldn’t it be nicer to just notice the patterns, and adjust the entire code of f(x) at once?   We look for patterns in how the points on the graph are transformed, like this:

Graphical transformation

(I’ve put the answers in the graph in red font).  Here you can already see the backwards thinking in effect: although the fact that 0.25 < 1 might make you think that f(0.25x) is compressed, in fact it’s stretched.  And you understand the logic–you need a larger value of x, because the 0.25 is going to shrink it, and you’re still trying to come out to the same letters of A, B, and C.

What I really like about this approach is that it also makes clear why the order of operations is reversed for horizontal function transformations.  For example, in the function y=sin(2x+pi), you first shift everything to the left by pi, and then compress by a factor of 0.5.  Why are these transformations applied in the reverse order of operations?  Because in essence, you’re solving an equation, just like we solved (x+2)/3 = 19, above…and solving an equation always involves “undoing” the expression in the reverse order in which it was built up.

Students apply that reasoning, first by encoding the letters of the word “JAMES” using the function f(0.5x + 6), which has two transformations.  Then, looking at the pattern of the steps they took in each case (first subtracting 6 from both sides of an equation, and then multiplying by 2 to cancel out the 0.5), they apply those transformations graphically to another word, “BOND”, in the same order.  First subtracting 6 from each x-value means moving it to the left 6, and then multiplying by 2 means stretching it horizontally by a factor of 2.  Students know that the order matters here, because of some exploration we did in the intro section of this activity.


Students then translate their understanding to actual mathematical graphs:


This should set my students up nicely to understand period and phase shift.  Think again about the function y=sin(2x+pi).  When transforming graphs, it’s usually easier to stretch/compress before you translate.  Since order of operations is reversed, if we want to compress first, we have to rewrite it as y=sin[2(x + pi/2)].  This way, you compress by 0.5 first, and then just shift everything by pi/2 to the left. 

And, if you’ve gotten all the way down to this part of the blog post, here’s a link to the activity: https://www.dropbox.com/s/gam9uzwqrnl91l8/Introduction%20to%20Functions%20and%20Cryptography.pdf

For some context, see Kate Nowak’s related musings: http://function-of-time.blogspot.com/2013/09/building-functions-clarified.html

Teaching is hard

I’m in my 10th year as a teacher. I’m a much better teacher than I was in my first couple years, and a much better teacher than I was before I found the MTBoS. But I struggle with keeping a growth mindset–I keep wondering when I will finally be the teacher I want to be. Part of this is that my standards go up every year. Lessons that seem okay this year would have struck me as awesome 5 years ago. And part of it is that this is only the second year ever that I haven’t had either a brand new prep or a new school to adjust to. But still, teaching is hard. So for all of us out here busting our butts over summers, weekends, etc…keep it up, and remember that a growth mindset is the most important thing.

Rough idea of how I want to use Penny Circle

Been trying to figure out how to use Penny Circle, by Dan Meyer and Desmos.  I think the activity is terrific, but like L Hodge (in the comments on Dan’s blog), I think the activity does some of the intellectual work for students, so they won’t be challenged to think as hard as I’d like.  (However, it’s a great first experience of the IDEA of modeling, so I’m not criticizing it…I just want to take it further).

I only have a second right now (will flesh this out later), but I think I’m going to have students do the activity as intended, and then ask them to create an equation for diameter of the penny circle after Dan has been putting down pennies for t seconds.  The Desmos activity generates the function n(d), where n is the number of pennies and d is the diameter of the circle. My students will have to find the function d(n), and then use some video of Dan putting pennies down to find n(t).  Then they can use composition of functions to get d(t).  I think there is some video of Dan putting pennies down in the 3-Act.  Anyways, back to work, but if anyone has comments or especially criticisms/warnings, I’d love to hear them.

Frustration in trying to use Khan Academy exercises

The teacher reporting tools just don’t work.  So it’s taking me FOREVER to check students’ progress.  Literally, there are exercises on their site, which I have assigned to students, that are not in their list of exercises in the teacher reporting tools.  To check student progress, I have to go to each student, click on their name, and CONTROL + F to search the page for the name of the exercise, then look at their score, and finally type it into my gradebook.  And, because student proficiency changes constantly based on the mini retention quizzes KA gives (called mastery challenges), I will have to re-check student scores on each exercise every couple weeks.  KA, you say Shipping Beats Perfection doesn’t mean you ship things that are broken, but this is clearly broken!

UPDATE: Khan Academy Lead Developer Ben Kamens comments on this bug and on their current focus in building out more teacher reporting tools.  See comments section.

UPDATE: I found a new workaround, and I’m pretty sure it wasn’t there yesterday.  Perhaps KA fixed something already.  You can use CTRL + F to search the table when you view by table.  Exercises that weren’t in the table yesterday seem to be there today.  You can’t type their names into the search box above the table, but you can use your browser’s search function.  That’s a start.

UPDATE 9/22/13: Now that searching in the grid is working, what’s not working is searching in a student’s skill progress while viewing their profile.  So, for example, when a student has completed “Multiplying Expressions 0.5”, I can see their work in the grid or in the coach report by student, but if we view the student’s “skill progress” from his/her own profile, Multiplying Expressions 0.5 is not listed as an exercise.  In essence, KA’s back-end system for tracking performance seems to be having a hard time keeping up with its front end improvements this summer. so teachers should be prepared to potentially spend extra time figuring things out.

UPDATE 9/29/13: The issues with the teacher reporting have been completely fixed, and–even better–they have added a feature which lets you recommend exercises to students as a whole group rather than one student at a time.  Nice job, KA.  Exactly what I was hoping for.

How I’m Covering the Distributive Property in Precalc

I teach a class called Math Analysis/Trigonometry, which is close enough to Precalc that students who get A’s in my class are allowed to skip to calc. I’m spending the first several class periods covering a combination of Carol Dweck mindset stuff and the idea of distributing.  Distributing?  In Math Analysis/Trigonometry?  Yes.  I actually think many students, even post-Algebra 2, don’t understand it that well, and I think it’s what leads them to make some of their most common mistakes.


Later in this post, I’ll describe how I’m trying to talk about distribution in a way that’s conceptual and allows for Accountable Talk–I certainly don’t want to lead off the year with a bunch of drill-and-kill.  And to my surprise, distribution seems to be fertile ground for conjecture and discussion.  But first, here are the top 2 mistakes.  Recognize them?

  1. In (x + 2)/x , you can cancel out the x’s.
  2. (x + 5)2 is just x2 + 25

With mistake #1, students just don’t see that it’s distribution.  If it said x(x+2), they wouldn’t give the answer of x2 + 2, but they don’t see that distribution happens with division, too.  Somewhere in their brains, they know it, but that knowledge must be only loosely connected to everything else.

With mistake #2, students mistakenly think that you can “distribute” an exponent across addition.

There are other big mistakes relating to distribution, too.  Have you ever seen someone try to distribute the 2 in 2(3 * x)?  Usually, they won’t do it for 2(3x), but if I put a multiplication sign and some big spaces inside the parentheses, I usually get more than a handful who distribute.  This whole topic is closely connected to the ability to see whether an expression has one term or many, but the number-of-terms idea doesn’t unify it completely, because you can “distribute” the square in (3x)2, even though 3x is a single term.  So let me share what I’ve done.  Curious how you would have done it–surely other teachers must have to do battle with the same misconceptions.  And you may object to the use of the term “distributing” when talking about an exponent.  I used to discourage students from talking about it like that, but I now think allowing it as long as they have a deep understanding of distribution is better.

Accountable Talk Discussion Plan

Discussion was grounded in this student handout. The question of why the distributive property works led to some crickets chirping in class, so I had to take a student’s example from #1 — the example was 3(x + 2) — and ask WHY does it equal 3x + 6.  The first 3 students to respond gave variations on the theme that you have to multiply the 3 by both terms inside.  Right, but WHY is that rule correct?  I think I ended up priming the pump by writing “3x = x + x + x” and asking how you could write a similar statement for 3(x + 2).  A student explained that (x + 2) + (x + 2) + (x + 2) = 3x + 6.  Still, lots of students were either tuning out (“Isn’t it obvious that it’s 3x + 6?  I mean, you just distribute.”  Grrr) or not getting it.  After 3 more students paraphrased and small groups checked in with each of their members, everyone seemed to understand the point we were making.

So then, the set-up for the big discussion: #3 on the handout.  To clarify the question, I said, “There are other situations where you can distribute.  For example, what could you put inside the parentheses in the expression (     )2 so that you could distribute the square.  And 90-100% of the class wrote something like (x + y)2.  Success!  This was exactly the misconception I wanted to address, and instead of just presenting it from out of nowhere, we’ve created a natural context in which to debate it.  I grabbed a bunch of student responses and put them under the document camera.  One student actually wrote out his work: (x + 5)2 = x2 + 52 = x2 + 25,  I showed that response last, and a handful of students recognized that it was incorrect, but the bell was about to ring.

Bell rings, end of class, next class begins.  We picked apart what was wrong with (x + 5)2 = x2 + 25, and I asked them to try #3 again.  What can you put in the parentheses so you can distribute the square in (     )2.  Now a couple of students tried a monomial in the parentheses.  We proved that it worked using associativity and commutativity, and I asked several students to summarize.  So far, the class thinks the rule is that you can distribute an exponent across different things as long as there’s no addition or subtraction between those things.  That’s not the rule I want them to eventually come up with, but it’s a good enough start to let us move on to #4.

Check out #4 on the handout (here it is again).  I really hope we get some good mistakes there, and some good debate.  The rule I want them to generalize is that you can distribute what’s outside the parentheses if the operation/function outside is the repeated version of the operation inside.  Since raising to a power is repeated multiplication, an exponent distributes across multiplication but not addition.  Since multiplication is repeated addition, it distributes across addition (or subtraction) but nothing else.

Actually, I’m not sure if this is technically correct.  Distribution is a property of rings from abstract algebra, and there may be some exotic versions of rings where the “multiplication” operation doesn’t represent repeating the “addition” operation.  I got some help on Twitter from Dave Radcliffe (@daveinstpaul), who was telling me about power sets forming a commutative ring.  If anyone can help me understand whether I’m making a big mistake talking about distribution this way, I’d really appreciate it!

Khan Academy for Review Practice?

Meanwhile, my students have a bunch of factoring and quadratics exercises to do on Khan Academy.  (Yes, on Khan Academy.  These kids won’t be harmed by boring videos about factoring.  They learned factoring years ago.  They just need to gain fluency, and I need data on who needs remediation with what.)

UPDATE 9/11/13: When we did #4a today, students’ initial answers on whether to distribute in 2(3 * x) were evenly split.  Half the students got 12x, and half the students got 6x, until we discussed it.  Should get to the final summarization of the principle I’m looking for tomorrow.  (We don’t spend all class on this stuff, so it’s getting spread out over more days).

How I want to teach horizontal function transformations

Horizontal function transformations are tough to teach.  Sure, you can just tell students how they work (“everything is backwards inside the parentheses, so minus makes you go to the right”), but that just seems arbitrary and pointless to students.  Slightly better would be to let students discover by a process of induction that directionality is reversed inside the parentheses, and then get into the big discussion of Why?  My sense is that this is Dan Meyer’s preferred approach, based on his comment in this post by Kate Nowak, who is struggling with teaching this concept in the quadratic function family.  And I have nothing against Dan’s approach–letting the induction come first is fine, but it’s really the Why? that’s tough about this idea.

First of all, it’s not just the directionality that’s reversed–it’s also the order of operations.  So in sin(2x + pi), everything is shifted left by pi first, and then contracted by a factor of 1/2.  The “+ pi” is dealt with before the “2*”.  How are students supposed to discover that fact by induction, and even if they do discover it, how will that set them up to make meaning out of it?  I am not a fan of induction just for the sake of having students discover things when that discovery does not build connections or make meaning.  I mean, you could theoretically have students “discover” the rules for integer operations by letting them play around with a calculator and look for patterns, but that would still be an awfully shallow learning experience.

To return to the question of horizontal transformations, here’s an idea I’ve been cooking for a few years: teach the concept in the context of encryption.  We all know the idea of encoding by letting 1=A, 2=B, etc.  But as Julius Caesar found, it’s better to introduce a shift into your code, e.g., 1=Y, 2=Z, 3=A, 4=B, …  This shift works in the same way as a horizontal translation.  Pow!  A real-world, intuitive example (I hope) of an otherwise difficult concept.  The example even generalizes pretty easily to show why order of operations is backwards as well.  Here’s how I’d use this hook in class:

  • First, give students an encoded version of a sentence (with no shift applied), and ask them what they think it says.  I’d say something like, “Ender’s Game is Better than the Hunger Games”.  So “Ender’s” would be encoded as “5  14  4  5  18 ‘ 19”  Shouldn’t take long before someone cracks the code.
  • Then discuss why a shift would be nice–both you and the message recipient would have to agree on the shift in advance, but it would be more secure.  Maybe I’d show the video linked above about Caesar and his cipher.
  • The class would agree on a small shift, say -3.  Now, to read a message, you subtract 3 from each number, and then decode.
  • I’d ask each student to write a message and encode it so it can be decoded with a shift of -3.  They’d check their work by decoding their own message before exchanging with other people and decoding each others’.
  • Here’s the point: to encode when the shift is -3, you have to start by adding 3 to each number, since the recipient will then subtract 3.  So far, this should all be fun–little complaining about “when are we ever going to use this.”
  • Then I can introduce a graphical representation of the code.  It might look something like this:













That’s the graph of the original code.  I’d ask students for the graph of the new code, with the shift.  The shape would be shifted to the right by 3.

  • Of course, this concept would work with any original code–it wouldn’t have to be 1=A, 2=B,… to start.  It could be 1=Z, 2=Y, … or any one-to-one function to start.  This lets us talk about functions versus relations in an intuitive and meaningful way in the same lesson.
  • I developed the connection to inverse order of operations a little bit in my first attempt at this lesson last year.  Here it is on dropbox.  It was a success in the sense that students finally understood that between reversing direction and order of operations with horizontal transformations is just an application of “undoing” an expression to solve an equation.  But I thought the lesson design and delivery were a little clunky.  I plan to revise it for future use.  But if someone else likes the idea and revises it first, and then shares it with the rest of us, well, I won’t object…

That’s it!  So, what do you think?

UPDATE: the formatting seems all messed up in my dropbox .docx link, so here’s a pdf version

FURTHER UPDATE: Oops, in my original post, I had the grid showing the “graph” of the original code transposed.  That is now fixed.

From the comments:

Dan Meyer: Great to see you blogging on the regular, Kevin. We seem to be on the same page here that discovery without building connections between the discoveries and axioms is kind of useless. But using a real-world context here seems risky also. Is it possible that the cryptography connection will obscures mathematical meaning? How do you help students make the leap from Caesar Augustus to periods and amplitudes?

(My response to Dan’s question is in the comments below).