Almost Variation with Inequalities

I‘ve started thinking about Modules 0 for Grade 6. And I’ve written my first sequence for inequalities, which I’ll show below. Although I tried to design the sequence using ideas from variation theory, I found that the specific goal I had for this sequence—writing inequalities of the form x < c and c < x from number line models—did not make it easy to think of a boatload of questions I could ask, each slightly different from the previous one. Plus, I had some slightly more robust instructional goals in mind. Still, I found that it paid off to even just try thinking about variation.

So, I start with the video below, which serves as the first (and only) instructional worked example in the sequence.

I use the Silent Teacher method, wherein I essentially show the worked example twice, the second time with my voice annotating what I’m seeing, doing, and thinking as I write the inequality to represent the two models. In the lesson, I include a brief reminder to students above the video what the inequality symbols mean and what the equals sign means.

My assumptions with regard to this content are that students have seen and used inequality symbols for a long time before they get to Grade 6, though primarily with positive numbers and not variables or negatives. So, this represents a kind of “start-again” topic, which is one reason why I include the block models along with the number line model. It is a compromise between extending the concept and reviewing it: so I do a bit of both.

Another reason I include the block models is because they make a solid, albeit abstract, connection to the use of inequalities with algebraic expressions to express relative values in situations where we don’t know one of the values. We know that q above represents a number greater than x, but we can’t mark q on the number line because we don’t know its exact value. This is what the thinking question below the video is hopefully getting at. It’s numbered in case an instructor wants to assign the sequence to a student.

The Sequence

After the video, there is a sequence of a mere 8 questions. The first of these, shown at the right, is not a typical “Your Turn” type of question, where the student tries out a technique on a very similar problem. Here we unpack the other ways to express the inequalities shown in the video—it’s important to constantly make the point that there is almost always a few different ways of looking at mathematical relationships—and we include the equation, in part because research tells us that comparing the equals sign with other relational operators reinforces the correct relational view of the equals sign.

Next up is a more typical Your Turn, with a block model and number line model both closely mirroring the models shown in the video.

Students can write n or 1 to represent the single block (or the point labeled with both n and 1 on the number line). Doing so helpfully reinforces a slightly better meaning of “variable,” which is a letter that represents any quantity, known or unknown.

And here, for the first time (in a thinking question), I ask students to relate the number line model to the blocks model.

The next question in the sequence is an example of some minimal variation. What’s different here is that the m and n block towers switch sides in the illustration, and the inequality model on the number line shifts to the right. Everything else stays the way it was.

We could continue in this way, adding or subtracting blocks, switching sides, etc., but this kind of model has limitations that don’t allow for examining more of the variation space. But we can hint at the fact that adding the same number to both sides of an inequality doesn’t change the direction of the inequality.

And that’s what we do in the next exercise in the sequence. Here also, the known number is moved along the number line. The thinking question I ask here is:

Would adding 1 block to each tower change the direction of the inequality? Why or why not?

I phrase the question as a hypothetical because, strictly speaking, it’s not evident from the diagram that I added exactly 1 block to tower m.

And Now for a Big Change

Now we see how this isn’t really a sequence of minimal variation. One reason for the change-up is that I realized too late that the model I started with could only show the greater quantity as the unknown quantity. I thought about changing to a different model, one which could show the full range of variation, but I couldn’t think of a situation that worked.

This example, in which the larger quantity (the greater height) is the known, was too good to pass up. And it gave me a context to foreshadow subtracting both sides of an inequality by the same number, which is what (kind of) happens in the next exercise.

Here, though—and again—it was not plausible to hit this balance of operations idea directly (plus, it’s outside of the scope anyway). We only hint at it. But we still ask the thinking question—again, as a hypothetical—about whether subtracting the same value from both quantities changes the direction of the inequality.

The height examples, and perhaps all of the items in the sequence, lie somewhere between minimal variation and maximal variation. At some point while designing it, I had to stop searching for more perfect examples and just run with it.

The final two items in the sequence present two more (more or less abstract) situations where inequalities seem to fit.

The first, shown at the right, is the “swarm,” which contains too many items to count, though we can know for sure that the number is a greater value than 6. Here too is an example situation that better fits with the idea of a larger unknown that couldn’t be handled by the earlier block models.

In this example, I’ve switched up the labels on the number line for a small taste of minimal variation within all the macro variation going on.

Finally, there’s temperature and a quick example showing negative numbers.

What we get at here, also, is that we haven’t left the universe of comparing numbers just because we’re introducing a little algebra. Plus, I’ve eliminated the number line model here, just for a little flavor—and it’s too close in appearance to the thermometer levels. I didn’t want that confusion creeping in.

Sicklied O’er












Knowledge Is Essential for Cognitive Flexibility





Makin’ Copies






Social Learning Strategies Tournament



Some Key Findings





Why Copying Is Effective








Variation and Example Spaces

I‘ve been thinking a lot about Craig Barton’s wonderful book How I Wish I’d Taught Maths and have been scanning three of his new websites, Variation Theory, Same Surface, Different Deep Problems, and Maths Venns, as well as some research and other books on variation, and a lot of online commentary, in anticipation of starting to implement these ideas in some way.

Writing Algebraic Expressions

As I was reading the last page of Mr Barton’s Book, I was working on instruction around writing algebraic expressions, so this is the topic kind of hovering next to me wherever I go, waiting for when I have time to dig in. This topic is a little more fraught than the purely procedural examples that have been circulating, so it’s worth exploring how variation can be applied to something a little looser.

What does writing algebraic expressions involve (for a beginner)? Well, if I force myself to ignore what other people think writing algebraic expressions involves (essentially ignoring standards and any written material on the topic), then I would say that writing algebraic expressions means to write something like s + 2 or 2 + s when presented with a question like “How old in years will Sam be in exactly two years?”1

This, then, I would call the first example in my example space. Or, rather, an example of an example in the example space—because, if this example is any good, then I will use it as an instructional example to start and leave it out of variation work, which is about PRACTICE, not instruction.2 So, something like this, with the brilliantly simple Silent Teacher method, mentioned in Barton’s book (and a few other places), though without the natural pauses and instructions for students to copy down the correct worked example used during a normal classroom implementation of this.

Try This One

Write an algebraic expression to model the situation.

How old in years was Sam exactly 10 years ago?

I would include a follow-up to this process, here involving a discussion around (a) the idea that the resulting algebraic expression represents an answer to the question of how old Sam will be—it’s just that one part of that expression is not known, (b) asking students to check that the answer makes sense, here by substituting different values for s and comparing the result to the situation, (c) the idea that any letter can be chosen for the variable, and (d) perhaps drawing a visual model of the result (an annotated number line). Some of these could be packaged into the instruction and question above, of course—or perhaps I’ll decide to split this up even more, considering how much “in addition to” I’ve now done about this—but I think that, in general, leaving room for a stepping back step at the end of this is a good idea, to catch the kind of overflow that is difficult to squeeze into expositions like this.

And Now Enters Variation

The paired problem here has opened up a dimension of variation—using addition or subtraction in the expression, so we can play with that during Intelligent Practice (really love that phrase). Technically, the instruction was open to all four operations, but I think it makes sense to focus exclusively on addition and subtraction, leaving multiplication and division expressions for another round.

Here’s what I cooked up.

  1. How much money in dollars did Sam have if he got exactly 10 dollars?
  1. How much money in dollars did Sam have if he got exactly 10 cents?
  1. How much money in dollars did Sam have if he got exactly 2 dollars?
  1. How much money in dollars did Sam have if he lost exactly 10 cents?
  1. How much money in dollars did Sam have if he got exactly 1 dollar?
  1. How much money in dollars did Sam have if he lost exactly 1 dollar?
  1. How much money in dollars did Sam have if he got exactly 50 cents?
  1. How much money in dollars did Sam have if he lost exactly 2 dollars?
  1. How much money in dollars did Sam have if he got exactly 25 cents?
  1. How much money in dollars did Sam have if he didn’t lose or gain any money?

After this, it might be good to have students cut out the strips and place them on a number line.

It’s interesting how much my experience and training rebels against this process. What I want to get to, right away, are the difficult and ambiguous situations. In particular, I started with, and then rejected, a variation sequence involving height: How tall in inches will Sam be if he grows 2 inches? The subtraction variation is bound to confuse: How tall in inches was Sam if he grew 2 inches? That’s tricky.

But knowing about and looking out for those tricky and ambiguous and interesting situations can serve you well creating instructional routines like this. It shows you where you’re going—and your example space can be richer and broader. And if you’re serious about implementing minimally different variation like this, it shows you how far away your knowledge really is from a beginner’s. You just have to learn to have more sympathy for learners who are encountering mathematics for the first time that you’ve seen a gazillion times.

  1. It’s important to me—at the moment, at least—that the examples in this example space should also involve identifying the correct unknown, rather than simply recording the unknown, as would happen with a question like, “Sam is s years old. How old will he be in 2 years?” or with an exercise of the form “2 more than a number.” In both of these cases, the unknown is entirely exposed.
  2. This is an important aspect of variation that I worry will be lost on U.S. teachers. Intelligent practice can’t happen, beneficially, until some acquisition has happened. In 20 years, I haven’t seen a robust public discussion about acquisition. The rhetoric around instruction in the States treats it as just one long assessment, though almost no one realizes that’s what it has become.

Mr Barton’s Book











Do-It-Ourselves Education

When I lived for a year in Germany in high school as a foreign exchange student, I picked up, among many other things, a great quote from my host father: “Die Vorbereitung ist alles.” When said in my first language, it sounds fairly banal: “(The) preparation is everything.”

In both languages, though, the understood meaning of the phrase has a teleological ring: preparation is all-important for the goal you want to accomplish or some particular end you have in mind to achieve or realize. But I prefer a more extreme interpretation of the quote, in particular for education: that there is no achievement track, only a preparation track (multiple tracks in reality).


How do we get to that achievement above from, say, the middle of the preparation track? We don’t (in general). We move along the preparation track until we are in close enough proximity to the achievement to grab it. We don’t, in fact, keep our eyes on the prize. We keep our eyes on the preparation needed to move us within striking distance of the prize. Indeed, from way back in the middle of the track, the prize may look more tempting than it will appear close up (and it may be a mirage). And we may not be able to grasp it until we’re a little past it in our preparation.

Stay on the Right Track

The goal or achievement can be anything, really. So, for example, just cruise Twitter for a bit to find some quotable goal for education. The Feynman quote on the right is a good example. It is part of a quotation from a letter to a student in 1976, in which Feynman refers to himself in the third person, from The Quotable Feynman:

Just because Feynman says he is pro-nuclear power, isn’t any argument at all worth paying attention to because I can tell you (for I know) that Feynman doesn’t know what he is talking about when he speaks of such things. He knows about other things (maybe). Don’t pay attention to “authorities,” think for yourself.

Okay, great. Sincerely, that’s a great goal. I definitely would like to help students be appropriately mistrustful of authority—to the extent that it stimulates constructive thinking, not just having temper tantrums about authority. Who wouldn’t? So, let’s talk about distrusting authority as a goal for education.

The usefulness of the above interpretation about preparation is that now we must find the image of that goal somewhere along the preparation track and work out how we will connect the beginning and middle of the track to the point where that goal can be attained. Almost instantly we will see that we need to define what we (society) want for students. (For example, students have a lot of authority figures in their lives. Will they interpret the pro-skepticism message in a way that makes them start ignoring what their parents tell them? Does skepticism just mean that they have an ability to say, “I don’t think that’s right” and then never follow up?) But more importantly, we need to think about the steps along the path: What do students need to know first to understand skepticism and how to wield it appropriately? How does that ability progress over time? What knowledge is involved?

I don’t know about you, but when I deliberate on that simple Feynman quote for a while, I think of dozens of different sub-steps I would want to put in place along the preparation track from the goal back to the starting point. And these would probably break down into hundreds of smaller steps. Balancing appropriate skepticism—actionable skepticism, not armchair, consequence-free questioning—with the absolute necessity in modern life of trusting experts and authorities is lifelong work for adults who take on that challenge. If we want it to be an explicit goal for students—and not just a slogan we pass around on social media—then it will require a lot of work and technical planning.

My wish for 2018 and beyond is that, in addition to wanting these kinds of big things for students, we realize the hard, technical, scientific work involved in doing those things ourselves. Let’s leave behind the childish idea that, in order for students to achieve X, they just have to do X. That works for small things, not for anything worth having.


–Richard Feynman