Teach the Controversy

Here is how your unit on linear equations might look:

  1. Writing linear equations.
  2. Solving linear equations.
  3. Applying linear equations.
  4. Graphing linear equations.
  5. Special linear equations.
  6. Systems of linear equations.
  7. Etc.

On the one hand, this looks totally normal. The study of the linear functions unit should be all about linear functions.

But a few recent posts have reminded me that the linear functions unit needs also to teach not linear functions, that good instruction in [x] means helping students differentiate [x] from not [x].

Ben Orlin offers a useful analogy here:

If I were trying to teach you about animals, I might start with cats and dogs. They’re simple, furry, familiar, and lots of people have them lying around the house. But I’d have to show you some other animals first. Otherwise, the first time you meet an alligator, you’re gonna be like, “That wet green dog is so ugly I want to hate it.”

Michael Pershan then offers some fantastic prompts for helping students disentangle rules, machines, formulas, and functions, all of which seem totally interchangeable if you blur your eyes even a little.

Not all rules that we commonly talk about are functions; not all functions are rules; not all formulas have rules; not all rules have machines. Pick two: not all of one is like the other. A major goal of my functions unit to help kids separate these ideas. So the very first thing I do is poke at it.

And then I was grateful to Suzanne von Oy for tweeting the question, “Is this a line?” a question that is both rare to see in a linear functions unit (where everything is a line!) and important. Looking at not lines helps students understand lines.

So I took von Oy’s question and made this Desmos activity where students see three graphs that look linear-ish. The point here is that not everything that glitters is gold and not everything that looks straight is linear. Students first make their predictions.

Then they see the graphs again with two points that display their coordinates. Now we have a reason to check slopes to see if they’re the same on different intervals.

Finally, we zoom out to check a larger interval on the graph.

I’m sure I will need this reminder tomorrow and the next day and the next: teach the controversy.

BTW. In addition to being good for learning, controversy is also good for curiosity.

Bonus. Last week’s conversation about calculators eventually cumulated in the question:

“Calculators can perform rote calculations therefore rote calculations have no place on tests.” Yay or nay?

I’ve summarized some of the best responses — both yay and nay — at this page. (I’m a strong “nay,” FWIW.)

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


  1. We work on scatter plots throughout our “linear” unit and the other day we came across one where the line of best fit made a whole lot of sense for the first 10 data points or so, but quickly realized our predictions were way off after that. My first instinct was to just say, “Well, that’s extrapolation, kids!” and move on to another example. I’m not sure why, but I didn’t do that this time.

    Useful anecdote!

    I thought I had made a huge mistake when I allowed my students to explore other possible models. I guess I was worried it would only confuse them about lines. While none of them found any models that really fit the data, they did figure out how to “make the line bend/curve” and how to make a parabola. The idea that there is a world outside of linear equations has made the world of lines so much easier for them to understand. Not only that, but I imagine there will be so much less confusion when we oh so casually move into quadratic equations.
  2. Chester Draws

    March 31, 2017 - 12:33 pm -

    To me it depends entirely where the students are in their knowledge.

    If you are talking about students just starting out with lines, then introducing non-lines is a really bad idea. It is likely to confuse them about what they are doing — like talking about how French grammar works would be a bad idea in a junior English grammar class. The first time round the students need to get the ideas into some semblance of order first.

    Otherwise, the first time you meet an alligator, you’re gonna be like, “That wet green dog is so ugly I want to hate it.” A terrible analogy, because it assumes that the person teaching about dogs and cats does not start off with defining a dog or cat. If you teach lines without showing students what a line is and how you recognise one, then you are doing it wrong.

    If the student has seen lines before, then I agree the next time around it is time to mix it up a bit. In these cases I would always mix lines and non-lines because for most students who have the basic concepts grounded the problem is not knowing which skills to use in which situations. Which is why I explicitly teach my senior students how to recognise lines from other curves as the first step of any unit on lines.

    • Chester Draws

      March 31, 2017 - 1:06 pm -

      Which is why I explicitly teach my senior students how to recognise lines from other curves as the first step of any unit on lines.

      Just to make this clear this is the first step for senior students, who will need to deal with both lines and curves during the year.

      It’s not relevant to junior students, who only meet lines.

    • I linked to this author in the original post.

      Sometimes I find online teacher disagreements are just misunderstandings, two people with broadly similar views emphasizing different aspects of a complex job. Here, though, I suspect we really disagree!

      Let me posit a goal: for students to understand that graphs are pictures of relationships. That graphs help us visualize how two numbers (x and y) relate.

      Now, I find lots of teachers (including some colleagues I love) espouse a theory you might call “And Later It All Clicks Into Place.” They believe a student should climb a ladder of examples, carefully chosen for increasing technical sophistication, until eventually – once they have procedural mastery – they can look back and reach a holistic understanding. These teachers might share my desire for students to employ graphs to gain insight about a relationship (e.g., what happens to one variable as the other grows very large?) but they believe this kind of understanding will take years of gradual technical progress before the insight “clicks” into place.

      I think this theory is an accurate representation of how many of us learned math; perhaps some aspects can only be learned this way. But I don’t think the nature of graphs is one of them; I think there’s a better, surer path.

      Before starting in on linear relationships with my 12-year-old students, I like to spend a week on tasks like this: “On your graphs, mark points where… (1) The coordinates are the same as each other. (2) The second coordinate is double the first. (3) The coordinates add to five. (4) The coordinates multiply to twelve. (5) The second coordinate is the square of the first.” And so on: some linear, some nonlinear.

      I give them a few minutes for each, and then we have a conversation: have you gotten them all? Do the “in-between” points fit the rule? If the first coordinate is really big–say, 100–how does that affect the second coordinate? What about negative coordinates? etc.

      I find that students can develop a sense of what graphs *are* – what they’re for, how their visual language works, what they tell us about relationships – before we ever narrow our focus to lines.

      Like anything in teaching, this approach has costs. It defers technical competence; open-ended conversation-driven lessons can get noisy and hard to manage; students can grow frustrated if it breaks their expectations of what learning math is about. But I find “students get confused about linear relationships” is not one of the costs!

      (BTW, Dan, I love the zoom-out punchline to this activity; seems like a striking visual that a kid might be able to draw on when running into “linear approximations” in calculus class a few years later.)

    • Chester Draws

      April 1, 2017 - 4:07 pm -

      Paul, I love your honesty but “It defers technical competence” is not acceptable to me. I belong to the school of thought that says teaching Maths without technical competence is largely useless. And definitely unfair on the teachers who have to patch any gaps I leave.

      I shall have a go this year at trying teaching graphing using what you have suggested and see how it pans out. I don’t see that it need to defer competence — they can’t even start what you suggest until they can plot accurately for a start.

      What concerns me, and I suppose I shall find out, is when I ask “The second coordinate is the square of the first” whether I am going to get graphs — or looks of blankness. I suspect your intake is somewhat better than mine if you are prepared to get into that at the start of a unit. I think I may alter that to “the second co-ordinate is the first one times itself”.

    • Paul Hartzer

      April 1, 2017 - 4:53 pm -

      “for students to understand that graphs are pictures of relationships.”

      One thing I’ve been thinking about recently is how “linear equation” is a bit of a misnomer if we’re coming from this perspective. The name “linear equation” puts the horse before the cart, as it were, in that it says that the equation is the relationship of the object “line”, whereas the thrust lately (as in, the last century or so) has been as Ben says it here: Geometric objects on coordinate planes are representations of relationships, not vice versa.

      I think this is the correct perspective, despite my current (preferred) status as a geometry teacher, but I do wonder how much the name gets in the way of the proper understanding.

    • Hi Chester, I’m largely with you on the centrality of technicality competence. In this approach to the material, I find that certain skills get delayed by about two weeks; a tangible cost, but (except under circumstances of dire test-prep urgency) probably an acceptable one. I should mention that my current school has two weird features: a very strong intake (it’s a selective private school), and fairly little instructional time (2.5 hours of math per week).

  3. Here’s my own perspective on the “nay” for avoiding rote calculation problems: There are situations where a calculator isn’t just a good idea, it’s by far the best approach. This week, I introduced trigonometric ratios. Overwhelmingly, the students tried to find some short cut, some way to mash legs of length 15 and 12 together to get an angle measure of 37-ish degrees. Why? Because they’ve had it bashed into their heads that doing things without calculators is Superior Mathematics and people who use calculators are weak snivelers who deserve to fail.

    Of course, you could look it up on a table (and one of my best students asked for a copy of the table because finds table lookups less work for the rote problems), but without a table, a slide rule (which I showed them, because I wanted to impress upon them that calculators are not the work of Satan), or a calculator, it’s nigh on impossible to move between triangle side lengths and triangle angle measures.

    Common Core’s basic standards include “Use appropriate tools strategically”, and students do need to know there are times when a calculator is the appropriate tool. If we avoid test questions that can best be answered using a calculator, we continue to reinforce that Real Mathematicians don’t use calculators. Of course they do, when it’s appropriate.

    It occurs to me writing this that we could easily modify traditional problems to both require calculators AND push critical thinking. Instead of “In right triangle ABC, BC = 7 and m∠A = 40°. How long is the hypotenuse?”, how about “A right triangle has a side length of 7 and an angle of 40°. What are the possible lengths of the other two sides?”

  4. Chester Draws

    March 31, 2017 - 1:00 pm -

    Here is how your unit on linear equations might look:

    Writing linear equations.
    Solving linear equations.
    Applying linear equations.
    Graphing linear equations.
    Special linear equations.
    Systems of linear equations.

    I don’t think this structure’s problems relate to the problem that you talk about later in your piece Dan. The issues with a structure you show are quite different from those relating to whether we mix lines and not lines in our unit. Adding in “talking about non-lines” would not solve that structure.

    I would structure an introduction to graphing lines for a class of novices or near novices as:

    — Reminder what an equation is. How some equations link two variables — x and y in our case.
    — What equations we are dealing with in this unit, and what they look like.
    — Substitution, to get into the whole business of getting a y for an x.
    — Revising plotting on Cartesian co-ordinates
    — Plotting the results of our substitutions (OMG — a line!)
    — Figuring out the equations from the lines
    — Using gradient-intercept to write the equation
    — Now using gradient-intercept to plot equations as revision on plotting
    — Linking how linear patterns (1, 5, 9, …) work to lines.
    — Revising figuring out gradients, because that’s always the weak point
    — Higher level stuff to keep the better students engaged
    — Mixed practice alternating between skills of drawing, plotting, etc at their various levels

    What that does mean, and this is very deliberate, is that plotting and writing equation of lines is kept entirely separate from solving linear equations. Solving linear equations is done when you do exponents, expanding and factorising etc.

    Beginning students do not see the two as linked — one is a physical process with a visual result, and the other is an algebraic process that yields a number. Just because we as teachers realise the link doesn’t mean students do, no matter how much we tell them.

    Only years later, once the students are properly familiar with lines and properly familiar with linear solving, should the two be formally linked. To do so earlier is to merely confuse. To add in anything other than a passing reference to non-lines in those early years would just make it worse.

    This is hardly personal fancy. In New Zealand the syllabus recognises that beginning students to not see the links between drawing lines and solving linear algebra very easily. The exams we do mostly separate those tasks at Year 11 — solving linear equations is in a separate exam from drawing lines.

  5. Jessica Breur

    March 31, 2017 - 6:03 pm -

    Words to live by.

    A mentor of mine once gave me similar advice: always give examples that are and those that are not.

    When doing proportional show non-proportional. When doing square roots show cube roots. When doing rational show irrational. This has really needed up my discussions and has allowed students to do e more deeply and be curious when they encounter an unknown. We also talk about the fact that very few things are truly linear. Thanks for the great post.

  6. Nice extension.

    That Desmos activity is great – but when I saw it, my mind immediately went to Calculus, where of course we work this problem backwards: Every (differentiable) curve, when you “zoom in” far enough, looks like a line – but which line is it? I’m going to file this activity away not only for the linear function unit in Algebra 1, but also for the introduction to tangent lines in Calculus. Thanks Dan!
    • Thanks, Jen. The visual setup tickled the calculus region of my region but I couldn’t articulate why until your comment. Let me know if you’d like any changes to the activity or visuals.

  7. Very nice translation of those ideas into an activity, Dan! I’m always looking for connections and patterns even across domains, and since I’m a science gal, I immediately think of ideas that relate to science.

    Love this application of domain.

    For example, looking at climate change over a short period of time gives one picture, but enlarging the frame to geological scale shows great fluctuations in temperature. This argument becomes “fuel” for people who claim that global warming is not a problem – yet the current dramatic increase (sometimes called the hockey stick) convinces many people (myself included) that action is needed. The question becomes – how can we accurately model the past and attempt to extrapolate forward in science? What if something that looks like a linear relationship in our “slice” of time is actually non linear? These types of controversies highlight how mathematical modeling and scientific ideas are intertwined, and could spark additional conversations in the classroom.

    This activity that can also become a great opportunity to add context as well as controversy to mathematical ideas.
    Thanks Dan!

    To see temperature in geologic time see — https://climate.nasa.gov/evidence/
    An example of a refutation of climate change — https://www.heartland.org/topics/climate-change/
    Evidence of trends from NOAA – https://www.ncdc.noaa.gov/monitoring-references/faq/indicators.php

    • For example, looking at climate change over a short period of time gives one picture, but enlarging the frame to geological scale shows great fluctuations in temperature.

      Oo. Let’s load that up in an activity where students frame climate data in different ways to draw different conclusions. What could we do with this?

  8. Michael Pershan’s post reminds me of the time I tried to get someone to tell me what, specifically, was the difference between functions and equations. https://educationrealist.wordpress.com/2015/05/24/functions-vs-equations-fx-is-y-and-more/ This led to a really interesting twitter conversation with a bunch of math professors who were stunned to realize how relatively little time high school math teachers spend on the meaning of functions. I”m also pretty happy with my decoding activity, which defines functions, one to ones, and inverses all at one swoop. https://educationrealist.wordpress.com/2015/03/30/illustrating-functions/

    I start linear equations with modeling, although I don’t bring in other shapes. By algebra 2, they either know what a line is, think they know, or are capable of remembering with prodding what they know. So I’m more interested in getting them to realize the many ways to describe them. Also, why is it that you’d never model “Alan has 3x as much money as Tom” in the same way you’d say “Tom has $50 and wants to spend all his money on some tacos and burritos at the concession stand.” And notice that you’d never use point-slope to model an application from its description, unless the description was two points. https://educationrealist.wordpress.com/2013/02/16/modeling-linear-equations-part-3/

    I am in the process of totally revamping my functions units to describe everything in terms of function operations, but I’m not teaching A2 right now, so it’s on hold. But I had gotten as far as getting my kids to think of a parabola in three ways:

    1. Take a line, square it, move it up and down the line of symmetry. (vertex form)
    2. Take two lines and multiply them. (factored form)
    3. My new one, which is really cool: Take a parabola at the origin and add a line to it. (standard form). This has been really tremendous in terms of explaining how to graph from standard form, convert it to vertex without ever having to complete the square, and introduce quadratic formula again without completing the square. Not that I object to completing the square, but my algebra two kids just aren’t capable of grasping it all at once.


    I am definitely mulling this “what it is, and what it’s not” ever since I read Ben’s post last week. Very interesting.

  9. This post makes me think of the visual patterns we do in 7th grade. According to the standards this is when students are introduced to proportional lines…the need to recognize the graphs are linear and pass through the origin.


    But how boring would it be if the only visual patterns we did were proportional?? Students at this age are beyond capable of creating an equation for a non-proportional or even non-linear relationships and graphing them (by hand or using tech.) I like to bring in the idea of “proportional or no?” into our pattern discussions. It’s amazing to hear 7th graders making these predictions and discoveries about what graphs look like.

    I agree that teaching non-examples is incredibly important when learning a new concept.

    • Nice! At what point in the visual patterns tasks do you have them discuss if it’s proportional or not?

    • My routine is typically:
      1. How many items in the next couple steps? How do you see the pattern growing?
      2. How many items would be in stage 46? What would it look like?
      3. How many items would be in any stage? How could you describe what any stage would look like?
      4. Do you think this is a proportional relationship? Why?— At this point students usually determine if there’s a constant of proportionality or they create a step 0 to see if it would have 0 items. Some notice that the graph doesn’t go up at a constant rate so “the graph must be curved.” By the end of the year almost all students recognize when there’s a squared or cubed in the problem it’s going to be a curved graph.

      5. Check by graphing— In the past I had students create their own graph in a shared google doc, but this year I’ve taken to projecting desmos on the screen. I create a table with the points we know and then graph all of the equations students came up with to see if they are the same equation and if they pass through all of the points.

  10. Dwight Williams

    April 2, 2017 - 3:47 am -

    I enjoy this line of conversation. Can I use the graphs above? I think they could lead to some intellectual conversations among my students

  11. I’d like to suggest the source of confusion around visual results and algebraic processes results from the Cartesian connection between number and location. Students first see a magic marriage of number and point in a “number line”, and then the magic continues with a “coordinate plane”. Lines and planes are geometric objects; they do not come with numbers. Number lines and coordinate planes are temporary crutch concepts. We know better, but we do not know how to explain it to children.

    I maintain the missing concept is a function that takes a number (or two for a plane) from a number system and produces a point in a space. It is often said a function maps from one to the other. Elaborated and expanded in the last 150 years, the function concept has come to play a central role in mathematics, but nothing has come along to introduce the idea into school education. The result is not pretty: with nothing to get at the heart of the matter, education is left to explain what has happened, not show how to make it happen.

    Let me explain a “function” that all us can understand, one that we probably have now. It maps a number to a point in a straight-line space, or the other way around; it is really both a function and its inverse.

    A tape measure, or a measuring tape, displays numbers, each associated with its own tick mark. When the tape measure is aligned with the line, and with its tab anchored at the origin mark that I put on the line, a tick mark associated with a number points to a point to the line. It maps one to the other; It is “functioning” in plain sight for all to see, and understand. Extending a tape, one way or the other from the origin provides a function from the signed rational number to all the points on the line that can be located by measurement.

    A coordinate system on a line is a function, so is a coordinate system on a surface/plane. A student can put a put one on a line or a plane, can see the association of point and number as physical. In fact a student can put two separate coordinate systems on a line and perform all all of signed rational number arithmetic as with measurement on a line.

    There is more to say, but I think one can see how an early introduction of function concept and use can clear a path to more effective teaching and learning. Anyone interested in computer animation of functioneering?

  12. Cool post. Though, correct me if I’m wrong, I believe you reversed graphs 2 and 3 for the graphs that zoom out.