If Graphing Linear Inequalities Is Aspirin, Then How Do You Create The Headache?

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This Week’s Skill

Here is the first paragraph of McGraw-Hill’s Algebra 1 explanation of graphing linear inequalities:

The graph of a linear inequality is the set of points that represent all of the possible solutions of that inequality. An equation defines a boundary, which divides the coordinate plane into two half-planes.

This is mathematically correct, sure, but how many novices have you taught who would sit down and attempt to parse that expert language?

The text goes on to offer three steps for graphing linear inequalities:

  1. Graph the boundary. Use a solid line when the inequality contains ≤ or ≥. Use a dashed line when the inequality contains < or >.
  2. Use a test point to determine which half-plane should be shaded.
  3. Shade the half-plane that contains the solution.

The text offers aspirin for a headache no one has felt.

The shading of the half-plane emerges from nowhere. Up until now, students have represented solutions graphically by plotting points and graphing lines. This shading representation is new, and its motivation is opaque. The fact that the shading is more efficient than a particular alternative, that the shading was invented to save time, isn’t clear.

We can fix that.

What a Theory of Need Recommends

My commenters save me the trouble.

Chris Hunter:

Ask students to find two numbers whose sum is less than or equal to ten (or, alternatively, points that satisfy your 2x + y < 5 above). The headache is caused by asking them to list ’em all. The aspirin is another way to communicate all of these points — the graph determined by the five steps listed above. Rather than present the steps, have students plot their points as a class.

Bowen Kerins:

One problem I like is having each kid pick a point, then running it through a “test” like y > x2. They plot their point green or red depending on whether or not it passes the test — and a rough shape of the graph emerges.

John Scammell writes about a similar approach. Nicole Paris offers the same idea, and adds hooks into later lessons in a unit.

Great work, everybody. My only addition here is to connect all of these similar lessons with two larger themes of learning and motivation. One large theme in Algebra is our efforts to find solutions to questions about numbers. Another large theme is our efforts to represent those solutions as concisely and efficiently as possible. My commenters have each knowingly invited students to represent solutions using an existing inefficient representation, all to prepare them to use and appreciate the more efficient representation they can offer.

They’re linking the new skill (graphing linear inequalities) to the old skill (plotting points) and the new representation (shading) to the old representation (points). They’re tying new knowledge to old, strengthening both, motivating the new in the process.

Next Week’s Skill

Proofs. Triangle proofs. Proving trigonometric identities. If proof is aspirin, then how do you create the headache?

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.

18 Comments

  1. Firstly, “solving” a linear inequality is finding the set of points satisfying the inequality. I guess that nowadays “solving” a linear equation in two variables is “plotting its graph”. In each case we are finding the solution SET (of points, ordered pairs, whatever). The complete disregard of the CCSSM for anything to do with sets is one of the great losses to conceptual understanding, and this stuff is one of the victims.
    Secondly, after all this stuff on “solving”, how many students can figure out which side of a line a point lies, and how many can find out how far a point is from a line.
    I still think that this obsession with linear inequalities is a leftover from the days when “linear programming” was flavour of the month.

  2. One more thing, about shading. It is more sensible to shade the part that does not satisfy the inequality, as if you have to find the part that satisfies two inequalities at the same time then the solution region is the part left unshaded.

  3. Next week’s skill:proof. I immediately think of this excerpt:

    “But you can’t prove it,” said Susan.
    Hawk smiled his warm meaningless smile, “provin’ don’t matter to me, knowin’s enough.”
    “I want it all”
    From Pale Kings and Princes, as quoted on PROMYS problem set #7

    The next thing I think of are non-proofs and what we gain by discriminating between proofs and non-proofs. In particular, mathematics is a game, but we have to write down the rules. Proofs are a way of testing to see if we have gotten a usefully complete set of rules. When we don’t use all our axioms in a proof, then we have identified a pattern that holds more generally, like a conjecture about integers that actually is true for all rings. When our axioms are insufficient or we are called to assume something that feels artificially technical, we can open a new world to explore by making a different assumption.

    A prime example of the latter is non-euclidean geometry. This came up recently when we were rediscovering a simple proof of the sum of interior angles of a triangle: proof for 5 yr old.

  4. I’ll be very interested to see your write-up on proofs. Proofs are one of those topics that could be so great, but all the fun has been sucked out over time. As soon as you say “proof”, the students moan and say, “Those are too hard!”, even before the unit starts.

    Funny story, this year I had a student come to me after school for some help with a proof, and he confided, “I don’t know why no one likes these. I actually kind of think they’re fun, but everyone else hates them.” I would wager a fair number of my students secretly enjoy proofs, but the stigma against them is so strong.

    We played logic games this year as an introduction to indirect proof/proof by contradiction. I didn’t use the word “proof” until the very end. We started with this one:

    Use two red pieces of paper, and one white piece. Get two student volunteers. Show everyone, including the volunteers, all three papers. Have the volunteers stand close together, facing the class. Without letting the students see, keep the white paper and tape one red paper to each volunteer’s back. Forbid the volunteers from speaking to each other or removing their own paper. Tell the volunteers to look at the other’s back without letting the classroom see. (There’s a few ways to do this, with paper, a ball held behind the back, etc)

    Now, ask the volunteers, “What color is the paper on your back?” Ask the class as well. “What color paper is on [student name]’s back? What color is on [other student]’s back?Hopefully, someone in the class will solve the puzzle. Give them about 60 seconds to think. This leads to an interesting discussion. “The two test subjects each had MORE information than the class, they knew the other’s color. The rest of the class had NO information. How could someone in the audience figure it out before the test subjects?”

    I enjoy this activity a lot because when the students hear the solution, there’s an audible “Ohhhhh” followed by everyone talking to each other about the solution. Very rarely can I get 25+ students all talking about the solution to a math problem.

    This leads us to the difference between direct and indirect problem solving. “What’s the direct solution? Tear the paper off your back and look. But that wasn’t possible. So what do you do?” I usually ask the class who enjoys watching crime shows or reading mysteries. Our time with indirect proof is very “detective”-themed, which has done well for me in engaging the students and making the proof fun again.

    Actual teenager quote: “AB can’t be congruent to BC, that’s a contradiction! Awww yeah, bro!”

    I can safely say nothing like that ever happened while using the textbook examples.

  5. I wrote this up a few year ago: Modeling Linear Equalities. I’ve sculpted it a bit over the years, and it’s not as “step right on through” as it looks in the write up.

    I’m not much on aspirin per se, but as I mention in the entry, I had previously exempted inequalities from my curriculum design, giving it short shrift. When I committed to redo the sequence, I really gave some thought for how to make it meaningfully different from linear equations.

    My next goal is to take the Pizza party and make it less formulaic. Right now, I’m having them figure out the equations and then shade the solution region. Would be more interesting to come up with another t/f series and have them work it backwards.

  6. I like the four approaches that you mention, but I notice that they all start at an abstract level. There’s nothing wrong with that, but here is an approach that I’ve been using to introduce linear inequalities to kids who are a little weak on jumping right to the abstract.

    I describe details of the lesson below, but in summary, here’s the core need/headache-inducing idea: don’t mention shading for several days. Start with simple numbers and concrete word problems (first quadrant only) and always have the students “graph every integral pair of numbers that is both on the grid and in the relationship.” As a bonus, demand this without ever suggesting that they ‘draw a dividing line’ as a guide.

    Here are more details on how I lay that out.

    When we start, I do NOT tell students the topic is Linear Inequalities. Instead, I tell them they have $15 to buy chips and subs. Chips are $2 and subs are $5. I tell them to make a table with every possible purchase combination. You might think they’d start with (0,0), then (1, 0), etc., but they have just been steeped in linear equations for several weeks, so by now that topic has become THE fixed notion in their heads. I amp this by building them up to a ‘statement’ of:

    2X + 5Y 15

    Funny thing, most students assume I’ve mandated spending all the money, so they make tiny tables {(0, 3), (5, 1)} and then sit and wait. Good things eventually happen. Maybe someone asks: “do we have to spend all the money?” or someone notices that the group in the corner is still working or I say to a group “you’re missing a few” or I eventually announce “make sure you get all 18 pairs of numbers.” Lots of ‘ah-ha’ moments.

    Next they come up with the rule, introducing the inequality symbol. After that, we finally graph, but by then it is sort of obvious: just a visual representation of all those pairs of numbers. NOT a shaded region: rather, a bunch of points.

    We practice with computer-generated multi-part word problems. The graphing part does NOT have a shading option — students have to click on every point in the relationship. The program does not require them to draw a ‘dividing line,’ although they can sketch one as a guide if (when) they figure out how helpful that is. After a while, the program quietly opens up a shading option for students who have made a lot of progress, but if they begin to make mistakes, it takes the option away.

    Feel free to use the ppt that backs the discovery here:

    http://getmoremath.com/LinearInequalitiesWP.pptx

    You can also use the computer generated-problems as much as you’d like here:

    http://getmoremath.com/Mobile/linearinequalities.html

    I love this lesson. It is one of those pivotal days when a bunch of prior concept building (linear equeations) couples with a simple new challenge to spring a whole new concept into place in ‘hey-presto.’ When we do the abstract problems a few days later, concepts are strong and everyone gets it rapidly. Same deal with systems.

    Regards,

    Josh Britton

  7. I like to use a “notice and wonder” approach to proofs in geometry. For example, I might give students a diagram of a parallelogram with diagonals and ask them what they notice and wonder. My goal is to get them to “wonder” if whatever they “notice” will always hold true. I often use GeoGebra for this step and give them a chance to explore, but perhaps doing some measuring by hand would create more of a “headache”. Once they’ve determined their conjecture works for many specific examples, we move on to trying to determine if it will always hold true which, if it does, then leads to a formal proof. I also like to throw in problems where a conjecture appears to be true for many specific cases but then students realize it won’t always hold true.

  8. Shannon McClintock

    July 17, 2015 - 2:03 pm -

    The classic example that I like to use in my class is the number of regions formed when connecting various numbers of points on a circle. Connect each point to every other point on the circle and count the number of regions formed. (Make sure you place the points on the graph 1 point = 1 region. 2 points = 2 regions. 3 points = 4 regions. 4 points = 8 regions. 5 points = 16 regions. But when you get to 6 points, there’s only 31 regions. Conjecture busted.

    Most of them are sure it works after just 4 points, and then doing 5 points confirms their idea. Then 6 points absolutely blows their minds. Does this example get them totally excited about doing proofs? No. But they at least have a small taste of why proofs are important and necessary, and they are slightly more willing to do them.

  9. A disclaimer. I am opposed to placing the ‘need’ in the student (i.e., causing the ‘headache’ to produce the ‘aspirin’). I have already written about this on my own weblog; rather than reiterating my criticism here, I offer the following suggestion.

    Background. I have found that, once students are convinced that the sum of interior angles for a triangle is 180 degrees, it is not hard to bring them on board for two methods to compute a convex n-gon’s sum of interior angles (methods of Proof which can be unpacked, compared/contrasted, have their assumptions questioned, and so forth):

    1. Connect one vertex to each non-adjacent vertex, thereby creating n-2 triangles, which each have interior angles summing to 180 degrees. Summing across all constructed triangles gives the sum of the convex n-gon’s interior angles; i.e., (n-2)180 degrees.

    2. Construct a point in the interior of the convex n-gon, and connect all n vertices to it, thereby creating n triangles, which each have interior angles summing to 180. Summing across all triangles gives n180 degrees; but we have overcounted by the 360 degrees around the constructed point, so we end up with a total of: n180 – 360 degrees.

    Luckily, the two formulae agree: (n-2)180 = n180 – 360.

    Foreground. A number of students (at many ages and stages!) quite like to doodle. I have noticed that few of them are able to freehand draw a 7-point star (at the least – few have seen it done) and many become suddenly curious about such a figure’s construction.

    Here is a rough picture of how to draw one: http://i.imgur.com/70npKaT.jpg

    I have used colors to indicate how the pen[cil] or marker moves; in order: red, orange, yellow, green, blue, purple, black.

    In the past, my students have gravitated towards sketching this picture [give them a chance to try and freehand it on their own, first!] and this may pique their interest in stars.

    This is a good opportunity to explore stars. As far as the next topic mentioned (“triangle proofs”) a nice question is, what is the sum of the interior angles of the 7-star?

    In fact, maybe it is best to step back and examine a 5-star. Then a 6-star, a 7-star… and can they figure out the formula for an n-star? (There are, of course, opportunities to use technology here: perhaps graphing software that uses angle measures to draw a shape, which means computing by hand certain interior angles for a particular star.)

    Note that stars are not convex figures (why?) so we ought not assume the previous techniques will necessarily* work in finding their sum of internal angles. Still, method 2 does work quite nicely: Draw a central point, and connect it to each vertex; how many triangles does this make? How does this help? Etc.

    All of this business gives a potential opportunity to talk about proofs more generally (what are our assumptions? – e.g., that the sum of any triangle’s interior angles is 180 degrees, or that all stars are “star convex”; how are we reasoning from one step to the next in the proof for a fixed n?; how are we reasoning in providing the formula for general n?; why did we start with a 5-star and not, say, a 4-star? etc).

    I cannot say this is the right task for All Students, but it can be a good direction in which to press further their knowledge of interior angles (which might already involve at least the two methods above, though this is usually pressed further by asking for the interior angle of a regular n-gon or some such thing) while emphasizing important components in proving (besides assumptions: strategies/heuristics, pitfalls such as overcounting, generalizing- to name a few) and one can certainly take it even further (e.g., examining Koch snowflakes).

    Though I recognize the technology component could be well-implemented, I might reiterate that the doodling aspect appears to stay with [some] students. (Notebooks and scrap-paper tell no lies…) Once their interest in a geometric figure is real and personalized, exploring the shape and playing with its properties become more meaningful.

    MQ

    *An n-sided polygon always has interior angles summing to (n-2)180 degrees. (See the classic, and funnily titled, “Polygons Have Ears” by G.H. Meisters.) A nice task is to draw a concave figure C for which neither method 1 nor 2 above can be used directly to find the sum of interior angles; for curious students, a reasonable follow-up question could be: What is the fewest number of sides C can have – and why?

  10. Good ideas here for linear inequality. The top image makes me wonder about a guessing game a la Battleship. Take turns guessing points, first to get two points on the line or guess the equation wins. That’s different than creating the headache. But it might be motivation for making the aspirin.

    Proofs… @Joshua: Never thought to see a Spenser quote on a math blog. This gets at the problem for me: Hawk doesn’t need proof, and a lot of our students don’t either. Maybe they can all appreciate proof, though?

    I’ve approached it a lot like Sandy, extension of noticing and wondering. The headache almost has to be their own conjecture. The problem (if it is) with dynamic geometry is that the examples are so convincing. It’s basically a million examples (usually literally uncountable) that all verify. How much proof do you need?

    So what do mathematicians get out of it? For me, it’s a top level of knowing and understanding. If you can prove it, usually you really understand it.

  11. I think I’m going to jump in later with some thoughts on proofs…but…my thoughts on inequalities.

    My 1st year teaching, I essentially did this. Choose three points that make y>2x+5 true. Graph them. Write them down. Share them out with the whole group.

    The problem was, this took a lot of time (and we had 55 minute classes at the time). This seems like something that could be accomplished through the use of technology (and taking textbooks out of “airplane mode”). Have all the kids graph 2 points that work and two points that don’t. Automatically display the graph with all the students’ points. See if students notice anything. This is a case where technology increases sharability and speed of getting to the WTF…

  12. Howard Phillips:

    I still think that this obsession with linear inequalities is a leftover from the days when “linear programming” was flavour of the month.

    Maybe. Certainly a bunch of people on Twitter recommended teaching linear inequalities only in the context of linear programming. But from another angle, it’s part of establishing the coherence of secondary math. Up to this point, we’ve graphed inequalities in one dimension and equalities in two. Apart from the existence of standards, I’m still going to wonder what happens when you mash those two up.

    Sandy:

    I like to use a “notice and wonder” approach to proofs in geometry. For example, I might give students a diagram of a parallelogram with diagonals and ask them what they notice and wonder. My goal is to get them to “wonder” if whatever they “notice” will always hold true. I often use GeoGebra for this step and give them a chance to explore, but perhaps doing some measuring by hand would create more of a “headache”. Once they’ve determined their conjecture works for many specific examples, we move on to trying to determine if it will always hold true which, if it does, then leads to a formal proof. I also like to throw in problems where a conjecture appears to be true for many specific cases but then students realize it won’t always hold true.

    If proof is the aspirin, then doubt is the headache. Doubt is generally absent from proofs that read, “Prove the diagonals of a parallelogram bisect each other.” Students know they bisect each other now. The proof is a contrivance. With your method, students have a moment to wonder if what they’re seeing between their parallelogram and their neighbors will be true of everyone’s.

    Mr Ruppel:

    The problem was, this took a lot of time (and we had 55 minute classes at the time). This seems like something that could be accomplished through the use of technology (and taking textbooks out of “airplane mode”).

    Seems like John Golden’s idea could similarly benefit from some crowdsourced acceleration. Fans of both.

    Maya Quinn, Shannon McClintock, Josh Britton, ed, & Jennifer:

    As much as I like the lesson ideas you all are suggesting, I’m more interested here in the underlying theory of task design motivating each of them. Tasks don’t generate other tasks; task design frameworks do.

  13. Anyone who remembers the first time they encountered the Monty Hall problem or has tried it with students or others unfamiliar with it, knows that there are times when people are DESPERATE for a proof. There’s no question but that there’s a headache here. This problem also demonstrates that it’s not necessarily the case that knowing what you’re trying to prove keeps you from wanting to prove it. I think the relevant thing about the Monty Hall problem in the context of this headache/aspirin question is that it is an example of a problem with (i) no barrier to initial engagement and (ii) an answer that is so counter-intuitive that people simply need to know why it’s true (or are certain that they are going to be able to show that the purported answer is wrong).

    I’ve found that I can use trig identities to provide precalculus students with this experience of NEEDING to know what the heck is going on then and discovering that they having the tools to figure it out for themselves. (Eventually, when they’re all done, I can point out that they’ve done a proof.):

    Once they’re very familiar with the unit circle and the graphs of the six trig functions I ask them to type a set of complicated-looking equations into Desmos (The ones I’ve used most recently are at https://www.desmos.com/calculator/vj2budpxkh). They expect crazy graphs and are quite surprised to see horizontal lines (with the possible exception of some weird very narrow spikes which tend add to the motivation for figuring out what’s happening) and the graphs of trig functions they recognize well. It’s helpful in this case that the insights required to make progress are accessible enough (e.g., tan x = sin x/cos x) that someone always thinks of them and no one says “how did you ever think of that”? Also, the fact that we’ve begun with multiple puzzles which turn out to be solvable using similar but not identical insights means that even those who need help from the group the first or second time through eventually experience the satisfaction of proving something for themselves and different people find the keys to different problems.

    This year, someone yelled out in the midst of a frenzy of proving, “Whoa, is there some whole other kind of crazy arithmetic of trig functions?” A vast improvement over “When are we ever going to use this?”!

    (As a bonus, in this approach where students start by looking at graphs, it becomes completely obvious to them that there are actually values of x for which the identities don’t hold. I find the idea that I harp on during consideration of holes in rational functions, i.e., that two expressions can be equal almost–but not quite–everywhere, gets great reinforcement and is finally driven home.)

  14. I’m more interested here in the underlying theory of task design motivating each of them. Tasks don’t generate other tasks; task design frameworks do.

    I wouldn’t have put it that way, but the task-based approach is why I’m redesigning my inequalities section this year.

  15. If anything in math deserves to be social, it is proofs.

    I do massive whiteboarding 4×2 in groups. For me this unit is all about communication and fun. Students do work in groups as well as quizzes in groups, but there is a caveat…. Groups change frequently. 3 choices, they choose, I choose and random.

    Our goal is making them better at seeing another viewpoint, right? What better way then encouraging them to listen and share frequently? Especially if they experience competition and discussion about these ideas.

    How about they pick groups. They get 4 problems. 5 mins… No writing. Then. Groups change. All must be proven, writing only, except each one must be in a different persons handwriting. Then groups change and these groups participate In a group quiz.

    Or. Then. Groups change and all different solutions or methods are discussed and verified or disproven and discussed regarding strengths and weaknesses. I do still like the group quiz idea.

    I even test with a hybrid method with part done individually and part done either in a group or individually.

    Regarding the interior angles discussion. (Polygons). I am convinced the usual exploration method is flawed. Build the polygons and the formula comes much more organically.

  16. I kind of like this thought experiment around the headache/aspirin metaphor.

    I wrote a blog post about Triangle proofs, centering around this idea:

    ————
    So here’s my scratchwork for a lesson idea: work from the opposite case, can we make a student doubt that different triangles exist? We infuse doubt by assigning students to find non-congruent triangles. When they run up against their sandbox’s boundary– the conditions that cause some triangles to be automatically congruent– they can all of a sudden doubt that triangles can always be made differently.

    more here at my shameless plug: http://scottfarrar.com/blog/if-triangle-proofs-are-the-aspirin-what-is-the-headache/