Read more here: If Math Is the Aspirin, Then How Do You Create the Headache?
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Let’s try to create headaches for the concepts below.
Ask yourself, “If this skill is aspirin, then what is the headache and how do I create it?”
Ask yourself, “How would a mathematician’s life be worse if they didn’t have this skill?”
Let that guide you in your design of an activity for students.
* Calculating limits. This group from San Diego, CA, creates a headache for efficient methods of calculating limits by first asking students to calculate limits using the very inefficient method of evaluation. Instead of teaching L’hopital’s method straight away to find the limit as x -> 2 of some curve with a hole at x = 2, they ask students to evaluate lots of numbers that get very close to 2 first. “Now how could we have known this would be the limit without doing all that work?”
* Calculating integrals. What shapes can be used to approximate an integral? This Norristown, PA, invites students to guess estimate the area of an irregular shape, estimating the amount of time it’ll take to search a wooded area for a missing person.
* Using the chain rule. Laura Baucom writes: “Students have a background knowledge of the product rule. Ask students to find the derivative of quantities raised to a power, such as f(x)=(3x-5)2. Have them find the derivative as a product of (3x-5)(3x-5). Then ask them to do f(x)=(3x-5)3. Continue with f(x)=(3x-5)4, continuing to push them to see the pattern. Students began to say things like, ‘Just do the power rule times the derivative of the inside.’ It was awesome! Then I put up a problem like f(x)=(3x-5)29 and asked them to find the derivative.”
* Finding a best fit line. This group from San Diego, CA, has students first collect some personal data. Easy. No problem there. Then they ask the students just to estimate a line of best fit. Just draw it right on there. Still no headache yet. The headache comes when they ask students to compare their lines with other students. “We all have different lines. They can’t all be best. How do we settle this?”
* Calculating measures of variation. We came up with measures of variation to help us rank sets of numbers. Rather than jump straight into an explanation of how to calculate variances (for example) this group from Norristown, PA, asks us to have students guess which contestant in a dance-off should win given each of their four ratings.
* Calculating trig ratios. If these ratios didn’t exist (like sine 30° = .5) mathematicians lives would be a drag because they’d have to calculate different ratios for every right triangle ever. The aspirin is that if you know the ratio for one right triangle, you know the ratio for every right triangle with that angle. So like these Pasco & Yakima, WA, folks suggest give each student a piece of paper that has lots of right triangles with the same non-right angle. Like 30°-60°-90° with different side lengths. Ask them to calculate opposite / hypotenuse for all of them. This is easy, but tedious. Eventually they may notice a fact that will help them go much faster.
* Modeling data using trig functions. Give students two data points from a trig function. Like daylight hours over time. Or temperature over time. Ask them to model it with a line and tell you what some point way off in the future means. The more ridiculous the conclusion the better. (ie. “This model says there will be two minutes of sunlight per day in two years.”) Then we have established the need for a better model – like a sinusoid. [More.]
* Proving trigonometric identities. According to Harel, proof satisfies the need for certainty. So we give students something simple to do. “Pick an angle. Any angle. Now calculate sin^2 + cos^2 of that angle.” We ask them to compare with their neighbors. That’s … weird. We all have the same answer. “Okay, that had to be a coincidence. Evaluate a different angle. That can’t happen all the time … can it.” The headache is this strange coincidence and the aspirin is our proof. (See also: Vancouver, WA, with proving triangle theorems.)
* Calculating inverse trigonometric ratios. Seattle, WA, offers us a useful task here asking the very guessable, very headachey question, “Which bike ramp would be worst to bike up?”
* Calculating solutions of trigonometric equations.
How do we motivate the fact that the solution set of this equation can be written as "x = π/6 + 2πn or x = 5π/6 + 2πn where n is an integer"? It seems like needless notation. Instead of proceeding directly to that solution in a worked example, ask students to tell you any solution to sin x = 1/2. Start collecting them. Ask "is that ALL of them?" It will become tedious to come up with all of them. "Can anyone give me a concise summary of ALL of them at once?" Take whatever natural language your students offer you, clunky but not incorrect, and show them how mathematicians write the same answer concisely.
* Translating between point-slope and standard forms of linear equations. Give students ten linear equations. Tell them to graph any six of them by plotting two points for each. What your students don’t know but you do is that all ten equations have the same linear graph. That fact will slowly emerge for your students who may wonder if there is a way to know in advance if the graphs are the same or not. Explain that this is why we choose a standardized representation for the equation of lines and show them how to use that representation.
* Writing an equation of a line and using it to determine if points are solutions. Our colleagues from upstate New York invite every student to write down a point and pick a partner. Everyone puts their coordinate on the board and, by estimation alone, decides which ones are and aren’t on the same line as their two points. This may create a “need for computation,” through which you can then help your students understand how to construct the linear equation that defines what’s true about every point that is on their line.
* Adding / subtracting rational expressions. Our colleagues in Brentwood, CA, have tackled a very challenging standard, the kind that can make teachers and students alike feel like washing their hands of the whole math education project. But is there a point to the symbolic manipulation? Why would someone prefer the simplified form of a lengthy sum of two rational expressions with unlike denominators. Yes, all things being equal, mathematicians prefer the simple to the complex, the elegant to the ungainly. But students may not find the elegance worth the simplicity. So instead we recognize that the simplified form allows for easier, faster operations, operations like graphing and evaluating.
* Graphing. Show the unsimplified form of a sum of rational expressions. Ask students to pick three numbers and evaluate them. Ask them to graph those answers as a function of their original numbers. Give every student three circular stickers to put up on the board to represent each of their three answers. Then ask them to graph the same for the simplified version, though don’t tell them how the two functions relate. The similarity of the two graphs may provoke a headache (“How is that possible?”) and in the following conversation, you can show your students how one form becomes the other.
* Evaluation. Ask the class collectively to select three numbers from 0 to 10. Then show them the unsimplified form of the sum of rational expressions. Ask them to evaluate those numbers. As they labor through the evaluation, you write “Answers” on the board and below them you conceal the three answers. This doesn’t seem possible. You’re a math teacher, of course, and perhaps quicker with the evaluation than most. But the students chose those numbers. You didn’t know what they were in advance. That headache can provoke your lesson. Happily, the evaluation work your students do, in addition to provoking their headache, will prepare them to learn from your lesson. (You: “Here you guys were creating like denominators using numbers, it’s the same here with variables.”)
* Constructing inverse functions. Ask everybody to pick a number from 0 to 10. Tell them to write it down. Now show them the function f(x) = 2x - 10. Have them evaluate their x into f(x). Now ask them, “Did anybody get the number I wanted?” Write 50 on the board. No one did. “Ask them to try again. And again. And again.” It’s easy, but it’s also tedious. Those are great conditions for an explanation about algebra.
* Graphing inequalities. When we’re graphing one-variable inequalities (like x < 3) and two-variable inequalities (like x + y < 5) mathematicians have methods for showing a picture of all the solutions where writing all the solutions down would be impossible. We’d like students to feel how annoying it would be to manually mark every single solution to these inequalities on a number line or coordinate plane. Have them pick a number (or two numbers for the two-variable version). Then show them the inequality. “Does your number work? If so, put a dot. If not, put an x. Great. Have we found all the solutions. Try another number.” They keep doing this until we throw our hands up and say, “This is miserable. We know where all the solutions are. Is there a faster way to show them all?” Glad you asked. Great work from Pasco & Yakima, WA, here. I don’t even think the context adds a lot after they nailed the headache.
* Factoring quadratics with integer roots. Great work from two Wenatchee, WA, teams here. I can’t offer anything more. Similar from San Diego, CA.
* Using the distributive property. The distributive property makes collecting and adding like terms easier and faster. Since you know the property and your students don’t, challenge them to a race. In this example from Pasco & Yakima, W