Category Archives: Pedagogy

Deep Unit Circle Understanding

Chris Bolognese started a new Honors Trigonometry unit a few days ago and posted a new problem set, welcoming feedback.  Clicking on the document image gives you access to Chris’ problem set.

This post is my feedback.  I wrote it as a conversation with Chris.  it

FEEDBACK…

Question 2 suggests you have already discussed special angles (or your students recalled them from last year’s algebra class).  What I love about this problem set is how you shift the early focus away from these memorized special angles and onto the deep symmetry underlying the unit circle (and all of trigonometry).  Understanding deep structure grants always far more understanding than rote memorizations ever does!

HINTS OF IDENTITIES TO COME…

The problem set’s initial exploration of trig symmetry starts in parts a-d of Question 1, asking students to justify statements like $sin(x+\pi )=sin(x)$ and $cos(2\pi -x)=cos(x)$.  The symmetry is nice, and I hope you refer back to this problem set when your class turns to its formal exploration of trig identities, helping them see that not all proofs of identities require algebraic justifications.

CONSIDERATION 1:  Put some creative power in your students’ hands.  Challenge them to discover additional algebraic/transformational identities like Questions 1a-1d.  For example, what is the relationship between $sin \left( \frac{\pi}{2} + x \right)$ and $sin \left( \frac{\pi}{2} - x \right)$ ?  There are LOTS of symmetry statements they could write.

CONSIDERATION 2:  I would shift Question 1e to Question 4, as 1e is the basis of the Pythagorean identities you explore in the later group.

CONSIDERATION 3:  I suggest dropping Question 4d and asking students to discover another equation showing another Pythagorean relationship between trig functions.  That cotangent and cosecant have not yet been used is hopefully a loud, silent hint.

CONSIDERATION 4:  They’re not ready for this one until they see sinusoidal graphs, but another trig pre-identity I love is graphing $y=cos^2 (x)$ and asking students to write an equivalent equation using only translations and dilations. Unlike the symmetric relationships in 1a-1d, I don’t think you can actually KNOW it is true without algebraic relationships.  I use problems like this to set up and justify the transition to identities.

VERY CLEVER SUMMATIONS …

I don’t recall ever seeing something like your Question 3 before.  In my opinion, is the gold nugget in the assignment, especially with your students’ prior exposure to special angles.

CONSIDERATION 5:  In some ways the large number of addends in gives away that there must be a simpler approach than the problem suggests on its surface.  What about something that suggests a direct solution if you don’t invoke the symmetry, like ?

CONSIDERATION 6:  Closely related to this, why not shamelessly take advantage of the memorized values to see if students notice the symmetry to notice a simpler approach?  I suggest , with sine or cosine.

CONSIDERATION 7:  My prior two examples tweaked your initial problem.  Like Consideration 1, why not challenge your students to develop their own summations?  I bet they can develop some clever alternatives.

In Question 3, you had some nice explorations using degrees.  Unfortunately, I’m not aware of many equally clever early questions involving radians that aren’t degree-oriented in disguise.  Here are my final suggestions to address this gap.

CONSIDERATION 8:  Without using any technology, rank

sin(1), cos(2), tan(3), cot(4), sec(5), csc(6)

in ascending order.  Note that all angles are expressed in radian measures.  One pair of expressions is very difficult.  (A softer version of this question ranks only sin(1), cos(2), & tan(3).)

CONSIDERATION 9:  One pair of expressions in Consideration 8 is very difficult to rank without technology.  Which pair is this and why is it so difficult to rank?  Exchange angles or functions to change this question in a way that makes it easier to rank.

Thanks for the fun and thoughtful problem set.

Computers vs. People: Writing Math

Readers of this ‘blog know I actively use many forms of technology in my teaching and personal explorations.  Yesterday, a thread started on the AP-Calculus community discussion board with some expressing discomfort that most math software accepts sin(x)^2 as an acceptable equivalent to the “traditional” handwritten $sin^2 (x)$.

From Desmos: Some AP readers spoke up to declare that sin(x)^2 would always be read as $sin(x^2)$.  While I can’t speak to the veracity of that last claim, I found it a bit troubling and missing out on some very real difficulties users face when interpreting between paper- and computer-based versions of math expressions.  Following is an edited version of my response to the AP Calculus discussion board.

MY THOUGHTS:

I believe there’s something at the core of all of this that isn’t being explicitly named:  The differences between computer-based 1-dimensional input (left-to-right text-based commands) vs. paper-and-pencil 2-dimensional input (handwritten notation moves vertically–exponents, limits, sigma notation–and horizontally).  Two-dimensional traditional math writing simply doesn’t convert directly to computer syntax.  Computers are a brilliant tool for mathematics exploration and calculation, but they require a different type of input formatting.  To overlook and not explicitly name this for our students leaves them in the unenviable position of trying to “creatively” translate between two types of writing with occasional interpretation differences.

Our students are unintentionally set up for this confusion when they first learn about the order of operations–typically in middle school in the US.  They learn the sequencing:  parentheses then exponents, then multiplication & division, and finally addition and subtraction.  Notice that functions aren’t mentioned here.  This thread [on the AP Calculus discussion board] has helped me realize that all or almost all of the sources I routinely reference never explicitly redefine order of operations after the introduction of the function concept and notation.  That means our students are left with the insidious and oft-misunderstood PEMDAS (or BIDMAS in the UK) as their sole guide for operation sequencing.  When they encounter squaring or reciprocating or any other operations applied to function notation, they’re stuck trying to make sense and creating their own interpretation of this new dissonance in their old notation.  This is easily evidenced by the struggles many have when inputting computer expressions requiring lots of nested parentheses or when first trying to code in LaTEX.

While the sin(x)^2 notation is admittedly uncomfortable for traditional “by hand” notation, it is 100% logical from a computer’s perspective:  evaluate the function, then square the result.

We also need to recognize that part of the confusion fault here lies in the by-hand notation.  What we traditionalists understand by the notational convenience of sin^2(x) on paper is technically incorrect.  We know what we MEAN, but the notation implies an incorrect order of computation.  The computer notation of sin(x)^2 is actually closer to the truth.

I particularly like the way the TI-Nspire CAS handles this point.  As is often the case with this software, it accepts computer input (next image), while its output converts it to the more commonly understood written WYSIWYG formatting (2nd image below).  Further recent (?) development:  Students have long struggled with the by-hand notation of sin^2(x) needing to be converted to (sin(x))^2 for computers.  Personally, I’ve always liked both because the computer notation emphasizes the squaring of the function output while the by-hand version was a notational convenience.  My students pointed out to me recently that Desmos now accepts the sin^2(x) notation while TI Calculators still do not.

Desmos: The enhancement of WYSIWYG computer input formatting means that while some of the differences in 2-dimensional hand writing and computer inputs are narrowing, common classroom technologies no longer accept the same linear formatting — but then that was possibly always the case….

To rail against the fact that many software packages interpret sin(x)^2 as (sin(x))^2 or sin^2(x) misses the point that 1-dimensional computer input is not necessarily the same as 2-dimensional paper writing.  We don’t complain when two human speakers misunderstand each other when they speak different languages or dialects.  Instead, we should focus on what each is trying to say and learn how to communicate clearly and efficiently in both venues.

In short, “When in Rome, …”.

Tell a Friend

I’ve been in several conversations over these first couple weeks of school with colleagues in our lower and middle schools about what students need to do to convince others they understand an idea.

On our first pre-assessments, some teachers noted that many students showed good computation skills, but struggled when they had to explain relationships.  Frankly, I’m never surprised by revelations that students find explanations more difficult than formulas and computations.  That’s tough for learners of all ages.  But, in my opinion, it’s also the most important part about developing a way to communicate mathematically.

In the other direction, I frequently hear students complain that they just don’t know what to write and that teachers seem to arbitrarily ask for “more explanation”, but they just can’t figure out what that means.

SOLUTION?:

Just like writing in humanities classes, a math learner needs to seriously consider his “audience”.  Who’s going to read your solution?  I think too many write for a classroom teacher, expecting him or her to fill in any potential logical gaps.

Instead, I tell my students that I expect all of their explanations to be understandable by every classmate. In short,

Don’t write your answer to me; write it to a friend who’s been absent for a couple days.

If a random classmate who’s been out a couple days can get it just based on your written work, they you’re good.

Graphing Ratios and Proportions

Last week, some colleagues and I were pondering the difficulties many middle school students have solving ratio and proportion problems.  Here are a few thoughts we developed to address this and what we think might be an uncommon graphical extension (for most) as a different way to solve.

For context, consider the equation $\displaystyle \frac{x}{6} = \frac{3}{4}$.

(UNFORTUNATE) STANDARD METHOD:

The default procedure most textbooks and students employ is cross-multiplication.   Using this, a student would get $\displaystyle 4x=18 \longrightarrow x = \frac{18}{4} = \frac{9}{2}$

While this delivers a quick solution, we sadly noted that far too many students don’t really seem to know why the procedure works.  From my purist mathematical perspective, the cross-multiplication procedure may be an efficient algorithm, but cross-multiplication isn’t actually a mathematical function.  Cross-multiplication may be the result, but it isn’t what happens.

METHOD 2:

In every math class I teach at every grade level, my mantra is to memorize as little as possible and to use what you know as broadly as possible.  To avoid learning unnecessary, isolated procedures (like cross-multiplication), I propose “fraction-clearing”–multiplying both sides of an equation by common denominatoras a universal technique in any equation involving fractions.  As students’ mathematical and symbolic sophistication grows, fraction-clearing may occasionally yield to other techniques, but it is a solid, widely-applicable approach for developing algebraic thinking.

From the original equation, multiply both sides by common denominator, handle all of the divisions first, and clean up.  For our example, the common denominator 24 will do the trick. $\displaystyle 24 \cdot \frac{x}{6} = 24 \cdot \frac{3}{4}$ $4 \cdot x = 6 \cdot 3$ $\displaystyle x = \frac{9}{2}$

Notice that the middle line is precisely the result of cross-multiplication.  Fraction-clearing is the procedure behind cross-multiplication and explains exactly why it works:  You have an equation and apply the same operation (in our case, multiplying by 24) to both sides.

As an aside, I’d help students see that multiplying by any common denominator would do the trick (for our example, 12, 24, 36, 48, … all work), but the least common denominator (12) produces the smallest products in line 2, potentially simplifying any remaining algebra.  Since many approaches work, I believe students should be free to use ANY common denominator they want.   Eventually, they’ll convince themselves that the LCD is just more efficient, but there’s absolutely no need to demand that of students from the outset.

METHOD 3:

Remember that every equation compares two expressions that have the same measure, size, value, whatever.  But fractions with differing denominators (like our given equation) are difficult to compare.  Rewrite the expressions with the same “units” (denominators) to simplify comparisons.

Fourths and sixths can both be rewritten in twelfths.  Then, since the two different expressions of twelfths are equivalent, their numerators must be equivalent, leading to our results from above. $\displaystyle \frac{2}{2} \cdot \frac{x}{6} = \frac{3}{3} \cdot \frac{3}{4}$ $\displaystyle \frac{2x}{12} = \frac {9}{12}$ $2x=9$ $\displaystyle x = \frac{9}{2}$

I find this approach more appealing as the two fractions never actually interact.  Fewer moving pieces makes this approach feel much cleaner.

UNCOMMON(?) METHOD 4:  Graphing

A fundamental mathematics concept (for me) is the Rule of 4 from the calculus reform movement of the 1990s.  That is, mathematical ideas can be represented numerically, algebraically, graphically, and verbally.  [I’d extend this to a Rule of 5 to include computer/CAS representations, but that’s another post.]  If you have difficulty understanding an idea in one representation, try translating it into a different representation and you might gain additional insights, or even a solution.  At a minimum, the act of translating the idea deepens your understanding.

One problem many students have with ratios is that teachers almost exclusively teach them as an algebraic technique–just as I have done in the first three methods above.  In my conversation this week, I finally recognized this weakness and wondered how I could solve ratios using one of the missing Rules: graphically.  Since equivalent fractions could be seen as different representations of the slope of a line through the origin, I had my answer.

Students learning ratios and proportions may not seen slope yet and may or may not have seen an xy-coordinate grid, so I’d avoid initial use of any formal terminology.  I labeled my vertical axis “Top,” and the horizontal “Bottom”.  More formal names are fine, but unnecessary.  While I suspect most students might think “top” makes more sense for a vertical axis and “bottom” for the horizontal, it really doesn’t matter which axis receives which label.

In the purely numeric fraction in our given problem, $\displaystyle \frac{x}{6} = \frac{3}{4}$, “3” is on top, and “4” is on the bottom.  Put a point at the place where these two values meet.  Finally draw a line connecting your point and the origin. The other fraction has a “6” in the denominator.  Locate 6 on the “bottom axis”, trace to the line, and from there over to the “top axis” to find the top value of 4.5. Admittedly, the 4.5 solution would have been a rough guess without the earlier solutions, but the graphical method would have given me a spectacular estimate.  If the graph grid was scaled by 0.5s instead of by 1s and the line was drawn very carefully, this graph could have given an exact answer.  In general, solutions with integer-valued unknowns should solve exactly, but very solid approximations would always result.

CONCLUSION:

Even before algebraic representations of lines are introduced, students can leverage the essence of that concept to answer proportion problems.  Serendipitously, the graphical approach also sets the stage for later discussions of the coordinate plane, slope, and linear functions.  I could also see using this approach as the cornerstone of future class conversations and discoveries leading to those generalizations.

I suspect that students who struggle with mathematical notation might find greater understanding with the graphical/visual approach.  Eventually, symbolic manipulation skills will be required, but there is no need for any teacher to expect early algebra learners to be instant masters of abstract notation.

I’m teaching Algebra 2 this summer for my school.  In a recent test on quadratic functions, I gave a question I thought would be a little different, but still reachable for those willing to make connections or exert a little creativity.

Write a system of quadratic functions that has exactly one solution:  (1,1).

Their handheld graphing calculators were allowed.  Some students definitely had difficulty with the challenge, some gave a version of the answer I expected, and one adopted a form I knew was possible, but doubted anyone would actually find during a test situation.

I show my students’ solutions below.  But before you read on, can you give your own solution?

WHAT I EXPECTED

We’ve had many discussions in class about the power of the Rule of 4–that math ideas can be expressed numerically, graphically, algebraically, and verbally.  When you get stumped in one representation, being able to shift to a different form is often helpful.  That could mean a different algebraic representation, or a different Rule of 4 representation altogether.

The question is phrased verbally asking for an algebraic answer.  But it asks about a solution to a system of equations.  I hoped my students would recall that the graphical version of a system solution is equivalent to the point(s) where the graphs of the equations intersected.  In my mind, the easiest way to do this is to write quadratic functions with coincident vertices.  And this is most easily done in vertex form.  The cleanest answer I ever got to this question was A graphical representation verifies the solution. Another student recognized that if two parabolas shared a vertex, but had different “slopes”, their only possible point of intersection was exactly the one the question required.  Here’s a graphical version of her answer. From these two, you can see that there is actually an infinite number of correct solutions.  And I was asking them for just one of these!  🙂

WHAT I KNEW, BUT DIDN’T EXPECT

Another way to solve this question makes use of the geometry of quadratic graphs.  If two quadratics have the same leading coefficients, they are the same graph, intersect exactly once, or never intersect.  This is a very non-trivial idea for most students.  While I’m not convinced the author of the following solution had this in mind when he answered the question, his solution works because of that fact.  Here’s what J wrote on last week’s test and its graph.  J used more equations than he needed, but had he restricted himself to just two equations, I’m not sure the lovely pattern would have been so obvious.

This is a very different (and super cool) answer than what I expected my students to produce.  Lesson re-learned:  Challenge your students, give them room to express creativity and individuality, and be prepared to be amazed by them.

NEXT STEPS

J’s answer actually opens the door to other avenues of exploration.

1. Can you generalize the form of all of J’s equations, essentially defining a family of quadratics?  Can you prove that all members of your generalization satisfy the question posed and that no other answers are possible?
2. Can you find forms of other generalized families of quadratic functions whose only solution is (1,1)?
3. Notice that there were two types of solutions above:  A) those with coincident vertices and different lead coefficients and B) those with identical lead coefficients and different vertices.  Are these the only types of quadratics that can answer this question?  That is, is there a system of quadratics with (1,1) as the only solution that have identical vertices and lead coefficients?  Could both be different and (1,1) be the only solution?
4. If I relax the requirement that the quadratics be functions, what other types of quadratics are possible?  [This could be a very nice calculus question!]

For my part, I’m returning to some of these questions this week to stretch and explore my student’s creativity and problem-solving.

I’d love to hear what you or your students discover.

Math Play and New Beginnings

I’ve been thinking lots lately about the influence parents and teachers have on early numeracy habits in children.  And also about the saddeningly difficult or traumatic experiences far too many adults had in their math classes in school.  Among the many current problems in America’s educational systems, I present here one issue we can all change.  Whether you count yourself mathphobic or a mathophile, please read on for the difference that you can make for yourself and for young people right now, TODAY.

I believe my enthusiasm for what I teach has been one of the strongest, positive factors in whatever effectiveness I’ve had in the classroom.   It is part of my personality and therefore pretty easy for me to tap, but excitement is something everyone can generate, particularly in critical areas–academic or otherwise.  When something is important or interesting, we all get excited.

In a different direction, I’ve often been thoroughly dismayed by the American nonchalance to innumeracy.  I long ago lost count of the number of times in social or professional situations when parents or other other adults upon learning that I was a math teacher proclaimed “I was terrible at math,” or “I can’t even balance my own checkbook.”   I was further crushed by the sad number of times these utterances happened not just within earshot of young people, but by parents sitting around a table with their own children participating in the conversation!

What stuns me about these prideful or apologetic (I’m never sure which) and very public proclamations of innumeracy is that NOT A SINGLE ONE of these adults would ever dare to stand up in public and shout, “It’s OK.  I never learned how to read a book, either.  I was terrible at reading.”  Western culture has a deep respect for, reliance upon, and expectation of a broad and public literacy.  Why, then, do we accept broad proclamations of innumeracy as social badges of honor?  When an adult can’t read, we try to get help.  Why not the same of innumeracy?

I will be the first to admit that much of what happened in most math classrooms in the past (including those when I was a student) may have been suffocatingly dull, unhelpful, and discouraging.  Sadly, most of today’s math classrooms are no better.  Other countries have learned more from American research than have American teachers (one example here).  That said, there are MANY individual teachers and schools doing all they can to make a positive, determined, and deliberate change in how children experience and engage with mathematical ideas.

But in the words of the African proverb, “It takes a village to raise a child.”  Part of this comes from the energetic, determined, and resourceful teachers and schools who can and do make daily differences in the positive mindsets of children.  But it also will take every one of us to change the American acceptance of a culture of innumeracy.  And it starts with enthusiasm.  In the words of Jo Boaler,

When you are working with [any] child on math, be as enthusiastic as possible. This is hard if you have had bad mathematical experiences, but it is very important. Parents, especially mothers of young girls, should never, ever say, “I was hopeless at math!”  Research tells us that this is a very damaging message, especially for young girls. – p. 184, emphasis mine

Boaler’s entire book, What’s Math Got to Do With It? (click image for a link), but especially Chapter 8, is an absolute must-read for all parents, teachers, really any adult who has any interactions with school-age children. I suspect some (many?  most?) readers of this post will have had an unfortunate number of traumatic mathematical experiences in their lives, especially in school.  But it is never, ever too late to change your own mindset.  While the next excerpt is written toward parents, rephrase its beginning so that it applies to you or anyone else who interacts with young people.

There is no reason for any parent to be negative about the mathematics of early childhood as even the most mathphobic of parents would not have had negative experiences with math before school started.  And the birth of your own children could be the perfect opportunity to start all over again with mathematics, without the people who terrorized you the first time around.  I know a number of people who were traumatized by math in school but when they started learning it again as adults, they found it enjoyable and accessible. Parents of young children could make math an adult project, learning with their children or perhaps one step ahead of them each year. -p. 184

Here’s my simple message.  Be enthusiastic.  Encourage continual growth for all children in all areas (and help yourself grow along the way!).  Revel in patterns.  Make conjectures.  Explore. Discover.  Encourage questions.  Never be afraid of what you don’t know–use it as an opportunity for you and the children you know to grow.

I’ll end this with a couple quotes from Disney’s Meet the Robinsons.  “Math Play” Presentation for Early Childhood teachers

Even though my teaching experiences are all middle and high school, as a PreSchool-12 math chair and father of 3 young children, I’m intensely interested in how math is presented to very young people.

As a result, I’m presenting ideas for teaching math through fun and exploration to about 55 Cleveland area pre-school through kindergarten teachers this morning.  My handout is on Scribd and should show below.  Math is  about Play and Curiosity.  Teach it that way.

Which came first: Math Ability or Computational Speed ?

I’ve claimed many times in conversations over the last two weeks that I believe many parents and educators misconstrue the relationship and causality direction between being skilled/fluent at mathematics and being fast at computations.  Read that latter as student accomplishment defined by skill on speed testing as done in many, many schools.  Here is a post from Stanford’s Jo Boaler on math anxiety created by timed testing.

Here’s my thinking:  When we watch someone perform at a very high level in anything, that person appears to perform complex tasks quickly and effortlessly, and indeed, they do.  But . . . they are fast because they are good, and NOT the other way around.  When you learn anything very well and deeply, you get faster.  But if you practice faster and faster, you don’t necessarily get better.

I fear too many educators and parents are confusing what comes first.  From my point of view, understanding must come first.  Playing with ideas in different contexts eventually leads to recognizing that the work one does in earlier, familiar situations eventually informs your understanding in current, less familiar settings.  And you process more quickly in the new environment precisely because you already understood more deeply.

I think many errantly believe they can help young people become more talented in mathematics by requiring them to emulate the actions of those already accomplished in math via rapid problem solving.  I worry this emphasis is placed in exactly the wrong place.  Asking learners to perform quickly tasks which they don’t fully understand instills unnecessary anxiety (according to Boaler’s research) and confuses the deep thinking, pattern recognition, and problem solving of mathematics with rapid arithmetic and symbolic manipulation.

Jo Boaler’s research above clearly addresses the resulting math anxiety in a broad spectrum of students—both weak and accomplished.  My point is that timed testing–especially timed skill testing–at best confuses young students about the nature of mathematics, and at worst convinces them that they can’t be good at it.  No matter what, it scares them.   And what good does that accomplish?

Traveling Dots, Parabolas, and Elegant Math

Toward the end of last week, I read a description a variation on a paper-folding strategy to create parabolas.  Paraphrased, it said:

1. On a piece of wax paper, use a pen to draw a line near one edge.  (I used a Sharpie on regular copy paper and got enough ink bleed that I’m convinced any standard copy or notebook paper will do.  I don’t think the expense of wax paper is required!)
2. All along the line, place additional dots 0.5 to 1 inch apart.
3. Finally, draw a point F between 0.5 and 2 inches from the line roughly along the midline of the paper toward the center of the paper.
4. Fold the paper over so one of the dots on line is on tope of point F.  Crease the paper along the fold and open the paper back up.
5. Repeat step 4 for every dot you drew in step 2.
6. All of the creases from steps 4 & 5 outline a curve.  Trace that curve to see a parabola. I’d seen and done this before, I had too passively trusted that the procedure must have been true just because the resulting curve “looked like a parabola.”  I read the proof some time ago, but I consumed it too quickly and didn’t remember it when I was read the above procedure.  I shamefully admitted to myself that I was doing exactly what we insist our students NEVER do–blindly accepting a “truth” based on its appearance.  So I spent part of that afternoon thinking about how to understand completely what was going on here.

What follows is the chronological redevelopment of my chain of reasoning for this activity, hopefully showing others that the prettiest explanations rarely occur without effort, time, and refinement.  At the end of this post, I offer what I think is an even smoother version of the activity, freed from some of what I consider overly structured instructions above.

CONIC DEFINITION AND WHAT WASN’T OBVIOUS TO ME

A parabola is the locus of points equidistant from a given  point (focus) and line (directrix). What makes the parabola interesting, in my opinion, is the interplay between the distance from a line (always perpendicular to some point C on the directrix) and the focus point (theoretically could point in any direction like a radius from a circle center).

What initially bothered me about the paper folding approach last week was that it focused entirely on perpendicular bisectors of the Focus-to-C segment (using the image above).  It was not immediately obvious to me at all that perpendicular bisectors of the Focus-to-C segment were 100% logically equivalent to the parabola’s definition.

1. I knew without a doubt that all parabolas are similar (there is a one-to-one mapping between every single point on any parabola and every single point on any other parabola), so I didn’t need to prove lots of cases.  Instead, I focused on the simplest version of a parabola (from my perspective), knowing that whatever I proved from that example was true for all parabolas.
2. I am quite comfortable with my algebra, geometry, and technology skills.  Being able to wield a wide range of powerful exploration tools means I’m rarely intimidated by problems–even those I don’t initially understand.  I have the patience to persevere through lots of data and explorations until I find patterns and eventually solutions.

I love to understand ideas from multiple perspectives, so I rarely quit with my initial solution.  Perseverance helps me re-phrase ideas and exploring them from alternative perspectives until I find prettier ways of understanding.

In my opinion, it is precisely this willingness to play, persevere, and explore that formalized education is broadly failing to instill in students and teachers.  “What if?” is the most brilliant question, and the one we sadly forget to ask often enough.

ALGEBRAIC PROOF

While I’m comfortable handling math in almost any representation, my mind most often jumps to algebraic perspectives first.  My first inclination was a coordinate proof.

PROOF 1:  As all parabolas are similar, it was enough to use a single, upward facing parabola with its vertex at the origin.  I placed the focus at $(0,f)$, making the directrix the line $y=-f$.  If any point on the parabola was $(x_0,y_0)$, then a point C on the directrix was at $(x_0,-f)$. From the parabola’s definition, the distance from the focus to P was identical to the length of CP: $\sqrt{(x_0-0)^2-(y_0-f)^2}=y_0+f$

Squaring and combining common terms gives $x_0 ^2+y_0 ^2-2y_0f+f^2=y_0 ^2+2y_0f+f^2$ $x_0 ^2=4fy$

But the construction above made lines (creases) on the perpendicular bisector of the focus-to-C segment.  This segment has midpoint $\displaystyle \left( \frac{x_0}{2},0 \right)$ and slope $\displaystyle -\frac{2f}{x_0}$, so an equation for its perpendicular bisector is $\displaystyle y=\frac{x_0}{2f} \left( x-\frac{x_0}{2} \right)$. Finding the point of intersection of the perpendicular bisector with the parabola involves solving a system of equations. $\displaystyle y=\frac{x_0}{2f} \left( x-\frac{x_0}{2} \right)=\frac{x^2}{4f}$ $\displaystyle \frac{1}{4f} \left( x^2-2x_0x+x_0 ^2 \right) =0$ $\displaystyle \frac{1}{4f} \left( x-x_0 \right) ^2 =0$

So the only point where the line and parabola meet is at $\displaystyle x=x_0$–the very same point named by the parabola’s definition.  QED

Proof 2:  All of this could have been brilliantly handled on a CAS to save time and avoid the manipulations. Notice that the y-coordinate of the final solution line is the same $y_0$ from above.

MORE ELEGANT GEOMETRIC PROOFS

I had a proof, but the algebra seemed more than necessary.  Surely there was a cleaner approach. In the image above, F is the focus, and I is a point on the parabola.  If D is the midpoint of $\overline{FC}$, can I conclude $\overline{ID} \perp \overline{FC}$, proving that the perpendicular bisector of $\overline{FC}$ always intersects the parabola?

PROOF 3:  The definition of the parabola gives $\overline{FI} \cong \overline{IC}$, and the midpoint gives $\overline{FD} \cong \overline{DC}$.  Because $\overline{ID}$ is self-congruent, $\Delta IDF \cong \Delta IDC$ by SSS, and corresponding parts make the supplementary $\angle IDF \cong \angle IDC$, so both must be right angles.  QED

PROOF 4:  Nice enough, but it still felt a little complicated.  I put the problem away to have dinner with my daughters and when I came back, I was able to see the construction not as two congruent triangles, but as the single isosceles $\Delta FIC$ with base $\overline{FC}$.  In isosceles triangles, altitudes and medians coincide, automatically making $\overline{ID}$ the perpendicular bisector of $\overline{FC}$.  QED

Admittedly, Proof 4 ultimately relies on the results of Proof 3, but the higher-level isosceles connection felt much more elegant.  I was satisfied.

TWO DYNAMIC GEOMETRY SOFTWARE VARIATIONS

Thinking how I could prompt students along this path, I first considered a trace on the perpendicular lines from the initial procedure above (actually tangent lines to the parabola) using to trace the parabolas.  A video is below, and the Geogebra file is here.

http://vimeo.com/89759785

It is a lovely approach, and I particularly love the way the parabola appears as a digital form of “string art.”  Still, I think it requires some additional thinking for users to believe the approach really does adhere to the parabola’s definition.

I created a second version allowing users to set the location of the focus on the positive y-axis and using  a slider to determine the distances and constructs the parabola through the definition of the parabola.  [In the GeoGebra worksheet (here), you can turn on the hidden circle and lines to see how I constructed it.]  A video shows the symmetric points traced out as you drag the distance slider.

A SIMPLIFIED PAPER PROCEDURE

Throughout this process, I realized that the location and spacing of the initial points on the directrix was irrelevant.  Creating the software versions of the problem helped me realize that if I could fold a point on the directrix to the focus, why not reverse the process and fold F to the directrix?  In fact, I could fold the paper so that F touched anywhere on the directrix and it would work.  So, here is the simplest version I could develop for the paper version.

1. Use a straightedge and a Sharpie or thin marker to draw a line near the edge of a piece of paper.
2. Place a point F roughly above the middle of the line toward the center of the paper.
3. Fold the paper over so point F is on the line from step 1 and crease the paper along the fold.
4. Open the paper back up and repeat step 3 several more times with F touching other parts of the step 1 line.
5. All of the creases from steps 3 & 4 outline a curve.  Trace that curve to see a parabola.

This procedure works because you can fold the focus onto the directrix anywhere you like and the resulting crease will be tangent to the parabola defined by the directrix and focus.  By allowing the focus to “Travel along the Directrix”, you create the parabola’s locus.  Quite elegant, I thought.

As I was playing with the different ways to create the parabola and thinking about the interplay between the two distances in the parabola’s definition, I wondered about the potential positions of the distance segments. 1. What is the shortest length of segment CP and where could it be located at that length?  What is the longest length of segment CP and where could it be located at that length?
2. Obviously, point C can be anywhere along the directrix.  While the focus-to-P segment is theoretically free to rotate in any direction, the parabola definition makes that seem not practically possible.  So, through what size angle is the focus-to-P segment practically able to rotate?
3. Assuming a horizontal directrix, what is the maximum slope the focus-to-P segment can achieve?
4. Can you develop a single solution to questions 2 and 3 that doesn’t require any computations or constructions?

CONCLUSIONS

I fully realize that none of this is new mathematics, but I enjoyed the walk through pure mathematics and the enjoyment of developing ever simpler and more elegant solutions to the problem.  In the end, I now have a deeper and richer understanding of parabolas, and that was certainly worth the journey.

The Value of Counter-Intuition

Numberphile caused quite a stir when it posted a video explaining why $\displaystyle 1+2+3+4+...=- \frac{1}{12}$

Doug Kuhlman recently posted a great follow-up Numberphile video explaining a broader perspective behind this sum.

It’s a great reminder that there are often different ways of thinking about problems, and sometimes we have to abandon tradition to discover deeper, more elegant connections.

For those deeply bothered by this summation result, the second video contains a lovely analogy to the “reality” of $\sqrt{-1}$.  From one perspective, it is absolutely not acceptable to do something like square roots of negative numbers.  But by finding a way to conceptualize what such a thing would mean, we gain a far richer understanding of the very real numbers that forbade $\sqrt{-1}$ in the first place as well as opening the doors to stunning mathematics far beyond the limitations of real numbers.

On the face of it, $\displaystyle 1+2+3+...=-\frac{1}{12}$ is obviously wrong within the context of real numbers only.  But the strange thing in physics and the Zeta function and other places is that $\displaystyle -\frac{1}{12}$ just happens to work … every time.  Let’s not dismiss this out of hand.  It gives our students the wrong idea about mathematics, discovery, and learning.

There’s very clearly SOMETHING going on here.  It’s time to explore and learn something deeper.  And until then, we can revel in the awe of manipulations that logically shouldn’t work, but somehow they do.

May all of our students feel the awe of mathematical and scientific discovery.  And until the connections and understanding are firmly established, I hope we all can embrace the spirit, boldness, and fearless of Euler.