## A Teacher’s Prayer

At the end of this week, I’m leaving Westminster, my school of 23 years to pursue a phenomenal opportunity for my family and me at the Hawken School in Cleveland, OH.  As you can imagine, my farewells this past week and now into my last have been both bitter and sweet.  In an attempt to craft a fitting acknowledgement for the gratitude and respect for all I have learned here, I penned a Teacher’s Prayer inspired by a prayer accredited to and adapted by Mother Teresa.  I hope you find something of value in it.

# A Teacher’s Prayer

My students are often scared they’re not going to understand.  They fear failure and disappointment.  Even when they hold back, help me continue to encourage and challenge them anyway.

Sometimes my students don’t do all they can to maximize their learning.  Help me continue to meet them where they are anyway.

Some of my students don’t care for my classes’ content and my lessons might not be well received. Help me continue to be passionate about every moment anyway.

Sometimes none of the explanations I know make an idea stick.  Even when I already may have offered enough, help me continue to graciously find another approach anyway.

If my lessons break free from where I’ve planned, some may not be able to keep up, and others may outrun me or even catch me in a mistake.  Help me continue to always take chances and run with them anyway.

Even when I think I know every connection and approach a student might offer, help me continue to listen to every “new” idea as if it were the best insight any person has ever made, because to the one brave enough to share, it was.

New technologies and teaching ideas threaten how and what I teach; new tools may force me to redevelop or abandon favorite lessons I’ve spent years perfecting.  But a new idea also may be just what I need to reach someone who’s been beyond my grasp, so help me continue to embrace innovation anyway.

I sometimes despair I won’t be able to reach the ambitious goals I set for my students or me.  Even though I will sometimes fall short, help me continue to be ambitious anyway.

Grant me strength to give more than I think I have, determination to overcome what I didn’t think I could achieve, a love deeper than I believe exists, a hunger to make the world a better place even if only in a small way and against all odds, grace to learn from my inevitable mistakes, and courage to never fear growth.  After all, that’s what we’ve always asked of our students anyway.  Amen.

Wolakota, Chris

I leave you with a favorite song:  Lee Ann Womack’s I Hope You Dance.

## Unexpected Math Creativity Lessons

This is the second of two posts on my recent experiences with a Four 4s activity.  As I explained in my first post, I’ve used this activity for over a decade, but was re-inspired by a recent Math Munch post about an IntegerMania page playing a  Four 4s variation using Ramanujan’s 1729 taxi cab number.

What struck me was IntegerMania’s use of an exquisiteness level which I included in my recent Four 4s activity, calling it a “complexity scale” for my students.  I thought it a nice external measure of the difficulty of student constructions, but the scale drove several unexpected lessons.

Explaining Exquisiteness: Many students wanted to know why the mathematical functions and operations were leveled the way they were.  Hypothesizing the intent of the scale’s original author(s), I explained them as what one might expect to encounter as one’s mathematical understanding grew.

• Level 1.0 involves only single-digit 4s and the most basic math operations:   +,  -,  *,  and  /.
• Students bridge to Level 2.0 when they concatenate single digits (44 & 4.4) and use percentages.
• Level 3.0 introduces exponents and roots (which are really thinly-veiled exponents) and factorials.
• Level 4.0 opens high school math:  logarithms, trigonometry (circular and hyperbolic) and their inverses.

Mathematical Elegance:  I honestly thought my students would stop there.  While the formulation of the scale and “surcharges” (or ‘penalties’ as my students called them) were debatable and something I will work out as a group rather than imposing the next time I use this, they did reinforce some of what I’ve always discussed with my students.

• Any solution is better than no solution,
• Long or complicated solutions sometimes provide valuable insights and alternative perspectives on problems, and
• Once mathematicians begin to get a solid grasp on a situation, brief, elegant, often “minimalist” solutions that get directly to the core of an idea become the desired goal.

For these reasons, solutions with the lowest total “complexity” would be the solutions listed first on our collective Four 4s bulletin board.  My students called the replacement of any solution with a less complex solution sniping.  I thought their group goal would be to get solutions for all integers before sniping.  I was wrong.  They focused much more intently on sniping higher level solutions until we were down to fewer than 10 missing integers at which point there was a definite push to finish the list.  3-4 weeks after the activity started, our integer board is completed, and students continue to snipe existing solutions.

Unexpected Complexity:  Three of my students (juniors P and JP, & senior T) became absolutely entranced with some of the higher-level functions.  IntegerMania’s complete exquisiteness list contains more functions, but here are the ones these three primarily used, along with links to deeper explanations if needed.

• - They loved the Level 5.0 gamma function.  (For what it’s worth, I argue $\Gamma(4)=3!=6$ should be a higher level function because it ultimately relies on integral calculus, and IntegerMania lists derivatives as Level 6.0.)
- One even leveraged a matrix determinant to create a 61–a solution I pose below.
• Level 6.0 included
- $p_a$ as the $a^{th}$ number in the list of prime numbers ($p_4=7$),
- $f_a$ as the $a^{th}$ Fibonacci Number ($f_4=3$),
- $\pi (a)$, the Prime Counting Function which conveniently is a Wolfram Alpha function,
- $d(a)$, the number of divisors of a,
- $\sigma(a)$, the sum of the divisors of a,
- Euler’s totient function, $\phi (a)$, “the number of positive integers less than or equal to a that are relatively prime to a“–also a Wolfram Alpha function, and
- the derivative from calculus, allowing a convenient way to lose an extra 4 because 4′=0.
• Finally, some Level 7.0 favored functions:
- Double factorials with $4!!=4\cdot 2=8$,
- the Lucas Numbers, $L_a$$L_4=7$, and
- the Triangular Numbers, $T_a$, a sort of stealthy use of combinations where $T_4=10$.

Strategizing:  A couple days into the activity, P and JP set themselves a goal of writing every integer from 0-25 with a single 4.  Enamored with the possibility of using their newfound functions, they realized that if they could accomplish this goal, they could write every integer 1-100 on the board with four 4s.  It didn’t matter to them that the complexity levels would be high, they wanted to prove to themselves that every answer could be found without actually finding each–in short, they sought a form of an existence proof long before all answers were posted.  I didn’t anticipate this, but loved their approach.

Here’s a reproduction of their list:

• $0 = 4'$
• $1 = \Gamma \left( \sqrt{4} \right)$
• $2 = \sqrt{4}$
• $3 = f_4$, T made huge use of this one.
• $4 = 4$
• $5 = p_3=p_{d(4)}$
• $6 = \Gamma(4)$
• $7 = L_4$
• $8 = 4!!$
• $9 = \pi(24)=\pi(4!)$
• $10 = T_4$
• $\displaystyle 11 = L_5 = L_{p_{d(4)}}$
• $12 = \sigma(6) = \sigma \left( \Gamma(4) \right)$
• $13 = \sigma(9) = \sigma \left( \pi (4!) \right)$
• $14 = \pi(45) = \pi \left( T_9 \right) = \pi \left( T_{ \pi (4!) } \right)$
• $15 = \sigma(8) = \sigma(4!!)$
• $16 = \pi(55) = \pi(f_{10}) = \pi \left( f_{T_4} \right)$
• $17 = p_7 = p_{f_4}$
• $18 = \sigma(10) = \sigma(T_4)$
• $19 = p_8 = p_{ (4!!) }$
• $20 = \phi(25) = \phi(\pi(\sigma(\phi(\phi(p_{(T_{(f_4)})})))))$
• $21 = f_8 = f_{4!!}$
• $22 = \phi(23) = \phi( p_{ \pi(4!!) } )$
• $23 = p_9 = p_{ \pi(4!) }$
• $24 = 4!$
• $25 = \pi(98) = \pi(\sigma(52)) = \pi(\sigma(\phi(106))) = \pi(\sigma(\phi(\phi(107))))$
$= \pi(\sigma(\phi(\phi(p_{28})))) = \pi(\sigma(\phi(\phi(p_{(T_7)})))) = \pi(\sigma(\phi(\phi(p_{(T_{(f_4)})}))))$

That 25 formulation is a beast (as is the 20 that depends on it), but P and JP accomplished their goal and had proven that the entire board was possible.

Now, all that remained for the class was to find less complex versions.

A Creative Version of 61:  As my sign-off, I thought you might enjoy JP’s use of a determinant and some Level 6.0 functions to create his 61.  He told me he knew it would be sniped, but that wasn’t the point.  He just wanted to use a determinant.

$\pi (4!)=\pi (24)=9$ because there are 9 primes less than or equal to 24, and $p_9=23$ because the 9th prime number is 23.  With $f_4=3$ from above, the remainder of the determinant is easily handled.  The prime number functions were a base Level 6.0, and the surcharges for each of them, the factorial, the implied 2 on the root, and the Fibonacci function raised this to a Level 7.0.

A little over a week later, JP’s determinant was sniped by a student who isn’t even in my classes, N, whose Level 3.4 construction follows.

I hope you can have some fun with this, too.

## Gender and Expectations Lessons from Research

A couple reports from NPR yesterday have me thinking about some of the articles I’ve compiling in my Diigo library about what I’ve learned about gender and expectations differences in parenting and teaching.  I don’t have anything particular to tie together here, but I thought these four resources were more than I could comfortably tie together in a coherent Tweet thread, so I thought I’d gather them into an impromptu ’blog post.

Girls May Get More ‘Teaching Time’ From Parents Than Boys Do via @NPR.  Excerpts:
… “ ’How often do you read with your child?’ or ‘Do you teach them the alphabet or numbers?’ … Systematically parents spent more time doing these activities with girls.”
… “Since parents say they spend the same amount of time overall with boys and girls, Baker’s analysis suggests that if parents are spending more time with girls on cognitive activities, they must be spending more time with boys on other kinds of activities.”
… “The costs of investing in cognitive activities is different when it comes to boys and girls. As an economist, he isn’t referring to cost in the sense of cash; he means cost in the sense of effort.”

Gender Gap Disappears in School Math Competitions via sciencedaily.com .  Excerpt:
… “Most school math contests are one-shot events where girls underperform relative to their male classmates. But a new study by a Brigham Young University economist presents a different picture.  Twenty-four local elementary schools changed the format to go across five different rounds. Once the first round was over, girls performed as well or better than boys for the rest of the contest.”
… ”It’s really encouraging that seemingly large gaps disappear just by keeping [girls] in the game longer.”

A broader look at school expectations leading to enhanced math performance:  What Distinguishes a Superschool From the Rest via ideas.time.com .
… “The difference seems to lie in whether a school focuses on basic competence or encourages exceptional achievement. While almost all the schools saw it as their responsibility to cover the math knowledge necessary to do well on the SATs, the authors noted that “there is much less uniformity in whether schools encourage gifted students to develop more advanced problem solving skills and reach the higher level of mastery of high school mathematics.”
… “The fact that the highest achieving girls in the U.S. are concentrated in a very small set of schools, the authors write, indicates ‘that almost all girls with the ability to reach high math achievement levels are not doing so.’ ”

Girls, Boys And Toys: Rethinking Stereotypes In What Kids Play With via @NPR.  Excerpts:
… Some toy companies are re-thinking gender-specific marketing and branding.
… ”I think what they were worried about was causing gender identification needlessly — to turn off passive learning, passive expression down the road, even passive economic opportunity for girls or boys if they felt they couldn’t do something because of societal norms,”
… ”It’ll be interesting to see how this changes the attitudes of parents and of kids over time or whether or not it does. There may be some hard-wired differences,”

## Teaching Creativity in Mathematics

This will be the first of two ‘blog posts on an activity that could promote creativity for elementary, middle school, and high school students.  A suggestion for parents and teachers is in the middle of this post.

ABOUT A DECADE AGO, I first discovered what I call the Four 4s activity.  In brief, the game says that using exactly four 4s (no more, no less, and no other digits) and any mathematical operation you want, you can create every integer from 1 to 100.  Two quick simple examples are $\displaystyle 3= \frac{4+4+4}{4}$ and $\displaystyle 16= 4\cdot 4+4-4$.

As for mathematical operations, anything goes!  The basic +, -, *, / along with exponents, roots, decimals (4.4 or .4), concatenation (44), percentages, repeating decimals ($.\overline{4}$), and many more are legal.

At the time, I was teaching a 7th grade prealgebra course with several students who were struggling to master order of operations–that pesky, but critical mathematical grammar topic that bedevils some students through high school and beyond.  I thought it would be a good way to motivate some of my students to 1) be creative, and 2) improve their order of operations abilities to find numbers others hadn’t found or to find unique approaches to some numbers.

My students learned that even within the strict rules of mathematical grammar, there is lots of room for creativity.  Sometimes (often? usually?) there are multiple ways of thinking about a problem, some clever and some blunt but effective.  People deserve respect and congratulations for clever, simple, and elegant solutions.  Seeing how others solve one problem (or number) can often grant insights into how to find other nearby solutions.  Perhaps most importantly, they learned to a small degree how to deal with frustration and to not give up just because an answer didn’t immediately reveal itself.  It took us a few weeks, but we eventually completed with great communal satisfaction our 1-100 integer list.

PARENTS and TEACHERS:  Try this game with your young ones or pursue it just for the fun of a mental challenge.  See what variations you can create.  Compare your solutions with your child, children, or student(s).  From my experiences, this activity has led many younger students to ask how repeating decimals, factorials, and other mathematical operations work.  After all, now there’s a clear purpose to learning, even if only for a “game.”

I’ve created an easy page for you to record your solutions.

A FEW WEEKS AGO, I read a recent post from the always great MathMunch about the IntegerMania site and its additional restriction on the activity–an exquisiteness scale.  My interpretation of “exquisiteness” is that a ‘premium’ is awarded to solutions that express an integer in the simplest, cleanest way possible.  Just like a simple, elegant explanation that gets to the heart of a problem is often considered “better”, the exquisiteness scale rewards simple, elegant formulations of integers over more complex forms.  The scale also includes surcharges for functions which presume the presence of other numbers not required to be explicitly written in common notation (like the 1, 2, & 3 in 4!, the 0 in front of .4, and the infinite 4s in $.\overline{4}$.

In the past, I simply asked students to create solutions of any kind.  I recorded their variations on a class Web site.  Over the past three weeks, I renamed exquisiteness to “complexity” and re-ran Four 4s across all of my high school junior and senior classes, always accepting new formulations of numbers that hadn’t been found yet, and (paralleling Integermania’s example) allowed a maximum 3 submissions per student per week to prevent a few super-active students from dominating the board.  Also following Integermania’s lead, I allowed any new submission to remain on the board for at least a week before it could be “sniped” by a “less complex” formulation.  I used differently colored index cards to indicate the base level of each submission.

Here are a few images of my students’ progress.  I opted for the physical bulletin board to force the game and advancements visible.  In the latter two images, you can see that, unlike Integermania, I layered later snipes of numbers so that the names of earlier submissions were still on the board, preserving the “first found” credit of the earliest formulations.  The boxed number in the upper left of each card is the complexity rating.

The creativity output was strong, with contributions even from some who weren’t in my classes–friends of students curious about what their friends were so animatedly discussing.  Even my 3rd grade daughter offered some contributions, including a level 1.0 snipe, $\displaystyle 5=\frac{4\cdot 4+4}{4}$ of a senior’s level 3.0 $\displaystyle 5=4+\left( \frac{4}{4} \right)^4$.  The 4th grade son of a colleague added several other formulations.

When obviously complicated solutions were posted early in a week, I heard several discussing ways to snipe in less complex solutions.  Occasionally, students would find an integer using only three 4s and had to find ways to cleverly dispose of the extra digit.  One of my sometimes struggling regular calculus students did this by adding 4′, the derivative of a constant. Another had already used a repeating decimal ( $. \overline{4}$), and realized she could just bury the extra 4 there ( $.\overline{44}$).  Two juniors dove into the complexity scale and learned more mathematics so they could deliberately create some of the most complicated solutions possible, even if just for a week before they were sniped.  Their ventures are the topic of my next post.

AFTERTHOUGHTS:  When I next use Four 4s with elementary or middle school students, I’m not sure I’d want to use the complexity scale.  I think getting lots of solutions visible and discussing the pros, cons, and insights of different approaches for those learning the grammar of mathematical operations would be far more valuable for that age.

The addition of the complexity scale definitely changed the game for my high school students.  Mine is a pretty academically competitive school, so most of the early energy went into finding snipes rather than new numbers.  I also liked how this game drove several conversations about mathematical elegance.

One conversation was particularly insightful.  My colleague’s 4th grade son proposed $\displaystyle 1=\frac{44}{44}$ and argued that from his perspective, it was simpler than the level 1.0 $\displaystyle \frac{4+4}{4+4}$ already on the board because his solution required two fewer operations.    From the complexity scale established at the start of the activity, his solution was a level 2.0 because it used concatenated 4s, but his larger point is definitely hard to refute and taught me that the next time I use this activity, I should engage my students in defining the complexity levels.

1) IntegerMania’s collection has extended the Four 4s list from 1 to well past 2000.  I wouldn’t have thought it possible to extend the streak so far, but the collection there shows a potential arrangement of Four 4s for every single integer from 1 to up to 1137 before breaking.  Impressive.  Click here to see the list, but don’t look quite yet if you want to explore for yourself.

As a colleague noted, it would be cool for those involved in the contest to see how their potential solutions stacked up against those submitted from around the world.  Can you create solutions to rival those already posted?

2) IntegerMania has several other ongoing and semi-retired competitions along the same lines including one using Four 1s, Four 9s, and another using Ramanujan’s ‘famous’ taxi cab number, 1729.  I’ve convinced some of my students to make contributions.

Play these yourself or with colleagues, students, and/or your children.  Above all, have fun, be creative, and learn something new.

It’s amazing what can be built from the simplest of assumptions.  That, after all, is what mathematics is all about.

## Arrangements Connections for Young Students

Mathematics is not arithmetic.

The latter is a set of symbol manipulation rules that dominates most of what we teach in school.  Mathematics, on the other hand, is a science of patterns.  It is a way of logical thinking, making sense of forms and arrangements–sometimes applied and sometimes purely imagined.  It involves looking at the implications of what we know and pushing that knowledge as far as we can to see what else can be learned based solely upon connections we can make from our assumptions.

Within the last few weeks, I’ve discovered a great daily ‘blog run by @Five_Triangles ”for (but not limited to) school years 6-8.”  I’d argue that those posts are great for a broader range of ages. I gave my 3rd grade daughter one of the puzzles during breakfast.  We had some great conversations then and on the way to school.  I share those below.  Another offering extends that thinking in a way that may not be immediately obvious to young people.

Here’s the part of the post I used at breakfast.

For my daughter, I saw this problem presenting two different possibilities–the obvious arithmetic problem and a mathematics extension.  The arithmetic requires very basic subtraction facts and wee bit of trial-and-error (a GREAT mathematics skill!) to tease out a solution.  Part of the mathematics here, in my opinion, involves asking a “What if?” question.

I posed this problem to my 3rd grade daughter and after randomly dropping in some numbers at first and seeing some frustration, I said to her, “I wonder what sorts of numbers subtract to give 3.”  Her frustration evaporated as she started making a list of several possibilities for such digits. She noted that there were far more possibilities for these difference than space in the problem allowed.  I encouraged her to keep trying.  We never explicitly discussed the problem’s set up with a four-digit number subtracted from a five-digit number, but I saw her try a couple different first digits before realizing that the first character of the five-digit number clearly had to be “1″.  A little more experimentation and she had an answer.

She thought the puzzle was over–after all, school has trained her to think that once she had “an” answer, she must have found “the” answer.

That’s when I prompted some mathematics.  I asked if she could find another answer.  A few other prompts and she had found 6 different solutions.  I asked her how she found them.  ”Easy,” she replied.  ”You just put the number pairs in different orders.”  She found through trial-and-error that the five-digit number always started “12…” and therefore the four-digit number started “9…”.  Checking her list of differences leading to 3 left no other possibilities.  Everything else was flexible, thus her six different answers.

• Can you explain why the five-digit number must start “12…”?
• Once I had the “12…” and “9…”, I knew there were at least 6 solutions  before I had found even the first one.  My daughter wasn’t ready for this thought, but can you explain why this is true?
• Can you find all 6 answers?
• Better: Can you explain why there cannot be any more?

The second part of the problem (with the same rules and a different result) is definitely tougher.

You can quickly conclude that the first digit of the five-digit number must be 4 or 3, but it’s definitely more challenging to tease out the rest.  Rather than dealing with the entire problem at once, I suggest another great mathematics strategy:  Simplify the problem.  Using only the digits 1 to 9, can you find all possibilities that would result in the beginning of the problem?

If this is part of an answer, the six digits not used in those three boxes must have an arrangement that subtracts to 333.  Unfortunately, none of these actually pan out.  Convince yourself why this must be true.  Students need to learn that not finding an answer is OK.  Knowing that there’s not a solution is actually a solution–you’ve learned something.

Extending the beginning of the problem to

eventually shows that the five-digit number could start “412..” with the four-digit number starting “79..”.  That means the remaining four digits must have exactly two arrangements for precisely the same reasons that the first problem had six solutions.  Can you find the two arrangements that satisfy the 33333 problem?  In case you want to check, I list the answers at the end of this post.

The next week provided another puzzle using the arrangements idea.

The problem doesn’t yield a straightforward solution that can be solved.  Instead, laying out all possible finishing arrangements and testing the veracity of the claims leads to a solution.  Again, there are three entries, so this problem is (mathematically) just like the 3333 subtraction problem above–both have six possible arrangements.  Helping a young person see this connection would be a great thought achievement.

Start by listing the six possible 1st, 2nd, and 3rd place arrangements of the letters A, B, and C:  A-B-C, A-C-B, B-A-C, B-C-A, C-A-B, & C-B-A.   As an example, if the boys finished A-B-C, all three boys would have told the truth, so that finish doesn’t satisfy the problem requirement of one false statement.  Comparing each arrangement to the boys’ statements eventually shows that only one of these arrangements satisfies the problem’s requirement that exactly one of the three boys made a false statement.

A good mathematical extension would be to see if there are any other questions that could be asked from the boys’ statements.  Is it possible that all three told the truth?  Is it possible that only one was truthful?  Are there any other possible outcomes?  Do any of these have unique outcomes given the boys’ statements, or do some have multiple possibilities?

CONCLUSION:  I fear that too often school and students stop at a single answer and don’t explore other possibilities.  Asking “What if” is a critical question in all of science and mathematics.  It inspires creativity, wonder, and exploration.  It doesn’t always yield results, so it also helps motivate stamina.  Convincing yourself that there are no (more) solutions is itself an intellectual accomplishment.

We need more of this.

SOLUTIONS:

• 3333 solutions: 12678-9345, 12687-9354, 12768-9435, 12786-9453, 12867-9534, & 12876-9543.
• 33333 solutions: 41268-7935 & 41286-7953.
• Competition solution:  A-C-B

## A Student’s Powerful Polar Exploration

I posted last summer on a surprising discovery of a polar function that appeared to be a horizontal translation of another polar function.  Translations happen all the time, but not really in polar coordinates.  The polar coordinate system just isn’t constructed in a way that makes translations appear in any clear way.

That’s why I was so surprised when I first saw a graph of $\displaystyle r=cos \left( \frac{\theta}{3} \right)$.

It looks just like a 0.5 left translation of $r=\frac{1}{2} +cos( \theta )$ .

But that’s not supposed to happen so cleanly in polar coordinates.  AND, the equation forms don’t suggest at all that a translation is happening.  So is it real or is it a graphical illusion?

I proved in my earlier post that the effect was real.  In my approach, I dealt with the different periods of the two equations and converted into parametric equations to establish the proof.  Because I was working in parametrics, I had to solve two different identities to establish the individual equalities of the parametric version of the Cartesian x- and y-coordinates.

As a challenge to my precalculus students this year, I pitched the problem to see what they could discover. What follows is a solution from about a month ago by one of my juniors, S.  I paraphrase her solution, but the basic gist is that S managed her proof while avoiding the differing periods and parametric equations I had employed, and she did so by leveraging the power of CAS.  The result was that S’s solution was briefer and far more elegant than mine, in my opinion.

S’s Proof:

Multiply both sides of $r = \frac{1}{2} + cos(\theta )$ by r and translate to Cartesian.

$r^2 = \frac{1}{2} r+r\cdot cos(\theta )$
$x^2 + y^2 = \frac{1}{2} \sqrt{x^2+y^2} +x$
$\left( 2\left( x^2 + y^2 -x \right) \right) ^2= \sqrt{x^2+y^2} ^2$

At this point, S employed some CAS power.

[Full disclosure: That final CAS step is actually mine, but it dovetails so nicely with S's brilliant approach. I am always delightfully surprised when my students return using a tool (technological or mental) I have been promoting but hadn't seen to apply in a particular situation.]

S had used her CAS to accomplish the translation in a more convenient coordinate system before moving the equation back into polar.

Clearly, $r \ne 0$, so

$4r^3 - 3r = cos(\theta )$ .

In an attachment (included below), S proved an identity she had never seen, $\displaystyle cos(\theta) = 4cos^3 \left( \frac{\theta }{3} \right) - 3cos \left( \frac{\theta }{3} \right)$ , which she now applied to her CAS result.

$\displaystyle 4r^3 - 3r = 4cos^3 \left( \frac{\theta }{3} \right) - 3cos \left( \frac{\theta }{3} \right)$

So, $\displaystyle r = cos \left( \frac{\theta }{3} \right)$

Therefore, $\displaystyle r = cos \left( \frac{\theta }{3} \right)$ is the image of $\displaystyle r = \frac{1}{2} + cos(\theta )$ after translating $\displaystyle \frac{1}{2}$ unit left.  QED

Simple. Beautiful.

Obviously, this could have been accomplished using lots of by-hand manipulations.  But, in my opinion, that would have been a horrible, potentially error-prone waste of time for a problem that wasn’t concerned at all about whether one knew some Algebra I arithmetic skills.  Great job, S!

S’s proof of her identity, $\displaystyle cos(\theta) = 4cos^3 \left( \frac{\theta }{3} \right) - 3cos \left( \frac{\theta }{3} \right)$ :

## Fun with Series

Two days ago, one of my students (P) wandered into my room after school to share a problem he had encountered at the 2013 Walton MathFest, but didn’t know how to crack.  We found one solution.  I’d love to hear if anyone discovers a different approach.  Here’s our answer.

PROBLEM:  What is the sum of $\displaystyle \sum_{n=1}^{\infty} \left( \frac{n^2}{2^n} \right) = \frac{1^2}{2^1} + \frac{2^2}{2^2} + \frac{3^2}{3^3} + ...$ ?

Without the $n^2$, this would be a simple geometric series, but the quadratic and exponential terms can’t be combined in any way we knew, so the solution must require rewriting.  After some thought, we remembered that perfect squares can be found by adding odd integers.  I suggested rewriting the series as

where each column adds to the one of the terms in the original series.  Each row was now a geometric series which we knew how to sum.  That  meant we could rewrite the original series as

We had lost the quadratic term, but we still couldn’t sum the series with both a linear and an exponential term.  At this point, P asked if we could use the same approach to rewrite the series again.  Because the numerators were all odd numbers and each could be written as a sum of 1 and some number of 2s, we got

where each column now added to the one of the terms in our secondary series.  Each row was again a geometric series, allowing us to rewrite the secondary series as

Ignoring the first term, this was finally a single geometric series, and we had found the sum.

Does anyone have another way?

That was fun.  Thanks, P.