Love, Forgiveness, and Kindness

Each day, students in my classroom are peppered with what I call life lessons; ideas that I think about when trying to live a happier and more helpful life.  Some ideas are part of the greater whole, only recognized as such later. For example,"Grades are Private," is a common phrase in my class, but is part of the larger idea of, "Err in the direction of Kindness." (George Saunders' convocation speech.)

One of the clarifying exercises I hold in my mind is the following.

If I were to be asked to speak in a graduation ceremony, what would I say?  Would it be different for college students than for high school?  What would I want these young people -- of either category -- to carry away?  The best graduation speech I've ever heard was given by a writer for the Grateful Dead, John Perry Barlow. His "Just Say Yes" speech still rings in my ears for both his command of the language, his eloquence, and his singular message.  Oh, if I could ever have his working vocabulary, his calm confidence, his intelligence, and his creativity, then wouldn't that be wonderful!  As those of you who have had me in a class for any of the last six years knows, I ask you to read or listen to a graduation speech that gives another important message -- more on that later -- but graduation speeches are peoples' attempts to summarize what's important in some kind of meaningful way to others who are just entering the world.  I've listened in person to in excess of 50 graduation speeches -- no kidding -- and only a short handful stay with me.

In the middle of the night last night, I had a moment of clarity.  Of all the small tidbits I offer to my students about life "things,"  three run really clear to me as the most important of all the messages.  No one would be surprised to hear these come from me; I've said these things so often before in class, in hallways, on the path on the hill, and whenever I run into people.  But what's new is the clarity that these three are the most important.  These three hang together as the strongest of life lessons that I've ever wanted to impart to people I've seen in my classroom each day. 1. Love. 2. Forgive. 3. Be kind.  So here's my graduation speech that I'd give to people I've seen in my classroom. 

1. I love you. Unconditionally. I love who you are, how you think, your process of becoming.  I want to applaud your accomplishments, share in your griefs.  I want to help you become the people you want to become.  I want you to know I'm here for you.  It's so important to me that you know this.  Equally important is that you know I'm not the only one; I'm just your math teacher.  There are others who love you. I'm not sure most of you fully understand this, but sometimes, I get the idea that some people I've known in the classroom truly understand the depth of my affection for each of you.  A former student named Seren found me and called me over 20 years after she sat in my classroom.  "I need advice," she said.  "I don't have a partner or a partner prospect. I'm 39 years old and want to have a baby. What should I do? "  While flattered to receive this very personal call, I suggested she discuss this with, perhaps, her mom, or a sibling or a therapist. She then launched into a narrative about how she knew I always loved her, wanted the best for her, supported her in her decision making process, and that I would be unbiased. She wasn't sure all these things were the case for anyone else in her life.  I remember taking a deep breath when I realized that she truly understood why I go to school every day. She understood my love. (BTW, she has two beautiful children and is raising them successfully as a single parent. What a joy.) But again, I'm not the only one who loves Seren, just as I'm not the only one who loves you.  If you think about it a minute, I'm sure you can find others who love you. If you don't think you can, then find other people you know you love and let them know you love them.

2. You (and me and your friends and your parents and your younger siblings) are allowed to grow up, to change, and to become new people.  Indeed, let's hope we all do.  Being stagnant or frozen as a 12 year old or 15 year old or even as a 55 year old would not be a good thing. I often see teenagers write in yearbooks, "You're so sweet, don't ever change."  Yipes, really?  Being stuck in teenager-hood would not be the best for most folks.  The corollary: You (and me and your friends and your parents and your younger siblings) need to allow each other to grow up and to change, and to become new people. Many of you might remember my story about Mark.  Each time I tell the story, this young man's name changes; I've been trying to keep his identity anonymous. The story starts at a twenty-year high school reunion for Mark, as he is introducing me to his family. I re-print it below, for those readers who have not seen or heard this: 

 He stood in front of me with his well-coiffed wife and well-scrubbed pair of children: one boy, one girl.  All were smiling as they chatted with me, Mark's former math teacher at a boarding school.  The chit-chat was lovely all around, but could have gone astray when his wife said, "Please tell us some stories about Mark in high school."

I glanced at Mark, then did a double take.  He stepped back, letting his eyeballs protrude from their sockets, and shaking his head with a spasmotic "no."  I noted that my mouth was frozen in a vertical "O" shape, which I changed immediately into a smile with my head cocked to one side.

I'm good with platitudes and veiled references; she giggled when I said, "Times in the library were so fun, it was like a party." The reality: Mark had stolen a faculty member's keys, somehow acquired a keg and was host to a keg party in the library one Saturday night/Sunday morning.  Mark appeared relieved when I made it clear I would not reveal anything unsavory about his times as a teenager.  We all have moments in our past that embarrass us; we are allowed to grow up, to change, and to become better people. Indeed, let's hope we do; no one should be stuck as a 15-year-old for his whole life.

Regardless, the most dramatic story I could not reveal filled my consciousness.  But of course I'll reveal it here, in full.  I learned that I could take 14 students skiing at the local ski area and receive a free pass. This was a big deal; my monthly income was less than $600 per month; a ski pass was in excess of 10% of my monthly salary and thus considered unaffordable.   The only hitch in the "free pass" arrangement was that I was responsible for these youngsters, which sometimes meant a trip or two to the local ER.  During one ski trip, Mark was in my group of 14.  I had just made sure each teenager had his or her hat, ski pass, and skis, when I made my way to the slopes.  After one run and at 9:00 am, I got the message that someone was in the medical tent and needed me.  So I went.  And there was Mark, with a clearly broken arm and a smirk on his face.  On the way to the ER, I asked him how it happened.  "I was challenging myself," he said.  That's not always a bad thing, I thought.  He continued, "By going down a black diamond slope."  That's ok, I know Mark to be a good skiier.  "With my eyes closed." Something clicked in me: my skiing day was to be ruined by Mark skiing a black diamond with his eyes closed?  Then he continued: "Backwards."  

I had to dig deep to be compassionate in the Emergency Room.  When he was in the x-ray room, I fiddled with his ski poles and noted that the tops were kind of funny.  Then I saw they unscrewed and held a clear liquid that could have done a great job cleaning the bathroom floor.  Now we had another issue to which I needed to alert the medical staff.  The results of his next test determined that while his blood alcohol level was still below the legal driving level, but he was only 16 and not a legal drinker.  

So let's return to the scene where I'm standing with Mark and his lovely family.  The story revealed above flooded me as the prime example of his teenaged antics that I will not share with his well-coiffed wife and well-scrubbed children. Silently, to myself, I'm repeating the mantra, People are Allowed to Grow Up.  "So Mark," I begin.  "What do you do now?"

He clears his throat briefly and looks down, so I'm sure he's thinking of the same moment in his teenage history that I remember clearly, then says, "I'm a member of the clergy."

So it's always good to recognize that each of us is on our own journey of becoming.  We can chose to forgive and move forward and hope that others will do the same for us. So really, the second 

3.  Be kind.  To everyone.  Be excessively kind.  Be genuinely, honestly, clearly, thoroughly, glamorously, and intentionally kind.  Be kind at all times and without exceptions.  No one is ever sorry for being too kind.  Whether or not you actually remember, if you were in my class since about 2011, you made a commitment to being kind in your first blog post.  You read a convocation speech by George Saunders and responded. Maybe some of you wrote in an effort to "please the teacher," (yep, I know that happens in the first few days of class particularly), but you still each wrote about the importance of being kind.  If you're not an alum of my classroom and don't know this speech, it's worth listening to. The link given here,  Err in the Direction of Kindness, will direct you to my blog that introduced the speech and gives the citations to George Saunders' speech. Over many years, I copied and printed quotes from your posts and posted them around the classroom.  You all were so eloquent; I was impressed.  Let me share with you some of your comments:

Kindness is essential to our everyday lives. Being kind can help you make more friends. Isabelle Baca

 Kindness isn’t naïve. It takes real courage to be kind.  Lily Johnson

One day in the near future, we will all learn to respect and appreciate each other’s differences. Hopefully, one day in the near future, we will all learn to respect and appreciate each other’s differences and reconsider our actions and words toward other people from foreign parts of the world. We should learn to appreciate all walks of life and realize that every culture is unique and valuable to the human race. No one is superior to another person, we are all just, simply, different. Brigid Contreras- Ridlington

 Kindness is something so valuable (yet)it is an endless resource.  Anyone can act kind be to truly be kind is something entirely different.  Little things here and there are all you need to be kind. Alex Monka

Kindness…puts me and others in a better mood.  Destyn Gaffney

Doing kind things (even if it is ultimately selfish) is worth spending time doing.  Sophia Brito

 (Even a) simple encounter (can be) an act of kindness…and very significant. Sophie Edwards

 Kindness is shown to be an essential human trait.  Colleges like Harvard are now emphasizing kindness on their applications. There are many ways to be kind, but overall it requires treating others how you want to be treated and going behind that mantra. With a generalregard for others, and a little bit of selflessness, being kind is easy and can lead to great relationships and satisfaction in life.   Jodham Fine.

 “One act of kindness can change someone’s life, can change the world. That all it takes.” Courtney Moore, quoted by Jack Lambie

 Kindness is a basic human instinct like survival. Liam Lambie

Kindness is something that is very valuable to everybody. (There are) many ways to practice kindness including simple things such as a gentle smile, a compliment, writing a note, supporting a loved one, giving a hug and more. Zoe Calott-Wang

Just be kind to people and everything will be ok. Being kind isn't always easy to do, but it is the right thing to do. Nico Carson

 Helping a neighbor do something or holding the door for someone or even telling someone that they have a nice outfit can go a long way. Kindness is key for a healthy community. Waite Dawkins

 Making eye contact and smiling can boost someone’s happiness. Gavin Blevins

I believe everyone to have the capacity to truly exhibit the courage (of kindness). Deana Chefchis

 Every day we are faced with opportunities to help others.  If everyone erred in the direction of kindness the world would slowly begin to help itself.  Aidan McKinley

 It's always important to be kind. Aaron Parmeter

 Kindness is such a simple task. Yasmine Torres

No one has an excuse to be unkind because (kindness) benefits both parties. Cynthia Viets

All of kindness is good for everyone. Noah Vigil

Selflessness, tolerance, and empathy are the most important qualities (related to kindness) that I believe are vital in our society. Bridegette Watkins

If you have a positive outlook on life and are more kind, people will think better of you. Kiedis Connell

Kindness makes the world a better place. Lea Benevides

If  you  have  a choice   of  actions … choose  the kind-er  one.  Kaeli Kampschmidt

Kindness is caring about others, being aware of the things around you, and being loving. Hannah Topf

…people are inspired by kindness…Will Drescher

Kindness is such an important thing in today's world because we are shown a lot of news that is not supporting kindness. Aidan Burdge

Being kind to someone can easily change how they perceive themselves. Ethan Richardson 

Brushing over the opportunity to be kind to anyone is a failure. Adison Romano 

It's the unmade choices that are regretted the most.  We are blinded in the darkness, and when the opportunity to shine light comes, we mustn't hesitate to grab it by the arms and use it.  We must use our abilities to the fullest extent: it is needed now more than ever that we use our ability to edify and to make what is good become great.  Jacob Roberts-Baca

Nothing good ever happens in a situation unless you act upon what you feel is right.Life's too short to just do what everyone else does. Kirby Myers

Kindness is not solely an action, but rather a soulful assertion of the heart over the mind.  Jesse Sanchez

While a selfish lifestyle will only bite you in the tail, a life of kindness will be a gift that never stops giving. Michael Weber

Kindness can be one of the easiest things to achieve.Kindness brings us together and is a reminder that what unites us can be far greater than what divides us.  Stephen Chu

Being nice and accepting is so important because some people never realize how much of an impact words or an action could have on a single person. Daisy Canales 

Agape: unconditional love. This is the way we should treat others. Nathan Roberts

It is impossible for anyone to take the brightness that makes you tick from you. If we don’t respect ourselves enough to treat other people as human beings, then we are being less than human. Rachel Templeton

Kindness is never making someone feel poor about who they are or what they stand for.  Willa Wiese

A true sense of purpose can be obtained just by putting in a little more kindness into the world. Deana Chefchis

There are still wonderful hearted people that care for us and want us to be the best we can be, but of course while heading in the direction of kindness.  This makes me want to head in that direction as well. Joshua Guillen

Take a minute to look around from a new perspective, and you will realize something new each time. Ethan Klein 

A lack of kindness is still wrong and regrettable. Evie McIndoo

Kind actions can result in a happier, healthier society with a much stronger sense of community and positive relationships.The world today would be a much better place if it encompassed a culture of kindness. Lily Yatskowitz

I believe in being kind to everyone.  Alex Lee

 People can grow, change, and learn empathy and kindness. Sydney Bruner 

The key to happiness, the key to being that “someone,” is to be kind and accepting to everyone. Maxwell Dryfoos

 Kindness is one of the most important values and one of the easiest to implement. Siddhi Gardner 

Strangers may become your best friends. It is better to try to help or share some kindness than to regret it later. Gabby Nemer

Kindness, even in small acts, can change the world. Cynthia Viets

Everyone is looking for kindness, and I think if we had sympathy for others, everyone would be a little kinder in return. Riley Burns

If you manage to spread more kindness around yourself and your loved ones, then your life will be much better.     The more kindness you give, the more you will receive, thus making your life a massive ball of joy.The people you unconsciously avoid could have very interesting personalities and life stories. Diego Moore

Openness towards people who are different is one of the most valuable things a human can have. Marwa Chohan

 For those of you who do wonder what the first step towards being kind is, it's this: if you see that kid who looks a little different from the rest of the class, who wears the glasses that are just a bit too big, whose skin is a shade darker, or who has crushes on the same gender, accept them. Marwa Chohan

If I want to have a conversation with someone who thinks differently from me, then the best way to do it is in a calm and respectful manner, and make sure I’m listening. Sowang  Kundeling

Kindness can be shown in small acts; it does not take large actions to be kind.     Kai Seidenberg

 Kindness is something that should be practiced whenever possible. Lucas Anderson

The harm caused by bias in today’s world is immeasurable and the answer to healing is education and kindness. Maya Benavides

if everyone is nicer to people we meet then everyone is happier. Stephen Clem

I'd say that kindness is the single most human virtue there is. Marceline Corbin

Learning how to be kind to ourselves helps us to live more beautiful, truthful, and meaningful lives. Likewise, if we can learn to appreciate acts of kindness when they are presented to us, we will be able to better grasp the meaning of happiness and fulfilment. Kiera Friedman

Kindness is an act of helping people understand that they can be themselves, something that makes a person feel comfortable in their own skin, something that makes people light up and be joyful and happy. Gigi Galles

I think that everyone deserves kindness, and everyone should give kindness to others. Caroline Klein

 Being ignorant towards kindness and shallowness can be just as harmful as perpetuating the opposites. Sophia Liem

Kindness is vital to our society. Laurel Nash-Jarecky

Humans are very vulnerable. We are fleshy, weak, slow and clumsy. As a result, we have to look out for each other to survive. (Note: especially in a mosh pit.) Attacus Peknik

By doing simple acts of kindness it can become a habit, and that habit could lead you to do great things for someone and yourself. Tino Muneri

Do an act of kindness each and every day.     Grace Jorgensen

I need to take every opportunity that I can, no matter how small, to be the kindest and most encouraging person that I can be.  Matt McNaney

It doesn’t matter who you are, where you have come from, or what your backstory is. Everyone can be kind to one another. Even when things suck, you must have it in you to remain kind.Kindness comes from your actions, your body language, your views on life and your visions of the future. Teagun Glenn

The feeling you get when that light starts shining inside you is more than enough reason to be kind. Rebecca Torrez

And, perhaps my all-time favorite student quote: 

Luckily, I always keep kindness in my utility belt. Wyatt Blevins

So in your utility belts, keep the following three things: Love, Forgiveness, and Kindness. 

"Grades are Private"

 I am pretty insistent that grades are between you, me, and the lamppost. Sharing your grades is so rarely a good idea, so rarely supports the concept of friendship, that I don't allow grade sharing in my classroom. Grade sharing has the potential of being distinctly unkind.   I know that you all will do what you will do in the hallways, over your phones and whatnot, but before you volunteer your grades, let me share with you what may be the consequences of grade sharing. 

1. You both get the same grades.  How common would this be?  Not very common.  If you share grades, this is the best scenario. You can hug each other and either celebrate or commiserate. This won't happen often, and you certainly can't count on this happening. 

2. Your grade is lower than your classmate's (and your friend's) grade.  You share grades.  Think about how each of you feels.  I function under the assumption that each person wants to earn an A; each person wants to do his/her best and be successful.  So in this scenario, you feel stinky and your friend feels awkward.  Or perhaps they gloat, and you feel even worse. At that point you might think, dang. I wish I never shared my grade. 

3. Your grade is higher than your classmate's (and your friend's) grade. Then the responses are reversed. And you might think, dang. I wish I never shared my grade. 

Because the most common scenario is case 2 or 3 above, you likely will end up feeling stinky or awkward. 

So let's say you get your test back, and you find that you did well. You know that you can't share your specific grade, but you are excited. So you pump the air, scream YES, YES, YES, and perhaps stand up so you could include a couple pelvic thrusts for good measure, then people who didn't do well might end up having some less-than-ideal feelings toward you.  Your best bet is to be reserved and share your joy with me outside of class and with your parents over the dinner table. 

So let's say you get your test back, and you find you didn't achieve your goal.  We are all human and sometimes this happens.  Bad days are a reality for human beings of all abilities. Moaning loudly, ripping up your test into small pieces, and eating it for good measure (yes, I've had someone do that), makes a very awkward situation in the classroom. Your best bet is to be reserved and to share your concerns with me outside of class and with your parents at the dinner table.  I'm here for you to help you achieve your goals; let's work together privately to see what we can do to to help you learn the material and successfully demonstrate that learning. 

The reality is that your friends, acquaintances, and classmates might just ask you in the hall or over lunch how the test went for you.  Let's brainstorm responses you could have to that request. Some suggestions follow; if those don't work for you, then let's think together about responses that would work for you. 

1. Ms Peaches said we shouldn't share grades, and she always offers good advice in our best interest and from a position of love and wisdom, so I won't share my grades. 

2. I don't feel comfortable sharing my grade with anyone. Hey, did you see the Bachelor last night? 

3. Yes, that was a test, wasn't it?  And we each received grades, you're right.  Did you see the Bachelor last night? 

4. Let's do our math homework together.  It sounds like it will be a fun and interesting assignment.

5. I like you too much to create tension in our friendship. Let's not talk about grades. 

6. Can we discuss the topics instead?  I'd like to hear what you have to say about the applications of right triangle trigonometry to the pitch of a roof. 

Being kind, aware of others, and self-aware are always good practices.  Making a choice not to share grades displays these good practices. 

As you navigate your life, you'll find decisions about how to respond to an event are so much easier if you ask yourself if a response is kind, necessary, and truthful. 

Why are We So Illiterate in Mathematics?

(Originally published January 11, 2016 and periodically updated)

And why does our common culture perpetuate the illiteracy?  As a math teacher, it makes me crazy to see our common culture supporting bizarre impressions of numbers and shapes and to see how students, made victims of these notions, can sometimes struggle with what used to be the most basic mathematical ideas.

Take, for example, the ice "cube." A cube is a regular hexahedron: a polyhedron that has six congruent faces, each of which is a square (think dice).  I don't know about you all, but the freezer of my youth held rectangular plastic pieces that froze water in the shape of roughly hexahedrons, with all edges that are line segments that are either parallel, perpendicular , or skew at right angles to each other.  Now, mind you, my own freezer mocks me in my passion for mathematics by producing a solid with two edges that are arcs and  two pairs of edges that are parallel line segments; these solids are definitely not cubes. Therefore it is no wonder that current students are confused by the word "cube."

Also take the coach, well meaning that she or he is, encouraging players to give 120% (or even a larger percentage) effort.  How does that happen?  When I fill my glass to the brim, it is 100% full. When I try to pour more water (or orange juice) into the glass, it overflows. I cannot add more than 100%.  Thus it is with athletics. How can you possibly give more than all you have?  No wonder students struggle with the concept of percentages in elementary school or middle school or in high school with the idea of probabilities summing to 1 or 100%.

Willy Wonka played with the idea of more than 100% in the following scene. It's worth a look -- at least one of the characters says, "Hey, that's more than 100%"

My son, as a young child, participated in many sports, but particularly loved basketball.  He played in a pee-wee kind of league in Colorado Springs that played wonderful games, but clearly someone in the program wasn't a math teacher.  The teams played 3 quarters in a game.  Yes, instead of playing 4 quarters to make a whole game, they played 3 twenty minute sessions that they called "quarters" and the game was meant to be over.   When I asked the ref when we'd finish the game as we had only finished 3/4 of the game, he looked at ME as if I was the crazy one.  If they played 5 twenty minute sessions, I don't think they would have liked to play fifths.  That word has other implications.

Now ok, I'm contemplating these kind of challenges I face each day in the classroom as I walk my dog, Karma (as in "good Karma") around my neighborhood in Albuquerque and I start noticing the numbers on mailboxes.  I live at 1198.  To our left as we face the street live our wonderful neighbors at 1196.  Across the street from them is 1197; order is maintained with the folks across the street from us: 1199.  But much to my consternation, the wonderful folks to our right live at not 1200; they live at 11100.  Is this a problem unique to Albuquerque? 

Seems no one in the US is immune to the "I can't do math," syndrome.  Check out this ad, then do the math.  Is $19.99 the value you'd get if you take 85% off of the original price of $169.99?

I enjoy a good math joke as well as any other football player; people recognize this as they link comics, cartoons, and what-not to me on facebook; I do delight in the humorous additions to my day.  But I squirm at the one that had one triangle talking to another triangle: "You are so obtuse, you wouldn't know an isosceles triangle if it bit you in the hypotenuse."  Now stop right there.  The cartoon has a cute one-liner and anthropomorphizes triangles, but only a right triangle (and not an obtuse triangle) has a hypotenuse.

 My hairs stand up when I see a joke that implies that there are no points contained in a circle:

And no wonder people have trouble telling the difference between an isosceles right triangle and an equilateral triangle when this image is used to help people, "Maintain social distancing": 

 

And of course, there's always arithmetic:



I'll let you do the correction.  I'm not sure that each person would even get enough for a Big Mac. How much would each person get?  And a serious question: why is this so difficult for people?

November 29, 2020, during a year that is going down in infamy, the front page of the Sunday New York Times covered the return to school (really the start of school) for a set of kindergarteners.  The article says, "Ms. Hellman, 26, dodged the triangular desks, spaced 6 feet apart..." The following photo accompanied the text.  The desks are clearly not triangular, as any kindergartener would know. I don't know of any classroom desks that are triangular. These are in the shape of isosceles trapezoids.

And while we are on the topic of the global pandemic and '6 feet apart,' let's return to the image about maintaining social distancing.

If it's been a while since you've been in 10th grade geometry: if you have a quadrilateral with sides of equal length 6 feet, the diagonals are 6 times root 2 feet, or approximately 8.5 feet.  

While we may find this funny, there's a bigger concern.  It gets expensive to not know or understand mathematics.  Staying alert and savvy can help in a variety of places, but specifically while calculating tip.  A restaurant, which shall remain nameless, provided the following inaccurate guide to calculating tips.  It seems they are vigorously and inaccurately encouraging you to be extra-generous to the servers (to the tune of 47% instead of 22%, for example). I do support encouragement of generous tipping, but I also support awareness of your personal tipping practices. I also support ethically providing accurate information. When called out for the error, the staff said, "It's up the the customer to make their own choices about tipping," failing to recognize responsibility for grossly inaccurate information. I'll leave it to the reader to calculate what the receipt should report as a "tip guide."

Here's something interesting.  In India, where many young people have arranged marriages and meet their betrothed on their wedding day, parents gather information about the partner they are choosing for their son/daughter.  Sometimes they receive erroneous information.  In Spring of 2015, one young woman was given information about her husband's education which made her skeptical. On her wedding day, she asked him a question about math:  What is 15+6?  When he couldn't answer it correctly, she walked out.   Math literacy is a predictor of future success; she knew it.

So perhaps I need more to do with my life so I'm not so concerned with such trivia, but I already fill 100% of my time between the triangle of 1198, math classes, and a tonic with ice cubes. Real ones, mind you.

             In this New York Times column, Nicholas Kristof quotes several eighth-grade math problems on which U.S. students did worse than those in other countries:

-    What is the sum of three consecutive whole numbers with 2n as the middle number?

Only 37 percent of American eighth-graders answered correctly, lagging behind South Korea, Iran, Indonesia, and Ghana.

-    How many degrees does a minute hand of a clock turn through from 6:20 a.m. to 8 a.m. on the same day?

Only 22 percent of American students answered correctly, behind Palestinians, Turks, and Armenians.

-    A piece of wood was 40 centimeters long. It was cut into 3 pieces. The lengths in centimeters are 2x -5, x +7, and x +6. What is the length of the longest piece?

Only 7 percent of Americans got it right, compared to 53 percent in Singapore.

“This is the generation that will be in the labor force for the next half-century,” says Kristof, “struggling to compete with citizens of other countries. It’s not just that American results are dragged down by poverty. Even American millennials with graduate degrees score near the bottom of international ranks in numeracy.”  Indeed, one of my high school friends serves on the "Common Council" political leadership board in Middletown, Connecticut.  Over dinner one evening, it became clear that she could not calculate 20% of 400. She laughed this off, as did everyone else present because it's a "geeky thing" to have that skill.  Keep in mind this board discusses tax rates in percentages that include not only far more difficult numbers in calculations (i.e, not multiples of 10) but calculations often in the abstract.  Without those skills, how could she make informed and intelligent decisions that would well serve my hometown? 

            What’s going on here? Kristof believes we have a societal attitude problem with numerical literacy – and we need to get over it. “Numeracy isn’t a sign of geekiness,” he says, “but a basic requirement for intelligent discussions of public policy. Without it, politicians routinely get away with using statistics, as Mark Twain supposedly observed, the way a drunk uses a lamppost: for support rather than illumination… Statistical literacy should be part of every citizen’s tool kit.” And maybe we should deemphasize calculus.

            Yes, some people are exceptionally gifted at math. “But let’s not use that as an excuse to hide from the rigor of numbers,” says Kristof. “Countries like Singapore manage to impart extraordinary math skills in ordinary children because they work at it.”

For your own blogs, dear students, you have a choice.  1. you write on your theory about why U.S. Citizens aren't very savvy mathematically or 2.  you find some mathy idea to write about that interests you. or 3. Find something in the paper, a publication, an ad, social media, anything, that includes a gross mathematical error.  Point out the error and explain how it's wrong or how it could be considered correctly. Remember to submit it appropriately. Remember to make this blog your own, yet cite sources.

it's a Dog's Life: Piecewise Functions

Have you heard that one year of a dog's life is equal to seven years of a person's life?  

This would imply the following equation: y = 7x where x is the dog's age (in years) and y is the human age (in years). According to this site, "One explanation for how this formula got started is that the 7:1 ratio seems to have been based on the statistic that people lived to about 70, and dogs to about 10."

Using graph paper with your x scaled from 0-12 and y scaled from 0-84 (think about why these values for domain and range make sense), draw this line ( y = 7x) to represent a MODEL of the relationship between dog-years and human-years. 

(An option other than graph paper is to use Desmos.  Consider: you can plot human years on the x- axis and dog years on y-axis -- this is the inverse of the functions given here.  Desmos can have you switch the x's and y's quite nicely should you want to do this. ) 

However, there's more to it than that.  Dog development is different from human development.  Even different breeds of dog develop at different rates. Below are two different additional models that compare a dog's age to a human's age.

This site has the following conversion: "For the first two years, a dog year is equal to 10.5 human years. After that, each dog year equals 4 human years. This calculation is based on studies that indicate dogs, except maybe larger breeds, develop more quickly in the first two years of life."

This implies a piecewise function to MODEL the relationship between the age of a dog and the age of a person: one function for 0<x<2 and a second function for 2<x, where x is the age of the dog. In the language and symbology of math:

 

Consider: where did the "13" come from? Remember this MUST be a continuous and increasing function -- otherwise there's a serious breach in logic: a dog can't suddenly become younger or the dog can't suddenly jump to be a dramatically different age. 

Using the graph paper from the first model (with your x scaled from 0-12 and y scaled from 0-84 and containing the linear function already graphed), graph this piecewise function. This is a second, likely improved, model of the relationship between dog-years and human-years.

Yet the  American Veterinary Medical Association has yet another MODEL of the relationship between a dog's age and a human's age:

• 15 human years equals the first year of a medium-sized dog’s life.
• Year two for a dog equals about nine years for a human.
• And after that, each human year would be approximately five years for a dog.

Your task for this blog: 

1. Identify the piecewise function that models the American Veterinary Medical Associations description of the relationship between a dog's age and a human's age (as described above in green with bullets). Again, don't let there be a breach in logic: this must be a continuous function. 
2.  Add this function to the graph paper that contains the other two models. Then choose three different dog ages and find how many "human years" is the age of the dog for each of the three models.  (You'll be turning in this graph on paper on the due date for this blog; this is where desmos could really help you.)  
AND, because there are always overachievers, you can expand your blog to include the following (or you can make your blog be the following):  
3. Find another piecewise function in the Real World.  It can be a different age conversion for dogs (or for cats).  It can be how the power company bills for services.  It can be a discussion of the examples used by Khan Academy or another site (DO credit your source.) Describe and explain the piecewise function and how it models what you've chosen  Graph it, if that's necessary in your explanation,  As always, find something that's fun for you. 
OF COURSE, you'll need to (a) add this to your current blogsite and not create a new one.  (b) turn in the blog accurately and carefully through canvas. Be sure I do NOT need a password to access the URL you give me.  To do this, be sure you click on "view blog" prior to copying the URL.  (c) If I send you a message through canvas that I can't read your blog, you MUST respond within 3 days or your grade will remain a 0

It's a Dog's Life: FSK

The following equations graph each of the three models connecting the age of a dog with a relative age of a human. X value, or horizontal axis, is the real years, the years a dog has been alive.  the Y value, or vertical axis, is a corresponding human age.

The green represents the simple "7 human years = 1 dog year" linear relationship.
The red represents the two-piece piecewise function.
The black represents the three-piece piecewise function.

They graph as follows:

Pi. where does it occur?

Pi in collisions:

https://youtu.be/HEfHFsfGXjs

Transformations: Mathematics in Animations

Mathematics, and more specifically, geometry, transformations and coding are used in computer (and other) animations.  This blog is to help you explore the mathematics of animations, particularly the use of transformation in animations.

https://www.youtube.com/watch?v=IjsbdeRJl5E

Here  are a couple things to think about with the applications of transformations to computer animation. You'll note that some of the transformations are isometries (preserve congruence, like when Mr Incredible is working out....) and some are homotheties (preserve similarity, like when Mr. Incredible is falling from the sky) and some are neither, but are still transformations.

If you look carefully in this fun you-tube video, you'll be able to see that there's some 3-d geometry here along with programming and, of course, art. Have you seen Madagascar? You'll like this one.

Five types of animation are shown in this you-tube video.  You'll see how coding is used to create a computer animation (type 3 of animation). Type 5 animation (Stop Motion) you can do a little of at the museum downtown, Explora.

And, of course, there's a Ted Talk by a Pixar animator that discusses science with the animation that Pixar does.

Your task (the "prompt") here is to find some element of mathematics in animation then discuss it in your blog.  Do a search, see what you find.  Watch one (or more) of the fun video links I've included here.  You all have some great searching skills; find some additional information about how art and math are combined in animation.  Have fun!