2015 Calculus Presentations

Today is the birthday of James Hutton, born 1726, in Edinburgh, Scotland. Hutton is famous for his theory of the age of the Earth.

The math quote of the day is by Rene Descartes who said, "Perfect numbers like perfect men are very rare."

Every year the final for my calculus classes consists of finding a partner, agreeing on a topic of interest, finding the mathematics that exists in it, and creating a presentation on that topic. This year's classes formed twenty presentations.

I have linked the title of each presentation to its Youtube site or the reader may visit my Youtube site: Charles Kruger.

The topics are: Aquariums, Aviation, Car Audio, Computer Animation, Dance 1, Dance 2, Discus, Farm, Golf, Guitars, Hydroelectric Power, Luminance in Art, Natural Disasters, Photosynthesis, Rollerblading, Soccer, Swimming, Tangled, Theory of Relativity, and Track & Field.

I enjoy these presentations for many reasons. Students take ownership of their presentations. The students and I usually learn something new. The students enjoy finding the mathematics in their topics. The presentations allow the students to be creative. The presentations are a showcase or a capstone of what the students have learned. The students use skills they have learned in other classes.

Enjoy!

Math In A Pea Cart

Today is the birthday of Abraham de Moivre born 1667 in Vitry-le-Francois, Champagne, France. de Moivre was instrumental in the development of Analytical Geometry and Theoretical Probability.

The math quote of the day is by Theodore Von Karman who said, "The scientist describes what is, the engineer creates what never was."

When I was sixteen I started my employment with Lakeside Packing Company. This company was a cannery in my home town that processed peas and corn. I worked every summer from the age of sixteen to twenty-one. My hours of work were from 6 am to 6 pm, seven days a week for the entire summer. There were a few days during the "pea pack" that the combines could not enter into the fields due to a large amount of rainfall, but generally speaking, I worked every day. The greatest and perhaps only incentive for working at Lakeside was its pay structure. I was paid overtime, time and half, for each hour over 48 hours in a work week. On the rare occasion that the peas were canned with carrots (a commodity provided by the federal government), work after 40 hours was considered overtime. My base pay was the current minimum wage of $2.10/hour and with overtime pay the hourly wage was boosted to $3.15/hour. My gross weekly wage was: 12 hours x 7 = 84 hours; 48 x $2.10 = $100.80; 36 x $3.15 = $113.40; $100.80 + $113.40 = $214.20.

At Lakeside, I was allowed to go to the restroom when I needed to, take morning, lunch and afternoon breaks, and most importantly, socialize. What I wasn't allowed to do was think. My entry-level job was to push pea carts into lines that were sorted by pea's size and density (determined by the pea's ability to float in a brine). 

A pea cart was a trapezoidal prism on rusted casters whose momentum was often impeded by B6's, peas the size of #Triball 12 buckshot with the hardness of diamonds. As a side note, when I asked who ate these type of peas, I was told unsuspecting institutions such as schools and nursing homes. These carts were filled with peas from tubes extending from the ceiling. The tubes were the recipients of peas garnered from trucks that had dumped their cargo into washers. From the washers, the peas had made a journey to the attic of the factory where they were washed, sorted, and flung down the appropriate tubes to the carts below. On the image above, there would be a small door on the side opposite of the slanted side. Once the carts were filled, I would push the cart to an assigned line and another employee would roll the cart to a predetermined position, lift up the door, and feed the peas through the rectangular opening into a small, swirling, bubbling vortex of steaming water. This vortex would take the peas to the roof of the factory where the peas coursed their way through a series of pipes. This corkscrew of rides was called the blanching process that all vegetables went through prior to being canned.

Every day, 12 hours each day for about 3 months, I pushed a cart. When I went back to school, my mind was like undercooked oatmeal. Constructing a thought was like wading through wet cement, and as a result of this mental decay, my mathematical processes were slowed. Also, my vocabulary was, well... not creative. I had heard a particular word so often that its usage had to be considered a fundamental rite of passage in this work environment. I had heard this word used as a noun, a verb, an adjective, and an adverb in one sentence. With this one word, employees described their frustrations and their joys. This monolexemic sentence was not conducive in a family or academic setting. This poisonous mixture of mental atrophy and constricted diction reduced me to a zombie with a hunger for abstract thought.

My second year at Lakeside Packing Company, I was promoted to the job of emptying the peas from the carts into the boiling whirlpool of churning water. As much as I had hated last year's job, I hated this worse. The temperature was hot and humid and no one wanted to spend any time discussing teenage topics within my restricted work area.

I was concerned about my deteriorating mental state and I devised a plan to solve math problems. I wanted to know how many A3 peas were in my cart. I first measured the dimensions of my cart and the radii of randomly selected of peas. I averaged those radii and formed an estimate of the volume of an A3 pea.  My initial answer was computed by dividing the volume of the cart by the volume of the pea. I knew my quotient was higher than the actual number due to the empty space between the peas.

I devised a new plan based on average rates. My peas eventually made it to the "fillers". The "fillers" were machines that actually placed the peas into a can, sealed the cans, and sent the canned peas to the "cookers". "Cookers" were a holding location where the contents were "cooked" for an appropriate time and temperature. At the beginning of my shift, my "filler" had no peas. I emptied my cart and stopped. When my assigned "filler" received the peas, it started up but stopped again when it ran out of peas. I timed the start and stop interval of the "filler". I knew the number of cans per minute that the machine sealed. Quality control randomly selected five cans from the "filler" to examine their contents. I had them count the number of peas in each can and determined an average number of peas per can. My formula: Minutes x Cans/Minute x Peas/Can = Total Peas in a Cart. I was pleased with the results. Both methods resulted in close estimates with the latter being slightly lower than the former.

I can't remember how long this problem took me to solve. I don't think it took more than a couple of days. I was hoping for a problem that I could work on for the entirety of canning season. I do know I was sad when I completed it. I believe I worked on mental math problems the rest of the summer but those activities did not give me the same satisfaction. One of the reasons I enjoyed this problem is that I involved other workers. I had the people working on the fillers giving me information about their machine and taking the risk of stopping and starting their machine which was not a popular idea among the mechanics and our superiors. I also had to work in concert with the quality control and convince them of doing the added work of counting peas. Perhaps, the problem temporarily broke everyone from the constant boredom that encased their daily routine. Many people were curious on how many peas were in a cart during that brief moment in the summer of 1975. Yes, for a brief time, doing math made me cool :-)

And so it begins, again

Today is the birthday of Henry Dudeney born 1857 in Sussex, England. Dudeney was interested in mathematical inquiries and past times. His publications about these topics lead to major mathematical research.

Today's quote is from Jospeh-Louis Lagrange who said, "As long as algebra and geometry have been separated, their progress have been slow and their uses limited; but when the two sciences have been united, they have lent each mutual forces, and have march together towards perfection."

This is my tenth year of teaching AP Calculus. I retired last year and was hired back part time. This year was also the year I had my official teacher observation. During the follow up conversation with my principal, I remarked that I would like to teach again next year if the opportunity arose. He was very open to the idea but did inquire why I didn't look to teaching opportunities in a different educational setting such as in a collegiate environment. I paused and then confessed that I was touched by the relationships that I formed with my students. He smiled and said, "Your not such a tough guy."

My classes are currently reviewing all the topics we have covered and preparing for the AP Calculus Exam that will be in May. I also had Parent-Teacher Conferences this past week and some serious discussions with parents on the probability that their child will pass the exam and receive college credit. I analyze the statistical data I have gathered on each individual student and have a fairly accurate predication model on that probability. I have a conservative model. By that I mean, most students due better that I predict but for me personally, this time of year is heart wrenching.

My students have worked hard, possibly harder than any math class they have taken previously. Many have given up the luxury of an A or B with little work and had to adjust to struggling to get a C. What has happened as I have become older, I agonize on how I can get them to squeak out 1 or 2 extra points on a question or how to see that problems they struggle with are related to the problems with which they solve with ease. The parents really want their child to succeed and not feel the pain of perceived failure.

I try to assure the parents that even if their child scores a 1 or a 2, they will be fluent in calculus when and if their child takes it again but the parents want more, they want assurance. I understand that, I'm a parent.

I have been with these students the entire year and every year it happens, I fall in love with them. They are delightful. They make me laugh, they make me think, they challenge me, and they teach me. I become the student. If I could, I would give them all 5's but that is not going to happen.

Bob Dylan wrote in the song "If You See Her, Say Hello" these lyrics: "Either I'm too sensitive or I'm getting soft. Sundown, yellow moon, I replay the past. I know every scene by heart, they all went by so fast." The years and classes do. I wish I could slow them down.

The Sands of Time

Today is the birthday of Clifford Dowker born 1912 in Parkhill, Ontario. Dowker made contributions in the area of Knot Theory.

Today's quote is from Joseph Serret who said "Algebra is, properly speaking, is the Analysis of equations."

"Like the sands through the hourglass so are the days of our lives." Quoted by MacDonald Carey in the opening credits of 'Days of Our Lives.'

"Sand is pouring into a conical pile of sand, at the constant rate of 6 m^3/second. Suppose the side of the cone makes a 60° angle to the cone's base. How fast is the height of the sand pile changing when the volume of the cone is 180 m^3?"

"Oil is filling a cylindrical tank at a rate of 2 gallons/minute. If the radius of the tank is 2.5 ft, at what rate is the height changing when the tank half full?"

These are the problems that I was thinking of at a recent wedding that I attended. I sometimes lose my focus and start thinking of mathematics at inappropriate times. The couple was pouring sand into a container. The colors of the sand were blue and white. The pouring of sand replaced the tradition of the unity candle. Although the pouring of sand was a new wedding experience for me, I have found out it is not unique in wedding ceremonies.

During my son and my daughter-in-law's wedding, the parents of the couple lit the candles the groom and bride used to light the unity candle. I found that act a moving experience.

As I watched this couple pour each of their respective sands into a truncated cone (often called a tapered glass/i.e pint glass), I noticed that there were times in which one paused to let the other pour. The pause that allowed the other their moment of time, their moment to pour.  Within the wait and pour, a unique formation of sand was formed.

I believe in marriage, each partner is required to pause. Whether the pause is to allow the other a moment in the sun, a moment of respite, a moment of encouragement, or a moment of solitude. Such is marriage, a series of give and takes that is formed by each individual grain of sand, each waiting their turn, each supporting the other, and forming its unique beautiful structure.

The Probability of the Fabulous Five

Today is the birthday of Agnes Scott born 1894 in Jersey. Scott graduated from Edinburgh University and taught mathematics at Raffles' Girls School in Singapore.

Today's quote is from Alan Turing who said "Mathematical reasoning may be regarded rather schematically as the exercise of a combination of two facilities which we may call intuition and ingenuity."

A parent function is the simplest function that contains all the characteristics of the family that can be formed from it. Examples of equations that form parent functions are: y = x (linear); y = x^2 (parabolic); y = a^x (exponential), and y = |x| (absolute value). The examples I have listed are just a small sampling of the parent functions that exist. The "children" that are formed from these parents are formed by transformations which are translations, rotations, reflexions, dilations, or some combination of thereof. In spite of the transformation that may occur to the parent function to create its "child", the offspring still retains the basic characteristics that defines its parent.

An indefinite integral creates a general antiderivative which when graphed forms a family of antiderivatives. On the graph below, I have depicted a sample of a family of antiderivatives on a slope field. 

I have four siblings, one sister and three brothers. My sister refers to us as the Fabulous Five. I have an older brother and sister. My brother is nine years older than I and my sister is seven years older. My younger brothers are identical twins and are a year and a half younger than me. I am sure that my parents remark, as my wife and I do, on the differences that exist among their children and on the remarkable, similar personality traits related to themselves.

I was contemplating the probability of the order in which the Fabulous Five were born; M, F, M, M, M. This probability is the product of each individual probability since each birth can be considered an independent event. I thought that the probability of gender would be equally likely so the P(Female) = P(Male) = 0.5 which would form a theoretical probability of (0.5)^5 = 1/32 = 0.03125. However, nature (empirical probability) adapts accordingly. With some variance due to environmental changes the P(Male) = 0.51 and the P(Female) = 0.49.  There are several theories on why the probabilities are not equal. One of the most interesting to me is that nature adjusts the probability slightly due to the survival rate of males past the first 6 months of birth and the historical early demise of males before their female counterparts. Using nature's probability, the probability of Russ, Marilyn, Chuck, Bill, and Bruce is approximately 0.03315. 

Another analyzation of family is birth order. The rationalization for the importance of birth order in the development of an individual's personality is a result of the changing interaction that parents have with each successive child.  The basic traits of birth order are defined for a family of three with adjustments for number of siblings, gender, and age differences. The basics are defined as follows: The oldest is a perfectionist, a list-maker, well-organized, self-sacrificing, conservative, critical, serious, scholarly, reliable, conscientious, structured, cautious, controlling, loyal, and an achiever. The middle child is a people-pleaser, somewhat rebellious, thrives on friendships, has a large social circle, and is a peacemaker. The youngest is fun-loving, uncomplicated, manipulative, outgoing, an attention-seeker, and self-centered.

Gender, number of siblings, and age differences play an important role in our birth order structured according to "The Birth Order Book" by Dr. Kevin Leman. The first born of any gender is likely to take on first born characteristics. Our family had very traditional role assignments. My oldest brother was assigned outdoor jobs such as lawn mowing, painting, and helping my father. My sister was assigned duties that helped my mother such as cleaning, cooking, and "taking care of the babies". These roles cemented the first born characteristics for each of them.

Age difference is another important factor in my family's structure.  When there is a span of more than five years between children, "a second family" is formed. I am the oldest of this "second family" and took on the roles of the oldest, the middle child, and the youngest. 

The final spice that added flavor and complexity to our family structure was the birth of twin boys. Usually twins place additional pressure on any sibling close to their age. As twins, my brothers have many characteristics of the youngest. They have a confidence and "swagger" that I admire and envy. 

The last part of dynamic that I am focusing on is called "The Sibling Effect". I recently watched a Ted Talk entitled "The Sibling Bond" in which the speaker, scientist Jeff Kluger, makes a pitch on how interactions between siblings model us to what we are today. There are 21 possible dyadic relationships in my family. These relationships will probably be the longest relationships I will have. How impactful were these relationship in forming my personality and how do they impact me now? I don't know. Kluger gives very little science in his talk or his book. I do agree that squandering my sibling relationships is foolish. Recently, my siblings and I have come together in managing my parents' affairs. I found that I leaned on the knowledge and expertise that they have garnered. Our love for our parents is apparent.

Desmond Tutu said, "You don't choose your family. They are God's gift to you as you are to them."

The Voice in Mathematics

Today is the birthday of Oronce Fine born 1494 in Briancon, France. Fine published major works in mathematics and astronomy.

Today's quote is by Raoul Bott who said "There are two ways to do great mathematics. The first is to be smarter than everyone else. The second way is to be stupider than than everybody else - but persistent."

I am of the belief that mathematics and written language are more similar than different. Each has its structure and its form of creativity. I have touted this belief to my English teaching colleagues using author, John Greene, and his views on the education continuum as a supportive example. I often hear from some of these colleagues that they do not possess the "math gene". I am not a believer in a math gene. I am not any more a genetic math mutant than I am a lobster fisherman. Keith Devlin states in the prologue of his book, The Math Gene, How Mathematical Thinking Evolved and Why Numbers Are Like Gossip, "One of my aims in this book is to convince you of just how remarkable and powerful - and uniquely human - language and mathematics are. Let me once again quote Neil Armstrong. When the lunar module broke free from the command ship that would remain in orbit above the moon during the course of the moon walk, Armstrong declared that "The Eagle has wings." The acquisition of language and mathematics gave humanity the wings to soar above our fellow creatures. My other aim is to argue that these two faculties are not separate: both are made possible by the same feature of the human brain."

My first educational love was books. Each day as a youth I visited the library and each day my imagination soared with the descriptive voice of the author. I was transported from my sleepy hometown to bustling cities and exotic locales snuggled in a world that were filled with intrigue and adventure that I was sure did not exist in my community and most definitely, in my life. During my high school years, I took classes that were titled Short Stories and Novels which were taught by Bill Nems and with his facilitation, my perception of what I read deepened.

As my formal education continued, I gained a mistress. Her name was Mathematics, the queen of sciences. She offered me structure, organization, predictability, and the allure of a right answer. She eventually introduced me to my drug of choice, solving complex mathematics problems. I had my first initial rush from this drug in my 8th grade algebra class. I do not recall the problem but I do remember solving it in my sleep. This was my first dream in which I awoke with the solution in hand. I was ecstatic. The thrill was overwhelming and I have been addicted ever since. I have continued to solve math problems in varied states of consciousness.

Language is not complete without the written word. I started to appreciate writing as I was completing my master's degree. Within the composition of papers that detailed the educational practices of teaching mathematics and examinations of mathematical curriculum, I discovered my voice. I could be insightful and humorous, sarcastic and tactful, gregarious and prudent. I became a logophile, a lover of words. Writing has offered me creativity in structure and the tantalizing mystism of solving the problem of organizing my thoughts and voicing my passion.

As I stated earlier, I constantly discuss with the English teachers in my school, the parallels that exist in mathematics and English. Recently, an English teacher stopped me in the hallway, approached me resolutely, and stated "Where is the voice in mathematics?" I was delivered a knock out punch. I stood speechless and wandered away muttering to myself, "Where is the voice of in mathematics?"

My first question to address is what is meant by voice in writing. I found two sites that offered descriptions of an author's voice. Understanding Voice and Tone in Writing by Julie Wildhaber and Voice in Writing: Developing a Unique Writing Voice by Cris Freese. Freese states "A writer's voice is something uniquely their own. It makes their work pop, plus readers recognize the familiarity." Wildhaber defines voice as "the distinct personality, style, or point of view of a piece of writing or any other creative work." She elaborates, "Many musicians have played the 'Star-Spangle Banner,' for instance, but there's a world of difference between the Boston Pop's performance and Jimi Hendrix's, even though the basic melody is the same."

I asked my English colleagues their opinions of voice.

Beth Gadola: "In writing, voice can be expressed when a writer puts him or herself 'into' the words, providing a sense that a real person is speaking and cares about the message. When a writer is personally engaged with the topic, he/she imparts a personality to the piece that unmistakably his/hers alone. And it is that individual personality - different from the personalities of all others - that we call voice. Voice is the distinct personality of a piece of writing."

Stan Berg: "For me, it's a sense of authenticity, the idea that a real person is letting his/her real self come through in the writing so that it doesn't sound like a canned response that could have been written by anyone."

Lyndsy Schwantes: "I would say that a writer's voice is what makes the writing stand out to me. It's not only their word choice, but the way their words work together to tell a story and create characters the reader can connect with. When I read a story by an author I consider to have a good "voice" I don't think about the words or why their choosing their words, but I become lost in the story."

Maria Burnham: "Voice, simply put, is the thing that breathes life into writing. It's the formula of writing that tells my brain's inner voice how to read something. Is the writing factual and simply stated? There is voice in that style of writing. Is the writing full of vivid description and flowery language? There's voice in that style as well. Is the writing full of questions followed by answers? You've got it, there's voice in that, too.

For what it's worth, the reason I majored in English in college was that I fell in love with voice in writing. I loved that the choice in words and the way in which they were arranged changed how the writing sounded. In writing there's cadence, emotion, meaning, all because of an author's choice.

I find myself gravitating toward specific kinds of voices, particularly those of poets and essayists like Whitman and Thoreau. But I also love humor, sappy love, and even the simplicity of technical writing.

Letters and words in isolation have meaning but little depth unless put together by an artist, a wordsmith, a writer."

A common thread winds through these definitions. Voice, like beauty, is in the eye of the reader. Even a boring voice in one's mind may conjure up an image of Ben Stein lecturing in Ferris Bueller's Day Off . A voice that paints images, resonates emotion, and is layered with complexity may only be appreciated by a few experienced readers. I watched a rich and vivid dance performance recently. A strong female voice emanated from that performance but I knew that my perception only heard whispers of her messages.

Mathematics has beauty. "Mathematics, rightly viewed, possesses not only truth, but supreme beauty - a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show." Bertrand Russell,  The Study of Mathematics

A February 2014 article by James Gallagher in the BBC News entitled Mathematics: Why the brain sees maths as beauty, discusses the brain activity of emotion that is triggered equally by a beautiful equation, a great painting, or a classical piece of music. The equation that is believed to be the most beautiful is e^(iπ) + 1 = 0. This equation is referred to as Euler's Identity. The proof of the identity resonates in me wonderment and awe.

Does beauty in a medium indicate a voice? I don't necessarily think so. A rose can be appreciated for its beauty but it remains silent.

I was at a loss in describing a voice in mathematics until a recent Math League practice. I sat and listened to my students work on various problems. I believe these three problems spoke to my students.

1.

I was asked my students where to start on this problem and told them to rationalize the denominator. As one student proceeded, she paused, "Oh, cool! This is a cute problem!"

2. 

As we worked on this problem, several students remarked how this was a "sneaky" problem. When I asked them to define "sneaky", they described problems that first appear to relatively direct to approach but as those problems are solved, they entail twists, turns, and surprises that require persistence and flexibility by the solver.

3.

In this problem, the graph is defined as the second derivative and students were asked to draw the first derivative and the original function from which the second and first derivative were formed. Students described this problem as a simply stated but multilayered in its complexity. I compare their description of this problem to a single stanza poem. A poem that offers new insight to a reality that surrounds us each time it is read.

As I contemplate what specific mathematical ideas speak to me, I have a few the quickly come to mind.

The proof that the square root of 2 is an irrational number has always evoked a voice of sarcasm. I can hear my son, Sam saying,"So you don't believe it to be true? Let's assume its not and see where that leads us!"

The Pythagorean Theorem speaks to me in a voice of security, loyalty, and strength. The Pythagorean Theorem is in algebra, geometry, trigonometry, and construction. When I am stuck on a problem, I can hear John Wayne lean in and say "Hey, little pilgrim have you tried the Pythagorean Theorem?"

I can hear the imaginary i screaming at mathematicians, "You thought that negative numbers were imaginary and you found application. You define me as the square root of a negative one and call me imaginary, yet you found application for me! What is it with you guys?!"

Finally, the controversial number 0, the number that apparently does nothing in addition but is the mighty destroyer in multiplication. The number that can be divided but is not allowed to the dividing. The number that was considered the null and void and repulsed by religions but without it we would not have calculus. The number that says, "Go ahead, make my day."

The voice in mathematics exists but it is different than the voice in writing, the voice in music, the voice in dance, and the voice in art. Mathematics is creative. Its voice may be more restrained and its voice may take more work to hear but the voice exists, nonetheless.

Deja Vu All Over Again

Today is the birthday of Franz Aepinus born in Rostock, Germany, 1724. Aepinus made contributions in the area of electricity and magnetism.

Today's quote is by Niccolo Tartaglia. This quote is a poem written to Jerome Cardan. In this poem, Tartaglia reveals to Cardan the secret to solving a cubic equation.

"When the cube and the things together

Are equal to some discrete number,

Find two other numbers differing by this one,

Then you will keep this as a habit,

That their product will always be equal,

Exactly to the cube of a third of the things.

The remainder then as a general rule

Of their cube roots subtracted

Will be equal to your principal thing."

In my calculus classes, I talk about my dog, Rosie, and she has become quite a celebrity. Last year and again this year, I showed them a graph that I entitled My Walk With Rosie. I assigned them the task of detailing the calculus concepts that are embedded in the graph. I have listed those responses. The responses are both mathematical and creative. One student included cartoons of Rosie and myself.

Rosie patiently waiting for a walk.

Rosie spending some quality time with my son.

The graph of My Walk With Rosie.

1) The limit of f(t) as t approaches 45 from the right is 150.

2) At 45 minutes there is a relative maxima (vertex), where the derivative is zero. This represents the point at which you turned around to return home because for a moment you were not moving away from or toward you house.

3) There are x-intercepts at (0, 0) and (60, 0). These points represent times when you were at your house. Because there are only two points, we can infer that you left at (0, 0) and returned at (60, 0) and did not go to your house at anytime between those two points.

4) There are two points on this graph (30, 50) and (45, 150). We can find a secant line with those two points. The slope of this secant line is 20/3 or 6 2/3 feet per minute. Because his rate (slope) is positive we can tell that you’re moving away from your house.

5) After coming home from a grueling day of teaching, Kruger came home and decided to take his dog, Rosie, on a walk. Starting at the end of his driveway, he started walking at an average rate of 3 1/3 feet per minute^* for a duration of 15 minutes. This took him a total of 50 feet. Being too tired to continue, he had to stop and sit down on the curb. They sat on the curb for 15 minutes when Rosie got up and started chasing a passing car. Alarmed, Kruger got up and pursued his runaway dog at the slow average rate of 6 2/3 feet per minute^*. Even though this was double his previous pace you would think that if he was concerned about Rosie, he would pick up the pace. He grew discouraged after 100 feet and 15 minutes and turned around to go home to call the police and report Rosie missing. On the return trip of 150 feet he averaged a speed of 10 feet per minute^* and arrived home after 15 minutes. When he got to his front door he found Rosie there waiting for him.

^* Secant lines can be used to find his average speeds at different intervals of his walk. The part of the graph with a slope of zero means he is not walking.

6) I can tell you took a break from 15 minutes to 30 minutes because you stayed at 50 feet the whole time. The slope is zero because the tangent line is a horizontal.

7) I can tell your started off slow and once you got closer to 15 minutes you moved faster. The slope of the tangent line is steeper towards the end.

8) 150 feet is the furthest you walked. This shown at the top of the parabola.

9) It took you 45 minutes to walk Rosie the first half of your walk and 15 minutes to walk her back. The graph is increasing the it starts decreasing that's how I can tell.

10) It was a find day outside an on October 12, 2014. The sun was out and the chill of autumn was in the air. Being that math nerd that he was, Mr. Kruger decided to make a very well drawn graph of his little adventure. With his handy dandy calculator in hand and his sidekick Rosie by his side, Kruger began his adventure.

Mr. Kruger started off his walk at a good pace. After 15 minutes, he had already gone 50 feet! This fast pace was due to the fact that Rosie saw a squirrel ahead and she would stop at nothing to chase it. Well, Rosie did stop for the squirrel ran up an oak tree to seek refuge.

Rosie's sudden outburst almost caused Mr. Kruger to drop his calculator right down the storm drain near by. Thankfully, Rosie was jumping after the squirrel at the same time the calculator air born and their trajectories collided and the calculator deflected off of Rosie's side. The calculator then soared back into Mr. Kruger's waiting hands.

Mr. Kruger wept at the sight of the mathematical miracle that had happened here. He tried to go on with his walk but the heat from his tears had caused the poor man's glasses to fog to the point where he couldn't see. While waiting for his glasses to defog, he decided to figure out the equation for the first 15 minutes of his walk. After fiddling with this calculator for 12 minutes, he did some math like this: Start point (0, 0), End point (15, 50), m (slope) = (50 - 0)/(15 - 0) = 50/15 = 10/3, 50 = 10/3(15) + b, 50 = 50 which means the secant line of line for the first 15 minutes of his journey was: y = 10/3x. 

Kruger was about to continue on his walk when he realized he had just wasted 15 minutes standing still, finding no challenge in determining an equation to the next section of his walk. The equation was y = 50. For those 15 minutes he was nothing but a sad constant.

Mr. K quickly got over the fact that he was a constant and continued on his merry way. After another 15 minutes, Kruger stopped to pick up a coin on the ground - after all it was heads up - and stuck the coin in his pocket. Little did he know that this move would alter the equation of his walk for the day.

Mr. Kruger, being the math genius he is, quickly figured out the equation to this piece of his walk was y = -2/9(x - 45)² + 150.

Kruger then started to increase his speed. After all, he had been walking Rosie for 45 minutes now. Mr. Kruger pushed with all his might to go another 15 minutes. He was out of breath and huffing and puffing and wheezing and squeezing and Rosie may have had to drag hime a bit but he made it a whole hour.

Mr. Kruger sat in his kitchen wondering what else he could conclude from his walk. He noticed his graph was continuous. Since time never stops, there was no holes or asymptotes. Mr. Kruger then decided to dig a little deeper and find the derivative portions of his graph.