Brain Based Strategies
Rational and Irrational Numbers Project
Below is a story that Willis, a graduate student created for his class to engage and excite, changing up his presentation of math. I have posted it with his permission. Also I have cut and pasted suggestions I gave to Willis for presenting this story to his class.
How you present this strategy to them will be important. I would prepare your students ready for a surprise in the action asking them to guess what you will be presenting that has to do with Math. Prediction is powerful and incorporating 20 questions in preparation for this story would pique curiosity and wonder. Everything and everyone encompassess a story in ourlives and that puts us all on the level playing field of intuitive connection as Dan Meyer stated in the video.
As they listen, you, as their teacher, will come to life before their eyes because you trusted them to listen and I would emphasize this as you present to them that when any of us share, it is a privilege to be a listening participant. This truly sets the stage for mutual and respectful community in your classroom!
You could also spend a few minutes brainstorming with them for a couple of days, ideas for their stories because blank sheets of paper are scary or whatever mode they decide to present in or from can be intimidating before one begins. As a class you could make lists of all the ideas for setting, characters, content, and plots or story lines; that way each of them will have a list to pull from. I hope this is helpful, as you play and have fun with this idea! Many teachers might think content gets lost in such an activity, but quite the opposite occurs. Content, rention of content, and learning thrives well.
I would love to post some of your stories and experiences when you try out this strategy in the days and weeks to come.
FROM WILLIS
Rational and Irrational Numbers
On a crisp Saturday morning in late November, an unusually dense marine layer gradually blanketed the hillsides of Burbank, then Pasadena on its way towards conquering the skies of Los Angeles. As the invasion of aquatic fog continued its march inland, I stood beneath its imposing shadow of dullish gray. A swift and frigid draft swept across Baldwin Avenue and the sea of spectators packed on the sidewalk all braced their bodies in response to the sudden chill. Not a single muscle on my body moved as I looked towards the sidewalks, and saw the people in the crowd tighten their jackets and windbreakers. My thick uniform jacket and tight-fitting helmet sheltered me from the autumn gusts. To me, the draft felt like a gentle and comforting breeze as it splashed my face, washing away some of the tension and anxiety that had been building up for the last five minutes.
The fatigue of supporting the weight of my 30-pound tenor saxophone solely on my right arm began permeating my thoughts, but I disciplined my mind to ignore it as I continued standing at attention. In just four minutes from now, I will have completed the final performance of a parade I have trained every day after school for since school began in late August. I recall sweeping my peripherals with my eyes, and seeing my fellow band mates lined up in perfect rows and diagonals. The two hundred of us had woken up at 5:00 AM in the morning and boarded buses that transported us 115 miles from our school in San Diego to the quiet and tranquil L.A. suburb of Arcadia, where we annually compete in the most renown marching band festival in Southern California. Our school had won the championship every year for the last years. Adorned in our school colors as we fielded white jackets with neatly polished buttons, military hats with bleached feathery plumes, and royal blue plants with parallel white stripes streaking vertically down to our white marching shoes, we were poised to bring home the 6-foot tall trophy for a fifth consecutive year.
“Tweeeeeeett. Tweet, Tweet, Tweet!” After captivating the audience with his baton twirl, the drum major signaled to band that the stage was ours. The orchestrated cadence of drums and cymbals counted down the final seconds before our performance and then, on a syncopated off-beat, we played our first note of the march and a split second later, our band of 200 stepped off in unison to the low brass opening of “Washington Greys.” Two measures later, the woodwinds joined in, and then, the fanfare of trumpets cut through the air. Down the street we marched to the fast-tempo tune in the key of F-minor. I marveled at how impeccably straight and precise the rows, columns, and diagonals of our parade block remained as we glided down the street, perfectly harmonious in both sound and motion. Just two months earlier, my eyes frequently witnessed the vision of the exact opposite.
When the marching season began, our band was seemingly years away from the orderly and rationally cohesive performance ensemble we had shaped into by the end of November. Our first few practices were chaotic and confusing. Musicians who had not yet memorized the music kept playing off-beat, often influencing their entire instrument section to play in disagreement with the rest of the band. Our marching was greatly flawed, and I remember the ends of some rows would curve off by an entire yard from the center of the row. On extreme cases, we even had musicians bump into each other and trip then fall down when we were practicing our halftime field shows for football games. At times, the various horn sections of the band were playing 2 to 3 seconds apart from where the band was supposed to be during a song… Our band was a disorderly and chaotic mess on some days, unable to perform together rationally as part of a unified ensemble. Our band director held us to a high level of excellence, and had no tolerance for the disorganized display of confused and unorganized musicianship. So, he cracked down hard on us for the irrational mistakes we made, as he increased practice times and frequencies of practices. His hard and uncompromising approach towards subpar and irrational mistakes reminded me the Greek mathematicians Pythagoras and Hippasus.
Thousands of years ago, Pythagoras taught the brightest minds in Ancient Greece on the most advanced topics in math of that era. One of the mathematical ideas that Pythagoras advanced was the idea of rational numbers. Pythagoras believed that all numbers are rational, meaning that they can be written as a ratio of two integers. For example, the number 7.5 is a rational number, because it can expressed as a ratio of the numbers 15, and 2 if we write 15 and two as a fraction: = 7.5. Likewise, the number 2.33333333….., even though it is a repeating decimal is also a rational number because it can be written as the ratio of two integers. In the case of 2.333333, the two numbers are 7 and 3. If we place 7 on the numerator, and 3 on the denominator, then we come up with the fraction . Dividing 7 by 3, we end up with the answer: 2.3333333…..proving that 2.33333 is a rational number.
However, one of his students named Hippasus started creating disorder, and began disturbing the logical organization that all numbers can be expressed a ratio of two integers. Hippasus said that some numbers, like the square root of 2 (, cannot be expressed as a ratio of two integers. The value of is 1.414213562373095048801688724209698078569671875 37694 807317667973799…and so it continues without any pattern forever. When Hippasus tried to write the value of as a fraction of two integers, he found there is no solution. He could come close to the answer by solving for 99/70, but the answer was merely a close approximation, not an exact match. Other number values, such as the commonly used Pi, , are also irrational. is equal to 3.14159265…and so the numbers continue without any pattern. Thus, Pi cannot be written as a ratio of one integer over another.
Pythagoras was deeply angered and disappointed in his student Hippasus. Just like my band director, Pythagoras would not tolerate a single student who operated out of alignment and in disagreement with the rest of the group. When Hippasus created disharmony within Pythagoras’ teaching by pursuing the idea of irrational numbers, Pythagoras realized that he could not tolerate Hippasus’ disorderly conduct. Since Pythagoras viewed the idea of an irrational number to be illogical and harmful towards his orderly organization of number theory, he knew that Hippasus could no longer be a member of his group. Pythagoras invited Hippasus out a on a sailing trip one afternoon, and threw Hippasus into the sea, drowning his student.
Although our band director was often mean and yelled at us for even the most minor errors in our musical performances, we knew that we were fortunate students because he would work with us on improving our techniques until we could perform as a rationally cohesive group rather than just drown us like Pythagoras did Hippasus when Hippasus discovered the existence of irrational numbers. After all, the ocean was just a few minutes away from our school…
Video link: http://www.youtube.com/watch?v=nuX_6KLyj6Y
Math Lab Assignment
(Due Monday, September 20th)
This assignment is worth 20 points in the Labs/Projects category.
Write your own story, relating it to our lesson on rational and irrational numbers. In your story, you must:
1) Define what rational and irrational numbers are
2) State at least one example of a rational number and one of an irrational number
3) State the criteria for rational numbers.
Hint: Use your notes from class. Also, your story can be fictional. Your stories do not have to be tied to a life experience or a historical event like parts of mine are. In fact, I look forward to reading stories that are brilliant works of fiction.
Remember: You have complete creative licensing over this assignment. Your story can be set in outer space, the Triassic period, or a fictional universe. You can personify numbers into characters of your story if you’d like…maybe even create historical satire where one society is depicted as rational numbers, and another society as irrational numbers and explore the differences between the societies… or a league of rational numbered super heroes that is recruiting new members only if they meet the criteria for being a rational number… The possibilities are endless!
Be creative, and let your imagination take you to new places!!!
Learning the nine rules of comma usage with secondary students.
Joel Thomas, Teach for America Secondary Language Arts
All the Girls will use them, comma.
Everybody pay attention, this right here is my grammar boy swag.
Grammar boy Swag (4)
Get out the wayy
Commas are commin through
Lists of three or more, like “knife, fork, and spoon.”
Independent clauses use them with their conjunctions
Or if Dependent comes first, yeah that’s another function.
I’m lookin’ for a yellow bone, long haired star
2 adjectives about one noun like “big, old car.”
Or with places and dates in case you have to go far
Grammar boy take off in 5.4.3.2.1
This right here is my comma.
All the Girls will use them, comma.
Everybody pay attention, this right here is my grammar boy swag.
Grammar boy Swag (4)
Comma Songs- Reflection from Joel Thomas
To try and enhance the learning experience of my students, I designed a project that would let them branch out a little, while helping them remember nine rules for comma usage. The assignment required them to put the nine rules in a song, rap, or poem, memorize it, and deliver it to me or the whole class. In order to make sure I had 100% participation I did allow an “opt out” assignment for students that don’t think they’re creative and are too shy to perform for the class. Students fitting this mold could simply memorize the nine rules and orally present them to me. The number of students who took the “opt out” assignment varied greatly from my 9th to 12th grade classes (the older the student, the more likely he/she was to perform a rap). My 9th and 10th grade classes each had only one person do the rap, while the rest chose the “opt out” assignment, while my 11th and 12th grade classes had nearly 100% of the students do the full project. My students seemed to really enjoy the project over all. Many said that this was the only fun thing we had ever done in here. After memorizing the nine rules, students were tested over them, and the difference is clear: students who put their rules to music performed significantly higher than those that just tried to memorize the rules. This is certainly a project I will resurrect in the future, thought I will probably require that students not “opt out” next time, even if that means putting the rules to Mary Had a Little Lamb. After the project was finished, I did a comma song of my own. I even enjoyed doing the project, so I understand why my kids did.
I’m Comma boy swaggin’ when I contrast or pausen
Or when the end’s ‘bout the beginning, I’ll avoid confusion
Tatted on my neck, yeah that’s extra information
You don’t need to hear so I’ll save humiliation.
Took her to a speaker, let the dude talk
When we quoted him later, we shot commas like a glock
Man I like the way, those commas really rock
She’s writin’ lots of papers, while I comma boy swag.
James Spencer
Marian University – Teach For America
Emmanuel, the Emperor Penguin, Saves the Species!
http://news.softpedia.com/news/Extinct-Penguin-Species-Found-in-New-Zealand-98265.shtml
“Oh No!” cried Momma, “the ice is melting too quickly for all of us to reach the safety of the frozen tundra of our habitat, the Antarctic! Little Elllie and Edward waddle so slowly, with such little feather cover to keep them protected from the burning sun and are in terrible need of food! I just know we are all going to perish, especially our chicks! ”
“Ellie was only 1/2 the weight of her brother, Edward, at birth. Edward, who now weighs 12 pounds, has gained 4 times his birth weight. We must get her food and protection quickly!” sobbed Aunt Elena.
“Our original rookery of 96 Emperor penguins has been searching for frozen passages for five long days! Today is our sixth day. On days 1, 2, and 3 we traveled an average of 18 miles, especially exhausted after our 9-mile journey of day 3. Unfortunately, our colony decided to travel equally shorter distances on our first two days, “recalled Momma.
“Thank goodness for a democratic vote to spend day 4 resting. Our heart rates, normally 60 beats per minute, were twice their normal rhythm,” declared President Papa. “I was worried I would have to veto the decision if ½ + 1 of the adults hadn’t agreed with this decision. 2/3 of our rookery are adults and certainly needed a rest. I understand Yellow-Eyed New Zealand Penguin ruled his species as a dictator, perhaps a reason for their unfortunate ending.
“Recalling the tragedy of day 5 makes me so upset,” replied Momma. “Losing 1/6 of the original rookery to the ravenous predators, Killer Whale and Leopard Seal, causes me to feel faint! As we paddled through the broken ice to the safety of frozen ice, only 4 miles, the especially hungry Killer Whale consumed three times as much as the Leopard Seal. Our poor, exhausted group must find help soon!”
“Perhaps we need another leader, not me, to guide our weak rookery to our destination, 7 miles away. Perhaps our ice will not melt as quickly once we reach our goal,” declared President Papa.
“You are probably right,” sighed Uncle Emmanuel. “For the last four years we have depended on you to get our rookery to its winter home. I am afraid this time we need a miracle. Today I will volunteer to dive to the deep depths of the sea in search of fish, krill, and squid to nourish our starving group. Lucky for us, our species can store oxygen in our body for up to 22 minutes while we dive to a depth of 1,800 feet with no special equipment! How proud and different we are from other birds! That species, man, we see taking pictures of us, could never withstand the cold, dark, deep pressure of this food search! Man can only survive about 4 minutes without oxygen! I wonder what man would do if its world continued to get warmer and warmer. If only there was a super hero among us!”
22 Minutes Later……
Emmanuel emerged from the deep, deep depth of the sea, 1,800 feet, with a bounty of fish, krill, and squid. The rookery flapped their wings in excitement! Momma and Aunt Elena gobbled their food and regurgitated it to their starving youngsters’ open mouths, especially Little Ellie. The children, who had formed a crèche or small group in the center of the adults, squawked with delight.
“Why Emmanuel…I declare you are our SUPER HERO!” proudly beamed Aunt Elena.
“We shall gain enough nourishment from your fine catch to make it safely across to solid ice today, our sixth day! I declare that tomorrow, our seventh day, will be Emmanuel the Emperor Penguin Day!” proclaimed President Papa. “You have saved our species from extinction!”
The exhausted, yet satisfied rookery of Emperor Penguins safely made their way to their final, safe destination. Once the crèche was encircled by the adults, President Papa prompted another cheer to Emmanuel. He directed the remaining adult penguins to listen to his idea. “Should the rookery choose a new leader?”
100% of the penguins flapped their wings in agreement! Little Edward continued flapping his wings and shouted, “I believe I am so happy I could fly to the sky!”
The adults laughed, like donkeys braying, and told the naïve, flightless bird that will never be a possibility. However, they gave all the crèche permission to gleefully slide across the rock-solid ice on their tiny webbed feet.
Aunt Elena beamed with pride as her husband puffed out his white chest. “Tomorrow, wear your finest tuxedos and celebrate with us at a feast to inaugurate Emmanuel, our democratic leader!”
Standards:
Math
Science
Social Studies
Reading
Writing
Math Questions related to the story:
How much did Ellie weigh at birth?
Answer: 12/4 = 3 pounds (Edward); Ellie weighed ½ of 3 = 1.5 pounds
How many miles were traveled on days 1 and 2?
Answer: Day 1 + Day 2 + Day 3 = 18 miles
Day 1 + Day 2 + 9 miles = 18 miles
½ (9) x 2 + 9 = 18 miles
4.5 miles + 4.5 miles + 9 miles = 18 miles
What was the average heart rate on day 4?
Answer: 60 beats per minute x 2 = 120 beats per minute
How many adults placed a democratic vote to rest on day 4?
Answer: 2/3 of 96 = 64.
½ of 64 = 33 adults
What fraction of the colony are chicks, baby penguins?
Answer: 2/3 are adults, then 1/3 are chicks.
How many penguins perished on day 5?
Answer: 1/6 of 96 = 16 penguins.
How many penguins did each predator, Killer Whale and Leopard Seal, consume?
Answer: 3x + x = 16 penguins
4x = 16
X = 4; Leopard Seal consumed 4 penguins; Killer Whale consumed 12 penguins
How many penguins remain in the colony?
Answer: 96 – 16 = 80 penguins
How many miles did the Emperor Penguins travel in all?
Answer: Day 1: 4.5
Day 2: 4.5
Day 3: 9
Day 4: 0
Day 5: 4
Day 6: 7
TOTAL: 29 miles
What was the mode of distance?
Answer: 4.5 miles
What was the mean of the distance traveled?
Answer: 29 divided by 6 = 4.83 miles
What is the median of the distance traveled?
Answer: 0, 4, 4.5, 4.5, 7, 9; 4.5 is the median
Make a bar graph to show the distances traveled during the six day journey.
How many feet in all did Emmanuel dive to get food for his rookery?
Answer: 1, 800 down + 1,800 up = 3,600 feet
What was the average distance he dove in 22 minutes?
3,600 feet divided by 22 minutes = 163.6 feet per minute
Science Questions Related to the Story
Are Emperor Penguins carnivores?
Answer: Yes (eat fish, krill, squid)
Which animals are predators of the Emperor Penguin?
Answer: Killer Whale and Leopard Seal are predators of this species.
How do baby penguins get nourishment?
Answer: Parents regurgitate their food and drop it in the mouth of the chick.
Can penguins or humans survive without oxygen for a longer period of time?
Answer: Penguins can store oxygen for up to 22 minutes.
What animal does a penguin sound like when it makes noise?
Answer: An Emperor Penguin resembles the sound of a donkey breying.
Are penguins able to fly?
Answer: Penguins are flightless birds.
Social Studies Questions Related to the Story
Which continent is home of the Emperor Penguins located?
Answer: Antarctic
Does a dictatorship allow its citizens have a say in decisions?
Answer: No
Which form of government allows its citizens to vote?
Answer: Democracy
Reading Questions Related to the Story
What is the problem in the story?
Answer: The penguins need to reach their habitat of Antarctic.
Put these events in order 1, 2, 3, 4, and 5.
President Papa and the other penguins vote for Emmanuel to be the next leader.
Mama and Aunt Elena are concerned for the survival of the chicks.
The penguins finally reach the safety of the frozen Antarctic.
Frozen land is melting.
Emmanuel dives deep in the sea to get fish for the hungry penguins.
Answer: 1-Frozen land is melting.
2-Mama and Aunt Elena are concerned for the survival of the chicks.
3- Emmanuel dives deep in the sea to get fish for the hungry penguins.
4- The penguins finally reach the safety of the frozen Antarctic.
5- President Papa and the other penguins vote for Emmanuel to be the next leader.
Which is the name for the entire group of penguins? Rookery or crèche
Answer: Rookery
What is the name for a young penguin? Rookery or chick
Answer: Chick
What human characteristics were given to the penguins? Name at least three characteristics.
Answer: Fear, Appreciation, Cheerfulness, Bravery
Writing Questions Related to the Story
How does the writer show humor in the last paragraph?
Answer: Humor is shown when Aunt Elena tells the penguins to wear their finest tuxedos.
Explain why you think the ice is melting?
Answer: The story referred to the sun’s heat so I think that is why the ice is melting.
Did you enjoy reading “Emmanuel the Emperor Penguin”? Provide at least three reasons to support your response.
Create a new event or ending to the story.
