New section: to be written
(1) If a person is falling off a bridge, can they save themselves by throwing a hammer ahead of them to break the surface tension of the water prior to their own impact?
(Also called the factor-label method)
Dimensional analysis is a super useful technique used for many different reasons:
scaling up recipes to much larger quantities
Converting English-to-metric units (or vice-versa)
Physics problems, e.g. involving speed and distance
Chemistry problems (how much product is made from how much starting materials?)
Dimensional analysis is just a a trick in which we use conversions factors to get from the info we have, to the answer we need.
Sometimes we need to string several conversion factors together to get the answer that we need.
What is a conversion factor?
Two things that are exactly equivalent
We can always write 2 equivalent things as a fraction
The fraction can be written with either term on top, or bottom:
Conversion factor for money
Conversion factor for distance
Example: How do we convert inches to cm?
Convert days to seconds
2016 Massachusetts Science and Technology/Engineering Curriculum Framework
Science and Engineering Practices: 5. Using Mathematics and Computational Thinking:
Apply ratios, rates, percentages, and unit conversions in the context of complicated measurement problems involving quantities with derived or compound units (such as mg/mL, kg/m 3, acre-feet, etc.).
NSTA: The efficiency and effectiveness of the metric system has long been evident to scientists, engineers, and educators. Because the metric system is used in all industrial nations except the United States, it is the position of the National Science Teachers Association that the International System of Units (SI) and its language be incorporated as an integral part of the education of children at all levels of their schooling.
Kinematics is the study of objects in motion. It allow us to study displacement, velocity, time and acceleration. Check out our introductory lesson on this study of motion here:
The full article is available here: The Early Catastrophe: The 30 Million word gap by age 3. and at http://www.aft.org/periodical/american-educator/spring-2003
Betty Hart and Todd Risley entered the homes of 42 families from various socio-economic backgrounds to assess the ways in which daily exchanges between a parent and child shape language and vocabulary development. Their findings showed marked disparities between the sheer number of words spoken as well as the types of messages conveyed.
After four years these differences in parent-child interactions produced significant discrepancies in not only children’s knowledge, but also their skills and experiences with children from high-income families being exposed to 30 million more words than children from families on welfare.
Follow-up studies showed that these differences in language and interaction experiences have lasting effects on a child’s performance later in life.
The Early Catastrophe
Betty Hart & Todd R. Risley
Betty Hart and Todd Risley were at the forefront of educational research during the 1960’s War on Poverty. Frustrated after seeing the effects of their high quality early intervention program aimed at language skill expansion prove unsuccessful in the long-term, they decided to shift their focus. If the proper measures were being taken in the classroom, the only logical conclusion was to take a deeper look at the home.
What difference does home-life make in a child’s ability to communicate? Why are the alarming vocabulary gaps between high school students from low and high income environments seemingly foreshadowed by their performance in preschool? Hart and Risley believed that the home housed some of these answers.
Hart and Risley recruited 42 families to participate in the study including 13 high-income families, 10 families of middle socio-economic status, 13 of low socio-economic status, and 6 families who were on welfare. Monthly hour-long observations of each family were conducted from the time the child was seven months until age three. Gender and race were also balanced within the sample.
The results of the study were more severe than the researchers anticipated. Observers found that 86 percent to 98 percent of the words used by each child by the age of three were derived from their parents’ vocabularies.
Furthermore, not only were the words they used nearly identical, but also the average number of words utilized, the duration of their conversations, and the speech patterns were all strikingly similar to those of their caregivers.
After establishing these patterns of learning through imitation, the researchers next analyzed the content of each conversation to garner a better understanding of each child’s experience. They found that the sheer number of words heard varied greatly along socio-economic lines. On average, children from families on welfare were provided half as much experience as children from working class families, and less than a third of the experience given to children from high-income families.
In other words, children from families on welfare heard about 616 words per hour, while those from working class families heard around 1,251 words per hour, and those from professional families heard roughly 2,153 words per hour. Thus, children being raised in middle to high income class homes had far more language exposure to draw from.
In addition to looking at the number of words exchanged, the researchers also looked at what was being said within these conversations. What they found was that higher-income families provided their children with far more words of praise compared to children from low-income families. Conversely, children from low-income families were found to endure far more instances of negative reinforcement compared to their peers from higher-income families.
Children from families with professional backgrounds experienced a ratio of six encouragements for every discouragement. For children from working-class families this ratio was two encouragements to one discouragement. Finally, children from families on welfare received on average two discouragements for every encouragement. Therefore, children from families on welfare seemed to experience more negative vocabulary than children from professional and working-class families.
The authors conclude:
We learned from the longitudinal data that the problem of skill differences among children at the time of school entry is bigger. more intractable. and more important than we had thought. So much is happening to children during their first three years at home, at a time when they are especially malleable and uniquely dependent on the family for virtually all their experience. that by age 3, an intervention must address not just a lack of knowledge or skill, but an entire general approach to experience…
…Estimating, as we did, the magnitude of the differences in children’s cumulative experience before the age of 3 gives an indication of how big the problem is. Estimating the hours of intervention needed to equalize children’s early experience makes clear the enormity of the effort that would be required to change children’s lives. And the longer the effort is put off, the less possible the change becomes. We see why our brief, intense efforts during the War on Poverty did not succeed. But we also see the risk to our nation and its children that makes intervention more urgent than ever.
A summary from “The Early Catastrophe: The 30 Million Word Gap by Age 3” by University of Kansas researchers Betty Hart and Todd R. Risley. (2003). American Educator. Spring: 4-9, which was excerpted with permission from B. Hart and T.R. Risley (1995). Meaningful Differences in the Everyday Experiences of Young American Children. Baltimore, MD: Brookes Publishing.
Prof. Howard Wiseman, a physicist at Griffith University in Brisbane, Australia, along with his collaborators Dr. Michael Hall, also of Griffith University, and University of California, Davis mathematician Dr. Dirk-Andre Deckert, published their new “many interacting worlds” (MIW) theory in the journal Physical Review X.
They posited that other universes are real, exist in vast numbers and exert influence on each other.
“The idea of parallel universes in quantum mechanics has been around since 1957,” Wiseman said in a statement. “In the well-known ‘Many-Worlds Interpretation’, each universe branches into a bunch of new universes every time a quantum measurement is made. All possibilities are therefore realised – in some universes the dinosaur-killing asteroid missed Earth. In others, Australia was colonised by the Portuguese.”
“But critics question the reality of these other universes, since they do not influence our universe at all,” he added. “On this score, our “Many Interacting Worlds” approach is completely different, as its name implies.”
There are three main points to the MIW theory, according to the Griffith statement. First, that the universe we live in is just one of an unknown “gigantic” number of worlds, some of which are “almost identical to ours,” but most are “very different.”Second, all of the worlds are “equally real,” existing continuously through time with precisely defined properties.Third, quantum phenomena arise from “a universal force of repulsion between ‘nearby’ (i.e. similar) worlds, which tends to make them more dissimilar.”
“All quantum effects arise from, and only from, the interaction between worlds,“ the physicists explained in their abstract.
Hall said the radical new theory may even create the extraordinary possibility of testing for the existence of other worlds.
“The beauty of our approach is that if there is just one world our theory reduces to Newtonian mechanics, while if there is a gigantic number of worlds it reproduces quantum mechanics,” he said in the statement. “In between it predicts something new that is neither Newton’s theory nor quantum theory. We also believe that, in providing a new mental picture of quantum effects, it will be useful in planning experiments to test and exploit quantum phenomena.”
American theoretical physicist Richard Feynman once noted: “I think I can safely say that nobody understands quantum mechanics.” And the MIW group admits that their theory is a bit out there.
“Any explanation of quantum phenomena is going to be weird, and standard quantum mechanics does not really offer any explanation at all ‒ it just makes predictions for laboratory experiments,” Wiseman told the Huffington Post in an email.“Our new explanation… is that there are ordinary [non-quantum] parallel worlds which interact in a particular and subtle way.”
Motherboard asked if the theory suggests that humans might someday be able to interact with other universes.
“It’s not part of our theory,” Wiseman replied. “But the idea of [human] interactions with other universes is no longer pure fantasy.”
Others in the quantum mechanics field ranged from skepticism to excitement, Huffington Post reported, noting there is no consensus on whether “many interacting worlds” exist or interact.
“There are some who are completely happy with their own interpretations of QM, and we are unlikely to change their minds,”Wiseman said in his email. “But I think there are many who are not happy with any of the current interpretations, and it is those who will probably be most interested in ours. I hope some will be interested enough to start working on it soon, because there are so many questions to answer.”
The Atlantic, October 2012
By Peg Tyre
in 2009, when Monica DiBella entered New Dorp, a notorious public high school on Staten Island, her academic future was cloudy. Monica had struggled to read in early childhood, and had repeated first grade. During her elementary-school years, she got more than 100 hours of tutoring, but by fourth grade, she’d fallen behind her classmates again. In the years that followed, Monica became comfortable with math and learned to read passably well, but never seemed able to express her thoughts in writing. During her freshman year at New Dorp, a ’70s-style brick behemoth near a grimy beach, her history teacher asked her to write an essay on Alexander the Great. At a loss, she jotted down her opinion of the Macedonian ruler: “I think Alexander the Great was one of the best military leaders.” An essay? “Basically, that wasn’t going to happen,” she says, sweeping her blunt-cut brown hair from her brown eyes. “It was like, well, I got a sentence down. What now?” Monica’s mother, Santa, looked over her daughter’s answer—six simple sentences, one of which didn’t make sense—with a mixture of fear and frustration. Even a coherent, well-turned paragraph seemed beyond her daughter’s ability. An essay? “It just didn’t seem like something Monica could ever do.”
For decades, no one at New Dorp seemed to know how to help low-performing students like Monica, and unfortunately, this troubled population made up most of the school, which caters primarily to students from poor and working-class families. In 2006, 82 percent of freshmen entered the school reading below grade level. Students routinely scored poorly on the English and history Regents exams, a New York State graduation requirement: the essay questions were just too difficult. Many would simply write a sentence or two and shut the test booklet. In the spring of 2007, when administrators calculated graduation rates, they found that four out of 10 students who had started New Dorp as freshmen had dropped out, making it one of the 2,000 or so lowest-performing high schools in the nation. City officials, who had been closing comprehensive high schools all over New York and opening smaller, specialized ones in their stead, signaled that New Dorp was in the crosshairs.
And so the school’s principal, Deirdre DeAngelis, began a detailed investigation into why, ultimately, New Dorp’s students were failing. By 2008, she and her faculty had come to a singular answer: bad writing. Students’ inability to translate thoughts into coherent, well-argued sentences, paragraphs, and essays was severely impeding intellectual growth in many subjects. Consistently, one of the largest differences between failing and successful students was that only the latter could express their thoughts on the page.
If nothing else, DeAngelis and her teachers decided, beginning in the fall of 2009, New Dorp students would learn to write well. “When they told me about the writing program,” Monica says, “well, I was skeptical.” With disarming candor, sharp-edged humor, and a shy smile, Monica occupies the middle ground between child and adult—she can be both naive and knowing. “On the other hand, it wasn’t like I had a choice. I go to high school. I figured I’d give it a try.”
New Dorp’s Writing Revolution, which placed an intense focus, across nearly every academic subject, on teaching the skills that underlie good analytical writing, was a dramatic departure from what most American students—especially low performers—are taught in high school. The program challenged long-held assumptions about the students and bitterly divided the staff. It also yielded extraordinary results. By the time they were sophomores, the students who had begun receiving the writing instruction as freshmen were already scoring higher on exams than any previous New Dorp class. Pass rates for the English Regents, for example, bounced from 67 percent in June 2009 to 89 percent in 2011; for the global-history exam, pass rates rose from 64 to 75 percent. The school reduced its Regents-repeater classes—cram courses designed to help struggling students collect a graduation requirement—from five classes of 35 students to two classes of 20 students.
…[Why were the students previously failing?]
…. New Dorp students were simply not smart enough to write at the high-school level. You just had to listen to the way the students talked, one teacher pointed out—they rarely communicated in full sentences, much less expressed complex thoughts… Scharff, a lecturer at Baruch College, a part of the City University of New York, kept pushing, asking: “What skills that lead to good writing did struggling students lack?” …
Maybe the struggling students just couldn’t read, suggested one teacher.
A few teachers administered informal diagnostic tests the following week and reported back. The students who couldn’t write well seemed capable, at the very least, of decoding simple sentences. A history teacher got more granular. He pointed out that the students’ sentences were short and disjointed. What words, Scharff asked, did kids who wrote solid paragraphs use that the poor writers didn’t? Good essay writers, the history teacher noted, used coordinating conjunctions to link and expand on simple ideas—words like for, and, nor, but, or, yet, and so. Another teacher devised a quick quiz that required students to use those conjunctions. To the astonishment of the staff, she reported that a sizable group of students could not use those simple words effectively. The harder they looked, the teachers began to realize, the harder it was to determine whether the students were smart or not—the tools they had to express their thoughts were so limited that such a judgment was nearly impossible.
The exploration continued. One teacher noted that the best-written paragraphs contained complex sentences that relied on dependent clauses like although and despite, which signal a shifting idea within the same sentence. Curious, Fran Simmons devised a little test of her own. She asked her freshman English students to read Of Mice and Men and, using information from the novel, answer the following prompt in a single sentence:
“Although George …”
She was looking for a sentence like: Although George worked very hard, he could not attain the American Dream.
Some of Simmons’s students wrote a solid sentence, but many were stumped. More than a few wrote the following: “Although George and Lenny were friends.”
A lightbulb, says Simmons, went on in her head. These 14- and 15-year-olds didn’t know how to use some basic parts of speech. With such grammatical gaps, it was a wonder they learned as much as they did. “Yes, they could read simple sentences,” but works like the Gettysburg Address were beyond them—not because they were too lazy to look up words they didn’t know, but because “they were missing a crucial understanding of how language works. They didn’t understand that the key information in a sentence doesn’t always come at the beginning of that sentence.”
Some teachers wanted to know how this could happen. “We spent a lot of time wondering how our students had been taught,” said English teacher Stevie D’Arbanville. “How could they get passed along and end up in high school without understanding how to use the word although?”
…The Hochman Program, as it is sometimes called, would not be unfamiliar to nuns who taught in Catholic schools circa 1950. Children do not have to “catch” a single thing. They are explicitly taught how to turn ideas into simple sentences, and how to construct complex sentences from simple ones by supplying the answer to three prompts—but, because, and so. They are instructed on how to use appositive clauses to vary the way their sentences begin. Later on, they are taught how to recognize sentence fragments, how to pull the main idea from a paragraph, and how to form a main idea on their own. It is, at least initially, a rigid, unswerving formula. “I prefer recipe,” Hochman says, “but formula? Yes! Okay!”
…Within months, Hochman became a frequent visitor to Staten Island. Under her supervision, the teachers at New Dorp began revamping their curriculum. By fall 2009, nearly every instructional hour except for math class was dedicated to teaching essay writing along with a particular subject. So in chemistry class in the winter of 2010, Monica DiBella’s lesson on the properties of hydrogen and oxygen was followed by a worksheet that required her to describe the elements with subordinating clauses—for instance, she had to begin one sentence with the word although.
Although … “hydrogen is explosive and oxygen supports combustion,” Monica wrote, “a compound of them puts out fires.”
Unless … “hydrogen and oxygen form a compound, they are explosive and dangerous.”
If … This was a hard one. Finally, she figured out a way to finish the sentence. If … “hydrogen and oxygen form a compound, they lose their original properties of being explosive and supporting combustion.”
As her understanding of the parts of speech grew, Monica’s reading comprehension improved dramatically. “Before, I could read, sure. But it was like a sea of words,” she says. “The more writing instruction I got, the more I understood which words were important.”
Classroom discussion became an opportunity to push Monica and her classmates to listen to each other, think more carefully, and speak more precisely, in ways they could then echo in persuasive writing.
PEG TYRE is the director of strategy at the Edwin Gould Foundation and the author of The Good School: How Smart Parents Get Their Kids the Education They Deserve.
Physicist Werner Heisenberg said, “When I meet God, I am going to ask him two questions: why relativity? And why turbulence? I really believe he will have an answer for the first.” As difficult as turbulence is to understand mathematically, we can use art to depict the way it looks.
Natalya St. Clair illustrates how Van Gogh captured this deep mystery of movement, fluid and light in his work.
Natalya St. Clair, Educator
Avi Ofer , Animator
Addison Anderson, Script Editor