October 10th 2017 HELP! Since so many people find this blog helpful, would you please show your appreciation by helping me save my car from possession? Save Madeleine by making a financial contribution at the following link: Save Madeleine
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F.O.I.L. I think this technique is supposed to be used on binomials and polynomials and PEMDAS is for shorter equations. I’ll verify my assumptions with a math teacher, of course.
Examples provided by Encore’s Math Advantage (when it was available for Macintosh…)
Smurfy site with art maths!
Cusped epicycloids develop—the cardioid, nephroid, and ranunculoid.
It’s nice to know their names at last!
More interesting things. The difference between geometry and trig is that although geometry is a broader area of study, trigonometry is used for astronomy because it concerns infinite measurements.
“Other than geometry being a lot broader, the main difference is that trigonometry is computational.was developed after geometry for the purposes of astronomy.”
“Trigonometry is a branch of Geometry. Perhaps a better question is: What is the difference between Trigonometry today and Geometry as practiced by Euclid?
It’s hard for us to understand now, but Euclid was not concerned with absolute measures of lengths and angles. With angles there were a few special cases, parallel and perpendicular, but otherwise geometry was about relative sizes. These sides are congruent. This line is a perpendicular bisector of that side. These angles are congruent. You can only do relative measurements with a straightedge and a compass. The real numbers aren’t involved. There are no infinite processes.”
“Geometry is an overarching term to describe shapes and their related measures of size, length, angles, etc.
More technically, geometry refers more to local measurements in contrast to topology which concerns itself with global measures.
Trigonometry is the mathematics of the circle.
The definitions of the trig functions come from various measures of a circle:
The geometric picture of relationships of the points of a circle yields the trig graphs:
In summary, trigonometry is the algebra and geometry of a circle, while geometry describes any kind of shape.”
“Trigonometry is about triangles, relation between their sides, opposite angles etc. sine, cosine and tangent are few of the trigonometric functions.
Geometry is a broader term dealing with points, lines, surfaces, shapes etc.
It has specific formulas for these elements, for example, distance between two lines, area of a rectangle, volume of a cube etc.”
I’ve made some infographics of pre-algebra’s Order of Operations mnemonic PEMDAS. I transcribed the lessons from Encore’s Math Advantage (which is no longer available for macs!) It’s has been the most helpful tutorials for algebra that I’ve found in 30 years! I don’t even remember my teacher mentioning either PEMDAS or FOIL. As much as I like my teacher, he never even wrote it on the board. There should’ve been infographics all over the classroom, but no…
I hope these infographics aid the visually oriented. PEMDAS is the first in a three part series on the Order of Operations. For inner-city public school kids, the Order of Operations is the more formal way of your teacher’s annoying note: “Show Your Work“. Yeah, like anyone can look up “Show Your Work” in the library or internet!
If math is a language, then Algebra II is English 120: the goal is sentence structure, the result is secondary.
For example: Sailing the Seas of Cheese must not be written in algebraic symbols as “Sailing-with Sea-a-Cheese”. That’s why we use the Order of Operations. Also, the student must keep all of the methods straight by memorizing their distinctions.
For example: to simplify the equation, Sailing the Sea of Cheese, you would write “Sailing a Cheese Sea”—but with algebraic symbols.
Could an equation be simply a sentence? I’ll verify my interpretation with my favorite physics teacher, of course. However, for my mind to understand algebra in how it’s taught in high school, particularly algebra II, I must let go of results oriented basic math and pre-algebra ( and algebra 1) and understand that the equations are not supposed to solve anything at this point. The student is merely practicing structuring “mathematical sentences” and taking care not get methods of writing them mixed up!
So, at last I found a way to understand how math tells a story—for the rest of us! The Right Brain is finally communicating with the Left Brain….it only took 30 years!
It’s 1987. I’m 15 years old, I’ve flunked algebra II twice already and I am obsessed with the Transformers.
As a pnuemonic. I named the different methods after the Autobots. The first batch being the scientists, the middle being the annoying ones (is it any wonder an entire hemisphere of my mind shorted out?), the last being the Autobot officers and aficionados who guide the student through tough but fun methods of structuring “algebraic sentences”.
Fun Scientist Autobots
1. Wheeljack: Graphing Equations
2. Ratchet: Slope of a Line
3.Perceptor: Writing Equations Of Lines
4.Hound: Graphing Inequalities
5. Teletran1: Solving Systems of Equations by Graphing
6. Huffer: Solving Systems of Equations by Substitution
7. Braun: Solving Systems of Equations by Addition
8. Cliffjumper: Solving Systems of Equations in Three Variables
Tough but Interesting Challenges the Following Autobots Will Valiantly Help You Solve.
9. Prowl: Simplifying Radical Expressions
10. Jazz: Add/Subtract Radical Expressions
11. Bluestreak: Multiply/Divide Radical Expressions
12. Smokescreen:Solving Equations With Radicals
13. Red Alert: Fractional Exponents
14. Hoist: Solving Polynomial Equations
15. Oppie: The Quadratic Formula
So this is what saves you from the Tracking System B-List in public high school: organizing these distinct methods of solving math problems in head and not mixing them up! Well, one can do that with the humanities as well, but since weapons are our county’s largest export, rather than culture, math gets the most power…the power to crush the spirit of a bright student and send them into despair.
How many gang members has the tracking system created? “You can learn the humanities unless you pass algebra, so you’re not worth the investment!” says the school system. A 15 year old inner-city faces the problems critical thinking skills learned in the humanities everyday: social hierarchies, sexual selection, poverty via the division of labor. Algebra leads to economic stability, but it awfully hard to concentrate on the abstract world when a kid is facing considerable difficulties in the material world! The humanities are far more useful and fascinating to the marginalized. History is far more interesting to those who have been a victim of it.
Math is very important, but it shouldn’t be used as a litmus test to gauge whether a student will do well in other subjects or be responsible adult.
If your teenager is doing well in every other subject, but failing algebra, then the teacher is using the wrong approach. Frankly, with some subjects it can take going through several instructors to find one who can communicate with the visually oriented. Try seeing math as the language it is as stated above, but also, for a recap on forgotten lectures, try this DVD:
Algebra II Tutor. It’s boring like any math lecture, but at least it will help your teenager make sense of their math lecture notes when they settle down to do their homework. I would also recommend FILMING high school math lectures!
There. So ends thirty years of feeling inadequate as a problem solving intelligent being. I feel so much better! Now I can write the story of Jasian Genetic and concentrate on the math that I really like!
It’s been a rough year so far. It came in with a bang of prosperity to job loss at the midpoint. The rest of the year doesn’t look good. So… to remind myself that I have quite the battle ahead, I painted my own tarot to brace myself for the coming months.
2017: Year of Challenges and Changes
February: The Devil I should not have took out a credit account for my tv….a safe always falls out of the sky and land on my head when I use credit instead of cash.
March: Queen of Swords (reversed) Ok, I opened up a can of whoop-ass on someone (probably David). I don’t remember what for…
April: Seven of Swords (reversed). What hit me? Which one of y’all hit me? Come ‘ere! We were only paid one paycheck this month.
May: Ten of Chalices (reversed) Job Loss: the buying partner won’t may overhead during merger. No paychecks. We were a great crew. We loved puzzles and most of all afternoon tea and discourses.
June: The Magician (reversed) I’ve had to be sketchily clever this month. Thanks, Mom.
July: Three of Swords (reversed) I am very scared.
August: Seven of Swords Now what am I going to try to get away with? I’d better make sure I don’t inadvertently make away with someone else’s golden ticket.
September: Ace of Wands. Hopefully a new job doing something new and more profitable.
October: Ten of Pentacles (reversed). I’m going to lose someone and an inheritance. This month is gonna suck.
November: Four of Chalices. Breathe. Just Breathe. Jinkies what an ordeal.
December: The World. What ever happens this year, all will be completed. I’d better make damn sure my second graphic novel has been completed and released during all of the involuntary time off!
The Year: Eight of Pentacles:
New Enterprise or skillset even if its at the apprentice stage. Well, I hope I become a junior producer after this right of passage. SHEESH!
Upon screening a review of Spielberg’s films. I learned that the Color Purple was not well received by the African-American community. I took that information with a grain of salt, because as we all know, when speaking of a group, our patriarchal society always asks the opinion of men. Ahem! The Color Purple is a story for WOMEN and the experiences women endure. I was pleased to find out that Black women found the film theraputic. As a teenage “Celie”, I sure as hell did. In fact the film helped me void many traps that I could’ve fallen into and never climbed out of. For example: leave men alone of you are not their ideal.
”No media vehicle since ‘Roots’ has caused this kind of dialogue,” observed Chuck Sutton, host of a program on the subject last week on WLIB radio in New York. ‘One Woman’s Story’
The movie, directed by Steven Spielberg and based on the Pulitzer Prize-winning novel by Alice Walker, tells the story of Celie, a black woman who was physically and sexually abused by men most of her life. For her portrayal of Celie the actress Whoopi Goldberg received a Golden Globe Award this weekend.
Many black women defend the film, saying that it accurately reflects their own experience or the experience of women they know. Many black men, and some black women, contend that it gives a misleading picture of blacks in America and distorts their history.
Oprah Winfrey, host of a television talk show in Chicago who played Sophia, Celie’s stepdaughter-in-law, in the movie, said: ”This movie is not trying to represent the history of black people in this country any more than ‘The Godfather’ was trying to represent the history of Italian-Americans. In this case, it’s one woman’s story.”
Many women say it is also their story. Eartis Thomas, a telephone company employee in Chicago, said she knew many Celies when she was growing up in Sunflower County, Mississippi. She, her mother and her aunts were all beaten and otherwise brutalized by their husbands, Miss Thomas said. The movie, she said, ”just lifted a burden.”
Miss Thomas added: ”Black women should not be sacrificed for black men’s pride. Let the film roll.” Arguments Heard Before
Some of the arguments being heard now are similar to those heard in 1977 about Ntozake Shange’s play, ”For Colored Girls Who Have Considered Suicide/When the Rainbow Is Enuf,” and in 1979 about Michele Wallace’s book, ”Black Macho & the Myth of the Superwoman.”
These works confront the often uneasy relationship between black men and black women. Eldridge Cleaver, in his book ”Soul on Ice,” described it as a war. Miss Wallace described it as ”a profound distrust, if not hatred.”
”It’s a very touchy subject between folks, always has been,” said Haywood E. McDuffie, a Cleveland lawyer who said he recently argued about the movie with several women friends.
”The most frightening thing for me as a male was to look into their eyes,” Mr. McDuffie said. ”I wondered, ‘Is that really what they think of us?’ It’s as if there were an element of cruelty implicit in black men that all black women seem to identify.” ‘Never Showed the Good’
Willis Edward, president of the Beverly Hills chapter of the National Association for the Advancement of Colored People, argued in an interview that the movie ”never showed the good” about black men.
But Clarence Page, a columnist for The Chicago Tribune, asserted, ”This movie is no more destructive of black men than ‘The Burning Bed’ was destructive of white men.” He referred to a television drama about abuse by spouses.
Others argue that the movie distorts black history and appears to blame the victims of racism for a host of social problems, including a preponderance of broken families and a high incidence of teen-age pregnancy.
Lerone Bennett, a historian who is senior editor of Ebony magazine, said, speaking of Mr. Spielberg: ”He doesn’t show us the strong black women who nursed the sick and cared for the orphans and organized clubs and, in general, exercised a leadership role denied white women by white society. What Steven Spielberg doesn’t show us, in short, is the color black.” Actor Defends Film
Vernon Jarrett, a columnist for The Chicago Sun-Times, contended: ”Mr. Reagan, and his Attorney General, Mr. Meese, have decided that they are going to turn back the clock on us. The purpose of movies like this is to make it acceptable to you.”
One movie cannot do that, insisted Danny Glover, the actor who played the role of Mister. Mr. Glover, who is appearing in a Chicago production of ”A Lesson from Aloes,” a play by Athol Fugard, said more people probably see ”The Cosby Show” on television each week than will see the film and can just as easily formulate their opinions of blacks on that basis.
Black men are perhaps no different from other men in their attitudes toward women, he said. But he added, ”Lots of times we sweep our own problems under the rug under the justification of upholding black history and the black man.”
Lee K. Richardson, the artistic director of the Crossroads Theater Company, in New Brunswick, N.J., said: ”It’s not the artist’s responsibility to right the wrongs. There is not just one side to our black experience.”
As the debate continued, the movie had grossed nearly $29 million through Jan. 19, drawing audiences that are 72 percent white, and the book, according to a spokesman for the publisher Simon & Shuster, has become ”the fastest-selling paperback in a long time.”
photo of Whoopi Goldberg
Yeah, Black women don’t trust Black men. I am not angry, just wary. It also helps that I’ve never been attracted to Black man to begin with (I prefer men of the Mediterranean). It is a male tendency to conquer the unknown, whatever is different and when living in someone else’s society as Blacks do, one becomes conditioned by the dominant cultures standards of beauty. Black women are the polar opposite of blond hair and blue eyes, heck long straight hair ( naturally), so Black man go for it. That’s fine. Spread those genes just as every other race has. Black women will move on and we have. Once more of us abandon the princess cancer strain, we won’t breed before we’re educated anymore and hopefully, lift our communities of poverty as a result
Don’t be victimized Celie, ladies. Be triumphant Celie. You may not be Shug or be a Scandinavian blonde, but you’re talented and have a history to draw upon for strength. Be the Celie who inherits the property her body and tears paid for and run your own businesses. Leave men who don’t want you and don’t feel the need to work for you alone. Don’t fall for the duplicitous words of heat seeking bachelors. The Matriarchal African American community will rise to power, changing our negative fiscal situation one woman at a time, one family at a time to be admirable communities…in time.
…and I am so sorry…
Despite a tremendous effort, you were defeated. By the smallest of margins and by the technicality of the electoral collage. You missed Wisconsin in your campaign and it cost you dearly. What an upset. A bigoted demagogue has won the White House and largely because the emerging industries had not reached the regions of White working class voters yet and everyone blamed the establishment. Well, I blame them!
Some say you weren’t likable. I like you. Even love you. If anyone is unlikable it’s the White working class that has hatred bubbling right under the surface of their smiling faces. Not all businesses are established by White males. If I were Asian ( which many tech company owners are), I would be afraid to establish a tech firm in Idaho, so where my core employees would suffer under a limited quality of life due to a hostile environment. You did it to yourselves rural America. The “Strange Fruit” you cultivated in the South and Easy Rider lawless in the same region has told every non-Occidental person that it’s not safe to near you.
Hillary the Warmonger, the Millennials call you. Part of your defeat was taking an opportunity you saw in the Arab Spring: capturing Quadaffi
Muamar Quadaffi, who no one seems to remember was our baddie of the 80’s. He may have been dormant, but no were near pacified. He had a yen for for American weapons and dispatched terror cells to acquire them. He was such a threat he ended up in pop culture. Watch Back to the Future. It was one of Quadaffi’s operatives who shot Doc Brown and was the catalyst of the story.
TO add to this: by controlling Afghanistan and omitting Quadaffi, we weaken Al Queada/ISIS. They now have no established power to tell the how to establish and hold a caliphate and no established power to secure weapons for them. These were important steps to weaken them to a glorified, international street gang and we’re still whittlin them down. However, if we let Putin do whatever he wants (claim bits of Europe that he wants) we’re going to have a superpower to deal with on our hands.
This corrupt because big risk politics is complicated) war monger was needed for the good of world security.Thank you third party voters 😡 We took out Destro Cobrala and Zartan, just so Russia can move in an take over the region. If they’re successful, they set their sights on Europe which had been Russia’s ambition since Catherine the Great. War is not desirable for anyone, but sometimes you have to do a little evil to do a lot of good.
I love just about anything Russian. Mikka is Russian. However, I do not like their imperial ambitions. Putin claims that he does not want the former Soviet empire back, however he has an unhealthy admiration of previous Russian leaders, Peter and Catherine the Great. Catherine the Great vowed that India would be annexed, China humbled and all of Europe will be governed under the scepter of Russia. She tried her best to claim Istanbul from the Turks. To Sultan’s relief, she died before she could fulfill this effort. However, in the modern era it is very possible that Putin will pick up where she left off.
NPR reported a few years ago, that scores of men in Russia fell into despair after the collapse of the Soviet Union. Understandable. Yet, it creates a problem from one of the tenets of government: order. What do you with Bezerker personality types? Their energy has to be channeled somewhere or they’ll destroy themselves and others like a live grenade. With so many people in the vast expanse that is Russia, Putin and his populace live in the very real danger a standing army that’s about to bring it’s simmering frustration to a boil.
I fear that energy may be unleashed on the Middle East. A region, as a whole. that has not fully recovered from WWI. That is suffering from the lack of the pooled resources of the Ottoman Empire and is in real danger of living under yet another foreign power. This was not the plan of the West. We may want their oil, but we definitely want the Arab states to rule themselves. All we want is their governments to be stable so trade will flow without interruption. Now that Hillary has lost the election, that may never happen.
A have friends who feared that Hillary would bring on WWIII. Perhaps. We all know it was going to to happen. However, with Trump giving a looser reign to Putin, instead of the entire West striving to protect the Middle East, the West , with the U.S protecting it as an ally, will have have to save itself from Russia, should the mood strike Putin’s fancy.
This is politics. The great quagmire of desires and solutions striving to find order under an elected or unwanted power. It takes a great deal of experience to wield this power well, and the West and Muslim states of the Mediterranean and West Asia has just lost its chance to contain the chaos with someone who someone who was busy at work whittling the problem away, one warlord at a time.
Stay in touch, Hillary Rodham Clinton
Dear, ‘Rebecca’, Beth Cirami, and ’Woman’ of the wealthy Francis Howell school district,
I don’t think you are racist at heart, but unfortunately, your rhetoric against the student body of the Normandy school district IS. A better argument to make against the wrong side of the tracks moving into your district would be that you WELCOME students with good academic and disciplinary records, but you are concerned about those who have contempt for school and have poor disciplinary records. Your original statement paints an entire community as worthless and dangerous. The revision I provided acknowledges those who doing well and have great potential.
Do you know that there is a large group of minorities who try their darndest to be the exemplar of society? No, I am not speaking of the model minority that is the Asian community, but the very communities that are widely disdained: Black and Brown. Yes, we’re being systematically moved to your community to enjoy the resources you’re property taxes pay for, but you know what? Our children will pay you back by being the doctors, lawyers and bankers that may help your child in the future as a peer. Seriously, the U.S. can use all the STEM graduates it can get.
Who care where they come from???
Do you know that the curriculum in wealthy communities like yours are more kinesthetically oriented? Learning by Doing and interactivity is the most universal form of teaching, however it’s also the most expensive. Only 20% of the population are auditory learners—guess which learning style is taught in under funded public schools? Yeah, expect only 20% of the student body pass algebra II in an underfunded school. Such a small percentage is not going to boost the test scores of those schools, therefore the college recruiters won’t visit. Forget it.
As a predominantly visual learner, I became an artist. Artist aren’t in high demand, but I am good at what I do and earn a respectable living as an animator. When I transferred into a wealthy school when I was 11 years old in 1983, I was one of those poor kids who inspired my wealthy peers with my ability to turn any lecture or project (save math) into a comic strip. One right-side-of-the-tracks classmate even said that she would not have gotten through trigonometry without me. I was an asset. I AM an asset. So, please. Give those poor Black and Brown kids a chance! They WILL surprise you.
Yes, there will be problems with adjusting, but be patient and most of all—DON’T LEAVE!!! White Flight, as This American Life said, is the other half o the problem. Civilizations advance when cultures mix, not when they’re isolated. It may look like Northern Europe has made every advancement on their own, but no, much of the prowess comes from China, North Africa (through the Romans), the Ottomans and the riches plundered via dependency theory on the New World. White hegemony came through European expansion during the Renaissance. War, technology and administration( you don’t need to be rocket scientist smart to organize desperate people) learned from other nations and innovated into Dutch East/West India trading companies et al. Humans. We all need each other.
Please reconsider running away from the absorption of the Normandy school district. As I said, it will be uncomfortable for a while, but if the community gels together and develops a great network of parents, your fears of school violence and under achievement won’t be as disproportionately overwhelming as you fear.
I have always liked the storyline of Tiamat (The Dragon). Thus, I appropriated it as an element for Pirsia in Superficial’s story line. It isn’t an exact copy, of course, but it got the gears in my then 15 year old head, to consider dragon’s signifying time and space.
So, in the name of research, here’s a great clip from a website Starry Skies on Draco the Constellation.
This page is copyright of Starry Skies writers. I in no way claim the information as my own. I am only copying and pasting the info here, so that it will always be available. Sites die and take their wonderful information with them. I don’t want Starry Skies’ info nearly lost like that of the great pedagogy website Making Sense of Maths.
Draco is a circumpolar constellation visible all night from northern latitudes. The constellation winds around the little dipper. Its’ stars are not very bright, containing only three stars above magnitude 3.0. At one time Draco was quite a bit larger when the ancient Mesopotamians gave the dragon large wings which wound around Ursa Major. The Greek philosopher Thales lopped off the wings in the sixth century BC.
The Pharaoh Khufu ruled ancient Egypt around 2550 BC and was buried in the largest of the Giza pyramids when he died. During his time, Thuban was the pole star, (because of Earth’s precession) around which all other stars revolved. Khufu’s burial chamber was fashioned deep inside the Great Pyramid. Two skinny shafts bore outward from the chamber.
For decades, scholars thought they were airshafts. But in the 1960s, astronomers found that they have an astronomical purpose. It was found that one of the shafts pointed directly towards Thuban. The other was aimed at the belt of Orion, which symbolized Osirus.
The stars close to the pole never set. The Egyptians described these stars as “imperishable” or “undying.” Khufu expected that when he died, he would join not only with the Sun, but with Thuban as well, maintaining order in the celestial realm, just as he had on Earth.
During the time that Draco’s star Thuban was the pole star, it would have appeared to ancient sky watchers that the Earth revolved around Draco. Dragons and other similar creatures often played a role in creation myths. In these stories the gods would often battle such creatures for control of the Earth. When defeated, the dragons were flung up into the skies.
Roman myth calls this dragon Ladon and he guarded the golden apples on a tree in a garden tended by the Hesperides, the daughters of Atlas. Hercules was sent to obtain the apples while under pledge to Eurystheus. He learned from Nereus that he could not pluck the apples himself, but must get help from Atlas. Hercules shot and killed Ladon with an arrow, making way for Atlas to enter and pluck the golden apples. The goddess Hera was greatly distressed by the death of Ladon and placed the dragon in the heavens.
A Greek legend tells the story of Draco as a horrible dragon that guarded a sacred spring and slew the soldiers of Cadmus (first king of Thebes) who had been sent to gather water. Cadmus then fought the dragon and won,. After the dragon died, Athena appeared and told Cadmus to sow the ground with the creature’s teeth. The teeth immediately sprang up as a group of armed soldiers who helped Cadmus found Thebes.
A Babylonian creation story tells of Tiamat, who turned herself into a dragon but was later defeated and split into two parts. One part became the heavens and the other, the Earth.
A Chinese tale sees the stars as the dragon who eats the Sun or Moon (possible represented by the north star Polaris) in an eclipse. During a real eclipse, ancient Chinese would make as much noise as possible, banging on pots and pans to try and scare away the dragon which was eating the Sun or Moon.
A Norse creation myth tells of a dragon who gnaws at the roots of Ygdrasil, the tree that covers the world.
Because Thuban was the pole star 5000 years ago the ancient Egyptians keenly observed it. Some of Draco’s stars were part of their constellation of Hippopotamus and some were of the Crocodile. They appear on the planisphere of Denderah and the walls of the Ramesseum at Thebes. The hieroglyph for the Hippopotamus was used for the heavens in general while the constellation is supposed to have been a symbol of Isis Rathor, Athor, or Athyr, the Egyptian Venus. Draco’s stars were also said to represent the falcon headed god Horus.
Around 800 BC, the prehistoric Adena people who lived in the Ohio area of the United States created Serpent Mound which is believed to mirror the constellation Draco. This huge mound is nearly a quarter mile long.
The Persians have regarded Draco as a man-eating serpent called Azhdeha.
In early Hindu worship, Draco is given the form of an alligator known as Shi-shu-mara.
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Kathy Miles, Author, and Chuck Peters, Systems Administrator