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ACTIVITY - insert brackets to make this equation true. (ie: both sides EQUAL)

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Issue2 - National Council of Teachers of Mathematics

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COMMON DENOMINATORS - In order to accurately compare fractions, we need a common denominator.  Until this is achieved, it is like comparing Euros and Dollars.  Which is more? 50 euros or 42 dollars?  We need a common currency in order to understand how these numbers relate to one another.  Similarly, we need fractions to have the same denominators to accurately compare them, order them, and even add or subtract them.

Adding/subtracting fractions

Multiplying/dividing fractions

“OF” and multiplication: When you take a “fraction of” something, you are taking a small peice of it.  You may have heard the saying “at a fraction of the cost”.  This means you are multiplying the cost by a fraction.

The nice thing about multiplying fractions is there is no catch.  Simply multiply the numerators by the denominators!

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Monte Python Holy Grail Witch Sketch

http://www.annenbergclassroom.org/page/monty-python-and-the-quest-for-the-perfect-fallacy

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3^6 + 2^6 same as (3+2)^6?

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An exponent applies to its immediate base. ex: 2+5^3 vs (2+5)^3.

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Everyone's got quirks. Me, I like finding new ways to think about problems, and I've started seeing numbers in a new way. Today I'm bringing you along for the ride.

And why do you care?

You'll see math as one of several approaches. We're still improving our number system.

You'll start noticing core relationships and apply them to diverse areas

You'll have a way to approach "weird" concepts like division by zero and imaginary numbers.

Math is Software For Your Brain

Though our favorite encyclopedia describes math as "the body of knowledge centered on such concepts as quantity, structure, space, and change" I think there's more to it than that.
Math is software for your brain:

Your brain is a raw computer.

You learn new math (install "Counting 1.0" and "Algebra XY Pro") and suddenly you can solve new types of problems.

Sometimes math has bugs. "Roman Numerals I" was ok, but Decimals 2.0 was a much-needed upgrade. But we still have a few issues, like dividing by zero.

It's a strange analogy, but I'm a bit strange, so I think it works out.

But Math Has Its Limits

Quick quiz: Can you multiply two Roman numerals? No cheating, no converting to decimal: I'm talking about "IX times XXXIV". Ready, set, multiply!

...

Having fun yet? It's CCCVI.

Does this horrendous experience mean multiplication is "hard"? Or are we thinking about multiplication in the wrong way, using the wrong mental software?

If you upgrade your brain from "Roman Numerals I" to "Decimals 2.0", you'll find that 9 times 34 is a much easier question: after some work you'd get 306. Same problem + different mental software = drastic difference.

Painting With Notepad

Yes, you could squeeze multiplication into Roman Numerals. But it's neither fun nor easy, and don't get me started on long division.
Our difficulties are often due to our approach, not the concept.
Think of it like trying to draw in Notepad. It's a nice tool, and you can even "type" the Mona Lisa, but the software just wasn't built with images in mind.

Similarly, Roman Numerals were built when we were still learning to count - zero wasn't even invented yet! Math is a software system that gets better over time, and Roman Numerals were due for an upgrade.

But before we get too high-and-mighty, realize our current number system is a patchwork of new features and bug fixes, used to improve our understanding of the universe.

And when we hit difficulties (What's 1/0? The square root of -1?) we need to wonder if we're hitting universal "laws" or walls of our own making. Like the Romans trying to multiply, let alone do fractions, my money's on the latter.

From Ug to Infinity

Our number system developed over time. We started counting on our fingers, moved to unary (lines in the sand), Roman Numerals (shortcuts for large numbers) and Arabic Numerals (the decimal system) with the invention of zero.

Along the way we found "bugs" in our number system and had to invent new ways around it. Again, the bug was in our thinking (our mental software).
Ugware
Ugware is the counting system devised by Ug the caveman: counting on your fingers and toes. Ug's bug was that he was limited to 20 items!

The fix was to abstract the need for physical objects: you don't need 20 cows to count 20 cows. You can make 20 lines in the sand. Or take shortcuts like C for 100.
Unary and Roman Numerals
Having numbers represented abstractly let us do cool things like add and subtract, even fairly large numbers. I + II = III. X + XX = XXX. Not bad.

But there was still a few "bugs" - what is III - III?
Zero
What a fantastic, beautiful invention: using the symbol 0 to represent nothingness! It's a mind-bending and useful idea: we can keep track of "no" cows at all!

This development led to our familiar positional number system: 204 means two "hundreds", zero "tens" and four ones.

Integer division and multiplication became possible in ways the Romans (and Ug) had never imagined. You could work out 1234 × 5678 if given enough time. What a great feature!
Negatives
But zero didn't solve everything; subtraction still had problems. What happens when we take 5 from 3? One solution is to throw up our hands and say "it's a bug and it's undefined", but we'll do better.

We can think about the problem differently, and entertain the possibility that a number can be "negative" - a number that is less than nothing! (Pretty mind-bending, no?).

There are many interpretations (a lack of cows, a debt of cows) and negatives were invented to handle this "bug" in subtraction. Of course, it took a few thousand years to accept this new feature - negative numbers were still controversial in the 1700s!
Rational Numbers
Division introduced bugs as well. 8/4 is fine, but what is 3/4? It's a bug!

The fix is to find a way to represent "numbers between numbers". 3/4 is really 75/100, or "0.75".

We invented the decimal point to handle the crazy idea of a number more than zero but less than one. Wow! Pretty wild, but we included these crazy types of numbers to make our mental software better. Lo and behold, fractions have their uses. The average family can have 2.3 kids and we know what it means.
Irrationals Make Greeks Angry
Here we are, minding our own business when we see a right triangle:

The sides are 1 and 1. And there, staring us square in the face, is the square root of 2. It taunts us, asking to be written down. We can't - it's an infinite, non-repeating decimal number that can't be expressed as a fraction! And yet it's right there on paper.

It's more than a conundrum - it's madness! The guy who discovered irrationals got thrown off a boat.

Luckily, irrationals are at least "algebraic" in that they are the solution to some algebra equation. We can consider sqrt(2) as shorthand for "the solution to the equation x^2 = 2". We often forget sqrt(9) is really both 3 and -3, don't we? Convention implies the positive root.
Complex Numbers
Now some a smart aleck asks, "Ok bub, what number is the solution to the equation x^2 = -1?".

What to do? Declare this to be impossible and non-sensical, just like zero, fractions, rationals and irrationals were once "impossible and non-sensical"? Or do we accept that maybe, just maybe, our human understanding of the universe is not complete and we have more to learn. You know where my money lies.

Imaginary numbers are just as "realistic" as other numbers (or equally contrived, depending on your viewpoint). But, we don't have an intuition for them because they're often "explained": Oh, you don't have an Electrical Engineering degree? Didn't learn about complex impedance? No intuitive imaginary numbers for you!

I've been thinking about these numbers and plan to address this issue. But not yet - have patience.

Why .9999... = 1, and why you should care

Our number system is a way of thinking, but it still has a few gaps. We're not quite sure how to deal with infinity and infinitely small numbers.

Here's a brain-buster for you:

You: What's 1/3?

Me: Um, .33 repeating.

You: Ok. What's 1/3 + 1/3?

Me: 2/3 You: Sure, but do it in decimals.

Me: Um, .33 repeating plus .33 would be... .666 repeating.

You: Great. Now what's 1/3 + 1/3 + 1/3, in decimal?

Me: Uh... .33 repeating plus .33 repeating plus .33 repeating... is .99 repeating.

You: But doesn't 1/3 + 1/3 + 1/3 = 1? So .99 repeating = 1.

Me: What manner of trickery is this? (Pushes you off blog).

Try that argument on a kid (or adult) - it's fun to see people's reactions. Clearly, 1/3 + 1/3 + 1/3 = 1, but somehow when we try to "add it in decimal" the result seems a bit strange. Again, is the strangeness due to the concept, or our thinking?

What is .33 repeating, anyway? Is it a monkey writing 3's until the end of time? Is it a number beyond our notation that we're hopelessly trying to approximate, like the square root of 2? If we simply switch to base 3, the problem goes away: 1/3 = .1 in base 3, so .1 + .1 + .1 = 1 (again, in base 3).

And why do you care? It may be time for a number system upgrade. Discussing infinity with our current numbers is like drawing in notepad. It's crude and feels "tacked on" (like saying 1/0 = infinity. What about 2/0 or 0/0?).

Mathematicians are working on new number systems where infinity is built-in, but there's still unsolved problems about how to "count" infinity.

Let's seek the "a ha!" insights that made zero, fractions and negative numbers understandable, not just the results of manipulating equations. We've been able to overcome every previous mathematical roadblock.

Going Forward

The goals of this article were simple:

Show how math is like mental software that improves over time

Explain that "nonsense" like zero or negative numbers can start as a paradox and become intuitive as we adapt our approach.

Today, we still have trouble with ideas like infinity (or at least I do). It's ok to admit it.

This is a way to think about math; combine it with your own understanding. Don't stand in a daze, unable to add because you're unsure what 1/3 really means.

Insights deepen our understanding, but sometimes only emerge with use. Newton didn't have a "formal" understanding of infinitesimals when he invented calculus, but it seemed to work fine for him (equations got solved). I don't advocate plug-and-chug, but for certain ideas you need to hammer away before the insights come.

But enough philosophy. Upcoming articles will show real, concrete ways to think about arithmetic and complex numbers, which can aid the "mechanical" understanding we have today. Happy math.

PS. If you're curious, there's more on .999... and division by zero on wikipedia.

Other Posts In This Series

1. Techniques for Adding the Numbers 1 to 100

2. Rethinking Arithmetic: A Visual Guide

3. Surprising Patterns in the Square Numbers (1, 4, 9, 16...)

4. Fun With Modular Arithmetic

5. Learning How to Count (Avoiding The Fencepost Problem)

6. A Quirky Introduction To Number Systems

7. Another Look at Prime Numbers

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Everyone's got quirks. Me, I like finding new ways to think about problems, and I've started seeing numbers in a new way. Today I'm bringing you along for the ride. And why do you care? You'll see math as one of several approaches. We're still improving our number system.

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Born in England in 1819, novelist and poet George Eliot is best remembered for writing classic books like Middlemarch and Silas Marner. Despite the time period she wrote in, the author-whose real name was Mary Anne (or Marian) Evans-was no stuffy Victorian. She had a famously scandalous love life and, among other linguistic accomplishments, is responsible for the term pop music. Here are 15 things you might not know about the beloved British writer.

1. SHE WAS BORN ON THE ESTATE WHERE HER FATHER WORKED.

Eliot was born on the grounds of Arbury Hall and Estate, a sprawling mansion in Warwickshire, England with hundreds of acres of surrounding gardens and farmland. Her father, Robert Evans, worked for the estate's owners, the Newdigate family, as a manager and agent. His job entailed collecting rents from tenant farmers and overseeing the property's coal mine.

2. HER RURAL UPBRINGING INSPIRED HER LATER NOVELS.

Eliot was just an infant when her family moved from Arbury Hall to a home in a nearby town. But Arbury and the Warwickshire countryside left their mark on her. In Scenes of Clerical Life (1858), Eliot's collection of three short stories, she wrote about the area and drew inspiration from real places and people. And some of her stories mirrored reality pretty closely. For instance, she turned Arbury Hall into Cheverel Manor, and Sir Roger Newdigate, Arbury's owner, into Sir Christopher Cheverel.

3. SHE EDITED A JOURNAL FOR PROGRESSIVE THINKERS.

In the early 1850s, Eliot wrote for The Westminster Review, a London-based periodical founded by philosophers Jeremy Bentham and James Mill, contributing essays and reviews using the name Marian Evans. She soon became the de facto editor of the progressive journal, though her role was anonymous. Years later, other writers reviewed Eliot's own pseudonymous works in the journal she once edited.

4. SHE WORKED AS A TRANSLATOR.

Throughout her life, Eliot put her language skills to work translating foreign works into English. She translated books like David Friedrich Strauss's Das Leben Jesu ( The Life of Jesus), a highly controversial German treatise that argued that Jesus Christ was a real person, but not divine. (Upon reading her translation, one English nobleman called it "the most pestilential book ever vomited out of the jaws of hell.") Eliot also translated The Essence of Christianity by German philosopher Ludwig Feuerbach and the Latin Ethics by Benedict de Spinoza, incorporating facets of these philosophical and religious ideas into her own writing.

5. SHE WASN'T A FAN OF MOST WOMEN WRITERS OF HER DAY.

Eliot was by no means a misogynist, but she did have some harsh words for fellow women writers. In an anonymous essay titled "Silly Novels by Lady Novelists," Eliot lamented the frivolous characters and unrealistic plots that she argued were nearly ubiquitous features of novels written by women at the time. Published in The Westminster Review in 1856, Eliot's essay asserted that these books, full of cliches and improbable romantic endings, made educated women look foolish. She also criticized the writing style of other women of her time, saying they mistook "vagueness for depth, bombast for eloquence, and affectation for originality." However, she did allow that not every book written by a woman fell into this trap, praising writers like Currer Bell ( Charlotte Brontë) and Elizabeth Gaskell.

6. SHE WAS NOT CONSIDERED CONVENTIONALLY ATTRACTIVE ...

Eliot's appearance was a source of avid discussion during her lifetime, and her looks continue to fascinate readers today. Eliot herself joked about her ugliness in letters to friends, and the novelist Henry James once described her in a letter to his father as "magnificently ugly, deliciously hideous." He went on to say that the "horse-faced" writer had a "vast pendulous nose," a low forehead, and bad teeth, among other physical flaws.

7. ... BUT MEN LOVED HER.

Despite her plain appearance, men were drawn to Eliot. In the same letter where he called her "deliciously hideous," James explained his counterintuitive attraction towards her like this: "Now in this vast ugliness resides a most powerful beauty which, in a very few minutes, steals forth and charms the mind, so that you end, as I ended, in falling in love with her."

After various dalliances and a marriage proposal that she turned down, she spent more than two decades with the philosopher and critic George Lewes. But Lewes was already married, and as a result, many in Eliot's social circle (including her brother) shunned her. Though Lewes couldn't obtain an official divorce from his estranged wife, he and Eliot lived together as partners until his death in 1878, and she referred to herself as Mrs. Marian Lewes.

8. HER PEN NAME PAID HOMAGE TO HER LOVER.

In 1856, both to avoid the sexism of the publishing industry and distance her literary work from her scandalous romantic situation, she adopted the pen name George Eliot, a male nom de plume that paid homage to Lewes. In addition to adopting his first name, some historians have also suggested that "Eliot" derives from "To L(ewes), I owe it."

9. SHE MARRIED A MAN TWO DECADES HER JUNIOR ...

After Lewes's death, Eliot channeled her grief by editing his writing and spending time with her lawyer and accountant, John Cross. Although Eliot was 60 and Cross was just 40, the two friends fell in love and married at London's St. George's Church in the spring of 1880.

10. ... BUT THEIR HONEYMOON TOOK A DARK TURN.

After their wedding, the pair traveled to Venice, Italy for their honeymoon. Although Cross wrote a letter to his sister indicating that he was having a delightful time, Eliot knew something was wrong. Her new husband was depressed, agitated, and losing weight. She called a doctor to their hotel room and was speaking with him when Cross jumped off the balcony into the Grand Canal.

Cross was rescued by a hotel worker and the personal gondolier the couple had hired to take them around the waterways. The newlyweds eventually continued on their trip, and they remained married until Eliot's death later that year. Historians continue to speculate about the reason for his jump, and whether it was a suicide attempt-Cross may have had a personal and family history of mental illness-or some kind of heat-induced delirium. The mysterious incident was recently turned into a novel.

11. SHE INVENTED THE TERM POP ...

You probably don't associate George Eliot with Lady Gaga, but the Oxford English Dictionary credits the Victorian novelist with coining the term pop to refer to popular music. In November 1862, Eliot wrote in a birthday letter to a friend, "We have been to a Monday Pop. this week to hear Beethoven's Septet, and an amazing thing of Bach's played by the amazing Joachim. But there is too much 'Pop.' for the thorough enjoyment of the chamber music they give."

12. ... AND A NEW MEANING OF THE WORD BROWSER.

Eliot coined a number of other now-common terms in her writing. For instance, she was the first to use the word browser in the modern sense of someone who is casually looking around (like a browser in a bookstore). According to the Oxford English Dictionary, in the 16th century, the word browser meant "a person who cuts the leaves and twigs of trees to use as food for animals in winter." Later, it came to mean an animal that searched for leaves and twigs to eat. Eliot's historical novel marked the first recorded time the word was used to mean a person generally surveying something. In it, she describes several friends of Florentine politician Bartolomeo Scala as "amiable browsers in the Medicean park."

13. SHE WAS ALSO A POET.

Although Eliot was most famous for her novels, she also produced two volumes of poetry. Her first published piece of writing was a poem called "Knowing That Shortly I Must Put Off This Tabernacle." Published in The Christian Observer in 1840, the poem refers to the Bible and imagines a person who is about to die saying goodbye to Earth. In a later poem, "O May I Join the Choir Invisible," Eliot argues that improving the world during one's lifetime is the only way to achieve permanence.

14. VIRGINIA WOOLF ADMIRED HER WRITING.

Author Virginia Woolf praised Middlemarch's mature prose, referring to it as "the magnificent book which with all its imperfections is one of the few English novels written for grown-up people." And modern readers seem to agree. In 2015, a BBC poll of 82 book critics from around the world named Middlemarch the greatest British novel of all time. Several authors, including Julian Barnes and Martin Amis, have also listed the book as one of the greatest English novels ever written.

15. HER FORMER HOME IS NOW A STEAKHOUSE.

Griff House, where Eliot lived as an infant until her early twenties, still exists, but it's now home to a steakhouse and hotel. Called the Griff House Beefeater & Nuneaton Premier Travel Inn, the spot also features a pond, gardens, and a play area for kids.

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12 Mind Blowing Number Systems From Other Languages

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Today is a big day for lovers of the number 12, and no one loves 12s more than the members of the Dozenal Society. The Dozenal Society advocates for ditching the base-10 system we use for counting in favor of a base-12 system.

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Powers of ten investigation, can be combined with fractions base 10,100 etc.

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Patterns in multiplying by 1/10, 1/100, etc

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Do we want to include this type of thing at all? Too long a topic but posting in case we want at least an image if nothing more.

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Idea of using images to help remember - Natural (sesame street, the count or someone counting, thermometer for integers, etc)

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Square one video of battling positives and negatives (... blue and red monsters to illustrate that they cancel each other out and whatever is left is your answer

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Can't attach! Will save to google drive

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Adding and Subtracting Fractions with historical context

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Powers with negative base investigation?

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Number systems

Integers - in chapter 0? I recall this being a teeth pulling topic for many

Squares and square roots? (Cube roots? Exponents in general?)

Order of operations

Equivalent fractions

Adding/subtracting fractions

Multiplying/dividing fractions

Converting to decimals

Operations with decimals

Understanding percentages

Converting to percent

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Bedmas/bodmas/pemdas/christmas?

Take a number, double it and add 6. Take the same number, add 6 then double it. Does order matter? What is the only case these would be equal

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Why do we need common denominators? (Hope to end confusion about WHEN we need them)

Visual-shade in half of a third, 1/4 of a fifth, etc. What do you notice?

Government spending

Producing a movie, how would you divide up budget (among actors, ads, props, etc). Use at least three different demonimators l. What must your fractions add up to?