x^{2 }– 3x – 4 = 0

x^{2} – 3x + 2 = 6

(x – 3/2)^{2} = 6

(x – 3/2) = √6 or – (x – 3/2) = √6

x = 3/2 + √6 or x = 3/2 – √6

x^{2 }– 3x – 4 = 0

x^{2} – 3x + 2 = 6

(x – 3/2)^{2} = 6

(x – 3/2) = √6 or – (x – 3/2) = √6

x = 3/2 + √6 or x = 3/2 – √6

Filed under algebra, Uncategorized

j giambrone

facebook, and the rest, they almost led me astray

So it goes with the fiction we read, the movies we watch, the music we listen to and, scarily, the ideas we subscribe to. They’re not challenged. They’re validated and reinforced. By bookmarking given blogs and personalizing social-media feeds, we customize the news we consume and the political beliefs we’re exposed to as never before. And this colors our days, or rather bleeds them of color, reducing them to a single hue.

…“Facebook allows people to react to each other so quickly that they are really

afraidto step out of line,” he said.

I could add plenty, but won’t. Am I afraid of stepping out of line here?

*Gilding the Allegory (Free)*

Free -e-Book

Filed under Uncategorized

The same old mess, again.

Andy Hargreaves, Professor at Boston College and recipient of many honors, including the Grawemeyer Award, writes here about the problems of English schools, which he attributes to its reckless pursuit of free-market policies, akin to those now dominant in the U.S. In this article, which appeared in the *Times Education Supplement* (U.K.), Hargreaves blames the free-market strategy of “reform,” which demoralizes teachers and damages the profession.

He writes:

*Britain has a teacher recruitment crisis. But it is not truly British. The complaint is much more spectacular in England. In Scotland, teaching is an attractive profession and while recruitment levels are disappointing, the issue is not as profound. The Scottish system is creaking; the English system has fallen over. What explains the difference?*

*The answer is simple. Scotland values a strong state educational system run by 32 local authorities that is staffed by well-trained and highly valued professionals who stay and…*

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Filed under Uncategorized

, and it’s not just california.

Filed under Uncategorized

Any use would be fair use at the end of the testing period.

I think the students in our leadership and education policy classes at California State University Sacramento (scholarly and academic purposes) and the readers of Cloaking Inequity (news reporting) will be very interested in this new, ongoing case study where a PARCC, a testing company, is trying to limit the fair use of copyrighted material. Here is a case study that was contributed to by several anonymous and on-the-record authors that I believe is fair use for research, scholarship and news reporting:

Celia Oyler, professor at Teachers College, Columbia University, posted a biting commentary by an anonymous teacher about the flaws of PARCC. She received a letter from PARCC threatening legal action unless she removed the post because it contained copyrighted material —and divulged the name of the author. Oyler left the post on her blog but removed anything that might be copyrighted. She has not given up the name of the author. Many…

View original post 2,895 more words

Filed under Uncategorized

I don’t know the name of this one. The flower is about 2 inches high. The plant is a bush.

Eleconia. The plants grow to about 10 ft.

The fancy bromelia. The pink parts are leaves.

Flowers from my neighbor’s garden.

Mini sweet pimiento plant. It just appeared and is now about 3 ft tall.

Oregano in a pot.

A tree bromelia which fell out of the tree. Now it is soon going to flower. About 4 ft high.

A Puertorrican Name (nee-amey) with a new shoot. Too late to cook this one.

And a visiting beetle. Yesterday we had a visit from a 4-5 ft boa constrictor.

That’s all. folks.

Filed under caribbean, Puerto Rico, tropical garden, Uncategorized

What exactly are negative numbers?

A reference , from Wikipedia:

In A.D. 1759, Francis Maseres, an English mathematician, wrote that negative numbers “darken the very whole doctrines of the equations and make dark of the things which are in their nature excessively obvious and simple”.

He came to the conclusion that negative numbers were nonsensical.[25]

A minus times a minus is a plus

Two minuses make a plus

Dividing by a negative, especially a negative fraction !!!!

(10 – 2) x (7 – 3) = 10 x 7 – 2 x 7 + 10 x -3 + 2 x 3, really? How do we know?

Or we use “the area model”, or some hand waving with the number line.

**It’s time for some clear thinking about this stuff.**

Mathematically speaking, the only place that requires troublesome calculations with negative numbers is in algebra, either in evaluation or in rearrangement, but what about the real world ?

Where in the real world does one encounter negative x negative ?

I found two situations, in electricity and in mechanics:

1: “volts x amps = watts”, as it it popularly remembered really means “voltage drop x current flowing = power”

It is sensible to choose a measurement system (scale) for each of these so that a current flowing from a higher to a lower potential point is treated as positive, as is the voltage drop.

Part of simple circuit A———–[resistors etc in here]————–B

Choosing point A, at potential a, as the reference, and point B, at potential b, as the “other” point, then the potential drop from A to B is a – b

If b<a then a current flows from A to B, and its value is positive, just as a – b is positive

If b>a then a current flows from B to A, and its value is negative, just as a – b is negative

In each case the formula for power, voltage drop x current flowing = power, must yield an unsigned number, as negative power is a nonsense. Power is an “amount”.

So when dealing with reality minus times minus is plus (in this case nosign at all).

The mechanics example is about the formula “force times distance = work done”

You can fill in the details.

Now let’s do multiplication on the number line, or to be more precise, two number lines:

Draw two number lines, different directions, starting together at the zero. The scales do not have to be the same.

To multiply 2 by three (3 times 2):

1: Draw a line from the 1 on line A to the 2 on line B

2: Draw a line from the 3 on line A parallel to the first line.

3: It meets line B at the point 6

4: Done: 3 times 2 is 6

Number line A holds the multipliers, number line B holds the numbers being multiplied.

To multiply a negative number by a positive number we need a pair of signed number lines, crossing at their zero points.

So to multiply -2 by 3 (3 times -2) we do the same as above, but the number being multiplied is now -2, so 1 on line A is joined to -2 on line B

The diagram below is for -2 times 3. Wow, it ends in the same place.

Finally, and you can see where this is going, we do -2 times -3.

Join the 1 on line A to the -3 on line B, and then the parallel to this line passing through the -2 on line A:

and as hoped for, this line passes through the point 6 on the number line B.

Does this “prove” the general case? Only in the proverbial sense. The reason is that we do not have a proper definition of signed numbers. (There is one).

Incidentally, the numbering on the scales above is very poor. The positive numbers are **NOT NOT NOT** the same things as the unsigned numbers 1, 1.986, 234.5 etc

Each of them should have a + in front, but mathematicians are **Lazy**. More on this another day.

Problem for you: Show that (a-b)(c-d) = ac – bc – ad + bd without using anything to do with “negative numbers”

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**References.**

Wikipedia:

Reference direction for current

Since the current in a wire or component can flow in either direction, when a variable I is defined to represent

that current, the direction representing positive current must be specified, usually by an arrow on the circuit

schematic diagram. This is called the reference direction of current I. If the current flows in the opposite

direction, the variable I has a negative value.

Yahoo Answers: Reference direction for potential difference

Best Answer: Potential difference can be negative. It depends on which direction you measure the voltage – e.g.

which way round you connect a voltmeter. (if this is the best answer, I hate to think of what the worst answer is)

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Filed under algebra, arithmetic, definitions, education, geometrical, math, meaning, negative numbers, Number systems, operations, subtraction, teaching, Uncategorized