huge increases that globally exist for accessing the Internet over the past
eight years. Statistics follow.
the manner in which our world has shrunk due to emerging technologies and the
‘twitch’ factor of today’s learners (and me) meaning ‘I want the information now
and fast’ with precious little tolerance for even a minimal delay in
exchange? How about the fact that today’s students will have around 12 to
15 careers in their life spans – requiring skills for adapting and flexing to be
successful in those shifts and within unique environments? These
students/workers will be ‘producers’ of content and knowledge – not passive
recipients. The entrepreneurial spirit will be required for successful
innovation and accommodation within the work world.
The move to student-centric instead of teacher-centric classrooms is
required. Teachers become facilitators, organizers – really orchestra
directors
developing the shared vision, supporting and empowering, today’s leader must
construct opportunities for creating
Fourth, the classroom has no walls

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DIGITAL LEARNING ENVIRONMENTS: Tools and Technologies for Effective Classrooms

Designing an Evaluation Plan

Tags: grants on 2008-09-25 -All Annotations (6) -About

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Good evaluation plans are tightly integrated into the proposal and links the
evaluation to your project goals and objectives
what do you want to learn? What information will you
need to measure your effectiveness? What purpose will the evaluation serve? Who
is the audience for the evaluation information? From what sources should you
collect the information? In what format will you collect your data? What
resources will you need to collect the data?
there are over five million hits on Google for "designing an evaluation plan."
For each major objective you should be able to measure the effectives of your
activities/implementation. I like to keep the numbering of the evaluation in
alignment with the rest of my proposal
build benchmarks into your proposal to keep your activities on track
assessment to mean a measurement tool. If the measurement occurs over a
specified time, it is known as a longitudinal study. Evaluators speak of
qualitative and quantitative data. The main difference is that qualitative data
usually involves the collecting of opinion surveys and anecdotal stories from
participants. Quantitative data relies on numbers. Qualitative collects "soft"
data while quantitative collects "hard" data.

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DIGITAL LEARNING ENVIRONMENTS: Tools and Technologies for Effective Classrooms

strategies for teaching students information literacy

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more fromwww.guide2digitallearning.com

netTrekker or Web directories like Awesome Library.
the ability to access, evaluate, synthesize, and build upon information and
media are crucial skills.
Teach them to search
Left to their own devices, students will depend on natural language to search
rather than analyze keywords that would be more effective. They also tend to
rely entirely on a single search tool such as Yahoo or Google for obtaining
information.
Challenge students to search using a variety of strategies and tools (see
"21st-Century Literary Terms and Definitions") and report back on the most and
least effective search approaches.
require them to include a number of keywords and search options they used along
with their traditional, footnoted attributions
inaccuracies found in the Wikipedia Web site and other collaboratively created
online sources, prompting certain educational organizations to ban their use for
research. Why not treat the site itself as a subject of study
Do they find any misleading, inaccurate, or missing information in Wikipedia?
How does it compare to overviews they find elsewhere?
What makes a source viable?
look at the advantages and disadvantages of various resources—not only with
regard to the accuracy issues discussed earlier but also in terms of the
fluidity and speed at which information is updated.
National Council of Teachers of English literacy recommendation: the need for
students to learn to manage "multiple streams of simultaneous information
Copyright is a huge topic
important for students living and interacting online to have a clear
understanding of the legal issues involved in copying and redistributing the
work of others. Some key concepts worth reiterating here
include:
    The creator of an original work—whether a student
or a professional artist—automatically owns all rights to its use, with certain
exceptions, including the exception for "fair use."
    Fair
use allows people to use copyrighted materials, without paying or getting
special permission, if they are using the materials for the purpose of
education, review, satire, or journalism, and are taking into consideration the
following criteria:
    the purpose and character of the use,
including whether such use is of commercial nature or is for nonprofit
educational purposes;
    the nature of the copyrighted
work;
    the amount and substantiality of the portion used in
relation to the copyrighted work as a whole; and
    the
effect of the use upon the potential market for or value of the copyrighted
work.
    Barring some drastic redefinition or legal
precedent, fair use does not apply to educational materials posted on the public
Internet for others to access and redistribute at will.

The copyright holder can always choose to grant to others some or all rights to
their work.
An understanding of Creative Commons not only allows students to determine the
conditions under which they want to share their own work but also gives them an
understanding of the legal and ethical issues involved in reusing the work of
others in situations that do not qualify as fair use
digital materials circulated via e-mail or posted at sites such as YouTube
frequently lack adequate information about the copyright holder, CC
licensing—with the attribution requirement that typically accompanies it—is
raising new awareness about the importance of identifying and citing one's
sources
The ease by which we all cut and paste these days raises many questions about
the definition of the word "plagiarism." But by expecting students to provide
attribution to the best of their ability, and discussing the challenges they
encounter as they try to do this, the education world can help redefine what it
means to be an ethical and active participant in collaborative authoring
ventures

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ISTE | Technology Leadership Standards

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offer a variety of professional development opportunities that facilitate the
ongoing development of knowledge, skills, and understanding of concepts related
to technology
Design developmentally appropriate learning opportunities that apply
technology-enhanced instructional strategies to support the diverse needs of
learners.
Apply current research on teaching and learning with technology when planning
learning environments and experiences
Apply current research on teaching and learning with technology when planning
learning environments and experiences. Candidates
Identify and locate technology resources and evaluate them for accuracy and
suitability. Candidates:
Identify and locate technology resources and evaluate them for accuracy and
suitability. Candidates
Use technology to support learner-centered strategies that address the diverse
needs of students
Use current research and district/state/national content and technology
standards to build lessons and units of instruction
Use technology resources to collect and analyze data, interpret results, and
communicate findings to improve instructional practice and maximize student
learning
Use technology resources to collect and analyze data, interpret results, and
communicate findings to improve instructional practice and maximize student
learning. Candidates:
Continually evaluate and reflect on professional practice to make informed
decisions regarding the use of technology in support of student learning
Follow procedures and guidelines used in planning and purchasing technology
resources
investigate purchasing strategies and procedures for acquiring adminstrative and
instructional software for educational settings
develop and utilize guidelines for budget planning and management procedures
related to educational computing and technology facilities and resources
Apply effective group process skills. Candidates:

1. discuss issues related to building collaborations, alliances, and
partnerships involving educational technology initiatives.
use evaluation findings to recommend modifications in technology implementations
develop curriculum activities or performances that meet national, state, and
local technology standard
use strategic planning principles to lead and assist in the acquisition,
implementation, and maintenance of technology resources
plan, develop, and implement strategies and procedures for resource acquisition
and management of technology-based systems, including hardware and software

44Expand

101 uses of a quadratic equation: Part II

Tags: College_Algebra on 2008-09-13 and saved by3 people -All Annotations (44) -About

more fromplus.maths.org

so $i^2=-1$ . So,
$i$ can't
be a real number and because of this it is called an imaginary number.
Notice also that

$(-i)^2 = -1 \times i \times -1 \times i = i\times i = -1.$
So $x=-i$ is
also a solution to the equation $x^2=-1$ .
Kepler discovered was that the planets did not go round the Sun in circles,
instead they went round in ellipses. Kepler's rules then fitted the
observations perfectly. At last the conic sections that we saw in the last
article came into their own, 1500 years after they were discovered
such as certain comets, were found to move along hyperbolic orbits.
Quadratic equations not only described the orbits along which the planets moved
round the Sun, but also gave a way to observe them more closely
Galileo's telescope used lenses, the shape of which was formed by two
intersecting hyperbolae. The reflecting telescope, invented by Newton (see
later) has a mirror for which each cross section takes the shape of a parabola!
The same parabolic shape works just as well for the bowl of a giant radio
telescope, a shaving mirror and a satellite TV dish
quadratic equations lie at the heart of modern communications.
Usually the particle has a force acting on it, such as gravity for a rugby ball
or friction in the brakes of a car
In particular, if the particle starts at the position $x=0$ then
the position $s$ at the
time $t$ is
given by

$s = ut + \frac{1}{2} at^2.$
suppose that we know the braking force applied to a car: then this formula
allows us to work out either how far we travel in a time $t$ , or
conversely, solving for $t$ , how
long it takes to travel a given distance.
find the stopping distance of a car travelling at a given velocity $u$
if a constant deceleration $-a$ is
applied to slow a car down from speed $u$ to
speed 0, then solving for $t$ and
substituting gives the stopping distance $s$ :

$s = \frac{u^2}{2a}.$
The reason that this result is so important for all of us is that it predicts
that doubling your speed quadruples, rather than doubles, your stopping distance
see stark evidence as to why we should slow down in urban areas, as a small
reduction in speed leads to a much larger reduction in stopping distance
the basis of the science of ballistics, which looks at the way that objects move
under gravity. In this case, an object falls in the $y$
direction with a constant acceleration $g$
the position at time $t$ is
given by

$x = ut \quad \mbox{and} \quad y = vt - \frac{1}{2} g t^2.$
Put another way, we have

$y = \left( \frac{v}{u} \right) x - \left( \frac{g}{2u^2}\right) x^2,$
yet another quadratic equation, this
time relating $x$ to
$y$ . What
was remarkable was that the resulting shape of the trajectory was, of course, a
parabola.
Suppose now that you are in the final minute of a rugby match and you have to
kick a perfect drop goal. To do this you must kick the ball at the correct angle
and velocity so that when it travels a distance x to the goal it is at
the right height y to go over the goalposts.
the parabolic equation for the particle trajectory - with modifications to allow
for air resistance, the spin of the projectile and also the spin of the Earth -
serves as the basis for artillery calculations
he started watching a chandelier swing to and fro - and made a remarkable
discovery: the time taken for a swing of the chandelier was independent of its
amplitude. This discovery led to the invention of the pendulum and various
timepieces such as the Grandfather clock, but at the time Galileo could not
explain it.
he described the fundamental law of gravitation, which was that two masses were
attracted to each other by a force inversely proportional to the square of the
distance between them
tremendous fluke that the inverse square law led to orbits which could be
explained in terms of known curves!)
The best shape for the mirror, to bring all points into focus, was none other
than the parabola, leading to the reflecting telescopes we saw earlier
The fundamental device in the application of calculus to the real world is the
differential equation, which relates the change in the conditions of an
object to (for example) the forces acting on them. Differential equations are at
the heart of nearly all modern applications of mathematics to natural phenomena,
from understanding how heat flows through a bar to the way that animal coat
patterns develop
the motion of the pendulum which so interested Galileo. This motion can be
described in terms of a differential equation
Solving it requires finding the solution to a quadratic equation!
It is possible to find approximate solutions to equations such as this by using
a computer, and this is the approach generally used for the very complex
differential equations encountered in modern technology
different types of solution of the quadratic equation lead to quite different
solutions of the differential equation. If b²>4ac,
then the quadratic equation has two real solutions
The same differential equation has a solution looking like the diagram to the
right. Physically this solution corresponds to a pendulum with a lot of friction
(or a pendulum moving in a liquid such as water
if b²<4ac, then the same differential equation has
oscillating solutions which look like the diagram to the left. These are
more like the motions of the pendulum that we are familiar with.
in the second case, the solutions of the quadratic equation are complex
and involve the square root of -1
the solutions of the resulting quadratic equation tell us whether the solutions
are likely to grow, stay the same size, or get smaller. This is very important
to engineers who are trying to design safe structures and machines. In these
structures, small disturbances which grow will rapidly lead to structural
failure (called instability).
often it is by solving the above quadratic equation and finding whether the
roots w have certain properties that a safe machine can be designed
link between quadratic equations and second order differential equations is no
coincidence: it is all tied up with the link between force and acceleration
described in Newton's second law.
Sophisticated versions of these laws (called the Navier-Stokes and related
partial differential equations) are solved on large computers to forecast the
weather
key ingredients in the discovery of the basic principles of flight. The
consequences of this have been immeasurable and are linked (as ever) with a
quadratic equation called the Bernouilli equation.
if you look at the steady flow of air with speed $u$ and
pressure $P$ , and
an air particle is moving at a height $h$ , then
there is a constant $E$ (the
energy of the air particle) so that

$\frac{u^2}{2} + P = h.$
what happens when we square a number: that is to say, when we take $x$ and
calculate $x^2$ . One
thing we notice is that, no matter what value we take for $x$ , $x^2$ is
always non-negative. As a consequence, $x^2=-1$ cannot
have a solution
The letter $i$ is
used to represent a solution to

$x^2=-1,$
Historically, imaginary numbers first came to light when trying to solve cubic
equations, rather than quadratics. What was most perplexing was that in using
these subtle and imaginary numbers it was possible to solve cubic equations. In
fact, the case that needed imaginary numbers during the calculation turned out
to have real solutions
Using a combination of real and imaginary numbers, known as complex
numbers, turns out to be sufficient to solve virtually all mathematical
problems! The first person to really use imaginary numbers with confidence was
Leonhard Euler, who lived from 1707 to 1783,
Euler discovered that

$e^{it}=\cos (t)+i\sin (t).$

Both $\cos (t)$ and
$\sin (t)$ are
oscillating terms, which is to say they repeat periodically. This formula
provides an insight into how the differential equation which modelled the damped
pendulum, which has a solution of the form $e^{wt}$ , can
have oscillating solutions. If $w$ is
imaginary, or complex, Euler’s formula allows the exponential term to be
rewritten as a combination of $\sin (t)$ and
$\cos (t)$ .
oscillating behaviour can be described using $i$ . The
fundamental equation of quantum theory which is used to calculate the "wave
number" of a quantity (the probability of it being in a particular location) is
Schrödinger’s equation. This is a (partial) differential equation involving $i$ , which
can be written as

$i \frac{\partial u}{\partial t} + \nabla ^2u +v(x)u=0.$
By using it to predict the motion of the elections and holes in semi-conductors
it is possible to design integrated circuits with huge numbers of components
which can perform amazingly complex tasks. Such circuits are at the heart of
much modern technology, including computers, cars, DVD players and mobile phones
a mobile phone works by converting your speech into high frequency radio waves
and the behaviour of these waves can then be calculated using further formulae
involving $i$ . So we
can say with justification that without the simple quadratic equation $x^2=-1$ the
mobile phone would never have been invented.
goal, grandfather clocks, rabbits, areas, singing, tax, architecture, sundials,
stopping, electronics, micro-chips, fridges, sunflowers, acceleration, paper,
planets, ballistics, shooting, jumping, asteroids, quantum theory, chaos,
windows, tennis, badminton, flight, radio, pendulum, weather, falling, shower,
differential equations, telescope, golf

3Expand

When are complex numbers such as imaginary numbers used in the real world? - Yahoo! Answers

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Question Corner -- Complex Numbers in Real Life

applications and when you would use complex numbers

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more fromwww.math.toronto.edu

Here are some examples of the first kind that spring to mind. In electronics,
the state of a circuit element is described by two real numbers (the voltage
V across it and the current I flowing through it). A circuit
element also may possess a capacitance C and an inductance L that
(in simplistic terms) describe its tendency to resist changes in voltage and
current respectively.
These are much better described by complex numbers. Rather than the circuit
element's state having to be described by two different real numbers V
and I, it can be described by a single complex number z = V
+ i I. Similarly, inductance and capacitance can be thought of as the
real and imaginary parts of another single complex number w = C +
i L. The laws of electricity can be expressed using complex addition and
multiplication
electromagnetism. Rather than trying to describe an electromagnetic field by two
real quantities (electric field strength and magnetic field strength), it is
best described as a single complex number, of which the electric and magnetic
components are simply the real and imaginary parts
talk about a beam of light passing through a medium which both reduces the
intensity and shifts the phase, and how that is simply multiplication by a
single complex number.
Much more important is the second kind of application of complex numbers, and
this is much harder to get across. I'm inclined to do this by analogy. Think of
measuring two populations: Population A, 236 people, 48 of them children.
Population B, 1234 people, 123 of them children. You might say that the fraction
of children in population A is 48/236 while the fraction of children in
population B is 123/1234, and that 48/236 (approx. 0.2) is much less than
123/1234 (approx. 0.1), so population A is a much younger population on the
whole.
Now point out that you have used fractions, non-integer numbers, in a problem
where they have no physical relevance. You can't measure populations in
fractions; you can't have "half a person", for example.
electrical engineers, electronic circuit designers, and also anyone in a
profession where differential equations need to be solved. Besides, of course,
mathematicians and physicists!