What is Differentiating: Definition and 269 Discussions
Differentiated instruction and assessment, also known as differentiated learning or, in education, simply, differentiation, is a framework or philosophy for effective teaching that involves providing all students within their diverse classroom community of learners a range of different avenues for understanding new information (often in the same classroom) in terms of: acquiring content; processing, constructing, or making sense of ideas; and developing teaching materials and assessment measures so that all students within a classroom can learn effectively, regardless of differences in their ability. Students vary in culture, socioeconomic status, language, gender, motivation, ability/disability, learning styles, personal interests and more, and teachers must be aware of these varieties as they plan in accordance with the curricula. By considering varied learning needs, teachers can develop personalized instruction so that all children in the classroom can learn effectively. Differentiated classrooms have also been described as ones that respond to student variety in readiness levels, interests, and learning profiles. It is a classroom that includes and allows all students to be successful. To do this, a teacher sets different expectations for task completion for students, specifically based upon their individual needs.Differentiated instruction, according to Carol Ann Tomlinson, is the process of "ensuring that what a student learns, how he or she learns it, and how the student demonstrates what he or she has learned is a match for that student's readiness level, interests, and preferred mode of learning." Teachers can differentiate in four ways: 1) through content, 2) process, 3) product, and 4) learning environment based on the individual learner. Differentiation stems from beliefs about differences among learners, how they learn, learning preferences, and individual interests (Algozzine & Anderson, 2007). Therefore, differentiation is an organized, yet flexible way of proactively adjusting teaching and learning methods to accommodate each child's learning needs and preferences to achieve maximum growth as a learner. To understand how students learn and what they know, pre-assessment and ongoing assessment are essential. This provides feedback for both teacher and student, with the ultimate goal of improving student learning. Delivery of instruction in the past often followed a "one size fits all" approach. In contrast, differentiation is individually student centered, with a focus on appropriate instructional and assessment tools that are fair, flexible, challenging, and engage students in the curriculum in meaningful ways.
1. I have this equation: y= 2.52e^-0.8472x which I need to differentiate.
2. I am unsure whether to use the product or chain rule>
3. I honestly don't know what to do...
I was thinking 2.52e^u?
I'm trying to differentiate 2^(x^2), but I'm getting a factor of two out and can't figure out why. I approached the question as follows..
y=2^(x^2) , so y=(2^x)^x
u=2^x y=u^x
du/dx = (2^x)ln2
dy/du = xu^(x-1)
= x(2^x)^(x-1)
= x(2)^((x^2)-x)
So dy/dx =...
Homework Statement
let z=f(x,y) be a differentiable function. If we change to polar coordinates, we make the substitution x=rcos(θ), y=rsin(θ), x^2+y^2=r^2 and tan(θ) = y/x.
a. Find expressions ∂z/∂r and ∂z/∂θ involving ∂z/∂x and ∂z/∂y.
b. Show that (∂z/∂x)^2 + (∂z/∂y)^2 = (∂z/∂r)^2 +...
Homework Statement
Prove that d/dt[r.(vxa)] = r.(vxda/dt)
Homework Equations
r, v, a are position, velocity and acceleration vectors.
..r.(v.. is the dot product.
..vxa.. is the cross product
The Attempt at a Solution
I expand the equation using the product rule for dot and...
Homework Statement
Differentiate twice: z = sinx
Homework Equations
Product rule
Chain rule
The Attempt at a Solution
dy/dx = dy/dz * dz/dx
dy/dx = dy/dz * cosx
Using the product rule:
d^2y/dx^2 = d^2y/dz^2 * cosx - dy/dz * sinx
According to the answer in the book...
Homework Statement
I need to derive Euler-Lagrange equations and natural boundary conditions for a given model. I've worked out and broken down the model into the following 5 parts:
J1 = ∫ {ϕ>0} |f(x) − u+(x)|^2dx
J2 = ∫ {ϕ<0} |f(x) − u-(x)|^2dx
J3 = ∫ Ω |∇H(ϕ(x))|dx
J4 = ∫ {ϕ>0}...
Quick question on typical integration/differentiation. What is the justification for differentiating some integrals with respect to constants in order to obtain results for them, i.e. \frac{∂}{∂α}\int e^{-αx^{2}}dx=\int-x^{2}e^{-αx^{2}}dx? It seems to me α is a constant, so it seems a little...
Homework Statement
The Attempt at a Solution
When the textbook says: Differentiating both sides of x^3 + y^3 = 6xy with respect to x, regarding y as a function of x, and using the Chain Rule on the y^3 term and the Product Rule on the 6xy term, we get
I really don't get what...
Homework Statement
1. a)
A rope 25m long is cut into two pieces. One piece is bent into the shape of a square, and the other into the shape of a circle. How should the rope be cut to maximise the total area enclosed by the pieces? And how should it be cut in order to minimise the total...
Even though I've taken 3 semesters of calc, some of the terms are used interchangeably and are a bit obfuscated (at least in my mind lol). I understand how to take the derivative of a function with respect to 1 or more variables, but I'm reading Mary L Boas Math methods in physical science and...
Hi,
I'm not quite sure how to differentiate indexed equations in a quick way, and I'm not sure that the way I use is correct. Does anyone have any tips/methods/resources that I could use to do these kind of operations.
By indexed equations I mean equations like
\frac{\partial F^i}{\partial...
Homework Statement
Verify the identity:
arctanx + arctan(1/x)=∏/2
using calculus theory.
(Hint: Differentiate the left hand side of the identity)
Homework Equations
?
The Attempt at a Solution
is this correct?
tan(arctanx + arctan(1/x))
= [tan(arctan(x)) +...
Hi,
I found on youtube a video showing the steps to differentiate tan(x). I can follow the steps quiet easily but I'm not sure which rule is being used when substituting u for cos(x) and du for sin(x)dx.
So which rule is being used and how do I know which is u and which is du.
Thanks
Homework Statement
Given here is an Operational Amplifier, its feeding Voltages are +-10.
A) Calculate Vin to get a Vout of -2V, point out the direction of the current in this case (From a to b or b to a)
http://img72.imageshack.us/img72/2550/needheed.jpg The Attempt at a Solution...
k guys.. i'd like to try to figure out how to get to the derrivatives on my own so really just looking for a right or wrong.
1. f(x) = e^(3x^2+x) find f'(2)
f'(x) = e^(3x^2+x) * (6x+1)
f'(2)= e^(3(2)^2+2) * [6(2)+1]
f'(2)=13e^14 ... correct?2) find slope...
1. my function is: f(x)=log 1.5 (-.76x+305). f(x)= log base 1.5 of -.76x+305
3. How do i differentiate it? here is what i have so far: (1/((-.76x+305)ln1.5))*(-.76/dx)
Homework Statement
A funny car accelerates from rest through a measured track distance in time T with the engine operating at a constant power P. If the track crew can increase the engine power by a differential amount dP, what is the change in the time required for the run? (Use any variable...
find the derivatives
of differentiation of trigonometric functions
1. y=cos(3x^2+8x-2)
2. y=tan^3 2x
3. y=sin5x sin^5 x
4. y=Square root of 4sin^2x+9cos^2x
help here please..
i can't understand trigonometric functions
sorry admin or moderator, i just search the net on how...
Homework Statement
if y = K sin(Gx + 5)
what is dy/dx
Homework Equations
y=sin x
dy/dx = cos x
The Attempt at a Solution
y = K sin(Gx + 5)
dy/dx = KG cos (Gx +5) (1)
or
dy/dx = KG cos GX + 5 (2)
i think it is (1)
is this correct or it is non of these...
Hey everybody!
Physicists have no problem differentiating a function of many variables - in flat space R^n.
But I don't like how many books don't give examples of how this done in a manifold- even if it may be easy when one finally understands it.
For example, how do I differentiate a...
Hi there,
I should probably know this (attempting to do a PhD in physics!) but is there a way to differentiate between longitudinal and shear acoustic waves in a solid?
I know that seismologists know which is which by using the time of flight difference for the two types of waves and the...
Homework Statement
Differentiate, f(x)=71+x/1-x
Homework Equations
The Attempt at a Solution
Working this out, hoping it's right as I was going along, my final answer is 71+x/1-x * (2ln7)/1-x
Homework Statement
I'm doing a optimisation question and I get to a point where I have to verify a maximum using a double derivative and I need to differentiate -5sin^2(x)
Homework Equations
-5sin^2(x)
The Attempt at a Solution
-10cos(x)sin(x) I am not sure if the answer is positive...
differentiating for "y"
Homework Statement
I don't quite understand the reason why one can \frac{d}{dx} y2 and get 2y \frac{dy}{dx} when using implicit differentiation. Why can one even go as far as to change y2 to 2y? Homework Equations
The Attempt at a Solution
I see the chain rule is...
Differentiate the following:
u = 2 (x-t2 / 3)et/3
Here is my attempt
I have to find du/dx and du/dt
du/dx = 2 (x-t2 / 3)et/3
du/dt = -4t/9 (x-t2 / 3)et/3
Homework Statement
Differentiate the following function: f(x) = x\sqrt{x^{3}+2x}
Homework Equations
N/A
The Attempt at a Solution
Here's my attempt:
First I rewrite it as:
f(x) = x(x^{3}+2x)^{\frac{1}{2}}
Now, I differentiate using chain rule:
f(x) =...
Homework Statement
Does this look correct? How do I know when to stop simplifying things? Sometimes it comes out to a nice little expression, and other times it's a long solution. In the latter, I spend too much time trying to simplify it further!
Homework Equations
The Attempt at a...
Homework Statement
Calculate the integral
I = \int (t^x - 1)/ln(t) dt, boundaries: 0 \leq t \leq 1, x \geq 0
by differentiating first with respect to x.
Homework Equations
-
The Attempt at a Solution
I have no idea how to solve this, but it's on our sample exam and there are no...
Homework Statement
By differentiating the summation, show that ∂/∂b₀ ƒ(b₀ , b₁) = -2 ∑_(i=1)^n (y₁ - b₀ - b₁x₁)
Homework Equations
the ‘derivative of the sums’ equals ‘the sum of the derivatives’:
The Attempt at a Solution
How would we partially differentiate a summation...
Find the general solution to:
dy/dx = ex+y
Im not sure if I am doing this right or not. i tried saying ex+y = ex x ey and using partial differentiation to solve it but keep getting the same as the question: ex+y
i know differentiating ex gives ex so is it the same in this case?
Hello!I'm getting confused when differentiating summations.
I understand that if you differentiate an expression and it gives a kroneker delta, that then sums over the appropriate summation and it disappears. But in my notes it has
\frac{\partial}{\partial p_{i}} [-k \sum_{i=1}^{r} p_{i} ln...
Hello!
I am very VERY confused!
Would anyone please be kind enough to point me in the right direction.
I read that, in general, the derivative of a tensor is not a tensor.
What do you find when you differentiate a tensor, then?
I thought that you wanted to find the acceleration, to...
I am not sure whether there is any difference between differentiating complex and real numbers... I am just trying to differentiate:
e^(2+3i)x = (2+3i)e^(2+3i)x
Is this correct? I have a feeling its not this simple.
If r and s are vectors that depend on time, prove that the product rule for differentiating products applies to r.s, that is that:
d/dt (r.s) = r. ds/dt + dr/dt .s
--
I'm not entirely sure how I'm supposed to go about proving this, can anyone point me in the right direction, please...
Homework Statement
How would you differentiate, (v^3-2v*squareroot v)?
Homework Equations
The Attempt at a Solution
would it look like...3v^2 - v^-1/2 ?
Homework Statement
y = -5x3 + 4/x5 + 1.7pi
Homework Equations
Differentiate each Function.
Derivatives
The Attempt at a Solution
I have done question like these before but this is the first time using pi.
I used all derivative rules none of them give me the right answer.
The answer I am...
Homework Statement
Differentiate.
y=x^2 + 4x +3/(square root x)
and
H(x)=(x+x^-1)^3
Homework Equations
The Attempt at a Solution
For the first one I got x^3/2 + 4x^1/2 + 3x^-1/2
But I don't know if that is correct or where to go from here?
The second I really...
Hey All,
Here's a stupid and probably ridiculously easy question, but I want to make sure that I have it right.
Let G be a Lie group with Lie algebra \mathfrak g . Assume that H \in \mathfrak g and \phi \in \mathfrak g^* the algebraic dual. Assume that X(t) is an integral curve...
Basic question on scalar filed theory that is getting on my nerves. Say that we have the langrangian density of the free scalar (not hermitian i.e. "complex") field
L=-1/2 (\partial_{\mu} \phi \partial^{\mu} \phi^* + m^2 \phi \phi^*)
Thus the equations of motion are
(\partial_{\mu}...
Homework Statement
I am working on a problem that asks to use the integral form of the continuity equation (for a steady flow) and show that it can equal this (by taking the derivative of it): dr/r + dV/V + dA/A = 0 where V is Velocity and r is the density.
Homework Equations
What would...
My question is just a concept that I don't understand.
When differentiating a power series that starts at n=0 we bump that bound up to n=1.
My question is do we always do that?
or
Do we only do that when the first term of the power series is a constant and thus when it is...
Hi,
I am trying to do the multipole expansion of a point charge away from the origin, I can't understand Legendre polynomials so want to do it in Cartesian but when I try to Taylor expand
the \left|\frac{1}{\vec{r}-\vec{r'}}\right| I am not sure how to handle the modulus.
Any help...
Homework Statement
Let's say A is a 7x7 matrix which is defined as [a b c 0 0 0 0; b a 0 d 0 0 0; c 0 a b e 0 0; 0 d b f 0 e 0; 0 0 d 0 f b g; 0 0 0 d b f h; 0 0 0 0 0 0 0] where semicolon (;) represent a new row and a space is a new column.Homework Equations
If y = expm (A*t), where expm...
Hello :smile:
I'm currently using past papers to revise for January exams, and I've found a bit of a problem with something I thought I was okay with.
Homework Statement
The position at time t of a particle undergoing damped oscillations is given by:
x = 2e^{-t}\sin(3t).
Express...