"Algebraic" view of energy/work concepts for High School teaching I'm trying to describe to high school students the concepts behind Work and Energy. I need a sinple approach to this topic, and I thought to follow this path:

*

*Work is nothing but Energy transfer

*If body A makes Work on body B it transfers energy to it

*When we talk about energy owned by a body we refer only to kinetic energy

*When we talk about potential energy of a body, we're really talking about energy owned by the system including the body and the gravitational or magnetic/electrical fields or by the body and the spring for elastic forces; in the body the energy is only... potential; it will become real (kinetic) only when the system (body + field or body + spring) will give it to the body

Then, I'd do an example: I lift a book on a table from the floor. In this case there are two forces acting doing work on the book: by raising the book I make positive work, while the gravitational field makes negative work on it. Starting from rest and ending at rest, there is no net change in kinetic energy. The work I did has been converted in the grativational potential energy of the system. Then, I make the book fall from the table into the floor: the potential energy is fully converted to kinetic energy during the free fall, if I neglect dissipation due to air drag and other non-conservative contributions.
In case of not-conservative forces there are no potential energy associated with them, so the energy trasnferred by them must be taken into account separately. Where does this contribution goes? Can I say that it goes to internal (thermal) energy of the system, and then dissipated by heat transfer?
Let me summarize my doubts in few answers:

*

*is this process fine for High School teaching?

*how can I formalize this with simple algebraic equations?

*while the kinetic energy is directly associated with the motion of the system, can I say that the potential energy is owned by the system as well? Or who owns the Mechanical Energy?

(Edited to get a cleaner question)
 A: I sympathize with your desire to define work for your students in terms of energy transfer. Most textbooks define it simply as $W=Fd$, and they typically don't provide any sufficient motivation for this magic formula. It is indeed better, IMO, to define it in terms of energy transfer. However, doing so is a little tricky, and I don't think the way you're going about it works.
One problem is that in your strategy of dealing with potential energy, you simply zoom out to encompass the whole system, and then you say there was no work done, because the system was isolated. But this makes it impossible to talk about many of the most interesting and important examples of work.
I don't think it's possible to give a useful and coherent account of work from this point of view without getting into some of the nitty gritty distinctions: (1) distinguishing between thermal energy and macroscopic KE; (2) distinguishing between a force that acts at an identifiable point of contact and one that doesn't; (3) distinguishing between macroscopic and microscopic forces (such as the ones in interatomic collisions that transfer heat); and (4) distinguishing between the work-KE theorem (which only deals with center-of-mass KE and has d as the distance the c.m. travels) and other applications in which there is a well-defined point of contact (such as spinning a wheel by applying forces whose sum is zero).
Note that your proposed strict distinction between KE and PE doesn't really work in practical applications when you deal with microscopic energy. For example, we want to say that when a book sliding across a wooden table comes to a stop, the table does negative work on the book by the work-KE theorem. But we will never be easily able to say whether the thermal energy gained by the table is KE or PE. In fact, some of it is KE and some is PE, but we can't say how much of each without a detailed thermodynamic model of wood at the microscopic level.
A: First, some comments/suggestions about some of your statements:



*If body A makes Work on body B it transfers energy to it


Suggest "If body A does positive work on body B it transfers energy to it". Reason: Work can be positive or negative. Negative work on a body takes energy away from the body. An example is work done by kinetic friction.



*When we talk about energy owned by a body we refer only to kinetic energy


True at the macroscopic level, because potential energy is a system property and not the property of a body by itself. However, internal energy (energy at the microscopic level) is the sum of kinetic and potential energy, where potential energy refers to that associated with intermolecular forces and separations.



*When we talk about potential energy of a body, we're really talking about energy owned by the system ...


True, unless we are talking about internal energy, as discussed above.

Let's make an example: I lift a book on a table from the floor. In
this case on the book there are two kinds of Work that are done: I
make positive Work but the gravitational field makes negative Work.
The total Work is null, so the total energy transferred to the book is
null and there is no new Energy (kinetic).

Yes, but only because the book begins and ends at rest so that the change in kinetic energy is zero.

When the book fall from the table into the floor, there is no a
transformation of Energy from potential to kinetic (as stated in some
Physics books), but simply the system makes Work on the body so there
is an Energy transfer from the system to the body.

Whether or not you choose to call this "transformation of energy from potential to kinetic", the fact is positive work done by gravity on the falling book gives the book kinetic energy at the expense of the gravitational energy of the book-earth system.
Now, regarding your final question:

If this is all correct, how can Mechanical Energy be described, as the
kinetic energy is owned by the body and the potential energy is owned
by the system? Who owns the Mechanical Energy?

The system, in this case the earth-book system, owns the gravitational potential energy, while the book alone owns the kinetic energy.  Total mechanical energy of the system is the sum of its kinetic and potential energies at the macroscopic level.
For an isolated system, such as the book-earth system, total mechanical energy is conserved if there are no dissipative (e.g., friction) forces internal to the system. For the book-earth system when the book falls off the table the total mechanical energy (macroscopic kinetic and gravitational potential energy) is conserved as it falls, if we can ignore air resistance (a form of friction).
If we include air resistance, total mechanical energy is not conserved because the kinetic energy of the book when it reaches the ground will be less than the gravitational potential energy it possessed on the table. The difference equals the increase in the internal energy (kinetic  energy of the atoms and molecules of the book and air as reflected in an increase in their temperatures). So while total mechanical energy is not conserved, total energy is.
Hope this helps.
A: It's very hard to teach Physics in High Schools, since students don't have all the Mathematical tools for in-depth treatment of the topics.
Let me split the answer to your quite a long question, and answer using basic algebraic equations, that should be fine for High School students.
Preliminary definitions
Let's start with some definitions:

*

*total energy $E^{tot}$ of a system can be interpreted as the ability of the system of doing work on the environment;

*kinetic energy $K$ of a system is the contribution of energy of the system related to the macroscopic motion of the system;

*work of external forces acting on the system $W^e$ is a way to transfer energy to/from the system from/to the environment (the other way is heat transfer $Q^e$, but if your are teaching Mechanics it's likely your doing Thermodynamics next year).

Energy balance equations
After these definitions, you can introduce two energy balance equations:

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*$\Delta E^{tot} = W^e$, i.e. the First Principle of Thermodynamics without heat transfer (even if you're teaching Mechanics, you can present it as a principle, i.e. a mathematical expression that agrees with experimental evidence and can't be proved analytically).

*Neglecting heat transfer and internal energy, you can write the total energy as the sum of the kinetic energy and the work of internal forces $\Delta E^{tot} = \Delta K - W^{i}$, telling the students that if you apply external forces to a system, you can alter the motion of the system or have a (opposed) response of the system through internal forces (and thus the minus sign). Putting everything together and rearranging the terms, you can get the balance equation for the kinetic energy, $\Delta K = W^{tot} = W^e + W^i$.
(This is the Theorem of Kinetic
Energy, a direct consequence of the Principles of Dynamics, not a
physical principle, but I don't know if the students are able to
appreciate this level of detail. When you write $\Delta E^{tot} = \Delta K - W^{i}$ you're telling the students a little lie, because you're hiding internal energy, and assuming that $W^{i}$ can be somehow identified with an energy contribution, that is not true if you have non conservative elements - see below)
Conservative force fields and potential energy
Assuming that you have already explained what a force field is, the work done by a force or a force field, it's possible to introduce the definition of a conservative force field saying that the work done by a conservative force $W^c$ only depends on the final and initial configuration of the system, through the difference of a scalar function, $W^c = - \Delta V$. Writing the work as the sum of conservative and non-conservative contributions, $W = W^c + W^{nc} = -\Delta V + W^{nc}$
$\Delta K + \Delta V^e + \Delta V^i = W^{e, nc} + W^{i, nc}$
putting together contributions from internal and external forces, for the potential energy $V = V^e + V^i$, and for the non conservative contributions $W^{nc} = W^{e, nc} + W^{i, nc}$
$\Delta K + \Delta V = W^{nc}$
and defining the mechanical energy $E^{mec} = K + V$ "of the system" (you can attribute the energy to the system, but you have to keep in mind that the contribution of the potential energy of the external forces is due to the interaction of the system with the external field of force, like a gravitational field),
$\Delta E^{mec} = W^{nc}$ .
With this last equation, you can easily state the conservation of the mechanical energy of a system in absence of non-conservative forces.
You need to mention that internal non-conservative contributions and frictions do negative work on the system (you need that for the Second Principle of Thermodynamics), so that they make mechanical energy of the system decrease.
Examples
Now, it's time for some examples. I'd do:

*

*your book example, with mechanical energy $E^{mec} = \frac{1}{2} m v^2 + m g h$

*an example with mass and spring, with $E^{mec} = \frac{1}{2} m v^2 + \frac{1}{2} k x^2$

*an example combining both gravity and elastic potential energy, stressing the fact that you can add all the contributions to mechanical energy.

