Instructor's Manual: The Physics as Proportionality Trilogy

J. Rogers, SE Ohio

Introduction: A New Pedagogy for Physics

This manual provides a detailed guide for delivering a transformative three-lesson series in introductory physics. The curriculum is designed to reframe the entire subject for students, moving them away from the rote memorization of complex, constant-laden formulas toward a deep, intuitive understanding of physics as a system of simple, elegant proportions. It is a revelation designed to dismantle a mythology.

The pedagogical journey across the three lessons is a carefully orchestrated arc of deconstruction, reconstruction, and generation:

1. Lesson 1 (Deconstruction): The first lesson systematically dismantles the mythology surrounding dimensional constants. Students learn that constants like G, c, and h are not fundamental mysteries of nature, but are merely "bookkeeping" artifacts created by our arbitrary human systems of measurement (meters, kilograms, seconds).
2. Lesson 2 (Reconstruction): With the "bookkeeping" set aside, the second lesson reveals where the "real physics" lies. Students learn to "un-compile" familiar SI formulas to uncover the simple, invariant, dimensionless ratios that represent the true physical laws.
3. Lesson 3 (Generation): The final lesson synthesizes the previous insights, demonstrating that all these simple ratios are part of a single, interconnected, and predictive grid—a "Periodic Table of Physics." Students learn not just to analyze laws, but to generate them.

The overarching teaching strategy is to shift the student's identity. They begin as passive "explorers" who are told about disconnected laws discovered by others. They finish as active "cartographers" who possess a map of the entire structure of physical law, able to navigate its connections and predict what they will find in unexplored territory.

This guide provides the detailed scripts, pedagogical insights, and materials needed to execute the first lesson in this series.

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Part 1: Lesson Plan - The Constant Mythology

This foundational lesson is the most critical component of the entire series. Its strategic purpose is to create profound cognitive dissonance by challenging a sacred cow of physics—the fundamental nature of constants. The goal is to frame the standard explanation as a "mythology" and present this lesson as the "revelation" that explains the myth. By showing that these "mysteries" are solvable through a simple algorithm, you make students receptive to a completely new way of thinking about the subject.

Category	Details
Objective	Students will be able to derive any dimensional constant in physics from first principles and explain the difference between a unit-dependent scaling factor and a true, dimensionless constant of nature.
Time Allotment	50 minutes
Prerequisites	Introductory Physics (First Year Undergrad) / Advanced High School Physics
Materials	Whiteboard or Projector; Handout with "Dimensional Imbalance Table" and "Practice Problems".

1.1 (0:00 - 5:00) | The Setup: A Deliberate Deception

Pedagogical Goal The purpose of this opening is to intentionally create cognitive dissonance. You will frame the standard explanation of constants as a "mythology," immediately positioning yourself as the instructor who will reveal the hidden truth, thereby capturing the students' full attention.

Teacher's Script & Actions

Begin by writing on the board and asking the class directly:

"Why does G have this bizarre value and these insane units? Where does this number come from?"

After fielding a few standard answers ("it's measured," "it's a fundamental property of the universe"), acknowledge them as the textbook explanations. Then, deliver the premise for the day:

"Today, I'm going to show you that everything we just said is wrong. G is not a mystery. It's not fundamental. And by the end of this class, you will be able to calculate it, and every other constant, from scratch. The 'mystery' is a historical accident."

Whiteboard Layout

• The formula for Newton's Law of Universal Gravitation: F = G · m₁m₂ / r²
• The accepted value for G: G ≈ 6.674 × 10⁻¹¹ m³ kg⁻¹ s⁻²

Instructor's Insight: By calling the units "insane," you validate the students' unspoken intuition that the standard form is weird and arbitrary. This builds trust and positions you as someone who will finally make it make sense, rather than someone who will just ask them to memorize it.

1.2 (5:00 - 15:00) | The Algorithm: The Key to the Kingdom

Strategic Purpose The goal here is to replace a profound "mystery" with a simple, mechanical recipe. The power of this segment lies in reframing constants from immutable properties of nature to practical properties of our human measurement system.

The 4-Step Algorithm for Finding Any Constant

1. Step 1: Write the formula without the constant. Start with the pure proportionality that Newton identified: F = m₁m₂ / r².
2. Step 2: Check the dimensional balance. Analyze the units on both sides of the equation.
   • Left side (Force): [M L T⁻²]
   • Right side: [M² L⁻²]
   • They do not match.
3. Step 3: Find the imbalance. Determine the exact dimensional factor needed to multiply the right side by to make it dimensionally equivalent to the left side: [L³ M⁻¹ T⁻²].
4. Step 4: Fill the imbalance with the fixes. Introduce the Planck units (l_P, m_P, t_P). Frame them not as mystical entities, but as the practical fixes for our mis-scaled rulers—the error bars between how we scale our measurements and the actual ratios of the universe. The constant is simply the combination of these fixes that corrects our man-made formula. For G, this is G = l_P³ / (m_P · t_P²).

Teacher's Script & Actions

Erase the constant from the original formula.

"Let's start from what Newton actually wrote: a simple, clean proportionality. This part, F ~ m₁m₂ / r², is the PHYSICS. Everything else is just bookkeeping. Let's do the books."

After working through the 4-step algorithm, deliver the triumphant conclusion:

"That's G. That's the entire secret. It's the ratio of the fixes for our mis-scaled rulers, arranged to fix the imbalance in our man-made formula. The weird number 6.674 × 10⁻¹¹ is just what this ratio equals when you use meters, kilograms, and seconds."

Instructor's Insight: Present this as a simple trick, not a complex proof. The framing is crucial; concepts like "Physics vs. Bookkeeping" are powerful analogies. Demystify Planck units by presenting them as a practical "parts bin" used to build conversion factors.

1.3 (15:00 - 30:00) | The Pattern: Applying the Algorithm Everywhere

Pedagogical Goal The objective is to build overwhelming, undeniable proof for the algorithm's universality. This is achieved through speed and repetition. You must move quickly through these derivations to create the sense that this is a universal tool, not a special case. This culminates in the "punchline"—the analysis of the Fine Structure Constant, α—which proves the algorithm's honesty.

Rapid-Fire Derivations Move quickly through the following derivations on the board to demonstrate the algorithm's power and simplicity:

• Speed of Light (c): Start with l ~ t. The imbalance is [L T⁻¹]. The fix is l_P / t_P. Done.
• Planck's Constant (h): Start with E ~ f. The imbalance is [M L² T⁻¹]. The fix is m_P · l_P² / t_P. Done.
• Boltzmann's Constant (k_B): Start with E ~ T. The imbalance is [M L² T⁻² K⁻¹]. The fix is m_P · l_P² / (t_P² · T_P). Done.

The Critical Contrast: The Fine Structure Constant (α) Apply the same algorithm to the dimensionless Fine Structure Constant, α.

1. Ask the class what the dimensional imbalance is. The answer is [1]—it's already balanced.
2. Ask, "How many Planck units do we need to fill a null imbalance?" The answer is zero.
3. Deliver the key insight: "This is why α is different. It's not a bookkeeping constant. It's a real piece of physics. It's a number the universe actually cares about."

Teacher's Script

After the rapid-fire derivations, pause for effect and pose a rhetorical question to the class:

"Four different 'fundamental mysteries' of the universe. Four applications of the same 60-second algorithm. Are we seeing a pattern?"

Instructor's Insight: The α example is the punchline. By showing a case where the algorithm yields "nothing," you prove that the algorithm is honest. It cleanly separates the unit artifacts from the true, dimensionless constants. This is the moment the entire lesson clicks for the students.

1.4 (30:00 - 45:00) | The Practice: Students Use the Tool

Strategic Importance This is the most critical segment of the lesson. Students must use the algorithm themselves to transfer ownership of the concept. The act of personal discovery is what achieves the "Aha!" moment, cementing the idea far more effectively than any lecture. This is no longer something they were told; it's something they can do.

Classroom Activity

1. Distribute the worksheet with practice problems.
2. Instruct students to work in pairs to apply the 4-step algorithm to derive the constants for:
   • Coulomb's Law (k_e)
   • Wien's Displacement Law (b)
3. Circulate and guide the students, prompting them through the steps as needed.
4. Once most pairs are finished, review the answers on the board, having students walk the class through their own derivation steps.

1.5 (45:00 - 50:00) | The Conclusion: The New Worldview

Purpose The conclusion brings the lesson full circle. It directly answers the question posed in the opening and empowers students with a new, critical worldview about the nature of physics. You are giving them a new and more honest way to do science.

Teacher's Script & Final Takeaway

To close the lesson, deliver a clear and memorable summary.

"So, let's go back to our original question. Why does G have that crazy value? Because our rulers—the meter, the kilogram, the second—are ridiculously scaled compared to the universe's natural rulers. The constant isn't telling us about gravity; it's telling us about the awkwardness of our own tools."

End with a powerful call to action:

"The physics is in the simple proportions. The complexity is in the constants we invent to translate those proportions into our human measurement systems. Your job, as the next generation of scientists, is to learn to see past the bookkeeping and find the simple physics underneath. Don't let anyone tell you a conversion factor is a mystery."

With this new framework established, students are now prepared to explore where the real physics has been hiding.

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Part 2: Lesson Plan - The Real Physics

Having deconstructed the "bookkeeping" of dimensional constants, this lesson's strategic purpose is to reconstruct a new understanding. This is the promised revelation: a glimpse of the elegant, simple physical laws that were hidden beneath the complex SI formulas. It definitively answers the question, "If constants are just conversion factors, where is the real physics?"

Category	Details
Objective	Students will be able to derive a physical law in its invariant, dimensionless form and explain that the fundamental content of physics lies in these pure ratios.
Time Allotment	50 minutes
Prerequisites	"The Constant Mythology" lesson
Materials	Whiteboard or Projector; Handout with a list of Planck unit equivalents.

2.1 (0:00 - 5:00) | The Recap & The Next Question

Pedagogical Goal This opening validates the students' intellectual curiosity by building directly on the previous lesson's conclusion. It recaps the "bookkeeping" concept and immediately poses the next logical question, framing this lesson as the anticipated answer.

Teacher's Script & Whiteboard Actions

Start with a quick, powerful recap. While speaking, write the Planck unit definitions for the major constants on the board.

"Last time, we proved that every dimensional 'constant' is a conversion factor... a property of our rulers, not a property of the universe. This is the bookkeeping."

G = l_P³ / (m_P · t_P²) c = l_P / t_P h = m_P · l_P² / t_P

Then, pose the central question for the day:

"So, if this is all just bookkeeping... where is the PHYSICS? What are the actual, real laws of the universe, if not these formulas?"

2.2 (5:00 - 15:00) | The Algorithm for Finding the Real Physics

Strategic Purpose The goal is to provide a new algorithm that acts as a powerful tool to "un-compile" complex SI formulas. This process reveals the simple, elegant "source code" of the physical law that was obscured by the constants.

The 3-Step Algorithm for Finding the Invariant Law Use the dramatic example of Hawking Radiation to demonstrate the process.

1. Step 1: Start with the familiar SI formula. Write the intimidating formula on the board: T = ħc³ / (8πGk_B M).
2. Step 2: Substitute the Planck unit definitions for every constant. Systematically replace ħ, c³, G, and k_B with their equivalents in terms of m_P, l_P, t_P, and T_P. The resulting equation should look like a "horrifying mess" of Planck units. This is an intentional and theatrically effective step.
3. Step 3: Cancel everything. The algebra will reveal the truth. Carefully and systematically cancel out all the Planck units. After the algebraic dust settles, the complex mess will reduce to a strikingly simple expression: T = T_P / (8πM).

Teacher's Script & The Big Reveal

After completing the cancellation, rewrite the result in its normalized, proportional form: T_nat ~ 1 / M_nat.

"This is the REAL physics. The actual law of Hawking radiation is that temperature is inversely proportional to mass. That's it. The entire mess of ħc³ / Gk_B was nothing but the composite conversion factor needed to say that simple sentence in the clumsy language of kilograms and Kelvin."

Instructor's Insight: You are teaching students to "un-compile" the code of physics. They are learning to reverse-engineer the complex "machine code" (the SI formula) to find the simple, elegant "source code" (the invariant law). The mess of canceling Planck units is theatrical and effective—it's like a magician revealing the mundane mechanics behind a grand illusion.

2.3 (15:00 - 30:00) | The Invariant Laws of Nature

Pedagogical Goal The objective here is to build an overwhelming case for this new perspective by creating a "greatest hits" list. By revealing the simple, one-line truths behind the most famous and intimidating equations in physics, you demonstrate the universal power of this method.

Table of Examples

SI Formula	Invariant Law	The Law in Simple English
E = mc²	E_nat = m_nat	"Energy and mass are the same thing."
r_s = 2GM/c²	r_nat = 2 m_nat	"The size of a black hole is proportional to its mass."
λ = h/p	λ_nat = 1 / p_nat	"Wavelength is inversely proportional to momentum."

2.4 (30:00 - 45:00) | The Practice: Finding the Real Law

Strategic Importance This puzzle-like activity reinforces the "un-compiling" skill. It forces students to look past the surface complexity of a formula to find its essential proportional core, solidifying their understanding through active problem-solving. They are now actively doing the work of a natural philosopher.

Classroom Activity

1. Distribute a worksheet with two columns: one listing constant-laden SI formulas and the other listing simple, invariant proportionalities.
2. Instruct students, working in groups, to match each SI formula to its corresponding invariant law.
3. They must use the 3-step algorithm to prove the connection. Example pairings include:
   • Newton's Gravitation (F = G m₁m₂/r²) matches F_nat ~ m₁_nat m₂_nat / r_nat².
   • The Stefan-Boltzmann law (j = σT⁴) matches j_nat ~ T_nat⁴.

2.5 (45:00 - 50:00) | The Conclusion: What is a Physicist?

Purpose The goal of this conclusion is to redefine the identity of a physicist for the students. You will shift their perception of the discipline from one of mastering complex calculations to one of finding simple, underlying patterns, giving them a new and more exciting mission.

Teacher's Script & Final Takeaway

Point to the list of simple invariant laws on the board.

"For the last 100 years, we have trained physicists to be expert bookkeepers. They are masters of the SI formulas, the constants, the conversion factors. They are incredibly good at calculating with our human rulers."

"But the real job of a physicist, the job of Newton and Einstein, is not to be a bookkeeper. It is to be a ."

"The universe is not complex. It is breathtakingly simple. The complexity is an illusion created by our tools. Your job is not to memorize the illusions. Your job is to find the simple patterns they are hiding."

This new mission statement provides the perfect motivation for the final lesson, where students learn to use these patterns to build the entire map of physics.

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Part 3: Lesson Plan - The Periodic Table of Physics

This capstone lesson transforms students from analysts of existing laws into generators of new ones. Its strategic purpose is to demonstrate that all of physics is a single, interconnected, and predictive system. By organizing the simple ratios from Lesson 2 into a coherent grid, students learn to navigate and even predict the form of physical laws.

Category	Details
Objective	Students will use the Planck Equivalence Chain to systematically derive all pairwise relationships between fundamental physical quantities and predict the form of "undiscovered" physical laws.
Time Allotment	50 minutes
Prerequisites	"The Constant Mythology" & "The Real Physics" lessons
Materials	Projector; Handout with a blank version of the periodic grid.

3.1 (0:00 - 10:00) | From Ratios to a Unified Chain

Pedagogical Goal This segment builds the final conceptual bridge, moving students from a collection of individual invariant ratios to a single, unified "Chain of Equivalence." This chain is presented as the foundational "source code" from which all of physical law is derived, a moment of profound simplification.

Teacher's Script & Whiteboard Actions

Recap the key invariant laws from the previous lesson by writing them on the board.

E_nat = m_nat E_nat = f_nat m_nat = 1/l_nat

"But look at this. If Energy is equivalent to Mass, and Energy is equivalent to Frequency, what does that imply about Mass and Frequency?"

Guide the class to state the transitive property: m_nat = f_nat.

"This means they are all part of a single, unified Chain of Equivalence. This is the deep structure of the universe. Every fundamental law in physics is just a statement comparing two links in this chain."

Write the full chain on the board:

T/T_P = f·t_P = m/m_P = l_P/l = E/E_P = p/p_P = ...

3.2 (10:00 - 25:00) | Introducing the Grid: The Periodic Table of Law

Strategic Purpose The goal is to introduce the central, powerful metaphor of the "Periodic Table of Physics." This provides a compelling visual for the organization, structure, and, most importantly, the predictive power of physical law.

Revealing the Grid

1. Display the periodic grid on the projector. Explain its structure: the rows and columns represent fundamental quantities (T, f, m, l, E, p, F), and each cell contains the law connecting them.
2. "This is the Periodic Table of Physics. It's not an analogy; it's a precise description of the structure of physical law."

Demonstration Process

• First, show known laws to build credibility. Point to the intersection of the Mass row and the Energy column. "What connects mass and energy? We look at the cell." Highlight E=mc². "It's right there. The grid organizes what we already know." Do the same for Length and Momentum, showing the de Broglie relation.
• Then, dramatically reveal a "predicted" law to create the hook. Point to an empty-looking cell, like the intersection of Temperature and Mass. "The grid is telling us there must be a direct relationship between Temperature and Mass. The structure demands it. Has anyone heard of the 'Temperature-Mass Law'?" They haven't. This creates the intrigue needed for the next step.

Instructor's Insight: The dramatic reveal of the "predicted" but unnamed laws is the hook. It shows the students that the grid is not just a filing cabinet for old laws, but a machine for generating new ones.

3.3 (25:00 - 40:00) | The Generative Algorithm: Filling in the Blanks

Pedagogical Goal This active discovery phase positions students as physicists who are generating new laws. It proves that derivation is not an act of creative genius but a mechanical process of extracting information from the system's inherent structure.

The 4-Step Generative Algorithm Demonstrate the algorithm using the Temperature-Mass relationship as an example.

1. Step 1: Write the Equivalence. From the unified chain, we know T/T_P = m/m_P.
2. Step 2: Solve for the Target. Isolate the target variable: T = m · (T_P / m_P).
3. Step 3: Find the Constant. The term in parenthesis is the constant. "We can find this in two ways. The messy way is to substitute all the Planck unit definitions and cancel them out. The easy way is to use dimensional analysis. What are the units of T_P / m_P? That's [K/kg]. What combination of c, h, G, and k_B gives us [K/kg]? The answer is c²/k_B."
4. Step 4: Write the Final Law. Combine the pieces: T = m · c²/k_B.

Classroom Activity

1. Distribute the handout with the blank grid.
2. Assign student pairs a specific "predicted" cell to solve, such as the relationship between Force and Frequency.
3. Have them use the 4-step generative algorithm to derive the law for their cell and fill it into their grid.
4. Ask teams to present their findings to the class.

3.4 (40:00 - 50:00) | The Conclusion: The Map of Physics

Purpose This final conclusion reframes the history of science and solidifies the students' new role. They are no longer explorers stumbling upon disconnected facts but "cartographers" who possess a complete map of the system.

Teacher's Script & Final Takeaway

Display the full, completed grid on the screen.

"Look at this grid. Out of 21 fundamental relationships, we only gave famous names to about 8 of them. The rest were always there, implied by the structure, but we never bothered to look because we were focused on the 'known' laws as separate discoveries."

"A hundred years ago, we were explorers wandering in a forest, stumbling upon amazing discoveries like E=mc² and E=hf and thinking they were all unique. Today, you have seen the satellite map of the entire forest. You see that all the paths are connected. You see the structure."

Deliver the final, empowering message that completes the three-lesson arc:

"Your job is no longer to be an explorer. Your job is to be a cartographer. To map this structure, understand it, and use it to navigate anywhere you want to go."

Instructor's Insight: The "Explorer vs. Cartographer" analogy is a powerful redefinition of the student's role. It gives them a sense of mastery and higher perspective. They are not just learning the old discoveries; they are learning the system that generates all discoveries, past and future.

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Appendix: Reproducible Instructor & Student Materials

This section outlines the essential handouts required to execute the lessons effectively.

A.1: Handout for Lesson 1

Dimensional Imbalance Table A table should be provided here listing the fundamental dimensions (M, L, T, K) for common physical quantities like Force, Energy, Momentum, etc., to speed up Step 2 of the algorithm.

Practice Problems

1. Coulomb's Law: The electrostatic force between two charges is proportional to the product of the charges and inversely proportional to the square of the distance between them (F ~ q₁q₂ / r²). Using the 4-step algorithm, derive the dimensional constant k_e.
2. Wien's Displacement Law: The peak wavelength of emitted radiation from a black body is inversely proportional to its temperature (λ_max ~ 1/T). Using the 4-step algorithm, derive the dimensional constant b.

A.2: Handout for Lesson 2

Planck Unit Equivalents A handout should be provided listing common physical quantities (Energy, Mass, Length, Time, Temperature) and their corresponding Planck unit equivalents (e.g., M_nat = M_si / m_P, E_nat = E_si / E_P). This will be used in Step 2 of the "un-compiling" algorithm.

Matching Worksheet A two-column matching exercise should be provided. The left column lists several constant-laden SI formulas. The right column lists their corresponding simple, invariant forms. Students must draw lines connecting the pairs. Examples include:

SI Formula	Invariant Form
j = σT⁴ (Stefan-Boltzmann)	j_nat ~ T_nat⁴
F = G m₁m₂/r² (Gravitation)	F_nat ~ m₁_nat m₂_nat / r_nat²
... (add more pairs)	...

A.3: Handout for Lesson 3

The Blank Periodic Table of Physics A handout should be provided containing a blank grid. The rows and columns should be labeled with the following fundamental physical quantities: Temperature (T), Frequency (f), Mass (m), Length (l), Energy (E), Momentum (p), and Force (F). Students will use this worksheet to fill in the laws they derive during the generative classroom activity.