🧭 3D-Compass: Navigating the Spherical World

Hands-on STEM Workshop for Middle/High School Math & Science

Diagram illustrating the spherical coordinate system (rho, theta, phi) relative to the X, Y, Z axes.

🎯 Alignment with EBR/Louisiana Student Standards (LSSS)

This workshop directly addresses key standards in Mathematics (High School), Earth and Space Science, and Engineering Design, critical to the Louisiana Student Standards framework.

Standards Alignment Matrix

Workshop Component Discipline LSSS Code & Description
Phase 2 & 3: Coordinate Conversion Mathematics (F-TF) **LSSS: F-TF.A.2 (Trigonometric Functions):** Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions...
Activity D: 3D Pathfinding Mathematics (N-VM) **LSSS: N-VM.A.1 (Vector Quantities):** Recognize vector quantities have both magnitude and direction, represent vector quantities by directed line segments...
Activity C: Error Analysis Mathematics (S-ID) **LSSS: S-ID.A.4 (Interpreting Data):** Use the mean and standard deviation of a data set to fit it to a normal distribution... (Focus on measurement precision).
Activity F: Celestial Targeting Science (HS-ESS) **LSSS: HS-ESS1-4 (Earth's Place in the Universe):** Use mathematical or computational representations to determine how the orbital motions of celestial bodies... are predictable.
Activity E: Great Circle Distance Science & Math **LSSS: HS-ESS1-2 (The Universe and Its Stars):** Construct an explanation of the Big Bang theory based on astronomical evidence... (Applies spherical geometry to Earth).
Phase 1, 3, & Activity A Engineering (ETS) **LSSS: HS-ETS1-2 (Engineering Design):** Design a solution to a complex real-world problem by breaking it down into smaller, more manageable problems...
Activity B: Code Conversion Technology (CS) **LSSS: K-12.CS.A.2 (Abstraction):** Use and apply abstraction to manage complexity. (Coding for coordinate conversions).

Reinforcement of Science and Engineering Practices (SEP)

All modules reinforce core SEP skills vital for success in STEM:

  • **Developing and Using Models:** The 3D-Compass acts as a physical model for abstract coordinates.
  • **Using Mathematics and Computational Thinking:** Direct application of trigonometry and vector math.
  • **Constructing Explanations and Designing Solutions:** Core problem-solving throughout the workshop.
  • **Analyzing and Interpreting Data:** Used in error analysis and precision measurement.

🎒 Materials & Resources (Per Team)

  • **1 x 3D-Compass Instrument** (The main device)
  • **1 x Scientific Calculator** (or device access for trigonometric functions)
  • **1 x Ruler/Tape Measure** for distance ($\rho$) scaling
  • **1 x Set of "Target Drone" Coordinate Cards** (unique mission data)
  • **String/Yarn** (to represent the vector radius $\rho$ during testing)
  • **Worksheet** with formula space and conversion tables
  • Access to a scaled, 3D grid display (e.g., a large classroom corner or foam model)

🛠️ Workshop Structure (Approx. 90 Minutes)

Phase 1: Explore & Define (15 Minutes)

Goal: Understand the need for 3D coordinate systems.

  • **Introduction:** Introduce the 3D-Compass and the challenge of location in 3D space.
  • **Problem Definition:** Students define the problem: Converting a spherical target location ($\rho, \theta, \phi$) into rectangular coordinates ($x, y, z$) for recovery vehicle navigation.

Phase 2: Learn & Practice (35 Minutes)

Goal: Master instrument use and trigonometric conversion.

  • **Hands-On Operation:** Teams practice setting Azimuth ($\phi$) with the compass and Declination ($\theta$) with the sliding arrow.
  • **Math Application:** Review the conversion formulas (using the astronomical/Dec convention):

Spherical ($\rho, \theta, \phi$) to Rectangular ($x, y, z$)

$$ \begin{align*} x &= \rho \cos(\theta) \cos(\phi) \\ y &= \rho \cos(\theta) \sin(\phi) \\ z &= \rho \sin(\theta) \end{align*} $$

Note: $\theta$ here is **Elevation/Declination** from the horizon ($xy$-plane).

Phase 3: Design & Test (30 Minutes)

Goal: Apply EDP to solve the unique mission problem.

  • **Design & Calculate:** Teams take their "Target Drone" coordinates and perform the full trigonometric conversion to calculate $x, y, z$.
  • **Build & Test:** Teams physically set the calculated direction on the 3D-Compass and use the string/grid to visually confirm the calculated $x, y, z$ location.

Phase 4: Share & Reflect (10 Minutes)

Goal: Evaluate solutions and connect to careers.

  • **Reflection:** Teams share their calculated results and discuss any challenges encountered (e.g., sign conventions in different quadrants).
  • **STEM Connection:** Discuss how aerospace engineers, cartographers, and programmers use these coordinate systems daily for mapping, navigation, and robotics.

🌟 Challenge Extensions & Advanced Activities

For longer workshops (120+ minutes) or advanced students, incorporate these activities focused on deeper engineering and computational thinking.

Activity A: The "Field Mount" Engineering Challenge

**Focus:** Engineering Design, Materials Science, Stability

**Task:** Design and prototype a stable, level mount or tripod system for the 3D-Compass that can be deployed quickly on uneven ground (simulated using classroom materials like straws, tape, paperclips, etc.). Teams must apply structural engineering principles to maximize stability and minimize deployment time. The system should maintain the protractors perpendicularity and the compass level during use. This directly engages the **Engineering Design Process** by requiring iteration, materials testing, and optimization.

Activity B: The Code Conversion Challenge

**Focus:** Technology, Computational Thinking, Abstraction

**Task:** Students often encounter two major conventions for spherical coordinates. Using **pseudocode** or a programming language like **Python**, create a function that takes coordinates in one convention (e.g., Astronomical $\rho, \theta_{dec}, \phi_{ra}$) and outputs the coordinates in the other convention (Physics $\rho, \theta_{zenith}, \phi_{azimuth}$). This requires students to identify the mathematical relationship between the two polar angles ($\theta_{zenith} = 90^\circ - \theta_{dec}$) and write logical, executable steps, bridging the math concept to computer science.

Activity C: Precision and Error Analysis

**Focus:** Mathematics, Data Analysis, Engineering Tolerance

**Task:** The instructor sets a single, unknown target. Each team takes three independent measurements of the spherical coordinates ($\rho, \theta, \phi$). Teams must calculate the **mean** coordinate set, the **standard deviation** for each variable, and determine the **angular error** in their device's accuracy. Discuss how precision impacts real-world applications (e.g., landing a rover, aiming a telescope). This reinforces the importance of significant figures and measurement consistency.

Activity D: Sequential 3D Pathfinding

**Focus:** Technology, Kinematics, Vector Analysis

**Task:** Teams are given a list of 5 sequential spherical coordinates representing a drone's flight path (Waypoint 1 to Waypoint 5). They must first convert all 5 waypoints to rectangular coordinates. Then, they calculate the **distance and direction vectors** between consecutive points (Waypoint 1 to 2, 2 to 3, etc.). Plotting the path on the 3D grid helps visualize the sequence of movements required for autonomous navigation, connecting coordinate systems to vector mathematics.

Activity E: Global Navigation and Great Circle Distance

**Focus:** Geography, Trigonometry, Real-World Navigation

**Task:** Use the spherical coordinates of two major cities (e.g., Baton Rouge, LA and Paris, France, where latitude is $\theta$ and longitude is $\phi$). Teams must calculate the shortest flight path distance along the Earth's surface using the **Spherical Law of Cosines**. This requires students to look up coordinates, recognize the Earth as a sphere ($\rho = 6371 \text{ km}$), and apply the final trigonometric formula below. This demonstrates the critical math used by pilots and shipping navigators.

Great Circle Distance ($D$)

$$ D = R \cdot \Delta\sigma $$ Where $R$ is Earth's radius and $\Delta\sigma$ is the angular separation: $$ \Delta\sigma = \arccos(\sin(\theta_1)\sin(\theta_2) + \cos(\theta_1)\cos(\theta_2)\cos(\phi_2 - \phi_1)) $$

Note: All angles must be in radians for calculation.

Activity F: Celestial Targeting: Finding a Star

**Focus:** Astronomy, Earth & Space Science, Frame of Reference

**Task:** Provide teams with the **Celestial Equatorial Coordinates** (Right Ascension, Dec) for a bright star (e.g., Polaris: Dec $\approx +89^\circ$, RA $\approx 2.5 \text{ hours}$). Teams set the Dec angle ($\theta$) on the 3D-Compass. **Challenge:** Discuss why the Azimuth (RA/$\phi$) angle set on the compass will only point to the star at a specific moment in time. This introduces the concept of the **Local Sidereal Time (LST)**, which is the missing variable needed to convert static Celestial Coordinates (RA/Dec) to the observer's dynamic **Horizontal Coordinates** (Azimuth/Altitude), used for real-time telescope targeting.

Specific Example: Pointing to Betelgeuse

**Scenario:** Observing from Baton Rouge, LA ($\text{Lat} \approx 30.45^\circ$) on November 7th at 9:00 PM CST.

  • **Star's Fixed Equatorial Coordinates:**
    • **Declination (Dec):** $\delta \approx +7.4^\circ$
    • **Right Ascension (RA):** $\alpha \approx 5^h 55^m$
  • **Required 3D-Compass Settings (Horizontal Coordinates):**
    • **Declination ($\theta$):** Set the Dec protractor arrow to **$+25^\circ$ (Altitude)**.
    • **Azimuth ($\phi$):** Rotate the compass half to **$105^\circ$ (East-Southeast)**.
  • **Key Takeaway:** The RA of $5^h 55^m$ must be offset by the **Local Sidereal Time** and the observer's latitude to calculate the required Azimuth of $105^\circ$. This highlights the challenge of dynamic coordinate conversions required for sophisticated astronomical navigation.

✅ Assessment Rubric Criteria

This rubric assesses student performance across the core workshop objectives, aligning with LSSS Mathematical (F-TF, N-VM) and Engineering (ETS1) standards.

Criterion (Focus SEP) Exceeds Expectations (4 pts) Meets Expectations (3 pts) Needs Improvement (1-2 pts)
1. Mathematical Accuracy (LSSS: F-TF.A.2) All spherical-to-rectangular conversions are calculated correctly with proper notation and unit consistency (radians/degrees). Conversions are mostly correct (1-2 minor arithmetic errors) and the correct formulas are applied. Incorrect formulas are used, or major calculation errors result in an unusable coordinate solution.
2. Model Application & Precision (SEP: Developing Models) The 3D-Compass is set with high precision (within $\pm 1^\circ$ of target) and the resulting vector is accurately plotted on the 3D grid. The 3D-Compass is set correctly (within $\pm 3^\circ$) and the target direction is clearly established on the grid. Angles are set outside of reasonable tolerance ($\pm 5^\circ$) or the instrument is used incorrectly (e.g., compass misalignment).
3. Engineering Design (LSSS: HS-ETS1-2) Prototype mount is robust, stable, and maintains instrument level on uneven terrain. Team documents iteration and optimization choices. Prototype mount provides basic stability and is functionally usable, but lacks durability or quick-deployment features. Prototype fails to maintain stability or level, or the design process does not reflect organized testing and iteration.
4. Conceptual Understanding (SEP: Explanations) Team clearly articulates the difference between fixed (RA/Dec) and dynamic (Azimuth/Altitude) coordinate systems and explains the role of the rotation offset. Team correctly identifies the need for two different coordinate systems but the explanation of the rotational offset is vague or incomplete. Team fails to differentiate between the celestial and horizontal coordinate frames, treating all angles as static measurements.