Shadow Stalkers Slides Project Launch
Shadow Stalkers
Mastering the math of indirect measurement and geometric similarity.
The Mission
How do you measure a flagpole, a building, or a tree without a ladder or a giant tape measure?
01
Identify the target unreachable height.
02
Use a known object (a "gnomon") to create a shadow.
03
Apply Similarity Theorems to calculate the unknown.
Unknown (H) Shadow (S)
Why does it work?
Angle-Angle (AA) Similarity Postulate
The Sun's Rays
The Sun's rays hit the Earth at the same angle for objects near each other.
\( \angle \text{Sun}_1 \cong \angle \text{Sun}_2 \)
The AA Proof
If two angles of one triangle are congruent to two angles of another, the triangles are SIMILAR (\(\sim\)).
Proportionality follows similarity.
Beyond Shadows
SSS and SAS Applications
SSS
Side-Side-Side
All corresponding sides must be proportional.
\[ \frac{a}{A} = \frac{b}{B} = \frac{c}{C} = k \]
SAS
Side-Angle-Side
Two sides proportional and the included angle is congruent.
\[ \frac{a}{A} = \frac{b}{B} \text{ and } \angle C \cong \angle Z \]
Project Specs
Outdoor Lab
Measure shadows for 3 objects: One known (Gnomon), two unknown.
Calculations
Determine the Scale Factor (\(k\)) and solve for height (\(H\)).
The Proof
Write a formal proof justifying your similarity method.
Sun Sighting Pack Sun Sighting Pack
Indirect Measurement Project
Name: ____________________
Date: __________________
Project Objective
Use Similar Triangles to calculate heights. Identify a "Gnomon" (known height), measure its shadow, and use the Scale Factor (k) to solve for unreachable objects.
Toolkit
• Meter Stick
• "Gnomon" (Known Object)
• Sunshine / Flat Ground
• Calculator
Part 1: The Field Data
Object Description Height (H) Shadow (S) Ratio (H/S) Gnomon (Control) KNOWN MEASURED ________ Object A: ____________ X MEASURED ________ Object B: ____________ Y MEASURED ________
Site Sketch: Label Right Angles and Sun Angle (\(\theta\))
Part 2: Proportional Math
1. Calculate the Scale Factor (k)
Use your Gnomon data: k = Height / Shadow
2. Solve for Object A Height
Show your proportion setup and the final value of X.
3. Solve for Object B Height
Show your proportion setup and the final value of Y.
Part 3: Formal Proof
Prove that the two triangles formed by the Sun's rays are similar using the Angle-Angle (AA) Postulate .
Statements Reasons 1. Height \(\perp\) Ground; Gnomon \(\perp\) Ground GIVEN 2. \(\angle \text{Base}_1 \cong \angle \text{Base}_2 = 90^\circ\) 3. Sun's rays hit Earth in parallel lines GIVEN (Distant Source) 4. \(\angle \text{Elevation}_1 \cong \angle \text{Elevation}_2\) CORR. ANGLES POST. 5. \(\triangle \text{Object} \sim \triangle \text{Gnomon}\)
Analysis Challenge
If you measured Object A's shadow at 10 AM but didn't measure the Gnomon until 2 PM, why would your height calculation be wrong? Use Geometric Reasoning to explain the change in similarity.
Similarity Logic Worksheet Similarity Logic
Practice: AA, SSS, and SAS Proofs
NAME: __________________ PER: _____
Task 1: Identify and State
State the similarity criteria (AA, SSS, SAS) and the similarity statement.
A B C 90° 40° D E F 90° 40°
Postulate: ___________
\(\triangle ABC \sim \triangle\) ___________
X Y Z 60° 8 10 P Q R 60° 4 5
Postulate: ___________
\(\triangle XYZ \sim \triangle\) ___________
Task 2: SSS Verification
Compare the triangles below. Show the simplified ratios for all sides.
Triangle 1
S1=6, S2=8, S3=12
Triangle 2
S1=15, S2=20, S3=30
Ratio 1 (15/6)
Ratio 2 (20/8)
Ratio 3 (30/12)
Similar? ________ Scale Factor: ________
Task 3: Indirect Logic
A surveyor uses nested triangles to find a width. Complete the proof.
A B C D E
Given: DE || BC
Statements Reasons 1. DE 2. ∠ADE ≅ ∠ABC 3. ∠AED ≅ ∠ACB 4. ΔADE ~ ΔABC
Project Appraisal Rubric Project Appraisal Rubric
Teacher Resource • Shadow Stalkers Project
Scores are assigned per criterion based on accuracy, logic, and presentation.
Criterion Exemplary (10) Proficient (8) Developing (6) Beginning (4) Field Log All data recorded with units. Measurements are precise. All data recorded. Minor unit errors. Incomplete data. Poor measurement units. Significant missing data or impossible values. Math Setup Proportions set up correctly. Scale factor is accurate. Proportions correct. Minor arithmetic error. Incorrect proportion setup for one object. Substantial errors in logic and setup. Proof Logically sound. Correct criterion (AA, SSS, SAS) used. Logic is sound but lacks minor formal details. Identifies criteria but fails to prove logic steps. Incorrect criterion or no formal proof attempted. Drawing Sketch shows triangles, right angles, and rays. Well-labeled. Shows triangles and labels most parts. Triangles drawn but lack labels or context. Diagram is missing or geometrically incorrect. Analysis Examines how time affects elevation and similarity. Recognizes time matters but lacks geometric detail. Vague response with no geometric reference. Response is missing or factually incorrect.
Final Grade Scale
A (90%+) 45-50
D (60%+) 30-34
B (80%+) 40-44
F (<60%) 0-29
C (70%+) 35-39
Total Score & Feedback
/ 50
