Slope Cipher Challenge Activity
Alpine Cartography Unit • Early Finisher Extension
Slope Cipher Challenge
Independent Mission
Navigator:
Date:
Station / Period:
Expedition Directive:
A mountain expedition team transmitted scrambled distress coordinates. Calculate the slope \(m = \frac{y_2 - y_1}{x_2 - x_1}\) for each trail segment to decode the scrambled letters, then unscramble the word to reveal their rendezvous location!
1 Part I: Ridge Slope Decoder • Anagram Challenge
Find each slope, match letter, then unscramble below
Trail 1: K → L (-2, 8) to (1, 2)
\(m =\) Letter: ___
Trail 2: D → E (5, -2) to (5, 7)
\(m =\) Letter: ___
Trail 3: Z → P (3, -4) to (-1, 8)
\(m =\) Letter: ___
Trail 4: A → B (-4, 3) to (2, -1)
\(m =\) Letter: ___
Trail 5: S → W (-6, -1) to (4, 4)
\(m =\) Letter: ___
Trail 6: G → T (-3, -5) to (6, -5)
\(m =\) Letter: ___
Cipher Key:
\(-2\) = M Undef. = U \(-3\) = T \(-\frac{2}{3}\) = S \(\frac{1}{2}\) = I \(0\) = M
Scrambled Letters (1–6):
→ Unscrambled Destination:
2 Part II: The Collinear Hazard Test
Points lie on one straight line if and only if \(m_{AB} = m_{BC}\)
Trail Assessment Alpha Ridge Path
Waypoints: \(A(-3, -7)\), \(B(1, 1)\), and \(C(3, 5)\)
Slope \(m_{AB}\):
Slope \(m_{BC}\):
Are waypoints collinear?
YES NO
Trail Assessment Beta Crevasse Zone
Waypoints: \(D(-2, 6)\), \(E(2, 0)\), and \(F(6, -5)\)
Slope \(m_{DE}\):
Slope \(m_{EF}\):
Are waypoints collinear?
YES NO
Alpine Survey Protocol 104-B Page 1 of 2 • Turn over for Cabin Plotting & Slope Types
Slope Cipher Challenge • Phase II
Mountain Cabin Plotting & Slope Types
Cartographer Level: Advanced
3 Part III: Plotting Mountain Cabins & Analyzing Slope Types
Plot coordinates, connect cabin trails, and classify slope types
x y 2 4 6 -2 -4 -6 2 4 6 -2 -4 -6
Plot, label each cabin letter, and connect each pair with a straight edge
Cabin Registry Coordinates:
Trail 1: Pine → Eagle
Cabin A(-4, -3) • Cabin B(2, 1)
Trail 2: Bear → Timber
Cabin C(-2, 5) • Cabin D(4, -1)
Trail 3: Glacier → Valley
Cabin E(-5, -2) • Cabin F(3, -2)
Trail 4: Granite → High Peak
Cabin G(3, -4) • Cabin H(3, 4)
| Trail Pair |
Visual Observation (Direction / Line Look) |
Type of Slope (Positive, Negative, Zero, or Undefined) |
Slope Value \(m = \frac{y_2 - y_1}{x_2 - x_1}\) |
| Cabin A → Cabin B (-4, -3) to (2, 1) |
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| Cabin C → Cabin D (-2, 5) to (4, -1) |
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| Cabin E → Cabin F (-5, -2) to (3, -2) |
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| Cabin G → Cabin H (3, -4) to (3, 4) |
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Cartographer's Quick-Recognition Strategy:
How can you instantly tell whether a trail between two cabins has a zero slope versus an undefined slope just by glancing at their coordinate numbers, before doing any math?
Mission Complete • Submit to Lead Navigator / Guest Educator Page 2 of 2
Slope Cipher Answer Key
Substitute & Teacher Reference Guide
Slope Cipher Answer Key
Master Solution Sheet
Alpine Cartography Extension
Substitute Teacher Quick Guide:
This extension is self-directed. In Part I, students find slopes that decode to scrambled letters: M - U - T - S - I - M. They must unscramble these letters to find the secret rendezvous destination: S - U - M - M - I - T. If a student's letters do not match, have them check their negative signs or check if they inverted rise and run.
1 Part I: Ridge Slope Decoder • Worked Solutions
Scrambled: MUTSIM → Unscrambled: SUMMIT
1. Trail K → L Letter: M
\(m = \frac{2 - 8}{1 - (-2)} = \frac{-6}{3}\)
\(m = -2\)
Simplifies to integer \(-2\).
2. Trail D → E Letter: U
\(m = \frac{7 - (-2)}{5 - 5} = \frac{9}{0}\)
\(m =\) Undefined
Division by zero = vertical cliff.
3. Trail Z → P Letter: T
\(m = \frac{8 - (-4)}{-1 - 3} = \frac{12}{-4}\)
\(m = -3\)
Steep downhill slope.
4. Trail A → B Letter: S
\(m = \frac{-1 - 3}{2 - (-4)} = \frac{-4}{6}\)
\(m = -\frac{2}{3}\)
Watch out for \(-(-4) = +4\).
5. Trail S → W Letter: I
\(m = \frac{4 - (-1)}{4 - (-6)} = \frac{5}{10}\)
\(m = \frac{1}{2}\)
Positive fractional slope.
6. Trail G → T Letter: M
\(m = \frac{-5 - (-5)}{6 - (-3)} = \frac{0}{9}\)
\(m = 0\)
Zero numerator = horizontal flat.
2 Part II: Collinear Hazard Test • Verification
Confirming straight paths vs. hazardous bends
Trail Alpha: \(A(-3, -7), B(1, 1), C(3, 5)\) COLLINEAR (YES)
Slope \(m_{AB}\): \(\frac{1 - (-7)}{1 - (-3)} = \frac{8}{4} = 2\)
Slope \(m_{BC}\): \(\frac{5 - 1}{3 - 1} = \frac{4}{2} = 2\)
Conclusion: Since \(m_{AB} = m_{BC} = 2\) and they share common point \(B\), all three waypoints lie on the exact same straight line.
Trail Beta: \(D(-2, 6), E(2, 0), F(6, -5)\) NOT COLLINEAR (NO)
Slope \(m_{DE}\): \(\frac{0 - 6}{2 - (-2)} = \frac{-6}{4} = -\frac{3}{2} = -1.5\)
Slope \(m_{EF}\): \(\frac{-5 - 0}{6 - 2} = \frac{-5}{4} = -1.25\)
Conclusion: Since \(-\frac{3}{2} \ne -\frac{5}{4}\), the slopes are different. The trail bends at point \(E\), indicating a hazard.
Section 1 & 2 Verified Page 1 of 2 • Cabin Plotting Solutions Overleaf
Slope Cipher Answer Key • Page 2 Solutions
Mountain Cabin Plotting & Slope Analysis
Sub Quick-Reference
3 Part III: Cabin Trail Slope Classification • Master Table
4 Key Slope Types Represented
| Trail Pair |
Visual Observation |
Slope Type |
Slope Calculation \(m\) |
| Cabin A → B (-4, -3) to (2, 1) |
Rises / slants upward from left to right (uphill trail). |
POSITIVE |
\(m = \frac{1 - (-3)}{2 - (-4)} = \frac{4}{6} = \mathbf{\frac{2}{3}}\) |
| Cabin C → D (-2, 5) to (4, -1) |
Falls / slants downward from left to right (downhill trail). |
NEGATIVE |
\(m = \frac{-1 - 5}{4 - (-2)} = \frac{-6}{6} = \mathbf{-1}\) |
| Cabin E → F (-5, -2) to (3, -2) |
Completely flat horizontal line (plateau / valley floor). |
ZERO |
\(m = \frac{-2 - (-2)}{3 - (-5)} = \frac{0}{8} = \mathbf{0}\) |
| Cabin G → H (3, -4) to (3, 4) |
Straight up and down vertical line (vertical cliff wall). |
UNDEFINED |
\(m = \frac{4 - (-4)}{3 - 3} = \frac{8}{0} \rightarrow \mathbf{Undefined}\) |
Cartographer's Quick-Recognition Strategy (Reflection Answer):
Zero Slope vs. Undefined Slope by Coordinate Inspection:
- Zero Slope: The \(y\)-coordinates are identical (\(y_1 = y_2 = -2\)). Because rise is \(y_2 - y_1 = 0\), the numerator is zero (\(\frac{0}{\Delta x} = 0\)), making it a flat horizontal line.
- Undefined Slope: The \(x\)-coordinates are identical (\(x_1 = x_2 = 3\)). Because run is \(x_2 - x_1 = 0\), the denominator is zero (\(\frac{\Delta y}{0}\)), which is mathematically undefined (vertical line).
Substitute Diagnostic • Common Student Traps to Spot:
1. Confusing 0 and Undefined:
Students often call Trail 3 "no slope" and Trail 4 "zero slope." Clarify: horizontal has slope = 0 (walking on flat ground); vertical has UNDEFINED slope (impossible cliff to walk).
2. Double Negatives in Formula:
In Cabin A to B: \(1 - (-3) = 1 + 3 = 4\) and \(2 - (-4) = 2 + 4 = 6\). Remind students to write parentheses around negative coordinates.
3. Plotting Coordinates (x, y):
Students who plot \((-4, -3)\) as 4 down and 3 left inverted \(x\) and \(y\). Remind: "Run on the ground (\(x\)) before you take the elevator (\(y\))."
Sub Master Key Complete • Store in Sub Folder / Lesson Binder Page 2 of 2
