Commander Teacher Guide Current Commander
Teacher Facilitation Guide
Objective
Apply vector concepts (relative velocity) to write, solve, and graph one- and two-variable algebraic inequalities.
Duration
45-50 Minutes
Materials
Slide Deck, River Map, Student Worksheet, Calculators
Lesson Timeline
05 min
Warm-up: Escalator Physics
Introduce the concept of "Net Speed" using the downward escalator thought experiment. Establish the formula: \( \text{Net Speed} = \text{Person Speed} - \text{Escalator Speed} \).
10 min
Video & Direct Instruction
Watch the "Leilani Boat" segment (4:16-6:29). Use the slides to pause and discuss the inequality \( b - c \leq 30 \). Discuss the "ground speed" vs. "engine speed" distinction.
25 min
Main Activity: Current Events
Distribute the River Map. Students analyze 4 river sections. They assume a boat max speed and determine which sections are "Safe Zones" based on local speed limits. They must write an inequality for each section.
10 min
Wrap-up & Extension
Compare results. Discuss the extension: how does the inequality change when moving *with* the current? (Addition vs. Subtraction).
Teaching Tips & Keys
Key Misconception: The "Subtraction" Logic
Students often think moving "against" a faster current is more dangerous for speed limits. In reality, a fast current *decreases* your ground speed, making it easier to stay under the limit but harder to actually progress forward. Clarify that "Speed Limit" refers to ground speed (velocity relative to the shore).
Mathematical Answer Key (Main Activity)
Section 1: The Narrows (Current: 12 mph, Limit: 15 mph)
Boat Max: 25 mph. Inequality: \( 25 - 12 \leq 15 \Rightarrow 13 \leq 15 \). (TRUE: Safe Zone)
Section 2: Serpent's Bend (Current: 4 mph, Limit: 15 mph)
Boat Max: 25 mph. Inequality: \( 25 - 4 \leq 15 \Rightarrow 21 \leq 15 \). (FALSE: Violation Zone)
Section 3: Iron Gorge (Current: 20 mph, Limit: 10 mph)
Boat Max: 25 mph. Inequality: \( 25 - 20 \leq 10 \Rightarrow 5 \leq 10 \). (TRUE: Safe Zone, but barely moving!)
Section 4: The Delta (Current: 2 mph, Limit: 25 mph)
Boat Max: 25 mph. Inequality: \( 25 - 2 \leq 25 \Rightarrow 23 \leq 25 \). (TRUE: Safe Zone)
Differentiation Strategy
Support: Provide a scaffolded inequality template: \( (\text{Boat}) - (\text{Current}) \leq (\text{Limit}) \).Extension: Ask students to find the *exact* maximum boat throttle allowed for a section if the current is fixed.
Commander Slides CURRENT COMMANDER
Navigating Relative Velocity & Inequalities
WARM-UP: The Escalator
Imagine you are walking UP an escalator that is moving DOWN.
If you walk faster than the escalator, you go up.
If you walk slower, you go down.
NET SPEED = YOUR SPEED - ESCALATOR SPEED
VIDEO: Boat Problems
Embedded media
Watch For:
How does Leilani know she might get a ticket?
Key Variable:
Actual Speed (Ground Speed)
THE PHYSICS MODEL
Engine Speed (\(b\))
How fast the motor is spinning the propeller.
-
Current Speed (\(c\))
How fast the water is moving against you.
GROUND SPEED = \(b - c\)
MISSION: CURRENT EVENTS
You are the pilot of a supply boat. Your max throttle speed is 25 mph. Use the map to determine which sections of the river are legal to travel in without exceeding local speed limits.
Analyze the Map
Write Inequalities
Solve & Shade
"What if we turn around?"
If you are traveling WITH the current (downstream), how does your Ground Speed inequality change?
GROUND SPEED = \(b\) ? \(c\)
River Run Map Handout Navigation Chart: Sector 4-B
Upper Rogue River Basin
Scale: 1:50,000
Projection: Mercator
Legend
Safe Zone
Violation Zone
Current Direction
Vessel Type
Type-S Supply Boat
Max Throttle (\(b_{max}\))
25 MPH
Warning
Violations \(\geq \$500\)
1. The Narrows
Against Current (\(c\)): 12 mph
Speed Limit: 15 mph
2. Serpent's Bend
Against Current (\(c\)): 4 mph
Speed Limit: 15 mph
3. Iron Gorge
Against Current (\(c\)): 20 mph
Speed Limit: 10 mph
4. The Delta
Against Current (\(c\)): 2 mph
Speed Limit: 25 mph
MISSION: Navigate upstream from The Delta to the basin headwaters.
Current Events Worksheet CURRENT EVENTS
Relative Velocity & Inequality Analysis
Pilot:
Date:
Part 1: The Escalator Effect
You walk 3 mph up an escalator that is moving 2 mph down. How fast are you actually moving relative to the floor?
Equation
3 - 2 = 1
Net Speed
1 mph
Direction
Upward
Define "Net Speed" in your own words:
Part 2: Boat Patrol
Watch the video segment. Leilani is moving against a current.
Write the inequality used in the video:
≤
Explain why Leilani’s actual speed is her Boat Speed minus the Current Speed :
Part 3: River Run Analysis
Your supply boat has a Max Engine Speed (\(b\)) of 25 mph . You are traveling upstream (against the current). For each section on your Map, write and solve an inequality to determine if you are in a Safe Zone or a Violation Zone .
River Section Inequality (\(25 - c \leq \text{Limit}\)) Result Status 1. The Narrows
|
Safe
Violation |
| 2. Serpent's Bend |
|
|
Safe
Violation |
| 3. Iron Gorge |
|
|
Safe
Violation |
| 4. The Delta |
|
|
Safe
Violation |
Extension: Turning Around
A storm is coming. You must turn around and travel with the current back to the Delta. If you keep your engine at 25 mph, what is your new Ground Speed through Section 1?
Are you still in the "Safe Zone" for Section 1 if you travel with the current? Why or why not?
