Slope Detectives Teacher Guide Slope Detectives
Teacher Facilitation Guide
8th Grade Math
Objective
Students will clarify misconceptions regarding negative slope versus positive speed/rate of change. By the end of this lesson, students should be able to explain why a rate of change can be positive (magnitude) even when its slope is negative (direction/decreasing).
Materials Needed
Slope Detectives Slides
Direction Detectives Activity
Scissors & Glue (for cards)
Slope Detectives Exit Ticket
Instructional Routine
5 min Warm-Up
"Can you drive a car at -60 mph?"
Expected Student Response: "No, you'd just be going 60 mph in reverse."
Teacher Point: In physics and real-world math, the negative sign often tells us direction or decreasing amounts, while the speed remains a positive magnitude.
10 min Video
Video: "Using Equations from Word Problems"
Focus on the Runner Problem (1:47 - 2:37)
Pause at 2:10: Ask: "The equation shows a -5. Does that mean she is running backwards? How should we describe her speed?"Key Takeaway: Distance is decreasing (negative slope), but speed is 5 m/s (positive rate).
20 min Activity
Direction Detectives Card Sort
Students work in pairs to sort real-world scenarios into "Positive Slope" (increasing) or "Negative Slope" (decreasing).
Teacher Prompt during activity:
"If the water is leaving the pool at 10 gallons per minute, why do we write the slope as -10 in an equation?"
10 min Reflection
Exit Ticket: The Burning Candle
Students analyze \(h = 10 - 2t\) to identify speed vs. slope. This checks for the primary misconception one last time before students leave.
Red Flags to Watch For
The Error
Student says the speed of the runner is "-5 m/s".
The Fix
Remind them that speed is "how fast," which is always positive. The negative just means the distance to the goal is getting smaller.
Slope Detectives Slides Case File: 8th Grade Math
Slope
Detectives
Investigating the mystery of negative rates and positive speeds.
Warm-Up Inquiry
"Can you drive a car at -60 mph?"
Think & Discuss
What does a negative sign mean in math?
What does it mean in the real world?
The Mission
Today, we are uncovering a common math trap:
The Misconception:
"If the slope is negative, the speed must be negative too."
Learning Goals
1 Identify slope as a rate of change.
2 Understand that the sign shows direction while the value shows speed.
Case Study: The Runner
Video Evidence
The Scenario:
Kayla is running a 400m race. Her distance from the finish line is modeled by:
\(d = -5t + 400\)
Watch for:
How does the narrator explain the "-5"?
Embedded media
Source: "Using Equations from Word Problems" by Justin
Analysis of Evidence
Slope: -5
The distance from the finish line is decreasing by 5 meters every second.
Speed: 5
The runner is moving at a positive rate of 5 meters per second.
The Verdict
"The negative sign tells us the direction (getting closer). The number itself tells us the speed ."
Direction Detectives
Activity Instructions:
1
Sort
Organize the scenario cards into Positive or Negative slopes.
2
Identify
Write the specific "Rate of Change" units for each case.
3
Justify
Explain why a scenario has a negative slope but a positive speed.
Direction Detectives Activity Direction Detectives
Case Activity: Scenario Sort
Agent Name & Date
Your Assignment
Cut out the evidence cards on the following pages. Sort them into Positive Slope (the value is increasing) or Negative Slope (the value is decreasing). For each card, determine the Rate of Change (units).
Positive Slope
The total amount is INCREASING
Negative Slope
The total amount is DECREASING
Evidence Cards
✂️ Cut along the dotted lines
Case #001
A garden hose fills a 50-gallon tub at a rate of 5 gallons every minute.
Units (Rate of Change):
Case #002
Your phone battery starts at 100% and loses 12% charge for every hour of gaming.
Units (Rate of Change):
Case #003
An airplane at 30,000 feet begins its descent, dropping 1,500 feet every minute.
Units (Rate of Change):
Case #004
A teenager earns $15 for every hour they spend mowing lawns.
Units (Rate of Change):
Case #005
You have $200 in your savings account and spend $8 each day on lunch.
Units (Rate of Change):
Case #006
A hiker climbs a mountain, gaining 400 feet of elevation for every mile walked.
Units (Rate of Change):
Slope Detectives Exit Ticket The Final Verdict
Exit Ticket
Lead Detective
Date
Case File: The Burning Candle
The equation for a candle's height (\(h\)) in inches after (\(t\)) hours is:
\(h = 10 - 2t\)
1. According to the equation, what is the speed of burning?
2 inches per hour
-2 inches per hour
2. Justify your answer using evidence from today's investigation.
Explain why you chose that value. What does the negative sign tell us?
Case Closed. Return this ticket to the facilitator.
✂️ Cut Here for Two Copies ✂️
The Final Verdict
Exit Ticket
Lead Detective
Date
Case File: The Burning Candle
The equation for a candle's height (\(h\)) in inches after (\(t\)) hours is:
\(h = 10 - 2t\)
1. According to the equation, what is the speed of burning?
2 inches per hour
-2 inches per hour
2. Justify your answer using evidence from today's investigation.
Explain why you chose that value. What does the negative sign tell us?
Case Closed. Return this ticket to the facilitator.
