Roller Coaster Design Worksheet Roller Coaster Blueprint
Pre-Calculus: Intervals of Increase & Decrease
Name: __________________________
Date: ___________________________
Part 1: Track Design
Instructions: Using your assigned Criteria Card , draw your 2D roller coaster track on the coordinate plane below. Use a single, continuous function. Ensure your "start" and "end" points match the domain on your card.
y
x
-10-8-6-4-20246810
1086420-2-4-6-8-10
Part 2: Interval Analysis
Reviewer: __________________________
Swap papers with your partner. Analyze their roller coaster track and identify the intervals below using Interval Notation . Remember: use x-values only!
Intervals of Increase (Going Up)
Intervals of Decrease (Going Down)
Intervals of Constant Behavior (Flat Track)
Verification Check
Look at the intervals you wrote above. If you add up the lengths of all these intervals, should it equal the total domain of the coaster? Why or why not?
Roller Coaster Criteria Cards Coaster Specs
Engineering Design Criteria Cards
Blueprint Set A
The "Mega Drop"
Domain: \([-8, 8]\)Requirement 1: Must increase from \(x = -8\) to \(x = -4\).Requirement 2: Must include a "Big Drop" (decrease) for exactly 6 units of x.Requirement 3: Must finish with a flat (constant) section for at least 2 units of x.
Blueprint Set B
The "Camel Hump"
Domain: \([-10, 6]\)Requirement 1: Start with a constant loading platform from \(x = -10\) to \(x = -7\).Requirement 2: Must have two separate intervals of increase.Requirement 3: The highest peak must occur at \(x = 2\).
Blueprint Set C
The "Switchback"
Domain: \([-6, 10]\)Requirement 1: Must decrease on the interval \([-6, -2]\).Requirement 2: Must have exactly one interval of constant behavior that is 4 units long.Requirement 3: Must increase for all values of \(x > 4\).
Blueprint Set D
The "V-Valley"
Domain: \([-5, 5]\)Requirement 1: Must have a relative minimum at \(x = 0\).Requirement 2: Decreasing on \((-5, 0)\).Requirement 3: Increasing on \((0, 5)\).
Engineer Note:
When drawing your coaster, ensure the y-values (height) are realistic for a roller coaster. The intervals must be strictly followed based on the x-axis values provided.
Roller Coaster Slides Pre-Calculus Unit 1
Roller Coaster
Calculus
Increase • Decrease • Constant
Warm-Up: Notation Check
5 Minutes
Inequality vs. Interval
x > 5 \((5, \infty)\)
-2 < x ≤ 10 \((-2, 10]\)
x ≤ -1 \((-\infty, -1]\)
The Golden Rule
When describing WHERE a function increases or decreases, we use Interval Notation representing X-VALUES .
"Open dots and infinity always get parentheses. Closed dots get brackets."
Video Analysis
5:00 - 6:46
Embedded media
Focus Question:
"The narrator emphasizes using X-values. Why don't we use Y-values to describe these intervals?"
Roller Coaster Design
25 Minutes
1
Design
Grab your Criteria Card and draw your coaster on the grid. Follow the specs exactly!
2
Trade
Swap papers with your partner. You are now the safety inspector.
3
Analyze
Identify all intervals of Increase , Decrease , and Constant behavior.
Safety Check: Common Errors
Error #1: The Y-Trap
Students often use the height (y) to describe the interval. Remember: Intervals are chunks of the domain (x).
Error #2: Bracket Confusion
At a "peak" or "valley," is the graph increasing? No. It's changing. Many mathematicians use parentheses \(( \text{ } )\) for these intervals to show the point of change is excluded.
Roller Coaster Teacher Guide Engineer's Handbook
Teacher Facilitation Guide
Grade 11 | Pre-Calculus | Unit 1
Lesson Objective
Students will define and represent intervals of increase, decrease, and constant behavior using correct interval notation. They will demonstrate the ability to translate visual graph features into algebraic domain intervals.
Key Vocabulary
• Interval Notation
• Union Symbol (\(\cup\))
• Relative Extrema
• Constant Behavior
Instructional Flow
00-05
Warm-Up: Notation Review
Project Slide 2. Quickly poll students on the difference between brackets and parentheses. Ensure they recall that \(\infty\) always takes a parenthesis.
05-15
Video Viewing & Discussion
Watch Problem 6 (5:00-6:46).
Teacher Prompt: "The narrator mentioned using x-values only. Why is it technically incorrect to say a function increases on the interval \([y_1, y_2]\)?" (Answer: We are describing the domain over which the behavior occurs, not the height itself.)
15-40
Activity: Roller Coaster Design
Hand out Criteria Cards and Worksheets . Students design for 10-12 mins, then swap for analysis.
Monitoring Tips: Look for students using "curvy" lines for increase/decrease but "straight/horizontal" lines for constant. Remind them it must be a function (Vertical Line Test).
40-45
Closure: Common Pitfalls
Discuss Slide 5. Emphasize that at the exact peak, the derivative is zero (though don't use that term unless ready)—the graph is neither increasing nor decreasing, hence parentheses are preferred at endpoints of these intervals.
Criteria Card "Ideal" Intervals
Set A (Mega Drop)
Possible Intervals:
Inc: \((-8, -4)\)
Dec: \((-4, 2)\)
Const: \([6, 8]\)
Set B (Camel Hump)
Possible Intervals:
Const: \([-10, -7]\)
Inc: \((-7, -3) \cup (0, 2)\)
Dec: \((-3, 0) \cup (2, 6)\)
Differentiation Strategies
Scaffolding (Struggling Learners)
Provide a "Trace and Color" version of the worksheet where they color increase intervals in green and decrease in red before writing the notation.
Extension (Advanced Learners)
Challenge students to write the piecewise equations for their constant and linear portions of the coaster.
