Wave Master Slides
Wave Watchers
Modeling Periodic Phenomena
Small Group Intervention: Algebra 2 / Functions
The Periodic Pattern
Many things in the world repeat over time:
- Ocean Tides (High to Low)
- Ferris Wheel Height
- Sound Waves & Heartbeats
"Periodic" means it happens in cycles.
1 The Midline
The Horizontal Center
The horizontal line that runs exactly halfway between the maximum and minimum values.
Midline (y) = [Max + Min] / 2
In our equation: \(y = a \sin(bx) + \mathbf{d}\)
MIDLINE
2 Amplitude
The Vertical Distance
The distance from the midline to the peak (or the midline to the trough).
Amplitude (a) = Max - Midline
Amplitude is always positive!
In our equation: \(y = \mathbf{a} \sin(bx) + d\)
AMPLITUDE
3 Period
The Cycle Length
The horizontal distance for the graph to complete one full cycle before repeating.
B = \(2\pi\) / Period
Think: "How long until I see the same exact spot again?"
In our equation: \(y = a \sin(\mathbf{b}x) + d\)
ONE PERIOD
Building the Signal
Step-by-Step Guide
- 1 Identify the Max and Min heights.
- 2 Find the Midline (Vertical Shift, \(d\)).
- 3 Find the Amplitude (\(a\)).
- 4 Find the Period and calculate \(b\).
The Equation Blueprint
\(y = \mathbf{a} \sin(\mathbf{b}x) + \mathbf{d}\)
Write this down in your guided notes!
Signal Decoder Worksheet
Signal Decoder
Modeling Periodic Graphs
Name:
Date:
Midline (\(d\))
The "Average" Height
(Max + Min) / 2
Amplitude (\(a\))
Height from Midline
Max - Midline
Frequency (\(b\))
Cycle Constant
\(2\pi\) / Period
1
Phase 1: Component Breakdown
\(2\pi\) \(\pi\) 3 -3
Max / Min: 3 / -3
Midline (\(d\)):
Amplitude (\(a\)):
Period:
4 2 10 5 0
Midline (\(d\)):
Amplitude (\(a\)):
Period:
Value of \(b\):
2
Phase 2: Assemble the Equation
Use the parameters you found in Phase 1 to write the full equation: \(y = a \sin(bx) + d\)
Problem 3: The Ocean Tide
A buoy in the harbor moves up and down with the waves. The high tide is 12 feet, and the low tide is 4 feet. It takes 6 hours to go from one high tide to the next.
Midline (\(d\)):
Amplitude (\(a\)):
Period:
\(b\) value:
Final Equation:
y =
Problem 4: Ferris Wheel Spin
You board a Ferris wheel at the midline. The maximum height is 50 meters, and the minimum height is 10 meters. The wheel makes one full rotation every 40 seconds.
Midline:
Amplitude:
Period:
\(b\) value:
Final Equation:
y =
Frequency Check Exit Ticket
Frequency Check
Exit Ticket • Tier 2 Progress Monitoring
Student Name
Date
1. Examine the periodic graph below and identify the parameters.
8 5 2 40 80
Midline (\(d\))
Amplitude (\(a\))
Period (\(P\))
2. Write the final equation for this graph in the form \(y = a \sin(bx) + d\).
(Hint: Remember to calculate \(b = 2\pi / P\))
y =
How confident do you feel with these steps?
Lost
Getting it
Mastered
Tidal Trends Teacher Guide
Tidal Trends
Teacher Facilitation Guide
Topic: Trig Modeling (HS.F-TF.B.5)
Learning Objective
Students will model periodic phenomena by identifying the amplitude, period, and midline from visual representations and translating those parameters into a trigonometric equation.
Group Focus
- Tier 2 Intervention
- Scaffolded Practice
- 3-5 Students
Common Misconceptions
Peak-to-Peak vs. Amplitude
Students often think the total vertical distance (max to min) is the amplitude. Remind them amplitude is half that distance (from the center).
Period vs. B-Value
Students often put the Period directly into the equation. Emphasize that \(b = 2\pi / \text{Period}\). If the period is 10, the equation uses \(2\pi/10\) or \(\pi/5\).
Suggested Flow (30-40 Minutes)
0-5 min
Hook: Wave Discovery
Use Slide 2 to discuss real-world cycles. Ask: "What happens if a Ferris wheel turns faster?" (Period changes).
5-15 min
Guided Instruction
Walk through Slides 3-6. Use color coding on a white board if possible (Red for midline, Green for amplitude) to match the slide theme.
15-30 min
The Signal Decoder Activity
Students work on the worksheet. Focus on Problem 2 and 3—ensure they are subtracting Max and Min correctly to find the midline.
30-35 min
Exit Ticket & Check
Collect the Frequency Check. If students struggle with the \(b\) calculation, do a quick "lightning round" of \(2\pi/P\) examples.
Scaffolding Strategies
Visual Aids
Have students physically trace one cycle of the graph with a highlighter before measuring the period. Seeing where it "restarts" is the biggest hurdle.
Verbal Prompts
Instead of asking "What is the midline?", ask "Where is the elevator floor if the ride goes up to 10 and down to 0?" (Midpoint is 5).
Quick Key: Worksheet
Prob 1: d=0, a=3, P=\(2\pi\), y=3sin(x)
Prob 2: d=5, a=5, P=4, b=\(\pi\)/2, y=5sin(\(\pi\)/2 x)+5
Prob 3 (Buoy): d=8, a=4, P=6, b=\(\pi\)/3
Prob 4 (Wheel): d=30, a=20, P=40, b=\(\pi\)/20
