Tidal Trends Teacher Guide Tidal Trends Facilitation Guide
High School Functions | Tier 2 Intervention | CO HS.F-TF.B.5
Objective
Students will be able to identify the amplitude, midline, and frequency of a periodic context and use these values to construct a trigonometric function model \(y = a \sin(b(x-c)) + d\) or \(y = a \cos(b(x-c)) + d\).
Key Vocabulary
• Amplitude
• Midline
• Period
• Frequency (\(b\))
• Periodic
Facilitation Steps
1
Connect to Context (5 mins)
Slide 1-3
Introduce the concept of periodic data using tides or sound waves. Ask: "What repeats in this data? How high does it go? Where is the center?"
2
The Component Breakdown (10 mins)
Slide 4-6
Explicitly model how to find the Midline (\(k\)) and Amplitude (\(a\)) first. These are the most concrete. Use the formula: \(k = \frac{max + min}{2}\) and \(a = max - k\).
3
Solving for Frequency (10 mins)
Slide 7-9
This is often the hardest step for intervention students. Emphasize that \(b\) is not the period, but a calculation: \(b = \frac{2\pi}{Period}\). Practice identifying one full cycle on a graph.
4
Guided Model Building (15 mins)
Model Builder Worksheet
Hand out the Model Builder Worksheet. Students use the step-by-step checklist to build their first model from tidal data. Check for accuracy at the "Midline/Amplitude" stage before they move to frequency.
5
Peer Review Progress Monitoring (10 mins)
Peer Review Comparison Card
Students swap models and use the review criteria to verify a partner's work. Use the feedback to address common misconceptions immediately.
Common Misconceptions
Confusing Period and Frequency: Students often plug the period directly in for \(b\). Remind them: "Period is the time for one wave; \(b\) is how we adjust the math to fit that time."Amplitude from Midline vs. Total Height: Students may think amplitude is Max - Min. Remind them it is the distance from the center to the peak. Tidal Trends Slides Tidal Trends
Modeling Periodic Data
Tier 2 Intervention | High School Functions
What is Periodic?
Events that repeat at regular intervals.
Ocean Tides
Ferris Wheels
Breathing Rate
Wave Anatomy
Midline (\(k\))
The horizontal line halfway between the peak and the trough.
\(k = \frac{max + min}{2}\)
Amplitude (\(a\))
The distance from the midline to either the max or min.
\(a = max - k\)
Frequency (\(b\))
The number of cycles over \(2\pi\). Found using the period.
\(b = \frac{2\pi}{\text{Period}}\)
The Step-by-Step Goal
\(y = {\color{#60a5fa} a} \sin({\color{#60a5fa} b}(x)) + {\color{#60a5fa} k}\)
Step 1
Find Midline
Step 2
Find Amplitude
Step 3
Calculate Frequency
The Trap!
Wait! The Period and Frequency (\(b\)) are NOT the same thing.
Example:
If the wave repeats every 12 hours...
\(b = \frac{2\pi}{12} = \frac{\pi}{6}\)
Period
How long for 1 wave
Frequency (b)
The math value in our model
Tidal Trends Worksheet Model Builder
Tidal Trends
Tier 2 Intervention
Student:
Date:
Modeling Checklist
1. Find Midline (\(k\)) Halfway between max & min
2. Find Amplitude (\(a\)) Distance from midline to peak
3. Calculate Frequency (\(b\)) Use \(b = \frac{2\pi}{\text{Period}}\)
4. Write Model Plug it all in!
The Scenario: Crescent Bay
At Crescent Bay, the high tide reaches a height of 14 feet and the low tide drops to 2 feet. The time between one high tide and the next is exactly 12 hours.
Max Height
14 ft
Min Height
2 ft
Period
12 hrs
Step 1: The Midline (\(k\))
Formula: \(\frac{\text{Max} + \text{Min}}{2}\)
\(k =\)
Step 2: The Amplitude (\(a\))
Formula: \(\text{Max} - k\)
\(a =\)
Step 3: Calculating Frequency (\(b\))
We know the Period is 12 hours. Use that to find \(b\).
Equation Value
\(b =\) _____
Final Model
Substitute your values into the sine function below:
\(y =\)
\(a\)
\(\sin(\)
\(b\)
\(x) +\)
\(k\)
\(y =\) __________________________________________________
Tidal Review Card Model Review Card
Progress Monitoring & Peer Feedback
Tidal Trends
Reviewer:
Partner being reviewed:
Partner's Equation
\(y = \) __________________________________________________
Quality Control Check
k
The Midline
Correct
Needs Fix
Did they use \(\frac{max + min}{2}\)? Is it 8?
a
The Amplitude
Correct
Needs Fix
Did they find the distance from center (8) to top (14)? Is it 6?
b
The Frequency
Correct
Needs Fix
Did they divide \(2\pi\) by 12? Is it \(\pi/6\)?
Great Work:
One Suggestion:
Student is ready for the exit challenge!
Teacher Signature: _________________
