Trends and Tides Slides Unit 1: Eco-Data Lab
TRENDS AND
TIDES
Visualizing the heartbeat of our planet through periodic data.
The Keeling Curve
Global CO2 concentrations since 1958.
The Big Question
There is a clear upward trend. But what causes the "zigzag" every year?
Linear Trend: The long-term rise.Periodic Oscillation: The seasonal cycle.
Identifying Periodic Data
Predictable Cycles
Periodic data repeats its values at regular intervals. If you know one cycle, you can predict the next.
Environmental Examples
Average Monthly Temperature
Daylight Hours
Ocean Tide Heights
Anatomy of the Curve
Maximum Minimum Cycle Length (Period)
1. The Extremes
Identify the highest (peak) and lowest (trough) values in your data set.
2. The Midline
The horizontal "average" that the curve oscillates around.
3. The Period
How long it takes to complete one full cycle before repeating.
Lab Mission: Sketching the Trend
Open your "Trends and Tides" worksheet. We will be plotting temperature data for Fairbanks, Alaska.
Step 1: Plot Data
Step 2: Identify Peaks
Step 3: Sketch Smooth Curve
Trends and Tides Worksheet Lab Report: Trends and Tides
Eco-Data Lab // Lesson 01: Periodic Foundations
Researcher:
Date:
Mission Objective
Analyze the relationship between time and temperature in Fairbanks, Alaska. Identify periodic trends and visually estimate the parameters of the environmental cycle.
Phase 1: Raw Data Collection
The following table shows the average high temperature (°F) for Fairbanks, Alaska, by month (Month 1 = January).
Month
1
2
3
4
5
6
7
8
9
10
11
Temp (°F)
-1
11
25
44
61
71
73
66
55
32
11
Month
12
13
14
15
16
17
18
19
20
21
22
Temp (°F)
1
-1
11
25
44
61
71
73
66
55
32
Phase 2: Visual Modeling
1. Plot the data points on the grid below. 2. Sketch a smooth periodic curve (a "best-fit" sine wave) that passes through or near the points.
Temperature (°F)
Time (Months)
80
40
0
-40
0
6
12
18
24
Phase 3: Data Analysis
1. Extremes: Estimate the Maximum and Minimum values from your sketch.
Maximum High:
Minimum Low:
2. Midline: Draw a horizontal line through the "middle" of your wave. What temperature is it at?
3. Periodicity: How many months pass between one peak and the next? Explain how you found this.
Investigator Note: Environmental data is rarely "perfect." Your curve might not hit every point exactly, but it should represent the general pattern. This is called a "best-fit" model.
4. Conclusion: Why do you think Fairbanks has such a wide temperature range compared to a coastal city like Miami?
Trends and Tides Teacher Guide Teacher Guide: Trends and Tides
Eco-Data Lab // Lesson 01: Foundations
Duration: 60-90 min
Lesson Objective
Students will distinguish between linear trends and periodic oscillations in real-world data. They will practice plotting discrete data points and sketching continuous curves to represent environmental cycles, visually identifying the maximum, minimum, midline, and period.
Key Vocabulary
Periodic Oscillation Midline Amplitude Period Extrema
Materials Needed
• Trends and Tides Slides
• Student Worksheets
• Colored Pencils (2 colors)
• Rulers
01 The Hook: The Keeling Curve (10 min)
Show Slide 2. Don't reveal the cause of the oscillation yet. Ask students: "Why does the CO2 level drop and rise every single year while still going up overall?"
Teacher Note: The "zigzag" is caused by seasonal photosynthesis. In the Northern Hemisphere spring/summer, forests (the "lungs" of the Earth) breathe in CO2, lowering the concentration. In winter, they breathe out. This introduces the idea of environmental cycles.
02 Concept: Identifying Periodic Data (15 min)
Discuss Slide 3. Ask students to brainstorm other things that repeat. Guide them toward environmental variables like tide heights, temperature, and daylight. Contrast this with non-periodic data like stock market prices or population growth.
03 Lab Activity: Sketching the Curve (30 min)
Distribute the worksheet. Have students plot the Fairbanks data points.
Encourage students to use a pencil for the first draft of the curve.
Remind them that the curve should be smooth—not "connect the dots" with straight lines.
Point out that the data spans 22 months, so they should see nearly two full cycles.
Misconception Alert
Linear vs. Periodic
Students often try to draw a straight line through periodic data if the trend is strong (like CO2). Emphasize that we are looking for the residuals or the "wiggle."
"Sharp" Waves
Students may draw zig-zags with sharp points at the max/min. Explain that natural processes usually change gradually, resulting in a smooth sine-like wave.
Expected Student Results (Fairbanks Data)
Estimated Max
~73°F
Estimated Min
~ -1°F
Midline
~36°F
Height and Midline Slides Lesson 02: Parameters
Height and
Midline
Moving from visual sketches to algebraic precision.
The Low Tide Trap
"Captain, when will the water be exactly in the middle?"
Mission Brief
A ship needs to know the "average" water level to safely clear a reef. You only have the High Tide and Low Tide measurements for the day.
High Tide 12.4 ft
Low Tide 2.2 ft
How do we find the exact...
Middle Point?
Total Height Range?
Distance from Middle?
THE MIDLINE (k)
The horizontal axis about which the graph oscillates.
\[k = \frac{\text{Max} + \text{Min}}{2}\]
It is the Average of the extremes. It shifts the entire graph vertically.
AMPLITUDE (A)
The vertical distance from the midline to a peak or trough.
\[A = \frac{\text{Max} - \text{Min}}{2}\]
!
Important: Amplitude is always positive! It measures distance, not direction.
Think of it as the "Stretch" of the wave. A large amplitude means high peaks and deep valleys.
Building the Model
y = A sin(...) + k
Vertical Stretch
Amplitude (A)
Vertical Shift
Midline (k)
Height and Midline Worksheet Lab Report: Height & Midline
Eco-Data Lab // Lesson 02: Vertical Parameters
Researcher:
Date:
Midline Formula
\[k = \frac{\text{Max} + \text{Min}}{2}\]
The vertical shift (center line) of the oscillation.
Amplitude Formula
\[A = \frac{\text{Max} - \text{Min}}{2}\]
The vertical stretch (distance from midline).
01 Case Study: San Francisco Tides
You are monitoring the water level at the Golden Gate Bridge. Below is the recorded data for high and low tides on October 12th.
High Tide (Max)
5.82 ft
Low Tide (Min)
-1.04 ft
Mission:
Calculate the Midline and Amplitude for this data set. Show your work.
A. Calculate Midline (k)
k =
B. Calculate Amplitude (A)
A =
02 Comparative Analysis
Calculate the parameters for these different environmental locations.
Death Valley, CA
Max Temp: 116°F | Min Temp: 38°F
Midline (k)
Amplitude (A)
Reykjavik, Iceland
Max Temp: 55°F | Min Temp: 28°F
Midline (k)
Amplitude (A)
Analysis Question:
Which location has a higher "vertical shift" (higher average temperature)? Which location has a larger "vertical stretch" (more extreme seasonal change)? Use your calculations to justify your answer.
Critical Thinking:
In Lesson 1, we visually estimated the midline for Fairbanks. How does having an algebraic formula change the way you trust your model?
Height and Midline Teacher Guide Teacher Guide: Height and Midline
Eco-Data Lab // Lesson 02: Vertical Parameters
Focus: Midline & Amplitude
Lesson Objective
Students will calculate the vertical parameters (midline and amplitude) of a trigonometric model using extreme values from a data set. They will apply these calculations to real-world scenarios including tidal data and regional temperature variations.
Skills Focus
• Calculating Midline (k)
• Calculating Amplitude (A)
• Handling negative values
• Interpreting "Vertical Shift"
High-Impact Teaching Tip
In the San Francisco Tides activity, the Low Tide is negative (-1.04) . Many students will struggle with the subtraction in the amplitude formula:
\[A = \frac{5.82 - (-1.04)}{2} = \frac{5.82 + 1.04}{2}\]
Remind students that they are finding the distance between two points on a vertical line. Subtraction is the tool we use to find that distance.
Case Study 1: San Francisco Tides
Midline (k):
k = (5.82 + -1.04) / 2 = 2.39 ft
Meaning: The average water level is 2.39 feet above the reference point.
Amplitude (A):
A = (5.82 - -1.04) / 2 = 3.43 ft
Meaning: The tide swings 3.43 feet above and below the average.
Case Study 2: Comparative Analysis
Location Midline (k) Amplitude (A) Death Valley 77°F 39°F Reykjavik 41.5°F 13.5°F
Discussion Prompt:
"Look at Death Valley and Reykjavik. Which city has a 'hotter' midline? Which city has a more 'extreme' amplitude? How do these numbers tell a story about the climate in each location?"
Direct Instruction: 15 min Guided Practice: 20 min Independent Analysis: 25 min
Shift and Scale Slides Lesson 03: Horizontal Parameters
SHIFT AND
SCALE
Mastering the X-axis: When and how cycles repeat.
The Sunrise Challenge
"Why does summer feel longer in Seattle than in San Diego?"
Both cities follow a 365-day cycle . Their graphs look like waves, but their peaks happen at the same time.
What defines how "fast" or "slow" a cycle repeats? How do we slide the graph to line up with the first day of the year?
Mission: Determine the Horizontal Scaling Factor (B) and the Phase Shift (h).
P
Period
The physical time for one cycle.
B
Scale Factor
How we shrink/stretch the x-axis.
h
Phase Shift
Moving the start of the wave.
The Period (P)
Natural Cycles
Annual Temp: 12 Months
Ocean Tides: ~12.4 Hours
Moon Phases: ~29.5 Days
Period is the unit of time on the ground. But math speaks in radians .
\[B = \frac{2\pi}{P}\]
Building the Model (Part 2)
y = A sin( B (x - h)) + k
The Phase Shift (h) tells us where the starting point of the sine wave is. For environmental data, this is often the time when the value is at its Midline and rising.
Investigator Pro-Tip
Use Sine If:
You want to start your model at the Midline (the "average" day).
Use Cosine If:
You want to start your model at the Peak (the "hottest" or "highest" day).
In environmental modeling, we often use Cosine because we can easily identify the day of the year with the maximum temperature or tide.
Shift and Scale Worksheet Lab Report: Shift & Scale
Eco-Data Lab // Lesson 03: Horizontal Parameters
Researcher:
Date:
Horizontal Calibration
Scaling Factor (B)
\[B = \frac{2\pi}{P}\]
Phase Shift (h)
Location of First Peak
(If using Cosine model)
01 Case Study: Seattle Daylight
Seattle, WA, experiences its longest day of the year (Summer Solstice) on June 21st (Day 172). On this day, there are 16 hours of daylight. The shortest day (Winter Solstice) occurs on December 21st with 8.4 hours of daylight.
Phase 1: Vertical Recap
Max (Daylight) 16 hrs
Min (Daylight) 8.4 hrs
Midline (k)
Amplitude (A)
Phase 2: Horizontal Scale
What is the Period (P) in days?
Calculate the B value:
Leave your answer in terms of \(\pi\).
02 Aligning the Model (Phase Shift)
We will use a Cosine Model since we know exactly when the Peak occurs (Day 172). In a standard cosine graph, the peak is at \(x = 0\). We need to shift it to \(x = 172\).
What is the Phase Shift (h)?
Phase 3: The Complete Equation
Assemble all your parameters into a single model for Seattle's daylight:
Final Model Construction
y = _____ cos(_____ (x - _____)) + _____
Predictive Check:
If you plug in x = 172 (June 21), what should the result for y be? Verify this with your equation.
Investigator Reflection:
If we wanted to model the daylight hours for a city in the Southern Hemisphere (where the seasons are flipped), how would our Phase Shift (h) change?
Shift and Scale Teacher Guide Teacher Guide: Shift and Scale
Eco-Data Lab // Lesson 03: Horizontal Parameters
Focus: Period & Phase Shift
Lesson Objective
Students will determine the horizontal scaling factor (B) using the period (P) of environmental cycles. They will learn to apply phase shifts (h) to align their trigonometric models with specific dates/times, specifically using the Cosine model for peak-data alignment.
Skills Focus
• Solving for B using 2π/P
• Identifying Phase Shift (h)
• Assembling the full model
• Sine vs. Cosine selection
Instructional Tip: The Cosine Shortcut
When modeling environmental data, it is almost always easier to use Cosine . Why? Because the phase shift is simply the x-coordinate of the maximum point.
"Students: If you know the day of the year with the most daylight (Day 172), and you use Cosine, your phase shift is just 172. If you tried to use Sine, you'd have to calculate exactly when the day was perfectly average and rising—which is much harder to find in a data table!"
Case Study: Seattle Daylight Key
Vertical Parameters:
Midline (k): (16 + 8.4) / 2 = 12.2
Amplitude (A): (16 - 8.4) / 2 = 3.8
Horizontal Parameters:
Period (P): 365 days
B-value: 2π / 365
Phase Shift (h): 172 (June 21 peak)
Final Model:
\(y = 3.8 \cos(\frac{2\pi}{365}(x - 172)) + 12.2\)
Misconception Alert
The "h" Sign Trap
Students often write \( (x + 172) \) for a shift to the right. Remind them that the formula is \( (x - h) \), so a shift of \( +172 \) looks like subtraction in the parentheses.
B-Value confusion
Ensure students understand that \( B \) is NOT the period. It is the scale factor that ensures one cycle finishes by \( x = P \).
Southern Hemisphere Reflection:
In the Southern Hemisphere, the peak daylight occurs in December (around Day 355). The phase shift would change to ~355, or the amplitude would become negative if we kept the June start date.
Digital Regression Slides Lesson 04: Computational Modeling
DIGITAL
REGRESSION
Letting the machines do the math: Accuracy and residuals.
Hand-Built vs. High-Tech
"Is our human model good enough?"
Your Model
Uses only the Maximum and Minimum values. It captures the overall shape but ignores the "noise" in between.
Sinusoidal Regression
Uses EVERY data point simultaneously. It calculates the line that minimizes the total distance from all points.
Sinusoidal Regression
01
Input the Data
List 1 (Time) and List 2 (Data Value).
02
Run the Algorithm
The calculator searches for the best A, B, h, and k.
03
Analyze Error
Compare the regression model to your manual sketch.
Calculator Output
y = a * sin(bx + c) + d
------------------
a = 4.218...
b = 0.017...
c = -2.941...
d = 12.155...
Measuring "Goodness of Fit"
Residuals
The vertical distance between an actual data point and the predicted value on the curve.
Residual
Smaller residuals = Better Model
Your Mission
Can we predict the high temperature for your next birthday? We'll use 10 years of historical data and run a regression model to find out.
Step 1: Gather Data
Step 2: Sinusoidal Regression
Step 3: Forecast!
Digital Regression Worksheet Lab Report: Digital Regression
Eco-Data Lab // Lesson 04: Computational Modeling
Researcher:
Date:
Mission Objective
Use technology to generate a sinusoidal regression model. Compare your "Extreme-Point" manual model to the "Global-Fit" digital model and calculate the residuals.
01 Local Temperature Data
Record the average monthly temperatures for your city (or use the provided regional data).
Jan (1)
Feb (2)
Mar (3)
Apr (4)
May (5)
Jun (6)
Jul (7)
Aug (8)
Sep (9)
Oct (10)
Nov (11)
Dec (12)
02 Battle of the Models
Manual Model (Extreme Points)
Equation:
Regression Model (Full Set)
SinReg Result:
03 Residual Investigation
Pick one month (e.g., April) and compare the actual data to the prediction made by both models.
Actual Data
Manual Prediction
Digital Prediction
Calculation: Calculate the Residual for both models.
(Residual = Actual - Predicted)
Manual Residual:
Digital Residual:
Investigator Conclusion:
"Why might a regression model be slightly off for a specific data point, even if it is considered the 'best fit' model for the entire year?"
Digital Regression Teacher Guide Teacher Guide: Digital Regression
Eco-Data Lab // Lesson 04: Computational Modeling
Focus: Technology & Error Analysis
Lesson Objective
Students will use graphing technology to perform sinusoidal regressions on real-world data sets. They will contrast the output of the regression algorithm with their manual models built from extreme points and quantify the model's accuracy using residuals.
Tech Specs
• Desmos or TI-84 (SinReg)
• Residual = Actual - Predict
• Model Optimization
The Core Insight
The Manual Weakness
The manual model only "trusts" two points (Max and Min). If those points are outliers or slightly irregular, the entire model is skewed.
The Digital Strength
Regression averages the error across all points. It might not pass exactly through the Max/Min, but it represents the "true" center of the data.
Technology Guide (Desmos)
// Step 1: Create a table with L1 and L2
// Step 2: Use the following regression command:
y1 ~ a sin(b(x1 - h)) + k
// Note: You may need to provide 'initial guesses' for a, b, h, and k if the regression fails to converge.
Misconception Alert
R-squared vs. Residuals
Students often think a "best-fit" line should go through every point. Remind them that environmental data has "noise" (unpredictable weather, extra-high tides). A residual of 0 is almost impossible in nature.
Radian vs. Degree Mode
Check that students' technology is in Radian Mode . Trigonometric modeling of continuous periodic data is fundamentally a radian-based calculation.
Data Gathering: 15 min Regression Tutorial: 15 min Analysis & Residuals: 30 min
Future Forecasts Slides Lesson 05: Extrapolation
FUTURE
FORECASTS
Using models to solve real-world problems and make safe decisions.
The Harbor Master's Dilemma
"When is it safe to bring the ship in?"
Scenario
A large cargo ship requires at least 10 feet of water to enter the harbor without hitting the seafloor.
Model & Solve
The Math Task
Given the model:
Find all x where y ≥ 10
This is a trigonometric inequality.
Phase 1: Prediction
Question:
"What will the water level be at 4:00 PM (x = 16)?"
Direct Evaluation
Simply plug in the value for x and solve for y .
Phase 2: Inversion
The Graphical Method
Graph your model in Y1 .
Graph your threshold (e.g., 10) in Y2 .
Find the Intersections .
Identify the intervals above the line.
Window of Opportunity
Deliver Your Verdict
Based on your model, you must provide the Captain with the exact time windows for a safe harbor entry. lives and millions of dollars depend on your accuracy.
Eco-Data Lab: Mission Complete
Future Forecasts Worksheet Lab Report: Future Forecasts
Eco-Data Lab // Lesson 05: Final Assessment
Lead Scientist:
Date:
Mission: The Harbor Entry
A ship carrying critical medical supplies is approaching Echo Bay . The ship's hull is deep; it requires at least 12 feet of water to navigate the channel safely. Using the following tidal model (where \(x\) is hours past midnight), determine the safety windows for today.
\[y = 5.4 \cos(\frac{\pi}{6}(x - 4)) + 10.5\]
01 Environmental Analysis
Before solving, analyze the model to understand the daily conditions in Echo Bay.
A. What is the Maximum water level?
B. What is the Minimum water level?
C. How many hours pass between high tides?
D. At what time (AM) does the first High Tide occur?
02 Solving the Inequality
The ship can enter when \(y \geq 12\). Use the grid below to sketch the model and the threshold line. Label all intersection points (use your calculator to find precise values).
Depth (ft)
Time (Hours past Midnight)
0
6
12
18
24
Calculated Safe Windows:
Identify the intervals where the water depth is at least 12 feet. Provide your answers in HH:MM format.
Morning:
to
Afternoon:
to
Final Scientific Recommendation:
The captain wants to arrive at 2:00 PM and stay for 4 hours to unload. Based on your forecast, is this a safe plan? Justify your answer using the data and model above.
Future Forecasts Teacher Guide Teacher Guide: Future Forecasts
Eco-Data Lab // Lesson 05: Final Assessment
Focus: Application & Mastery
Lesson Objective
Students will apply their modeling skills to a scenario-based challenge. They will interpret a given trigonometric tidal model, calculate extremes and period, and solve a trigonometric inequality graphically to determine safe windows for ship navigation.
Mastery Check
• Extracting A, B, h, k
• Solving f(x) ≥ threshold
• Unit conversions (Hours → Time)
• Contextual Justification
Model: \(y = 5.4 \cos(\frac{\pi}{6}(x - 4)) + 10.5\)
Max: 10.5 + 5.4 = 15.9 ft
Min: 10.5 - 5.4 = 5.1 ft
Period: 2π / (π/6) = 12 hours
High Tide 1: 4:00 AM (x = 4)
Graphical Solution (y ≥ 12)
Students should set \(Y1 = 5.4 \cos(\frac{\pi}{6}(x - 4)) + 10.5\) and \(Y2 = 12\).
Intersections & Windows:
• Morning Window: x ≈ 1.54 to x ≈ 6.46 (approx 1:32 AM to 6:28 AM)
• Afternoon Window: x ≈ 13.54 to x ≈ 18.46 (approx 1:32 PM to 6:28 PM)
The Recommendation Verdict
"The captain wants to arrive at 2:00 PM and stay for 4 hours (until 6:00 PM)."
Correct Verdict: YES, it is safe. The afternoon window starts at 1:32 PM and ends at 6:28 PM. Since 2:00 PM to 6:00 PM falls entirely within this window, the ship will have enough water.
Differentiation:
Support: Provide a pre-graphed version of the model and ask students to highlight the "safe" areas visually before calculating.
Extension: Ask students how the windows would change if the ship was heavier and required 14 feet of water instead of 12.
