Line Fit Challenge Worksheet
Algebra I • Linear Modeling & Regression
Name: Date: Period:
TEKS S(7)(B) The student is expected to transform a linear parent function to determine a line of best fit.
- A meteorology research aircraft climbs through atmospheric layers to record ambient outside temperatures. Flight safety regulations state that the automated wing thermal de-icing system must be engaged if the predicted outside air temperature at 32,000 feet drops below −40.0°F.
| Altitude (ft) | Temperature (°F) |
|---|
| 4,000 | 58 |
| 8,000 | 44 |
| 12,000 | 29 |
| 16,000 | 14 |
| 20,000 | −1 |
| 24,000 | −16 |
| 28,000 | −30 |
Using data showing the outside temperature at each altitude during the first 28,000 feet, predict whether or not the flight crew will need to engage the wing de-icing system at 32,000 feet.
A. No, because at 32,000 feet the predicted temperature would be −34.8°F.
B. No, because at 32,000 feet the predicted temperature would be −45.1°F.
C. Yes, because at 32,000 feet the predicted temperature would be −52.6°F.
D. Yes, because at 32,000 feet the predicted temperature would be −45.1°F.
Work Space Show all calculations and justification
Algebra I • Linear Modeling & Regression
Name: Date: Period:
TEKS S(7)(B) The student is expected to transform a linear parent function to determine a line of best fit.
- A municipal water authority monitors usable water volume in an alpine reservoir during an extended summer drought. Conservation rules dictate that the city must declare a Stage 3 Emergency if usable reservoir storage drops below 210 thousand acre-feet by Day 150.
| Drought Day | Storage
(1,000 acre-feet) |
| --- | --- |
| 15 | 742 |
| 30 | 683 |
| 45 | 625 |
| 60 | 560 |
| 75 | 495 |
| 90 | 437 |
| 105 | 378 |
Using data showing reservoir storage during each of the first 105 days, predict whether or not the city will need to declare a Stage 3 Emergency by Day 150.
A. Yes, because by Day 150 the reservoir storage would be 192.7 thousand acre-feet.
B. Yes, because by Day 150 the reservoir storage would be 248.3 thousand acre-feet.
C. No, because by Day 150 the reservoir storage would be 192.7 thousand acre-feet.
D. No, because by Day 150 the reservoir storage would be 216.5 thousand acre-feet.
Work Space Show all calculations and justification
Line Fit Answer Key
Teacher Solution Guide • TEKS S(7)(B)
Problem 1: Atmospheric Flight Sounding
Correct Key: Option D
Scenario Context: Aircraft recording ambient temperatures from 4,000 to 28,000 ft.
Trigger Rule: Temp < −40.0°F at 32,000 ft
1 Graphing Calculator Linear Regression (LinReg ax+b) L1 = Altitude (ft), L2 = Temp (°F)
Slope (\(a\)): −0.003696
\(y\)-Intercept (\(b\)): 73.1429
Correlation (\(r\)): −0.9997
Determination (\(r^2\)): 0.9994
Alternative (thousands of feet): When \(x = 4, 8, \dots, 28\), \(a \approx -3.6964\) and \(b \approx 73.143\).
2 Line of Best Fit Equation
\(y = -0.0037x + 73.14\)
(or \(y = -3.70x + 73.14\) in 1,000s)
3 Predict for 32,000 Feet
\(y = -0.003696(32,000) + 73.143\)
\(y = -45.14^\circ\text{F} \approx \mathbf{-45.1^\circ\text{F}}\)
4 Threshold & Decision Evaluation
The safety protocol requires engaging de-icing if the temperature is below −40.0°F. Since −45.1°F < −40.0°F (−45.1°F is colder than −40.0°F), the condition is met and the crew YES, must engage the wing de-icing system.
Student Error Diagnostic & Distractor Rationale
A. Incorrect: Miscalculated value (−34.8°F) from improper entry or reversed slope and concluded "No".
B. Incorrect: Correct prediction of −45.1°F, but flawed negative number comparison (mistakenly thinking −45.1 is greater than −40.0).
C. Incorrect: Concluded "Yes" but used an exaggerated cooling rate (−52.6°F), showing computational inaccuracy.
D. Correct: Accurately predicts −45.1°F and correctly interprets that −45.1°F is below −40.0°F.
Pedagogical Note: Remind students that with negative temperatures, numbers with larger absolute values represent colder, lower temperatures (e.g., −45.1 is less than −40.0).
Teacher Solution Guide • TEKS S(7)(B)
Problem 2: Municipal Reservoir Drought Management
Correct Key: Option A
Scenario Context: Reservoir storage tracking across drought days (Days 15–105).
Trigger Rule: Storage < 210 thousand ac-ft by Day 150
1 Graphing Calculator Linear Regression () L1 = Drought Day, L2 = Storage (1,000 ac-ft)
