Rate Workshop Student Worksheet
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Rate & Intercept Workshop
Connecting Context, Tables, and Linear Equations \( (y = mx + b) \)
Name:
Date:
Period:
Directions: Calculate the rate of change (\(m\)), identify the initial value (\(b\)), write the equation, and interpret both in context.
Problem 1
Drone Altitude Ascent
A wildlife research drone launches from a hill platform and climbs steadily. The table records altitude \(y\) (meters) after \(x\) seconds of flight.
\(x\) (sec) \(y\) (meters)
-2 5
0 15
4 35
8 55
↓ Write 2 new \((x, y)\) pairs above ↓
1. Slope (\(m\)): Show \(\frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}\)
\(m = \frac{\Delta y}{\Delta x} =\) \(=\)
Meaning: The drone ascends meters every .
2. \(y\)-Intercept (\(b\)):
\(b =\)
Point: \((0,\) \()\)
Meaning: At time \(0\), the drone's starting altitude was .
3. Linear Equation & Prediction: \(y = mx + b\)
\(y =\)
Find altitude at \(x = 20\) seconds: \(y =\) (show work)
Problem 2
Delivery Van Fuel Depletion
A regional courier van begins its daily route with a full fuel tank. The table records gasoline \(y\) (gallons) remaining after driving for \(x\) hours.
\(x\) (hrs) \(y\) (gal)
0 18
2 14
4 10
6 6
↓ Write 2 new \((x, y)\) pairs above ↓
1. Slope (\(m\)): Notice sign of change: \(\Delta y < 0\)
\(m = \frac{\Delta y}{\Delta x} =\) \(=\)
Meaning: The van burns gallons of gas per .
2. \(y\)-Intercept (\(b\)):
\(b =\)
Point: \((0,\) \()\)
Meaning: Before driving (\(x=0\)), the tank started with .
3. Linear Equation & Prediction: \(y = mx + b\)
\(y =\)
When the tank is empty (\(y = 0\)), hours driven \(x =\)
Algebra 1 • Rate of Change & Initial Value Page 1 of 2
P2
Rate & Intercept Workshop | Continued
Name: Fractional Rates & Zero Crossings
Problem 3
Community Garden Rain Cistern
During a steady shower, a greenhouse collects rainwater in a cistern that already held water. The table tracks total volume \(y\) (gallons) after \(x\) minutes.
\(x\) (min) \(y\) (gal)
0 4
4 7
8 10
16 16
↓ Write 2 new \((x, y)\) pairs above ↓
1. Slope (\(m\)): Simplify fraction or write decimal
\(m = \frac{\Delta y}{\Delta x} =\) \(=\)
Meaning: The cistern fills at a rate of gal per minute.
2. \(y\)-Intercept (\(b\)):
\(b =\)
Point: \((0,\) \()\)
Meaning: Before the storm (\(x=0\)), the cistern held .
3. Linear Equation & Prediction: \(y = mx + b\)
\(y =\)
Predict total water after \(24\) minutes: \(y =\) gal
Problem 4
Mountain Weather Station Freeze
A high-altitude weather observatory monitors falling evening temperatures. The table records ambient temperature \(y\) (\(^\circ\text{C}\)) after \(x\) hours past sunset.
\(x\) (hrs) \(y\) (\(^\circ\text{C}\))
0 8
3 2
6 -4
9 -10
↓ Write 2 new \((x, y)\) pairs above ↓
1. Slope (\(m\)): Calculate with signed integers
\(m = \frac{\Delta y}{\Delta x} =\) \(=\)
Meaning: Temperature drops by \(^\circ\text{C}\) each .
2. \(y\)-Intercept (\(b\)):
\(b =\)
Point: \((0,\) \()\)
Meaning: At sunset (\(x=0\)), the temperature was .
3. Linear Equation & Prediction: \(y = mx + b\)
\(y =\)
Time at freezing point (\(y = 0^\circ\text{C}\)): \(x =\) hours
Algebra 1 • Rate of Change & Initial Value Page 2 of 2
Rate Workshop Answer Key
✓
Rate & Intercept Workshop
Teacher Answer Key
Model Solutions, Pedagogical Notes & Common Student Misconceptions
Form: \(y = mx + b\)
Watch Out: Ensure students compute \(\frac{\Delta y}{\Delta x}\) and do not invert it as \(\frac{\Delta x}{\Delta y}\). The \(y\)-intercept always corresponds strictly to \(x=0\).
Problem 1
Drone Altitude Ascent • Positive Slope
Research drone launched from hill platform; altitude \(y\) (meters) vs. flight time \(x\) (seconds).
\(x\) (sec) \(y\) (meters)
-2 5
0 15
4 35
8 55
6 (ex) 45
10 (ex) 65
✓ Any valid pair on line \(y=5x+15\)
1. Slope (\(m\)): \(\Delta y / \Delta x\)
\(m = \frac{35 - 15}{4 - 0} =\) \(\frac{20}{4}\) \(=\) \(5\)
Meaning: The drone ascends 5 meters every 1 second.
2. \(y\)-Intercept (\(b\)):
\(b =\) \(15\)
Point: \((0, 15)\)
Meaning: At time \(0\), the drone's starting altitude was 15 meters above ground.
3. Linear Equation & Prediction: \(y = 5x + 15\)
\(y =\) \(5x + 15\)
At \(x = 20\): \(y = 5(20) + 15 = 100 + 15 =\) 115 meters.
Problem 2
Delivery Van Fuel Depletion • Negative Rate
Delivery courier van fuel burn; gallons remaining \(y\) vs. driving time \(x\) (hours).
\(x\) (hrs) \(y\) (gal)
0 18
2 14
4 10
6 6
8 (ex) 2
9 (ex) 0
✓ Any valid pair on line \(y=-2x+18\)
1. Slope (\(m\)): \(\Delta y < 0 \implies m < 0\)
\(m = \frac{14 - 18}{2 - 0} =\) \(\frac{-4}{2}\) \(=\) \(-2\)
Meaning: The van burns 2 gallons of gas per 1 hour driven.
2. \(y\)-Intercept (\(b\)):
\(b =\) \(18\)
Point: \((0, 18)\)
Meaning: Before driving (\(x=0\)), the tank held a full 18 gallons.
3. Linear Equation & Prediction: \(y = -2x + 18\)
\(y =\) \(-2x + 18\)
When \(y = 0\): \(0 = -2x + 18 \implies 2x = 18 \implies\) \(x = 9\) hours.
Rate Workshop Teacher Answer Key Page 1 of 2
P2
Rate & Intercept Key | Problems 3 & 4
