Slope Decoded

Write linear model equation:

\(B(h) =\)

14 Graphic Design Freelance Contract

Modeling

Devon charges a flat consultation fee of $75 plus $45 per hour for website design.

a) Write equation for total cost \(C\) after \(x\) hours:

\(C(x) =\)

b) If a project took 6 hours, what was the total cost?

15 Deep Thinking: Zero Rate vs Undefined Rate

A car's odometer reading is graphed over time while parked at a grocery store. What is the slope of this line and what does it represent in reality?

Steep Stories • Page 2: Real-World Applications Algebra I • Linear Functions Unit

1. Identify

Find \(m\) & \(b\) from \(y = mx + b\)

2. Transform

Solve standard form for \(y\)

3. Justify

Explain student error

Duration: 8-10 minutes • Independent Work Slide 5 of 8

Part 2: Real World Rate of Change

In the Real World, Slope is Rate of Change

Math Definition \(m = \frac{\Delta y}{\Delta x}\)

⇒

Real-World Meaning \(\frac{\text{Change in Dependent Variable}}{\text{Change in Independent Variable}}\)

The Universal Interpretation Frame:

“For every 1 [unit of \(x\)], the [variable \(y\)] increases / decreases by [rate with units].”

Always include units: miles per hour, dollars per month, gallons per minute! Slide 6 of 8

Part 2: Real World Guided Model

Case Study: Draining a Water Tank

Scenario Data:

A tank has \(1,200\) gallons at \(t = 0\).
After \(5\) hours, \(600\) gallons remain.

\(m = \frac{600 - 1200}{5 - 0} = \frac{-600}{5} = \mathbf{-120}\)

Rate: \(-120\text{ gal/hr}\)

Decoding the Linear Model:

\(W(t) = -120t + 1200\)

• • \(-120\): loses 120 gallons every hour
• • \(1200\): initial water before draining

Turn to your Steep Stories Practice Sheet to explore more scenarios! Slide 7 of 8

Lesson Wrap-Up Minutes 70–100

Putting It All Together

1. Steep Stories Practice

Work with your partner on Section B (Problems 12-15). Ensure every rate of change answer includes proper units!

Pacing: 20 minutes

2. Final Check: DOL 2

Independent Exit Ticket: Municipal pool scenario. Solve for the rate, write the equation, and predict time to empty.

Pacing: 10 minutes

“Slope tells the story of how fast things change.”

Turn in DOL 1 and DOL 2 at the end of class! Slide 8 of 8

Support / Intervention

Provide index cards with the \(m = -\frac{A}{B}\) shortcut and sentence frames pre-printed. Have students circle the \(x\)-term in blue and \(y\)-term in red before moving terms.

English Learners (EL)

Emphasize cognates (constante, pendiente, razón). Have students chorally read the interpretation sentence frame: “For every 1 [unit], the [variable] changes by [rate].”

Extension / Challenge

Prompt advanced learners to investigate perpendicular line slopes (\(m_1 \cdot m_2 = -1\)) and explain why horizontal and vertical lines are mathematically perpendicular.

Slope Decoded • Facilitation & Lesson Plan Page 1 of 2

Master Keys

Complete Solutions & Keys

Teacher Reference All Handouts & DOLs

Guided Notes Solutions

Ex A: \(m = -3/4\), \(b = 5\)

Ex B: \(m = -2\), \(b = 6\)

Ex C: \(m = 1\), \(b = 0\)

Model 1: \(2y = -3x + 12 \implies y = -\frac{3}{2}x + 6\), \(m = -3/2\)

Model 2: \(-5y = -4x + 20 \implies y = \frac{4}{5}x - 4\), \(m = 4/5\)

Checkpoint 1: 1) \(0\); 2) undefined; 3) \(-2\); 4) \(3\)

Scooter: \(m = \$0.35/\text{min}\), \(b = \$2.50\) (unlock fee). Cost rises $0.35 per min.

Tank: \(m = -120\text{ gal/hr}\). Negative because water is being removed. Model: \(W(t) = -120t + 1200\).

DOL 1: Equation Slope Key

Q1: Slope \(m = -4/7\); \(y\)-intercept \(b = 9\) (or \((0,9)\)).

Q2: a) \(y = -6 \implies m = 0\); b) \(x = 3 \implies m = \text{undefined}\).

Q3: \(2y = -5x + 14 \implies y = -\frac{5}{2}x + 7\).
Slope \(m = -5/2\).

Q4: Marcus is incorrect. The equation is in standard form, not slope-intercept form. Solving for \(y\): \(-4y = -3x + 12 \implies y = \frac{3}{4}x - 3\). The true slope is \(3/4\).

Steep Stories Practice Sheet Key (Problems 1 – 15)

1. \(m = 7\), \(b = -4\)

2. \(m = -2/3\), \(b = 12\)

3. \(m = -1\), \(b = 8\)

4. \(m = 1/5\), \(b = 0\)

5. \(y = -3x + 11 \implies m = -3\)

6. \(-2y = -8x + 14 \implies y = 4x - 7 \implies m = 4\)

7. \(-4y = -x + 16 \implies y = \frac{1}{4}x - 4 \implies m = 1/4\)

8. \(9y = -6x - 27 \implies y = -\frac{2}{3}x - 3 \implies m = -2/3\)

9. \(m = 4/3\) (directly from point-slope form)

10. \(y = -11 \implies m = 0\) (horizontal line)

11. \(x = 5 \implies m = \text{undefined}\) (vertical line)

12. a) \(m = -800\text{ ft/min}\), \(b = 36,000\text{ ft}\). b) For every 1 min, altitude drops 800 ft.

13. \(m = \frac{68 - 92}{2 - 0} = -12\%\text{ per hr}\). Equation: \(B(h) = -12h + 92\).

14. a) \(C(x) = 45x + 75\). b) \(C(6) = 45(6) + 75 = 270 + 75 = \$345\).

15. Slope is 0; odometer distance does not change while parked (0 miles/hour).

DOL 2: Real-World Slope Exit Ticket Key

Task 1 (Rate of Change): \(m = \frac{36000 - 48000}{4 - 0} = \frac{-12000}{4} = \mathbf{-3,000\text{ gallons per hour}}\). (Accept 3,000 gal/hr decrease).

Task 2 (Interpretation): “For every 1 hour that passes, the pool volume decreases by 3,000 gallons.”

Task 3 (Equation): \(V(t) = -3000t + 48000\) (or \(V = 48000 - 3000t\)).

Task 4 (Prediction): Set \(V = 0 \implies 0 = -3000t + 48000 \implies 3000t = 48000 \implies t = \mathbf{16\text{ hours}}\) to completely empty.

Reflection: Rate of change / unit rate with real-world units.

Slope Decoded • Master Keys & Solutions Page 2 of 2