Slope Climber Practice Worksheet
8th Grade Math Linear Functions • Front Page
Slope Climber Practice Worksheet
Mastering Rise & Run, the Slope Formula, and Multi-Step Calculations
Name:
Date:
Period:
Core Concept:
Slope \( (m) = \frac{\text{Rise}}{\text{Run}} = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1} \)
↗ Pos (+) ↘ Neg (−) → Zero (0) ↕ Undefined
1
Identify Rise and Run on a Graph
Count vertical change (Δy) and horizontal change (Δx), then simplify.
Problem 1 Positive Slope
x y A B
Vertical Rise: _____
Horizontal Run: _____
Slope (\(m\)): _____
Problem 2 Negative Slope
x y P Q
Vertical Rise: _____
Horizontal Run: _____
Slope (\(m\)): _____
Problem 3 Fractional Slope
x y M N
Vertical Rise: _____
Horizontal Run: _____
Slope (\(m\)): _____
2
The Slope Formula Studio: Show All Work
Use \( m = \frac{y_2 - y_1}{x_2 - x_1} \) • Show substitution • Simplify
Problem 4: A(2, 5) and B(6, 17) Integers
\( x_1 = \underline{\hspace{1.5cm}} \) \( x_2 = \underline{\hspace{1.5cm}} \)
\( y_1 = \underline{\hspace{1.5cm}} \) \( y_2 = \underline{\hspace{1.5cm}} \)
Substitute: \( m = \frac{\qquad\qquad - \qquad\qquad}{\qquad\qquad - \qquad\qquad} \)
Simplify: \( m = \frac{\qquad\qquad}{\qquad\qquad} \)
Slope \(m =\)
Problem 5: C(-3, 4) and D(1, -8) Watch Signs
\( x_1 = \underline{\hspace{1.5cm}} \) \( x_2 = \underline{\hspace{1.5cm}} \)
\( y_1 = \underline{\hspace{1.5cm}} \) \( y_2 = \underline{\hspace{1.5cm}} \)
Substitute: \( m = \frac{\qquad\qquad - \qquad\qquad}{\qquad\qquad - \qquad\qquad} \)
Simplify: \( m = \frac{\qquad\qquad}{\qquad\qquad} \)
Slope \(m =\)
Problem 6: E(-5, -2) and F(5, 4) Fraction Answer
\( x_1 = \underline{\hspace{1.5cm}} \) \( x_2 = \underline{\hspace{1.5cm}} \)
\( y_1 = \underline{\hspace{1.5cm}} \) \( y_2 = \underline{\hspace{1.5cm}} \)
Substitute: \( m = \frac{\qquad\qquad - \qquad\qquad}{\qquad\qquad - \qquad\qquad} \)
Simplify: \( m = \frac{\qquad\qquad}{\qquad\qquad} \)
Slope \(m =\)
Problem 7: G(-4, 6) and H(8, 6) Special Case
\( x_1 = \underline{\hspace{1.5cm}} \) \( x_2 = \underline{\hspace{1.5cm}} \)
\( y_1 = \underline{\hspace{1.5cm}} \) \( y_2 = \underline{\hspace{1.5cm}} \)
Substitute: \( m = \frac{\qquad\qquad - \qquad\qquad}{\qquad\qquad - \qquad\qquad} \)
Orientation: Horizontal or Vertical?
Slope \(m =\)
Page 1: Graphical & Coordinate Fundamentals Turn page for Method Connections and Real-World Rate of Change → Slope Climber Practice
Page 2 • Back Side Synthesis & Practical Application
Connecting Methods & Real-World Rates
Student Name:
3
Connect Multiple Methods: The Tri-Bridge Challenge
Compare visual rise/run with the formula for points \( J(-2, -3) \) and \( K(2, 5) \).
Method A: Graph & Rise/Run x y J K
Counted Rise (\(\Delta y\)): _____
Counted Run (\(\Delta x\)): _____
Slope \( m = \frac{\text{Rise}}{\text{Run}} \): _____
Method B: The Slope Formula
\( m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{5 - (\qquad)}{2 - (\qquad)} \)
\( m = \frac{\qquad\qquad}{\qquad\qquad} \)
Simplified Slope: \( m = \) _____
Mathematical Connection:
Explain how the numerator \( (y_2 - y_1) \) is the exact same distance as the vertical Rise counted on the triangle:
4
Real-World Slope & Rate of Change: Show Complete Steps
Rate of Change = \( \frac{\Delta \text{ Output (units of } y)}{\Delta \text{ Input (units of } x)} \)
Trail Elevation Gain
Scenario A
Elena hikes up Mt. Baldy. At mile 2, her altitude is 1,400 ft. At mile 6, her altitude reaches 3,200 ft.
Coordinates: \( (x_1, y_1) = (2, 1400) \), \( (x_2, y_2) = (6, 3200) \)
a. Show slope calculation with subtraction & units:
\( m = \frac{\qquad\quad - \qquad\quad}{\qquad\quad - \qquad\quad} = \frac{\qquad\quad}{\qquad\quad} \)
Rate = ft/mile
b. Practical meaning in context:
For every 1 horizontal mile hiked, Elena climbs feet in elevation.
EV Battery Charging
Scenario B
A car plugs into a supercharger. At 10 min, the battery is at 25%. At 35 min, it reaches 85%.
Coordinates: \( (x_1, y_1) = (10, 25) \), \( (x_2, y_2) = (35, 85) \)
a. Show slope calculation with subtraction & units:
\( m = \frac{\qquad\quad - \qquad\quad}{\qquad\quad - \qquad\quad} = \frac{\qquad\quad}{\qquad\quad} \)
Rate = %/min
b. Practical meaning in context:
The battery charges at a steady rate of % every 1 minute.
Page 2: Conceptual Connections & Applications Standards: CCSS.MATH.CONTENT.8.EE.B.5 & 8.F.B.4 Slope Climber Practice
