Grid Climber Worksheet
Grid Climber: Slope & Rate of Change
Finding Change in Y over Change in X (\(\frac{\Delta y}{\Delta x}\))
Name:
Date:
Key Rule
$$\text{Slope } (m) = \frac{\text{Change in } y}{\text{Change in } x} = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{\text{Vertical Change (Rise)}}{\text{Horizontal Change (Run)}}$$
1. Find Points → 2. Subtract → 3. Simplify
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Part 1: Find Coordinates on the Graph & Calculate Slope
Identify points \(A\) and \(B\), then compute \(\frac{\Delta y}{\Delta x}\)
Model Example Follow Steps
x y 0 2 4 6 8 2 4 6 \(\Delta x = 3\) \(\Delta y = 3\) A (1, 1) B (4, 4)
1. Coordinates: \(A(1, 1)\) and \(B(4, 4)\)
Vertical Change (\(\Delta y\)): \(y_2 - y_1 = 4 - 1 = \mathbf{3}\)
Horizontal Change (\(\Delta x\)): \(x_2 - x_1 = 4 - 1 = \mathbf{3}\)
Slope \((m) = \frac{\Delta y}{\Delta x}\): \(\frac{3}{3} = \mathbf{1}\)
Problem 1 Positive Slope
x y 0 2 4 6 8 2 4 6 A B
1. Find Coordinates: Point \(A(\underline{\hspace{18px}}, \underline{\hspace{18px}})\) Point \(B(\underline{\hspace{18px}}, \underline{\hspace{18px}})\)
\(\Delta y\) (Change in \(y\)):
\(y_2 - y_1 = \underline{\hspace{16px}} - \underline{\hspace{16px}} = \underline{\hspace{20px}}\)
\(\Delta x\) (Change in \(x\)):
\(x_2 - x_1 = \underline{\hspace{16px}} - \underline{\hspace{16px}} = \underline{\hspace{20px}}\)
Slope \((m) = \frac{\Delta y}{\Delta x}\): \(\frac{\underline{\hspace{18px}}}{\underline{\hspace{18px}}} = \)
Problem 2 Fractional Slope
x y 0 2 4 6 8 2 4 6 A B
1. Find Coordinates: Point \(A(\underline{\hspace{18px}}, \underline{\hspace{18px}})\) Point \(B(\underline{\hspace{18px}}, \underline{\hspace{18px}})\)
\(\Delta y\) (Change in \(y\)):
\(y_2 - y_1 = \underline{\hspace{16px}} - \underline{\hspace{16px}} = \underline{\hspace{20px}}\)
\(\Delta x\) (Change in \(x\)):
\(x_2 - x_1 = \underline{\hspace{16px}} - \underline{\hspace{16px}} = \underline{\hspace{20px}}\)
Slope \((m) = \frac{\Delta y}{\Delta x}\): \(\frac{\underline{\hspace{18px}}}{\underline{\hspace{18px}}} = \)
Problem 3 Downward Slope!
x y 0 2 4 6 8 2 4 6 A B
1. Find Coordinates: Point \(A(\underline{\hspace{18px}}, \underline{\hspace{18px}})\) Point \(B(\underline{\hspace{18px}}, \underline{\hspace{18px}})\)
\(\Delta y\) (Change in \(y\)):
\(y_2 - y_1 = \underline{\hspace{16px}} - \underline{\hspace{16px}} = \underline{\hspace{20px}}\)
\(\Delta x\) (Change in \(x\)):
\(x_2 - x_1 = \underline{\hspace{16px}} - \underline{\hspace{16px}} = \underline{\hspace{20px}}\)
Slope \((m) = \frac{\Delta y}{\Delta x}\): \(\frac{\underline{\hspace{18px}}}{\underline{\hspace{18px}}} = \)
Climber's Tip: \(y\) goes up and down (vertical), \(x\) goes left and right (horizontal). Always put change in \(y\) on top!
Part 2: Slope from Ordered Pairs & Rate of Change
Page 2 of 2 No Graph Needed
Steps for Ordered Pairs: 1. Label \((x_1, y_1)\) and \((x_2, y_2)\) • 2. Find \(\Delta y = y_2 - y_1\) • 3. Find \(\Delta x = x_2 - x_1\) • 4. Simplify \(\frac{\Delta y}{\Delta x}\)
Problem 4 \(A(2, 3)\) and \(B(6, 11)\)
\(x_1 = \underline{\hspace{14px}}\), \(y_1 = \underline{\hspace{14px}}\) \(x_2 = \underline{\hspace{14px}}\), \(y_2 = \underline{\hspace{14px}}\)
\(\Delta y\) (Change in \(y\)):
\(11 - 3 = \underline{\hspace{20px}}\)
\(\Delta x\) (Change in \(x\)):
\(6 - 2 = \underline{\hspace{20px}}\)
Slope \((m) = \frac{\Delta y}{\Delta x}\): \(\frac{\underline{\hspace{18px}}}{\underline{\hspace{18px}}} = \)
Problem 5 \(A(1, 5)\) and \(B(4, 14)\)
\(x_1 = \underline{\hspace{14px}}\), \(y_1 = \underline{\hspace{14px}}\) \(x_2 = \underline{\hspace{14px}}\), \(y_2 = \underline{\hspace{14px}}\)
\(\Delta y\) (Change in \(y\)):
\(y_2 - y_1 = \underline{\hspace{14px}} - \underline{\hspace{14px}} = \underline{\hspace{18px}}\)
\(\Delta x\) (Change in \(x\)):
\(x_2 - x_1 = \underline{\hspace{14px}} - \underline{\hspace{14px}} = \underline{\hspace{18px}}\)
Slope \((m) = \frac{\Delta y}{\Delta x}\): \(\frac{\underline{\hspace{18px}}}{\underline{\hspace{18px}}} = \)
Problem 6 \(A(3, 2)\) and \(B(9, 6)\)
\(x_1 = \underline{\hspace{14px}}\), \(y_1 = \underline{\hspace{14px}}\) \(x_2 = \underline{\hspace{14px}}\), \(y_2 = \underline{\hspace{14px}}\)
\(\Delta y\) (Change in \(y\)):
\(y_2 - y_1 = \underline{\hspace{14px}} - \underline{\hspace{14px}} = \underline{\hspace{18px}}\)
\(\Delta x\) (Change in \(x\)):
\(x_2 - x_1 = \underline{\hspace{14px}} - \underline{\hspace{14px}} = \underline{\hspace{18px}}\)
Slope \((m) = \frac{\Delta y}{\Delta x}\): \(\frac{\underline{\hspace{18px}}}{\underline{\hspace{18px}}} = \)
Problem 7 \(A(0, 10)\) and \(B(5, 0)\)
\(x_1 = \underline{\hspace{14px}}\), \(y_1 = \underline{\hspace{14px}}\) \(x_2 = \underline{\hspace{14px}}\), \(y_2 = \underline{\hspace{14px}}\)
\(\Delta y\) (Change in \(y\)):
\(0 - 10 = \underline{\hspace{20px}}\)
\(\Delta x\) (Change in \(x\)):
\(5 - 0 = \underline{\hspace{20px}}\)
Slope \((m) = \frac{\Delta y}{\Delta x}\): \(\frac{\underline{\hspace{18px}}}{\underline{\hspace{18px}}} = \)
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Part 3: Real-World Applications (Rate of Change)
Hiking Mountain Peak Elevation vs. Time
Maya records her elevation while hiking up a trail.
| Time (hours, \(x\)) |
1 |
2 |
4 |
| Elevation (feet, \(y\)) |
400 |
700 |
1,300 |
Pick any 2 points: \((\underline{\hspace{14px}}, \underline{\hspace{14px}})\) and \((\underline{\hspace{14px}}, \underline{\hspace{14px}})\)
Find \(\Delta y\) & \(\Delta x\): \(\Delta y = \underline{\hspace{18px}}\) \(\Delta x = \underline{\hspace{18px}}\)
Rate of Change with Units:
Maya climbs feet per hour.
Savings Goal Money Saved vs. Weeks
Leo is saving for a new skateboard. After 2 weeks, he has $30 saved. After 5 weeks, he has $75 saved.
Write ordered pairs \(( \text{weeks}, \text{dollars} )\):
Point 1: \((\underline{\hspace{16px}}, \underline{\hspace{16px}})\) Point 2: \((\underline{\hspace{16px}}, \underline{\hspace{16px}})\)
Calculate \(\frac{\Delta y}{\Delta x}\): \(\frac{75 - 30}{5 - 2} = \frac{\underline{\hspace{18px}}}{\underline{\hspace{18px}}} = \)
Meaning of the Slope:
Leo saves $ each week.
Self-Check: Did you remember to simplify your fractions and include negative signs when lines go down?
Ready for Quiz!
