Graph Steepness Slides Unit 1: Rise and Run
Lesson 1.1
Graph Steepness
Decoding Slope as the Rate of Change on the Coordinate Plane
Rise over Run • 4 Slope Types • Slope Triangles
Algebra 1 • Linear Foundations Ready to Climb
What is Slope?
Core Definition
Slope measures the steepness and direction of a line.
It tells us how fast the vertical value changes compared to the horizontal value.
The Golden Ratio of Slope
\[ \text{Slope } (m) = \frac{\text{Vertical Change}}{\text{Horizontal Change}} = \frac{\text{Rise}}{\text{Run}} \]
RISE (Change in \(y\))
Moving UP is positive (+), moving DOWN is negative (−).
RUN (Change in \(x\))
Moving RIGHT is positive (+), moving LEFT is negative (−).
💡 Rule of Thumb: Always read graphs from left to right , just like reading a sentence!
The Four Faces of Slope
Visual Identification
Positive Slope
Uphill from left to right
\(m > 0\)
Negative Slope
Downhill from left to right
\(m < 0\)
Zero Slope
Completely flat horizontal
\(m = 0\)
Undefined Slope
Vertical wall (cliff)
\(m = \text{undef}\)
⛷️ Ski Analogy: Positive = climbing up, Negative = skiing down, Zero = flat cross-country, Undefined = you fell off a cliff!
The Slope Triangle Method
3-Step Process
1
Find Grid Points
Locate two points where the line crosses grid intersections exactly (integers).
Example: \((1, 2)\) and \((4, 8)\)
2
Count Rise & Run
Draw a right triangle connecting the two points. Count vertical steps first, then horizontal.
Rise = \(+6\), Run = \(+3\)
3
Simplify Fraction
Write as \(\frac{\text{Rise}}{\text{Run}}\) and simplify to lowest terms. Keep as fraction or whole number.
\(m = \frac{6}{3} = 2\)
⚠️ Avoid Common Error: Do not flip the ratio! Rise is always on top (numerator). \(\frac{\Delta y}{\Delta x}\)
Practice Together
Class Check
Line A Points: \((0, 1)\) & \((3, 5)\)
+4 +3
Rise = +4 , Run = +3
\(m = \frac{4}{3}\)
Line B Points: \((1, 4)\) & \((5, 2)\)
−2 +4
Rise = −2 , Run = +4
\(m = \frac{-2}{4} = -\frac{1}{2}\)
Next: Grab your practice sheet and build slope triangles for each coordinate graph!
Graph Steepness Practice Sheet Unit 1: Rise and Run • Lesson 1.1
Graph Steepness Practice Sheet
Name: ____________________________ Date: ____________
Period: ______ Score: ______ / 24
1 Part 1: Slope Direction Quick Check
Classify each line as having a Positive , Negative , Zero , or Undefined slope.
Graph A
Graph B
Graph C
Graph D
2 Part 2: Slope Triangles on the Coordinate Grid
Draw a slope triangle connecting the two highlighted points. Calculate the rise, run, and simplified slope \(m\).
Problem 1
Points: \((0, 0)\) and \((3, 2)\)
Rise (\(\Delta y\)):
Run (\(\Delta x\)):
Slope (\(m\)):
Problem 2
Points: \((-2, 3)\) and \((1, -3)\)
Rise (\(\Delta y\)):
Run (\(\Delta x\)):
Slope (\(m\)):
Problem 3
Points: \((-3, 2)\) and \((2, 2)\)
Rise (\(\Delta y\)):
Run (\(\Delta x\)):
Slope (\(m\)):
Problem 4
Points: \((-1, -3)\) and \((1, 3)\)
Rise (\(\Delta y\)):
Run (\(\Delta x\)):
Slope (\(m\)):
Formula reminder: \(m = \frac{\text{Rise}}{\text{Run}} = \frac{y_2 - y_1}{x_2 - x_1}\) Page 1 of 2
Graph Steepness Practice Sheet (Continued)
Student Name: _______________________
3 Part 3: Special Cases & Fractional Slopes
Problem 5
Points: \((2, 2)\) and \((2, -2)\)
Rise: ________ Run: ________
Slope (\(m\)):
Is division by zero defined?
Problem 6
Points: \((-3, 3)\) and \((3, -1)\)
Rise: ________ Run: ________
Slope (\(m\)):
Simplify your fraction fully.
4 Part 4: Constructing Lines from a Point & Slope
Plot the given starting point, use the slope to plot at least two more points, and draw the line.
Problem 7
Point: \((-2, -1)\)
Slope: \(m = \frac{3}{2}\)
Move from point:+3 (up)+2 (right)
Problem 8
Point: \((0, 3)\)
Slope: \(m = -\frac{4}{3}\)
Graph Steepness Answer Key Teacher Resource • Answer Key
Graph Steepness Answer Key
Complete Worked Solutions
Unit 1: Rise and Run • Lesson 1.1
✓ Part 1: Quick Check Solutions
Graph A
Positive
Rises left to right
Graph B
Negative
Falls left to right
Graph C
Zero
Horizontal line
Graph D
Undefined
Vertical line
✓ Part 2: Slope Triangle Solutions
Problem 1 Solutions
Points: \((0, 0)\) and \((3, 2)\)
Rise: +2 (up 2)
Run: +3 (right 3)
\(m = \frac{2}{3}\)
Problem 2 Solutions
Points: \((-2, 3)\) and \((1, -3)\)
Rise: −6 (down 6)
Run: +3 (right 3)
\(m = \frac{-6}{3} = -2\)
Problem 3 Solutions
Points: \((-3, 2)\) and \((2, 2)\)
Rise: 0 (flat)
Run: +5 (right 5)
\(m = \frac{0}{5} = 0\)
Problem 4 Solutions
Points: \((-1, -3)\) and \((1, 3)\)
Rise: +6 (up 6)
Run: +2 (right 2)
\(m = \frac{6}{2} = 3\)
Teacher Insight on Left-to-Right Convention: Encourage students to always start at the leftmost point and travel right. This guarantees that Run (\(\Delta x\)) is always positive, meaning the sign of the slope depends solely on whether the Rise is positive (up) or negative (down).
Teacher Guide & Answer Key • Lesson 1.1 Page 1 of 2
Graph Steepness Answer Key (Page 2 Solutions)
Teacher Reference
✓ Part 3: Special Cases & Fractions
Problem 5: Undefined
Rise = \(4\), Run = \(0\)
\(m = \frac{4}{0} \rightarrow \text{Undefined}\)
Vertical lines have no horizontal run. Division by zero is mathematically undefined.
Problem 6: Fraction
Rise = \(-4\), Run = \(+6\)
\(m = \frac{-4}{6} = -\frac{2}{3}\)
Students must reduce fraction \(\frac{-4}{6}\) to lowest terms: \(-\frac{2}{3}\).
✓ Part 4: Line Construction Solutions
Problem 7 Line
Start: \((-2, -1)\), \(m = \frac{3}{2}\)
Plotted Points:
Formula Forge Slides Unit 1: Rise and Run
Lesson 1.2
Formula Forge
Calculating Exact Slope from Coordinate Points and Tables
The Slope Formula • Negative Traps • Rates of Change
Algebra 1 • Precision Algebra Beyond the Grid
From Graph to Formula
Conceptual Bridge
Why use a formula?
What if points are huge numbers like \((25, 140)\) and \((85, 380)\)? Counting grid squares is impossible!
The slope formula calculates the vertical distance and horizontal distance using pure arithmetic.
The Slope Formula
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Numerator:
\(\text{Rise} = y_2 - y_1\)
Denominator:
\(\text{Run} = x_2 - x_1\)
🔑 Memory Key: \(y\) goes in the sky (top), \(x\) runs across the ground (bottom).
The 4-Step Solution Protocol
Step-by-Step Method
1
Label Points
Write \(x_1, y_1\) and \(x_2, y_2\) directly above your coordinates.
\((x_1, y_1), (x_2, y_2)\)
2
Setup Formula
Substitute values using parentheses to protect negative signs.
\(m = \frac{(y_2) - (y_1)}{(x_2) - (x_1)}\)
3
Subtract
Compute numerator and denominator separately first.
\(m = \frac{\Delta y}{\Delta x} = \frac{-8}{4}\)
4
Simplify
Reduce the fraction or write as an integer. Never use mixed numbers.
\(m = -2\)
Note: It doesn't matter which point is Point 1 or Point 2, as long as you stay consistent!
Beware: The Double-Negative Trap!
#1 Most Common Error
The Fatal Mistake
Points: \((-4, 3)\) and \((2, -5)\)
\(m = \frac{-5 - 3}{2 - 4}\) ← Dropped negative!
The student wrote \(2 - 4\) instead of \(2 - (-4)\). This ruins the entire calculation!
The Bulletproof Fix
Points: \((-4, 3)\) and \((2, -5)\)
\(m = \frac{-5 - 3}{2 - (-4)} = \frac{-8}{2 + 4} = \frac{-8}{6} = -\frac{4}{3}\)
Subtracting a negative becomes addition : \(2 - (-4) = 2 + 4 = 6\).
💡 Pro-Tip: Always write the minus sign of the formula first , then insert the coordinates in parentheses.
Slope as a Real-World Rate of Change
Everyday Contexts
Formula Forge Practice Sheet Unit 1: Rise and Run • Lesson 1.2
Formula Forge Practice Sheet
Name: ____________________________ Date: ____________
Period: ______ Score: ______ / 25
The Slope Formula: \(m = \frac{y_2 - y_1}{x_2 - x_1}\)
Remember: Use parentheses when substituting negative numbers!
1 Part 1: Slope from Two Coordinates
Label each coordinate, substitute into the slope formula, show your algebraic work, and simplify your answer.
Problem 1 \((3, 7)\) and \((8, 17)\)
\(x_1 = \_\_\_\) \(y_1 = \_\_\_\) \(x_2 = \_\_\_\) \(y_2 = \_\_\_\)
Show Work:
Final Slope (\(m\)):
Problem 2 \((-2, 10)\) and \((4, -2)\)
\(x_1 = \_\_\_\) \(y_1 = \_\_\_\) \(x_2 = \_\_\_\) \(y_2 = \_\_\_\)
Show Work:
Final Slope (\(m\)):
Problem 3 \((-5, -3)\) and \((1, 5)\)
\(x_1 = \_\_\_\) \(y_1 = \_\_\_\) \(x_2 = \_\_\_\) \(y_2 = \_\_\_\)
Show Work:
Final Slope (\(m\)):
Problem 4 \((6, -1)\) and \((-2, -7)\)
\(x_1 = \_\_\_\) \(y_1 = \_\_\_\) \(x_2 = \_\_\_\) \(y_2 = \_\_\_\)
Show Work:
Final Slope (\(m\)):
Formula reminder: Keep fractions in simplest improper form (do not convert to mixed numbers). Page 1 of 2
Formula Forge Practice Sheet (Continued)
Student Name: _______________________
2 Part 2: Horizontal & Vertical Special Cases
Problem 5 \((4, 9)\) and \((-6, 9)\)
Slope:
Problem 6 \((-3, 5)\) and \((-3, -8)\)
Slope:
3 Part 3: Constant Rate of Change from Tables
Pick any two coordinate rows from the table, apply the slope formula, and state the unit rate.
Problem 7: Pool Drainage
<table class="w-full text-xs text-center border border-slate-200"><tbody><tr class="bg-slate-100 font-semibold"><th class="p-0.5 border border-slate-200">Time, \(x\) (hrs)</th><td class="p-0.5 border border-slate-200">2</td><td class="p-0.5 border border-slate-200">5</td><td class="p-0.5 border border-slate-200">8</td></tr><tr><th class="p-0.5 border border-slate-200 bg-slate-100 font-semibold">Water, \(y\) (gal)</th><td class="p-0.5 border border-slate-200">900</td><td class="p-0.5 border border-slate-200">675</td><td class="p-0.5 border border-slate-200">450</td></tr></tbody></table>
Points: \((\_\_\_, \_\_\_)\) and \((\_\_\_, \_\_\_)\)
Rate of change:
Problem 8: Road Trip Fuel
<table class="w-full text-xs text-center border border-slate-200"><tbody><tr class="bg-slate-100 font-semibold"><th class="p-0.5 border border-slate-200">Gas, \(x\) (gal)</th><td class="p-0.5 border border-slate-200">3</td><td class="p-0.5 border border-slate-200">7</td><td class="p-0.5 border border-slate-200">11</td></tr><tr><th class="p-0.5 border border-slate-200 bg-slate-100 font-semibold">Miles, \(y\)</th><td class="p-0.5 border border-slate-200">96</td><td class="p-0.5 border border-slate-200">224</td><td class="p-0.5 border border-slate-200">352</td></tr></tbody></table>
Formula Forge Answer Key Teacher Resource • Answer Key
Formula Forge Answer Key
Complete Algebraic Solutions
Unit 1: Rise and Run • Lesson 1.2
✓ Part 1: Two-Point Slope Calculation Solutions
Problem 1 Solution \((3, 7)\) and \((8, 17)\)
\(x_1=3, y_1=7, x_2=8, y_2=17\)
\(m = \frac{17 - 7}{8 - 3}\)
\(m = \frac{10}{5}\)
Final Slope: \(m = 2\)
Problem 2 Solution \((-2, 10)\) and \((4, -2)\)
\(x_1=-2, y_1=10, x_2=4, y_2=-2\)
\(m = \frac{-2 - 10}{4 - (-2)} = \frac{-12}{4 + 2}\)
\(m = \frac{-12}{6}\)
Final Slope: \(m = -2\)
Problem 3 Solution \((-5, -3)\) and \((1, 5)\)
\(x_1=-5, y_1=-3, x_2=1, y_2=5\)
\(m = \frac{5 - (-3)}{1 - (-5)} = \frac{5 + 3}{1 + 5}\)
\(m = \frac{8}{6} = \frac{4}{3}\)
Final Slope: \(m = \frac{4}{3}\)
Problem 4 Solution \((6, -1)\) and \((-2, -7)\)
\(x_1=6, y_1=-1, x_2=-2, y_2=-7\)
\(m = \frac{-7 - (-1)}{-2 - 6} = \frac{-7 + 1}{-8}\)
\(m = \frac{-6}{-8} = \frac{3}{4}\)
Final Slope: \(m = \frac{3}{4}\)
Pedagogical Watch-Point: Reversing Point Order
Students often ask if it matters which point is \((x_1, y_1)\) vs \((x_2, y_2)\). Demonstrate with Problem 1: calculating \(\frac{7 - 17}{3 - 8} = \frac{-10}{-5} = 2\) gives the exact same result! What matters is consistency : do not calculate \(y_2 - y_1\) in the numerator and then accidentally calculate \(x_1 - x_2\) in the denominator.
Teacher Guide & Answer Key • Lesson 1.2 Page 1 of 2
Formula Forge Answer Key (Page 2 Solutions)
Teacher Reference
✓ Part 2: Special Slopes
Problem 5: Horizontal Line
\(m = \frac{9 - 9}{-6 - 4} = \frac{0}{-10} = 0\)
Answer: Slope = 0 (Zero Slope)
Problem 6: Vertical Line
\(m = \frac{-8 - 5}{-3 - (-3)} = \frac{-13}{0}\)
Answer: Undefined (Division by 0)
✓ Part 3: Table Rate of Change Solutions
Problem 7: Pool Drainage
Points: \((2, 900)\) and \((5, 675)\)
Rate: \(-75\) gallons per hour
Problem 8: Road Trip Fuel
Points: \((3, 96)\) and \((7, 224)\)
Rate: \(32\) miles per gallon (mpg)
✓ Part 4: Real-World Scenario Solutions
a. Points:
\(P_1 = (3, 1200), P_2 = (8, 3700)\)
