Slope Connections Guided Notes
Algebra 1 Guided Notes Unit: Linear Relationships
Finding Slope: Graph vs. Formula
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Date: Period:
Definition of Slope (\(m\))
The rate of change or steepness of a line:
\(m = \frac{\text{Change in } y}{\text{Change in } x} = \frac{\text{Vertical}}{\text{Horizontal}}\)
Positive Uphill (L \(\to\) R) \(m > 0\)
Negative Downhill (L \(\to\) R) \(m < 0\)
Zero Flat / Horizontal \(m = 0\)
Undefined Vertical / Cliff \(x = c\)
Strategy 1 • Visual Method
Finding Slope on a Graph: Rise Over Run
\( m = \frac{\text{Rise}}{\text{Run}} \)
1
Pick 2 Points: Crosshairs where grid lines meet.
2
Count RISE: Up = +, Down = −.
3
Count RUN: Move right (always +).
I DO (Teacher Model) Ex 1
Draw slope triangle & count:
Points: (-2, -3) & (2, 3)
Rise (\(\Delta y\)):
Run (\(\Delta x\)):
Equation & Simplify:
\( m = \frac{\text{Rise}}{\text{Run}} = \frac{\quad\quad}{\quad\quad} = \quad\quad \)
Slope:
\(m = \)
WE DO (Guided) Ex 2
Draw slope triangle together:
Points: (-3, 3) & (3, -1)
Rise (\(\Delta y\)):
Run (\(\Delta x\)):
Equation & Simplify:
\( m = \frac{\text{Rise}}{\text{Run}} = \frac{\quad\quad}{\quad\quad} = \quad\quad \)
Slope:
\(m = \)
YOU DO (Solo Try) Ex 3
Draw triangle, count, & solve:
Points: (-1, -4) & (1, 2)
Rise (\(\Delta y\)):
Run (\(\Delta x\)):
Equation & Simplify:
\( m = \frac{\text{Rise}}{\text{Run}} = \frac{\quad\quad}{\quad\quad} = \quad\quad \)
Slope:
\(m = \)
Check Your Sign: Before calculating, look at the line from left to right. Is it uphill (\(+\)) or downhill (\(-\))? Does your sign match?
Slope Connections Guided Notes • Page 1 of 2 Turn over for Strategy 2 (Formula) and the Big Connection →
Strategy 2 • Algebraic Method
Finding Slope from Two Points: The Slope Formula
\( m = \frac{y_2 - y_1}{x_2 - x_1} \)
1
Label Points: Assign \((x_1, y_1)\) and \((x_2, y_2)\).
2
Subtract \(y\)'s on Top: "Y is in the sky!"
3
Subtract \(x\)'s on Bottom: Keep identical order!
I DO (Teacher Model) Ex 4
\((1, 3)\) and \((5, 11)\)
Step 1: Label Coordinates
\(x_1 = 1,\; y_1 = 3\) \(x_2 = 5,\; y_2 = 11\)
Step 2: Substitute & Calculate
\( m = \frac{y_2 - y_1}{x_2 - x_1} \)
\( m = \frac{\quad\quad - \quad\quad}{\quad\quad - \quad\quad} = \frac{\quad\quad}{\quad\quad} \)
Final Slope:
\(m = \)
WE DO (Guided) Ex 5
\((-2, 4)\) and \((3, -6)\)
Step 1: Label Coordinates
\(x_1 = \underline{\quad},\; y_1 = \underline{\quad}\) \(x_2 = \underline{\quad},\; y_2 = \underline{\quad}\)
Step 2: Substitute & Calculate
\( m = \frac{y_2 - y_1}{x_2 - x_1} \)
\( m = \frac{\quad\quad - (\quad\quad)}{\quad\quad - (\quad\quad)} = \frac{\quad\quad}{\quad\quad} \)
Final Slope:
\(m = \)
YOU DO (Solo Try) Ex 6
\((-4, -1)\) and \((2, 3)\)
Step 1: Label Coordinates
\(x_1 = \underline{\quad},\; y_1 = \underline{\quad}\) \(x_2 = \underline{\quad},\; y_2 = \underline{\quad}\)
Step 2: Substitute & Calculate
\( m = \frac{y_2 - y_1}{x_2 - x_1} \)
\( m = \frac{\quad\quad - (\quad\quad)}{\quad\quad - (\quad\quad)} = \frac{\quad\quad}{\quad\quad} \)
Final Slope:
\(m = \)
=
The Grand Connection: Why They Are the EXACT Same Thing!
Key Conceptual Takeaway
x y \((x_1, y_1)\) \((x_2, y_2)\) RUN: \(x_2 - x_1\) RISE: \(y_2 - y_1\) Subtraction computes physical grid distance!
When counting squares on a grid, you are physically measuring the distance between two coordinate values. In algebra, how do you find the distance between any two numbers? You subtract them!
Measurement On the Graph In the Formula
Vertical Change Count up/down (Rise) \(y_2 - y_1\)
Horizontal Change Count right (Run) \(x_2 - x_1\)
The Universal Bridge: The slope formula \(\frac{y_2 - y_1}{x_2 - x_1}\) is simply algebraic counting that works even when numbers are massive, negative, or fractions!
Slope Connections Guided Notes • Page 2 of 2 Both methods give the exact same slope!
Slope Mastery Worksheet
Independent Practice Algebra 1 • Slope Foundations
Slope Mastery Worksheet
Name:
Date: Score: / 24
Quick Reference: Graph: \(m = \frac{\text{Rise}}{\text{Run}}\) Formula: \(m = \frac{y_2 - y_1}{x_2 - x_1}\) Always simplify final ratios!
P1 Part 1: Finding Slope from a Graph
Draw a slope triangle, determine Rise and Run, and record slope \(m\).
- Uphill
Rise:
Run:
Slope \(m\):
- Downhill
Rise:
Run:
Slope \(m\):
- Steep
Rise:
Run:
Slope \(m\):
- Horizontal
Rise:
Run:
Slope \(m\):
P2 Part 2: Finding Slope from Two Points
Label coordinates, substitute into \(m = \frac{y_2 - y_1}{x_2 - x_1}\), and simplify.
5. \((2, 5)\) and \((6, 17)\) Standard
\(x_1 = \underline{\hspace{20px}},\; y_1 = \underline{\hspace{20px}}\)
\(x_2 = \underline{\hspace{20px}},\; y_2 = \underline{\hspace{20px}}\)
\(m = \frac{y_2 - y_1}{x_2 - x_1} = \)
\(m = \)
6. \((-3, 8)\) and \((5, -8)\) Negatives
\(x_1 = \underline{\hspace{20px}},\; y_1 = \underline{\hspace{20px}}\)
\(x_2 = \underline{\hspace{20px}},\; y_2 = \underline{\hspace{20px}}\)
\(m = \frac{y_2 - y_1}{x_2 - x_1} = \)
\(m = \)
7. \((-4, -2)\) and \((6, 3)\) Fraction
\(x_1 = \underline{\hspace{20px}},\; y_1 = \underline{\hspace{20px}}\)
\(x_2 = \underline{\hspace{20px}},\; y_2 = \underline{\hspace{20px}}\)
\(m = \frac{y_2 - y_1}{x_2 - x_1} = \)
\(m = \)
8. \((4, -3)\) and \((4, 7)\) Special Case
\(x_1 = \underline{\hspace{20px}},\; y_1 = \underline{\hspace{20px}}\)
\(x_2 = \underline{\hspace{20px}},\; y_2 = \underline{\hspace{20px}}\)
\(m = \frac{y_2 - y_1}{x_2 - x_1} = \)
\(m = \)
Slope Mastery Worksheet • Page 1 of 2 Turn over for Connection Challenges & Error Analysis →
P3 Part 3: The "Connect Both Worlds" Challenge
Proving that Algebra = Geometry
9. Given points \(A(-3, -1)\) and \(B(1, 5)\): Solve both ways to verify they match!
Method A: Slope Formula
Show formula substitution:
\(m = \frac{\quad\quad - \quad\quad}{\quad\quad - \quad\quad} = \frac{\quad\quad}{\quad\quad} = \)
Calculated \(m\):
Plot \(A\) & \(B\), draw triangle
Method B: Rise / Run
Counted Rise:
Counted Run:
Visual \(m\):
Explanation of Equality:
How does the subtraction in your numerator \(\mathbf{y_2 - y_1}\) directly explain why you counted that exact number of grid squares for your Rise?
10. Given points \(C(-1, 4)\) and \(D(3, -2)\): Notice the downward direction!
Method A: Slope Formula
Show formula substitution:
\(m = \frac{\quad\quad - \quad\quad}{\quad\quad - \quad\quad} = \frac{\quad\quad}{\quad\quad} = \)
Calculated \(m\):
Plot \(C\) & \(D\), draw triangle
Method B: Rise / Run
Counted Rise:
Counted Run:
Visual \(m\):
P4 Part 4: Error Analysis & Concept Check
Think deeply and critique reasoning
11. Spot the Mistake
Jordan was asked to find the slope between \((3, 2)\) and \((7, 10)\). Here is Jordan's work:
\( m = \frac{7 - 3}{10 - 2} = \frac{4}{8} = \frac{1}{2} \)
a. What mistake did Jordan make?
b. What is the correct slope?
Correct \(m = \)
12. Order Invariance Investigation
Does it matter which point is \((x_1, y_1)\) and which is \((x_2, y_2)\)? Test with \((2, 4)\) and \((6, 12)\):
Way 1: \(B - A\) \(m = \frac{12 - 4}{6 - 2} = \underline{\quad\quad}\)
Way 2: \(A - B\) \(m = \frac{4 - 12}{2 - 6} = \underline{\quad\quad}\)
Conclusion: What rule must you follow?
Hint: Watch negative ÷ negative!
Slope Mastery Worksheet • Page 2 of 2 Great job mastering slope!
