Coordinate Slope Coordinate Slope
Finding Slope: Rate of Change Using \(\text{Slope } (m) = \frac{y_2 - y_1}{x_2 - x_1}\)
Name:
Date:
Period:
The Slope Formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\)
Step 1: Label All 4 Label \(x_1, y_1\) and \(x_2, y_2\)
Step 2: Subtract Always place \(y_2 - y_1\) on TOP!
Part 1: Find Slope from Graphs (Label Points & Write in Fraction Boxes) 1. Label Coordinates → 2. Subtract → 3. Divide
Graph 1 x y 2 -2 2 -2 A B
Problem 1 Uphill = Positive
Step 1: Coordinates & Labels
Pt A: ( , ) \(x_1 =\) \(y_1 =\)
Pt B: ( , ) \(x_2 =\) \(y_2 =\)
Step 2: Calculate Differences
\(y_2 - y_1 =\) \(-\) \(=\)
\(x_2 - x_1 =\) \(-\) \(=\)
Step 3: Slope \(m = \frac{y_2 - y_1}{x_2 - x_1}\):
\(m =\)
= Slope \(m =\)
Graph 2 x y 2 -2 2 -2 C D
Problem 2 Downhill = Negative
Step 1: Coordinates & Labels
Pt C: ( , ) \(x_1 =\) \(y_1 =\)
Pt D: ( , ) \(x_2 =\) \(y_2 =\)
Step 2: Calculate Differences
\(y_2 - y_1 =\) \(-\) \(=\)
\(x_2 - x_1 =\) \(-\) \(=\)
Step 3: Slope \(m = \frac{y_2 - y_1}{x_2 - x_1}\):
\(m =\)
= Slope \(m =\)
Graph 3 x y 2 -2 2 -2 E F
Problem 3 Fraction Slope
Step 1: Coordinates & Labels
Pt E: ( , ) \(x_1 =\) \(y_1 =\)
Pt F: ( , ) \(x_2 =\) \(y_2 =\)
Step 2: Calculate Differences
\(y_2 - y_1 =\) \(-\) \(=\)
\(x_2 - x_1 =\) \(-\) \(=\)
Step 3: Slope \(m = \frac{y_2 - y_1}{x_2 - x_1}\):
\(m =\)
= Slope \(m =\)
Label \(x_1, y_1\) and \(x_2, y_2\) carefully before subtracting! Page 1 of 2 • Turn over for Part 2
Coordinate Slope Practice • Part 2
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| Page 2 of 2
RULE Formula: \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Label \(x_1, y_1\) and \(x_2, y_2\) before subtracting!
\(y_2 - y_1\) stays on TOP
Part 2: Find the Slope from Two Points (No Graph Given) Step 1: Label \(x_1, y_1, x_2, y_2\) → Step 2: Subtract → Step 3: Divide
4 Points: \((2, 4)\) and \((6, 12)\)
Guided Steps
1. Label All 4 Coordinates:
\(x_1 = \) , \(y_1 = \)
\(x_2 = \) , \(y_2 = \)
2. Find \(y_2 - y_1\) & \(x_2 - x_1\):
\(y_2 - y_1 = \) \(-\) \(= \)
\(x_2 - x_1 = \) \(-\) \(= \)
3. Slope \(m = \frac{y_2 - y_1}{x_2 - x_1}\):
\(m =\)
=
5 Points: \((2, 1)\) and \((5, 10)\)
Positive Whole Number
1. Label All 4 Coordinates:
\(x_1 = \) , \(y_1 = \)
\(x_2 = \) , \(y_2 = \)
2. Find \(y_2 - y_1\) & \(x_2 - x_1\):
\(y_2 - y_1 = \) \(-\) \(= \)
\(x_2 - x_1 = \) \(-\) \(= \)
3. Slope \(m = \frac{y_2 - y_1}{x_2 - x_1}\):
\(m =\)
=
6 Points: \((1, 9)\) and \((5, 1)\)
Watch the sign!
1. Label All 4 Coordinates:
\(x_1 = \) , \(y_1 = \)
\(x_2 = \) , \(y_2 = \)
2. Find \(y_2 - y_1\) & \(x_2 - x_1\):
\(y_2 - y_1 = \) \(-\) \(= \)
\(x_2 - x_1 = \) \(-\) \(= \)
3. Slope \(m = \frac{y_2 - y_1}{x_2 - x_1}\):
\(m =\)
=
7 Points: \((2, 3)\) and \((6, 6)\)
Keep as fraction
1. Label All 4 Coordinates:
\(x_1 = \) , \(y_1 = \)
\(x_2 = \) , \(y_2 = \)
2. Find \(y_2 - y_1\) & \(x_2 - x_1\):
\(y_2 - y_1 = \) \(-\) \(= \)
\(x_2 - x_1 = \) \(-\) \(= \)
3. Slope \(m = \frac{y_2 - y_1}{x_2 - x_1}\):
\(m =\)
=
Slope Detective: Spot the Mistake!
Jordan was given points \((1, 2)\) and \((4, 8)\) . He calculated:
\(m = \frac{4 - 1}{8 - 2} = \frac{3}{6} = \frac{1}{2}\)
What error did Jordan make with \(y_2 - y_1\)?
Correct Slope:
Slope Formula: \(m = \frac{y_2 - y_1}{x_2 - x_1}\) Great Job! Always double-check your subtraction!
Slope Starter Warmup Slope Starter: Spot the Steeper Line
Warm-Up • Slope Formula: \(m = \frac{y_2 - y_1}{x_2 - x_1}\)
Name:
Period:
Mission: Label \(x_1, y_1, x_2, y_2\), calculate \(y_2 - y_1\) over \(x_2 - x_1\), and compare steepness!
Line A x y 2 4 2 4
Pt 1: (1, 1) \(x_1 =\) \(y_1 =\)
Pt 2: (4, 4) \(x_2 =\) \(y_2 =\)
\(y_2 - y_1 = \) \(-\) \(= \)
\(x_2 - x_1 = \) \(-\) \(= \)
\(m_A = \frac{y_2 - y_1}{x_2 - x_1} =\)
Line B x y 2 4 2 4
Pt 1: (1, 1) \(x_1 =\) \(y_1 =\)
Pt 2: (2, 4) \(x_2 =\) \(y_2 =\)
\(y_2 - y_1 = \) \(-\) \(= \)
\(x_2 - x_1 = \) \(-\) \(= \)
\(m_B = \frac{y_2 - y_1}{x_2 - x_1} =\)
Conclusion: Which line is steeper? Line [ A / B ]
Why? The steeper line has a slope of , which is greater than .
CUT HERE (2 COPIES PER PAGE)
Slope Starter: Spot the Steeper Line
Warm-Up • Slope Formula: \(m = \frac{y_2 - y_1}{x_2 - x_1}\)
Name:
Period:
Mission: Label \(x_1, y_1, x_2, y_2\), calculate \(y_2 - y_1\) over \(x_2 - x_1\), and compare steepness!
Line A x y 2 4 2 4
Pt 1: (1, 1) \(x_1 =\) \(y_1 =\)
Pt 2: (4, 4) \(x_2 =\) \(y_2 =\)
\(y_2 - y_1 = \) \(-\) \(= \)
\(x_2 - x_1 = \) \(-\) \(= \)
\(m_A = \frac{y_2 - y_1}{x_2 - x_1} =\)
Line B x y 2 4 2 4
Pt 1: (1, 1) \(x_1 =\) \(y_1 =\)
Pt 2: (2, 4) \(x_2 =\) \(y_2 =\)
\(y_2 - y_1 = \) \(-\) \(= \)
\(x_2 - x_1 = \) \(-\) \(= \)
\(m_B = \frac{y_2 - y_1}{x_2 - x_1} =\)
Conclusion: Which line is steeper? Line [ A / B ]
Why? The steeper line has a slope of , which is greater than .
Coordinate Slope Answer Key TEACHER KEY
Coordinate Slope Answer Key
Complete Worked Solutions • Page 1: Graph Practice • Formula: \(m = \frac{y_2 - y_1}{x_2 - x_1}\)
Answer Key • Page 1 of 2 8th Grade Mathematics
SCORING Each problem (3 pts): 1 pt for correct coordinate labeling • 1 pt for subtraction \((y_2 - y_1)\) • 1 pt for simplified slope \(m\).
Total: 24 Points
Part 1: Graph Problems (Problems 1–3 Worked Solutions) Answers shown in RED BOLD
Graph 1 xy A B
Problem 1 Solutions Positive Slope
Step 1: Coordinates & Labels
Pt A: (0, 1) → \(x_1 = \) 0 , \(y_1 = \) 1
Pt B: (2, 5) → \(x_2 = \) 2 , \(y_2 = \) 5
Step 2: Calculate Differences
\(y_2 - y_1 = \) 5 \(-\) 1 \(= \) 4
\(x_2 - x_1 = \) 2 \(-\) 0 \(= \) 2
Step 3: Slope \(m = \frac{y_2 - y_1}{x_2 - x_1}\):
\(m =\) \(\frac{4}{2}\) = Final Answer: m = 2
Graph 2 xy C D
Problem 2 Solutions Negative Slope
Step 1: Coordinates & Labels
Pt C: (-1, 3) → \(x_1 = \) -1 , \(y_1 = \) 3
Pt D: (2, -3) → \(x_2 = \) 2 , \(y_2 = \) -3
Step 2: Calculate Differences
\(y_2 - y_1 = \) -3 \(-\) 3 \(= \) -6
\(x_2 - x_1 = \) 2 \(-\) (-1) \(= \) 3
Step 3: Slope \(m = \frac{y_2 - y_1}{x_2 - x_1}\):
\(m =\) \(\frac{-6}{3}\) = Final Answer: m = -2
Graph 3 xy E F
Problem 3 Solutions Fraction Slope
Step 1: Coordinates & Labels
Pt E: (-2, -2) → \(x_1 = \) -2 , \(y_1 = \) -2
Pt F: (2, 1) → \(x_2 = \) 2 , \(y_2 = \) 1
Step 2: Calculate Differences
\(y_2 - y_1 = \) 1 \(-\) (-2) \(= \) 3
\(x_2 - x_1 = \) 2 \(-\) (-2) \(= \) 4
Step 3: Slope \(m = \frac{y_2 - y_1}{x_2 - x_1}\):
\(m =\) \(\frac{3}{4}\) = Final Answer: m = 3/4
Teacher Note: Watch out for subtraction with negative numbers: \(1 - (-2) = 1 + 2 = 3\). Answer Key • Page 1 of 2
TEACHER KEY
Slope Stepper Answer Key • Page 2
Part 2 & Warm-Up Keys | Page 2 of 2
Part 2: Two-Point Coordinate Problems (Problems 4–7 Worked Solutions) Formula: \(m = \frac{y_2 - y_1}{x_2 - x_1}\)
4 Points: \((2, 4)\) and \((6, 12)\)
Answer: \(m = 2\)
1. Labeled:
\(x_1 = \) 2 , \(y_1 = \) 4
\(x_2 = \) 6 , \(y_2 = \) 12
2. Differences:
\(y_2 - y_1 = 12 - 4 = \) 8
\(x_2 - x_1 = 6 - 2 = \) 4
3. Slope: \(m = \frac{8}{4}\)
m = 2
5 Points: \((2, 1)\) and \((5, 10)\)
Answer: \(m = 3\)
1. Labeled:
\(x_1 = \) 2 , \(y_1 = \) 1
\(x_2 = \) 5 , \(y_2 = \) 10
2. Differences:
\(y_2 - y_1 = 10 - 1 = \) 9
\(x_2 - x_1 = 5 - 2 = \) 3
3. Slope: \(m = \frac{9}{3}\)
m = 3
6 Points: \((1, 9)\) and \((5, 1)\)
Answer: \(m = -2\)
1. Labeled:
\(x_1 = \) 1 , \(y_1 = \) 9
\(x_2 = \) 5 , \(y_2 = \) 1
2. Differences:
\(y_2 - y_1 = 1 - 9 = \) -8
\(x_2 - x_1 = 5 - 1 = \) 4
3. Slope: \(m = \frac{-8}{4}\)
m = -2
7 Points: \((2, 3)\) and \((6, 6)\)
Answer: \(m = \frac{3}{4}\)
1. Labeled:
\(x_1 = \) 2 , \(y_1 = \) 3
\(x_2 = \) 6 , \(y_2 = \) 6
2. Differences:
\(y_2 - y_1 = 6 - 3 = \) 3
\(x_2 - x_1 = 6 - 2 = \) 4
3. Slope: \(m = \frac{3}{4}\)
m = 3/4
Error Analysis Solution (Jordan's Mistake)
What error did Jordan make?
Jordan put \(x_2 - x_1\) on top and \(y_2 - y_1\) on the bottom (Run over Rise). The \(y\)-values must ALWAYS be in the numerator!
Correct Calculation: \(m = \frac{8 - 2}{4 - 1} = \frac{6}{3}\)
m = 2
Quick Key: "Spot the Steeper Line" Bellringer Warm-Up Bellringer Key
Line A: \(y_2 - y_1 = 4 - 1 = 3\)\(m_A = 1\)
Line B: \(y_2 - y_1 = 4 - 1 = 3\)\(m_B = 3\)
Conclusion: Line B is steeper
Teacher Answer Key Complete • Slope Stepper Lesson Page 2 of 2
