WEEK 4
Grade 9 Mathematics
Slope and Rate of Change
Name: ____________________ Date: ____________________ Score: _______ / 111
CCSS: CCSS.MATH.CONTENT.HSF.IF.B.6 · Difficulty: medium
Fast Track
22 min
Standard
30 min
Thorough
38 min

Learning Objectives (What You'll Master Today)

I Can... I can calculate slope using the formula m = (y2 - y1) / (x2 - x1)

(A) Mathematical Concepts - Self-Learning Guide

Big Idea: Key Concept

Slope measures the steepness and direction of a line. It is calculated as the ratio of vertical change (rise) to horizontal change (run) between any two points: m = (y2 - y1) / (x2 - x1). A positive slope means the line rises from left to right, a negative slope means it falls, a zero slope is a horizontal line, and an undefined slope is a vertical line. Slope is not just a number on a graph — it represents rate of change. When you drive 60 miles in 2 hours, the slope of the distance-time graph is 30, meaning you travel 30 miles per hour. Rate of change appears everywhere: population growth, temperature shifts, speed, and cost per unit.

PhET: Interactive Simulation

🧪 Interactive
Simulation

Finding Slope x y -4 -3 -2 -1 1 2 3 4 5 6 -2 -1 1 2 3 4 5 6 7 8 9 10 0 1 ① y = x + 3 (-2, 1) (4, 7)

How-To Methods

  1. Identify Two Points: Pick any two points on the line, (x1, y1) and (x2, y2)
  2. Calculate Rise: Subtract the y-coordinates: rise = y2 - y1
  3. Calculate Run: Subtract the x-coordinates: run = x2 - x1
  4. Divide Rise by Run: m = rise / run = (y2 - y1) / (x2 - x1)
  5. Check Sign: Positive slope = line goes up; Negative slope = line goes down
  6. Interpret: State the slope as a rate of change in context (e.g., 3 means '3 units up for every 1 unit right')

Key Formula/Rule:

Slope Formula: m = (y2 − y1) / (x2 − x1)

Worked Example:

Problem: Find the slope of the line passing through (-2, 1) and (4, 7).

Label the points: Let (x1, y1) = (-2, 1) and (x2, y2) = (4, 7)

Calculate rise: rise = y2 - y1 = 7 - 1 = 6

Calculate run: run = x2 - x1 = 4 - (-2) = 4 + 2 = 6

Divide: m = 6 / 6 = 1

Interpret: The slope is 1, meaning the line rises 1 unit for every 1 unit to the right.

Answer: The slope is m = 1.

Why This Matters

Every time you check 'miles per gallon,' compare price per pound, or read that a city's population grows by 2,000 people per year, you are working with slope. Understanding rate of change helps you predict trends, compare options, and make informed decisions in science, business, and everyday life.

(B) Basic Practice

QR Code
Problem 1 Apply ★ [8 pts]
What is the slope of the line passing through (1, 3) and (4, 9)?
Two Points x y -1 1 2 3 4 5 6 -1 1 2 3 4 5 6 7 8 9 10 11 0 (1, 3) (4, 9)

A. m = 2

B. m = 3

C. m = 1/2

D. m = 6

QR Code
Problem 2 Apply ★★ [10 pts]
A line passes through (-1, 5) and (3, -7). What is its slope?

A. m = -4

B. m = 3

C. m = -3

D. m = 4

QR Code
Problem 3 Apply ★★ [10 pts]
The table below shows values of x and y. What is the constant rate of change?
Input-Output Table x y 1 4 3 12 5 20 7 28

A. 2

B. 4

C. 8

D. 3

QR Code
Problem 4 Apply ★ [8 pts]
Which term best describes the slope of a horizontal line?

A. Positive

B. Negative

C. Undefined

D. Zero

(C) Problem Solving (Real-World Applications)

QR Code
Problem 5 Apply ★★ [10 pts]
A ramp rises 6 feet over a horizontal distance of 24 feet. What is the slope of the ramp?

A. 1/4

B. 4

C. 1/6

D. 6

QR Code
Problem 6 Apply ★★ [12 pts]
A car travels at a constant speed. After 2 hours it has gone 90 miles, and after 5 hours it has gone 225 miles. What is the car's speed (rate of change) in miles per hour?
Distance Traveled 247 185 123 61 0 0 2 5 Distance (mi) Time (hours)

A. 40 mph

B. 50 mph

C. 45 mph

D. 55 mph

QR Code
Problem 7 Apply ★★★ [12 pts]
The population of a town was 12,000 in 2010 and 18,000 in 2016. What was the average rate of change in population per year?

A. 500 people/year

B. 1,000 people/year

C. 1,500 people/year

D. 6,000 people/year

QR Code
Problem 8 Apply ★★★ [12 pts]
A line has slope m = -2 and passes through the point (3, 7). What is the y-coordinate when x = 6?

A. -1

B. 3

C. 5

D. 1

(D) Advanced Challenge (Higher-Order Thinking)

QR Code
Problem 9 Apply ★★★ [14 pts]
For the function f(x) = x² - 3x + 2, what is the average rate of change from x = 1 to x = 4?

A. 2

B. 3

C. 6

D. 1

QR Code
Problem 10 Apply ★★★★ [15 pts]
Line p passes through (1, 2) and (4, 8). Line q is perpendicular to line p. What is the slope of line q?
Line p x y -2 -1 1 2 3 4 5 6 -2 -1 1 2 3 4 5 6 7 8 9 10 0 1 ① Line p

A. m = 2

B. m = -2

C. m = -1/2

D. m = 1/2

(E) Mini Quiz - Check Your Understanding!

Mini Quiz

E1. What is the slope of the line through (0, 0) and (3, 12)?

A. m = 4

B. m = 3

C. m = 12

D. m = 1/4

E2. A temperature rises from 50°F to 74°F over 8 hours. What is the average rate of change?

A. 2°F per hour

B. 3°F per hour

C. 4°F per hour

D. 8°F per hour

E3. The slope of a line is 5/3. If the line rises 15 units vertically, what is the horizontal change?

A. 3

B. 5

C. 15

D. 9

Self-Reflection

How did you feel about this topic?
Confident / Okay / Confused - Circle one

What is one thing you must remember?


Problem-Solving Strategy (4-Step Process)

Challenge Problem: A candle is 12 inches tall when first lit. After 2 hours, it is 9 inches tall, and after 5 hours, it is 4.5 inches tall. Is the rate of change of the candle’s height constant? If so, predict when the candle will burn out completely.

1. Understand:


2. Plan:


3. Execute:



4. Check:


(F) Answer Key & Detailed Explanations

# Answer Type Solution
1 A Apply final_step: The slope is 2. Answer: A
step_1: Label points: (x1, y1) = (1, 3) and (x2, y2) = (4, 9)
step_2: m = (9 - 3) / (4 - 1) = 6 / 3 = 2
2 C Apply final_step: The slope is -3. Answer: C
step_1: Label points: (x1, y1) = (-1, 5) and (x2, y2) = (3, -7)
step_2: m = (-7 - 5) / (3 - (-1)) = -12 / 4 = -3
3 B Apply final_step: The constant rate of change is 4. Answer: B
step_1: Pick two rows: (1, 4) and (3, 12)
step_2: Rate of change = (12 - 4) / (3 - 1) = 8 / 2 = 4
step_3: Verify with another pair: (5, 20) and (7, 28) gives (28 - 20) / (7 - 5) = 8 / 2 = 4
4 D Apply final_step: The slope of a horizontal line is zero. Answer: D
step_1: A horizontal line has the same y-value at every point, so rise = 0
step_2: m = 0 / run = 0, regardless of the run
5 A Apply final_step: The slope of the ramp is 1/4. Answer: A
step_1: Slope = rise / run = 6 / 24
step_2: Simplify: 6/24 = 1/4
6 C Apply final_step: The car's speed is 45 mph. Answer: C
step_1: Change in distance = 225 - 90 = 135 miles
step_2: Change in time = 5 - 2 = 3 hours
step_3: Rate = 135 / 3 = 45 miles per hour
7 B Apply final_step: The average rate of change is 1,000 people per year. Answer: B
step_1: Change in population = 18,000 - 12,000 = 6,000
step_2: Change in time = 2016 - 2010 = 6 years
step_3: Rate = 6,000 / 6 = 1,000 people per year
8 D Apply final_step: The y-coordinate when x = 6 is 1. Answer: D
step_1: From x = 3 to x = 6, the change in x is 3
step_2: Change in y = slope × change in x = -2 × 3 = -6
step_3: New y = 7 + (-6) = 1
9 A Apply final_step: The average rate of change is 2. Answer: A
step_1: Calculate f(1) = (1)² - 3(1) + 2 = 1 - 3 + 2 = 0
step_2: Calculate f(4) = (4)² - 3(4) + 2 = 16 - 12 + 2 = 6
step_3: Average rate of change = (f(4) - f(1)) / (4 - 1) = (6 - 0) / 3 = 2
10 C Apply final_step: The slope of line q is -1/2. Answer: C
step_1: Find slope of line p: m = (8 - 2) / (4 - 1) = 6 / 3 = 2
step_2: Perpendicular slopes satisfy m1 × m2 = -1
step_3: 2 × m2 = -1, so m2 = -1/2

(G) Mini Quiz Answers

Multiple Choice:

E1. A

E2. B

E3. D

Self-Reflection:

I feel pretty good about calculating slope from two points using the formula. The part I need to be most careful about is keeping the subtraction order consistent — if I use (y2 - y1) on top, I must use (x2 - x1) on the bottom, not the reverse.

Problem-Solving Strategy (4-Step):

1. Understand: A candle is 12 inches tall when first lit. After 2 hours it is 9 inches tall, and after 5 hours it is 4.5 inches tall. We need to determine whether the rate of change is constant and, if so, predict when the candle will completely burn out.

2. Plan: Calculate the rate of change over two separate intervals (0 to 2 hours and 2 to 5 hours). If both rates are equal, the rate is constant. Then use the constant rate to find when the height reaches 0.

3. Execute: Rate (0 to 2 hr) = (9 - 12)/(2 - 0) = -3/2 = -1.5 in/hr. Rate (2 to 5 hr) = (4.5 - 9)/(5 - 2) = -4.5/3 = -1.5 in/hr. Both rates equal -1.5, so the rate is constant. Time to burn out: 12 / 1.5 = 8 hours after lighting.

4. Check: Verify: height at 8 hr = 12 + (-1.5)(8) = 12 - 12 = 0. Also spot-check: at 2 hr, 12 - 1.5(2) = 9 ✓; at 5 hr, 12 - 1.5(5) = 4.5 ✓. All consistent.

(H) Error Analysis Notebook (Growth Mindset Tool)

Problem # My Score Error Type What I Learned / How to Improve
#9 0/15 Concept "I forgot to check my answer. Next time I will verify my solution."

Error Types: Concept (개념), Procedure (절차), Calculation (계산), Reading (문제 이해), Careless (실수)

(I) Review Schedule

Review this worksheet on these dates for maximum retention:

Today (Initial Learning)
Tomorrow (Day 2)
Day 3 (Short-term consolidation)
Week 5 (1 week later)
Month 1 (Long-term retention)

Why this works: Ebbinghaus forgetting curve research shows spaced repetition increases retention by 200%+

학부모 가이드 (Parent Guide)

  1. 기울기와 변화율
  2. 아이에게 'When you walk up a hill, what makes one hill harder to climb than another?'라고 물어보며 기울기 개념을 시작해보세요.
  3. 자동차 속도를 예시로 'If we drive 120 miles in 3 hours, how do we find our speed?'라고 물어보세요.
  4. 계단을 오르며 'How much do you go up for each step forward?'라고 물어 rise/run을 체험시켜주세요.
  5. 그래프를 보며 'Does this line go up, go down, or stay flat — and how steep is it?'라고 물어 기울기의 방향과 크기를 이해시켜주세요.

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