Lesson Plan
Slope: Rise Over Run
Students will be able to define slope, calculate it from two given points using the slope formula, and interpret the meaning of slope in real-world scenarios.
Understanding slope is fundamental to describing rates of change in the world around us, from road inclines to financial growth, and is a critical building block for future math concepts.
Audience
8th Grade Students
Time
30 minutes
Approach
Direct instruction, guided practice, and independent application.
Materials
Whiteboard or Projector, Markers/Pens, Finding Slope Slide Deck, Finding Slope Worksheet, and Finding Slope Answer Key
Prep
Preparation
10 minutes
- Review the Finding Slope Slide Deck to familiarize yourself with the content and flow.
* Print copies of the Finding Slope Worksheet and Finding Slope Answer Key for each student.
* Ensure your projector or whiteboard is ready for presentation.
* Prepare markers or pens for writing.
Step 1
Introduction & Hook
5 minutes
- Begin by asking students: "What does it mean for something to be 'steep'? Where do we see steepness in our daily lives?"
* Discuss examples like hills, roofs, or ramps. Introduce the idea that in math, we call this 'slope.'
* Briefly explain that slope helps us measure how steep a line is. (Refer to Finding Slope Slide Deck - Slides 1-2)
Step 2
Defining Slope & Formula
10 minutes
- Use the Finding Slope Slide Deck (Slides 3-6) to introduce the formal definition of slope as 'rise over run.'
* Explain the slope formula: m = (y2 - y1) / (x2 - x1).
* Go through a step-by-step example of calculating slope from two points using the slide deck.
* Emphasize careful subtraction and attention to positive/negative values.
Step 3
Guided Practice
8 minutes
- Work through 1-2 additional examples together as a class, encouraging students to volunteer steps for calculation.
* Distribute the Finding Slope Worksheet. Have students work on the first few problems independently or with a partner while you circulate and provide support.
* Address common misconceptions or difficulties as they arise.
Step 4
Wrap-up & Independent Practice
7 minutes
- Review the answers to the first few problems on the worksheet as a class.
* Assign the remainder of the Finding Slope Worksheet for homework or independent practice.
* Conclude by reiterating the importance of slope in real-world contexts and as a building block for future math.
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Slide Deck
Slope: Rise Over Run
Understanding Steepness in Our World
- What does it mean for something to be 'steep'?
- Where do you see steepness in everyday life?
Welcome students and introduce the day's topic. Ask students what comes to mind when they hear the word 'steep' or 'slope.' Engage them with real-world examples. Consider adding an image of a steep mountain road or ski slope here.
What is Slope?
The Measurement of Steepness
- In mathematics, slope tells us how steep a line is.
- It measures the rate of change of the vertical distance (rise) to the horizontal distance (run).
- We often use the letter 'm' to represent slope.
Transition from everyday steepness to the mathematical concept of slope. Explain that slope gives us a way to measure this steepness numerically. Consider adding a diagram showing lines with varying degrees of steepness (flat, moderately steep, very steep, positive, negative).
Rise Over Run
Visualizing Slope
- Rise: The vertical change between two points (up or down).
- Run: The horizontal change between two points (left or right).
- Slope (m) = Rise / Run
Introduce the 'rise over run' concept visually. Explain that 'rise' is the change in y-values and 'run' is the change in x-values. Consider adding a diagram illustrating 'rise' and 'run' on a coordinate plane between two points, perhaps with arrows.
The Slope Formula
Calculating Slope from Two Points
To find the slope (m) between two points (x₁, y₁) and (x₂, y₂), we use the formula:
m = (y₂ - y₁) / (x₂ - x₁)
- (x₁, y₁) represents the coordinates of the first point.
- (x₂, y₂) represents the coordinates of the second point.
Present the formal slope formula. Emphasize the importance of consistency when labeling points (x1, y1) and (x2, y2).
Example 1: Let's Calculate!
Find the slope of the line passing through (2, 1) and (6, 9).
-
Label your points:
(x₁, y₁) = (2, 1)
(x₂, y₂) = (6, 9) -
Apply the formula:
m = (y₂ - y₁) / (x₂ - x₁)
m = (9 - 1) / (6 - 2) -
Simplify:
m = 8 / 4
m = 2
Walk through the first example step-by-step. Show how to label the points, substitute values into the formula, and simplify. Ask students for input at each step. Consider adding a graph with points (2, 1) and (6, 9) plotted and the rise/run clearly marked on the line connecting them.
Your Turn! Example 2
Find the slope of the line passing through (3, 7) and (5, 3).
- Hint: Remember to label your points carefully!
Present a second example for students to try independently or with a partner. Encourage them to follow the steps from the previous example. Review the answer afterwards. Consider adding a graph with points (3, 7) and (5, 3) plotted and the rise/run clearly marked on the line connecting them.
Practice Time!
Now, let's put your new slope skills to the test with a worksheet!
- Work independently or with a partner.
- Don't be afraid to ask questions!
Final slide to indicate the end of the direct instruction and the beginning of worksheet practice. Reiterate that practice makes perfect.
Worksheet
Slope Sleuths: Finding Your Way with Rise Over Run
Name: _____________________________
Instructions: For each pair of points, calculate the slope (m) of the line that passes through them. Show your work!
Remember: Slope (m) = (y₂ - y₁) / (x₂ - x₁)
(Visual Aid Suggestion: Imagine a small coordinate plane here, showing two points connected by a line, with a right triangle drawn to illustrate the 'rise' (vertical change) and 'run' (horizontal change) between them.)
-
(2, 3) and (6, 5)
-
(1, 8) and (3, 2)
-
(-4, 1) and (2, 7)
-
(5, -3) and (1, 5)
-
(-2, -6) and (0, 4)
-
(7, 2) and (7, 9)
-
(0, 0) and (4, -8)
-
(-3, 5) and (6, 5)
Answer Key
Slope Sleuths: Answer Key
Remember: Slope (m) = (y₂ - y₁) / (x₂ - x₁)
-
(2, 3) and (6, 5)
- (x₁, y₁) = (2, 3)
- (x₂, y₂) = (6, 5)
- m = (5 - 3) / (6 - 2)
- m = 2 / 4
- m = 1/2
-
(1, 8) and (3, 2)
- (x₁, y₁) = (1, 8)
- (x₂, y₂) = (3, 2)
- m = (2 - 8) / (3 - 1)
- m = -6 / 2
- m = -3
-
(-4, 1) and (2, 7)
- (x₁, y₁) = (-4, 1)
- (x₂, y₂) = (2, 7)
- m = (7 - 1) / (2 - (-4))
- m = 6 / (2 + 4)
- m = 6 / 6
- m = 1
-
(5, -3) and (1, 5)
- (x₁, y₁) = (5, -3)
- (x₂, y₂) = (1, 5)
- m = (5 - (-3)) / (1 - 5)
- m = (5 + 3) / -4
- m = 8 / -4
- m = -2
-
(-2, -6) and (0, 4)
- (x₁, y₁) = (-2, -6)
- (x₂, y₂) = (0, 4)
- m = (4 - (-6)) / (0 - (-2))
- m = (4 + 6) / (0 + 2)
- m = 10 / 2
- m = 5
-
(7, 2) and (7, 9)
- (x₁, y₁) = (7, 2)
- (x₂, y₂) = (7, 9)
- m = (9 - 2) / (7 - 7)
- m = 7 / 0
- m = Undefined (Vertical Line)
-
(0, 0) and (4, -8)
- (x₁, y₁) = (0, 0)
- (x₂, y₂) = (4, -8)
- m = (-8 - 0) / (4 - 0)
- m = -8 / 4
- m = -2
-
(-3, 5) and (6, 5)
- (x₁, y₁) = (-3, 5)
- (x₂, y₂) = (6, 5)
- m = (5 - 5) / (6 - (-3))
- m = 0 / (6 + 3)
- m = 0 / 9
- m = 0 (Horizontal Line)