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 Lesson Plan: What Is Slope?

 

Lesson Summary

This lesson introduces the concept of slope as a fundamental aspect of linear relationships in mathematics. Students will learn to define slope as the ratio of vertical change (rise) to horizontal change (run) between two points on a line. Through real-world examples, such as analyzing the steepness of staircases, students will explore how slope quantifies the steepness and direction of lines. The lesson emphasizes practical applications, enabling students to calculate and interpret slope in various contexts.

Lesson Objectives

  • Define slope as a ratio of the vertical change to the horizontal change between two points on a line.
  • Identify slope as a measure of the steepness or pitch of a line.
  • Calculate slopes of staircases using measurements for the rise over the run.

Common Core Standards

  • 8.EE.B.5 Graph proportional relationships, interpreting the unit rate as the slope
  • 8.EE.B.6 Use similar triangles to explain slope as the "rise over run"
  • HSF-IF.B.6 Calculate and interpret the average rate of change of a function

Prerequisite Skills

To ensure success in this lesson, students should have proficiency in the following areas:

  • Understanding of Ratios and Proportions: Ability to compare two quantities and comprehend their relationships.
  • Basic Coordinate Plane Navigation: Familiarity with plotting and identifying points on a Cartesian plane.
  • Fundamental Arithmetic Operations: Competence in performing addition, subtraction, multiplication, and division.
  • Measurement Skills: Capability to measure lengths and distances accurately.

Key Vocabulary

Multimedia Resources

 

 

Warm Up Activities

Choose from one or more activities.

Activity 1: Image Slide Show

Activity 2: Building Staircases with Cubes

Students will use cubes (such as snap cubes or LEGOs) to build different staircases of varying steepness. They will compare staircases visually and describe which ones are steeper and which are flatter. At this stage, students will focus on developing an intuitive sense of steepness without performing any calculations.

 

Slope

 

Activity 3: Exploring Ratios with Everyday Objects

Before introducing slope, students will practice working with ratios using familiar objects. Show pictures of different sports balls (e.g., basketballs, soccer balls, tennis balls) and ask students to compare their sizes by setting up ratios (e.g., the size of a basketball compared to a tennis ball). Other examples could include the number of red marbles to blue marbles in a jar or the number of chairs to tables in the classroom. This activity builds a foundational understanding of ratios, which will later be applied to slope.

 

Slope

 

 

Teach

Key Vocabulary

Concept Development

Show students the following video on slope. This video uses staircases as the context for introducing slope. It connects slopes to ratios and calculates slope as the rise over the run. This is a precursor to introducing the slope formula in subsequent lessons.

https://www.media4math.com/library/75378/asset-preview

Math Examples

Show the following examples for calculating the slope of different staircases using the values for the rist and the run.

https://www.media4math.com/library/75376/asset-preview

For each of these examples, point out to students that a calculation for the slope was made with a single step. Ask what assumptions are made about the calculated slope. (All steps have the same rise over run.)

Go back to each of the examples and calculate the total Rise and Run based on the measurements given for the individual stair. Point out to students that the slope calculations are identical. Ask them why this is the case. (The ratios are proportional.)

Use this set of images to have different students calculate the slopes for different staircases and compare their results:

https://www.media4math.com/library/75377/asset-preview

Ask students why their results are the same, even though different numbers were used in the slope calculations.

Example 1: Calculating the Slope of a Staircase

Given the rise and run of a staircase, students will calculate the slope using the slope formula:

\[ \text{slope} = \frac{\text{rise}}{\text{run}} \]

For example, if a staircase has a rise of 4 feet and a run of 12 feet:

  • Step 1: Identify rise = 4 feet and run = 12 feet.
  • Step 2: Apply the formula: \[ \frac{4}{12} \]
  • Step 3: Simplify the fraction: \[ \frac{1}{3} \]
  • Step 4: Interpret: The staircase has a slope of \( \frac{1}{3} \), meaning for every 3 feet of horizontal distance, the stairs rise 1 foot.

Example 2: Comparing the Steepness of Two Staircases

Given two staircases with the following measurements:

  • Staircase A: rise = 6 feet, run = 18 feet
  • Staircase B: rise = 5 feet, run = 10 feet

Calculate the slope of each staircase:

  • Staircase A: \[ \frac{6}{18} = \frac{1}{3} \]
  • Staircase B: \[ \frac{5}{10} = \frac{1}{2} \]

Since \( \frac{1}{2} \) is greater than \( \frac{1}{3} \), Staircase B is steeper than Staircase A.

Example 3: Determining Rise and Run from an Image

Students will analyze an image of a staircase with a total of 5 steps. Each step has a height (rise) of 1 foot and a horizontal distance (run) of 1 foot. The goal is to determine the total rise and run of the staircase and then calculate the slope.

  • Step 1: Count the number of steps: 5
  • Step 2: Multiply to get total rise: \[ 5 \times 1 = 5 \text{ feet (total rise)} \]
  • Step 3: Multiply to get total run: \[ 5 \times 1 = 5 \text{ feet (total run)} \]
  • Step 4: Apply the slope formula: \[ \frac{5}{5} = 1 \]

Final Answer: The staircase has a slope of 1, meaning for every 1 foot of horizontal distance, the stairs rise 1 foot.

 

Slope

 

Example 4: Comparing Two Staircases from an Image

Students will analyze an image of two staircases side by side, each with different dimensions. The goal is to calculate and compare their slopes.

SlopeSlope

Staircase A: (Same as Example 3)

  • Number of steps: 5
  • Each step’s rise: 1 foot
  • Each step’s run: 1 foot

Calculations:

  • Total rise: \[ 5 \times 1 = 5 \text{ feet} \]
  • Total run: \[ 5 \times 1 = 5 \text{ feet} \]
  • Slope calculation: \[ \frac{5}{5} = 1 \]

Staircase B: (Less steep staircase)

  • Number of steps: 5
  • Each step’s rise: \( \frac{1}{2} \) foot
  • Each step’s run: 1 foot

Calculations:

  • Total rise: \[ 5 \times \frac{1}{2} = \frac{5}{2} = 2.5 \text{ feet} \]
  • Total run: \[ 5 \times 1 = 5 \text{ feet} \]
  • Slope calculation: \[ \frac{2.5}{5} = \frac{1}{2} \]

Final Comparison: Staircase A has a slope of \( 1 \) while Staircase B has a slope of \( \frac{1}{2} \). Since \( 1 > \frac{1}{2} \), Staircase A is steeper.

Example 5: Calculating the Pitch of a Roof

The pitch of a roof is calculated similarly to slope. The formula is:

\[ \text{pitch} = \frac{\text{rise}}{\text{run}} \]

Example:

  • A roof rises 8 feet over a horizontal distance (run) of 24 feet.
  • Calculate the pitch: \[ \frac{8}{24} = \frac{1}{3} \]
  • Interpretation: For every 3 feet of horizontal distance, the roof rises 1 foot.

 

Slope

 

 

Review

Lesson Summary

In this lesson, we explored the concept of slope, which is a measure of the steepness of a line. We learned that slope is calculated as the ratio of the vertical change (rise) to the horizontal change (run). Using real-world examples like staircases, ramps, and roofs, we developed an understanding of how slope is applied in practical situations. By practicing calculations and comparisons, students gained the skills to determine and interpret slope in different contexts.

Review of Key Vocabulary

  • Slope: A measure of the steepness of a line, calculated as the ratio of the rise to the run.
  • Rise: The vertical change between two points on a line or staircase.
  • Run: The horizontal change between two points on a line or staircase.
  • Ratio: A comparison of two quantities by division.
  • Pitch: A term often used in construction to describe the slope of a roof.
  • Linear Relationship: A relationship between two variables that forms a straight line when graphed.
  • Coordinate Plane: A two-dimensional grid used to graph points and determine slopes of lines.
  • Proportional Relationship: A relationship where two quantities change at the same rate, maintaining a constant ratio.

Example 1: Calculating the Slope of a Staircase

Given a staircase with a total rise of 6 feet and a total run of 12 feet, calculate the slope.

  • Step 1: Identify the values:
    • Rise = 6 feet
    • Run = 12 feet
  • Step 2: Apply the slope formula: \[ \frac{6}{12} = \frac{1}{2} \]
  • Step 3: Interpretation: The staircase has a slope of \( \frac{1}{2} \), meaning it rises 1 foot for every 2 feet of horizontal distance.

Example 2: Comparing the Steepness of Two Staircases

Two staircases have the following measurements:

  • Staircase A: Rise = 5 feet, Run = 10 feet
  • Staircase B: Rise = 4 feet, Run = 6 feet

Calculate the slope of each staircase:

  • Staircase A: \[ \frac{5}{10} = \frac{1}{2} \]
  • Staircase B: \[ \frac{4}{6} = \frac{2}{3} \]

Since \( \frac{2}{3} \) is greater than \( \frac{1}{2} \), Staircase B is steeper.

Example 3: Calculating the Slope of a Ramp

A wheelchair ramp rises 8 feet over a horizontal distance (run) of 24 feet. Calculate the slope of the ramp.

  • Step 1: Identify the values:
    • Rise = 8 feet
    • Run = 24 feet
  • Step 2: Apply the slope formula: \[ \frac{8}{24} = \frac{1}{3} \]
  • Step 3: Interpretation: The slope of the ramp is \( \frac{1}{3} \), meaning for every 3 feet of horizontal distance, the roof rises 1 foot.

 

Slope

 

 

Quiz

Answer the following questions.

  1. Define slope in your own words.
  2. Which formula is used to calculate the slope of a line?
  3. A staircase has a rise of 8 feet and a run of 16 feet. What is the slope?
  4. Compare two staircases. Which staircase is steeper?
    Staircase A: Rise = 6 feet, Run = 9 feet
    Staircase B: Rise = 4 feet, Run = 10 feet
  5. Given a staircase where each step has a 7-inch rise and a 10-inch run, and the staircase has 5 steps, what is the total slope?
  6. You are given an image of a roof with a rise of 4 feet and a run of 12 feet. What is the pitch of the roof?
  7. A ramp rises 3 feet over a horizontal distance of 15 feet. What is its slope?
  8. A hiking trail gains 500 feet in elevation over a horizontal distance of 2000 feet. Express the slope as a simplified fraction.
  9. If the slope of a staircase is 1, what does that tell you about its rise and run?
  10. A set of stairs has a total rise of 9 feet and a total run of 18 feet. If you double both the rise and run, does the slope change? Why or why not?

Answer Key

  1. Slope is a measure of the steepness of a line, defined as the ratio of vertical change (rise) to horizontal change (run).
  2. The formula is: \[ \text{slope} = \frac{\text{rise}}{\text{run}} \]
  3.  \[ \frac{8}{16} = \frac{1}{2} \]
  4. Staircase A: \( \frac{6}{9} = \frac{2}{3} \)
    Staircase B: \( \frac{4}{10} = \frac{2}{5} \)
    Staircase A is steeper because \( \frac{2}{3} > \frac{2}{5} \).
  5. Total rise: \( 5 \times 7 = 35 \) inches
    Total run: \( 5 \times 10 = 50 \) inches
    Slope: \[ \frac{35}{50} = \frac{7}{10} \]
  6. \[ \frac{4}{12} = \frac{1}{3} \]
  7. \[ \frac{3}{15} = \frac{1}{5} \]
  8. \[ \frac{500}{2000} = \frac{1}{4} \]
  9. A slope of 1 means that for every 1 unit of horizontal distance, the staircase rises 1 unit. This creates a 45-degree incline.
  10. Answer:

Original slope: \( \frac{9}{18} = \frac{1}{2} \)

New rise: \( 9 \times 2 = 18 \), New run: \( 18 \times 2 = 36 \)

New slope: \( \frac{18}{36} = \frac{1}{2} \)

The slope remains the same because multiplying both the rise and run by the same factor does not change the ratio.