Lesson Plan
Graphing Linear Functions
Students will be able to accurately graph linear functions from equations, identify key features like slope and y-intercept, and interpret graphs in real-world contexts.
Understanding linear functions is crucial for interpreting data, making predictions, and solving real-world problems in science, finance, and engineering. This lesson builds foundational analytical skills.
Audience
11th Grade Students
Time
30 minutes
Approach
Interactive discussion, guided examples, and practical application.
Prep
Teacher Preparation
15 minutes
- Review the Graphing Linear Functions Slide Deck and practice presenting the content smoothly.
- Print copies of the Linear Functions Warm Up for each student.
- Print copies of the Graphing Practice Worksheet for each student.
- Have the Graphing Practice Answer Key ready for quick reference.
- Prepare copies of the Linear Functions Cool Down for each student.
Step 1
Introduction & Warm Up
5 minutes
- Begin by distributing the Linear Functions Warm Up.
- Instruct students to complete the warm-up independently. This should activate prior knowledge on plotting points and basic equation recognition.
- After 3 minutes, briefly review answers as a class, addressing any immediate misconceptions.
Step 2
Direct Instruction: What's a Line Got to Do With It?
10 minutes
- Use the Graphing Linear Functions Slide Deck to introduce linear functions.
- Slide 2: What's a Line Got to Do With It? Engage students with real-world examples of linear relationships (e.g., speed vs. distance, cost vs. quantity).
- Slide 3: Anatomy of a Linear Equation Explain the standard form y = mx + b, defining 'm' as slope and 'b' as y-intercept.
- Slide 4: Slope: Your Guide to Steepness Dive into slope, explaining it as rise over run. Provide a quick example.
- Slide 5: Y-Intercept: Where the Story Begins Explain the y-intercept as the starting point on the y-axis.
- Slide 6: Graphing Step-by-Step Model how to graph a linear function using the y-intercept and slope. Work through one example together.
Step 3
Guided Practice: Let's Draw Some Lines!
10 minutes
- Distribute the Graphing Practice Worksheet.
- Work through the first one or two problems on the worksheet together as a class, providing step-by-step guidance.
- Encourage students to ask questions and share their thought processes.
- Circulate around the room, offering support and clarifying instructions as students begin working independently on the remaining problems.
Step 4
Wrap-Up & Cool Down
5 minutes
- Bring the class back together. Quickly review one or two of the more challenging problems from the Graphing Practice Worksheet if time permits.
- Distribute the Linear Functions Cool Down.
- Have students complete the cool-down independently as an exit ticket.
- Collect the cool-downs to assess student understanding.
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Slide Deck
Graphing Linear Functions: Your Map to Lines!
Today, we're becoming architects of algebra! We'll learn how to draw lines from equations and understand what those lines tell us.
Let's get ready to plot some points and discover the power of lines!
Welcome students and introduce the day's topic: Graphing Linear Functions. Explain that by the end of this lesson, they'll be able to create their own graphs from equations.
What's a Line Got to Do With It?
Linear functions are everywhere! They help us understand:
- Distance traveled at a steady speed.
- The cost of buying multiple items.
- How much money you earn per hour.
They show us patterns of consistent change!
Engage students with relatable scenarios where linear relationships are important. Ask them to think about how lines can represent consistent change.
Anatomy of a Linear Equation
The most common form for a linear equation is:
y = mx + b
Where:
- y is the output (what we calculate).
- x is the input (what we start with).
- m is the slope (how steep the line is).
- b is the y-intercept (where the line crosses the y-axis).
Introduce the standard slope-intercept form. Emphasize that 'm' and 'b' are the crucial pieces of information we'll use.
Slope: Your Guide to Steepness!
The slope (m) tells us the direction and steepness of our line.
It's often described as:
Rise / Run
- Rise: How much the line goes up or down (vertical change).
- Run: How much the line goes left or right (horizontal change).
If m = 2, it means for every 1 unit you move right, you move 2 units up. (2/1)
Explain slope with 'rise over run'. Give a simple example or ask students to imagine walking on a slope.
Y-Intercept: Where the Story Begins
The y-intercept (b) is the point where our line crosses the y-axis.
It's like the starting block of our graph.
When x = 0, y = b. So, the y-intercept is always the point (0, b).
Explain the y-intercept as the starting point on the y-axis. This is where our line 'begins' for graphing purposes.
Graphing Step-by-Step!
Let's graph y = 2x + 1
-
Identify the y-intercept (b): Here, b = 1. So, plot the point (0, 1).
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Identify the slope (m): Here, m = 2 (or 2/1).
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Use the slope to find more points: From (0, 1), move up 2 (rise) and right 1 (run) to get a new point. Repeat!
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Draw your line! Connect the points to form a straight line.
Walk through a step-by-step example. Use y = 2x + 1 or similar. First plot (0,1), then use slope 2/1 to find the next points.
Warm Up
Linear Functions Warm Up
Instructions: Take a few minutes to complete these problems. Do your best to recall what you already know!
-
Plot the following points on a coordinate plane:
- A: (2, 3)
- B: (-1, 4)
- C: (0, -2)
- D: (3, 0)
-
For the equation
y = 3x - 2:- What is the slope (m)?
- What is the y-intercept (b)?
- What is the slope (m)?
-
If a line passes through the points (1, 5) and (3, 9), what is the
Worksheet
Graphing Practice Worksheet
Instructions: For each equation below, identify the slope (m) and y-intercept (b), then graph the line on the provided coordinate plane.
1. y = 2x + 3
- Slope (m):
- Y-intercept (b):
Graph:
2. y = -1/2x + 4
- Slope (m):
- Y-intercept (b):
Graph:
3. y = 3x - 1
- Slope (m):
- Y-intercept (b):
Graph:
4. y = -2x - 2
- Slope (m):
- Y-intercept (b):
Graph:
5. y = x + 0 (or y = x)
- Slope (m):
- Y-intercept (b):
Graph:
Answer Key
Graphing Practice Answer Key
1. y = 2x + 3
- Slope (m): 2 (or 2/1)
- Thought Process: The coefficient of x is 2, which represents the slope. As a fraction, it's 2/1, meaning a rise of 2 and a run of 1.
- Y-intercept (b): 3
- Thought Process: The constant term in the equation is 3, which is the y-intercept. This means the line crosses the y-axis at (0, 3).
Graph Description: Plot the y-intercept at (0, 3). From there, move up 2 units and right 1 unit to find another point (1, 5). Connect these points with a straight line.
2. y = -1/2x + 4
- Slope (m): -1/2
- Thought Process: The coefficient of x is -1/2, which is the slope. This means a rise of -1 (down 1) and a run of 2.
- Y-intercept (b): 4
- Thought Process: The constant term is 4, so the y-intercept is at (0, 4).
Graph Description: Plot the y-intercept at (0, 4). From there, move down 1 unit and right 2 units to find another point (2, 3). Connect these points with a straight line.
3. y = 3x - 1
- Slope (m): 3 (or 3/1)
- Thought Process: The coefficient of x is 3, representing the slope. As a fraction, it's 3/1, meaning a rise of 3 and a run of 1.
- Y-intercept (b): -1
- Thought Process: The constant term is -1, so the y-intercept is at (0, -1).
Graph Description: Plot the y-intercept at (0, -1). From there, move up 3 units and right 1 unit to find another point (1, 2). Connect these points with a straight line.
4. y = -2x - 2
- Slope (m): -2 (or -2/1)
- Thought Process: The coefficient of x is -2, representing the slope. As a fraction, it's -2/1, meaning a rise of -2 (down 2) and a run of 1.
- Y-intercept (b): -2
- Thought Process: The constant term is -2, so the y-intercept is at (0, -2).
Graph Description: Plot the y-intercept at (0, -2). From there, move down 2 units and right 1 unit to find another point (1, -4). Connect these points with a straight line.
5. y = x + 0 (or y = x)
- Slope (m): 1 (or 1/1)
- Thought Process: The coefficient of x is 1 (implied), representing the slope. As a fraction, it's 1/1, meaning a rise of 1 and a run of 1.
- Y-intercept (b): 0
- Thought Process: There is no constant term, implying the y-intercept is 0, so the line passes through the origin (0, 0).
Graph Description: Plot the y-intercept at (0, 0). From there, move up 1 unit and right 1 unit to find another point (1, 1). Connect these points with a straight line.
Cool Down
Linear Functions Cool Down
Instructions: Please answer the following questions to reflect on what you learned today.
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Describe, in your own words, the two key pieces of information you need from a linear equation (
y = mx + b) to graph it, and why each is important.
-
If you were given the equation
y = -3x + 5, what would be your very first step to graph it? Why?
-
Think about a real-world situation that could be represented by a linear function. Describe it and explain what the slope and y-intercept would represent in that context.