Painting the Picture: Constructing Linear Functions from Real-World Scenarios
Lesson Description
Video Resource
Constructing linear functions example 2 | Algebra I | Khan Academy
Khan Academy
Key Concepts
- Linear functions
- Slope-intercept form (y = mx + b)
- Rate of change (slope)
- Initial value (y-intercept)
- Function notation
Learning Objectives
- Students will be able to identify the rate of change and initial value from a word problem.
- Students will be able to write a linear equation in slope-intercept form to represent a real-world scenario.
- Students will be able to interpret the meaning of the slope and y-intercept in the context of the problem.
Educator Instructions
- Introduction (5 mins)
Begin by reviewing the concept of linear functions and slope-intercept form (y = mx + b). Briefly discuss how linear functions can be used to model real-world situations. - Video Viewing (10 mins)
Play the Khan Academy video "Constructing linear functions example 2" (https://www.youtube.com/watch?v=5c9N_1PEfHw). Encourage students to take notes on the key information and steps the narrator takes to solve the problem. - Guided Practice (15 mins)
Work through the example problem from the video together as a class. Emphasize the following steps: 1. Identifying the variables (A(t) = area to paint, t = time). 2. Determining the rate of change (slope) and initial value (y-intercept). 3. Writing the linear equation in slope-intercept form (A(t) = -8t + 52). 4. Interpreting the meaning of the slope and y-intercept in the context of the problem. (The slope -8 represents that Hiro paints 8 square meters per hour, and the 52 is the initial amount he has to paint.) - Independent Practice (15 mins)
Provide students with a similar word problem to solve on their own or in small groups. For example: A candle is burning at a rate of 0.5 inches per hour. After 2 hours, the candle is 7 inches tall. Let H(t) denote the height of the candle in inches as a function of time, t, in hours. Write the function's formula. Possible adaptation for struggling students: Provide a partially completed table to help them organize the information. - Wrap-up (5 mins)
Review the key concepts and learning objectives. Answer any remaining questions and assign homework.
Interactive Exercises
- Think-Pair-Share
Present students with a word problem and have them think individually about how to solve it. Then, have them pair up with a partner to discuss their solutions. Finally, have a few pairs share their solutions with the class. - Whiteboard Activity
Divide students into small groups and give each group a whiteboard. Present a word problem and have each group work together to solve it on their whiteboard. Then, have each group present their solution to the class.
Discussion Questions
- How can linear functions be used to model real-world situations?
- What do the slope and y-intercept represent in a linear function?
- How can you identify the slope and y-intercept from a word problem?
- Why is it important to define variables clearly?
Skills Developed
- Problem-solving
- Critical thinking
- Mathematical modeling
- Algebraic reasoning
Multiple Choice Questions
Question 1:
Which of the following represents the slope-intercept form of a linear equation?
Correct Answer: y = mx + b
Question 2:
In the equation y = mx + b, what does 'm' represent?
Correct Answer: slope
Question 3:
In the equation y = mx + b, what does 'b' represent?
Correct Answer: y-intercept
Question 4:
A painter charges $20 per hour plus a $50 initial fee. Which equation represents the total cost (y) as a function of hours worked (x)?
Correct Answer: y = 20x + 50
Question 5:
What is the rate of change also known as?
Correct Answer: slope
Question 6:
The height of a plant increases by 2 cm per week. If the plant was initially 5 cm tall, what linear equation represents the height (h) after w weeks?
Correct Answer: h = 2w + 5
Question 7:
What does A(t) mean in the context of the lesson?
Correct Answer: Area to paint as a function of time
Question 8:
Hiro paints at a rate of 8 square meters per hour. This rate is also known as the ____.
Correct Answer: slope
Question 9:
If a line has a slope of -3 and a y-intercept of 7, which equation represents the line?
Correct Answer: y = -3x + 7
Question 10:
If you know the slope and one point on a line, what form can you use to easily write the equation of the line?
Correct Answer: Point-slope form
Fill in the Blank Questions
Question 1:
The slope-intercept form of a linear equation is y = mx + ____.
Correct Answer: b
Question 2:
The rate of change in a linear function is also known as the _____.
Correct Answer: slope
Question 3:
The point where a line crosses the y-axis is called the ______.
Correct Answer: y-intercept
Question 4:
In the equation A(t) = -8t + 52, the number 52 represents the ______ amount to paint.
Correct Answer: initial
Question 5:
The process of representing real-world situations with mathematical equations is called mathematical ______.
Correct Answer: modeling
Question 6:
A constant rate of change indicates a ________ function.
Correct Answer: linear
Question 7:
When writing a linear function, it's important to clearly ______ the variables.
Correct Answer: define
Question 8:
The y-intercept can also be called the _____ value.
Correct Answer: initial
Question 9:
The area that Hiro has left to paint is a _____ of time.
Correct Answer: function
Question 10:
A(t) is an example of _______ notation.
Correct Answer: function
Educational Standards
Teaching Materials
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