Banana Proportionality: Exploring Linear Relationships

Algebra 1 Grades High School 2:56 Video
Print Lesson Plan Watch Video

Lesson Description

This lesson explores the concept of proportionality using a relatable scenario involving bananas. Students will learn to distinguish between proportional and non-proportional relationships by analyzing data, calculating ratios, and interpreting results within the context of a word problem.

Video Resource

Banana proportionality | Rated and proportional relationships | 7th grade | Khan Academy

Khan Academy

Duration: 2:56
Watch on YouTube

Key Concepts

  • Proportional relationship
  • Ratio
  • Constant of proportionality
  • Linear relationship

Learning Objectives

  • Students will be able to define a proportional relationship.
  • Students will be able to calculate ratios to determine if a relationship is proportional.
  • Students will be able to interpret the results of ratio calculations in the context of a real-world problem.
  • Students will be able to differentiate between proportional and non-proportional relationships.

Educator Instructions

  • Introduction (5 mins)
    Briefly review the concept of ratios and proportions. Ask students to provide real-world examples of proportional relationships (e.g., ingredients in a recipe). Introduce the 'Banana Proportionality' problem from the Khan Academy video.
  • Video Viewing and Discussion (10 mins)
    Play the Khan Academy video. After watching, facilitate a class discussion about the problem. Emphasize the importance of carefully defining the variables being compared (bananas left vs. bananas eaten).
  • Table Creation and Analysis (15 mins)
    Guide students to create a table similar to the one in the video, tracking the number of days, bananas left, and the ratio of bananas left to days passed. Have them calculate the ratios for several days (e.g., 1, 2, 3, 5 days). Analyze the ratios to determine if they are constant.
  • Alternative Scenario (10 mins)
    Present the alternative scenario discussed in the video: 'Is the number of bananas Nate has *eaten* proportional to the number of days that pass?' Have students create a new table and analyze the ratios. Discuss why this relationship *is* proportional.
  • Conclusion (5 mins)
    Summarize the key concepts of proportionality and its relationship to constant ratios. Reiterate the importance of carefully defining variables when analyzing relationships. Assign homework: practice problems involving proportionality.

Interactive Exercises

  • Proportionality Sort
    Provide students with a set of tables representing different relationships. Have them sort the tables into two groups: proportional and non-proportional. Justify their choices.
  • Create-a-Problem
    Have students create their own real-world problem involving proportionality. They should then solve the problem and explain why the relationship is (or is not) proportional.

Discussion Questions

  • What does it mean for two quantities to be proportional?
  • How can you determine if a relationship is proportional by looking at a table of values?
  • Why is it important to clearly define what quantities you are comparing when assessing proportionality?
  • Can you think of other real-world scenarios where you can apply the concept of proportionality?

Skills Developed

  • Analytical thinking
  • Problem-solving
  • Data interpretation
  • Mathematical reasoning

Multiple Choice Questions

Question 1:

Which of the following best describes a proportional relationship?

Correct Answer: A relationship where the ratio between two quantities is constant.

Question 2:

In a proportional relationship, if one quantity doubles, what happens to the other quantity?

Correct Answer: It doubles.

Question 3:

If the ratio between x and y is always 3, which equation represents this proportional relationship?

Correct Answer: y = 3x

Question 4:

Which table shows a proportional relationship between x and y?

Correct Answer: x: 1, 2, 3; y: 2, 4, 6

Question 5:

What is another name for the constant ratio in a proportional relationship?

Correct Answer: Constant of proportionality

Question 6:

Nate buys apples where each apple costs $2. Is the total cost proportional to the number of apples bought?

Correct Answer: Yes

Question 7:

A taxi charges a $3 flat fee plus $1 per mile. Is the total cost proportional to the number of miles driven?

Correct Answer: No

Question 8:

If y is proportional to x, and y = 10 when x = 2, what is the constant of proportionality?

Correct Answer: 5

Question 9:

Which of the following graphs represents a proportional relationship?

Correct Answer: A straight line passing through the origin

Question 10:

If the number of workers and the time it takes to complete a task have an inverse relationship, is it a proportional relationship?

Correct Answer: No

Fill in the Blank Questions

Question 1:

In a proportional relationship, the _________ between two quantities is constant.

Correct Answer: ratio

Question 2:

The constant ratio in a proportional relationship is also known as the constant of _________.

Correct Answer: proportionality

Question 3:

If y is proportional to x, then y = kx, where k is the _________ of proportionality.

Correct Answer: constant

Question 4:

A graph of a proportional relationship is a straight _________ that passes through the origin.

Correct Answer: line

Question 5:

If the ratio of boys to girls in a class is always 2:3, the number of boys is _________ to the number of girls.

Correct Answer: proportional

Question 6:

If the price of gas is $3 per gallon, the total cost is _________ to the number of gallons purchased.

Correct Answer: proportional

Question 7:

In the banana problem, the number of bananas Nate has _________ is proportional to the number of days that pass.

Correct Answer: eaten

Question 8:

A relationship that is not proportional is called a _________ relationship.

Correct Answer: non-proportional

Question 9:

The slope of a line representing a proportional relationship indicates the _________.

Correct Answer: rate

Question 10:

The x and y values in a table representing a proportional relationship will always have a _________ ratio.

Correct Answer: constant

Educational Standards

A1.F.1:Understand functions as descriptions of covariation (how related quantities vary together) in real-world and mathematical problems. A1.F.2.1:Distinguish between linear and nonlinear (including exponential) functions. Understand that linear functions grow by equal intervals (arithmetic) and that exponential functions grow by equal factors over equal intervals (geometric). A1.A.4.1:Analyze, use and apply mathematical models and other data sets (e.g., graphs, equations, two points, a set of data points) to calculate and interpret slope and the x- and y-intercepts of a line.

Teaching Materials

Download ready-to-use materials for this lesson:

Student Worksheet Classroom Slides Assessment Quiz
Add to Google Classroom

User Actions

Sign in to save this lesson plan to your favorites.

Sign In

Share This Lesson

Related Lesson Plans