Identifying and Calculating the Constant of Proportionality

Describing Proportional Relationships with Numbers

• The fundamental question addressed in this material is how a proportional relationship can be described using a specific numerical value.
• This is achieved by identifying the unit rate within a set of data, such as a table, which provides a consistent measure across different quantities.

Identifying Unit Rates from Tabular Data

• To determine if a relationship is proportional and to find the number that describes it, one must calculate the unit rate for each individual entry in a dataset.
• In the provided example, a table describes the relationship between the number of books sold and the corresponding price for the customer.
• The goal is to determine the price per book (the unit rate) for every row to ensure consistency.

Step-by-Step Calculation Example: Price per Book

• The analysis examines each row of data by placing the price in the numerator and the number of books in the denominator.

• Row 1: 4 Books for 24**\n * Calculation: \frac{24}{4} = 6\n * Unit Rate: 6\, \text{per book}\n\n* **Row 2: 5 Books for 30

  • Calculation: $\frac{30}{5} = 6$
  • Unit Rate: $6\, \text{per book}$

• Row 3: 8 Books for 48**\n * Calculation: \frac{48}{8} = 6\n * Unit Rate: 6\, \text{per book}\n\n* **Row 4: 9 Books for 54

  • Calculation: $\frac{54}{9} = 6$
  • Unit Rate: $6\, \text{per book}$

• Conclusion: Because the unit rates for each row are exactly the same ($6$), the number $6$ represents a proportional relationship between the price and the books.

The Constant of Proportionality

• A unit rate can be formally categorized as the constant of proportionality.
• Definition: The constant of proportionality is defined as the unchanging value of the ratio between two proportional quantities.
• In any proportional relationship, this value remains constant regardless of which specific set of quantities from the relationship is being measured.