• Students identify the same value relating the measures of x and the measures of y in a proportional relationship as the constant of proportionality and recognize it as the unit rate in the context of a given situation.
• Students find and interpret the constant of proportionality within the contexts of problems.
If a proportional relationship is described by the set of ordered pairs that satisfies the equation y = kx, where k is a positive constant, then k is called the constant of proportionality.
Example 1: National Forest Deer Population in Danger?
Wildlife conservationists are concerned that the deer population might not be constant across the National Forest. The scientists found that there were 144 deer in a 16 square mile area of the forest. In another part of the forest, conservationists counted 117 deer in a 13 square mile area. Yet a third conservationist counted 24 deer in a 216 square mile plot of the forest. Do conservationists need to be worried?
a. Why does it matter if the deer population is not constant in a certain area of the national forest?
b. What is the population density of deer per square mile?
c. Use the unit rate of deer per square mile to determine how many deer are there for every 207 square miles.
d. Use the unit rate to determine the number of square miles in which you would find 486 deer?
A constant specifies a unique number.
A variable is a letter that represents a number.
If a proportional relationship is described by the set of ordered pairs that satisfies the equation y = kx, where k is a positive constant, then k is called the constant of proportionality. It is the value that describes the multiplicative relationship between two quantities, x and y. The (x, y) pairs represent all the pairs of values that make the equation true.
Note: In a given situation, it would be reasonable to assign any variable as a placeholder for the given quantities. For example, a set of ordered pairs (t, d) would be all the points that satisfy the equation d = rt where is r is the positive constant or the constant of proportionality. This value for r specifies a unique number for the given situation.
Example 2: You Need WHAT???
Brandon came home from school and informed his mother that he had volunteered to make cookies for his entire grade level. He needed 3 cookies for each of the 96 students in 7th grade. Unfortunately, he needed the cookies for an event at school on the very next day! Brandon and his mother determined that they can fit 36 cookies on two cookie sheets.
Encourage students to make a chart to organize the data from the problem.
a. Is the number of cookies proportional to the number of sheets used in baking? Create a table that shows data for the number of sheets needed for the total number of cookies needed.
b. It took 2 hours to bake 8 sheets of cookies. If Brandon and his mother begin baking at 4:00 pm, when will they finish baking the cookies?
Example 3: French Class Cooking
Suzette and Margo want to prepare crepes for all of the students in their French class. A recipe makes 20 crepes with a certain amount of flour, milk, and 2 eggs. The girls know that they already have plenty of flour and milk but need to determine the number of eggs needed to make 50 crepes because they are not sure they have enough eggs for the recipe.
a. Considering the amount of eggs necessary to make the crepes, what is the constant of proportionality?
b. What does the constant or proportionality mean in the context of this problem?
c. How many eggs will be needed for 50 crepes?