Caroline is selling pies for Thanksgiving. She earns a commission of $2.25 for each apple pie and pumpkin pie she sells. Her goal is to earn $630 in commission. Write an equation that models the different numbers of pies she has to sell to meet her goal. Let x represent the number of apple pies and y represent the number of pumpkin pies.

To write an equation that models Caroline's goal, we need to consider the commission she earns per pie and her target commission. Since the commission is the same for both apple pies (x) and pumpkin pies (y), we can write a single equation to represent the total commission from both types of pies.

The equation to represent Caroline's total earnings (E) from selling x apple pies and y pumpkin pies is:

E = (Commission per apple pie * Number of apple pies) + (Commission per pumpkin pie * Number of pumpkin pies)

Given that the commission per pie (both apple and pumpkin) is $2.25, and her target commission is $630, we can replace E with $630 and the commission per pie with $2.25 to form the equation:

630 = 2.25x + 2.25y

This equation shows the relationship between the number of apple pies and pumpkin pies Caroline needs to sell to meet her commission goal of $630.

Extra: When it comes to explaining this concept to a school student, you can draw a parallel to simpler concepts they are already familiar with. Explain that an equation is like a recipe for a cake—the ingredients are the numbers and variables (like x and y, which in this case represent the apple and pumpkin pies, respectively), and the end result (the cake) is the commission Caroline wants to earn, which is $630.

In this case, since Caroline earns the same amount of money for each pie, we can think of x and y as different ingredients that can be mixed in different amounts to get the same cake each time. Just like how you can adjust the ratio of flour to sugar in a recipe as long as the total amount meets the recipe's requirements, Caroline can adjust the number of apple pies and pumpkin pies she sells as long as their total commission equals $630.

You could also introduce the student to the concept of a linear equation in two variables (x and y), which is graphically represented as a straight line on a plane. Each point on the line represents a possible combination of apple pies (x) and pumpkin pies (y) that would result in the same total commission of $630. In real-world terms, this translates to the idea that Caroline has many different combinations of pie sales that will allow her to reach her earning goal, not just one single combination.