Come up with and describe two scenarios one that models a direct variation situation and one that models an inverse variation situation do not state explicitly which scenario is which but provide at least four data pairs for each situation. Your classmates will have to determine which of the scenarios is a direct variation and which is an inverse variation, and the value of k for each.

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Did you know?

Did you know that mathematical relationships can be classified as either direct variation or inverse variation? In direct variation, two variables are directly proportional to each other, meaning as one variable increases, the other also increases by a constant factor. For example, consider the scenario where the number of hours spent studying is directly related to the grade obtained on a test. Four data pairs for this scenario could be: Hours Studied (x) | Test Grade (y) ------------------------------ 3 | 60 5 | 100 8 | 160 10 | 200 In this case, the direct variation equation would be y = kx, where k represents the constant of proportionality. On the other hand, inverse variation occurs when two variables are inversely proportional to each other, which means as one variable increases, the other decreases by a constant factor. Let's say we have another scenario where the intensity of light decreases as the distance from the light source increases. Four data pairs for this scenario could be: Distance (x) | Intensity of Light (y) ------------------------------ 2 | 80 4 | 40 6 | 26.67 8 | 20 In this inverse variation, the equation would be y = k/x, where k represents the constant of proportionality. Now, it's up to you to determine which scenario represents direct variation and which one represents inverse variation, and also figure out the value of k for each!