Three things come to mind as to why I wrote this activity...
First, students are petrified of word problems. I wrote this activity to give students experience with solving applications of the quadratic. Second, I want students to demonstrate their ability to use four different methods to solve quadratics. Third, I want students to be cognizant that they will need to decide which method is best given the structure of the quadratic equation.
There are 24 cards: 4 story cards, 4 equation cards, 4 method cards, and 4 answer cards. I put students in random pairs and allow them scratch paper, pencils, erasers, and graphing calculator. Each pair gets a zip lock bag with the 28 cards inside and each pair gets a blank answer sheet (grid on front, space for solutions on back). The activity usually takes about 30-60 minutes, depending on the ability level of the students.
This activity comes with the answer key and some ideas for modifications for ability of students / time factor issues.