Practice with all methods, picking the best one to use, and drawing connections between them
• Construct and solve quadratic equations by the most strategic method in various contexts (A-CED.A.1$^\star$, A-REI.B.4b).
• Express a quadratic function in the appropriate form for a given purpose (F-IF.C.8a$^\star$).
Now that students have a full toolbox for solving quadratic equations and expressing quadratic functions in different forms, their task becomes selecting which one to use in varying situations. In this section students see a variety of mathematical and real world problems, with the most strategic solution method varying from one problem to the next.
WHAT: Students construct and solve a quadratic equation in the context of finding braking distance. They first find an approximate solution using graphs, then find the exact solution using the quadratic formula. The quadratic formula is the most appropriate method given the complexity of the coefficients.
WHY: The purpose of this task is to give an application arising from a real-world situation in which a quadratic equation arises, and where it is natural to use a graphical method to find an approximate solution and the quadratic formula to find an exact solution (MP4).
WHAT: Students are given a quadratic function modeling the motion of a diver and the asked questions about the context, such as the height of the springboard, the highest point of the diver, and the time the diver hits the water.
WHY: The purpose of this task is to give students experience translating questions about a real world context into mathematical questions about a quadratic function, and then answering those questions using the methods they have learned (MP4), such as completing the square and applying the quadratic formula. It could also be used earlier in this unit to show the need for finding general algebraic methods.
WHAT: Students are given representations of two quadratic functions modeling the path of two different baseballs, one in the form of an equation and one in the form of a graph. They are asked to compare the maximum height of the two baseballs and the time each spends in the air.
WHY: One purpose of this task is to give students experience in making strategic choices about tools solve a problem (MP5). They can use either graphical or algebraic methods. The task also gives students practice in comparing characteristics of two quadratic functions represented in different ways.
WHAT: Students are asked to solve a quadratic equation using as many different methods as possible. Both sides of the equation are given as squares of linear expressions, so in addition to standard methods students have the opportunity to make use of this structure (MP7) to find a quicker method.
WHY: The purpose of this task is to activate flexible procedural knowledge in students. Students might initially use a standard method such as completing the square or using the quadratic formula. However, the form of the equation allows for an easy factorization if students use structure to express the equation as a difference of two squares on the left and zero on the right.