Mathematics Support

Prior to this unit, students have studied quadratic functions. They analyzed and represented quadratic functions using expressions, tables, graphs, and descriptions. Students also evaluated the functions and interpreted the input and the output values in context. They encountered the terms “standard form,” “factored form,” and “vertex form” and examined the advantages of each form. They also rewrote expressions from factored form and vertex form to standard form.

In this unit, students interpret, write, and solve quadratic equations. They see that writing and solving quadratic equations enables them to find input values that produce certain output values. Suppose the revenue of a theater is a function of the ticket price for a performance. At what ticket price would the theater earn $10,000? Previously, students were only able to solve such problems by observing graphs and estimating, or by guessing and checking. Here, they learn to answer such questions algebraically.

Students begin solving quadratic equations by reasoning. For instance, to solve x^2+9=25 , they think: Adding 9 to a squared number makes 25. That squared number must be 16, so x must be 4 or -4. Along the way, students see that quadratic equations can have 2, 1, or 0 solutions.

Next, students learn that equations of the form (x-m)(x-n)=0 can be easily solved by applying the zero product property, which says that when two factors have a product of 0, one of the factors must be 0. When the equations are not in factored form, students rearrange them so that one side is 0, and rewrite the expressions from standard form to factored form. Students soon recognize that not all quadratic expressions in standard form can be rewritten into factored form. Even when it is possible, finding the right two numbers may be tedious, so another strategy is needed.

Students encounter perfect squares and notice that solving a quadratic equation is pretty straightforward when the equation contains a perfect square on one side and a number on the other. They learn that we can put equations into this helpful format by completing the square, that is, by rewriting the equation such that one side is a perfect square. Although this method can be used to solve any quadratic equation, it is not practical for solving all equations. This challenge motivates the quadratic formula.

Once introduced to the formula, students apply it to solve contextual and abstract problems, including those that they couldn’t previously solve. After gaining some experience using the formula, students investigate how it is derived. They find that the formula essentially encapsulates all the steps of completing the square into a single expression. Just like completing the square, the quadratic formula can be used to solve any equation, but it may not always be the quickest method. Students consider how to use the different methods strategically.

Throughout the unit, students see that solutions to quadratic equations are often irrational numbers. Sometimes they are expressed as sums or products of a rational number and an irrational number (such as  or ). Students reason about whether such sums and products are rational or irrational.

Toward the end of the unit, students revisit the vertex form and recall that it can be used to identify the maximum or minimum of a quadratic function. Previously students learned to rewrite expressions from vertex form to standard form. Now they can go in reverse—by completing the square. Being able to rewrite expressions in vertex form allows students to effectively solve problems about maximum and minimum values of quadratic functions.

In the final lesson, students integrate their insights and choose appropriate strategies to solve an applied problem and a mathematical problem (a system of linear and quadratic equations).