Unit 8

Quadratic Equations

Unit Narrative

In this unit, students interpret, write, and solve equations algebraically.

Previously, students represented quadratic functions using expressions, tables, and descriptions. They connected important features of graphs to standard, factored, and vertex forms and expanded expressions into standard form.

Students begin with solving quadratic equations through reasoning without much algebraic manipulation. Then, they examine solving using the zero product property for quadratic equations that can be written in factored form. They notice patterns that help them rewrite quadratic expressions in factored form and recognize that not all of them are easily...

to access Pre-Unit Assessments.

Section A

Section Goals

Section Narrative

In this section, students are presented with activities that motivate the need for more algebraic methods for solving quadratic equations. Students write quadratic equations from situations, but recognize that they do not have the tools to accurately solve the equations unless they are in very specific forms, such as a factored form equal to zero or a perfect square equal...

Let’s find some new equations to solve.

Let’s try to solve some quadratic equations.

to access Section Checkpoints.

Section A

Let’s find some new equations to solve.

Let’s try to solve some quadratic equations.

Section B

Section Goals

Section Narrative

In this section, students look more deeply into solving quadratic equations that can be written as a factored form equal to zero. They formalize their understanding of the zero product property, then work on recognizing patterns that allow them to rewrite quadratic expressions from standard form to factored form. Students also begin to recognize that quadratic equations can have 0,...

Let’s find solutions to quadratic equations.

Let’s find solutions to equations that contain products that equal zero.

Let’s use graphs to investigate quadratic equations that have two solutions, one solution, or no solutions.

Let’s write expressions in factored form.

Let’s write some more expressions in factored form.

Let’s look closely at some special kinds of factors.

Let’s solve some quadratic equations that, before now, we could only solve by graphing.

Let’s transform more-complicated quadratic expressions into the factored form.

to access Section Checkpoints.

to access Mid-Unit Assessments.

Section B

Let’s find solutions to quadratic equations.

Let’s find solutions to equations that contain products that equal zero.

Let’s use graphs to investigate quadratic equations that have two solutions, one solution, or no solutions.

Let’s write expressions in factored form.

Let’s write some more expressions in factored form.

Let’s look closely at some special kinds of factors.

Let’s solve some quadratic equations that, before now, we could only solve by graphing.

Let’s transform more-complicated quadratic expressions into the factored form.

Section C

Section Goals

Section Narrative

In this section, students learn to complete the square as an additional strategy for solving quadratic equations. First, they begin by recognizing when quadratic expressions can be rewritten as perfect squares and how this form is useful for solving equations of the form . Then, students learn how to turn any quadratic equation into that form by completing the square....

Let’s see how perfect squares make some equations easier to solve.

Let’s learn a new method for solving quadratic equations.

Let’s solve some harder quadratic equations.

Let’s complete the square for some more complicated expressions.

Let’s find exact solutions to quadratic equations, even if the solutions are irrational.

to access Section Checkpoints.

Section C

Let’s see how perfect squares make some equations easier to solve.

Let’s learn a new method for solving quadratic equations.

Let’s solve some harder quadratic equations.

Let’s complete the square for some more complicated expressions.

Let’s find exact solutions to quadratic equations, even if the solutions are irrational.

Section D

Section Goals

Section Narrative

In this section, students encounter the quadratic formula. Initially, the formula is given to students, and they notice that the solutions match those found using other methods. After getting comfortable with using the formula, they examine a proof of the quadratic formula that uses completing the square.

An optional lesson in which students analyze common errors encountered using the quadratic...

Let’s learn a formula for finding solutions to quadratic equations.

Let’s use the quadratic formula to solve some problems.

Let’s use the quadratic formula and solve quadratic equations with care.

Let’s find out where the quadratic formula comes from.

Let’s consider the kinds of numbers we get when solving quadratic equations.

Let’s make convincing arguments about why the sums and products of rational numbers and irrational numbers are always certain kinds of numbers.

to access Section Checkpoints.

Section D

Let’s learn a formula for finding solutions to quadratic equations.

Let’s use the quadratic formula to solve some problems.

Let’s use the quadratic formula and solve quadratic equations with care.

Let’s find out where the quadratic formula comes from.

Let’s consider the kinds of numbers we get when solving quadratic equations.

Let’s make convincing arguments about why the sums and products of rational numbers and irrational numbers are always certain kinds of numbers.

Section E

Section Goals

Section Narrative

In this section, students revisit the vertex form of quadratic expressions. In a previous unit, students noticed that this form can be used to easily find the vertex of a quadratic function and identified that point on a graph. Here, students use the process of completing the square to rewrite expressions from standard form to vertex form, then classify the...

Let’s see what else completing the square can help us do.

Let’s find the maximum or minimum value of a quadratic function.

to access Section Checkpoints.

Section E

Let’s see what else completing the square can help us do.

Let’s find the maximum or minimum value of a quadratic function.

Section F

Section Narrative

The final section of this unit is a lesson where students have the opportunity to apply their thinking from throughout the unit. As this is a short section followed by an End-of-Unit Assessment, there are no section goals or checkpoint questions.

Let’s analyze a situation modeled by a quadratic equation.

to access Section Checkpoints.

Section F

Let’s analyze a situation modeled by a quadratic equation.

to access End-of-Unit Assessments.

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