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Unit 8, Lesson 18

Applying the Quadratic Formula (Part 2) (Optional)

Algebra 1

18.1

Warm-up

Standards Alignment

Building On

Instructional Routines

None

Materials

None

Activity Narrative

In this Warm-up, students evaluate variable expressions that resemble those in the quadratic formula. The aim is to preview the calculations that are the source of some common errors in solving equations using the quadratic formula, which students will analyze in the next activity.

Launch

None

Activity

None

Evaluate each expression for , , and

Student Response

to access Student Response.

Building on Student Thinking

Activity Synthesis

Invite students to share their responses and discuss any disagreement. For each expression, ask if they can think of an error someone might make when evaluating such an expression. Some possible errors:

: Forgetting that is really and the product of two negative numbers is positive.
: When is -5, evaluating , instead of .
: Forgetting that subtracting by is equivalent to adding , or neglecting to see that if is negative, is positive, not negative.
: Neglecting the negative in front of , or neglecting to see that the expression takes two different values. Tell students that in the next activity, they will spot some errors in solving quadratic equations.

18.2

Activity

Instructional Routines

MLR8: Discussion Supports

Materials

To Gather

Math Community Chart

Activity Narrative

Solving an equation with the quadratic formula involves multiple calculations, some of which may be prone to error. This activity acquaints students with some of the commonly made mistakes and encourages them to write and evaluate expressions carefully. Students critique the reasoning of the presented work and construct arguments for correcting the errors (MP3).

As students work, identify those who can clearly and correctly explain the errors in the worked solutions. Ask them to share their responses during the class discussion.

Launch

Math Community
Display the Math Community Chart for all to see. Give students a brief quiet think time to read the norms, or invite a student to read them out loud. Tell students that during this activity they are going to practice looking for their classmates putting the norms into action. At the end of the activity, students can share what norms they saw and how the norm supported the mathematical community during the activity.

Arrange students in groups of 2. Give students a few minutes of quiet work time to solve 1–2 equations using the quadratic formula and then a moment to discuss their response with their partner. Then, ask them to study the worked solutions for the same equations that they had solved and identify the errors.

If time permits, ask groups to identify the errors in the remaining equations. If they get stuck, suggest that they try solving the equations first and then look for the errors.

Provide access to calculators for numerical computations.

18.3

Activity

Instructional Routines

5 Practices

Materials

None

Activity Narrative

This activity prompts students to check the solutions to quadratic equations and to recognize that there is more than one way to do so. Different approaches include substituting solutions to see if the equation is true, solving using a different algebraic method, or solving by graphing.

Monitor for students who use these different strategies to verify the solutions to the equation . Here are some strategies they may use, from less efficient to more efficient.

Solve the equation using another algebraic method, and check if they get the same solutions. For instance, if they used the quadratic formula initially, they might choose to solve it again by completing the square or by rewriting the equation in factored form.
Graph both sides of the equation to find the value of the intersection, or subtract 47 from each side, graph , and check the horizontal intercepts of the graph.
Substitute the solutions for in the equation and check if the equation is true.

The algebraic strategies are more precise but could lead to errors if the computations are performed incorrectly. The graphing strategies can be accomplished quickly with the use of technology but may give only approximate solutions.

Making calculators and graphing technology available gives students an opportunity to choose appropriate tools strategically (MP5).

Lesson Synthesis

Invite students to name some familiar mistakes that are made when using the quadratic formula, not only those seen in the activities, but also mistakes that the students themselves might have made.

Then, to encourage students to think about ways of checking solutions, discuss questions such as:

“What might be some good ways to check if and are solutions to ” (Some possibilities:
- Substitute each solution into the equation, and see if the equation is true. For example, see if equals 0.
- Rewrite in factored form: , and apply the zero product property to solve.
- Graph and find the -intercepts—where the graph has a -coordinate of 0.)
“Can completing the square be used to check the solution?” (Yes, but it is not simple because the leading coefficient is not a perfect square and the linear coefficient is an odd number.)
“Suppose we want to verify if are correct solutions to . Is substituting the expressions in the equation an effective way to check?” (In this case, substituting for makes the checking cumbersome. The expression to be evaluated would be , which is not easy to evaluate.)
“What might be some good ways to check?” (We could:
- Solve by completing the square. The leading coefficient of is 1, and the linear term coefficient is an even number, which makes it fairly simple to complete the square.
- Approximate the value of the solutions with a calculator, then graph , and see if the solutions match the -intercepts.)

to access Cool-down.

Student Lesson Summary

The quadratic formula has many parts in it. A small error in any one part can lead to incorrect solutions.

Suppose we are solving . To use the formula, let's rewrite it in the form of , which gives .

Here are some things to keep in mind:

Use the correct values for , , and in the formula.

Nope! is -11, so is , which is 11, not -11.

That’s better!
Multiply 2 by for the denominator in the formula.

Nope! The denominator is , which is , or 4.

That’s better!
Remember that squaring a negative number produces a positive number.

Nope! is 121, not -121.

That’s better!

Remember that a negative number times a positive number is a negative number.

Nope! , and is .

That’s better!
Follow the properties of algebra.

Nope! Both parts of the numerator, 11 and , get divided by 4. Also, is not .

That’s better!

Let’s finish by evaluating correctly:

To make sure our solutions are correct, we can substitute each solution back into the original equation and see whether it results in a true equation.

Checking 6 as a solution:

Checking as a solution:

We can also graph the equation and find its -intercepts to see whether our solutions to are accurate (or close to accurate).

<p>Function on a grid. X axis from negative 2 to 8, by 1’s. Y axis from negative 12 to 2, by 2’s. Parabola opens upward and crosses the x axis at one half and 5.</p>

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Lesson 18

Applying the Quadratic Formula (Part 2) (Optional)

Warm-up

Standards Alignment

Building On

6.EE.A.2.c

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Activity

Instructional Routines

Materials

To Gather

Activity Narrative

Launch

Activity

Instructional Routines

Materials

Activity Narrative

Lesson Synthesis

Student Lesson Summary

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Launch

Activity

Student Response

Building on Student Thinking

Activity Synthesis

8.EE.A.2

Addressing

Building Toward

HSA-REI.B.4.b

Standards Alignment

Building On

Addressing

HSA-REI.B.4.b

Building Toward

Standards Alignment

Building On

Addressing

HSA-CED.A.1

HSA-REI.B.4.b

HSF-IF.A.2

Building Toward