Lesson 4

Equations and Their Zeros

These materials, when encountered before Algebra 1, Unit 7, Lesson 4 support success in that lesson.

4.1: Math Talk: Equations with Zero (5 minutes)

Warm-up

The purpose of this Math Talk is to elicit strategies and understandings students have for solving equations involving zero. These understandings help students develop fluency and will be helpful later in this lesson when students will need to be able to find zeros from factored quadratic equations.

Launch

Display one problem at a time. Give students quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all problems displayed throughout the talk. Follow with a whole-class discussion.

Student Facing

Mentally find a value for \(a\) that will make the equation true.

\(4a = 0\)

\(3 \boldcdot 0 = 14a\)

\(0 \boldcdot a = 0\)

\(ab = 0\) with \(b \neq 0\)

Student Response

For access, consult one of our IM Certified Partners.

Activity Synthesis

Ask students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking:

  • “Who can restate \(\underline{\hspace{.5in}}\)’s reasoning in a different way?”
  • “Did anyone have the same strategy but would explain it differently?”
  • “Did anyone solve the problem in a different way?”
  • “Does anyone want to add on to \(\underline{\hspace{.5in}}\)’s strategy?”
  • “Do you agree or disagree? Why?”

4.2: Evaluating Quadratic Equations (15 minutes)

Activity

In this activity, students evaluate quadratic functions for a given value and compare the results. The given value is a zero of the function. In the associated Algebra 1 lesson, students examine the zero product property to find zeros of quadratic functions. This activity gives students the chance to notice patterns that will show up in later lessons.

Student Facing

  1. Evaluate each function for \(x = 6\).
    1. \(f(x)=(x+4)(x-6)\)
    2. \(g(x)=(x-6)(x+6)\)
    3. \(h(x)=x^2-6x\)
    4. \(j(x)=2(\frac{2}{3}x+8)(x-6)\)
    5. \(k(x)= 0.5x^2 - 3x\)
  2. What do these functions have in common?
  3. For each function, find another value of \(x\) that would give the same output as you found earlier?

Student Response

For access, consult one of our IM Certified Partners.

Activity Synthesis

The purpose of this discussion is for students to notice some common features of functions that could indicate zeros of the function. Select students to share their solutions. Ask students,

  • “Both \(h(x)\) and \(k(x)\) have 2 of the same numbers that make the equation zero. What do you notice about them?” (They are equivalent functions. Multiplying \(x^2 - 6x\) by 0.5 results in \( 0.5x^2 - 3x\).)
  • “Notice that \(h(x)\) can be factored and written as \(h(x) = x(x-6)\) and \(k(x)\) can be written as \(k(x) = 0.5x(x-6)\). Distribute in these forms to check that they are the same. Compare these factored forms to the factored forms for all of the other functions. What do you notice looks the same? Do you think that has anything to do with the reason all the functions evaluate to zero?” (All of the forms have \(x-6\) as a factor. I do think that has to do with why all the functions evaluate to zero.)

4.3: Card Sort: Matching Equation Forms (20 minutes)

Activity

In this activity, students match equivalent quadratic equations in factored and standard form. In the associated Algebra 1 lesson, students will work with quadratic equations in factored form. The additional fluency students gain from this activity will help students when making sense of the equations they will see in Algebra 1 lessons.

A sorting task gives students opportunities to analyze representations, statements, and structures closely and make connections (MP2, MP7).

Launch

Arrange students in groups of 2.

Ask students, “What are some ways we can tell that \((2x-5)(3 + x)\) and \(x - 15 + 2x^2\) represent the same function?”

Listen for these strategies and consider presenting any that students do not share:

  • Use the distributive property
  • Use technology to graph each function on the same plane and notice that they coincide
  • Inspect a table of values for both and see that the same output results from any input

Distribute one set of cards to each group of students. Give students time to work with their partner, followed by a whole-class discussion

Student Facing

Your teacher will give you a set of cards. Match each function in standard form with an equivalent function in factored form.

Student Response

For access, consult one of our IM Certified Partners.

Activity Synthesis

Select groups to share their matches. Ask students to share how they found the matches and if any are more difficult than others.