Lesson 12

Changing the Equation

These materials, when encountered before Algebra 1, Unit 6, Lesson 12 support success in that lesson.

12.1: Math Talk: A Negative Input (5 minutes)

Warm-up

The purpose of this Math Talk is to elicit strategies and understandings students have for multiplying and squaring signed numbers, and to help them see the differences between evaluating \(x^2\) when \(x\) is a negative number, and evaluating \(\text-x^2\). These understandings help students develop fluency and will be helpful later in this lesson when students will need to be able to evaluate a quadratic expression for a negative value of the variable. In this activity, students have an opportunity to notice and make use of structure (MP7) because they can use the outcome of evaluating the earlier expressions to help reason about the later expressions.

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

Evaluate each expression when \(x\) is -5:

\(\text-2x\)

\(x^2\)

\(\text-2x^2\)

\(\text-x^2\)

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

As students share their work, draw attention to notational ways to differentiate evaluating \(x^2\) and \(\text-x^2\) when \(x\) is negative. For example, show a substitution using parentheses, such that the first is written \((\text-5)^2\) and the second is written \(\text-(\text-5)^2\). Sometimes it is also useful to think of this as \(\text-1 \boldcdot (\text-5)^2\).

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?”

12.2: Equations and Their Graphs (20 minutes)

Activity

This activity has two distinct parts. It may be desirable to debrief the first two questions with the whole class before launching the second two questions which require students to access graphing technology. When students evaluate the sample work and explain who they agree with and why, they are critiquing the reasoning of others (MP3). When students vary the parameters in a linear function and record their observations, they have an opportunity to attend to precision through their word choices (MP6).

Launch

First invite students to work quietly or with a partner on the first two questions. (This is an opportunity to apply what they noticed in the warm-up, so it should be possible to start right away.) Invite selected students to share their responses and debrief these with the whole class before launching the second two questions. Provide access to graphing technology, either so that each student has their own device, or so that each group of two students share one device. To ensure that all students can access the last two questions, either demonstrate graphing \(y=x\) and \(y=x+4\) on the same set of axes, or invite a student to demonstrate this.

Student Facing

  1. Two students are evaluating \(x^2+7\) when \(x\) is -3. Here is their work. Do you agree with either of them? Explain your reasoning. 

    Tyler:

    \(x^2+7\)

    \(\text-3^2+7\)

    \(\text-9+7\)

    -2

    Lin:

    \(x^2+7\)

    \((\text-3)^2+7\)

    \(9+7\)

    16

  2. Evaluate each expression when \(x\) is -4:

    1. \(x^2\)
    2. \(\frac12 x^2\)
    3. \(\text-\frac18 x^2\)
    4. \(\text-x^2-8\)
  3. Using graphing technology, graph \(y = x\). Then, experiment with the following changes to the function. Record your observations (include sketches, if helpful).

    1. Adding different constant terms to \(x\) (for example: \(x + 4\), \(x - 3\)).
    2. Multiplying \(x\) by different positive coefficients greater than 1 (for example: \(6x, 2.5x\)).
    3. Multiplying \(x\) by different positive coefficients between 0 and 1 (for example: \(0.25x, 0.1x\)).
    4. Multiplying \(x\) by negative coefficients of \(x\) (for example: \(\text-9x, \text-4x\)).
  4. Use your observations to sketch these functions on the coordinate plane, which currently shows \(y=x\)

    1. \(y =\text-0.5x + 2.1\)
    2. \(y = 2.1x - 0.5\)

      graph of y = x
      graph of y = x

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

Display a graphic organizer with categories corresponding to the moves in the activity, for example: 

move effect example equation example graph
Adding a constant term to \(x\)      
Multiplying \(x\) by positive coefficient greater than 1      
...      
...      

As students share their observations for each move, record the effect in the organizer, along with any example equations or graphs that they mention.

12.3: Match the Graphs (20 minutes)

Activity

The activity is an opportunity to practice evaluating quadratic expressions for a negative value of the variable, and choosing parameters of an equation that will result in its graph looking a certain way.

Launch

Arrange students in groups of 2. Give students two minutes of quiet work time on the first question, and ask them to check their answers with their partner before starting the second question. Call attention to the scale on the axes. Identify students who use the scale to select the parameters for their linear equation.

Student Facing

  1. Evaluate each expression when \(x\) is -3.
    1. \(x^2\)
    2. \(\text-x^2\)
    3. \(x^2+20\)
    4. \(\text-x^2+20\)
  2. For each graph, come up with an equation that the graph could represent. Verify your equation using graphing technology. 
    graphs A, B, C, D, E, F of 6 lines. 

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

Invite students to share various equations that each graph could represent. Use different but equally correct equations to help students generalize. For example, either \(y=x+15\) or \(y=x+12\) makes sense for graph C, because it looks like \(y=x\), and is shifted up by some number between 10 and 20. So we know that the equation this graph represents is in the form \(y=x+b\), where \(b\) is positive, even though we can’t tell its exact value from the given information.