Unit 7, Lesson 12
Changing the Equation
Extra Support Materials for Algebra 1
12.1
Warm-up
Standards Alignment
Building On
Addressing
7.NS.A.2.c
Apply properties of operations as strategies to multiply and divide rational numbers.
Building Toward
Instructional Routines
Worked ExampleMaterials
NoneActivity Narrative
The purpose of this activity is to analyze a method of evaluating an expression involving exponents and negative numbers.
This activity uses the Worked Example routine. In this routine, students analyze steps used to solve a problem and discuss specific steps with a partner and the whole class. During the discussions, students construct viable arguments to justify the steps taken by the problem solver (MP3). Later, students can apply these strategies to solve similar problems.
Launch
Arrange students in groups of 2. Tell them to close their books or devices (or to keep them closed). Reveal the prompt and the first step. Give students a brief quiet think time, and ask them to give a signal when they have read the step and are ready to see the next. Once most students have given the signal, reveal the next step. Repeat until all steps have been revealed.
If previous work indicates that most students will struggle to make sense of a step, pause and give students 1 minute to discuss with their partner and formulate a question about the step. Ask 1–3 groups to share their questions, inviting other groups to answer while recording responses next to the step for all to see. Once questions have been answered, continue to the next reveal.
Action and Expression: Internalize Executive Functions. To support working memory, provide students with sticky notes or mini whiteboards.
Supports accessibility for: Memory, Organization
Supports accessibility for: Memory, Organization
Activity
NoneEvaluate when is -3.
Step 1:
Step 2:
Step 3:
16
Activity Synthesis
The purpose of this discussion is to analyze a method of evaluating an expression involving exponents with negative numbers.
Tell students to open their books or devices. Display the following questions, and give students 1–2 minutes of quiet think time to consider their responses.
- “Why did the solver put the -3 where the was in the expression?” (The problem asked the solver to evaluate the expression when is -3, so they substituted -3 for , since they are equal.)
- “In step 2, why is everything positive? What happened to the negative?” (The -3 was squared, or multiplied by itself. A negative number multiplied by a negative number is positive. So .)
- “How could the solver check their work to make sure they are correct?” (They could write an equation with the expression on one side of the equals sign and their solution on the other side. Then they could solve for and see if , in which case they have the right answer. In this case, it would be . Subtracting 7 from each side gives . The final step is taking the square root of each side, which gives two possible answers, 3 and -3.)
Ask partners to compare their responses to each question and decide if either or both answers are correct. During this partner discussion, monitor to select groups who reason about the questions in different ways. Follow with a whole-class discussion, inviting 1–2 selected groups to share their answers for each question.
If time allows, consider displaying a second worked example next to the first that either:
- Shows a different method of solving a similar problem that students can compare and contrast with the first example (MP7).
- Shows the same method with an error that students can critique and correct (MP3). For example, solving the same problem but replacing the 9 in step 2 with -9.
12.2
Activity
Instructional Routines
MLR2: Collect and DisplayMaterials
To Gather
Graphing technologyActivity Narrative
In this activity, students use graphing technology to see how adjusting different parts of a linear equation affect its graph. This connects to the associated Algebra 1 lesson where students will look at similar effects to quadratic equations. 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).
This activity uses the Collect and Display math language routine to advance conversing and reading as students clarify, build on, or make connections to mathematical language.
Launch
Display the following problems. Give students 1 minute of quiet think time to solve the problems.
Evaluate each expression when is -4:
Invite students to share their answers and clarify any misconceptions, focusing on how the solution is affected by a negative number being squared compared to the negative being applied after squaring a number. If needed, rewrite the last two problems as -1 multiplied by the rest of the expressions, rather than just a negative sign.
Then, arrange students in groups of 2. Graphing technology is needed for every student or every group. To ensure that all students can access the last two questions, either demonstrate graphing and on the same set of axes, or invite a student to demonstrate this.
Use Collect and Display to create a shared reference that captures students’ developing mathematical language. Collect the language that students use to describe the changes to the graphs. Display words and phrases, such as “up,” “down,” “positive,” “negative,” “steeper,” and “less steep.”
12.3
Activity
Instructional Routines
NoneMaterials
To Gather
Graphing technologyActivity Narrative
This 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. Students’ repeated work with evaluating quadratic expressions involving negative values allows them to express regularity in repeated reasoning (MP8).
Launch
Arrange students in groups of 2. Graphing technology is needed for every student. 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.