This task contains two distinct parts. Depending on what students need and time constraints, you could decide to do either or both.

In the associated Algebra 1 lesson, students will need to evaluate the functions and for different input values. The purpose of the first part is to prepare them to avoid some common mistakes. Analyzing and correcting incorrect work is an opportunity to critique the reasoning of others (MP3).

The purpose of the second part is to revisit ways to tell whether a function is linear, quadratic, or exponential, and to practice the skills of generating a table and sketching a graph for a given function.

The goal of this discussion is to analyze and correct the mistakes in addition to clarifying the differences between linear, exponential, and quadratic functions.

For the first part, use Critique, Correct, Clarify to give students an opportunity to improve a sample written response to the incorrect solutions by correcting errors, clarifying meaning, and adding details.

• Display this first draft:
  Noah and Mai both made mistakes. Noah multiplied and Mai solved it backwards.
  Ask, “What parts of this response are unclear, incorrect, or incomplete?” As students respond, annotate the display with 2–3 ideas to indicate the parts of the writing that could use improvement.
• Give students 2–4 minutes to work with a partner to revise the first draft.
• Select 1–2 students or groups to slowly read aloud their draft. Record for all to see as each draft is shared. Then invite the whole class to contribute additional language and edits to make the final draft even more clear and more convincing.

For the second part, display the tables and graphs for all to see. Ask students to share some ways they can tell a function is linear, exponential, or quadratic using the different representations. For example:

• The graph of the linear function is a straight line.
• The table shows that the exponential function has a growth factor of 2. Each time increases by 1, the values increase by a factor of 2.
• The table shows that the quadratic function does not have a constant rate of change or a constant growth factor.

The goal is to allow students to practice solving expressions involving exponents. Much of the discussion will happen between partners. These expressions were chosen to highlight errors that people commonly make evaluating expressions. Here are a few specific conventions to draw students’ attention to:

• When an exponent appears in an expression, evaluate the exponent before multiplication, division, addition, or subtraction. For example, is , not .
• Squaring a negative number results in a positive number, for example, is 9. This is because means , and a negative times a negative is a positive.

Additionally, this may be a good opportunity to revisit mentally evaluating expressions that involve a fraction, for example:

• , because multiplying by is the same as dividing by 5.
• Squaring a fraction just means you are multiplying a fraction by itself, for example , which equals .