Students may initially believe there is no number that multiplies by itself to give 10. Prompt them to think outside of the whole numbers.

The purpose of this activity is for students to calculate areas of rectangles on a grid, organize that information in a table, and look for structure in the table and in an algebraic expression to create a rule that describes the relationship between scale factor and area.

As students determine the pattern that emerges over several scale factors, they are expressing regularity in repeated reasoning (MP8).

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

This is the first time Math Language Routine 3: Critique, Correct, Clarify is suggested in this course. In this routine, students are given a “first draft” statement or response to a question that is intentionally unclear, incorrect, or incomplete. Students analyze and improve the written work by first identifying what parts of the writing need clarification, correction, or details, and then writing a second draft (individually or with a partner). Finally, the teacher scribes as a selected second draft is read aloud by its author(s), and the whole class is invited to help edit this “third draft” by clarifying meaning and adding details to make the writing as convincing as possible to everyone in the room. Typical prompts are: “Is anything unclear?” and “Are there any reasoning errors?” The purpose of this routine is to engage students in analyzing mathematical writing and reasoning that is not their own, and to solidify their knowledge and use of language.

The goal of this discussion is to establish that scaling a rectangle by a factor of has the result of multiplying the area by .

Use Critique, Correct, Clarify to give students an opportunity to improve a sample written response, by correcting errors, clarifying meaning, and adding details. 

• Display the following first draft:
  “The original rectangle has an area of 10 square units. If the scale factor is , then the area of the larger rectangle is square units.”
  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 individuals or pairs to read their revised draft aloud slowly enough to record for all to see. Scribe as each student shares, then invite the whole class to contribute additional language and edits to make the final draft even more clear and more convincing.

Ask students, “How do the algebraic expression and the numbers in the table relate to each other?” (The algebra says that the area of a figure scaled by should change by . We saw that, for example, when , the area changed by a factor of ).

If students give an answer of 120 square units for the answer to the last question, ask them what happened when the rectangle in the previous activity was scaled by a factor of 3. Did the area also increase by a factor of 3?

The goal is to make sure students understand that their conclusion from the previous activity applies more broadly. Invite several students to state in their own words why if a rectangle is scaled by a factor of , the area scales by a factor of .