The goal of this discussion is to explore multiple ways of keeping equations equivalent while solving them. Invite students to share their explanations for Diego’s moves. Ask previously identified students to share the different moves they used to solve the second problem. If students would benefit from seeing the moves represented with a hanger diagram to help them justify why the equations stay “balanced,” identify different students to represent those moves with a hanger diagram, and talk about the whys and hows of each move. Ask students about what some of the acceptable moves are that keep equations equivalent. (Multiplying or dividing each side by the same (non-zero) amount, adding or subtracting the same amount from each side.)

In this activity, students get a chance to practice working with formulas and writing them in equivalent forms, or expressing regularity in repeated reasoning as they find the values of unknown quantities in the formulas (MP8).

The practice will pay off in the associated Algebra 1 lesson when they solve formulas and equations to isolate chosen variables.

Monitor for students who come up with steps or algorithms or even write rules or formulas that they use for each of the problems. Let students know that they will be asked to share their algorithm, rule, or process, and that they should be prepared to explain it to the class.

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 by 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 for students to learn from groups that worked with other formulas and to give each student the chance to be the expert and explain their process.

After students teach their new group about their equation, use Critique, Correct, Clarify to give students an opportunity to improve a sample written response to “how would you teach someone else to find the height of a cylinder using the patterns you noticed?” by correcting errors, clarifying meaning, and adding details.

• Display this first draft:
  They should find the height by getting by itself in the equation.
  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.
• Display and review these criteria:
  • Explanation for each step
  • Specific words and phrases, including “each side,” “,” or “”
  • Complete sentences
  • Words, such as “first,” “next,” “then”
• 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.

Ask students to share any rules, patterns, or steps they came up with for each of the scenarios.

Record students' shortcuts symbolically. For example, if a student says, “I always divided the perimeter by 2 and subtracted the width to find the length,” record that as , and ask the student to check that the record matches what they did.

Display the formulas for perimeter of a rectangle, area of a rectangle, area of a triangle, and volume of a cylinder next to students’ processes for isolating a variable in question. For example:

Students will have more opportunities to rewrite equations to isolate variables in their Algebra 1 class.