Solving Equations with Rational Numbers

5 min

Narrative

This Math Talk focuses on reminding students about opposites and reciprocals. It encourages students to think about how to get a sum of 0 or product of 1. The strategies elicited here will be helpful later in the lesson when students use the additive and multiplicative inverses to solve equations.

Launch

Tell students to close their books or devices (or to keep them closed). Reveal one problem at a time. For each problem:

  • Give students quiet think time, and ask them to give a signal when they have an answer and a strategy.
  • Invite students to share their strategies, and record and display their responses for all to see.
  • Use the questions in the Activity Synthesis to involve more students in the conversation before moving to the next problem. Keep all previous problems and work displayed throughout the talk.
Action and Expression: Internalize Executive Functions. To support working memory, provide students with access to sticky notes or mini whiteboards.
Supports accessibility for: Memory, Organization

Student Task

Solve each equation mentally.

  • 7⋅b=17 \boldcdot b = 1
  • c⋅d=1c \boldcdot d = 1
  • 11+f=011 + f = 0
  • g+h=0g + h = 0

Sample Response

  • b=17b=\frac17. Sample reasoning: One seventh of 7 is 1.
  • Answers vary. Sample reasoning: cc and dd can be any two numbers that are reciprocals of each other.
  • f=-11f=\text-11. Sample reasoning: Adding the opposite value will give a sum of 0.
  • Answers vary. Sample reasoning: gg and hh can be any two numbers that are opposites of each other.

Synthesis

To involve more students in the conversation, consider asking:

  • “Who can restate ‾\underline{\hspace{.5in}}’s reasoning in a different way?”
  • “Did anyone use 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?”
  • “What connections to previous problems do you see?”

If not brought up in students’ explanations, make these ideas explicit:

  • The sum of a number and its opposite is 0.
  • The product of a number and its reciprocal is 1.
  • If you want to find a number that you can add to something and get 0 as a sum, use its opposite.
  • If you want to find a number that you can multiply something by and get 1 as a product, use its reciprocal.
MLR8 Discussion Supports. Display sentence frames to support students when they explain their strategy. For example, “First, I ‾\underline{\hspace{.5in}} because . . . .” or “I noticed ‾\underline{\hspace{.5in}} so I . . . .” Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class.
Advances: Speaking, Representing
Standards
Building On
  • 7.NS.A·Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.
  • 7.NS.A·Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.
Building Toward
  • 7.EE.B·Solve real-life and mathematical problems using numerical and algebraic expressions and equations.
  • 7.EE.B·Solve real-life and mathematical problems using numerical and algebraic expressions and equations.

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10 min