Representing Contexts with Equations

5 min

Narrative

This Math Talk focuses on whether the solution to an equation will be positive or negative. It encourages students to think about the signs of the numbers and their relative magnitudes and to rely on what they know about operations with rational numbers to mentally solve problems. The strategies elicited here will be helpful later in the lesson when students find the solutions to different 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 sticky notes or mini whiteboards.
Supports accessibility for: Memory, Organization

Student Task

Decide mentally whether each solution is positive or negative.

  • (-8.7)(1.4)=a(\text-8.7)(1.4) = a
  • -8.7b =1.4\text- 8.7b = 1.4
  • -8.7 +c =-1.4\text-8.7 + c = \text- 1.4
  • -8.7− d =-1.4\text-8.7 - d = \text- 1.4

Sample Response

  • aa is negative. Sample reasoning: The product of a negative number and a positive number is negative.
  • bb is negative. Sample reasoning: In order for the product of two numbers to be positive, the signs of the two factors have to be the same.
  • cc is positive. Sample reasoning: A positive number needs to be added to -8.7 to make the sum greater than -8.7, since -1.4 is greater than -8.7.
  • dd is negative. Sample reasoning: Since a subtraction expression can be written as the sum of the additive inverse, the sign of dd will be opposite the sign of cc.

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?”
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.4.a·Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. <em>For example, the perimeter of a rectangle is 54 cm. Its length is 6 cm. What is its width?</em>
  • 7.EE.B.4.a·Solve word problems leading to equations of the form <span class="math">\(px + q = r\)</span> and <span class="math">\(p(x + q) = r\)</span>, where <span class="math">\(p\)</span>, <span class="math">\(q\)</span>, and <span class="math">\(r\)</span> are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. <span>For example, the perimeter of a rectangle is <span class="math">\(54\)</span> cm. Its length is <span class="math">\(6\)</span> cm. What is its width?</span>

15 min

15 min

15 min