Different Options for Solving One Equation

5 min

Teacher Prep
Setup
Display one equation at a time. Allow 30 seconds of quiet think time, followed by a whole-class discussion.

Narrative

This Math Talk focuses on seeing structure in equations of the form p(x+q)=rp(x+q)=r. It encourages students to see (x−3)(x-3) as a chunk in order to mentally solve equations. The understanding elicited here will be helpful later in the lesson when students compare and contrast solution methods.

To work with (x−3)(x-3) as an object, students need to look for and make use of structure (MP7).

Launch

Display one equation at a time. Give students 30 seconds of quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all problems displayed throughout the talk. Follow with a whole-class discussion.

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.

  • \begin {align} 100(x-3) &= 1,000\end{align}
  • \begin {align} 500(x-3) &= 5,000\end{align}
  • \begin {align} 0.03(x-3) &= 0.3 \end{align}
  • \begin {align} 0.72(x+2) &= 7.2 \\ \end{align}

Sample Response

  • x=13x=13. Sample reasoning: 100 times a value is 1,000, so that value must be 10. Since x−3=10x−3=10, the value of xx must be 13.
  • x=13x=13. Sample reasoning: 500 times a value is 5,000, so that value must be 10. Since x−3=10x−3=10, the value of xx must be 13.
  • x=13x=13. Sample reasoning: 0.03 times a value is 3, so that value must be 10. Since x−3=10x−3=10, the value of xx must be 13.
  • x=8x=8. Sample reasoning: 0.72 times a value is 7.2, so that value must be 10. Since x+2=10x+2=10, the value of xx must be 8.
Activity Synthesis (Teacher Notes)

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 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