How Many Solutions?

10 min

Narrative

This Math Talk focuses on using the zero product property correctly. It encourages students to use the structure of the equations to mentally solve problems. The understanding elicited here will be helpful later in the lesson when students solve quadratic equations in factored form. To recognize the key features of the zero product property, students need to look for and make use of structure (MP7).

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 whether each statement is true or false.

  • 3 is the only solution to x2−9=0x^2-9=0.
  • A solution to x2+25=0x^2+25=0 is -5.
  • x(x−7)=0x(x-7)=0 has two solutions.
  • 5 and -7 are the solutions to (x−5)(x+7)=12(x-5)(x+7)=12.

Sample Response

  • False. Sample reasoning: There are two solutions because there are two numbers that can be squared to get 9.
  • False. Sample reasoning:There is no solution to x2+25=0x^2+25=0 because no number can be squared to get -25.
  • True. Sample reasoning: There are two factors, 0 and 7, that could make the product 0.
  • False. Sample reasoning: The product of the two factors is not 0, so we cannot solve it using the zero product property (which gives 5 and -7 as solutions). If we evaluate the expression (x−5)(x+7)(x-5)(x+7) at those two values, the result in each case will be 0, not 12.
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 On
  • 6.EE.5·Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.
  • 6.EE.B.5·Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.
Building Toward
  • A-REI.4·Solve quadratic equations in one variable.
  • A-REI.4·Solve quadratic equations in one variable.
  • A-REI.4·Solve quadratic equations in one variable.
  • HSA-REI.B.4·Solve quadratic equations in one variable.

15 min

10 min