Standard Form and Factored Form

5 min

Narrative

This Math Talk focuses on the equivalence of subtracting a number and adding the opposite of the number. It encourages students to think about the meaning of subtraction and to rely on the structure of the equations to mentally solve problems. The understanding elicited here will be helpful later in the lesson when students reason about quadratics in factored form.

To solve the equations, 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.
  • 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

Solve each equation for nn, mentally.

  • 40−8=40+n40-8=40+n
  • 25+-100=25−n25+\text-100=25-n
  • 3−12=3+n3-\frac12=3+n
  • 72−n=72+672-n=72+6

Sample Response

  • -8. Sample reasoning: The left side is 32 which is less than 40. To get a value less than 40 on the right by adding, the value being added must be negative, and for the value on the right side to be 32, the number added must be -8.
  • 100. Sample reasoning: Adding a negative number is the same as subtracting the positive number with the same magnitude.
  • -12\text{-}\frac{1}{2}. Sample reasoning: The left side is 2122 \frac{1}{2}, so negative one half must be added to the right side to get the same value.
  • -6. Sample reasoning: The right side is 78, which is greater than 72. To get a number greater than 72 on the left by subtracting, we must use a negative number, and that number must be -6.
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?”

If not already made clear in students’ explanations, highlight that subtracting a number gives the same outcome as adding the opposite of that number.

Tell students that remembering that subtraction can be thought of in terms of addition can help us rewrite quadratic expressions such as (x−5)(x+2)(x-5)(x+2) or (x−9)(x−3)(x-9)(x-3), where one or both factors are differences.

MLR8 Discussion Supports. Display sentence frames to support students when they explain their strategy. For example, “First, I _____ because . . . .” or “I noticed _____ 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
Addressing
  • A-SSE.3·Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
  • A-SSE.3·Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
  • A-SSE.3·Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
  • HSA-SSE.B.3·Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

20 min

10 min