Estimating Areas

5 min

Teacher Prep
Setup
Display one problem at a time. 1 minute of quiet think time per problem, followed by a whole-class discussion.

Narrative

This Math Talk focuses on addition and subtraction problems that can be simplified through compensation. It encourages students to think about adjusting addends by 1 or 2 to mentally solve problems. The strategies elicited here are arithmetic analogues of the composition and decomposition techniques students will use in this lesson to calculate areas of shapes.

As they choose how to rewrite numbers in order to make finding their sum or difference as efficient as possible, 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

Find the value of each expression mentally.

  • 599+87599 + 87
  • 48+31348+313
  • 440−29440-29
  • 254−88254 - 88

Sample Response

  • 686. Sample reasoning: Taking one away from 87 and adding it to 599 turns it into 600+86600+86.
  • 361. Sample reasoning: Taking two away from 313 and adding it to 48 turns it into 50+31150 + 311.
  • 411. Sample reasoning: Instead of subtracting 29, it is easier to subtract 30. Since this is subtracting 1 more, 440 needs to be increased by 1, or 441−30441 - 30.
  • 166. Sample reasoning: In order to subtract 88, first subtract 54 and then subtract 34. 254−54=200254 - 54 = 200 and 200−34=166200 - 34 = 166.
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?”

The key takeaway to highlight is the idea of compensation: identifying numbers close to the given ones for which the calculation can be done more efficiently.

  • For 599 + 87, because 599 is only one away from 600 (a nice round number), it is natural to change 599 to 600. Adding 1 to 599 means that we need to subtract one from 87 to keep the sum the same. So the answer is 600 + 86, or 686.
  • For 254 - 88, students may identify 90 or 100 as a nice number near 88 which is simpler to subtract. Subtracting 100 would be subtracting 12 more than 88, so we need to add 12 to 254. So the answer is 266 - 100, or 166. Tell students that in this lesson they are going to use these kinds of strategies with geometric figures to find areas efficiently.
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
Building On
  • 5.OA.A·Write and interpret numerical expressions.
  • 5.OA.A·Write and interpret numerical expressions.

20 min

10 min