Applying Area of Circles

5 min

Teacher Prep
Setup
Remind students that the circular field is set into a square that is 800 m on a side. 2 minutes quiet work time followed by whole-class discussion.

Narrative

This Math Talk focuses on expressions with variables. It encourages students to think about equivalence and to rely on properties of operations to mentally solve problems. The understanding elicited here will be helpful later in the lesson when students express answers in terms of π\pi.

To generate an equivalent expression with fewer terms, 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

For each expression, find an equivalent expression with fewer terms.

  • a+a+a+aa + a + a + a
  • a+a+a+b+ba + a + a + b + b
  • 9x−x9x - x
  • 5+6x+75 + 6x + 7

Sample Response

  • 4a4a or equivalent. Sample reasoning: Adding aa four times gives the same results as multiplying aa by 4.
  • 3a+2b3a + 2b or equivalent. Sample reasoning: Adding bb twice gives the same result as multiplying bb by 2. Also, 4a4a and 2b2b are not like terms, so they can’t be combined.
  • 8x8x. Sample reasoning: By the distributive property, 9x−1x=(9−1)x9x-1x= (9-1)x .
  • 6x+126x + 12 or equivalent. Sample reasoning: By the commutative and associative properties, 5+6x+7=6x+(5+7)5 + 6x + 7 = 6x + (5 + 7). Also, 6x6x and 1212 are not like terms, so they can’t be combined.
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 is that applying properties of operations allows us to write equivalent expressions with fewer terms. This is often called "combining like terms."

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
  • 7.G.4·Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.
  • 7.G.B.4·Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.

25 min