Finding Cone Dimensions

5 min

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

Narrative

This Math Talk focuses on finding the unknown value in an equation. It encourages students to think about the fraction 13\frac{1}{3} and to rely on the structure of the equations to mentally solve problems. The strategies elicited here will be helpful later in the lesson when students are solving for the unknown length of the radius or height of a cone given its volume.

To solve each equation mentally, 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

Solve each equation mentally.

  • 27=13h27=\frac13h
  • 27=13r227=\frac13r^2
  • 12π=13πa12\pi=\frac13\pi a
  • 12π=13πb212\pi=\frac13\pi b^2

Sample Response

  • h=81h=81. Sample reasoning: I know 13⋅3=1\frac13\boldcdot3=1, so I multiplied each side by 3.
  • r=9r=9. Sample reasoning: I used the previous answer and that 81=9281=9^2.
  • a=36a=36. Sample reasoning: Since each side has π\pi, I know 12=13a12=\frac13 a, so aa must be 36.
  • b=6b=6. Sample reasoning: I used the previous answer and that 36=6236=6^2.
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.

15 min

10 min