Designing Simulations

5 min

Narrative

This Math Talk focuses on dividing sums. It encourages students to think about division problems with the same solution and to rely on the structure of previous solutions to mentally solve problems. The strategies elicited here will be helpful later in the lesson when students calculate means.

To divide sums with more terms, students need to look for and make use of structure (MP7).

In describing their strategies, students need to be precise in their word choice and use of language (MP6). 

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.

  • (4.2+3)÷2(4.2+3)\div2
  • (4.2+3.1+3.5)÷3(4.2+3.1+3.5)\div3
  • (4.2+3.1+3.5+3.6)÷4(4.2+3.1+3.5+3.6)\div4
  • (4.2+3.1+3.5+3.6+3.6)÷5(4.2+3.1+3.5+3.6+3.6)\div5

Sample Response

  • 3.6. Sample reasoning: 4.2÷2+3÷24.2 \div 2 + 3 \div 2 or 7.2÷27.2 \div 2.
  • 3.6. Sample reasoning: Compared to the first problem, there is an additional 3.6 in the sum (0.1 added to 3 and another 3.5), and the total is divided by 3, so it should have the same solution (or 10.8÷310.8\div 3).
  • 3.6. Sample reasoning: Because there is an additional 3.6 in the addends from the previous problem, and there are now four addends with a divisor of 4, the result is the same as in the previous problem (or 14.4÷414.4 \div 4).
  • 3.6. Sample reasoning: Again, 3.6 is added to the numerator, and the total is divided by 1 more than the previous problem, so it has the same solution (or 18÷518 \div 5).

Synthesis

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
  • 5.OA.1·Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.
  • 5.OA.A.1·Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.
  • 6.SP.5.c·Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered.
  • 6.SP.B.5.c·Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered.

10 min

20 min