Practice Solving Equations

5 min

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

Narrative

This Math Talk focuses on division of a number by a fraction. It encourages students to reason about the meaning of division and to rely on their knowledge of the division algorithm or what they know about the relationship between the dividend, divisor, and quotient to mentally solve problems (MP7). The understanding elicited here will be helpful later in the lesson when students solve equations of the form px=qpx = q where pp  and qq are fractions.

In explaining 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.

Representation: Internalize Comprehension. 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.

  • 10÷1510 \div \frac{1}{5}
  • 10÷2510 \div \frac{2}{5}
  • 1÷251 \div \frac{2}{5}
  • 110÷25\frac{1}{10} \div \frac{2}{5}

Sample Response

  • 50. Sample reasoning:
    • There are 5 groups of 15\frac{1}{5} in 1, so there are 10⋅510 \boldcdot 5 groups in 10.
    • 50⋅15=505=1050 \boldcdot \frac{1}{5} = \frac{50}{5} = 10
  • 25. Sample reasoning:
    • There are half as many groups of 25\frac{2}{5} in 10 as there are groups of 15\frac{1}{5}, and half of 50 is 25. 
    • Ten is 505\frac{50}{5}, and there are 25 groups of 25\frac{2}{5} in 505\frac{50}{5}.
  • 2.5. Sample reasoning:
    • The dividend 1 is one tenth of the dividend in the second problem while the divisor is the same, so the quotient is one tenth of 25, which is 2.5.
    • There are 2 full groups plus 12\frac{1}{2} group of 25\frac{2}{5} in 1.
  • 0.25. Sample reasoning:
    • The dividend 110\frac{1}{10} is one tenth of the dividend in the previous problem, so the quotient is one tenth of 2.5, which is 0.25.
    • 25\frac{2}{5} is 410\frac{4}{10}, which means there is 14\frac{1}{4} of 410\frac{4}{10} in 110\frac{1}{10}.
    • 110÷25=110⋅52\frac{1}{10} \div \frac{2}{5} = \frac{1}{10} \boldcdot \frac{5}{2}, which is 520\frac{5}{20} or 14\frac{1}{4}.
Activity Synthesis (Teacher Notes)

Ask students to share their strategies for each problem. Record and display their responses for all to see. 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?”
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
Addressing
  • 6.NS.3·Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.
  • 6.NS.B.3·Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.

15 min

15 min