Lesson 6

The purpose of this Number Talk is to elicit strategies and understandings students have for determining how the size of a quotient changes when the decimal point in the divisor or dividend moves. These understandings help students develop fluency and will be helpful later in this lesson when students will need to be able to check the reasonableness of their answers. While four problems are given in the first problem, it may not be possible to share answers for all of them. 

Arrange students in groups of 2. Display the first question. Give students 2 minutes of quiet think time and ask them to give a signal when they have an answer, and reasoning, to support their answer. Follow with a whole-class discussion. Display the second question and give students 1 minute of quiet think time.

Ask students to share where they placed the decimal point in the second question and their reasoning. After students share, ask the class if they agree or disagree. Ask selected students, who chose different problems to solve, to share quotients for the problems in the first question. Record and display their responses for all to see. To involve more students in the conversation, consider asking:

• “Who can restate ___’s reasoning in a different way?”
• “Did anyone have the same strategy but would explain it differently?”
• “Did anyone solve the problem in a different way?”
• “Does anyone want to add on to _____’s strategy?”
• “Do you agree or disagree? Why?”

Emphasize student reasoning based in place value that involve looking at the relationship between the dividend and divisor to determine the size of the quotient.

This activity requires students to work with larger numbers, which is intended to encourage students to use an equation and notice the efficiencies of doing so. It also emphasizes the interpretation of the constant of proportionality in the context. In this case, the constant represents the cost of a single ticket, and makes it easy to identify which singer would make more money for similar ticket sales in a concert series. Note that asking students to give the revenues for different ticket sales encourages looking for and expressing regularity in repeated reasoning (MP8). The last set of questions ask students to interpret the constant of proportionality as represented in an equation in terms of the context (MP2).

Monitor for students who solve the problems using the following strategies and invite them to share during the whole-class discussion.

• writing many calculations, without any organization
• creating a table to organize their results
• writing an equation to encapsulate repeated reasoning

Provide access to calculators. Consider using the names of actual performers to make the task more interesting to students.

Select student responses to be shared with the whole class in discussion. Sequence their explanations from less efficient and organized to more efficient and organized. Discuss how the solutions are the same and different, and the advantages and disadvantages of each method. An important part of this discussion is correspondences and connections between different approaches.

This activity is intended to further develop students’ ability to write equations to represent proportional relationships. It involves work with decimals and asks for equations that represent proportional relationships of different pairs of quantities, which increases the challenge of the task.

Students may solve the first two problems in different ways. Monitor for different solution approaches such as: using computations, using tables, finding the constant of proportionality, and writing equations. 

Arrange students in groups of 2. Provide access to calculators. Give 5 minutes quiet work time followed by sharing work with a partner.

If students have trouble getting started, encourage them to create representations of the relationships, like a diagram or a table. If they are still stuck, suggest that they first find the weight and dollar value of 1 can.

Select students to share their methods: using computations, using tables, finding the constant of proportionality, writing equations. If students did not use equations to solve the first two problems, ask them how they can use the equations they found later in the activity to answer the first two questions.

If time permits, highlight connections between the equations generated, illustrated by the sequence of equations below. \(\displaystyle r = 0.014c\) \(\displaystyle r = 0.014(62.5w)\) \(\displaystyle r = 0.875w\)