This Math Talk focuses on multiplication by a decimal. It encourages students to think about how they can use the result of one multiplication problem to find the answer to a similar problem with a different, but related, factor. The understanding elicited here will be helpful later in the lesson when students evaluate equations where the constant of proportionality is a decimal.
To recognize how a factor has been scaled and predict how the product will be affected, students need to look for and make use of structure (MP7).
Student Lesson in Spanish
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 access to sticky notes or mini whiteboards. Supports accessibility for: Memory, Organization
Activity
None
Find the value of each expression mentally.
Student Response
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Building on Student Thinking
Activity Synthesis
To involve more students in the conversation, consider asking:
“Who can restate ’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 ’s strategy?”
“Do you agree or disagree? Why?”
“What connections to previous problems do you see?”
The key takeaway to highlight is how we can use the structure of place value and properties of operations to find products involving decimals.
Speaking: 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
Math Community
After the Warm-up, display the Math Community Chart. Remind students that norms are agreements that everyone in the class shares responsibility for, so it is important that everyone understands the intent of each norm and can agree with it. Tell students that today’s Cool-down includes a question asking for feedback on the drafted norms. This feedback will help identify which norms the class currently agrees with and which norms need revising or removing.
6.2
Activity
Standards Alignment
Building On
Addressing
7.RP.A.2
Recognize and represent proportional relationships between quantities.
In this activity, students practice writing an equation to represent a proportional relationship in a new context. Students reason about quantities and prices, calculating several values before writing an equation. No table is provided for students to organize their thinking in order to encourage them to look for regularity in repeated reasoning (MP8) and to notice the efficiency of using an equation to express the relationship.
This activity also emphasizes the interpretation of the constant of proportionality in the context, as students may choose to express the relationship as 5 cents per bottle, 0.05 dollars per bottle, or 20 bottles per dollar. Students compare these different approaches during the whole-class discussion.
Monitor for students who:
Write many calculations, without any organization
Create a table to organize their work
Write an equation to record their repeated reasoning
Use 5 as the constant of proportionality
Use 0.05 as the constant of proportionality
Use 20 as the constant of proportionality
This activity uses the Compare and Connect math language routine to advance representing and conversing as students use mathematically precise language in discussion.
Launch
Provide access to calculators.
Select work from students with different strategies, such as those described in the Activity Narrative, to share later.
Engagement: Develop Effort and Persistence. Provide tools to facilitate information processing or computation, enabling students to focus on key mathematical ideas. For example, allow students to use calculators to support their reasoning. Supports accessibility for: Memory, Conceptual Processing
Activity
None
Answer the following questions. Be prepared to explain your reasoning.
In Iowa, collection centers pay 5¢ per bottle that is returned.
How much would 30 bottles be worth?
How much would 250 bottles be worth?
How much would 860 bottles be worth?
How many bottles would it take to earn \$100?
How many bottles would it take to earn \$2,750?
Write an equation that relates the number of bottles to the amount of money received when the bottles are returned. What do your variables represent?
Student Response
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Activity Synthesis
The two goals of this discussion are:
To highlight the efficiency of expressing the relationship as an equation
To contrast different ways the relationship could be expressed as an equation
Invite previously selected students to share how they found the amounts of money and the numbers of bottles. Use Compare and Connect to help students compare, contrast, and connect the different approaches. Here are some questions for discussion:
“How could we represent this reasoning with an equation? What does the constant of proportionality in this equation represent?”
“Why do the different approaches lead to the same outcome?”
“Are there any benefits or drawbacks to one approach compared to another?”
Here are some different strategies for finding the value of some number of bottles, along with one or more ways to record that reasoning with an equation.
sample repeated reasoning
possible equations
meaning of the constant of proportionality
multiply by 5 (and then divide by 100)
5 cents per bottle
multiply by 0.05
0.05 dollars per bottle
divide by 20*
or
dollar per bottle
or
20 bottles per dollar
*If students suggest an equivalent equation that uses division, such as , confirm that that equation is correct and also ask if they can think of a way to express that equation in the form .
The key takeaway is that defining variables and writing an equations can be an efficient way to describe the proportional relationship between two quantities. However, there is more than one way to represent a given situation with an equation. It is important to specify what each variable represents so others can interpret the equation.
6.3
Activity
Standards Alignment
Building On
Addressing
7.RP.A.2
Recognize and represent proportional relationships between quantities.
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. As students identify the constants of proportionality between each pair of quantities to represent the relationships with equations, they are reasoning quantitatively and abstractly (MP2).
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.
This activity uses the Three Reads math language routine to advance reading and representing as students make sense of what is happening in the text.
Launch
Arrange students in groups of 2. Provide access to calculators.
Use Three Reads to support reading comprehension and sense-making about this problem. Display only the problem stem, without revealing the questions.
For the first read, read the problem aloud then ask, “What is this situation about?” (weight of cans and the amount of money made from recycling). Listen for and clarify any questions about the context.
After the second read, ask students to list any quantities that can be counted or measured. (number of aluminum cans; total weight of aluminum cans, in kilograms; money earned, in dollars)
After the third read, reveal the questions: “A family threw away 2.4 kg of aluminum in a month. How many cans did they throw away? What would be the dollar value if they recycled those same cans?” and ask, “What are some ways we might get started on this?” Invite students to name some possible starting points, referencing quantities from the second read. (Calculate the weight of aluminum in 1 can and the amount of money earned from 1 can)
Give 5 minutes of quiet work time followed by partner discussion.
Representation: Internalize Comprehension. Represent the same information through different modalities by using tables. If students are unsure where to begin, suggest that they draw a table to help organize the information provided. Supports accessibility for: Conceptual Processing, Visual-Spatial Processing
Activity
None
Aluminum cans can be recycled instead of being thrown in the garbage. The weight of 10 aluminum cans is 0.16 kilograms. The aluminum in 10 cans that are recycled has a value of \$0.14.
A family threw away 2.4 kg of aluminum cans in a month.
How many cans did they throw away? Explain or show your reasoning.
What would be the dollar value if they recycled those same cans? Explain or show your reasoning.
Write an equation to represent the relationship between each pair of quantities:
the number of cans and their weight , in kilograms
the number of cans and their recycled value , in dollars
the weight of cans and their recycled value
Student Response
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Building on Student Thinking
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.
Activity Synthesis
Invite several students to share their methods for solving the first two problems, such as using computations, using tables, finding the constant of proportionality, and 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 following sequence of equations.
Lesson Synthesis
Share with students “Today we wrote equations to represent proportional relationships where no tables were given. We saw that it is important to state what the variables in the equation represent.”
Briefly revisit some equations from the activities. For each equation, ask students:
“In this equation, what did each variable represent?”
“What did the number mean?”
To help students generalize about equations of proportional relationships, consider asking students:
“What do all these equations have in common?” (They are of the form , where k is the constant of proportionality. They have two variables and one number. The constant of proportionality is being multiplied by one of the variables.)
“How did writing an equation help you solve the problems?” (It made it easier to see what number you should multiply or divide by to answer each question.)
Student Lesson Summary
Remember that if there is a proportional relationship between two quantities, their relationship can be represented by an equation of the form . Sometimes writing an equation is the easiest way to solve a problem.
For example, we know that Denali, the highest mountain peak in North America, is 20,310 feet above sea level. How many miles is that? There are 5,280 feet in 1 mile. This relationship can be represented by the equation
where represents a distance measured in feet and represents the same distance measured in miles. Since we know Denali is 20,310 feet above sea level, we can write
Solving this equation for gives , so we can say that Denali is approximately 3.85 miles above sea level.
Standards Alignment
Building On
5.NBT.B.7
Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.
