This Math Talk focuses on division that results in a decimal. It encourages students to think about how they can use the result of one division problem to find the answer to a similar problem with a different, but related, divisor or dividend. The understanding elicited here will be helpful later in the lesson when students calculate constants of proportionality.
To recognize how a divisor has been scaled and predict how the quotient 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.
645 ÷ 10
645 ÷ 100
645 ÷ 50
64.5 ÷ 50
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 the effects of multiplying or dividing numbers by powers of 10.
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
At the end of the Warm-up, display the Math Community Chart. Tell students that norms are expectations that help everyone in the room feel safe, comfortable, and productive doing math together. Using the Math Community Chart, offer an example of how the “Doing Math” actions can be used to create norms. For example, “In the last exercise, many of you said that our math community sounds like ‘sharing ideas.’ A norm that supports that is ‘We listen as others share their ideas.’ For a teacher norm, ‘questioning vs telling’ is very important to me, so a norm to support that is ‘Ask questions first to make sure I understand how someone is thinking.’”
Invite students to reflect on both individual and group actions. Ask, “As we work together in our mathematical community, what norms, or expectations, should we keep in mind?” Give 1–2 minutes of quiet think time and then invite as many students as time allows to share either their own norm suggestion or to “+1” another student’s suggestion. Record student thinking in the student and teacher “Norms” sections on the Math Community Chart.
Conclude the discussion by telling students that what they made today is only a first draft of math community norms and that they can suggest other additions during the Cool-down. Throughout the year, students will revise, add, or remove norms based on those that are and are not supporting the community.
3.2
Activity
Standards Alignment
Building On
Addressing
7.RP.A.2.b
Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.
This activity has two purposes. First, it involves an important context that can be represented by proportional relationships, namely, measurement conversion. Second, it introduces the idea that there are two constants of proportionality and that they are reciprocals (also known as multiplicative inverses). Students start to use “is proportional to” language to distinguish between the two constants of proportionality:
The length measured in millimeters is proportional to the length measured in centimeters, and the constant of proportionality is 10.
The length measured in centimeters is proportional to the length measured in millimeters, and the constant of proportionality is , or 0.1.
This activity uses the Collect and Display math language routine to advance conversing and reading as students clarify, build on, or make connections to mathematical language.
Launch
Use Collect and Display to direct attention to words collected and displayed from an earlier lesson. Collect the language students use to describe the proportional relationships. Display words and phrases such as “is proportional to,” “constant of proportionality,” “reciprocal,” “inverse,” etc.
Give students quiet work time followed by partner and whole-class discussion.
Action and Expression: Provide Access for Physical Action. Activate or supply background knowledge. Provide a display showing equivalence between dividing by a number and multiplying by its reciprocal for students to use as a reference. Supports accessibility for: Memory, Organization
Activity
None
There is a proportional relationship between any length measured in centimeters and the same length measured in millimeters.
A ruler. Centimeters on top. Millimeters on bottom. Centimeters, 0 to 5, by 1's. Millimeters, 0 to 50, by 10's. 1 centimeter = 10 millimeters, 2 centimeters = 20 millimeters, 3 centimeters = 30 millimeters, 4 centimeters = 40 millimeters, 5 centimeters = 50 millimeters.
There are two ways of thinking about this proportional relationship.
If the length of something in centimeters is known, its length in millimeters can be calculated.
Complete the table.
What is the constant of proportionality?
length (cm)
length (mm)
9
12.5
50
88.49
If the length of something in millimeters is known, its length in centimeters can be calculated.
Complete the table.
What is the constant of proportionality?
length (mm)
length (cm)
70
245
4
699.1
How are these two constants of proportionality related to each other?
Complete each sentence:
To convert from centimeters to millimeters, the value in centimeters is multiplied by ________.
To convert from millimeters to centimeters, the value in millimeters is divided by ________, or multiplied by ________.
Activity Synthesis
The goal of this discussion is to help students recognize how the structure of proportional relationships applies to this situation involving measurement conversion.
Direct students' attention to the reference created using Collect and Display. Ask students to share what they noticed about the two constants of proportionality, borrowing language from the display as needed. Next, ask students to suggest ways to update the display: “Are there any new words or phrases that you would like to add?” “Is there any language you would like to revise or remove?”
The key takeaway is that the two constants of proportionality are reciprocals. If needed, remind students that dividing by 10 is the same as multiplying by its reciprocal, . One way to explain why these two constants of proportionality are multiplicative inverses is to imagine starting with a measurement in centimeters, 15 centimeters for example. To convert 15 centimeters to millimeters, we multiply 15 by 10. So 15 centimeters = 150 millimeters. If we convert the measurement in millimeters back to centimeters, it will be 15, so the constant of proportionality we need to multiply by is .
3.3
Activity
Standards Alignment
Building On
Addressing
7.RP.A.2.b
Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.
This activity focuses on making connections between constant speed and proportional relationships, with special attention to the constant of proportionality. The numbers are chosen so that students are more likely to use unit rate rather than scale factors. The goal is for students to understand that when speed is constant, time elapsed and distance traveled are proportional. The constant of proportionality indicates the magnitude of the speed. For example, if distance is given in miles and time in hours, then the constant of proportionality indicates the speed in miles per hour. As students make these connections, they are reasoning abstractly and quantitatively (MP2).
Students might wonder why the route is not a straight line. If they ask about it, teachers can share some reasons why airplane routes are complex, for example the need to avoid congested areas and the fact that the shortest distances on the curved surface of Earth do not always correspond to lines on a map.
This is the first time Math Language Routine 5: Co-Craft Questions is suggested in this course. In this routine, students are given a context or situation, often in the form of a problem stem (for example, a story, image, video, or graph) with or without numerical values. Students develop mathematical questions that can be asked about the situation. A typical prompt is: “What mathematical questions could you ask about this situation?” The purpose of this routine is to allow students to make sense of a context before feeling pressure to produce answers, and to develop students’ awareness of the language used in mathematics problems.
This activity uses the Co-Craft Questions math language routine to advance reading and writing as students make sense of a context and practice generating mathematical questions.
Launch
Arrange students in groups of 2. Use Co-Craft Questions to give students an opportunity to familiarize themselves with the context, and to practice producing the language of mathematical questions.
Display only the problem stem and related image, without revealing the questions.
Ask students, “What mathematical questions could you ask about this situation?”
Give students 1–2 minutes to write a list of mathematical questions that could be asked about the situation before comparing questions with a partner.
As partners discuss, support students in using conversation and collaboration skills to generate and refine their questions, for instance, by revoicing a question, seeking clarity, or referring to their written notes.
Listen for how students use language about speed, distance traveled, and elapsed time.
Invite several students to share one question with the class and record for all to see. Ask the class to make comparisons among the shared questions and their own. Ask, “What do these questions have in common? How are they different?” Listen for and amplify questions that focus on the relationship between speed, distance traveled, and elapsed time.
Reveal the table and questions and give students 1–2 minutes to compare it to their own question and those of their classmates. Invite students to identify similarities and differences by asking:
“Which of your questions is most similar to or different from the ones provided? Why?”
“Is there a main mathematical concept that is present in both your questions and those provided? If so, describe it.”
Ask students to complete the questions.
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
Activity Synthesis
The goal of the discussion is for students to understand that when speed is constant, then distance traveled is proportional to elapsed time. The constant of proportionality is the speed. For example, if distance is given in miles and time in hours, then the constant of proportionality indicates the speed in miles per hour.
Begin by having a student share a table that is all in hours, miles, and miles per hour. Use this example to point out that every number in the first column can be multiplied by the speed to get the number in the second column. Ask:
“Which quantities are in a proportional relationship? How do you know?”
“What is the constant of proportionality in this case?”
If one or more students use minutes for time and miles per minute for speed in the table, the same questions can be asked for these units if time allows, but it is not required. In any case, summarize by making it explicit that when time and distance are in a proportional relationship, the constant of proportionality is the speed (or pace).
Lesson Synthesis
Share with students, “Today we looked at more tables of proportional relationships and found the constant of proportionality for each one.”
Briefly revisit the two contexts, reinforcing the use of new terms. Consider asking students:
“In the first activity, we examined the proportional relationship between millimeters and centimeters from two different perspectives and found two constants of proportionality. What were they?” (10 and )
“What is the relationship between the two constants of proportionality?” (They are reciprocals.)
“In the second activity, we examined a proportional relationship between the time a plane flies and the distance it travels. What was the constant of proportionality?” (550 or )
“What did the constant of proportionality represent in terms of the situation?” (the plane’s speed)
Student Lesson Summary
When something is traveling at a constant speed, there is a proportional relationship between the time it takes and the distance traveled.
The table shows the distance traveled and elapsed time for a bug crawling on a sidewalk.
We can multiply any number in the first column by to get the corresponding number in the second column. We can say that the elapsed time is proportional to the distance traveled, and the constant of proportionality is . This means that the bug’s pace is seconds per centimeter.
Table with 2 columns and 4 rows of data. The columns are: distance traveled (cm) and elapsed time (sec). The table has the ordered pairs (the fraction 3 over 2 comma 1), (1 comma the fraction 2 over 3), (3 comma 2) and (10 comma the fraction 20 over 3). Each pair has an arrow pointing from the value in the first column to the value in the second column. The arrow represents multiplying the first value by the fraction 2 over 3 to calculate the second value.
This table represents the same situation, except the columns are switched.
We can multiply any number in the first column by to get the corresponding number in the second column. We can say that the distance traveled is proportional to the elapsed time, and the constant of proportionality is . This means that the bug’s speed is centimeters per second.
Table with 2 columns and 4 rows of data. The columns are: elapsed time (sec) and distance traveled (cm). The table has the ordered pairs (1 comma the fraction 3 over 2), (the fraction 2 over 3 comma 1), (2 comma 3) and (the fraction 20 over 3 comma 10). Each pair has an arrow pointing from the value in the first column to the value in the second column. Times the fraction three over 2 is below the table.
Notice that is the reciprocal of . When two quantities are in a proportional relationship, there are two constants of proportionality, and they are always reciprocals of each other. When we represent a proportional relationship with a table, we say the quantity in the second column is proportional to the quantity in the first column, and the corresponding constant of proportionality is the number we multiply values in the first column by to get the values in the second.
Standards Alignment
Building On
5.MD.A.1
Convert among different-sized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multi-step, real world problems.
Some students may say that the constants of proportionality are both 10 since you can divide by 10 in the second table. Tell students, “The constant of proportionality is what you multiply by. Can you find a way to multiply the numbers in the first column to get the numbers in the second column?”
On its way from New York to San Diego, a plane flew over Pittsburgh, Saint Louis, Albuquerque, and Phoenix traveling at a constant speed.
Complete the table as you answer the questions. Be prepared to explain your reasoning.
A United States map with 5 segments representing the distance a plane flew. The segments are New York to Pittsburgh, Pittsburgh to Saint Louis, Saint Louis to Albuquerque, Albuquerque to Phoenix and Phoenix to San Diego.
segment
time
distance
speed
Pittsburgh to Saint Louis
1 hour
550 miles
Saint Louis to Albuquerque
1 hour 42 minutes
Albuquerque to Phoenix
330 miles
What is the distance between Saint Louis and Albuquerque?
How many minutes did it take to fly between Albuquerque and Phoenix?
What is the proportional relationship represented by this table?
Diego says the constant of proportionality is 550. Andre says the constant of proportionality is . Do you agree with either of them? Explain your reasoning.
Student Response
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Building on Student Thinking
Students who need support in understanding the context can trace the segments on the map, labeling the distances they know and putting question marks for unknown distances. An empty double number line could also be a useful tool in helping students reason about the context.
