options: Ratios & Proportional Relationships: students learn to recognize, describe, represent, and reason with ratios, rates, proportional relationships, and percent.

Counting & Cardinality: students learn to know number names and the count sequence, count to tell the number of objects, and compare numbers.

Operations & Algebra: students learn to solve problems using algebraic thinking and operations such as addition, subtraction, multiplication, and division. They may learn to identify and explain arithmetic patterns, evaluate or manipulate numerical expressions, and reason with equations and inequalities.

Statistics & Probability: students learn to analyze and produce data distributions, and build understanding of univariate and bivariate data. They learn to interpret data, make statistical inferences, justify conclusions, understand rules of probability, and use probability to make decisions.

Number Systems and Quantity: students learn to understand the complex number system and reason quantitatively.

Modeling: students learn to choose and use appropriate mathematics and statistics to analyze real-world empirical situations, improve decisions, and report on their conclusions and the reasoning behind them.

Geometry: students learn to classify geometric figures by their properties, and understand and compare the relationships between them. Students develop and use formulas to compute lengths, areas, and volumes, and they use transformations to generate new shapes and compare existing ones. Geometry can be studied with and without coordinates.

Measurement & Data: students learn to work with data and measure attributes such as time, money, length, area, and volume. They may learn to compare measurements to operations and convert between different units of measure.

Number & Operations - Fractions: students learn to understand fractions as numbers and may work with them using addition, subtraction, multiplication, and division.

Number & Operations in Base Ten: students learn to work with the base-ten system and build place value understanding.

Functions: students learn to define, use, and evaluate functions to model relationships between quantities.
problem_activity: Optional activity
In this activity, students continue their work comparing properties of functions represented in different ways. Students are given a verbal description and a table to compare and decide whose family traveled farther over the same time intervals. The purpose of this activity is for students to continue building their skill interpreting and comparing functions.
Launch
Give students 3–5 minutes of quiet work time, followed with whole-class discussion.
Student Facing
Elena’s family is driving on the freeway at 55 miles per hour.
Andre’s family is driving on the same freeway, but not at a constant speed.  The table shows how far Andre's family has traveled,
\(d\)
, in miles, every minute for 10 minutes.
{"table": [{"_attributes": {"border": "1", "cellpadding": "1", "cellspacing": "1", "style": "width:500px"}, "tbody": [{"tr": [{"th": [{"_attributes": {"scope": "row"}, "span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "166"}, "_value": "\\(t\\)"}]}], "td": [{"_value": "1"}, {"_value": "2"}, {"_value": "3"}, {"_value": "4"}, {"_value": "5"}, {"_value": "6"}, {"_value": "7"}, {"_value": "8"}, {"_value": "9"}, {"_value": "10"}]}, {"th": [{"_attributes": {"scope": "row"}, "span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "292"}, "_value": "\\(d\\)"}]}], "td": [{"_value": "0.9"}, {"_value": "1.9"}, {"_value": "3.0"}, {"_value": "4.1"}, {"_value": "5.1"}, {"_value": "6.2"}, {"_value": "6.8"}, {"_value": "7.4"}, {"_value": "8"}, {"_value": "9.1"}]}]}]}]}
How many miles per minute is 55 miles per hour?
Who had traveled farther after 5 minutes? After 10 minutes?
How long did it take Elena’s family to travel as far as Andre’s family had traveled after 8 minutes?
For both families, the distance in miles is a function of time in minutes. Compare the outputs of these functions when the input is 3.
Student Response
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Anticipated Misconceptions
Students may miss that Elena’s family’s speed has different units than what is needed to compare with Andre’s family. Point out to students that the units are important and possibly ask them to find out how many miles Elena’s family travels in 1 minute rather than 1 hour.
Activity Synthesis
The purpose of this discussion is for students to think about how they use a verbal description and table to answer questions related to the context. Ask students to share their solutions and how they used the equation and graph. Consider asking some of the following questions:
“How did you use the table to get information? How did you use the verbal description?”
“What did you prefer about using the description to solve the problem? What did you prefer about using the table to solve the problem?”
Engagement: Develop Effort and Persistence.
Break the class into small discussion groups and then invite a representative from each group to report back to the whole class.
Supports accessibility for: Language; Social-emotional skills; Attention
Writing, Speaking: MLR1 Stronger and Clearer Each Time.
Use this routine to give students a structured opportunity to reflect on their problem-solving strategies. Ask students to write an initial response to the prompt, “What did you prefer about using the description to solve the problem? What did you prefer about using the table to solve the problem?” Give students time to meet with 2–3 partners, to share and get feedback on their responses. Provide prompts for feedback that will help students strengthen their ideas and clarify their language (e.g., “What do you mean?”, “Can you give an example?”, and “Can you say that another way?”, etc.). Give students 1–2 minutes to revise their writing based on the feedback they received.​ This will help students produce a written generalization about interpreting and comparing functions.
Design Principle(s): Optimize output (for generalization)


correct_option_index: 10