options: 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.

Functions: students learn to define, use, and evaluate functions to model relationships between quantities.

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

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.

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

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.

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

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

Number Systems and Quantity: students learn to understand the complex number system and reason quantitatively.
problem_activity: Activity
This activity is intended to further students’ understanding of the graphs of proportional relationships in the following respects:
points on the graph of a proportional relationship can be interpreted in the context represented (MP2)
for these points, the quotient of the coordinates is—excepting
\((0,0)\)
—the constant of proportionality
if the first coordinate is 1, then the corresponding coordinate is
\(k\)
, the constant of proportionality.
Students explain correspondences between parts of the table and parts of the graph. The graph is simple so that students can focus on what a point means in the situation represented. Students need to realize, however, that the axes are marked in 10-unit intervals. The discussion questions are opportunities for students to construct viable arguments and critique the reasoning of others (MP3).
Launch
Arrange students in groups of 2. Give students 5 minutes of quiet work time followed by students discussing responses with a partner, followed by whole-class discussion.
Action and Expression: Internalize Executive Functions.
Chunk this task into more manageable parts to support students who benefit from support with organizational skills in problem solving. For example, present one question at a time and monitor students to ensure they are making progress throughout the activity.
Supports accessibility for: Organization; Attention
Speaking, Reading: MLR5 Co-Craft Questions.
To help students make sense of graphs of proportional relationships, start by displaying only the first line of this task (“Tyler was at the amusement park. He walked at a steady pace from the ticket booth to the bumper cars.”) and the graph. Ask students to write down possible mathematical questions that could be asked about the situation. Invite students to compare their questions before revealing the remainder of the questions. Listen for and amplify any questions involving correspondences between parts of the table and parts of the graph. This helps students produce the language of mathematical questions and talk about the relationship between distance and time.
Design Principle(s): Maximize meta-awareness; Support sense-making
Student Facing
Tyler was at the amusement park. He walked at a steady pace from the ticket booth to the bumper cars.
The point on the graph shows his arrival at the bumper cars. What do the coordinates of the point tell us about the situation?
The table representing Tyler's walk shows other values of time and distance. Complete the table. Next, plot the pairs of values on the grid.
What does the point
\((0, 0)\)
mean in this situation?
How far away from the ticket booth was Tyler after 1 second? Label the point on the graph that shows this information with its coordinates.
What is the constant of proportionality for the relationship between time and distance? What does it tell you about Tyler's walk? Where do you see it in the graph?
[IMG]
| time                         (seconds) | distance                         (meters) |
|----------------------------------------|-------------------------------------------|
|                   0                    |                     0                     |
|                   20                   |                    25                     |
|                   30                   |                   37.5                    |
|                   40                   |                    50                     |
|                   1                    |                                           |
Student Response
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Student Facing
Are you ready for more?
If Tyler wanted to get to the bumper cars in half the time, how would the graph representing his walk change? How would the table change? What about the constant of proportionality?
Student Response
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for free access to Extension Student Response.
Anticipated Misconceptions
These questions can be used for discussion or for students who need scaffolding.
"What quantities are shown in the graph?" (Distance in meters that Tyler is from the ticket booth and time elapsed in seconds since he started walking.)
"How far is the ticket booth from the bumper cars?" (50 meters, assuming that Tyler walked in a straight line.) This is an opportunity for attention to precision (MP6) and making explicit assumptions about a situation (MP4).
"Do the values in your table show a proportional relationship? How do you know?" (Based on prior lessons in this unit, students should identify the relationship as proportional because for every point the unit rate is the same.)
"What do the coordinates of the points on the graph show?" (The first coordinate gives amount of time in seconds that elapsed since Tyler started walking. Its corresponding second coordinate shows how many meters away from the ticket booth Tyler was at the corresponding time, assuming that Tyler walked in a straight line.)
Activity Synthesis
After students work on the task, it is important to discuss how the axis labels and the description in the task statement help us interpret points on the graph.
Consider asking these questions:
"What quantities are shown in the graph?" (Distance in meters that Tyler is from the ticket booth and time elapsed in seconds since he started walking.)
"How far is the ticket booth from the bumper cars?" (50 meters, assuming that Tyler walked in a straight line.) This is an opportunity for attention to precision (MP6) and making explicit assumptions about a situation (MP4).
"Do the values in your table show a proportional relationship? How do you know?" (Based on prior lessons in this unit, students should identify the relationship as proportional because for every point the unit rate is the same.)
"What do the coordinates of the points on the graph show?" (The first coordinate gives amount of time in seconds that elapsed since Tyler started walking. Its corresponding second coordinate shows how many meters away from the ticket booth Tyler was at the corresponding time, assuming that Tyler walked in a straight line.)
Ask students for the equation of this proportional relationship. Finally ask where students see
\(k\)
, the constant of proportionality, in each representation, the equation, the graph, the table, and the verbal description of the situation.


correct_option_index: 2