options: 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.

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.

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

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

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.

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

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.

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

Number & Operations in Base Ten: students learn to work with the base-ten system and build place value understanding.
problem_activity: Activity
In this activity, students investigate how the height of water in a graduated cylinder is a function of the volume of water in the graduated cylinder. Students make predictions about how the graph will look and then test their prediction by filling the graduated cylinder with different amounts of water, gathering and graphing the data (MP4).
Digital
Launch
Arrange students in groups of 3–4. Be sure students know how to measure using a graduated cylinder. If needed, display a graduated cylinder filled to a specific measurement for all to see and demonstrate to students how to read the measurement. Give each group access to a graduated cylinder and water.
Give groups 8–10 minutes to work on the task, follow with a whole-class discussion.
For classrooms with access to the digital materials or those with no access to graduated cylinders, an applet is included here. Physical measurement tools and an active lab experience are preferred.
Speaking: MLR8 Discussion Supports.
Display sentence frames to support small-group discussion. For example, “I think ____ , because _____ .” or “I (agree/disagree) because ______ .”
Design Principle(s): Support sense-making; Optimize output for (explanation)
Student Facing
Use the applet to investigate the height of water in the cylinder as a function of the water volume.
Before you get started, make a prediction about the shape of the graph.
Check Reset and set the radius and height of the graduated cylinder to values you choose.
Let the cylinder fill with different amounts of water and record the data in the table.
Create a graph that shows the height of the water in the cylinder as a function of the water volume.
Choose a point on the graph and explain its meaning in the context of the situation.
Student Response
Teachers with a valid work email address can
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for free access to Student Response.
Print
Launch
Arrange students in groups of 3–4. Be sure students know how to measure using a graduated cylinder. If needed, display a graduated cylinder filled to a specific measurement for all to see and demonstrate to students how to read the measurement. Give each group access to a graduated cylinder and water.
Give groups 8–10 minutes to work on the task, follow with a whole-class discussion.
For classrooms with access to the digital materials or those with no access to graduated cylinders, an applet is included here. Physical measurement tools and an active lab experience are preferred.
Speaking: MLR8 Discussion Supports.
Display sentence frames to support small-group discussion. For example, “I think ____ , because _____ .” or “I (agree/disagree) because ______ .”
Design Principle(s): Support sense-making; Optimize output for (explanation)
Student Facing
Your teacher will give you a graduated cylinder, water, and some other supplies. Your group will use these supplies to investigate the height of water in the cylinder as a function of the water volume.
Before you get started, make a prediction about the shape of the graph.
Fill the cylinder with different amounts of water and record the data in the table.
{"table": [{"_attributes": {"border": "1"}, "tbody": [{"tr": [{"th": [{"_attributes": {"scope": "row"}, "_value": "volume (ml)"}], "td": [{}, {}, {}, {}, {}, {}]}, {"th": [{"_attributes": {"scope": "row"}, "_value": "height (cm)"}], "td": [{}, {}, {}, {}, {}, {}]}]}]}]}
Create a graph that shows the height of the water in the cylinder as a function of the water volume.
[IMG]
Choose a point on the graph and explain its meaning in the context of the situation.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Activity Synthesis
Select groups to share the graph for the third question and display it for all to see. Consider asking students the following questions:
“What do you notice about the shape of your graph?”
“What is the independent variable of your graph? Dependent variable?”
“How does this graph differ from what you predicted the shape would be?”
“For the last question, what point did you choose, and what does that point mean in the context of this activity?”
“What would the endpoint of the graph be?” (There is a maximum possible volume for the cylinder. Once it’s filled, any extra water will spill out and not raise the water height.)
[IMG]
Ask students to predict how the graph would change if their cylinder had double the diameter. After a few responses, display this graph for all to see:Explain that each line represents the graph of a cylinder with a different radius. One cylinder has a radius of 1 cm, another has a radius of 2 cm, and another has a radius of 3 cm. Have students consider which line must represent which cylinder. Ask, “how did the slope of each graph change as the radius increased?” (As the radius is larger, the slope is less steep. This is because for a cylinder with a larger base, the same volume of water will not fill as high up the side of the cylinder.)


correct_option_index: 2