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.

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.

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.

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.

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

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

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.

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.

Ratios & Proportional Relationships: students learn to recognize, describe, represent, and reason with ratios, rates, proportional relationships, and percent.
problem_activity: Activity
The purpose of this activity is to use the new skills of multiplying and dividing rational numbers to represent and solve problems in a new context (MP4). In this activity students are using using multiplication and division of negatives and working with a proportional relationship with a negative constant of proportionality. Students should use what they know about proportional relationships to help them (for example; points lie on a straight line, line passes through zero, there is a point
\((1, k)\)
that lies on the line).
Digital
Launch
Remind students that we can model positions below the surface with negative values, so drilling 30 feet down is represented with -30 feet.
Ask students what they remember about proportional relationships. Examples:
Often represented with an equation in the form
\(y=kx\)
.
The constant of proportionality, often called
\(k\)
, is the change in
\(y\)
for a change by 1 in
\(x\)
.
A graph representing a proportional relationship is a line through (0,0) and
\((1,k)\)
Arrange students in groups of 3 during the discussion.
Representation: Internalize Comprehension.
Provide appropriate reading accommodations and supports to ensure students have access to written directions, word problems and other text-based content.
Supports accessibility for: Language; Conceptual processing
Student Facing
A water well drilling rig has dug to a height of -60 feet after one full day of continuous use.
Assuming the rig drilled at a constant rate, what was the height of the drill after 15 hours?
If the rig has been running constantly and is currently at a height of -147.5 feet, for how long has the rig been running?
Use the coordinate grid to show the drill’s progress.
At this rate, how many hours will it take until the drill reaches -250 feet?
[IMG]
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Print
Launch
Remind students that we can model positions below the surface with negative values, so drilling 30 feet down is represented with -30 feet.
Ask students what they remember about proportional relationships. Examples:
Often represented with an equation in the form
\(y=kx\)
.
The constant of proportionality, often called
\(k\)
, is the change in
\(y\)
for a change by 1 in
\(x\)
.
A graph representing a proportional relationship is a line through (0,0) and
\((1,k)\)
Arrange students in groups of 3 during the discussion.
Representation: Internalize Comprehension.
Provide appropriate reading accommodations and supports to ensure students have access to written directions, word problems and other text-based content.
Supports accessibility for: Language; Conceptual processing
Student Facing
A water well drilling rig has dug to a height of -60 feet after one full day of continuous use.
Assuming the rig drilled at a constant rate, what was the height of the drill after 15 hours?
If the rig has been running constantly and is currently at a height of -147.5 feet, for how long has the rig been running?
Use the coordinate grid to show the drill’s progress.
[IMG]
At this rate, how many hours will it take until the drill reaches -250 feet?
[IMG]
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Anticipated Misconceptions
Some students may struggle to get the length of time out of the phrase "one full day of continuous use," wondering whether the drill was running at night. Let them know the drill has been going for 24 hours.
Activity Synthesis
Have students to share their solutions with each other in groups of three and work to come to agreement.
To wrap up, emphasize that because we can now complete any calculation with any rational number, we can extend constants of proportionality
\(k\)
to include negative values. Because the operations are now well-defined, we can answer questions using such a constant.
Speaking: MLR8 Discussion Supports.
Use this routine to support whole-class discussion. Invite select groups to share their solution and reasoning with the class. After each group presents, ask students to restate and/or revoice what they heard using mathematical language. Consider providing students time to restate what they heard to a partner, before selecting one or two students to share with the class. Ask the original group if this accurately reflects their thinking. This will provide more students with an opportunity to describe what they have learned about proportional relationships with a negative constant of proportionality.
Design Principle(s): Support sense-making


correct_option_index: 8