standard_description: Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 5^(1/3) to be the cube root of 5 because we want (5^(1/3))^(3) = 5(^(1/3))^(3) to hold, so (5^(1/3))^(3) must equal 5.
problem_activity: Activity
The purpose of this task is for students to develop their understanding of the
\(|x|\)
notation in familiar contexts. They should build their understanding that
\(|x|\)
represents the distance from zero to
\(x\)
and that
\(|\text-x|\)
and
\(|x|\)
are equal.
Launch
Allow students 10 minutes quiet work time followed by whole-class discussion.
Student Facing
A part of the city of New Orleans is 6 feet below sea level. We can use “-6 feet” to describe its elevation, and “
\(|\text-6|\)
feet” to describe its vertical distance from sea level. In the context of elevation, what would each of the following numbers describe?
25 feet
\(|25|\)
feet
-8 feet
\(|\text-8|\)
feet
The elevation of a city is different from sea level by 10 feet. Name the two elevations that the city could have.
We write “
\(\text-5^\circ \text{C}\)
” to describe a temperature that is 5 degrees Celsius below freezing point and “
\(5^\circ \text{C}\)
” for a temperature that is 5 degrees above freezing. In this context, what do each of the following numbers describe?
\(1^\circ \text{C}\)
\(\text-4^\circ \text{C}\)
\(|12|^\circ \text{C}\)
\(|\text-7|^\circ \text{C}\)
Which temperature is colder:
\(\text-6^\circ \text{C}\)
or
\(3^\circ \text{C}\)
?
Which temperature is closer to freezing temperature:
\(\text-6^\circ \text{C}\)
or
\(3^\circ \text{C}\)
?
Which temperature has a smaller absolute value? Explain how you know.
Student Response
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Student Facing
Are you ready for more?
At a certain time, the difference between the temperature in New York City and in Boston was 7 degrees Celsius. The difference between the temperature in Boston and in Chicago was also 7 degrees Celsius. Was the temperature in New York City the same as the temperature in Chicago? Explain your answer.
Student Response
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Activity Synthesis
To summarize students’ work, consider displaying the following diagram and these four expressions. Give students a minute to study the diagram and match each letter on the diagram to an appropriate expression.
-4 feet
\(|{15}|\)
feet
\(|{\text -4}|\)
feet
15 feet
[IMG]
Representation: Develop Language and Symbols.
Create a display of important terms and vocabulary. Invite students to suggest language or diagrams to include that will support their understanding of: absolute value of a number, elevation, sea level, less than, greater than.
Supports accessibility for: Conceptual processing; Language; Memory
Representing, Speaking: MLR7 Compare and Connect
. To help students connect the numerical values with the diagram in the activity synthesis, provide students phrases that also connect to the situation (e.g., “4 feet below sea level,” “15 feet from sea level,” “4 feet from sea level,” and “15 feet above sea level”). Emphasize the language needed to say what the numerical values mean and how those values connect to the diagram and the phrases.
Design Principle(s): Maximize meta-awareness


aligns: no