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

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.

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.

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.

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.

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.

Number & Operations - Fractions: students learn to understand fractions as numbers and may work with them using addition, subtraction, multiplication, and division.
problem_activity: Activity
The purpose of this activity is for students to extend what they learned about square roots and
\(\frac12\)
exponents to cube roots and
\(\frac13\)
exponents.
Launch
Tell students that they will use the same reasoning as in the previous activity to make sense of numbers to the
\(\frac12\)
power or
\(\frac13\)
power. Give a few minutes of quiet work time followed by a brief, whole-class discussion.
Action and Expression: Internalize Executive Functions.
To support development of organizational skills, check in with students within the first 2–3 minutes of work time. Check that students are finding the exact values and remind them of how they could use the exponent rules to check their answers if needed. Invite students to show their thinking by rewriting the fractional exponents.
Supports accessibility for: Memory; Organization
Student Facing
Use the exponent rules and your understanding of roots to find the exact value of:
\(25^{\frac12}\)
\(15^{\frac12}\)
\(8^{\frac13}\)
\(2^{\frac13}\)
Student Response
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Activity Synthesis
The key takeaway is that extending exponent rules to these fractions means that
\(b^{\frac12}=\sqrt{b}\)
and
\(b^{\frac13}=\sqrt[3]{b}\)
. Select students to share their responses for
\(15^{\frac12}\)
and
\(2^{\frac13}\)
.
Conclude the discussion by reminding students that these expressions which involve the
\(\sqrt{}\)
symbol are often called
radical
expressions because the technical name for the symbol is the
radix
(which is also where the word root comes from). Expressions like
\(\sqrt{15}\)
and
\(\sqrt[3]{2}\)
are examples of radical expressions.
Speaking: MLR8 Discussion Supports.
As students share their responses for
\(15^{\frac{1}{2}}\)
and
\(2^{\frac{1}{3}}\)
, press for details by asking how they know that
\(15^{\frac{1}{2}}\)
is equal to
\(\sqrt{15}\)
and
\(2^{\frac{1}{3}}\)
is equal to
\(\sqrt[3]{2}\)
. Also ask students which properties of exponents they used to show that the exponential and radical expressions are equivalent. Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class. This will help students justify why
\(b^{\frac{1}{2}} = \sqrt{b}\)
and
\(b^{\frac{1}{3}} = \sqrt[3]{b}\)
.
Design Principle(s): Support sense-making; Optimize output (for justification)


correct_option_index: 2