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

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.

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.

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.

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.

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

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.

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

Ratios & Proportional Relationships: students learn to recognize, describe, represent, and reason with ratios, rates, proportional relationships, and percent.
problem_activity: Activity
In this activity, students match equivalent expressions involving exponents and radicals. This is the first time they see negative rational exponents, so expect a lot of discussion around those expressions. Students will have more chances to interpret negative exponents, so it is not crucial for them to show mastery at this time.
Monitor for groups who reason about the expressions with negative fractional exponents in different ways. For example, some students may see
\(7^{\text- \frac12}\)
as
\(\left(7^{\text-1}\right)^{\frac12}\)
, while others will think of the expression as
\(\left(7^{\frac12}\right)^{\text-1}\)
. Ask these groups to be ready to share their reasoning during the discussion.
Launch
Arrange students in groups of 2. Give students a few minutes of quiet think time before asking them to share their responses with their partners. Tell students that if there is disagreement, they should work to reach agreement. Follow with a whole-class discussion.
Conversing: MLR8 Discussion Supports.
Arrange students in groups of 2. Invite students to take turns finding a match and explaining their reasoning to their partner. Display the following sentence frames for all to see: “ ____ and ____ are equal because…” and “I noticed _____, so I matched….” Encourage students to challenge each other when they disagree. This will help students clarify their reasoning about equivalent expressions that involve exponents.
Design Principle(s): Support sense-making; Maximize meta-awareness
Student Facing
Match each exponential expression to an equivalent expression.
\(7^3\)
\(7^2\)
\(7^1\)
\(7^0\)
\(7^{\text-1}\)
\(7^{\text-2}\)
\(7^{\text-3}\)
\(7^{\frac{1}{2}}\)
\(7^{\text-\frac{1}{2}}\)
\(7^{\frac13}\)
\(7^{\text-\frac{1}{3}}\)
\(\frac{1}{49}\)
\(\frac{1}{343}\)
\(\sqrt7\)
\(\frac{1}{\sqrt[3]{7}}\)
\(\sqrt[3]{7}\)
49
\(\frac{1}{\sqrt7}\)
343
7
\(\frac{1}{7}\)
1
Student Response
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Student Facing
Are you ready for more?
How do we know without counting that the number of circles equals the number of squares? Because we can match every circle to exactly one square, and each square has a match:
[IMG]
[IMG]
We say that we have shown that there is a one-to-one correspondence of the set of circles and the set of squares. We can do this with infinite sets, too! For example, there are the same “number” of positive integers as there are even positive integers:
[IMG]
Every positive integer is matched to exactly one even positive integer, and every even integer has a match! We have shown that there is a one-to-one correspondence between the set of positive integers and the set of even positive integers. Whenever we can make a one-to-one matching like this of the positive integers to another set, we say the other set is
countable.
Show that the set of square roots of positive integers is countable.
Show that the set of positive integer roots of 2 is countable.
Show that the set of positive integer roots of positive integers is countable. (Hint: there is a famous proof that the positive rational numbers are countable. Find and study this proof.)
Student Response
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Anticipated Misconceptions
If students aren’t sure how to match the negative rational exponents, ask them if they could make an educated guess based on the other patterns they notice.
Activity Synthesis
Begin the discussion by selecting 2–3 groups to share how they matched the expressions with positive exponents. Next, invite previously identified groups to share how they matched the expressions with negative exponents. While students share their thinking, display any exponent rules referenced for all to see, such as
\(\left(b^m\right)^n = b^{m \boldcdot n}\)
or
\(b^{\text-n} = \frac{1}{b^n}\)
. It is important for students to understand that fractional exponents follow all the same exponent rules as integer exponents.


correct_option_index: 0