standard_description: Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.
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.


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