options: Apply and extend previous understandings of numbers to the system of rational numbers.

Represent complex numbers and their operations on the complex plane.

Perform operations on vectors.

Apply and extend previous understandings of multiplication and division to divide fractions by fractions.

Perform arithmetic operations with complex numbers.

Compute fluently with multi-digit numbers and find common factors and multiples.

Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.

Extend the properties of exponents to rational exponents.

Use properties of rational and irrational numbers.

Represent and model with vector quantities.

Reason quantitatively and use units to solve problems.

Use complex numbers in polynomial identities and equations.

Perform operations on matrices and use matrices in applications.

Know that there are numbers that are not rational, and approximate them by rational numbers.
problem_activity: Warm-up
This Math Talk encourages students to think about exponent rules and to rely on properties of exponents to mentally solve problems. The understandings elicited here will be helpful later in the lesson when students use graphs to approximate the value of
\(2^x\)
for various negative rational exponents.
The final question is the first time students see an exponent that is a negative fraction. To use their prior knowledge to evaluate this expression, students need to look for and make use of structure (MP7).
Launch
Display one problem at a time. Give students quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all problems displayed throughout the talk. Follow with a whole-class discussion.
Representation: Internalize Comprehension.
To support working memory, provide students with sticky notes or mini whiteboards.
Supports accessibility for: Memory; Organization
Student Facing
Evaluate mentally.
\(9^2\)
\(9^{\text-2}\)
\(9^{\frac12}\)
\(9^{\text-\frac12}\)
Student Response
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Activity Synthesis
Ask students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking:
“Who can restate
\(\underline{\hspace{.5in}}\)
’s reasoning in a different way?”
“Did anyone have the same strategy but would explain it differently?”
“Did anyone solve the problem in a different way?”
“Does anyone want to add on to
\(\underline{\hspace{.5in}}\)
’s strategy?”
“Do you agree or disagree? Why?”
If this strategy is not mentioned, consider asking students how they would use (9
1/2
)
-1
or (9
-1
)
1/2
to evaluate the last expression. If the results of the pre-assessment showed that students need extra practice with exponent rules, highlight any student strategies that leverage these rules and carefully record student thinking for all to see.
Speaking: MLR8 Discussion Supports.
Display sentence frames to support students when they explain their strategy. For example, "First, I _____ because…” or "I noticed _____ so I….” Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class.
Design Principle(s): Optimize output (for explanation)


correct_option_index: 7