few_shots: Example:
Problem/activity:
Find the general solution for:
$${\mathrm{cos}\left({\pi\over2}+x\right)=\mathrm{tan}{\pi\over4}}$$


Mathematical concept/skill options: 
A. Prove and apply trigonometric identities.
B. Model periodic phenomena with trigonometric functions.
C. Interpret functions that arise in applications in terms of the context.
D. Build a function that models a relationship between two quantities.
E. Define, evaluate, and compare functions.
F. Use functions to model relationships between quantities.
G. Understand the concept of a function and use function notation.
H. Analyze functions using different representations.
I. Extend the domain of trigonometric functions using the unit circle.
J. Interpret expressions for functions in terms of the situation they model.
K. Build new functions from existing functions.
L. Construct and compare linear and exponential models and solve problems.


Your response, in json format:
{
  "explanation": "This problem applies trigonometric identities, including the angle addition formula for cosine.",
  "answer": "A"
}

Now, assign the following problem or activity to one or more relevant mathematical concepts/skills it teaches.
shot_info: fl_problem_001894

HS
options: Analyze and solve linear equations and pairs of simultaneous linear equations.

Work with radicals and integer exponents.

Perform arithmetic operations on polynomials.

Work with addition and subtraction equations.

Use the four operations with whole numbers to solve problems.

Apply and extend previous understandings of arithmetic to algebraic expressions.

Rewrite rational expressions.

Gain familiarity with factors and multiples.

Use properties of operations to generate equivalent expressions.

Create equations that describe numbers or relationships

Use polynomial identities to solve problems.

Solve real-life and mathematical problems using numerical and algebraic expressions and equations.

Add and subtract within 20.

Understand and apply properties of operations and the relationship between addition and subtraction.

Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.

Understand solving equations as a process of reasoning and explain the reasoning.

Understand the connections between proportional relationships, lines, and linear equations.

Reason about and solve one-variable equations and inequalities.

Solve problems involving the four operations, and identify and explain patterns in arithmetic.

Multiply and divide within 100.

Write and interpret numerical expressions.

Work with equal groups of objects to gain foundations for multiplication.

Represent and analyze quantitative relationships between dependent and independent variables.

Analyze patterns and relationships.

Represent and solve equations and inequalities graphically.

Understand properties of multiplication and the relationship between multiplication and division.

Solve systems of equations.

Represent and solve problems involving addition and subtraction.

Understand the relationship between zeros and factors of polynomials.

Generate and analyze patterns.

Solve equations and inequalities in one variable.

Write expressions in equivalent forms to solve problems.

Represent and solve problems involving multiplication and division.

Interpret the structure of expressions.
problem_activity: Activity
The purpose of this activity is to extend the definition of exponents to include negative exponents. The meaning of zero exponents is also revisited. When students use tables to explore expressions with negative exponents, they are noticing and making use of structure (MP7).
Launch
Arrange students in groups of 2. Tell students to work on their own and then share their reasoning with their partner after each has had a chance to complete the problem. Follow with a whole-class discussion.
Student Facing
Complete the table.
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}}, {"_attributes": {"scope": "col"}, "_value": "exponential form"}, {"_attributes": {"scope": "col"}, "_value": "number form"}, {"_attributes": {"scope": "col"}, "_value": "calculations"}]}]}], "tbody": [{"tr": [{"td": [{}, {"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "1145"}, "_value": "\\(2^5\\)"}]}, {}, {}]}, {"td": [{}, {}, {"_value": "16"}, {}]}, {"td": [{"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "25037"}, "_value": "\\(\\frac{2^4}{2}=2^{4-1}=2^3\\)"}]}, {"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "1932"}, "_value": "\\(2^3\\)"}]}, {}, {}]}, {"td": [{"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "25045"}, "_value": "\\(\\frac{2^3}{2}=2^{3-1}=2^2\\)"}]}, {"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "1131"}, "_value": "\\(2^2\\)"}]}, {"_value": "4"}, {}]}, {"td": [{}, {}, {"_value": "2"}, {"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "25058"}, "_value": "\\(4 \\boldcdot \\frac12=2\\)"}]}]}, {"td": [{}, {}, {"_value": "1"}, {"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "failed-conversion"}, "_value": "\\(2 \u00a0\\boldcdot \\frac12=1\\)"}]}]}, {"td": [{}, {"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "8478"}, "_value": "\\(2^{\\text-1}\\)"}]}, {"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "497"}, "_value": "\\(\\frac{1}{2}\\)"}]}, {}]}, {"td": [{}, {}, {"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "451"}, "_value": "\\(\\frac{1}{4}\\)"}]}, {"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "failed-conversion"}, "_value": "\\(\\frac12 \u00a0\\boldcdot \\frac12 = \\frac14\\)"}]}]}, {"td": [{}, {"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "25113"}, "_value": "\\(2^{\\text-3}\\)"}]}, {}, {}]}, {"td": [{}, {"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "25133"}, "_value": "\\(2^{\\text-4}\\)"}]}, {}, {}]}, {"td": [{}, {}, {"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "2239"}, "_value": "\\(\\frac{1}{32}\\)"}]}, {}]}]}]}]}
Student Response
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Activity Synthesis
Display the completed table, and resolve any questions students have. Questions for discussion:
“How would you make sense of
\(2^{\text-6}\)
?” (I know
\(2^{\text-5}=\frac{1}{32}\)
, and I know that
\(2^{\text-6}\)
would be the result of multiplying by another
\(\frac12\)
. So,
\(2^{\text-6} = \frac{1}{64}\)
.
“How could we figure out the value of
\(7^{\text-2}\)
?” (Possible response: We could rewrite it as
\(\frac{7^0}{7^2}\)
, and then
\(\frac{1}{49}\)
.


correct_option_index: 1