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: Reason quantitatively and use units to solve problems.

Know that there are numbers that are not rational, and approximate them by rational numbers.

Perform operations on matrices and use matrices in applications.

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

Represent complex numbers and their operations on the complex plane.

Use properties of rational and irrational numbers.

Perform arithmetic operations with complex numbers.

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

Perform operations on vectors.

Represent and model with vector quantities.

Use complex numbers in polynomial identities and equations.

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.

Compute fluently with multi-digit numbers and find common factors and multiples.
problem_activity: Activity
The purpose of this activity is to introduce the idea that sums of complex numbers can be rewritten as a single complex number in the form
\(a+bi\)
, where
\(a\)
and
\(b\)
are real numbers. In previous lessons, students have used the complex plane to visualize complex numbers, and now they continue that work using the complex plane to visualize why the sum of two complex numbers is another complex number.
Launch
Display the image from the task statement for all to see.
Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice with their partner, followed by a brief whole-class discussion. Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the image.
Arrange students in groups of 2 and give them a few minutes of work time for the activity, encouraging students to discuss each problem and, if there is disagreement, to work to reach agreement. Follow up with a whole-class discussion.
Conversing, Writing: MLR5 Co-Craft Questions.
Use this routine to spark students’ curiosity about the sums of complex numbers. Display only the image and ask students to write down possible mathematical questions that could be asked about the situation. Invite students to compare their questions before revealing the activity’s questions. Listen for and amplify any questions involving how parts of the expression are represented on the diagram. This will help students create the language of mathematical questions before feeling pressure to produce solutions.
Design Principle(s): Maximize meta-awareness; Support sense-making
Student Facing
This diagram represents
\((2 + 3i) + (\text- 8 - 8i)\)
.
[IMG]
How do you see
\(2 + 3i\)
represented?
How do you see
\(\text- 8 - 8i\)
represented?
What complex number does
\(A\)
represent?
Add “like terms” in the expression
\((2 + 3i) + (\text- 8 - 8i)\)
. What do you get?
Write these sums and differences in the form
\(a+bi\)
, where
\(a\)
and
\(b\)
are real numbers.
\((\text- 3 + 2i) + (4 - 5i)\)
(Check your work by drawing a diagram.)
\((\text- 37 - 45i) + (11 + 81i)\)
\((\text- 3 + 2i) - (4 - 5i)\)
\((\text- 37 - 45i) - (11 + 81i)\)
Student Response
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Activity Synthesis
Select students to share their arithmetic results and connect them to arrows in the complex plane. It is important to discuss that doing operations with complex numbers results in another complex number. In order to see this, it is helpful to write the result in the form
\(a+bi\)
, where
\(a\)
and
\(b\)
are real numbers. When a complex number is written this way,
\(a\)
is called the
real part
of the number, and
\(bi\)
is called the
imaginary part
.
If time allows, ask students, “Is
\((2 + 3i) + (\text- 8 - 8i)\)
the same as
\((\text- 8 - 8i) + (2 + 3i)\)
?” (Yes. The diagrams will look different, but, for example, going right 2 then left 8 is the same as going left 8 and right 2.)
Representation: Internalize Comprehension.
Use color-coding and annotations to highlight connections between representations in a problem. For example, highlight connects between representations by showing the expressions
\(2 + 3i\)
and using one color to connect it to the arrows in the complex plane. Use a second color to highlight the connection between the expression
\(\text-8 - 8i\)
and the arrows in the complex plane.
Supports accessibility for: Visual-spatial processing


correct_option_index: 6