few_shots: 
Example 1:
Problem/activity:
Find the general solution for:
$${\mathrm{cos}\left({\pi\over2}+x\right)=\mathrm{tan}{\pi\over4}}$$


Concept/skill: 
Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.

Answer: 
No. This problem does not directly teach how side ratios in right triangles lead to definitions of trigonometric ratios. Instead, it involves solving a trigonometric function.

Example 2:
Problem/activity:
Task
Solve the quadratic equation
$$ x^2 = (2x - 9)^2 $$
using as many different methods as possible.


Concept/skill: 
Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).

Answer: 
No. This task does not involve an equation in two variables. There is only one variable, x.

Example 3:
Problem/activity:
Task
Suppose $A$ is an angle with vertex $P$, as pictured below:
[IMG]
Draw a circle with center $P$ and with  radius $r \gt 0$.  Explain why the circle
meets each ray of angle $A$ in a single point.  Label these points $Q$ and $R$ respectively.
Draw circles with centers $Q$ and $R$ respectively and radius $r$.  These circles meet at $P$ and a second point to be labelled $B$.
Show that ray $\overrightarrow{PB}$  bisects angle $P$.


Concept/skill: 
Construct a tangent line from a point outside a given circle to the circle.

Answer: 
No. This task does not involve drawing lines that are tangent to circles. The only line that may be drawn is the ray between points P and B.

Now, does the problem or activity shown below enable students to learn the full intent of the following concept or skill?

shot_info: fl_problem_001894_neg_0

HS_0

im_task_000870_neg_0

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im_task_001160_neg_0

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standard_description: Extend polynomial identities to the complex numbers. For example, rewrite x^(2) + 4 as (x + 2i)(x – 2i).
problem_activity: Activity
This info gap activity gives students an opportunity to determine and request the information needed to figure out which complex numbers were multiplied to produce another complex number. The challenge students will face here is that there will be two unknowns in each problem, a real part and an imaginary part, and two equations derived from the information students have about them. Once one of the unknowns is expressed in terms of the other and substituted into the other equation, students will see that they have to solve a quadratic equation. In following lessons, students will return to solving quadratics (with complex solutions), so these problems will be a bridge between complex numbers and quadratics.
The info gap structure requires students to make sense of problems by determining what information is necessary, and then to ask for information they need to solve it. This may take several rounds of discussion if their first requests do not yield the information they need (MP1). It also allows them to refine the language they use and ask increasingly more precise questions until they get the information they need (MP6).
Here is the text of the cards for reference and planning:
[IMG]
Launch
Tell students they will continue to investigate what happens when we combine complex numbers. In this activity, they will focus on multiplying. Explain the info gap structure, and consider demonstrating the protocol if students are unfamiliar with it.
Arrange students in groups of 2. In each group, distribute a problem card to one student and a data card to the other student. After reviewing their work on the first problem, give them the cards for a second problem and instruct them to switch roles.
Conversing:
This activity uses
MLR4 Information Gap
to give students a purpose for discussing information necessary to solve problems involving the multiplication of complex numbers. Consider providing questions or question starters for students who need a starting point such as: “Can you tell me…?” and “Why do you need to know…?”
Design Principle(s): Cultivate Conversation
Engagement: Develop Effort and Persistence.
Display or provide students with a physical copy of the written directions. Check for understanding by inviting students to rephrase directions in their own words. Keep the display of directions visible throughout the activity.
Supports accessibility for: Memory; Organization
Student Facing
Your teacher will give you either a problem card or a data card. Do not show or read your card to your partner.
If your teacher gives you the data card:
Silently read the information on your card.
Ask your partner “What specific information do you need?” and wait for your partner to ask for information. Only give information that is on your card. (Do not figure out anything for your partner!)
Before telling your partner the information, ask “Why do you need to know (that piece of information)?”
Read the problem card, and solve the problem independently.
Share the data card, and discuss your reasoning.
If your teacher gives you the problem card:
Silently read your card and think about what information you need to answer the question.
Ask your partner for the specific information that you need.
Explain to your partner how you are using the information to solve the problem.
When you have enough information, share the problem card with your partner, and solve the problem independently.
Read the data card, and discuss your reasoning.
Pause here so your teacher can review your work. Ask your teacher for a new set of cards and repeat the activity, trading roles with your partner.
Student Response
Teachers with a valid work email address can
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for free access to Student Response.
Activity Synthesis
After students have completed their work, share the correct answers and ask students to discuss the process of solving the problems. Here are some questions for discussion:
“What was a strategy you tried when you were working on the problems?”
“Did it seem at first like there wasn’t enough information to find an answer? If so, how did you figure out what you needed?”
Highlight for students that the real and imaginary parts of complex numbers always combine in the same way when we multiply them, similarly to the way the parts of linear functions combine when we multiply them. This means that we can separate those parts out again if we know enough about the result and what we started with.


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