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


Standard options: 
A. Prove the Pythagorean identity sin^(2)(θ) + cos^(2)(θ) = 1 and use it to find sin(θ), cos(θ),or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.
B. Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.


Thought: An initial step when solving for x in this problem requires the use of the addition formula for cosine on the left side of the equation. Solving this problem does not require use of the Pythagorean identity.
Answer: B

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


Standard options: 
A. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
B. Solve quadratic equations in one variable.


Thought: This task shows one quadratic equation with one variable, x.
Answer: B

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 $P$.


Standard options: 
A. Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.
B. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.


Thought: This task asks students to draw a ray that bisects an angle. The act of drawing circles and rays may involve compass and straightedge or dynamic geometric software.
Answer: A

Now, assign the following problem or activity to one or more relevant standards it aligns with.
shot_info: fl_problem_001894

HS_0

im_task_000870

HS_1

im_task_001160

HS_2
options: Use a pair of perpendicular number lines, called axes, to define a coordinate system, with the intersection of the lines (the origin) arranged to coincide with the 0 on each line and a given point in the plane located by using an ordered pair of numbers, called its coordinates. Understand that the first number indicates how far to travel from the origin in the direction of one axis, and the second number indicates how far to travel in the direction of the second axis, with the convention that the names of the two axes and the coordinates correspond (e.g., x-axis and x-coordinate, y-axis and y-coordinate).

Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation.
problem_activity: Narrative
The purpose of this activity is for students to describe coordinates for points. After playing a round of What’s the Point, students have an opportunity to write a description of the location of a point in the coordinate plane. The structure of this part of the activity mirrors what students did in the previous lesson when they were describing a rectangle. Some students may use the coordinates on the grid while others may use words to describe the location of the point.
Students make choices about how to revise their thinking based on what makes the description stronger and clearer to them. This activity not only supports students developing language to describe the location of a point but also develops a deeper understanding of the coordinate grid (MP6). Invite students to use language from the display from an earlier lesson if they find it helpful.
This activity uses
MLR1 Stronger and Clearer Each Time
. Advances: Reading, Writing.
Required Materials
Materials to Copy
What's the Point
Required Preparation
Create a set of cards from the blackline master for each group of 2.
Launch
Groups of 2.
Activity
10 minutes: partner work time
Monitor for students who:
revise their thinking
refer to the numbers on the coordinate grid to communicate the location of the point
MLR1 Stronger and Clearer Each Time
“Share your description of the location of the point on the grid with your partner. Take turns being the speaker and the listener. If you are the speaker, share your ideas and writing so far. If you are the listener, ask questions and give feedback to help your partner improve their work.”
3–5 minutes: structured partner discussion
Repeat with 1–2 different partners.
If needed, display question starters and prompts for feedback, such as:
“Can you use the numbers on the coordinate grid in your explanation?”
“Revise your initial draft based on the feedback you got from your partner.”
2–3 minutes: independent work time
Student Facing
Play 2 rounds of What’s the Point so each partner gets a chance to draw.
Sit back to back with your partner.
Partner A: Choose a card. Then, describe the location of the point to your partner.
Partner B: Draw the point on the blank coordinate grid.
Compare the card with your partner’s diagram.
Discuss: What’s the same? What’s different?
[IMG]
[IMG]
Use words to explain the location of the point on the grid.
[IMG]
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Advancing Student Thinking
If students do not reference the axes or the directionality of the gridlines, show them point
\((3, 4)\)
and point
\((4, 3)\)
on a grid and ask, “What is the same and different about these two points?”
Activity Synthesis
Select previously identified students to share how they revised their explanation for the location of the point in the last problem.
Display the image of the coordinate grid with a point from the student workbook.
“The point can be described using the
coordinates
\((3, 4)\)
. Where do you think these coordinates come from?” (They describe the location of the point. The point is on gridline 3 and gridline 4. The point is where these gridlines intersect.)


correct_option_index: 0