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: 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).

Graph the solutions to a linear inequality in two variables as a half plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.

Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.
problem_activity: Activity
The purpose of this activity is to give students practice writing and graphing equations from situations. In the next activity and in the supported Algebra 1 lesson, students will write and graph inequalities  Students then assess the meaning of points off of the line in the situation provided and interpret whether those points might be valid in the situation.
Student Facing
Write an equation that helps to answer the question about the situation. Then draw a graph that represents the equation.
Jada goes to an orchard to pick plums and apricots to make jam. She picks 20 pounds of fruit altogether. If she picks
\(a\)
pounds of apricots, how many pounds of plums does she pick?
[IMG]
Consider the point
\((5,16)\)
. Is it possible for the weight of the fruit to be represented by that point in this situation? Explain your reasoning.
In a video game, a character can run at a top speed of 30 miles per hour, but each additional pound that the character carries lowers the maximum running speed by 1 mile per hour. What is the maximum running speed of the character when they are carrying
\(w\)
pounds?
[IMG]
Consider the point
\((10,15)\)
. Is it possible for a character in this game to be represented by that point in this situation? Explain your reasoning.
Student Response
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Activity Synthesis
The purpose of the discussion is to examine points on and off the lines and determine what they might mean in the situations.
Ask students:
"What are the coordinates for a point on the line that represents the scooter situation? What do the coordinates mean in that situation?" (The point
\((1,22)\)
is on the line. This means that renting the scooter for 2 hours should cost
$
22.)
"What are the coordinates for a point above the line that represents the scooter situation? What do the coordinates mean in that situation? Is it realistic?" (The point
\((2,30)\)
is above the line. This means paying
$
30 for 2 hours with a scooter. It only makes sense if the person is overpaying for some reason like if they broke part of the scooter and have to pay extra.)
"The point
\((5,30)\)
is above the line that represents the maximum run speed for the videogame character. Describe a situation where this might be possible." (If the character has an item that ignores the carry weight or something that boosts their run speed, then maybe this is possible.)


correct_option_index: 0