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 a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).

Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ³ 1.
problem_activity: Activity
The purpose of this activity is to strengthen connections between zeros of a function (input(s) when the output is 0), the output when the input is 0, and the coordinates of the intercept(s) of a graph of the function.
Launch
To ensure that students understand the task before beginning, display one of the functions for all to see, and test an input value that is
not
a zero of the function. For example, evaluate
\(a(\text-5)=\text-5 - 5\)
. Is the output 0? No, the output is -10. So -5 is not an input that would give an output of 0. Draw students’ attention to the list of possible inputs, and tell them that the task is to find the input or inputs for each function that gives an output of 0.
Student Facing
Here are the definitions of some functions, followed by some possible inputs for the functions.
\(a(x)=x - 5\)
\(b(x)=x + 5\)
\(c(x)=x-3\)
\(d(x)=x+1\)
\(f(x)=3x - 6\)
\(g(x)=3x + 6\)
\(h(x)=(x+5)(x+3)\)
\(m(x)=(x+1)(x-3)\)
\(n(x)=(3x-6)(x-5)\)
Possible inputs: -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, and 5.
For each function, decide which input or inputs would give an output of 0.
Here are graphs of
\(b\)
,
\(f\)
, and
\(m\)
. Label each intercept with its coordinates, and be prepared to explain how you know.
[IMG]
Student Response
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Activity Synthesis
Display the correct responses to the first question and ask students to check their work. Invite a student to demonstrate how they know that
\(g(\text-2)=0\)
and
\(m(3)=0\)
(or choose a different function that any students struggled with). The important takeaway is not to look for shortcuts or patterns, but rather understand by evaluating the function why a given number is a zero of the function.
Invite students to share their responses and reasoning to the second question. If not mentioned, emphasize that:
The
\(y\)
-intercept of a graph corresponds to the output when the input is 0, so to find it, evaluate the function when
\(x\)
is 0. For example,
\(b(0)=5\)
, so the coordinates are
\((0,5)\)
of an intercept of the graph.
The
\(x\)
-intercepts correspond to the values found in the first question, the inputs that give an output of 0.
Any time you are dealing with an intercept, one of the coordinates must be 0. The coordinates of an
\(x\)
-intercept are
\((\text{something},0)\)
. The coordinates of a
\(y\)
-intercept are
\((0,\text{something})\)
.


correct_option_index: 1