few_shots: 
Example 1:
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.

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


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:
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).

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


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

Example 3:
Concept/skill: 
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.

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 $P$.


Answer: 
Yes. 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.

Now, determine whether the following problem or activity can enable students to understand the concepts or skills in the provided description.

shot_info: fl_problem_001894_neg_0

HS_0

im_task_000870_neg_0

HS_1

im_task_001160_pos_0

HS_2
standard_description: Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,” “not”).
problem_activity: Warm-up
This warm-up highlights the three forms of quadratic expressions students have seen so far. It reinforces how each form gives different insights into the graph of a quadratic function.
Launch
Arrange students in groups of 2. Give students a minute of quiet think time, and then ask them to share their thinking with a partner.
Student Facing
Expressions in different forms can be used to define the same function. Here are three ways to define a function
\(f\)
.
\(f(x)=x^2-4x+3\)
(standard form)
\(f(x)=(x-3)(x-1)\)
(factored form)
\(f(x)=(x-2)^2-1\)
(vertex form)
Which form would you use if you want to find the following features of the graph of
\(f\)
? Be prepared to explain your reasoning.
the
\(x\)
-intercepts
the vertex
the
\(y\)
-intercept
Student Response
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Activity Synthesis
Invite students to share their responses and explanations. Encourage students to name specific features of the graph of
\(f\)
. If time permits, consider asking students to use all of the info they can gather to sketch a graph.
If not already mentioned by students, point out that while the standard form allows us to find the
\(y\)
-intercept more readily, the
\(y\)
-intercept can always be found by evaluating the function at
\(x=0\)
.


aligns: no