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: 
Solve quadratic equations in one variable.

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


Answer: 
Yes. This task shows one quadratic equation with 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_pos_0

HS_1

im_task_001160_pos_0

HS_2
standard_description: Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.
problem_activity: Warm-up
The purpose of this warm-up is to ensure students understand that
they can infer the coordinates of a point based on knowing the coordinates of points on the same horizontal and vertical lines;
the length of a horizontal or vertical line segment can be determined based on the coordinates of its endpoints. Each of these is important background knowledge for this lesson.
Launch
Give students 2 minutes of quiet think time. Ask them to share their reasoning with a partner followed by a whole-class discussion.
Student Facing
[IMG]
Find each of the following and explain your reasoning:
The length of segment
\(BE\)
.
The coordinates of
\(E\)
.
Student Response
Teachers with a valid work email address can
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for free access to Student Response.
Anticipated Misconceptions
In order to find the length of segment
\(BE\)
, it is first necessary to find the lengths of
\(CD\)
,
\(AC\)
, and
\(AB\)
, and reason about side lengths in similar triangles. If students have trouble getting started, scaffold the work by suggesting they first find any
other
segment lengths in the diagram that they can find.
Activity Synthesis
There are three important things students should understand or recall as a result of working on the warm-up:
Points on the same vertical line have the same
\(x\)
-coordinate and points on the same horizontal line have the same
\(y\)
-coordinate.
In order to find the length of a vertical segment, you can subtract the
\(y\)
-coordinates of its endpoints.
The side lengths of triangle
\(ACD\)
are the same as the side lengths of the similar triangle
\(ABE\)
multiplied by the scale factor
\(AC \div AB\)
.
Display the image from the task. For each question, invite a student to share their reasoning. The displayed image should be used as a tool for gesturing. For example, if you start at the origin and want to navigate to point
\(D\)
, you would move 4 units to the right. You move the same distance to the right to navigate from the origin to
\(C\)
. Therefore,
\(C\)
must have the same
\(x\)
-coordinate as
\(D\)
.


aligns: yes