few_shots: 
Example 1:
Concept/skill: 
Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

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


Answer: 
Yes. 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.

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_pos_0

HS_0

im_task_000870_neg_0

HS_1

im_task_001160_pos_0

HS_2
standard_description: Analyze and compare two- and three-dimensional shapes, in different sizes and orientations, using informal language to describe their similarities, differences, parts (e.g., number of sides and vertices/“corners”) and other attributes (e.g., having sides of equal length).
problem_activity: Narrative
The purpose of this activity is for students to begin to distinguish rectangles from other shapes. By working with variants and non-examples of rectangles, students begin to develop their understanding of what makes a shape a rectangle. When they discuss which group a shape should be placed in, students informally describe the attributes of the shape.
MLR8 Discussion Supports.
Students should take turns finding a match and explaining their reasoning to their partner. Display the following sentence frames for all to see: “I noticed _____, so I matched . . . .” Encourage students to challenge each other when they disagree.
Advances Conversing, Listening
Required Materials
Materials to Copy
Rectangle Sort Cards
Required Preparation
Cut out rectangle cards from the blackline master. Each group of 4 needs 1 set of cards.
Launch
Groups of 4
Give each group a set of cards.
Draw or display a rectangle:
[IMG]
“Tell your partner two things you know about this shape.” (There are 4 sides. It looks like a door. It is a rectangle. It is long and flat.)
30 seconds: quiet think time
30 seconds: partner discussion
Share and record responses.
If needed, ask “What is the name of this shape?”
30 seconds: quiet think time
“This shape is a
rectangle
.”
Activity
“Work with your group to sort the shapes into two groups. Put the shapes that are rectangles on the left side of your page. Put the shapes that are not rectangles on the right side of your page. When you place a shape, tell your group why you think the shape belongs in that group.”
4 minutes: small-group work time
“Write a number to show how many shapes are in each group.”
1 minute: independent work time
“Which group has fewer shapes? How do you know?”
30 seconds: quiet think time
30 seconds: partner discussion
Student Facing
Let’s put the shapes into 2 groups.
Rectangle
[IMG]
[IMG]
[IMG]
Not a Rectangle
[IMG]
[IMG]
Student Response
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Advancing Student Thinking
If students put shapes other than rectangles into the group of rectangles, consider asking:
“How did you choose which group to put this shape in?”
“What is alike, or the same, about all of the shapes in this group?”
Activity Synthesis
Invite 1–2 groups to share how they sorted the shapes.
Draw or display a square:
[IMG]
“Which group did you put this shape in?” (We put it with the rectangles because it has 4 sides. We didn’t put it with the rectangles because it is a square.)
“This shape is a
square
, which is a special rectangle because all four sides are the same length.”


aligns: yes