few_shots: 
Example 1:
Problem/activity:
Find the general solution for:
$${\mathrm{cos}\left({\pi\over2}+x\right)=\mathrm{tan}{\pi\over4}}$$


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.

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


Concept/skill: 
Solve quadratic equations in one variable.

Answer: 
Yes. This task shows one quadratic equation with one variable, x.

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 $P$.


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.

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, does the problem or activity shown below enable students to learn the full intent of the following concept or skill?

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: Represent addition and subtraction with objects, fingers, mental images, drawings (Drawings need not show details, but should show the mathematics in the problem. (This applies wherever drawings are mentioned in the Standards.)), sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations.
problem_activity: Narrative
The purpose of this activity is for students to solve a Put Together/Take Apart, Both Addends Unknown story problem about dates stuffed with cheese or almonds in more than one way. In the activity synthesis, students share their solutions. As students share, record their drawings and solutions systematically, as shown:
1 cheese, 7 almond
2 cheese, 6 almond
3 cheese, 5 almond
4 cheese, 4 almond
5 cheese, 3 almond
6 cheese, 2 almond
7 cheese, 1 almond
Only record solutions that students find for the problem. Leave an empty space if a combination isn’t shared.
In the lesson synthesis, students discuss whether a solution of 3 dates stuffed with cheese and 5 dates stuffed with almonds is the same solution as 5 dates stuffed with cheese and 3 dates stuffed with almonds. The purpose of the question is for students to think about and discuss what decompositions of numbers mean in context (MP2, MP6). There is no correct answer to the question. Some students may determine that both solutions are the same because they both showed 5 and 3 while other students may determine that the solutions are different because there are different numbers of dates stuffed with cheese and dates stuffed with almonds (MP3).
Representation: Internalize Comprehension.
Synthesis: Reiterate the fact that in this problem there is more than one answer. Make a connection between the solutions that were found and that they all add up to the same total number of dates.
Supports accessibility for: Conceptual Processing
Required Materials
Materials to Gather
Connecting cubes or two-color counters
Launch
Groups of 2
Give students access to connecting cubes or two-color counters.
Read and display the task statement.
“Tell your partner what happened in the story.”
30 seconds: quiet think time
1 minute: partner discussion
“What are we trying to figure out?” (How many of the dates were stuffed with cheese and how many of the dates were stuffed with almonds?)
30 seconds: quiet think time
1 minute: partner discussion
“Show your thinking using drawings, numbers, words, or objects.”
Activity
2 minutes: quiet work time
2 minutes: partner discussion
“Write an expression to show how many of the dates were stuffed with cheese and how many were stuffed with almonds.”
1 minute: independent work time
“As you walk around, look to see if you can find other ways to solve the story problem.”
5 minutes: gallery walk
Student Facing
Andre and his older brother have 8 dates.
They stuff some of the dates with cheese.
They stuff the rest of the dates with almonds.
How many of the dates did they stuff with cheese?
Then how many of the dates did they stuff with almonds?
Expression: _______________________________
[IMG]
[IMG]
Student Response
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Activity Synthesis
Invite students to share their solution or solutions they found during the gallery walk. Record student solutions as explained in the activity narrative.
“What do you notice? What do you wonder?” (2 and 6 is there but not 6 and 2. When it’s 4 and 4, there are the same number of cheese dates and almond dates. The first numbers are getting bigger and the second numbers are getting smaller.)
“Are there any other ways that we can solve the problem? What makes you think that?”
Record each student solution with an equation.


aligns: yes