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


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

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


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).

Answer: 
No. This task does not involve an equation in two variables. There is only 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_pos_0

HS_0

im_task_000870_neg_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 learn stage 3 of the Shake and Spill center. Students practice counting to find the total of 2 groups of objects and representing addition with an expression.
After they participate in the center, students choose from any stage of previously introduced centers.
Number Race
Math Stories
Required Materials
Materials to Gather
Crayons
Materials from previous centers
Two-color counters
Materials to Copy
Shake and Spill Stage 3 Recording Sheet Kindergarten
Required Preparation
Each group of 2 needs 10 two-color counters.
Each group of 2 needs access to a yellow crayon and a red crayon.
Gather materials from:
Number Race, Stage 1
Math Stories, Stages 1 and 2
Launch
Groups of 2
Give each group of students two-color counters and red and yellow crayons. Give each student a recording sheet.
“We are going to learn a new way to play Shake and Spill. One partner will choose some counters, shake them, and spill them. Then draw a picture to show the red and yellow counters. Work together to figure out how to fill in an expression to show how many red and yellow counters. Take turns shaking and spilling the counters.”
Activity
8 minutes: partner work time
“Now you can choose another center. You can also continue playing Shake and Spill.”
Display the center choices in the student book.
Invite students to work at the center of their choice.
10 minutes: center work time
If time, invite students to choose another center.
Student Facing
Choose a center.
Shake and Spill
[IMG]
Number Race
[IMG]
Math Stories
[IMG]
Activity Synthesis
Display
\(4 + 3\)
.
“Andre and Tyler wrote
\(4 + 3\)
. What do you think their counters looked like?” (4 red counters and 3 yellow counters. 4 yellow counters and 3 red counters.)


aligns: yes