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: 
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_pos_0

HS_0

im_task_000870_pos_0

HS_1

im_task_001160_pos_0

HS_2
standard_description: Count to 100 by ones and by tens.
problem_activity: Narrative
The purpose of this activity is for students to count their collection in a way that makes sense to them and notice that the number of objects stays the same when a collection is counted multiple times (MP8). Students are provided with 10-frames and counting mats to help them organize their collections. Students use appropriate tools strategically as they choose which tools help them count their collections (MP5).
MLR8 Discussion Supports
. Display and read aloud the following sentence frames to support small-group discussion: “I agree because _____.” and “I disagree because _____.” Invite students to chorally repeat this sentence in unison 1–2 times.
Advances: Speaking, Conversing, Representing
Representation: Internalize Comprehension.
Synthesis: Use multiple examples and non-examples to emphasize the importance of counting every object one time. Demonstrate to students the non-example of counting one object twice in a collection.
Supports accessibility for: Conceptual Processing, Organization
Required Materials
Materials to Gather
10-frames
Collections of objects
Counting mats
Required Preparation
Each student needs a collection of 11–20 objects.
Launch
Groups of 2
Give each student a collection of objects and access to 10-frames and a counting mat.
“How many objects are in your collection?”
Activity
3 minutes: independent work time
“Tell your partner how many objects are in your collection.”
30 seconds: partner discussion
“Switch collections with your partner. Do you agree with your partner about how many objects are in the collection?”
3 minutes: independent work time
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Advancing Student Thinking
If students count each object more than once or do not count some of the objects, consider asking:
“How many objects do you have? How do you know?”
Display the counting mat or the 10-frame and ask “How can you use this mat to help you make sure that you count each object?”
Activity Synthesis
“Show your partner how you counted and tell your partner how many objects are in your new collection. Did you and your partner count the collection the same way? Did you agree about how many objects are in the collection?”
“We might count the objects differently than our partner, but we should get the same number if we count every object one time.”


aligns: no