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 following problem or activity 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: Compose simple shapes to form larger shapes. For example, “Can you join these two triangles with full sides touching to make a rectangle?”
problem_activity: Narrative
The purpose of this activity is for students to build and compare flat and solid shapes. Students are not expected to build precise shapes. Using clay to build the shapes helps students feel the difference between flat and solid shapes. The terms “flat” and “solid” are introduced in the activity synthesis. Students may benefit from getting to hold and see the solid shapes up close as they build them.
Required Materials
Materials to Gather
Clay
Required Preparation
Each student needs a piece of clay.
Gather a cone and cylinder to display.
Launch
Groups of 2
Give each student a piece of clay.
“Use your clay to make a shape that you know.”
1 minute: independent work time
“Share your shape with your partner. How are they the same? How are they different?” (The shapes are different. I made a circle and my partner made a triangle.)
30 seconds: quiet think time
1 minute: partner discussion
Share responses.
Activity
Display a cylinder.
“Make this shape with your clay.”
2 minutes: independent work time
Display a cone.
“Make this shape with your clay.”
2 minutes: independent work time
“Describe the shape that you made to your partner. What does it look like?” (It looks like an ice cream cone. It’s tall. It has a point on the bottom.)
30 seconds: quiet think time
2 minutes: partner discussion
Share responses.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Activity Synthesis
Display a group of flat shapes and a group of solid shapes that students created.
“What is different about the shapes that we made first and the rest of the shapes that we made?” (The shapes that we made first are all flat. The other shapes stick up.)
“The first shapes that we made were flat shapes, like circles and rectangles. Then we built some other shapes. What can we call these shapes?” (solid shapes, big shapes)
“These shapes are solid shapes.”
“Use your clay to make a flat shape.”
“Use your clay to make a solid shape.”


aligns: no