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: Draw a scaled picture graph and a scaled bar graph to represent a data set with several categories. Solve one- and two-step “how many more” and “how many less” problems using information presented in scaled bar graphs. For example, draw a bar graph in which each square in the bar graph might represent 5 pets.
problem_activity: Narrative
The purpose of this activity is for students to build any object they want to. In the next activity, students focus only ;on rectangular prisms. As students count, they will need to make sure to count each cube once and only once. Monitor for these strategies:
Touch each cube as it is counted
Count the cubes by layer (including when there are different numbers of cubes in each layer)
Use addition or multiplication when the cubes are organized in layers
Required Preparation
Each group of 2 needs 24 connecting cubes.
Launch
Groups of  2
Give 24 connecting cubes to each group.
“In this activity, you will use unit cubes to build objects and describe how you would measure the volume.”
10 minutes: partner work time
Activity
As students work, monitor for students who build rectangular prisms to share during the synthesis.
Student Facing
Partner A: Build an object using 8–12 cubes and give the object to Partner B.
Partner B: Explain how you would count the number of cubes in the object.
Partner A: Explain if you would count the cubes in the same way or in a different way.
Switch roles and repeat.
Which objects were easiest to count? Why?
[IMG]
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 are counting all of the cubes one at a time, ask, “where do you see cubes that could be counted as a group?”
Activity Synthesis
Discuss responses to the last problem.
“Which objects were most challenging to count?” (Objects where I had to count each individual cube because it was a weird shape.)
“Which objects were the simplest to count?” (Organized shapes like towers, rectangles, cubes, or boxes.)
Invite selected students to share the rectangular prisms they built and how they counted the number of cubes.
“How did you count the cubes?” (I knew that there were 4 cubes in each layer and 2 layers so that’s 8 cubes.)
“We call these objects
rectangular prisms
. What were some ways you used to count the cubes in rectangular prisms?” (I counted the cubes in groups and then skip counted. I used multiplication to find the total.)


aligns: no