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


Standard options: 
A. Prove the Pythagorean identity sin^(2)(θ) + cos^(2)(θ) = 1 and use it to find sin(θ), cos(θ),or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.
B. Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.


Thought: 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. Solving this problem does not require use of the Pythagorean identity.
Answer: B

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


Standard options: 
A. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
B. Solve quadratic equations in one variable.


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

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 $P$.


Standard options: 
A. 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.
B. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.


Thought: 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.
Answer: A

Now, assign the following problem or activity to one or more relevant standards it aligns with.
shot_info: fl_problem_001894

HS_0

im_task_000870

HS_1

im_task_001160

HS_2
options: Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume.

Recognize volume as an attribute of solid figures and understand concepts of volume measurement.

Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units.
problem_activity: Narrative
The purpose of this activity is for students to find the volume of a figure composed of two rectangular prisms and recognize that volume is additive. In the previous activity, students physically put two rectangular prisms together and then decomposed a figure into two rectangular prisms. In this activity, they are given an image of a figure composed of two rectangular prisms and asked to find the volume. They can do this in many ways. During the synthesis, connect the different strategies students used to the decomposition of the figure into two rectangular prisms.
As students experiment with different ways to group the cubes to efficiently count them, applying what they already know about the volume of rectangular prisms, they are looking for and making use of the structure of geometric objects (MP7). As students begin to generalize the idea that volume is additive, they are looking for and expressing regularity in repeated reasoning (MP8).
MLR7 Compare and Connect.
Synthesis: Invite students to prepare a visual display that shows the strategy they used to find the volume of each figure. Encourage students to include details that will help others interpret their thinking; for example, using specific language, or using different colors, shading, or arrows. Give students time to investigate each other’s work.
Advances: Representing
Launch
Groups of 2
Activity
5 minutes: individual work time
5 minutes: partner discussion
As students work, monitor for students who:
find two (or more) rectangular prisms that can be put together to make the figure and adding their volumes.
find the number of cubes in a base layer and multiplying by the number of layers, even though the base layer is not a rectangle (figure d).
Student Facing
Find the volume of each figure. Explain or show your reasoning.
a.
[IMG]
b.
[IMG]
c.
[IMG]
d.
[IMG]
Student Response
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Advancing Student Thinking
If students did not decompose the figures into rectangular prisms, encourage them to build the figures with connecting cubes and ask, “how can you decompose the figure into two rectangular prisms?”
Activity Synthesis
Display Figures c and d.
Ask selected students to share their way of splitting each figure.
“Who broke the figure up the same way? Who broke it up differently?”
“Can you think of other ways you could break up these figures?“ (I can cut them into several layers—3 horizontal layers for c, and 5 vertical layers for d. Each layer is a rectangular prism.)


correct_option_index: 0