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


Concept/skill: 
Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.

Answer: 
No. This problem does not directly teach how side ratios in right triangles lead to definitions of trigonometric ratios. Instead, it involves solving a trigonometric function.

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_neg_0

HS_0

im_task_000870_neg_0

HS_1

im_task_001160_pos_0

HS_2
standard_description: Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. For example, express the calculation “add 8 and 7, then multiply by 2” as 2 × (8 + 7). Recognize that 3 × (18932 + 921) is three times as large as 18932 + 921, without having to calculate the indicated sum or product.
problem_activity: Narrative
The purpose of this activity is for students to write equivalent expressions in order to find the volume of a figure composed of two right rectangular prisms. Students decompose the figure in two different ways, and write matching expressions to find the volume. For extra support, provide students with colored pencils to shade the two parts of the prism before finding the side lengths they need to calculate the volume.
Monitor and select a student with each of the following strategies to share in the synthesis:
decomposed the figure into a prism with the side lengths 4 ft by 4 ft by 3 ft and a prism with the side lengths 10 ft by 4 ft by 3 ft and wrote this expression (or one written in a different order) to represent the volume:
\((4 \times 4 \times 3) + (10 \times 4 \times 3)\)
decomposed the figure into a prism with the side lengths 4 ft by 8 ft by 3 ft and a prism with the side lengths 6 ft by 4 ft by 3 ft and wrote this expression (or one written in a different order) to represent the volume:
\((4 \times 8 \times 3) + (6 \times 4 \times 3)\)
Action and Expression: Internalize Executive Functions.
Invite students to plan a strategy, including the tools they will use, for finding the volume of the figures. If time allows, invite students to share their plan with a partner before they begin.
Supports accessibility for: Organization, Memory, Attention
Launch
Groups of 2
Activity
10 minutes: individual work time
5 minutes: partner discussion
As students work, consider asking, “Why did you choose to decompose the prism that way?”
Student Facing
Find the volume of the figure by decomposing the figure 2 different ways. Show your thinking. Organize it so it can be followed by others.
[IMG]
[IMG]
For each way you decomposed the figure, write an expression that represents the volume.
Mai used this expression to find the volume of the figure:
\((10 \times 8 \times 3) - (6 \times 4 \times 3)\)
.
Use the diagram to interpret Mai's expression. Show your thinking. Organize it so it can be followed by others.
[IMG]
Student Response
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Activity Synthesis
Ask the two selected students to display their work side by side for all to see.
“How are the diagrams the same? How are they different?”
“How do the expressions relate to the diagrams?”
Display:
\((10 \times 8 \times 3) - (6 \times 4 \times 3)\)
“How does this expression represent the volume of the prism?” (The larger rectangular prism has the side lengths
\(10 \times 8 \times 3\)
cubic feet. We can subtract a rectangular prism with the side lengths
\(6 \times 4 \times 3\)
cubic feet.)
“What is the value of
\((10 \times 8 \times 3) - (6 \times 4 \times 3)\)
?” (168)


aligns: yes