few_shots: 
Example 1:
Concept/skill: 
Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

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


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:
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).

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


Answer: 
No. This task does not involve an equation in two variables. There is only one variable, x.

Example 3:
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.

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 $P$.


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, determine whether the following problem or activity can enable students to understand the concepts or skills in the provided description.

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: 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 goal of this activity is to represent expressions as decompositions of a figure made of two non-overlapping right rectangular prisms. This gives students an opportunity to interpret parentheses in expressions while also checking their understanding of different ways to represent the volume of rectangular prism; namely, length times width times height and area of a base times the corresponding height.
Students work abstractly and quantitatively in this problem (MP2) as they relate abstract expressions to decompositions of figures composed of two rectangular prisms.
MLR7 Compare and Connect.
Invite students to think about details they can include that will help others understand their thinking. For example, specific language, using different colors, shading, arrows, labels, notes, diagrams or drawings. Give students time to investigate each others’ work. During the whole-class discussion, ask, “What kinds of details or language helped you understand your partner’s work?”
Advances: Representing, Conversing
Launch
Groups of 2
“Look at the first figure and think about how you might decompose it into 2 prisms.”
1 minute: quiet think time
“Now you will consider how different expressions can represent the volume of figures.”
Activity
5 minutes: independent work time
5 minutes: partner discussion time
Monitor for students who
use the numbers in the expressions to determine how to break up the figure
break up the figure and use this to identify the expressions.
Student Facing
Explain how each expression represents the volume of the figure. Show your thinking. Organize it so it can be followed by others.
\(((2 \times 3) \times4) + ((3 \times 3) \times 2)\)
\((5 \times 6) + (3 \times 4)\)
[IMG]
How does each expression represent the volume of the prism? Explain or show your thinking. Organize it so it can be followed by others.
\((5\times8\times6) + (5\times4\times9)\)
cubic inches
\((5\times4\times3) + (5\times12\times6)\)
cubic inches
[IMG]
Student Response
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Advancing Student Thinking
If students do not describe how the expressions represent the volume of the figure, show them part of the expression, the numbers inside the parentheses for example, and ask them to explain which part of the figure is represented.
Activity Synthesis
Invite students to share how the expressions for the first prism represent its volume.
“Why are there 3 factors in the expression
\((2 \times 3) \times 4)\)
but only 2 factors in the expression
\((5 \times 6)\)
?” (The 2, 3, and 4 are the length, width, and height of the prism in the back. The 5 is the height of the prism on bottom whose base area is 6.)
“In the expression
\((3 \times 3) \times 2\)
, what do the parentheses tell you?” (They tell me to first multiply 3 times 3, which gives me the area of the base, and then I multiply that by 2 to get the volume of the prism.)


aligns: yes