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: 
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: 
Construct a tangent line from a point outside a given circle to the circle.

Answer: 
No. This task does not involve drawing lines that are tangent to circles. The only line that may be drawn is the ray between points P and B.

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_neg_0

HS_0

im_task_000870_pos_0

HS_1

im_task_001160_neg_0

HS_2
standard_description: Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them). For example, the expressions y + y + y and 3y are equivalent because they name the same number regardless of which number y stands for.
problem_activity: Narrative
The purpose of this activity is for students to find the volume of a figure in different ways. The given figure can be decomposed in two ways into rectangular prisms by making different cuts. However, it can also be found using a single, larger rectangular prism by removing a smaller rectangular prism. This provides an opportunity to express its volume as a difference of volumes of rectangular prisms. Students may notice this feature, and it is highlighted in the activity synthesis.
When students decide whether or not they have the same expressions, they need to reason carefully about what “the same” means. They consider if the order of the factors is different, is it the same expression and if the order of the addends is different, is it the same expression. Students use what they know about volume, geometric figures, and the properties of operations to justify the equivalence of the expressions and critique their peers' reasoning (MP2, MP3, MP7).
MLR8 Discussion Supports.
Display sentence frames to support partner discussion: “First, I _____ because . . . .” , “_____ and _____ are the same because . . .”, “_____ and _____ are different because . . .”
Advances: Conversing, Representing
Launch
Groups of 2
“You are going to look for different expressions to calculate the volume of a figure.”
Activity
5 minutes: partner work time
Monitor for students who draw a vertical line to show where they decompose the figure to share during the synthesis.
Student Facing
[IMG]
Write an expression to represent the volume of the figure in unit cubes.
Compare expressions with your partner.
How are they the same?
How are they different?
If they are the same, try to find another way to represent the volume.
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 did not see the figure as a cube missing a rectangular prism, ask them to build the figure and the missing piece that can be added to make it a cube.
Activity Synthesis
Invite students to share different expressions for the volume of the figure.
Display expression:
\((3 \times 5 \times 5) + (2 \times 2 \times 5)\)
“How does the expression represent the volume of the figure?” (
\(3 \times 5 \times 5\)
represents the prism that is 3 by 5 by 5 cubes tall and the
\(2 \times 2 \times 5\)
represents the prism that is 2 by 2 by 5 cubes tall.)
If no student wrote the expression
\((5 \times 5 \times 5) – (2 \times 3 \times 5)\)
, display this expression.
“How does the expression show the volume of the figure in cubic units?” (There is a 5 by 5 by 5 cube and a piece has been taken away. The piece that is taken away measures 2 cubes by 3 cubes by 5 cubes.)


aligns: no