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: 
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 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

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im_task_001160_neg_0

HS_2
standard_description: Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.
problem_activity: Activity
In this activity, students continue the work on finding the volume of a right rectangular prism with fractional edge lengths. This time, they do so by packing it with unit cubes of different unit fractions for their edge lengths—
\(\frac13\)
,
\(\frac12\)
, and
\(\frac14\)
of an inch. They use these cubes to find the volume of the prism in cubic inches, decide which unit fraction works better to accomplish this goal and why, and explain whether cubes of different fractional edge lengths would lead to the same volume in cubic inches (MP3).
As students work, notice those who are able to clearly explain why cubes with a particular fractional edge length are preferable as a unit of measurement and why the volume in cubic inches will be the same regardless of the cubes used. Invite them to share later.
Launch
Arrange students in groups of 3–4. Give them 8–10 minutes of quiet work time, and then 5 minutes to discuss their responses with their group. Ask groups to be sure to discuss the third question. Encourage students to draw a sketch to help with reasoning, if needed.
Representation: Internalize Comprehension
. Activate or supply background knowledge about measurement. Students may benefit from watching a quick demonstration or video of packing cubes into a box. Review terms such as dimensions, volume, and cubic units.
Supports accessibility for: Memory; Conceptual processing
Student Facing
Diego says that 108 cubes with an edge length of
\(\frac13\)
inch are needed to fill a rectangular prism that is 3 inches by 1 inch by
\(1\frac13\)
inch.
Explain or show how this is true. If you get stuck, consider drawing a diagram.
What is the volume, in cubic inches, of the rectangular prism? Explain or show your reasoning.
Lin and Noah are packing small cubes into a larger cube with an edge length of
\(1\frac12\)
inches. Lin is using cubes with an edge length of
\(\frac12\)
inch, and Noah is using cubes with an edge length of
\(\frac14\)
inch.
Who would need more cubes to fill the
\(1\frac12\)
-inch cube? Be prepared to explain your reasoning.
If Lin and Noah each use their small cubes to find the volume of the larger
\(1\frac12\)
-inch cube, will they get the same answer? Explain or show your reasoning.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Anticipated Misconceptions
If students have trouble getting started, ask them to revisit their work with
\(\frac12\)
-inch cubes from a previous lesson. Remind them that a cube with
\(\frac 12\)
-inch edge length has a volume of
\(\frac18\)
in
3
(because we can fit 8 of such cubes in a 1-inch cube). Ask them to think about how many
\(\frac13\)
-inch cubes can fit into a 1-inch cube, or think about what the volume of a
\(\frac13\)
-inch cube is in cubic inches.
Some students may not be able to visualize and keep track of the measurements of the boxes in this task. Encourage students to draw and label the measurements of the boxes described in these questions.
Activity Synthesis
Select several students to share their responses and articulate their reasoning. Compare the different strategies students used for finding the volume of the rectangular prism. Ask students:
“Does it matter which fractional-unit cubes we use to find the volume? Why or why not?” (As long as the unit fraction can fit evenly into all three edge lengths of the prism, it doesn’t matter what unit fraction we use.)
“Do certain unit fractions work better as edge lengths of the small cubes than others?” (It helps to use as large a unit fraction as possible, since it means using fewer cubes and working with fractions that are closer to 1.)
“Is there another way of finding the volume of a rectangular prism with fractional edge length besides using these small cubes?” (Multiply the fractional edge lengths.)
Point out that it is helpful to use a unit fraction that is a common factor of the fractional edge lengths of the prism. Make sure students also recognize that multiplying the edge lengths of the prism is a practical way to find the volume of such a rectangular prism.


aligns: no