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: Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures.

Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments.

Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.
problem_activity: Activity
The purpose of this activity is for students to practice solving problems that involve volumes of pyramids and cones. Monitor for various problem-solving strategies—in particular, for the problem in which students find the radius or height of a cone given a particular volume. Some students may substitute values into a formula and solve the resulting equation, and others may follow a process of deductive reasoning.
Launch
Provide students access to scientific calculators. Arrange students in groups of 2. Ask students to discuss their thinking with their partner, and if they disagree, work to reach an agreement.
Representation: Internalize Comprehension.
Activate or supply background knowledge about the formulas for the area of a triangle, square, and circle. Allow students to use calculators to ensure inclusive participation in the activity.
Supports accessibility for: Memory; Conceptual processing
Student Facing
Calculate the volume of each solid. Round your answers to the nearest tenth if necessary.
A
[IMG]
B
[IMG]
C
[IMG]
D: height 12 cm; area of base 32 cm
2
[IMG]
A particular cone has radius
\(r\)
and height
\(h\)
.
Write an expression for the volume of this cone in terms of
\(r\)
and
\(h\)
.
What is the height of a cone whose volume is
\(16\pi\)
cubic units and whose radius is 3 units?
What is the radius of a cone whose volume is
\(16\pi\)
cubic units and whose height is 3 units?
Student Response
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for free access to Student Response.
Student Facing
Are you ready for more?
The Pyramid of Giza is 455 feet tall. The base is square with a 756-foot side length. How many Olympic-size swimming pool volumes of water can fit inside the Pyramid of Giza? Explain or show your reasoning.
Student Response
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click here to register or sign in
for free access to Extension Student Response.
Anticipated Misconceptions
For the pyramid with a triangular base, students may forget that the triangle area formula involves dividing by 2, resulting in a base area of 12 square units. Prompt them to draw and label the base prior to calculating its area.
Activity Synthesis
Ask students to share their strategies for volume calculations. For example, for the cone, did they calculate the area of the base first, and substitute it into the formula
\(V=\frac13 Bh\)
? Or, perhaps they wrote out
\(V=\frac13 \pi r^2 h\)
and performed the calculations all at once.
Select previously identified students to share their thought process on the “working backward” cone problem. If possible, find one student who solved an equation algebraically and another who used backward reasoning rather than formally solving an equation.
Representing, Conversing: MLR7 Compare and Connect.
Use this routine to enhance the whole-class discussion about strategies for volume calculation. After students calculate the volume of each solid and determine the radius or height of each cone, invite them to create a visual display of their work for one of the solids. Then ask students to quietly circulate and observe at least two other visual displays in the room. Give students quiet think time to consider what is the same and what is different about their strategies. Next, ask students to find a partner to discuss what they noticed. Listen for and amplify the language students use to compare and contrast strategies for volume calculation.
Design Principle(s): Cultivate conversation


correct_option_index: 2