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: Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).

Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).

Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).
problem_activity: In Class Launch
Use after Unit 3, Lesson 11
In this task, students will experiment with cups in order to see what shape they seem to trace out when they are rolled on a flat surface, and then refine their thinking in order to prove what the shape is. If possible, provide cups with a variety of sizes and shapes. Note that cylindrical cups will not curve as they are rolled, and so will not make the same kind of shape as cups with slanted sides.
Ask students to roll some cups around. Then ask them to guess what shape the cup traces out on the table or floor. Sketch the suggested shapes for all to see. Tell students their task is to try to prove what the shape is.
Teacher Instructions
Consider the types of cups to make available to students and how they will be able to manipulate the cups. For example, paper cups can be cut apart and unrolled, but plastic cups can’t. If materials are limited and need to be reused in multiple class periods, consider providing an intact cup and a cup that has already been cut apart so that students can examine both of them.
After students have completed the task, either invite them to share how they knew what shape the cup would trace out, or have a gallery walk so that students can see and comment on each other’s arguments.
Then invite them to share their responses to the last question: how did they make a cup that would trace out the largest possible area? If no student mentions it, ask what would happen if the sides of the cup weren’t slanted at all. (In theory, the cup will roll forever and trace out an infinite area, but in reality, it will run into something eventually and stop.)
Technology can be helpful when students are experimenting with different possible cup shapes. If technology is available, here is an applet students can use:
Student-Facing Statement
What shape does a cup trace out when it is rolled on a flat surface? Use words and pictures to explain how you know.
How can you locate the center and calculate the area of the shape that is traced out?
Using one sheet of 8.5 inch by 11 inch paper, how could you make a cup that will trace out the largest possible area?
Teacher Instructions
Consider the types of cups to make available to students and how they will be able to manipulate the cups. For example, paper cups can be cut apart and unrolled, but plastic cups can’t. If materials are limited and need to be reused in multiple class periods, consider providing an intact cup and a cup that has already been cut apart so that students can examine both of them.
If students see the questions in Part 2 before they have finished Part 1, it will give away the shape that the cup traces out, so hand out Part 2 after the discussion of Part 1. Part 1 can be discussed before all students have finished their explanations, as long as they all have some developed ideas about what the shape is. For the discussion, ask students to share what shape they think the cup traces out and why they think so. Students do not need to have fully developed proofs, but take the opportunity to press them to clarify their terminology and justify their assumptions.
After students have completed the task, invite them to share their responses to the last question: how did they make a cup that would trace out a larger area? (The bottom radius should be closer to the top radius so that the radius of the ring will be larger, or the cup should be taller so that the ring is thicker.) Then ask how they would make a shape that traces out the largest possible area. If no student mentions it, ask what would happen if the sides of the cup weren’t slanted at all. (In theory, the cup will roll forever and trace out an infinite area, but in reality, it will run into something eventually and stop.)
Technology can be helpful when students are experimenting with different possible cup shapes. If technology is available, here is an applet students can use:
Student-Facing Statement
Part 1:
What shape does a cup trace out when it is rolled on a flat surface? Explain or show your reasoning.
Pause for discussion.
Part 2:
The outside of this ring has a radius of 55 cm. The inside of the ring has a radius of 38 cm. Find the area of the ring.
[IMG]
Triangle
\(ABC\)
is isosceles. Find the length of side
\(AC\)
using the information given. The two vertical lines are parallel.
[IMG]
Here is the cross-section of a cup. It shows the width of the cup at the top and at the bottom, and the slant height of the cup. If we extend the sides of the cup down until they meet at a point, we will make a triangle. What would the slant height of that triangle be?
[IMG]
Measure the widths of the top and bottom of your cup, and measure its slant height. If you drew the cross-section of the cup and then extended the sides to make a triangle, what would the slant height of the triangle be?
If you rolled your cup in a circle to make a ring, what would the area of the ring be?
Give an example of a cup that would trace out a larger area than your cup would.


correct_option_index: 2