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 the Pythagorean Theorem to find the distance between two points in a coordinate system.

Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.

Explain a proof of the Pythagorean Theorem and its converse.
problem_activity: Activity
The purpose of this task is for students to use the Pythagorean Theorem to calculate which rectangular prism has the longer diagonal length. To complete the activity, students will need to picture or sketch the right triangles necessary to calculate the diagonal length.
Identify groups using well-organized strategies to calculate the diagonals. For example, for Prism K some groups may draw in the diagonal on the bottom face of the cube and label it with an unknown variable, such as
\(s\)
, to set up the calculation for the diagonal as
\(\sqrt{6^2+s^2}\)
. Students would then know that to finish this calculation, they would need to find the value of
\(s\)
, which is can be done using the edge lengths of the face of the prism
\(s\)
is drawn on.
The idea that the Pythagorean Theorem might be applied several times in order to find a missing length is taken even further in the extension, where students can apply it over a dozen times, hopefully noticing some repeated reasoning along the way, in order to find the last length.
Launch
Arrange students in groups of 2. Provide access to calculators. Ask students to read and consider the first problem and then give a signal when they have done so. If possible, use an actual rectangular prism, such as a small box, to help students understand what length the diagonal shows in the image. Once students understand what the first problem is asking, poll the class to see which prism they think has the longer diagonal and post the results of the poll for all to see. Select 2–3 students to share why they chose the prism they did. The conversation should illuminate a few key points: Prism K has one edge with length 6 units, which is longer than any of the edges of Prism L, but it also has a side of length 4 units, which is shorter than any of the edges of Prism L. Prism K also has a smaller volume than Prism L.
Give 1 minute of quiet think time for students to brainstorm how they will calculate the lengths of each diagonal. Ask partners to discuss their strategies before starting their calculations. Follow with a whole-class discussion.
Representation: Internalize Comprehension.
Provide access to 3-D models of rectangular prisms (for example, a tissue box) for students to view or manipulate. Use color coding to highlight and make connections between corresponding parts.
Supports accessibility for: Visual-spatial processing; Conceptual processing
Conversing, Writing: MLR5 Co-Craft Questions.
Before revealing the questions in this activity, display only the image of the rectangular prisms. Invite students to work with a partner to write mathematical questions that could be asked about the prisms. Select 1–2 groups to share their questions with the class. Listen for and amplify questions about the lengths of the diagonals. For example, “What is the length of the diagonal of Prism
\(K\)
?” and “Which rectangular prism has the longer diagonal length?” If no student asks these questions, ask students to adapt a question to align with the learning goals of this lesson. Then reveal and ask students to work on the actual questions of the task. This routine will help develop students’ meta-awareness of language as they generate questions about calculating the diagonal length of a rectangular prism.
Design Principle(s): Maximize meta-awareness
Student Facing
Here are two rectangular prisms:
[IMG]
Which figure do you think has the longer diagonal? Note that the figures are not drawn to scale.
Calculate the lengths of both diagonals. Which one is actually longer?
Student Response
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for free access to Student Response.
Activity Synthesis
Ask groups to share how they calculated the diagonal of one of the rectangular prisms. Display the figures in the activity for all to see and label the them while groups share.
If time allows and no groups pointed out how the diagonal length of the rectangular prisms are the square root of the sum of the squares of the three edge lengths, use Prism K and the calculations students shared to do so. For example, to calculate the diagonal of Prism K, students would have to calculate
\(\sqrt{6^2+\left(\sqrt{41}\right)^2}\)
, but
\(\sqrt{41}\)
is from
\(\sqrt{5^2+4^2}\)
which means the diagonal length is really
\(\sqrt{6^2+5^2+4^2}\)
since
\(\sqrt{6^2+\left(\sqrt{41}\right)^2}=\sqrt{6^2+\left(\sqrt{5^2+4^2}\right)^2}=\sqrt{6^2+5^2+4^2}\)
. So, for a rectangular prism with sides
\(d\)
,
\(e\)
, and
\(f\)
, the length of the diagonal of the prism is just
\(\sqrt{d^2+e^2+f^2}\)
.


correct_option_index: 1