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: Read and write numbers to 1000 using base-ten numerals, number names, and expanded form.

Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, and < symbols to record the results of comparisons.

Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. Understand the following as special cases:
- 100 can be thought of as a bundle of ten tens — called a “hundred.”
- The numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four, five, six, seven, eight, or nine hundreds (and 0 tens and 0 ones).

Count within 1000; skip-count by 5s, 10s, and 100s.
problem_activity: Narrative
The purpose of this activity is for students to use their understanding of place value to find the number that makes each equation true. Students must consider how units may be composed or decomposed to find the unknown number (MP7). The number choices intentionally emphasize the types of compositions and decompositions students may use to add and subtract within 1,000. The synthesis focuses on the equations that involve composing or decomposing two units.
Required Materials
Materials to Gather
Base-ten blocks
Launch
Groups of 2
Give students access to base-ten blocks.
Activity
5 minutes: independent work time
“Take turns sharing your thinking with your partner. Show or explain how you know your equation is true.”
3 minutes: partner discussion
For finding the number that makes 2 hundreds + 9 tens + 17 ones true, monitor for students who:
explain by using base-ten blocks or place value diagrams
explain how they noticed when they would need to compose or decompose more than one unit
Student Facing
Find the number that makes each equation true.
5 hundreds + _____ tens + 6 ones = 5 hundreds + 5 tens + 16 ones
1 hundred + 1 ten + 17 ones = 1 hundred + _____ tens + 7 ones
4 hundreds + 15 tens + 3 ones = _____ hundreds + 5 tens + 3 ones
4 hundreds + 3 tens + 7 ones = 3 hundreds + _____ tens + 7 ones
7 hundreds + 8 tens + 4 ones = 7 hundreds + 7 tens + _____ ones
[IMG]
6 hundreds + 9 ones = 5 hundreds + _____ tens + 9 ones
2 hundreds + 9 tens + 17 ones = _____ hundreds + 7 ones
3 hundreds + 1 ten + 5 ones = 2 hundreds + 10 tens + _____ ones
If You Have Time: Make your own equation with a missing value for your partner.
Student Response
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Advancing Student Thinking
If students create equations that are not true, prompt students to make the known side of the equation with base-ten blocks, consider asking:
“How can you compose or decompose units to show the equation is true?”
Activity Synthesis
Invite previously identified students to share their method for finding the unknown number for 2 hundreds + 9 tens + 17 ones = _____ hundreds + 7 ones.
For students who use base-ten blocks, draw place value diagrams to record their explanation.
Consider asking:
“How do you know both sides of the equation are equal?”
“Did you compose or decompose units? Did you compose or decompose more than one time?”


correct_option_index: 2