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.

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).

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

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 determine the number described in a riddle. Students then write the number of hundreds, tens, and ones, and represent the value as a three-digit number.
MLR8 Discussion Supports.
Display sentence frames to support partner discussion about each number written on the table: “I agree because . . .” or “I disagree because . . . .”
Advances: Conversing
Engagement: Provide Access by Recruiting Interest.
Provide choice. Invite students to decide the order in which they complete the work. They can start with any of the riddles, as long as they complete all of them.
Supports accessibility for: Attention, Social-Emotional Functioning
Required Materials
Materials to Gather
Base-ten blocks
Launch
Groups of 2
Give students access to base-ten blocks.
“I have 4 hundreds, 3 ones, and 2 tens.”
“Which of these shows the total value written as a three-digit number? Explain how you know.”
Display 432, 234, 423.
30 seconds: quiet think time
1 minute: partner discussion
Share responses.
Activity
“You are going to solve number riddles using base-ten blocks.”
As needed, demonstrate the task with a student.
“Take turns reading the clues, while your partner uses blocks to make the number.”
“Make sure you agree before adding each number to the table.”
10 minutes: partner work time
If students finish early, ask them to write their own riddles and trade them with other groups to solve.
Monitor for students who recognize they need a zero when writing the three-digit number in places where there were no tens or no ones.
Student Facing
Solve each riddle and write the three-digit number. Use the table to help you organize the digits.
| riddle | hundreds | tens | ones | three-digit number |
|--------|----------|------|------|--------------------|
|   1    |          |      |      |                    |
|   2    |          |      |      |                    |
|   3    |          |      |      |                    |
|   4    |          |      |      |                    |
|   5    |          |      |      |                    |
|   6    |          |      |      |                    |
I have 2 ones, 7 tens, and 6 hundreds.
I have 3 ones, 5 tens, and 2 hundreds.
I have 7 hundreds, 5 ones, and 3 tens.
I have 5 hundreds, no tens, and 9 ones.
I have 4 ones, 6 tens, and 3 hundreds.
I have 8 tens, 1 hundred, and no ones.
Student Response
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Activity Synthesis
Invite previously selected students to share what they noticed when writing 509 and 180.
As needed, write 81, 801, and 810 and ask, “What is the difference between 81, 801, and 810?”


correct_option_index: 0

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