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: Compare two numbers between 1 and 10 presented as written numerals.

Identify whether the number of objects in one group is greater than, less than, or equal to the number of objects in another group, e.g., by using matching and counting strategies (include groups with up to ten objects).
problem_activity: Narrative
The purpose of this activity is for students to compare numbers in a way that makes sense to them. Students can use physical objects or make drawings to represent each number (MP5), and match or count to compare. Students can also use their knowledge of the count sequence and understanding that each successive number refers to a quantity that is one more to compare the numbers. In the synthesis students describe how creating representations of numbers helps to compare numbers.
Action and Expression: Internalize Executive Functions.
After students decide which manipulative they will use for the activity, invite students to plan a strategy for how they will use their manipulatives to determine which number is more. If students choose to use counters, give them access to 5-frames to help with the counting.
Supports accessibility for: Visual-Spatial Processing, Conceptual Processing, Organization
Required Materials
Materials to Gather
Connecting cubes or counters
Launch
Groups of 2
Give students access to connecting cubes or counters.
“Work with your partner to figure out which number is more. Circle the number that is more.”
Activity
5 minutes: partner work time
Monitor for students who create representations of the numbers using cubes or a drawing and use these representations to compare.
Monitor for students who counted to figure out which number is more.
Student Facing
Circle the number that is more.
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Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Advancing Student Thinking
If students circle the number that is less instead of the number that is more, consider asking:
“What do you know about these two numbers? What are you trying to figure out?“
“How can you use the cubes to show each number? How can the cubes help you figure out which number is more?“
Activity Synthesis
Invite previously selected students to share the representations they made and how they used them to figure out which number is more.
“How did counting out the groups of objects or drawing pictures help you figure out which number is more?”
Invite previously selected students to share how counting helped them compare the numbers.
“We know 8 is more than 5. Compare these numbers using ‘less’.”


correct_option_index: 0