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: Find all factor pairs for a whole number in the range 1–100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1–100 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1–100 is prime or composite.
problem_activity: Narrative
In this activity, students solve problems that involve finding multiples that are shared by two different numbers: the number of hot dogs in a package and the number of hot dog buns in a package. They reason about how many of each package to get to make a certain number of servings of hot dogs. As multiple answers can be expected, the focus is on explaining why the solutions make sense (MP3).
To solve the problems, students may decontextualize the situation and reason about factors and multiples, and then recontextualize the solutions in terms of servings of hot dogs. As they do so, they practice reasoning quantitatively and abstractly (MP2). During the synthesis, analyze different solutions and discuss why numbers that are multiples both of 8 and 10 are useful in this situation.
Launch
Groups of 2–4
Read the opening paragraph as a class.
“What questions could you ask about the situation?”
1 minute: quiet think time
Share responses.
“Let’s solve problems about hot dogs and hot dog buns, which come in different-size packages.”
“For each problem, look for possible numbers of packages that are needed to serve hot dogs at a picnic.”
“Take a minute to read the problems and think about how you would solve them.”
1 minute: quiet think time
Activity
“Now work with your group to solve these problems.”
8–10 minutes: group work time
Monitor for students who:
use the term “multiple” in their explanations
try different possibilities for the number of packages to buy
try to find numbers of packages that would give the exact (or close to the exact) numbers of servings
Student Facing
Each package of hot dogs has 10 hot dogs. Each package of hot dog buns has 8 buns.
Lin expects to need 50 hot dogs for a class picnic.
How many packages of hot dogs should Lin get? Explain or show your reasoning.
Can Lin get exactly 50 hot dog buns? How many packages of hot dog buns should Lin get? Explain or show your reasoning.
Diego expects to need 72 hot dogs for a class picnic.
How many packages of hot dogs should Diego get? Explain or show your reasoning.
How many packages of hot dog buns should Diego get? Explain or show your reasoning.
Is it possible to buy exactly the same number of hot dogs and buns? If you think so, what would that number be? If not, explain your reasoning.
Student Response
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Advancing Student Thinking
If students are using guess and check, consider asking: “How are you choosing which numbers to try?” or “How can thinking about multiples help you solve the problem?”
Activity Synthesis
Invite students to share their responses to the problem about 72 hot dogs, including those who get 70 hot dogs and 72 buns or 80 hot dogs and 80 buns. Encourage students to highlight how they are using multiples in their reasoning.
“Why are there different answers to this question?” (The numbers do not work out exactly to get 72 hot dogs and 72 buns, so we need to look at different alternatives.)
“Is it possible to buy exactly the same number of hot dogs and buns? If so, how many of each?” (Yes, 40, 80.)
“Why is it possible to buy exactly the same number of hot dogs and buns when the packaging is different?” (Some multiples of 10 are also multiples of 8, such as 40 and 80.)


correct_option_index: 0