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: Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. For example, create a story context for (2/3) ÷ (3/4) and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that (2/3) ÷ (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general, (a/b) ÷ (c/d) = ad/bc.) How much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 3/4-cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with length 3/4 mi and area 1/2 square mi?
problem_activity: Activity
This activity allows students to practice using the algorithm from earlier to solve division problems that involve a wider variety of fractions. Students can use any method of reasoning and are not expected to use the algorithm. As they encounter problems with less-friendly numbers, however, they notice that it becomes more challenging to use diagrams or other concrete strategies, and more efficient to use the algorithm. As they work through the activity, students choose their method.
Monitor the strategies students use and identify those with different strategies— including those who may not have used the algorithm—so they can share later.
Launch
Keep students in groups of 2. Give students 5–7 minutes of quiet work time, followed by 2–3 minutes to discuss their responses with a partner.
Representation: Internalize Comprehension.
Activate or supply background knowledge. Provide students with access to blank tape diagrams. Encourage students to attempt more than one strategy for at least one of the problems.
Supports accessibility for: Visual-spatial processing; Organization
Student Facing
Calculate each quotient. Show your thinking and be prepared to explain your reasoning.
\(\frac 89 \div 4\)
\(\frac 34 \div \frac 12\)
\(3 \frac13 \div \frac29\)
\(\frac92 \div \frac 38\)
\(6 \frac 25 \div 3\)
After biking
\(5 \frac 12\)
miles, Jada has traveled
\(\frac 23\)
of the length of her trip. How long (in miles) is the entire length of her trip? Write an equation to represent the situation, and then find the answer.
Student Response
Teachers with a valid work email address can
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for free access to Student Response.
Student Facing
Are you ready for more?
Suppose you have a pint of grape juice and a pint of milk. You pour 1 tablespoon of the grape juice into the milk and mix it up. Then you pour 1 tablespoon of this mixture back into the grape juice. Which liquid is more contaminated?
Student Response
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click here to register or sign in
for free access to Extension Student Response.
Activity Synthesis
Select previously identified students to share their responses. Sequence their presentations so that students with the more concrete strategies (e.g., drawing pictures) share before those with more abstract strategies. Students using the algorithm should share last. Find opportunities to connect the different methods. For example, point out where the multiplication by a denominator and division by a numerator are visible in a tape diagram.
Conversing, Representing: MLR7 Compare and Connect.
As students consider the different strategies, invite them to make connections between the various representations and approaches. Ask, “What do each of the strategies have in common?”, “How are the strategies different?” and “Which strategy is more efficient? Why?” Listen for and amplify observations that include mathematical language and reasoning.
Design Principle(s): Maximize meta-awareness; Optimize output (for comparison)


correct_option_index: 0