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: Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.

Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., p^(2)). For example, by truncating the decimal expansion of Ö2, show that Ö2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations.
problem_activity: Activity
The purpose of this activity is for students to use rational approximations of irrational numbers and to match irrational numbers to equations they are solutions to. To do this, students sort cards into sets of three consisting of:
A square or cube root value
An equation of the form
\(x^2=p\)
or
\(x^3=p\)
that the value is a solution to
A number line showing the location of the value
Identify groups using clear explanations for how they chose to arrange their sets of 3, particularly when matching the cards with a value plotted on a number line.
Launch
Arrange students in groups of 2–4. Students should not use a calculator for this activity. Tell students that they will be given cards to sort into sets of three. Distribute 27 pre-cut slips from the blackline master to each group. Group work time followed by a whole-class discussion.
Representation: Internalize Comprehension.
Chunk this task into more manageable parts to differentiate the degree of difficulty or complexity by beginning with fewer cards. For example, give students a subset of the cards (using multiples of 3) to start with and introduce the remaining cards once students have completed their initial set of matches.
Supports accessibility for: Conceptual processing; Organization
Student Facing
Your teacher will give your group a set of cards. For each card with a letter and value, find the two other cards that match. One shows the location on a number line where the value exists, and the other shows an equation that the value satisfies. Be prepared to explain your reasoning.
Student Response
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Anticipated Misconceptions
Some students may mix up cube and square roots. Encourage these students to pay close attention to the notation. They may want to fill in the “2” for the square roots similar to how the “3” is used for cube roots.
Activity Synthesis
Select 3–4 previously identified groups to share one of their sets of three cards.
Also discuss:
“What was a match that was hard to make?” (Two of the number lines have values between 4 and 5, and it took some extra reasoning to figure out that the one with the value closer to 4 was
\(\sqrt{18}\)
while the one with the point closer to 5 was
\(\sqrt[3]{100}\)
.
“If another class was going to sort these cards, what is something you would recommend they have or do that you found helpful?” (I would recommend they have a list of perfect squares and perfect cubes to help think about where the different roots are plotted on the number line.)
Speaking, Writing: MLR8 Discussion Supports.
Display sentence frames to support students as they justify their reasoning for each match. For example, “The solution to _________ (
\(x^3 = 25\)
) is _________ (
\(\sqrt[3]{25}\)
) and it is between ______ (2 and 3) on the number line, because . . .” The helps students place extra attention on language used to engage in mathematical communication, without reducing the cognitive demand of the task. It also emphasizes the importance of justification.
Design Principle: Maximize meta-awareness


correct_option_index: 1