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: Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.
Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.

Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.
Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.
problem_activity: Narrative
This activity prompts students to compare multiple fractions and put them in order by size. The work gives students opportunities to look for and make use of structure (MP7) in each set of fractions and make comparisons strategically. For instance, rather than comparing two fractions at a time and in the order they are listed, students could first classify the given fractions as greater or less than
\(\frac{1}{2}\)
or 1, look for fractions with a common numerator or denominator, and so on.
If time is limited, consider asking students to choose two sets of fractions to compare and order.
Representation: Internalize Comprehension.
Synthesis: Invite students to identify which details were most useful when putting fractions in order. Display the sentence frame, “The next time I put fractions in order, I will pay attention to . . . .”
Supports accessibility for: Memory
Launch
Groups of 2
Activity
“Work independently on two sets. Then, discuss your work with your partner and complete the rest together.”
10 minutes: independent work time
Monitor for students who look for and make use of structure. Ask them to share during lesson synthesis.
3–4 minutes: partner discussion
Student Facing
Put each set of fractions in order, from least to greatest. Be prepared to explain your reasoning.
\(\frac{3}{12} \qquad \frac{2}{4} \qquad \frac{2}{3} \qquad \frac{1}{8}\)
\(\frac{8}{5} \qquad \frac{5}{6} \qquad \frac{11}{12} \qquad \frac{11}{10}\)
\(\frac{21}{20} \qquad \frac{9}{10} \qquad \frac{6}{5} \qquad \frac{101}{100}\)
\(\frac{5}{8} \qquad \frac{2}{5} \qquad \frac{3}{7} \qquad \frac{3}{6}\)
Student Response
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Advancing Student Thinking
Some students may try to write equivalent fractions with a common denominator for all four fractions in each set before comparing them but may be unable to do so. Encourage them to try reasoning about two fractions at a time, and to use what they know about the fractions to determine how they compare (to one another or to familiar benchmarks).
Activity Synthesis
See lesson synthesis.


correct_option_index: 1