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 gives students opportunities to practice explaining or showing whether two fractions are equivalent. Students may do so using a visual representation, by reasoning about the number and size of the fractional parts in each fraction, or by thinking about multiplicative relationships between the numbers in the given fractions.
Students participate in a gallery walk in which they generate equivalent fractions for the numbers on the posters. Students visit at least two of six posters (or as many as time permits)—at least one poster with two fractions (posters A–C) and one poster with three fractions (posters D–F). Here are the sets shown on the blackline master:
A
​​​​​
\(\frac{2}{10}\)
,
\(\frac{20}{100}\)
B
\(\frac{6}{4}\)
,
\(\frac{18}{12}\)
C
\(\frac{3}{5}\)
,
\(\frac{60}{100}\)
D
​​​​​​
\(\frac{1}{4}\)
,
\(\frac{3}{12}\)
,
\(\frac{30}{100}\)
E
\(\frac{15}{6}\)
,
\(\frac{7}{4}\)
,
\(\frac{30}{12}\)
F
\(\frac{7}{3}\)
,
\(\frac{21}{10}\)
,
\(\frac{28}{12}\)
Because at the posters with two fractions (A–C) students would need to generate an equivalent fraction that hasn’t already been written by others, generating equivalent fractions becomes more difficult as the activity goes on. Consider using this to differentiate for students who may need an additional challenge: start them at the posters with three fractions (D–F).
MLR8 Discussion Supports.
Synthesis. Display sentence frames to support whole-class discussion: “I agree because . . .” and “I disagree because . . . .”
Advances: Speaking, Conversing
Required Materials
Materials to Gather
Sticky notes
Materials to Copy
How Do You Know
Required Preparation
Each group needs 4 sticky notes.
Launch
Groups of 3–4
Give each group 4 sticky notes
Read the task statement as a class. Solicit clarifying questions from students.
Invite a couple of students to recap the directions in their own words or to demonstrate the process, if helpful.
Consider assigning each group a starting poster and giving directions for rotation.
Activity
10 minutes: gallery walk
Tell students who are visiting posters A–C that they could leave feedback about the fraction on a sticky note if they disagree that it is equivalent to the fractions on the poster. They should include their name and be prepared to explain how they know.
Student Facing
Around the room you will find six posters, each showing either two or three fractions.
With your group, visit at least two posters: one with two fractions and one with three fractions.
[IMG]
For the set of 2 fractions:
Explain or show how you know the fractions are equivalent.
Write a new equivalent fraction on a sticky note and add it to the poster. Think of a fraction that hasn’t already been written by someone else.
We visited poster __________, which shows __________ and __________.
New equivalent fraction: __________
For the set of 3 fractions:
Identify 2 fractions that are equivalent. Explain your reasoning.
We visited poster __________, which shows __________, __________, and __________.
Student Response
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Activity Synthesis
See lesson synthesis.


correct_option_index: 0