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: Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. For example, if a person walks 1/2 mile in each 1/4 hour, compute the unit rate as the complex fraction 1/2/1/4 miles per hour, equivalently 2 miles per hour.

Recognize and represent proportional relationships between quantities.

Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.
problem_activity: Activity
The purpose of this activity is to understand that discrete values in a table can be used as evidence that a relationship is proportional and can be used to know for sure that a relationship is not proportional, but can’t be used to know for sure whether a relationship is definitely proportional. This activity builds on previous ones involving constant speed, but analyzes pace (minutes per lap) rather than speed (laps per minute). Explaining why the information given in the table is enough to conclude that Han didn't run at a constant pace but is not enough to know for sure whether Clare ran at a constant pace requires students to make a viable argument (MP3).
Launch
Keep students in the same groups of 2. Give 5 minutes of quiet work time, followed by 5 minutes of students discussing responses with a partner, followed by a whole-class discussion.
Conversing, Reading: MLR5 Co-Craft Questions.
Use this routine to help students interpret the language of proportional relationships, and to increase awareness of language used to talk about proportional relationships. Display the first sentence of this problem (“Han and Clare were running laps around the track. The coach recorded their times at the end of laps 2, 4, 6, and 8.”) and the tables of Han and Clare’s run, and ask students to write down possible mathematical questions that could be asked about the situation. If needed, provide students with the question starter “Why is. . .?” Invite students to share their questions with the class before revealing the task questions. Listen for and amplify any questions involving proportional relationships or mathematical language, such as “constant pace,” “not constant,” “proportional,” or “constant of proportionality.” This will help students produce the language of mathematical questions and start noticing whether or not there is a proportional relationship in this task prior to starting the task.
Design Principle(s): Maximize meta-awareness; Support sense-making
Student Facing
Han and Clare were running laps around the track. The coach recorded their times at the end of laps 2, 4, 6, and 8.
Han's run:
| distance (laps) | time (minutes) | minutes per lap |
|-----------------|----------------|-----------------|
|        2        |       4        |                 |
|        4        |       9        |                 |
|        6        |       15       |                 |
|        8        |       23       |                 |
Clare's run:
| distance (laps) | time (minutes) | minutes per lap |
|-----------------|----------------|-----------------|
|        2        |       5        |                 |
|        4        |       10       |                 |
|        6        |       15       |                 |
|        8        |       20       |                 |
Is Han running at a constant pace? Is Clare? How do you know?
Write an equation for the relationship between distance and time for anyone who is running at a constant pace.
Student Response
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Anticipated Misconceptions
Students are likely to answer that Clare is running at a constant pace because the minutes per lap shown in the table are the same for each lap. Because we only have four data points in a table, spaced at 5-minute intervals, Clare could still be speeding up and slowing down between the recorded times. However, given the data, it is reasonable to assume Clare is running at a constant pace for the purpose of estimating times or distances.
Activity Synthesis
Invite students to explain why they think each person is or is not running at a constant pace. Point out to students that although the data points in the table for Clare are pairs in a proportional relationship, these four pairs of values do not guarantee that Clare ran at a constant pace. She might have, but we don't know if she was running at a constant pace between the times that the coach recorded.
Ask the following questions:
"Can you represent either relationship with an equation?" (The answer for Han is “no” and the answer for Clare is “yes, if she really ran at a constant pace between the points in time when the times were recorded.” Write the equation for Clare together:
\(t = 2.5 d\)
.)
“Are the pairs of values in the table for Clare's run still values from a proportional relationship if we calculate laps per minute instead of minutes per lap? How does that change the equation?” (Yes,
\(d = 0.4 t\)
.)


correct_option_index: 1