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: Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers. For example, recognize an incorrect result 2/5 + 1/2 = 3/7, by observing that 3/7 < 1/2.

Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators. For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12. (In general, a/b + c/d = (ad + bc)/bd.)
problem_activity: Narrative
The purpose of this activity is for students to think about different activities they might participate in during free time, to define a smaller set of categories, and to collect data. A significant part of this activity is the definition of the categories, which is an important aspect of mathematical modeling (MP4). Encourage students to convert the time for their activities from minutes to hours. Students may notice that some fractions are more meaningful in the context of time such as half, third, and quarter of an hour. Students will further explore the relationship between time and fractions in the activity synthesis.
If a student sees that a specific activity is not on the poster, have them consider if there is an umbrella activity. For example, “What category, that is on a poster, might talking to your friends go under?”
Engagement: Develop Effort and Persistence.
Invite students to generate a list of shared expectations for group work. Record responses on a display and keep visible during the activity.
Supports accessibility for: Social-Emotional Functioning, Attention
Launch
Groups of 4
“What are some things you do or like to do if you have free time on the weekend?” (sleep, play video games, play sports, spend time with family, social media, read, go to the movies)
Activity
“For this activity, you’ll think about how you might spend 2 hours of free time and then we’ll record the time we spend for some of those activities.”
“As you work, I will write down the different categories of your responses on the posters hanging around the room. You will record your times on the posters.”
As students work, monitor for 4–6 popular activities and write the titles of these activities on the large pieces of poster paper. If necessary, list some examples of the umbrella category.
8–10 minutes: independent work time
Monitor for students who record using multiples of
\(\frac{1}{2}\)
,
\(\frac{1}{4}\)
, and
\(\frac{1}{8}\)
of an hour.
Student Facing
Imagine on the weekend you have 2 hours of free time that you can spend any way you like.
How would you spend it? Record your answer in fractions of an hour. Show your reasoning.
Record the time for each activity from your list on the appropriate poster, if there is a category for it.
Student Response
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Activity Synthesis
“Look at the data of the different categories around the room. What do you notice?” (Most people recorded using halves and fourths. More students play games than read books. There is more data on one poster than on another. There are more small numbers on one poster, it’s hard to compare because there are so many numbers and they are not in any order.)
If it doesn’t come up, ask, “Why did most people record time as multiples of
\(\frac{1}{2}\)
and
\(\frac{1}{4}\)
?” (Because half of 60 minutes is 30 and half of that is 15. I can easily see that on the clock.)
“Does it make sense to talk about time using
\(\frac{1}{8}\)
or
\(\frac{1}{7}\)
of an hour?” (Yes but it’s not a whole number of minutes like
\(\frac{1}{2}\)
hour or
\(\frac{1}{4}\)
hour.)
“What are some ways we can organize the data?” (Bar graph, line plot, chart, list the numbers in order.)


correct_option_index: 0