few_shots: 
Example 1:
Concept/skill: 
Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

Problem/activity:
Find the general solution for:
$${\mathrm{cos}\left({\pi\over2}+x\right)=\mathrm{tan}{\pi\over4}}$$


Answer: 
Yes. 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.

Example 2:
Concept/skill: 
Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).

Problem/activity:
Task
Solve the quadratic equation
$$ x^2 = (2x - 9)^2 $$
using as many different methods as possible.


Answer: 
No. This task does not involve an equation in two variables. There is only one variable, x.

Example 3:
Concept/skill: 
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.

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 $P$.


Answer: 
Yes. 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.

Now, determine whether the following problem or activity can enable students to understand the concepts or skills in the provided description.

shot_info: fl_problem_001894_pos_0

HS_0

im_task_000870_neg_0

HS_1

im_task_001160_pos_0

HS_2
standard_description: 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.
problem_activity: Optional activity
This activity provides an opportunity for additional practice in solving equivalent ratio problems. Monitor for students solving the problems in different ways.
Launch
Give students 4 minutes of quiet work time and then have them discuss their solutions with a partner.
Representation: Internalize Comprehension.
Activate or supply background knowledge about equivalent ratios describing situations that happen at the same rate which can be displayed on double number lines. Allow students to use calculators to ensure inclusive participation in the activity.
Supports accessibility for: Memory; Conceptual processing
Writing: MLR5 Co-Craft Questions.
Before asking students to solve the given question, display only Lin's, Diego's and Elena's reading rates. Invite students think of possible mathematical questions about this situation. Keep in mind students do not need to answer their created questions. Students share and revise their questions with a partner and then with the whole class. Record questions shared with the class in a public space. This helps students connect mathematical language to get into the context of the problem as they reason about the three different quantities (i.e., number of pages read, number of pages in the book, and number of days reading).
Design Principle(s): Maximize meta-awareness​
Student Facing
Lin read the first 54 pages from a 270-page book in the last 3 days.
Diego read the first 100 pages from a 325-page book in the last 4 days.
Elena read the first 160 pages from a 480-page book in the last 5 days.
If they continue to read every day at these rates, who will finish first, second, and third? Explain or show your reasoning.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Student Facing
Are you ready for more?
The ratio of cats to dogs in a room is
\(2:3\)
. Five more cats enter the room, and then the ratio of cats to dogs is
\(9:11\)
. How many cats and dogs were in the room to begin with?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Activity Synthesis
Select students to present their solutions. Sequence solutions with diagrams first and then tables. Make sure students see connections between the different representations and ways of solving the problems.


aligns: no