few_shots: 
Example 1:
Problem/activity:
Find the general solution for:
$${\mathrm{cos}\left({\pi\over2}+x\right)=\mathrm{tan}{\pi\over4}}$$


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

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


Concept/skill: 
Solve quadratic equations in one variable.

Answer: 
Yes. This task shows one quadratic equation with one variable, x.

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 $P$.


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.

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, does the problem or activity shown below enable students to learn the full intent of the following concept or skill?

shot_info: fl_problem_001894_pos_0

HS_0

im_task_000870_pos_0

HS_1

im_task_001160_pos_0

HS_2
standard_description: Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change.
problem_activity: Optional activity
In this activity, students continue their work comparing properties of functions represented in different ways. Students are given a verbal description and a table to compare and decide whose family traveled farther over the same time intervals. The purpose of this activity is for students to continue building their skill interpreting and comparing functions.
Launch
Give students 3–5 minutes of quiet work time, followed with whole-class discussion.
Student Facing
Elena’s family is driving on the freeway at 55 miles per hour.
Andre’s family is driving on the same freeway, but not at a constant speed.  The table shows how far Andre's family has traveled,
\(d\)
, in miles, every minute for 10 minutes.
{"table": [{"_attributes": {"border": "1", "cellpadding": "1", "cellspacing": "1", "style": "width:500px"}, "tbody": [{"tr": [{"th": [{"_attributes": {"scope": "row"}, "span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "166"}, "_value": "\\(t\\)"}]}], "td": [{"_value": "1"}, {"_value": "2"}, {"_value": "3"}, {"_value": "4"}, {"_value": "5"}, {"_value": "6"}, {"_value": "7"}, {"_value": "8"}, {"_value": "9"}, {"_value": "10"}]}, {"th": [{"_attributes": {"scope": "row"}, "span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "292"}, "_value": "\\(d\\)"}]}], "td": [{"_value": "0.9"}, {"_value": "1.9"}, {"_value": "3.0"}, {"_value": "4.1"}, {"_value": "5.1"}, {"_value": "6.2"}, {"_value": "6.8"}, {"_value": "7.4"}, {"_value": "8"}, {"_value": "9.1"}]}]}]}]}
How many miles per minute is 55 miles per hour?
Who had traveled farther after 5 minutes? After 10 minutes?
How long did it take Elena’s family to travel as far as Andre’s family had traveled after 8 minutes?
For both families, the distance in miles is a function of time in minutes. Compare the outputs of these functions when the input is 3.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Anticipated Misconceptions
Students may miss that Elena’s family’s speed has different units than what is needed to compare with Andre’s family. Point out to students that the units are important and possibly ask them to find out how many miles Elena’s family travels in 1 minute rather than 1 hour.
Activity Synthesis
The purpose of this discussion is for students to think about how they use a verbal description and table to answer questions related to the context. Ask students to share their solutions and how they used the equation and graph. Consider asking some of the following questions:
“How did you use the table to get information? How did you use the verbal description?”
“What did you prefer about using the description to solve the problem? What did you prefer about using the table to solve the problem?”
Engagement: Develop Effort and Persistence.
Break the class into small discussion groups and then invite a representative from each group to report back to the whole class.
Supports accessibility for: Language; Social-emotional skills; Attention
Writing, Speaking: MLR1 Stronger and Clearer Each Time.
Use this routine to give students a structured opportunity to reflect on their problem-solving strategies. Ask students to write an initial response to the prompt, “What did you prefer about using the description to solve the problem? What did you prefer about using the table to solve the problem?” Give students time to meet with 2–3 partners, to share and get feedback on their responses. Provide prompts for feedback that will help students strengthen their ideas and clarify their language (e.g., “What do you mean?”, “Can you give an example?”, and “Can you say that another way?”, etc.). Give students 1–2 minutes to revise their writing based on the feedback they received.​ This will help students produce a written generalization about interpreting and comparing functions.
Design Principle(s): Optimize output (for generalization)


aligns: yes