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: Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.
problem_activity: Activity
The purpose of this task is for students to practice plotting coordinates in all four quadrants and find horizontal and vertical distances between coordinates in a puzzle. In past activities, students have been told the scale for the distance between grid lines, but here they must determine that each grid square has length 2 from the information given.
Digital
Launch
Arrange students in groups of 2. Tell students that they should not assume that each grid box is 1 unit. Give students 10 minutes quiet work time and 2 minutes for partner discussion. Follow with whole-class discussion.
Students using digital materials will be able to create a path through the maze by plotting points with an applet.
Representation: Internalize Comprehension.
Check in with students after the first 2–3 minutes of work time. Check to make sure students have selected appropriate coordinates for the first points of Andre’s route through the maze.
Supports accessibility for: Conceptual processing; Organization
Conversing: MLR5 Co-craft Questions.
Display only the picture of the maze and ask pairs of students to write possible mathematical questions about the situation. This is an opportunity for students to think about and relate to previous questions from previous lessons. Then, invite pairs to share their questions with the class. This helps students produce the language of mathematical questions and talk about the coordinate grid and the points on the maze.
Design Principle(s): Optimize output (for explanation); Maximize meta-awareness
Student Facing
The following diagram shows Andre’s route through a maze. He started from the lower right entrance.
[IMG]
What are the coordinates of the first two and the last two points of his route?
How far did he walk from his starting point to his ending point? Show how you know.
Jada went into the maze and stopped at
\((\text-7, 2)\)
.
Plot that point and other points that would lead her out of the maze (through the exit on the upper left side).
How far from
\((\text-7, 2)\)
must she walk to exit the maze? Show how you know.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Print
Launch
Arrange students in groups of 2. Tell students that they should not assume that each grid box is 1 unit. Give students 10 minutes quiet work time and 2 minutes for partner discussion. Follow with whole-class discussion.
Students using digital materials will be able to create a path through the maze by plotting points with an applet.
Representation: Internalize Comprehension.
Check in with students after the first 2–3 minutes of work time. Check to make sure students have selected appropriate coordinates for the first points of Andre’s route through the maze.
Supports accessibility for: Conceptual processing; Organization
Conversing: MLR5 Co-craft Questions.
Display only the picture of the maze and ask pairs of students to write possible mathematical questions about the situation. This is an opportunity for students to think about and relate to previous questions from previous lessons. Then, invite pairs to share their questions with the class. This helps students produce the language of mathematical questions and talk about the coordinate grid and the points on the maze.
Design Principle(s): Optimize output (for explanation); Maximize meta-awareness
Student Facing
The following diagram shows Andre’s route through a maze. He started from the lower right entrance.
[IMG]
What are the coordinates of the first two and the last two points of his route?
How far did he walk from his starting point to his ending point? Show how you know.
Jada went into the maze and stopped at
\((\text-7, 2)\)
.
Plot that point and other points that would lead her out of the maze (through the exit on the upper left side).
How far from
\((\text-7, 2)\)
must she walk to exit the maze? Show how you know.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Activity Synthesis
The key idea is that it is possible to find distances and describe situations involving movement using the coordinate plane. This abstraction is important to appreciate because it means we can use numbers (in this case,
pairs
of numbers in the coordinate plane) to model situations that involve distance or movement. This will play a key role in later studies.To highlight these ideas, consider asking:
How were you or your partner able to find the coordinates in the maze? Did you come up with any strategies or shortcuts?
How did you find the distances that Andre and Jada traveled?
What other situations involving movement could be represented with a coordinate plane?
Students may come up with examples like board games, maps, and perhaps even 3-dimensional examples.


aligns: yes