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: Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.
problem_activity: Optional activity
In this activity, the dots are distributed uniformly in the first square but not in the second square:
[IMG]
However, these dots are drawn so it is not too hard to see that if they were redistributed, each square inch would have 8 dots:
[IMG]
The fact that we have 8 dots per square inch means that if we distributed the dots uniformly throughout the square in the right way, we would see 8 dots in each square inch. This prepares students to be able to interpret the constant of proportionality in the next two activities when working with actual houses per square mile or people per square kilometer. When we speak of “500 houses per square kilometer,” we can think of this as, “If we took all of the houses in the region and spread them out uniformly, then we would see 500 houses in every square kilometer.”
Digital
Launch
Display the image of the two squares with dots. Ask students to describe what is the same and what is different about these squares.
Give students 2–3 minutes of quiet work time, followed by 4–5 minutes of partner work time and whole-class discussion.
Action and Expression: Internalize Executive Functions.
Chunk this task into more manageable parts to support students who benefit from support with organizational skills in problem solving. For example, pause to check for understanding after the first 2–3 minutes of work time.
Supports accessibility for: Organization; Attention
Writing, Conversing: MLR5 Co-craft Questions.
Display only the image of the four dot arrays without revealing any of the questions that follow. Give students 1–2 minutes to write a list of possible mathematical questions they could ask about the arrays. Invite students to share their questions with a partner, and then select 2–3 students to share their questions with the whole class. Highlight any questions that refer to how dots are “distributed,” even if students do not use that particular phrase. Finally, reveal the whole problem with text so that students may begin addressing the questions. This helps amplify language related to the distribution of dots.
Design Principle(s): Cultivate conversation; Maximize meta-awareness
Student Facing
The figure shows two arrays, each enclosed by a square that is 2 inches wide.
[IMG]
Let
\(a\)
be the area of the square and
\(d\)
be the number of dots enclosed by the square. For each square, plot a point that represents its values of
\(a\)
and
\(d\)
. Start by choosing an appropriate scale for each axis, using the sliders to adjust the intervals.
Draw lines from
\((0, 0)\)
to each point. For each line, write an equation that represents the proportional relationship.
What is the constant of proportionality for each relationship? What do the constants of proportionality tell us about the dots and squares?
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
Display the image of the two squares with dots. Ask students to describe what is the same and what is different about these squares.
Give students 2–3 minutes of quiet work time, followed by 4–5 minutes of partner work time and whole-class discussion.
Action and Expression: Internalize Executive Functions.
Chunk this task into more manageable parts to support students who benefit from support with organizational skills in problem solving. For example, pause to check for understanding after the first 2–3 minutes of work time.
Supports accessibility for: Organization; Attention
Writing, Conversing: MLR5 Co-craft Questions.
Display only the image of the four dot arrays without revealing any of the questions that follow. Give students 1–2 minutes to write a list of possible mathematical questions they could ask about the arrays. Invite students to share their questions with a partner, and then select 2–3 students to share their questions with the whole class. Highlight any questions that refer to how dots are “distributed,” even if students do not use that particular phrase. Finally, reveal the whole problem with text so that students may begin addressing the questions. This helps amplify language related to the distribution of dots.
Design Principle(s): Cultivate conversation; Maximize meta-awareness
Student Facing
The figure shows two arrays, each enclosed by a square that is 2 inches wide.
[IMG]
Let
\(a\)
be the area of the square and
\(d\)
be the number of dots enclosed by the square. For each square, plot a point that represents its values of
\(a\)
and
\(d\)
.
[IMG]
Draw lines from
\((0, 0)\)
to each point. For each line, write an equation that represents the proportional relationship.
What is the constant of proportionality for each relationship? What do the constants of proportionality tell us about the dots and squares?
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 goal of this discussion is for students to make sense of the constant of proportionality in the case where the dots are not uniformly distributed. Invite students to share their interpretations of the constants of proportionality.
Consider asking questions like:
“Why is one of the constants of proportionality larger than the other? How can we see this in the picture of the squares?” (There are more dots in the same area.)
“What are the units of the constants of proportionality?” (The number of dots per square inch).
Students will have more opportunities to think about this when working on the activities that follow.


aligns: no