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: 
Solve quadratic equations in one variable.

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


Answer: 
Yes. This task shows one quadratic equation with 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_pos_0

HS_1

im_task_001160_pos_0

HS_2
standard_description: Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.
problem_activity: Optional activity
This activity involves checking for a constant of proportionality in tables generated from simple equations that students should already be able to evaluate from their work with expressions and equations in grade 6. The purpose of this activity is to generalize about the forms of equations that do and do not represent proportional relationships. The relationships in this activity are presented without a context so that students can focus on the structure of the equations (MP7) without being distracted by what the variables represent.
Launch
Arrange students in groups of 2. Provide access to calculators. Give students 5 minutes of quiet work time followed by students discussing responses with a partner, followed by whole-class discussion.
Engagement: Internalize Self Regulation.
Chunk this task into more manageable parts to differentiate the degree of difficulty or complexity. Invite students to choose and complete tables for four of the equations. Encourage students to complete tables for the equations
\(y=\frac{x}{4}\)
and
\(y=4x\)
.
Supports accessibility for: Organization; Attention
Student Facing
Here are six different equations.
\(y = 4 + x\)
\(y = \frac{x}{4}\)
\(y = 4x\)
\(y = 4^{x}\)
\(y = \frac{4}{x}\)
\(y = x^{4}\)
[IMG]
Predict which of these equations represent a proportional relationship.
Complete each table using the equation that represents the relationship.
Do these results change your answer to the first question? Explain your reasoning.
What do the equations of the proportional relationships have in common?
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 might struggle to see that the two proportional relationships have equations of the form
\(y = kx\)
and to characterize the others as not having equations of that form. Students do not need to completely articulate this insight for themselves; this synthesis should emerge in the whole-class discussion.
Activity Synthesis
Invite students to share what the equations for proportional relationships have in common, and by contrast, what is different about the other equations. At first glance, the equation
\(y=\frac x4\)
does not look like our standard equation for a proportional relationship,
\(y = kx\)
. Suggest to students that they rewrite the equation using the constant of proportionality they found after completing the table:
\(y = 0.25x\)
which can also be expressed
\(y=\frac14 x\)
. If students do not express this idea themselves, remind them that they can think of dividing by 4 as multiplying by
\(\frac14\)
.
Reading, Writing, Speaking: MLR3 Clarify, Critique,
Correct. Before students share what they identified as similarities between the proportional and nonproportional relationships, present an incorrect explanation. For example, “There are no commonalities among the equations because they’re all different. ” Ask students to identify the error, critique the reasoning, and write a correct description. As students discuss with a partner, listen for students who clarify that equations
\(y = 4x\)
and
\(y=\frac14 x\)
are of the form
\(y = kx\)
and both represent proportional relationships. Invite students to share their critiques and corrected responses with the class. Listen for and amplify the language students use to describe the similarities and differences between equations that do and do not represent proportional relationships. This helps students evaluate, and improve upon, the written mathematical arguments of others, as they clarify how their understanding of proportional relationships and how they can be represented with equations.
Design Principle(s): Maximize meta-awareness


aligns: no