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: 
Construct a tangent line from a point outside a given circle to the circle.

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: 
No. This task does not involve drawing lines that are tangent to circles. The only line that may be drawn is the ray between points P and B.

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_neg_0

HS_2
standard_description: Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure.
problem_activity: Activity
Students observe that the angle formed by connecting endpoints of a diameter to a third point on the circle appears to be a right angle. They confirm that this is true for a few particular points, then write a conjecture. This conjecture will be generalized in an upcoming unit.
Monitor for different ways that students word the conjecture to highlight during the discussion. Listen for terms such as
diameter
,
endpoints
,
right triangle
, and
right angle
. The term
chord
will be defined in a subsequent unit. It is not necessary to define it here.
This activity works best when each student has access to devices that can use the embedded applet because students will benefit from seeing the relationship in a dynamic way. If students don’t have individual access, projecting the applet would be helpful during the synthesis.
Digital
Launch
Arrange students in groups of 4. Provide access to internet-enabled devices, ideally one per group. Distribute one index card with each internet-enabled device.
Conversing: MLR2 Collect and Display.
As students work on this activity, listen for and collect the language students use to justify why the angle formed by segments
\(BD\)
and
\(CD\)
is a right angle. Write the students’ words and phrases on a visual display. As students review the visual display, ask them to revise and improve how ideas are communicated. For example, a phrase such as, “The lines make 90 degrees because they have opposite slopes” can be improved by restating it as “Segments
\(BD\)
and
\(CD\)
are perpendicular because their slopes are opposite reciprocals.” This will help students use the mathematical language necessary to precisely justify why the measure of the angle formed by segments
\(BD\)
and
\(CD\)
is 90 degrees.
Design Principle(s): Optimize output (for justification); Maximize meta-awareness
Student Facing
Use the applet to answer the questions.
First, observe and describe the image:
What kind of line is
\(BC\)
in reference to the circle?
What does the measurement of angle
\(BDC\)
appear to be?
Move point
\(D\)
around. How does the measurement of angle
\(BDC\)
appear to change? Use a corner of the index card to compare the size of the angle as it moves.
Move point
\(D\)
to a location with integer coordinates. Each student in the group should choose a different spot for
\(D\)
. Calculate the slopes of segments
\(BC\)
and
\(BD\)
. What do your results tell you?
Now select the button labeled “slopes.” This will show the slopes of
\(BD\)
and
\(BC\)
and their product. Move point
\(D\)
around. What do you notice about the product of the slopes?
Based on the results, write a conjecture that captures what you are seeing.
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 4.
Conversing: MLR2 Collect and Display.
As students work on this activity, listen for and collect the language students use to justify why the angle formed by segments
\(BD\)
and
\(CD\)
is a right angle. Write the students’ words and phrases on a visual display. As students review the visual display, ask them to revise and improve how ideas are communicated. For example, a phrase such as, “The lines make 90 degrees because they have opposite slopes” can be improved by restating it as “Segments
\(BD\)
and
\(CD\)
are perpendicular because their slopes are opposite reciprocals.” This will help students use the mathematical language necessary to precisely justify why the measure of the angle formed by segments
\(BD\)
and
\(CD\)
is 90 degrees.
Design Principle(s): Optimize output (for justification); Maximize meta-awareness
Student Facing
The image shows a circle with several points plotted on the circle.
[IMG]
What kind of segment is
\(BC\)
in reference to the circle?
Choose one of the plotted points on the circle and call it
\(D\)
. Each student in the group should choose a different point. Draw segments
\(BD\)
and
\(DC\)
. What does the measure of angle
\(BDC\)
appear to be?
Calculate the slopes of segments
\(BD\)
and
\(DC\)
. What do your results tell you?
Compare your results to those of others in your group. What did they find?
Based on your group’s results, write a conjecture that captures what you are seeing.
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
Ask previously identified groups to share their conjectures. Challenge students to decide whether all the conjectures are saying the same thing, and if any of the conjectures need to be refined. Encourage students to respectfully disagree, ask for clarification, or point out contradicting information.
There are many ways to correctly word the conjecture. As students work to refine the wording, be sure that the word
diameter
is included, that it’s clear that 2 points are at the endpoints of the diameter, and that the third point is clearly somewhere else on the circle. Students may describe the result as 2 segments that form a right angle or 3 segments that form a right triangle.
Once the conjecture is finalized, ask students if they have proven it (they have not; they’ve just shown it’s true for a few particular cases). Tell students that we will look at a more general case of this assertion in an upcoming unit.
Action and Expression: Develop Expression and Communication.
Invite students to talk about their ideas with a partner. Display sentence frames, such as: “How did you develop your conjecture?” and “I chose this conjecture because. . . .” to support students when they explain their ideas.
Supports accessibility for: Language; Organization


aligns: no