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 Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.
problem_activity: Activity
The purpose of this task is for students to use the coordinates of a unit circle to determine information about coordinates on circles of different sizes. Returning to the context of clock hands, this activity focuses on the location of the end of a minute hand relative to the center of the clock.
While there are 3 different clock sizes in this activity, students taking advantage of the similarity between the circles (MP7) can use their solutions for the first clock to identify the solutions for the second clock that is 3 times the size of the first.
Monitor for students who express their answers using:
approximate values based on the unit circle display
exact values using
\(\cos(\theta)\)
and
\(\sin(\theta)\)
Launch
If needed, provide students with copies of separate clock faces.
Student Facing
Here is a clock face.
[IMG]
The length of the minute hand on a clock is 5 inches and the center of the clock is at
\((0,0)\)
on a coordinate plane. Determine the coordinates of the end of the minute hand at the following times. Explain or show your reasoning.
45 minutes after the hour
10 minutes after the hour
40 minutes after the hour
The minute hand on another clock, also centered at
\((0,0)\)
, has a length of 15 inches. Determine the coordinates of the end of the minute hand at the following times. Explain or show your reasoning.
45 minutes after the hour
10 minutes after the hour
40 minutes after the hour
At a certain time, the end of the minute hand of a third clock centered at
\((0,0)\)
has coordinates approximately
\((7.5,7.5)\)
. How long is the minute hand of the clock if each grid square is one inch by one inch? Explain or show your reasoning.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Student Facing
Are you ready for more?
The center of a clock is
\((0,0)\)
on a coordinate grid. Its hour hand is half the length of its minute hand. The coordinates of the end of the hour hand are about
\((3,0.5)\)
. What are the approximate coordinates of the end of the minute hand? Explain how you know.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Activity Synthesis
Select students to share their strategies for identifying the coordinates on the clocks with a minute hand of length 5 inches and 15 inches in the order listed in the Activity Narrative. If no students wrote their answers using exact values, such as
\(\left(5\cos\left(\frac \pi 6\right), 5\sin\left(\frac \pi 6\right)\right)\)
for the 5 inch minute hand 10 minutes after the hour, ask students to consider how they could write the exact coordinates. After a brief quiet think time, invite students to share their thinking, highlighting any callbacks to their previous work identifying exact values by using cosine and sine without approximating. If needed, here are some questions for discussion:
“What angle does the minute hand make, with the positive
\(x\)
-axis, at 10 minutes after the hour?” (
\(\frac \pi 6\)
radians.)
“If the minute hand was 1 foot long, could you use sine and cosine to express the coordinates of its tip?” (The coordinates, in feet, are
\(\left(\cos\left(\frac \pi 6\right), \sin\left(\frac \pi 6\right)\right)\)
.)
“What about when the minute hand is 5 inches long? 15 inches long?” (The coordinates, in inches, are
\(\left(5\cos\left(\frac \pi 6\right), 5\sin\left(\frac \pi 6\right)\right)\)
and
\(\left(15\cos\left(\frac \pi 6\right),1 5\sin\left(\frac \pi 6\right)\right)\)
.)
Conclude the discussion by selecting students to share how they calculated the length of the minute hand when the coordinates of the end of the minute hand are known. Make sure to highlight these strategies by inviting students to consider each and share a reason why they prefer one over the other:
use the Pythagorean Theorem
recognize the angle as
\(\frac{\pi}{4}\)
radians due to the
\(x\)
- and
\(y\)
-coordinates matching
Representation: Internalize Comprehension.
Use color and annotations to illustrate student thinking. As students share their strategies for identifying the coordinates on the clocks, scribe their thinking on a visual display. Highlight the exact coordinates (expressions using cosine and sine) and the approximate coordinates in the same color. Use annotation to draw attention to the scale factors of 5 and 15 in the expressions.
Supports accessibility for: Visual-spatial processing; Conceptual processing


aligns: no