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: 
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_neg_0

HS_1

im_task_001160_neg_0

HS_2
standard_description: Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. For example, estimate the population of the United States as 3 × 108 and the population of the world as 7 × 109, and determine that the world population is more than 20 times larger.
problem_activity: Activity
The purpose of this activity is to give students an opportunity to solve a relatively complicated application problem that requires an understanding of aspect ratio, the Pythagorean Theorem, and realizing that a good way to compare the sizes of two screens is to compare their areas. The previous activities in this lesson are meant to prepare students to understand the situation and suggest some strategies for tackling the problem.
Monitor for students using different approaches and strategies. Students may benefit from more time to think about this problem than is available during a typical class meeting.
Launch
Provide access to calculators that can take the square root of a number. The task statement is wordy, so consider using the Three Reads protocol to ensure students understand what the problem is saying and what it is asking.
We chose not to provide diagrams drawn to scale in the student materials, since it makes it somewhat obvious that the new phone design has a smaller screen area. However if desired, here is an image to show or provide to students:
[IMG]
Reading, Listening, Conversing: MLR6 Three Reads.
Use this routine to support reading comprehension of this word problem. Use the first read to orient students to the situation. Ask students to describe what the situation is about without using numbers (customers think that newly released smartphones have smaller screens). Use the second read to identify quantities and relationships. Ask students what can be counted or measured without focusing on the values (aspect ratios for phones manufactured before and after 2017). After the third read, invite students to brainstorm possible strategies they can use determine whether or not the new phones have a smaller screen, and by how much.
Design Principle(s): Support sense-making
Student Facing
Before 2017, a smart phone manufacturer’s phones had a diagonal length of 5.8 inches and an aspect ratio of
\(16:9\)
. In 2017, they released a new phone that also had a 5.8-inch diagonal length, but an aspect ratio of
\(18.5:9\)
. Some customers complained that the new phones had a smaller screen. Were they correct? If so, how much smaller was the new screen compared to the old screen?
Student Response
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Activity Synthesis
Invite students to share their ideas and progress with the class. If appropriate, students may benefit from an opportunity to clearly present their solution in writing.


aligns: no