few_shots: 
Example 1:
Problem/activity:
Find the general solution for:
$${\mathrm{cos}\left({\pi\over2}+x\right)=\mathrm{tan}{\pi\over4}}$$


Concept/skill: 
Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

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


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).

Answer: 
No. This task does not involve an equation in two variables. There is only one variable, x.

Example 3:
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 $P$.


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

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, does the problem or activity shown below enable students to learn the full intent of the following concept or skill?

shot_info: fl_problem_001894_pos_0

HS_0

im_task_000870_neg_0

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im_task_001160_neg_0

HS_2
standard_description: Recognize and represent proportional relationships between quantities.
problem_activity: Optional activity
In this activity, students measure lengths and determine possibilities for actual lengths. There are two layers of attending to precision (MP6) involved in this task:
Deciding how accurately the pencils can be measured, probably to the nearest mm or to the nearest 2 mm, but this depends on the eyesight and confidence of the student
Finding the possible percent error in the measurement chosen
Launch
Arrange students in groups of 2. Provide access to calculators. Give students 4–5 minutes of quiet work time, followed by partner and whole-class discussion.
Writing, Speaking: MLR8 Discussion Supports.
To support students as they respond to “How accurate are your estimates?”, provide a sentence frame such as: “My estimate is within ___ mm of the actual length because . . . ." Encourage students to consider what details are important to share and to think about how they will explain their reasoning using mathematical language. This will help students use mathematical language as they justify the accuracy of their estimates.
Design Principle(s): Optimize output (for explanation)
Student Facing
[IMG]
Estimate the length of each pencil.
How accurate are your estimates?
For each estimate, what is the largest possible percent error?
Student Response
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Anticipated Misconceptions
Some students may think that they can find an exact value for the length of each pencil. Because the pictures of the pencils are far enough away from the ruler, it requires a lot of care just to identify the “nearest” millimeter (or which two millimeter markings the length lies between). Prompt them to consider the error in their measurements by asking questions like, “Assuming you measured the pencil accurately to the nearest millimeter, what is the longest the actual length of the pencil could be? What is the shortest it could be?”
Some students may not remember how to calculate percent error. Ask them, “What is the biggest difference possible between the estimated and actual lengths? What percentage of the actual length would that be when the difference is as big as possible?”
Activity Synthesis
The goal of this discussion is for students to practice how they talk about precision.
Discussion questions include:
“How did you decide how accurately you can measure the pencils?” (I looked for a value that I was certain was less than the length of the pencil and a value that I was certain was bigger. My estimate was halfway in between.)
“Were you sure which mm measurement the length is closest to?” (Answers vary. Possible responses: Yes, I could tell that the short pencil is closest to 5.4 cm. No, the long pencil looks to be closest to 17.7 mm, but I’m not sure. I am sure it is between 17.6 cm and 17.8 cm.)
“Were the percent errors the same for the small pencil and for the long pencil? Why or why not?” (No. I was able to measure each pencil to within 1 mm. This is a
smaller
percentage of the longer pencil length than it is of the smaller pencil length.)
Other possible topics of conversation include noting that the level of accuracy of a measurement depends on the measuring device. If the ruler were marked in sixteenths of an inch, we would only be able to measure to the nearest sixteenth of an inch. If it were only marked in cm, we would only be able to measure to the nearest cm.


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