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 rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., p^(2)). For example, by truncating the decimal expansion of Ö2, show that Ö2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations.
problem_activity: Activity
The mathematical purpose of this activity is for students to collect, represent, and interpret data using a histogram. For most groups of high school students, the distribution of handspans should be approximately normal. If the distribution for your group is not approximately normal, display the distribution from the student response provided in these materials as the data from another class and ask students to comment on that distribution.
Making statistical technology available gives students an opportunity to choose appropriate tools strategically (MP5).
Launch
Tell students that they will measure the distance from their thumb to smallest finger when their hand is stretched, rounded to the nearest tenth of a centimeter, and then tell you the measurement.
Speaking, Reading: MLR5 Co-Craft Questions.
Use this routine to orient students to the situation. Display the image and the first sentence of this problem, without revealing the questions that follow. Ask students to create a list of mathematical questions that could be asked about the situation. Invite students to compare their questions before revealing the remainder of the task. Listen for and amplify any questions related to how far students’ hands can reach, along with those curious about organizing and interpreting data from the class. This will help students create the language of mathematical questions before feeling pressure to produce solutions.
Design Principle(s): Maximize meta-awareness; Support sense-making
Representation: Access for Perception.
Read the directions for taking the measurements aloud. Demonstrate measuring your own hand and how to estimate the measurement to the nearest tenth of a centimeter. Students who both listen to and read the information will benefit from extra processing time.
Supports accessibility for: Language; Conceptual processing
Student Facing
[IMG]
On many piano keyboards, the distance from one white key to the next is 2.39 centimeters. How many of your classmates could reach two notes that are 9 keys apart (21.5 cm) on a keyboard using only one hand?
Stretch your fingers apart as wide as you can and measure the farthest distance from your thumb to smallest finger. Round your measurement to the nearest tenth of a centimeter.
Your teacher will collect the measurements from the class. Draw a dot plot or histogram from the class data.
[IMG]
Describe the distribution you drew using terms such as: symmetric, approximately symmetric, skewed left, skewed right, approximately uniform, uniform, bell-shaped, or bimodal. Estimate the center of your distribution.
How would you use your distribution to determine how many people in the class can reach the two notes 9 keys apart?
Student Response
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Activity Synthesis
The purpose of this discussion is for students to recognize an approximately symmetric and bell-shaped histogram in context. Display a histogram representing the class data for all to see. If possible, display the distributions from other classes and combine the measurements from all the classes to see a distribution with more data. Select students to share their descriptions of the distributions they drew. Ask each student selected to explain their reasoning for that shape.
Here are some questions for discussion:
“What is the shape of the distribution?” (Approximately symmetric and bell-shaped.)
“What is the center of the distribution? Explain your reasoning.” (It is about 21 cm because the two middle bars appear in the center of the data and the middle of them is approximately 21 cm.)
“How did you estimate how many students could reach the keys?” (I found the total number of students with a measurement of 21.5 cm or more.)
“Do you think having a measurement of 21.5 cm or above would allow you to play the notes on the keyboard? Explain your reasoning.” (I do not think that everyone with a measurement of 21.5 cm would be able to play the notes because they might not have the same stretch while they are actually playing.)
“Does the data look like it could be modeled with a normal distribution?” (Yes. I think a normal distribution would model it well because the data is bell-shaped and approximately symmetric.)
Sketch the normal curve on the classroom data for all to see (either by hand or with technology). Ask, “Does the normal curve model the data well?” (Yes, it models the data well, but not exactly.) Note: for this discussion, the sample data provided should be sufficient. If you choose to include class data, you can do so by changing entries in Column A, but do not delete any.


aligns: no