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: Evaluate reports based on data.
problem_activity: Activity
The mathematical purpose of this activity is to give students a chance to practice finding data displays that represent the distribution of the same data set and using precise vocabulary for describing the shape of the distributions while taking turns matching cards. Students trade roles explaining their thinking and listening, providing opportunities to explain their reasoning and critique the reasoning of others (MP3).
Launch
Arrange students in groups of 2. Display the images of the dot plot and histogram. Ask students what they notice and wonder.
[IMG]
[IMG]
If it does not come up, help students notice that these two data displays show the same data in different formats. They can also be described as skewed right.
Give each group a set of cut-up cards. Explain that a match is two different displays that represent the distribution of the same set of data. Ask students to take turns: the first partner identifies a match, explains why they think it is a match, then describes the distribution while the other student listens and works to understand. Then they switch roles.
Student Facing
Take turns with your partner matching 2 different data displays that represent the distribution of the same set of data.
For each set that you find, explain to your partner how you know it’s a match.
For each set that your partner finds, listen carefully to their explanation. If you disagree, discuss your thinking and work to reach an agreement.
When finished with all ten matches, describe the shape of each distribution.
Student Response
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for free access to Student Response.
Anticipated Misconceptions
For students having trouble with the uniform distribution histograms, remind them that the lower bound for each interval is included and the upper bound is not. Ask them why this might change the last bar in each of these histograms. Some students may not know where to start to match data displays. You can tell them to look at the lowest and highest values as a starting point to finding similarities between two representations.
Activity Synthesis
Once all groups have completed the matching, discuss the following:
“Which matches were tricky? Explain why.” (The uniform distributions may be difficult.)
“What vocabulary was useful to describe the shape of the distribution?” (symmetric, skewed, uniform, bimodal, bell-shaped)
“Were there any matches that could be described by more than one of these vocabulary terms?” (Yes, symmetric or skewed can also be used with some of the other terms for some of the distributions.)
If necessary, ask students to revoice less formal descriptions of the shape of the distribution using formal language including:
Symmetric distribution
Skewed distribution
Uniform distribution
Bimodal distribution
Bell-shaped distribution
Speaking, Listening: MLR 7 Compare and Connect.
Ask students to prepare a visual display of their sorted cards. As students investigate each others’ work, ask students to share what worked well in a particular approach. Listen for and amplify any comments about the use of the words symmetric, skewed, bimodal, bell-shaped, and uniform to compare the two different displays. Then encourage students to make connections between the various ways to describe the different distributions. Revoice language students use to explain that the two different displays are equivalent. This will foster students’ meta-awareness and support constructive conversations as they compare and connect data displays that represent the same distribution.
Design Principle(s): Cultivate conversation; Maximize meta-awareness
Action and Expression: Internalize Executive Functions.
Provide students with a graphic organizer with the phrase “shapes of distributions” at the center, connecting to it all the related concepts mentioned during the discussion.
Supports accessibility for: Language; Organization


aligns: no