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: Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation.
problem_activity: Activity
In the previous activity, students commented on distributions on a dot plot and had opportunities to hear different ways to describe distributions. In this activity, they begin to think about how to characterize a distribution as a whole, in terms of its
center
and
spread
. Students learn that we can give a general description of a distribution by identifying a location that could be the center of the data, and by noting how alike or different the data points are.
Note that at this point “center” and “spread” can only be identified and described informally, via visual observation and intuitive reasoning about how data points are distributed. The lack of precise ways to identify center and spread helps to cultivate the need for more formal measures later.
In this activity, students evaluate and critique another's reasoning (MP3).
Launch
Keep students in groups of 2. Ask students to read the question in the stem, “How many hours do you generally spend on homework each week?” Then, ask students to explain why this is a statistical question. This is a statistical question because it can be answered by collecting data, and we can expect variability in the data.
Give them 6–7 minutes of quiet time to work on the first three questions, and then 4–5 minutes to share their responses with their partner and to complete the last two questions together. Be sure to leave at least 5 minutes for a whole-class discussion.
Representation: Internalize Comprehension.
Activate or supply background knowledge about calculating percentages. Allow students to use calculators to ensure inclusive participation in the activity.
Supports accessibility for: Memory; Conceptual processing
Student Facing
Twenty-five sixth-grade students answered the question: “How many hours do you generally spend on homework each week?”
This dot plot shows the number of hours per week that these 25 students reported spending on homework.
[IMG]
[IMG]
Use the dot plot to answer the following questions. For each, show or explain your reasoning.
What percentage of the students reported spending 1 hour on homework each week?
What percentage of the students reported spending 4 or fewer hours on homework each week?
Would 6 hours per week be a good description of the number of hours this group of students spends on homework per week? What about 1 hour per week? Explain your reasoning.
What value do you think would be a good description of the homework time of the students in this group? Explain your reasoning.
Someone said, “In general, these students spend roughly the same number of hours doing homework.” Do you agree? Explain your reasoning.
Student Response
Teachers with a valid work email address can
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for free access to Student Response.
Anticipated Misconceptions
Students may not recall how to find a percentage. Remind them that a percentage can be found if we know the size of a part and that of a whole. If needed, prompt them to determine the size of the entire data set.
Activity Synthesis
Ask a couple of students to share their thinking on what number of hours would be a good description of homework time for this group of students and why. Then, select a couple of other students to discuss how alike or different the lengths of homework time are.
Explain that sometimes it helps to describe a set of data generally, or to characterize it as a whole. One way to do that is by finding a value that describes the whole set reasonably well, or a value whose location can be considered the “center” of a distribution. Ask students:
“Where on the dot plot would you consider the center of the data? What is the value of that center?”
Another way to characterize a data set is by describing its “spread,” or the variability in the data points. The wider the spread (the more dispersed the data points are on a dot plot), the more variable or different they are. The narrower the spread (the more clustered together), the more alike they are. Ask students:
“Based on the dot plot, how would you describe the spread of the students’ homework time? Are the amounts of time they spend on homework alike or different?”
Tell students that we will continue to explore center and spread in upcoming lessons.
Representing, Writing: MLR7 Compare and Connect.
Ask pairs of students to prepare a display for a summary statement that generalizes the the number of hours spent on homework each week using the sentence frame, “Based on this data set, spending ___ hours on homework each week is typical for 6th graders because…”. Note a handful of sentences to highlight in the discussion. Give students opportunities to compare their statements with those shared and agree or disagree with each sentence, providing reasons for their agreement or disagreement supported by the data. Give students 1-2 minutes to revise their statements prior to collecting them for display. This will help students to develop conceptual understanding for “typical.”
Design Principle(s): Maximize meta-awareness, Cultivate conversation


aligns: no