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: 
Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.

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: 
Yes. This task asks students to draw a ray that bisects an angle. The act of drawing circles and rays may involve compass and straightedge or dynamic geometric software.

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_pos_0

HS_2
standard_description: Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.
problem_activity: Activity
In previous lessons, data was fit with a linear model, but is not appropriate in all cases. In this activity, students individually sort cards of different scatter plots then discuss with a partner. They then re-sort the cards based on linearity and then again by positive or negative association (MP4).
Launch
Arrange students in groups of 2, and give each student a set of cards. Give 5 minutes of quiet work time for the first question followed by partner discussion. For the final two questions, ask students to take turns sorting the cards by putting one card into one of the categories, then explaining their reasoning for putting it there. After both partners have agreed on the placement of that card, the other partner may repeat these steps to sort one more card. Repeat this process until all cards are sorted.
Tell the students that there is a third category for the last two parts of the activity: neither. It is possible for data to have absolutely no association and appear completely random which would not be a linear association or a non-linear association. It is also possible for data to not have a strictly positive or negative association if, for example, the scatter plot seems to increase and then decrease.
Representation: Internalize Comprehension.
Chunk this task into more manageable parts to differentiate the degree of difficulty or complexity by beginning with fewer cards. For example, give students a subset of the cards to start with and introduce the remaining cards once students have completed their initial set of matches.
Supports accessibility for: Conceptual processing; Organization
Speaking: MLR8 Discussion Supports.
To support students' explanations for sorting the scatter plots in the way they chose, display sentence frames for students to use when they are working with their partner. For example, “I think ____ because _____ .” or “I (agree/disagree) because ______ .”
Design Principle(s): Support sense-making; Optimize output for (explanation)
Student Facing
Your teacher will give you a set of cards. Each card shows a scatter plot.
Sort the cards into categories and describe each category.
Explain the reasoning behind your categories to your partner. Listen to your partner’s reasoning for their categories.
Sort the cards into two categories: positive associations and negative associations. Compare your sorting with your partner’s and discuss any disagreements.
Sort the cards into two categories: linear associations and non-linear associations. Compare your sorting with your partner’s and discuss any disagreements.
Student Response
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Activity Synthesis
To help clarify the meaning of linear and non-linear associations as well as positive and negative associations, consider asking some of the following questions:
“How did you originally group the cards?”
“Which graphs did you sort into the 'positive association' group?” (F and G)
“Are there any cards that did not fit with either positive or negative association?” (Yes, the points for card A form an arc that first goes up and then down.)
“Which graphs did you sort into the 'linear association' group?” (C and E)
“If a scatter plot has no association at all (neither linear nor non-linear), what might its scatter plot look like?” (A lot of random dots scattered all around the plot with no obvious trend.)


aligns: no