few_shots: 
Example 1:
Concept/skill: 
Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.

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


Answer: 
No. This problem does not directly teach how side ratios in right triangles lead to definitions of trigonometric ratios. Instead, it involves solving a trigonometric function.

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_neg_0

HS_0

im_task_000870_neg_0

HS_1

im_task_001160_pos_0

HS_2
standard_description: Understand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category. For example, all rectangles have four right angles and squares are rectangles, so all squares have four right angles.
problem_activity: Activity
The purpose of this activity is to show how a context can impose restrictions on the domain and range of a function. Given a description of a context, students determine the restrictions on the domain and range imposed by the context. Considering how a context restricts the domain or range of a function modeling it is an important part of mathematical modeling (MP4). This prepares students to interpret exponential functions and make sense of whether the graph should be discrete or continuous in the associated Algebra 1 lesson.
Launch
Remind students that domain refers to possible values of the independent variable, and range refers to possible values of the dependent variable. Use a few examples from the previous activity to illustrate. For example, for the cab ride, the domain is all numbers that are greater than 0, and the range is all numbers that are greater than 3.5. This presumes that a ride must have some distance (it can’t be 0 miles long), and can be any number of miles long. A modeler might decide not to consider any cab rides that are longer than, say, 100 miles, in which case the domain would be all numbers between 0 and 100. Sometimes restrictions on the domain are a decision made by the modeler. In contrast, for the function modeling the admission price to the park based on number of people, the domain only contains whole, positive numbers, since the number of people must be a whole, positive number.
Arrange students in groups of 2. After a few minutes of quiet work time, ask students to compare their responses to their partner’s and decide if they are both correct, even if they look different. Follow with a whole-class discussion.
Student Facing
To make sense in a given context, many functions need restrictions on the domain and range. For each description of a function
describe the domain and range
describe what its graph would look like (separate dots, or connected?)
weight of a puppy as a function of time
number of winter coats sold in a store as a function of temperature outside
number of books in a library as a function of number of people who live in the community the library serves
height of water in a tank as a function of volume of water in the tank
amount of oxygen in the atmosphere as a function of elevation above or below sea level
thickness of a folded piece of paper as a function of number of folds
Student Response
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Activity Synthesis
Much discussion takes place between partners. Invite students to share how they determined the domain and range of the function representing the context. Ask if other groups have different responses. Choose as many different groups as time allows. Attend to the language that students use to describe their pairing, giving them opportunities to describe their relationships more precisely.


aligns: no