few_shots: 
Example 1:
Problem/activity:
Find the general solution for:
$${\mathrm{cos}\left({\pi\over2}+x\right)=\mathrm{tan}{\pi\over4}}$$


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.

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:
Problem/activity:
Task
Solve the quadratic equation
$$ x^2 = (2x - 9)^2 $$
using as many different methods as possible.


Concept/skill: 
Solve quadratic equations in one variable.

Answer: 
Yes. This task shows one quadratic equation with one variable, x.

Example 3:
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 $P$.


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.

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, does the following problem or activity enable students to learn the full intent of the following concept or skill?

shot_info: fl_problem_001894_neg_0

HS_0

im_task_000870_pos_0

HS_1

im_task_001160_pos_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: Warm-up
The purpose of this warm-up is for students to jump back into recognizing functions and determining if two quantities are a function of each other. Students are asked questions similar to these throughout this lesson, and the discussion of this warm-up is meant to get students using the language of functions, which continues throughout the rest of the activities.
Launch
Give students 1–2 minutes of quiet work time, followed by a whole-class discussion.
Student Facing
Here is a graph of the amount of gas burned during a trip by a tractor-trailer truck as it drives at a constant speed down a highway:
[IMG]
At the end of the trip, how far did the truck drive, and how much gas did it use?
If a truck traveled half this distance at the same rate, how much gas would it use?
If a truck traveled double this distance at the same rate, how much gas would it use?
Complete the sentence: ___________ is a function of _____________.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Anticipated Misconceptions
If students struggle with the second and third questions, ask them if this is an example of a proportional relationship. Remind them what they learned previously about proportional relationships and how that can be used to answer these questions.
Activity Synthesis
Invite students to share their answers and their reasoning for why gas burned is a function of distance traveled. Questions to further the discussion around functions:
“Looking at the graph, what information do you need in order to determine how much gas was used?” (We need to know the number of miles traveled.)
“What is the independent variable? Dependent variable? How can you tell from the graph?” (The independent value is the distance traveled. The dependent value is the gas burned. By convention, the independent is on the
\(x\)
-axis and the dependent is on the
\(y\)
-axis.)
“What are some ways that we can tell from the graph that the relationship between gas burned and distance traveled is proportional?” (The graph is a line that goes through the origin, we can see a constant ratio between
\(y\)
and
\(x\)
in some points like
\((80,10), (160, 20)\)
, and
\((240, 30)\)
.)


aligns: no