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 problem or activity shown below 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: Multiply a vector by a scalar.
problem_activity: Task
Complete the table below. Is $\Delta x$ a constant?  If so, what constant is it? What do you notice about the 3rd column of the table?
{"table": [{"tbody": [{"tr": [{"th": [{"_value": "$x$"}, {"_value": "$f(x)=128\\cdot \\left({\\frac {1}{2}}\\right)^x$"}, {"_value": "Successive quotients"}]}, {"td": [{"_value": "0"}, {"_value": "128"}, {"_value": "----"}]}, {"td": [{"_value": "1"}, {"_value": "64"}, {"_value": "$\\frac{f(1)}{f(0)}=\\frac {64}{128} = \\frac{1}{2}$"}]}, {"td": [{"_value": "2"}, {}, {"_value": "$\\frac{f(2)}{f(1)}=$"}]}, {"td": [{"_value": "3"}, {}, {}]}, {"td": [{"_value": "4"}, {}, {}]}]}]}]}
Complete the table below. Is $\Delta x$ a constant?  If so, what constant is it? What do you notice about the 3rd column of the table?
{"table": [{"tbody": [{"tr": [{"th": [{"_value": "$x$"}, {"_value": "$f(x)=128\\cdot \\left({\\frac {1}{2}}\\right)^x$"}, {"_value": "Successive quotients"}]}, {"td": [{"_value": "0"}, {"_value": "128"}, {"_value": "---"}]}, {"td": [{"_value": "2"}, {"_value": "32"}, {"_value": "$\\frac{f(2)}{f(0)} = \\frac {32}{128} = \\frac{1}{4}$"}]}, {"td": [{"_value": "4"}, {}, {"_value": "$\\frac{f(4)}{f(2)}=$"}]}, {"td": [{"_value": "6"}, {}, {}]}, {"td": [{"_value": "8"}, {}, {}]}]}]}]}
Let $f(x)=a \cdot b^x$. Let $x_0$ be any particular $x$-value. Show that if $x_0$ is increased by a constant, $\Delta x$, the successive quotient  $$\frac{f(x_0+ \Delta x)}{f(x_0)}$$ is always the same no matter what $x_0$ is.
Is b) an example of the result of c)? Explain.


aligns: no