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


Concept/skill: 
Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

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


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).

Answer: 
No. This task does not involve an equation in two variables. There is only 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: 
Construct a tangent line from a point outside a given circle to the circle.

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, 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_pos_0

HS_0

im_task_000870_neg_0

HS_1

im_task_001160_neg_0

HS_2
standard_description: Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.
problem_activity: Problem 1
Write each rational number in the form
$${a\over{b}}$$
, where
$$a$$
and
$$b$$
are integers, and
$$b\neq0$$
.
a.
$${0.3333}$$
b.
$${-5.05}$$
c.
$${\sqrt{64}}$$
e.
$${\sqrt{4\over25}}$$
f.
$${\sqrt{1.21}}$$
Problem 2
A student wrote two repeating decimals as fractions using the strategy shown below.
{"table": [{"_attributes": {"class": ["table"], "style": "width:100%"}, "tbody": [{"tr": [{"td": [{"_attributes": {"style": "text-align:center"}, "p": [{"_value": "Given the decimal", "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$$0.\\overline{12}$$"}]}, {"_value": "Let", "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$$x=0.\\overline{12}$$"}]}, {"_value": "Then", "u": [{"span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$$100x=12.\\overline{12}$$"}]}]}, {"img": [{"_attributes": {"alt": "", "src": "https://d37su8u1z9ynie.cloudfront.net/uploads/2019/02/06/math8_u7_l5_ap2a.PNG", "style": "height:113px; width:126px"}}]}]}, {"_attributes": {"style": "text-align:center"}, "p": [{"_value": "Given the decimal", "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$$0.\\overline{123}$$"}]}, {"_value": "Let", "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$$x=0.\\overline{123}$$"}]}, {"_value": "Then", "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$$1000x=123.\\overline{123}$$"}]}, {"img": [{"_attributes": {"alt": "", "src": "https://d37su8u1z9ynie.cloudfront.net/uploads/2019/02/06/math8_u7_l5_ap2b.PNG", "style": "height:114px; width:132px"}}]}]}]}]}]}]}
a.   Describe the student’s strategy in your own words.
b.   Use the same strategy to write the two decimals below as fractions.
$${0.\overline{54}}$$
$${0.1\overline{23}}$$
Problem 3
What is the sum of
$${0.1}$$
and
$${0.\overline3}$$
written as a fraction?


aligns: yes