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: 
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_pos_0

HS_0

im_task_000870_pos_0

HS_1

im_task_001160_pos_0

HS_2
standard_description: Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For example, the function A = s2 giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line.
problem_activity: Optional activity
Previously, students made sense of the domain of functions in concrete contexts. This optional activity is an opportunity to reason about domain more abstractly. Students evaluate an expression that defines a function at some values of input and notice a value that produces an undefined output. They graph the function to examine its behavior, and then think about how to describe the domain of the function.
Launch
Provide access to graphing technology.
Action and Expression: Provide Access for Physical Action.
Provide access to tools and assistive technologies such as a graphing calculator or graphing software. Some students may benefit from a checklist or list of steps to be able to use the calculator or software.
Supports accessibility for: Organization; Conceptual processing; Attention
Student Facing
Consider the function
\(f(x)=\dfrac {6}{x-2}\)
.
To find out the sets of possible input and output values of the function, Clare created a table and evaluated
\(f\)
at some values of
\(x\)
. Along the way, she ran into some trouble.
Find
\(f(x)\)
for each
\(x\)
-value Clare listed. Describe what Clare’s trouble might be.
{"table": [{"_attributes": {"border": "1"}, "tbody": [{"tr": [{"th": [{"_attributes": {"scope": "row"}, "span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "50"}, "_value": "\\(x\\)"}]}], "td": [{"_value": "-10"}, {"_value": "0"}, {"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "238"}, "_value": "\\(\\frac12\\)"}]}, {"_value": "2"}, {"_value": "8"}]}, {"th": [{"_attributes": {"scope": "row"}, "span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "287"}, "_value": "\\(f(x)\\)"}]}], "td": [{}, {}, {}, {}, {}]}]}]}]}
Use graphing technology to graph function
\(f\)
. What do you notice about the graph?
Use a calculator to compute the value you and Clare had trouble computing. What do you notice about the computation?
How would you describe the domain of function
\(f\)
?
Student Response
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for free access to Student Response.
Student Facing
Are you ready for more?
Why do you think the graph of function
\(f\)
looks the way it does? Why are there two parts that split at
\(x=2\)
, with one curving down as it approaches
\(x=2\)
from the left and the other curving up as it approaches
\(x=2\)
from the right?
Evaluate function
\(f\)
at different
\(x\)
-values that approach 2 but are not exactly 2, such as 1.8, 1.9, 1.95, 1.999, 2.2, 2.1, 2.05, 2.001, and so on. What do you notice about the values of
\(f(x)\)
as the
\(x\)
-values get closer and closer to 2?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Activity Synthesis
Display a graph of the function for all to see. Invite students to share their observations of the behavior of function
\(f\)
based on the completed table and their graph. Solicit their ideas on what the problem might be with this function.
If no students mentioned division by 0 as the issue, bring this up. Ask questions such as:
"What happens when we divide a number by 0?" (The result is undefined.)
"In the expression
\(\dfrac {6}{x-2}\)
, what value or values of
\(x\)
would result in a denominator of 0?" (Only 2 gives a denominator of 0.)
"If 2 does not produce an output, is it a possible input for
\(f\)
?" (No)
Highlight that the domain of
\(f\)
includes all numbers except 2.


aligns: no