few_shots: 
Example 1:
Concept/skill: 
Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

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


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:
Concept/skill: 
Solve quadratic equations in one variable.

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


Answer: 
Yes. This task shows one quadratic equation with 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_pos_0

HS_0

im_task_000870_pos_0

HS_1

im_task_001160_pos_0

HS_2
standard_description: Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.
problem_activity: Optional activity
This activity is optional. Use this activity to allow students to practice analyzing graphs of exponential functions involving
\(e\)
and setting an appropriate graphing window on their graphing technology in order to produce useful graphs.
Students have graphed exponential functions and analyzed the graphs to solve problems prior to this point. Though the functions here use base
\(e\)
, the analytical work is fundamentally the same as that when the base is another number. Students will also have opportunities to graph exponential functions with base
\(e\)
later in the unit.
Launch
Provide access to devices that can run Desmos or other graphing technology.
Writing, Listening, Conversing: MLR1 Stronger and Clearer Each Time.
Use this routine to help students improve their written responses to the first question. Give students time to meet with 2–3 partners to share and receive feedback on their responses. Display feedback prompts that will help students strengthen their ideas and clarify their language. For example, “How did you get . . . ?”, “Why did you . . . ?”, and “How do you know . . . ?” Invite students to go back and revise or refine their written responses based on the feedback from peers. This will help students justify their reasoning for the population of bacteria 10 hours after the scientist put the initial bacteria in the dish.
Design Principle(s): Optimize output (for justification); Cultivate conversation
Student Facing
Use graphing technology to graph the function defined by
\(f(t) = 50 \boldcdot e^{(0.25t)}\)
. Adjust the graphing window as needed to answer these questions:
The function
\(f\)
models the population of bacteria in
\(t\)
hours after it was initially measured. About how many bacteria were in the dish 10 hours after the scientist put the initial 50 bacteria in the dish?
About how many hours did it take for the number of bacteria in the dish to double? Explain or show your reasoning.
Use graphing technology to graph the function defined by
\(p(t)=192 \boldcdot e^{(0.014t)}\)
. Adjust the graphing window as needed to answer these questions:
The equation models the population, in millions, in the U.S.
\(t\)
years after 1964. What does the model predict for the population of the U.S. in 1974?
In which year does the model predict the population will reach 300 million?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Student Facing
Are you ready for more?
Research what the population of the U.S. was in the year the model predicted 300 million people. How far off was the model? What factors do you think account for the actual population in that year being different from the prediction of the model? In what year did the U.S. actually reach 300 million people?
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
Focus the discussion on how students set the graphing window and how they used the graphs to answer the questions.
Depending on the graphing technology used, students may use the following strategies:
Visually estimate the value of the output given a certain input, or the input given an output.
Use a tracing tool to trace the graph and identify the coordinates when the input or output has a certain value.
To find an unknown input: graph a horizontal line at a given
\(y\)
-value (for example,
\(y=100\)
, a doubled bacteria population), find the intersection of that line and the graph of the exponential function, and identify the
\(x\)
-value at that intersection.
To find an unknown output: use a table or type something like
\(b(10)\)
after
\(b\)
was defined.
Some students may also notice that they could calculate the output value given an exponential equation and an input value, but had to rely on the graph to find an unknown input value because they were unsure how to solve for
\(x\)
when the equation has
\(e\)
as a base. Tell students that in a future lesson they will learn how to use logarithms to solve this type of equation algebraically similar to how they have used logarithms previously.


aligns: yes