few_shots: 
Example:
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.

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
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: Activity
This activity prompts students to again compare a linear function with an exponential one, but this time without a context, and the exponential function grows much more slowly over a long period of time. Even if students predict that the exponential function will grow more quickly because it is exponential, they still need to decide which one reaches 2,000 first. To do that, students may try:
continuing the table of values with increasingly larger values
using graphing technology
finding when
\(f(x) = 2,\!000\)
and substituting this value (
\(x=1,\!000\)
) into
\(g(x)\)
Making graphing and spreadsheet technology available gives students an opportunity to choose appropriate tools strategically (MP5).
Launch
Present the two equations that define two functions
\(f\)
and
\(g\)
:
\(f(x) = 2x\)
and
\(g(x) = (1.01)^x\)
. Give students a moment to share what they notice and wonder about the functions. Ask questions such as:
What is happening in the first function? What about in the second function?
Which function do you think will reach a value of 2,000 first?
Representing, Conversing: MLR7 Compare and Connect.
Ask students to prepare a visual display that shows their mathematical thinking in response to the question “Which function do you think will reach a value of 2,000 first?” Invite previously identified students to share their representations. In groups of 2, ask students to compare their strategies and discuss how the strategies are the same or different. Consider using the prompt “Which strategy do you believe is the most efficient in determining a solution?” Listen for and amplify observations of advantages and disadvantages to different approaches, such as using a particular approach for a certain time interval. This will help students connect different approaches and determine the effectiveness of each one through partner and whole-class discussion.
Design Principle(s): Cultivate conversation; Maximize meta-awareness
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
Complete the table of values for the functions
\(f\)
and
\(g\)
.
| \(x\) | \(f(x)\) | \(g(x)\) |
|-------|----------|----------|
|   1   |          |          |
|  10   |          |          |
|  50   |          |          |
|  100  |          |          |
|  500  |          |          |
|       |          |          |
|       |          |          |
|       |          |          |
Based on the table of values, which function do you think grows faster? Explain your reasoning.
Which function do you think will reach a value of 2,000 first? Show your reasoning. If you get stuck, consider increasing
\(x\)
by 100 a few times and record the function values in the table.
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?
Consider the functions
\(g(x)=x^5\)
and
\(f(x)=5^x\)
. While it is true that
\(f(7)>g(7)\)
, for example, it is hard to check this using mental math. Find a value of
\(x\)
for which properties of exponents allow you to conclude that
\(f(x)>g(x)\)
without a calculator.
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
Select students who used contrasting approaches to share. The exponential function
\(g\)
grows sufficiently slowly that it looks like it will not catch up with or ever overtake the linear function
\(f\)
. This is true for a table of values as well as using a graph. Invite students who used contrasting methods to share, in the order shown in the Activity Narrative.
Emphasize that the table needs to be continued for a long time to identify when the values of
\(g\)
become greater than those of
\(f\)
. Similarly, the window of the graph needs to be chosen carefully. A very efficient method is to identify that
\(f(1,\!000)=2,\!000\)
and then evaluate
\(g(1,\!000)\)
. Since
\(g(1,\!000)\)
is much greater than 2,000,
\(g\)
reaches 2,000 first.
If not already illustrated by students who used graphing, show this dynamic sketch and zoom out:
Though for quite a while it doesn't seem like the values of
\(g\)
would ever catch up with those of
\(f\)
, if the growth is allowed to take place long enough, exponential eventually surpasses linear.


aligns: yes