few_shots: 
Example:
Concept/skill: 
Fluently multiply multi-digit whole numbers using the standard algorithm.

Problem/activity:
Problem 1
Solve using the standard algorithm. If you get stuck, use an area model and/or the partial products algorithm to help. Then assess the reasonableness of your answer.
a.
$$5,803\times46$$
b.
$$63\times2,116$$
Problem 2
Dunkin' introduced a croissant doughnut in 2014. In the first three months of its debut, they sold 7,870 dozen of them per day. How many croissant doughnuts were sold in the first three months?


Answer: 
Yes. This set of problems involves multi-digit whole numbers because the numbers in each problem are four and two digit integers, and the mathematical calculations involved in both problems focus on multiplication. The first problem explicitly asks the student to solve its subproblems using the standard algorithm.

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_000674_pos_0

5
standard_description: Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
problem_activity: Task
An important example of a model often used in biology or ecology to model
population growth is called the logistic growth model.  The general
form of the logistic equation is
$$
P(t) = \frac{KP_0e^{rt}}{K+P_0(e^{rt}-1)}.
$$
In this equation $t$ represents time, with $t = 0$ corresponding to when the population in question is first measured; $K,P_0$  and $r$ are all real numbers
with $K$ being called the ''carrying capacity'' while $r$ is a growth rate and
is normally a positive number.
Explain why the value $P_0$ represents
the  population when it is first measured.
Explain why, as time elapses, the population stabilizes, approaching the value $K$.
Explain how the behavior of $P$ changes if the growth rate $r$ is increased or decreased.
Below is the graph of a particular logistic function $P$, showing the growth of a
bacteria population.  Using the graph, identify $P_0$ and  $K$.
[IMG]
Using the values of $P_0$ and $K$ from the previous part, sketch the graph of the logistic function $Q$ given by
$$
Q(t) =  \frac{KP_0e^{2rt}}{K+P_0(e^{2rt}-1)}.
$$
Note that $Q$ is the same as $P$ except that the growth rate $r$ has been doubled.


aligns: no