few_shots: Example:
Problem/activity:
For each pair of numbers, determine if the values are equivalent. Explain your reasoning for each one.
a.
$$-0.25$$
and
$$\frac{1}{-4}$$
b.
$$-\left ( \frac{1}{4} \right )$$
and
$$\frac{-1}{-4}$$
c.
$$\frac{-1}{4}$$
and
$$-\frac{1}{4}$$


Standard options: 
A. Solve real-world and mathematical problems involving the four operations with rational numbers. (Computations with rational numbers extend the rules for manipulating fractions to complex fractions.)
B. Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.
C. Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.


Your response:
Thought: This problem involves understanding fractions and division to divide rational numbers, such as 1, -4, 4, and -1. This problem also involves an understanding of multiplication: if p and q are integers, then -(p/q) = (-p)/q = p/(-q).
Answer: B

Now, assign the following problem or activity to one or more relevant standards it aligns with.
shot_info: fl_problem_001097

7
options: Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).

Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.

Distinguish between situations that can be modeled with linear functions and with exponential functions.

For exponential models, express as a logarithm the solution to ab^(ct) = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.
problem_activity: Task
Complete the table below. Is $\Delta x$ a constant?  If so, what constant is it? What do you notice about the 3rd column of the table?
| $x$ | $f(x)=128\cdot \left({\frac {1}{2}}\right)^x$ |               Successive quotients                |
|-----|-----------------------------------------------|---------------------------------------------------|
|  0  |                      128                      |                       ----                        |
|  1  |                      64                       | $\frac{f(1)}{f(0)}=\frac {64}{128} = \frac{1}{2}$ |
|  2  |                                               |               $\frac{f(2)}{f(1)}=$                |
|  3  |                                               |                                                   |
|  4  |                                               |                                                   |
Complete the table below. Is $\Delta x$ a constant?  If so, what constant is it? What do you notice about the 3rd column of the table?
| $x$ | $f(x)=128\cdot \left({\frac {1}{2}}\right)^x$ |                Successive quotients                 |
|-----|-----------------------------------------------|-----------------------------------------------------|
|  0  |                      128                      |                         ---                         |
|  2  |                      32                       | $\frac{f(2)}{f(0)} = \frac {32}{128} = \frac{1}{4}$ |
|  4  |                                               |                $\frac{f(4)}{f(2)}=$                 |
|  6  |                                               |                                                     |
|  8  |                                               |                                                     |
Let $f(x)=a \cdot b^x$. Let $x_0$ be any particular $x$-value. Show that if $x_0$ is increased by a constant, $\Delta x$, the successive quotient  $$\frac{f(x_0+ \Delta x)}{f(x_0)}$$ is always the same no matter what $x_0$ is.
Is b) an example of the result of c)? Explain.


correct_option_index: 2