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
$$-\frac{1}{4}$$


Concept/skill: 
Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.

Answer: 
Yes. 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).

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_001097_pos_0

7
standard_description: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
problem_activity: Activity
In this warm-up, students make sense of a linear function with a negative rate of change, set in a familiar context. Students have seen similar relationships between the amount of water in a tank and time in an earlier unit (on solving and graphing equations). Here, they think about the quantities in terms of input and output, interpret statements in function notation, and sketch a graph of the function.
The work here prepares students to find and interpret the inverse of functions written in function notation, and to use inverse functions to solve problems.
Launch
Arrange students in groups of 2. Provide access to scientific or four-function calculators, if requested.
Student Facing
A tank contained some water. The function
\(w\)
represents the relationship between
\(t\)
, time in minutes, and the amount of water in the tank in liters. The equation
\(w(t) = 80 - 2.5t\)
defines this function.
Discuss with a partner:
How is the water in the tank changing? Be as specific as possible.
What does
\(w(t)\)
represent? Is
\(w(t)\)
the input or the output of this function?
Sketch a graph of the function. Be sure to label the axes.
[IMG]
Student Response
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Activity Synthesis
Select students to share their responses, including their graph.
Make sure students recall that in an equation such as
\(w(t)=80-2.5t\)
,
\(w(t)\)
is the output when the input is
\(t\)
, that they can interpret the equation in terms of the situation, and see why the graph is a downward-slanting line.


aligns: yes