few_shots: 
Example:
Problem/activity:
Problem 1
The table below contains single equations in a single variable, single equations in two variables, and pairs of simultaneous equations in two variables. Check the appropriate box to show whether there is no solution, exactly 1 solution, or if there is an infinite number of solutions.
|                                                          | **           No solution          ** | **           Exactly 1 solution          ** | **           Infinite number of solutions          ** |
|----------------------------------------------------------|--------------------------------------|---------------------------------------------|-------------------------------------------------------|
|                       $${9=4x+7}$$                       |                                      |                                             |                                                       |
|                       $${y=3y+5}$$                       |                                      |                                             |                                                       |
|                    $${y={x\over2}}$$                     |                                      |                                             |                                                       |
|                     $${3x+6y-9=0}$$                      |                                      |                                             |                                                       |
|    $${x=y}$$                        $${1.25x=1.25y}$$    |                                      |                                             |                                                       |
| $${9.9=6x+8y}$$                        $${9x=2.5y-8.8}$$ |                                      |                                             |                                                       |
| $${11x-2y=1.5}$$                        $${11x-2y=2.5}$$ |                                      |                                             |                                                       |
Problem 2
Kim has a small container and a large container as shown.
[IMG]
It takes 16 of the small containers to fill the large container. Three small containers leave 1.95 gallons of space in the big container.
What is the size of each of the two containers?


Concept/skill: 
Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.

Answer: 
No. Completing neither of these problems requires replacing one equation by the sum of that equation and a multiple of the other equation.

Now, does the following problem or activity enable students to learn the full intent of the following concept or skill?

shot_info: fl_problem_001392_neg_0

8
standard_description: 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).
problem_activity: Task
Below is a table showing the approximate boiling point of water at different elevations:
| Elevation (meters above sea level) | Boiling Point (degrees Celsius) |
|------------------------------------|---------------------------------|
|                 0                  |               100               |
|                500                 |              98.2               |
|                1000                |              96.5               |
|                1500                |              94.7               |
|                2000                |              93.1               |
|                2500                |              91.3               |
Based on the table,
if we were to model the relationship between the elevation and the boiling point of water, would a linear function be appropriate? Explain why or why not.
Below are some additional values for the boiling point of water at higher elevations:
| Elevation (meters above sea level) | Boiling Point (degrees Celsius) |
|------------------------------------|---------------------------------|
|               5,000                |              83.2               |
|               6,000                |              80.3               |
|               7,000                |              77.2               |
|               8,000                |              74.3               |
|               9,000                |              71.5               |
Based on the table,
if we were to model the relationship between the elevation and the boiling point of water, would a linear function be appropriate? Explain why or why not.
When the information from both tables is combined, would a linear function be appropriate to model this data?  What kind of function would you use to model the data?  Why?


aligns: yes