few_shots: 
Example 1:
Concept/skill: 
Recognize and represent proportional relationships between quantities.

Problem/activity:
You’re mixing ingredients for cookies. The recipe says to combine 6 tablespoons, or ¾ stick, of butter with 1 cup of sugar. You accidentally mix in a whole stick of butter (8 tablespoons) with the cup of sugar. How can you fix this?


Answer: 
No. Usually problems that teach recognizing and representing proportional relationships between quantities involve deciding whether two quantities are in a proportional relationship or identifying the constant of proportionality, and then representing proportional relationships by equations or explaining proportional relationship using coordinate graphs.

Example 2:
Concept/skill: 
Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.

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


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).

Example 3:
Concept/skill: 
Analyze and solve pairs of simultaneous linear equations.

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?


Answer: 
Yes. The last three rows of the table in the first problem involves solving pairs of simultaneous linear equations. The second word problem can also be represented as two simultaneous linear equations.

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_000972_neg_0

6

fl_problem_001097_pos_0

7

fl_problem_001392_pos_0

8
standard_description: Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.
problem_activity: Problem 1
Determine the rate of change in each situation. Then describe the relationship using function language (for example, ____ is a function of ____ because _____.)
a.
|  Amt of watermelon in pounds,                                 $$x$$  |   |   |   |   |   |   |
|----------------------------------------------------------------------|---|---|---|---|---|---|
| Cost of watermelon in dollars,                                 $$y$$ |   |   |   |   |   |   |
b.
$${d=25t}$$
, where
$$d$$
is distance in miles and
$$t$$
is time in hours
c.    Giovanni can decorate
$$2$$
cakes,
$$c$$
, every
$$5$$
hours,
$$h$$
.
Problem 2
Determine the starting or initial output value in each situation.
a.
| **                Distance traveled in cab in miles,                *                   $$x$$                  * ** |  0   |  1   |  3   |   4   |
|---------------------------------------------------------------------------------------------------------------------|------|------|------|-------|
|    **                Cost of cab ride in dollars,                *                   $$y$$                  * **    | 2.50 | 4.75 | 9.25 | 11.50 |
b.
| **                Distance traveled in cab in miles,                *                   $$x$$                  * ** |  1   |  2   |   4   |   5   |
|---------------------------------------------------------------------------------------------------------------------|------|------|-------|-------|
|    **                Cost of cab ride in dollars,                *                   $$y$$                  * **    | 4.50 | 7.00 | 12.00 | 14.50 |
c.
$${p=16d+20}$$
where
$$p$$
is the total number of pages read in a book and
$$d$$
is the number of days
Problem 3
The function formula below shows the relationship between temperature in degrees Fahrenheit and degrees Celsius. °F is a function of °C.
$${F=\frac{9}{5}C+32}$$
a.   Determine what 28°C is in °F.
b.   Determine what 23°F is in °C.


aligns: yes