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: Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself. For example, given the rule “Add 3” and the starting number 1, generate terms in the resulting sequence and observe that the terms appear to alternate between odd and even numbers. Explain informally why the numbers will continue to alternate in this way.
problem_activity: Problem 1
a.   Estimate whether the following sums will be more or less than 1.
i.
$${{3\over4}+{2\over4}}$$
ii.
$${{5\over6}+{3\over6}}$$
b.   Solve for the actual sums in Part (a) above.
Problem 2
Cary computes
$$\frac{9}{10} + \frac4{10}$$
using a different strategy. Here is her reasoning:
[IMG]
“First I broke
$$\frac4{10}$$
into
$$\frac1{10}$$
and
$$\frac3{10}$$
. Then I added
$$\frac9{10}$$
and
$$\frac1{10}$$
to get to 1, then added the other
$$\frac3{10}$$
to get
$$1\frac3{10}$$
.
So
$$\frac{9}{10} + \frac4{10} = 1\frac3{10}$$
."
a.   Is Cary’s calculation correct? Explain.
b.   Will Cary’s strategy always work? If not, give an example of a calculation for which Cary can’t use this strategy.
c.   Use Cary’s strategy to find
$$\frac45+\frac35$$
.
Problem 3
a.   Estimate whether the following differences will be more or less than 1.
i.
$${1{1\over3}-{2\over3}}$$
ii.
$${1{3\over8}-{5\over8}}$$
b.   Solve for the actual differences in Part (a) above.
Problem 4
Corrado computes
$$1\frac15 - \frac25$$
using a different strategy. Here is his reasoning:
[IMG]
“First I broke
$$\frac25$$
into
$$\frac15$$
and
$$\frac15$$
. Then I subtracted
$$\frac15$$
from
$$1\frac15$$
to get to 1, then subtracted the other
$$\frac15$$
to get
$$\frac45$$
.
So
$$1\frac15 - \frac25 = \frac45$$
."
a.   Is Corrado’s calculation correct? Explain.
b.   Will Corrado’s strategy always work? If not, give an example of a calculation for which Corrado can’t use this strategy.
c.   Use Corrado’s strategy to find
$$1\frac2{12}-\frac5{12}$$
.


aligns: no