few_shots: 
Example 1:
Concept/skill: 
Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.

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: 
Yes. This problem solves a real-world problem involving mixing ingredients for cookies. A student doing this problem would need to reason about equivalent ratios of butter amounts to sugar amounts.

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: 
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.

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

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_pos_0

6

fl_problem_001097_pos_0

7

fl_problem_001392_neg_0

8
standard_description: Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.
problem_activity: Task
Some of the problems below can be solved by multiplying $\frac18\times\frac25$, while others need a different operation.  Select the ones that can be solved by multiplying these two numbers.  For the remaining, tell what operation is appropriate.  In all cases, solve the problem (if possible) and include appropriate units in the answer.
Two-fifths of the students in Anya’s fifth grade class are girls.  One-eighth of the girls wear glasses.  What fraction of Anya’s class consists of girls who wear glasses?
A farm is in the shape of a rectangle $\frac18$ of a mile long and $\frac25$ of a mile wide.  What is the area of the farm?
There is $\frac25$ of a pizza left.  If Jamie eats another  $\frac18$ of the original whole pizza, what fraction of the original pizza is left over?
In Sam’s fifth grade class, $\frac18$ of the students are boys.  Of those boys, $\frac25$ have red hair.  What fraction of the class is red-haired boys?
Only $\frac{1}{20}$ of the guests at the party wore both red and green.  If $\frac18$ of the guests wore red, what fraction of the guests who wore red also wore green?
Alex was planting a garden. He planted $\frac25$ of the garden with potatoes and $\frac18$ of the garden with lettuce. What fraction of the garden is planted with potatoes or lettuce?
At the start of the trip, the gas tank on the car was $\frac25$ full.  If the trip used $\frac18$ of the remaining gas, what fraction of a tank of gas is left at the end of the trip?
On Monday, $\frac18$ of the students in Mr. Brown’s class were absent from school.  The nurse told Mr. Brown that $\frac25$ of those students who were absent had the flu.  What fraction of the absent students had the flu?
Of the children at Molly’s daycare, $\frac18$ are boys and $\frac25$ of the boys are under 1 year old.  How many boys at the daycare are under one year old?
The track at school is $\frac25$ of a mile long.  If Jason has run $\frac18$ of the way around the track, what fraction of a mile has he run?


aligns: no