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: For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.
problem_activity: In Class Launch
Use after Unit 4, Lesson 8
Read the situation in the task statement aloud, or ask students to read it to themselves quietly.
Tell students to think about these questions:
“What might be some different ways of giving bonuses?”
“Does anyone have a job and has gotten a raise or a bonus? How are your allowances determined? Do they ever change?”
“What variables should be considered? What information do you need to know about the employees to compute bonuses for each of them?”
Give students a few minutes of quiet think time, then ask them to talk over their ideas with a partner. Finally, ask students to share their ideas with the whole class and record these ideas for all to see.
Ask students if they have any questions about the situation.
If students want to know what type of company it is and what jobs the employees are doing, tell them that they can make up a fake company with specific jobs.
Teacher Instructions
As students are considering the situation, prompt them with the following questions, as necessary:
"What are some variables you should consider?"
"Can you think of mathematically different methods to distribute the money?"
"What information would be helpful to know to compute the bonus for each employee?"
Tell students that they can make up the information they need to compute specific bonuses for 2 different methods and to apply them to the situation. In the proposal, students should compute the specific dollar amount each employee receives as a bonus.
If there is time, encourage students to try generalizing any values they invented to use variables instead.
Student-Facing Statement
A project at a large company was very successful, and the company made more money than expected as a result. Your boss has given you the task of coming up with different methods to distribute bonuses to the 5 employees that directly worked on the project. There is a total of
$
8,000 available to distribute.
The method for distributing the money will be shared with the entire company, so it is important that the employees feel the distribution is fair.
Make a proposal with at least two different methods for your boss to choose from. Describe the advantages and disadvantages of each method. Then give your recommendation and provide an argument in its support.
For each of the methods you propose, which of the five employees is most likely to complain about the method being unfair? How would you justify the method to this employee?
Teacher Instructions
As students are considering the situation, prompt them with the following questions, as necessary:
“Which of these variables should you consider?”
“Can you think of mathematically different methods to distribute the money?”
“What additional information would be helpful to know to compute the bonus for each employee?”
Tell students that they can make up any additional information they need to compute specific bonuses for 2 different methods and to apply them to the situation. In the proposal, students should compute the specific dollar amount each employee receives as a bonus.
If there is time, encourage students to try generalizing any values they invented to use variables instead.
Student-Facing Statement
A project in a large company was very successful, and the company made more money than expected as a result. Your boss has given you the task of coming up with different methods to distribute bonuses to the 5 employees that directly worked on the project. There is a total of
$
8,000 available to distribute.
Here is some information about the employees:
| employee | job description | hours working on project (per week) | annual salary | job experience |
|----------|-----------------|-------------------------------------|---------------|----------------|
|    A     |                 |                                     |               |                |
|    B     |                 |                                     |               |                |
|    C     |                 |                                     |               |                |
|    D     |                 |                                     |               |                |
|    E     |                 |                                     |               |                |
Make a proposal with at least two different methods for your boss to choose from. Outline the advantages and disadvantages of each method. Then give your recommendation and support your argument.
For each of the methods you propose, which of the five employees is most likely to complain about the method being unfair? How would you justify the method to this employee?


aligns: no