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: Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.
problem_activity: In Class Launch
Use after Unit 3, Lesson 9
The US Department of Education does a survey of colleges every year. Students will use this data to test a conjecture they have about colleges. Tell students that these are some of the characteristics we have information on:
college ownership: public, private non-profit, or private for-profit
average SAT scores of admitted students
acceptance rate
number of enrolled students
average cost of attendance
median earnings 10 years after graduation
Task Statement 1 has four more categories (highest degree awarded, completion rate, percent of white students, and median ACT score), which can also be included in this discussion if students will be using that task statement.
Ask students to identify what type of data each of these is (for example, categorical, percent, mean, or count). Once all the types are identified, ask "Which of these characteristics might be related to each other? How are they related?" Write down each idea for all to see. Ask students why they think a pair of categories is related in the way they propose, to increase investment in finding the answer. However, students do not need to have well-supported reasons at this time.
If students are having trouble generating ideas, or they do not bring up a wide enough variety of ideas, introduce a few more. Here are some examples:
The more selective the college is, the higher the average SAT scores of the people who are accepted.
Private for-profit colleges cost more on average, but also have a wider range of costs, because they need to make money but also attract lots of students.
Public schools are larger than private schools.
Once enough ideas have been suggested, ask “If these relationships exist, how could we see them in the data? How could we present the data to show that they’re true?” Give students quiet think time and then invite them to share one of their ideas with a partner. Then ask students to share their ideas with the class. Ensure that a variety of ways of presenting data are discussed. Here are some examples:
The more selective the college is, the higher the average SAT scores of the people who are accepted: Make a scatter plot of admission rate vs. average SAT score.
Private for-profit colleges cost more on average, but also have a wider range of costs: Make a box-and-whisker plot of costs for each type of college.
Public schools are larger than private schools: Make a two-way frequency table of school ownership (public or private) vs school size (small, medium, and large, defined by ranges of numbers of enrolled students).
Teacher Instructions
The data provided is a spreadsheet of 200 schools, not all of which have information about each of the ten categories. Students will need access to technology such as Geogebra or Excel that will allow them to analyze and display the relationships that interest them.
If desired, the complete data set can be downloaded from the Department of Education website, and data can be curated in a different way. Different categories can be chosen based on the interests of the class, or the range of colleges can be narrowed down to those in the state. Historical data is also available. A few local colleges could be compared across time. If a customized data set is created, modify the launch accordingly.
Student-Facing Statement
Here are the characteristics of colleges that we have data about:
Choose two that might be related. What do you predict the relationship between them is?
college ownership: public, private non-profit, or private for-profit
average SAT score of admitted students
acceptance rate
number of enrolled students
average cost of attendance per year
median earnings 10 years after graduation
highest degree awarded
completion rate
percent of students who self-identify as white
median ACT score of admitted students
Take a look at the data in the spreadsheet. Before calculating, do some estimation based on the data. Can you tell if your hypothesis seems reasonable?
Use an appropriate display to summarize the data.
Analyze your graph or table and do any other calculations needed. Do your results confirm your prediction? If not, what do you think explains the results?
Here are some headlines. What analysis do you think the article describes? Give some examples of evidence that would support the headline or contradict it.
“Students from Selective Colleges Tend to Succeed”
“Isolation at Larger Colleges Leads to Higher Dropout Rates”
“High Cost of Private Colleges Not Outweighed by Lifetime Earnings”
Teacher Instructions
The data provided is a spreadsheet of 100 schools for which there is information about each of the six categories. Students will need access to technology such as Geogebra or Excel that will allow them to analyze and display the relationships that interest them.
If desired, the complete data set can be downloaded from the Department of Education website, and data can be curated in a different way. Different categories can be chosen based on the interests of the class, or the range of colleges can be narrowed down to those in the state. Historical data is also available. A few local colleges could be compared across time. If a customized data set is created, modify the launch accordingly.
Student-Facing Statement
Here are the characteristics of colleges that we have data about:
Choose two that might be related. What do you predict the relationship between them is?
college ownership: public, private non-profit, or private for-profit
average SAT score of admitted students
acceptance rate
number of enrolled students
average cost of attendance per year
median earnings 10 years after graduation
Take a look at the data in the spreadsheet. Before calculating, do some estimation based on the data. Can you tell if your hypothesis seems reasonable?
Use an appropriate display to summarize the data.
Analyze your graph or table and do any other calculations needed. Do your results confirm your prediction? If not, what do you think explains the results?
Here are some headlines. What analysis do you think the article describes? Give some examples of evidence that would support the headline or contradict it.
“Students from Selective Colleges Tend to Succeed”
“Isolation at Larger Colleges Leads to Higher Dropout Rates”
“High Cost of Private Colleges Not Outweighed by Lifetime Earnings”


aligns: yes