few_shots: Example 1:
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?


Mathematical concept/skill options: 
A. Analyze proportional relationships and use them to solve real-world and mathematical problems.
B. Understand ratio concepts and use ratio reasoning to solve problems.


Your response:
{
  "explanation": "This problem focuses on the ratio between two ingredients. Students do not have to analyze proportional relationships in-depth, which is reserved for a higher grade level, but rather use ratios to find missing values.",
  "answer": "B"
}

Example 2:
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
$$-\frac{1}{4}$$


Mathematical concept/skill options: 
A. Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.
B. Reason quantitatively and use units to solve problems.
C. Apply and extend previous understandings of numbers to the system of rational numbers.
D. Represent complex numbers and their operations on the complex plane.
E. Perform arithmetic operations with complex numbers.
F. Compute fluently with multi-digit numbers and find common factors and multiples.
G. Represent and model with vector quantities.
H. Perform operations on vectors.
I. Use properties of rational and irrational numbers.
J. Use complex numbers in polynomial identities and equations.
K. Extend the properties of exponents to rational exponents.
L. Know that there are numbers that are not rational, and approximate them by rational numbers.
M. Perform operations on matrices and use matrices in applications.
N. Apply and extend previous understandings of multiplication and division to divide fractions by fractions.


Your response:
{
  "explanation": "This problem involves understanding fractions, multiplication, and division.",
  "answer": "A"
}

Example 3:
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.
{"table": [{"_attributes": {"align": "center", "class": ["table", "table-bordered"], "style": "table-layout:fixed; width:70%"}, "tbody": [{"tr": [{"td": [{"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}, "strong": [{"_value": "No solution"}]}, {"_attributes": {"style": "text-align:center"}, "strong": [{"_value": "Exactly 1 solution"}]}, {"_attributes": {"style": "text-align:center"}, "strong": [{"_value": "Infinite number of solutions"}]}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${9=4x+7}$$"}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${y=3y+5}$$"}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${y={x\\over2}}$$"}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"p": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${3x+6y-9=0}$$"}]}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"p": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${x=y}$$"}]}, {"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${1.25x=1.25y}$$"}]}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"p": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${9.9=6x+8y}$$"}]}, {"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${9x=2.5y-8.8}$$"}]}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"p": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${11x-2y=1.5}$$"}]}, {"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${11x-2y=2.5}$$"}]}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {}]}]}]}]}
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?


Mathematical concept/skill options: 
A. Write expressions in equivalent forms to solve problems.
B. Understand properties of multiplication and the relationship between multiplication and division.
C. Interpret the structure of expressions.
D. Solve real-life and mathematical problems using numerical and algebraic expressions and equations.
E. Use polynomial identities to solve problems.
F. Understand the relationship between zeros and factors of polynomials.
G. Write and interpret numerical expressions.
H. Work with equal groups of objects to gain foundations for multiplication.
I. Perform arithmetic operations on polynomials.
J. Create equations that describe numbers or relationships
K. Generate and analyze patterns.
L. Analyze and solve linear equations and pairs of simultaneous linear equations.
M. Understand solving equations as a process of reasoning and explain the reasoning.
N. Apply and extend previous understandings of arithmetic to algebraic expressions.
O. Rewrite rational expressions.
P. Represent and solve problems involving multiplication and division.
Q. Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.
R. Reason about and solve one-variable equations and inequalities.
S. Understand and apply properties of operations and the relationship between addition and subtraction.
T. Understand the connections between proportional relationships, lines, and linear equations.
U. Multiply and divide within 100.
V. Gain familiarity with factors and multiples.
W. Use the four operations with whole numbers to solve problems.
X. Work with addition and subtraction equations.
Y. Represent and solve problems involving addition and subtraction.
Z. Solve problems involving the four operations, and identify and explain patterns in arithmetic.
AA. Solve equations and inequalities in one variable.
AB. Solve systems of equations.
AC. Represent and analyze quantitative relationships between dependent and independent variables.
AD. Work with radicals and integer exponents.
AE. Represent and solve equations and inequalities graphically.
AF. Analyze patterns and relationships.
AG. Use properties of operations to generate equivalent expressions.
AH. Add and subtract within 20.


Your response:
{
  "explanation": "This set of problems involves solving linear equations and pairs of linear equations.",
  "answer": "L"
}

Now, assign the following problem or activity to one or more relevant mathematical concepts/skills it teaches.
shot_info: fl_problem_000972

6

fl_problem_001097

7

fl_problem_001392

8
options: Summarize, represent, and interpret data on two categorical and quantitative variables.

Use probability to evaluate outcomes of decisions.

Summarize, represent, and interpret data on a single count or measurement variable.

Use random sampling to draw inferences about a population.

Calculate expected values and use them to solve problems.

Investigate patterns of association in bivariate data.

Investigate chance processes and develop, use, and evaluate probability models.

Summarize and describe distributions.

Understand and evaluate random processes underlying statistical experiments.

Understand independence and conditional probability and use them to interpret data.

Use the rules of probability to compute probabilities of compound events in a uniform probability model.

Make inferences and justify conclusions from sample surveys, experiments, and observational studies.

Draw informal comparative inferences about two populations.

Develop understanding of statistical variability.

Interpret linear models.
problem_activity: Optional activity
This activity is a follow-up to the context used in the activities Three Different TV Shows and Who's Watching What?, but also follows the ideas of sample means from this lesson.
In this activity, students continue to look at how variability in the sample means can be used to think about the accuracy of an estimate of a population mean. If the means from samples tend to be very spread out, then there is reason to believe that the mean from a single sample may not be especially close to the mean for the population. If the means from samples tend to be tightly grouped, then there is reason to believe that the mean from a single sample is a good estimate of the mean for the population (MP7).
Launch
Arrange students in groups of 2.
Tell students that, although the dot plots seem to have a similar shape, attention should be given to the scale of the horizontal axis.
Allow students 3–5 minutes of quiet work time followed by partner and whole-class discussions.
Action and Expression: Internalize Executive Functions.
Chunk this task into more manageable parts. After students have solved the first 2-3 population mean estimate problems, check-in with either select groups of students or the whole class. Invite students to share the strategies they have used so far as well as any questions they have before continuing.
Supports accessibility for: Organization; Attention
Student Facing
The other day you worked with 2 different samples of viewers from each of 3 different television shows. Each sample included 10 viewers. Here are the mean ages for 100 different samples of viewers from each show.
[IMG]
[IMG]
[IMG]
For each show, use the dot plot to estimate the
population
mean.
Trivia the Game Show
Science Experiments YOU Can Do
Learning to Read
For each show, are most of the sample means within 1 year of your estimated population mean?
Suppose you take a new random sample of 10 viewers for each of the 3 shows. Which show do you expect to have the new sample mean closest to the population mean? Explain or show your reasoning.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Student Facing
Are you ready for more?
Market research shows that advertisements for retirement plans appeal to people between the ages of 40 and 55. Younger people are usually not interested and older people often already have a plan. Is it a good idea to advertise retirement plans during any of these three shows? Explain your reasoning.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Anticipated Misconceptions
Students may get confused about what the dot plots represent. Refer to the class dot plot of sample means from the reaction time activity to help them understand that a single dot on the dot plot represents a mean from a single sample. Each dot plot shows means from several different samples.
Activity Synthesis
The purpose of the discussion is to think about what the information in the given dot plots tells us about the accuracy of an estimate of a population mean based on the sample mean from a single sample.
Consider asking these questions for discussion:
“In the first dot plot for ‘Trivia the Game Show,’ what does the dot at 68 represent?” (There is a sample of 10 viewers that has a mean age of 68 years.)
“Based on the dot plot shown for ‘Learning to Read,’ do you think any 7 year olds were included in the data?” (Yes. Since one of the sample means is 6.8, there are probably a few 7 year olds and some 6 year olds in that sample.)
“Why is this kind of dot plot useful?” (It shows how variable the means from samples can be, so it gives a good idea of how accurate a sample mean might be as an estimate of a population mean.)
“Why might it be difficult to obtain a dot plot like this?” (Each dot represents a sample of 10. It might be hard to get 100 different samples of size 10.)
Representing, Speaking: MLR2 Collect and Display.
Create a visual display with the heading “Dot Plots of Sample Means.” As students respond to the discussion questions, write down the words and phrases students use on the visual display. Listen for and amplify words and phrases such as “each dot represents...,” “variability of the sample means,” and “population mean.” As students review the language collected in the visual display, encourage students to clarify the meaning of a word or phrase. For example, a phrase such as “each dot represents a sample” can be clarified by restating it as “each dot represents a sample mean.” This will help students make sense of dot plots of sample means as they draw conclusions about populations.
Design Principle(s): Support sense-making: Maximize meta-awareness


correct_option_index: 3