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:
Thought: 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:
Thought: 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.
|                                                          | **           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?


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:
Thought: 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: Interpret linear models.

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

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

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

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

Summarize and describe distributions.

Use probability to evaluate outcomes of decisions.

Understand and evaluate random processes underlying statistical experiments.

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

Develop understanding of statistical variability.

Investigate patterns of association in bivariate data.

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

Use random sampling to draw inferences about a population.

Calculate expected values and use them to solve problems.

Draw informal comparative inferences about two populations.
problem_activity: Optional activity
In the warm-up, students computed the mean of a sample. In this activity, a dot plot is created by the class that includes all of the calculated sample means. Students then compare that display to the data from the entire population to better understand the information that can be gained from a sample mean (MP7). In the discussion that follows the activity, students look at similar displays of sample means for samples of different sizes (MP8). Students should see that larger samples tend to more accurately estimate the population means than smaller samples.
Launch
Display the basis for a dot plot of sample means for all to see.
[IMG]
Allow students 5 minutes to create the class dot plot and complete the first set of questions.
Ask students to include their sample means from the previous task to the display for all to see (since the samples were originated in pairs, there will be many repeats. It is ok to include the repeats or ensure only 1 of the original partners includes their point on the display). They may do this by adding a dot to the display on the board, placing a sticky note in the correct place, or passing a large sheet of paper around the class.
After students have had a chance to answer the first set of questions, pause the class to display the dot plot of the the reaction times for the entire population of 120 seniors for all to see.
[IMG]
Provide students 5 minutes to complete the problems. Follow with a whole-class discussion.
Student Facing
Your teacher will display a blank dot plot.
Plot your sample mean from the previous activity on your teacher's dot plot.
What do you notice about the distribution of the sample means from the class?
Where is the center?
Is there a lot of variability?
Is it approximately symmetric?
The population mean is 0.442 seconds. How does this value compare to the sample means from the class?
Pause here so your teacher can display a dot plot of the population of reaction times.
What do you notice about the distribution of the population?
Where is the center?
Is there a lot of variability?
Is it approximately symmetric?
Compare the two displayed dot plots.
Based on the distribution of sample means from the class, do you think the mean of a random sample of 20 items is likely to be:
within 0.01 seconds of the actual population mean?
within 0.1 seconds of the actual population mean?
Explain or show your reasoning.
Student Response
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for free access to Student Response.
Anticipated Misconceptions
The center of the dot plot created by the class can be thought of as a mean of sample means. The phrase can be difficult for students to think through, so remind them what the dot they added to the dot plot represents and how they calculated that value. Consider displaying the class dot plot for the rest of the unit to help students remember this example to understand similar dot plots.
Activity Synthesis
Display the dot plot of sample means for all to see throughout the unit. It may be helpful to refer to this display when viewing dot plots of sample means in future lessons.
The purpose of the discussion is for students to understand that the centers of the two dot plots in the activity are close (if not the same), but the sample means have less variability and the shapes of the population distribution and the sample mean distribution are probably different. Additionally, students should use the dot plots shown here to see that larger samples have sample means that are still centered around the population mean, but have less variability than smaller samples. Therefore, larger samples should provide better estimates of the population mean than smaller samples do.
Items for further discussion:
“If the sample size were increased to 30 instead of 20, what do you think the distribution might look like?”
For a different set of data representing the wingspan in centimeters of a certain species of bird, compare the following dot plots.
Here is a dot plot of the population data.
[IMG]
Here is a dot plot of the sample means for 100 different random samples each of size 10
[IMG]
Here is a dot plot of the sample means for 100 different random samples each of size 30.
[IMG]
“What do you notice about the distributions of the sample means as the sample size increases?” (The variability decreases, but the center stays in about the same place.)
“Use the dot plots to explain why the sample mean from a random sample of size 30 would be a better estimate of the population mean than the sample mean from a random sample of size 10.” (Since the variability is less, the sample mean is probably closer to the true mean with size 30 than with size 10.)
Action and Expression: Internalize Executive Functions.
Provide students with a graphic organizer to support their participation during the synthesis. Invite students to describe what to look for to determine variability and shape of the population distribution and to include examples for each.
Supports accessibility for: Conceptual processing; Organization
Conversing: MLR8 Discussion Supports.
As students compare the wingspan dot plots, encourage students to use sentence frames such as: “When the sample size is 10, I noticed that the distribution . . .”, “When the sample size is 30, I noticed that the distribution . . .”, and “It’s better to have a larger/smaller sample size to estimate the population mean because . . . .” The listener should press for more detail and explanation by asking for evidence from the dot plots. This will help students compare the dot plots and make conclusions about the mean of the population through communicating with their partner.
Design Principle(s): Support sense-making; Cultivate conversation


correct_option_index: 12