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.


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.


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.


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

Interpret linear models.

Use random sampling to draw inferences about a population.

Investigate patterns of association in bivariate data.

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

Develop understanding of statistical variability.

Summarize and describe distributions.

Draw informal comparative inferences about two populations.

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

Understand and evaluate random processes underlying statistical experiments.

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

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

Calculate expected values and use them to solve problems.

Use probability to evaluate outcomes of decisions.

Investigate chance processes and develop, use, and evaluate probability models.
problem_activity: Activity
Now that students are more familiar with two-way tables showing relative frequency, they are ready to create their own segmented bar graphs. In this activity, students create two segmented bar graphs based on the same two-way table by considering percentages of the rows and columns separately. After creating the segmented bar graphs, they are analyzed to determine if there is an association present in the data.
Launch
Arrange students in groups of 2. After a brief introduction, give 5–10 minutes of quiet work time. Ask students to compare their answers with their partner and try to resolve any differences. Finish with a whole-class discussion.
Display the two-way table from the previous lesson's cool-down activity containing the data collected about the class's playing sports and musical instruments. If the data is unavailable, the data from this lesson's warm-up can be used.
Tell students they should work with their partners to each work on one of the graphs. One student should work on problems 1 and 2 while their partner should work on 3 and 4. After they have completed their graphs, they should work together to understand their partners graphs and complete the last problem together.
Action and Expression: Internalize Executive Functions.
Chunk this task into more manageable parts to support students who benefit from support with organization and problem solving. For example, present one question at a time. Some students may benefit from a checklist on how to create a segmented bar graph.
Supports accessibility for: Organization; Attention
Student Facing
Your teacher will give you a two-way table with information about the number of people in your class who play sports or musical instruments.
Complete this table to make a two-way table for the data from earlier. The table will show relative frequencies
by row
.
|                     | plays instrument | does not play instrument | row total |
|---------------------|------------------|--------------------------|-----------|
|     plays sport     |                  |                          |           |
| does not play sport |                  |                          |           |
Make a segmented bar graph for the table. Use one bar of the graph for each row of the table.
[IMG]
Complete the table to make a two-way table for the data from earlier. The table will show relative frequencies
by column
.
|                     | plays instrument | does not play instrument |
|---------------------|------------------|--------------------------|
|     plays sport     |                  |                          |
| does not play sport |                  |                          |
|    column total     |                  |                          |
Using the values in the table, make a segmented bar graph. Use one bar of the graph for each column of the table.
[IMG]
Based on the two-way tables and segmented bar graphs, do you think there is an association between playing a sport and playing a musical instrument? Explain how you know.
Student Response
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Anticipated Misconceptions
Students may draw the segmented bar graph incorrectly. Most likely, they will accidentally graph frequency instead of relative frequency. They may also graph relative frequencies, but without stacking them. Both segmented bars should go from 0 to 100.
Activity Synthesis
To clarify how to create and interpret segmented bar graphs, ask:
“What different information can be seen by the two segmented bar graphs?”
“Why are the numbers in the top left box in the two tables different? What do they mean?” (In the first table it represents the percentage who also play musical instruments out of all the people who play sports. In the second table it represents the percentage of people who also play sports out of all the people who play musical instruments.)
“Is there an association between the two variables? Explain or show your reasoning.” (The answer will depend on class data, but the reasoning should include an analysis of the relative frequencies within categories. There is an association if the percentages within one category are very different from the percentages in another category.)
If there is an association, ask what the segmented bar graphs would look like if there was no association. If there is not an association, ask what the segmented bar graphs would look like if there was one.
Writing, Speaking: MLR1 Stronger and Clearer Each Time.
Use this routine to give students a structured opportunity to revise and refine their response to the last question. Ask each student to meet with 2–3 other partners in a row for feedback. Provide students with prompts for feedback that will help them strengthen their ideas and clarify their language (e.g., “Why do you think there is a (positive/negative) association?”, “How do the relative frequencies help to answer this question?”, “How could you say that another way?”, etc.). Students can borrow ideas and language from each partner to strengthen the final product. They can return to the first partner and revise and refine their initial response.
Design Principle(s): Optimize output (for explanation)


correct_option_index: 3