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: Draw informal comparative inferences about two populations.

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

Investigate patterns of association in bivariate data.

Calculate expected values and use them to solve problems.

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

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

Develop understanding of statistical variability.

Understand and evaluate random processes underlying statistical experiments.

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

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

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

Use random sampling to draw inferences about a population.

Summarize and describe distributions.

Interpret linear models.

Use probability to evaluate outcomes of decisions.
problem_activity: Optional activity
The mathematical purpose of this activity is for students to collect, summarize, interpret, and draw conclusions from bivariate data using scatter plots, best fit lines, residuals and correlation coefficients. Students measure the approximate lengths of the humerus bone and height of their classmates to collect data and create a linear model. The model is then used to approximate the height of an ancient human based on the length of a found humerus bone.
Making graphing technology available gives students an opportunity to choose appropriate tools strategically (MP5). By collecting their own data and using a best fit line to find additional information, students are modeling with mathematics (MP4).
Launch
Arrange students in groups of 2 to 4. Present the task to students and ask students to brainstorm different ways that they could answer the question. After 2 minutes of quiet think time, ask students to share their ideas with the class. The remaining time should be used by students to collect, analyze, summarize, and interpret the data.
Speaking, Reading: MLR5 Co-Craft Questions.
Use this routine to help students further interpret and increase awareness of the language used to talk about scatter plots, best fit lines, residuals and correlation coefficients. Display the first sentence of this problem (“An anthropologist finds a fossilized humerus bone of an ancient human ancestor.”), and ask students to write down possible mathematical questions that could be asked about the situation. Invite students to compare their questions before revealing the remainder of the question. Listen for and amplify any questions involving comparing bivariate data.
Design Principle(s): Maximize meta-awareness; Support sense-making
Student Facing
[IMG]
An anthropologist finds a fossilized humerus bone of an ancient human ancestor. The humerus is an arm bone running from the shoulder to the elbow. It is 24 centimeters in length. Use data from your classmates to estimate the height of this ancient human.
Student Response
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Anticipated Misconceptions
Students may find it difficult to start to answer the question. Ask students what are the variables given in the situation. Ask students if there is a way we could collect information from people in the classroom to help answer the question.
Activity Synthesis
The purpose of this discussion is for students to communicate how they used mathematics to justify their findings.
Ask students:
“How confident are you in your answer? What information helped you determine your confidence?” (Not very confident. Since the correlation coefficient is near 0.6, there is only a moderate relationship between height and humerus length. Additionally, this ancient human ancestor may have a different anatomy—for example, apes tend to have proportionally longer arms than humans do.)
“How did you use mathematics to estimate the height of the ancient human?” (First, I collected data on the height and the approximate length of the humerus of my classmates. I then made a scatter plot to determine whether or not a linear model was appropriate, and then computed a line of best fit. I substituted 24 centimeters into my line of best fit and obtained my answer.)
“Do you think that the way you measured the humerus of your classmates impacted your choice of linear model? Explain your reasoning.” (Yes, I think that I probably overestimated the bone length, because I was measuring from the outside and not the inside. I think that my model likely overestimates height.)


correct_option_index: 9

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