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: Add and subtract within 20.

Write and interpret numerical expressions.

Interpret the structure of expressions.

Perform arithmetic operations on polynomials.

Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.

Generate and analyze patterns.

Reason about and solve one-variable equations and inequalities.

Represent and solve problems involving addition and subtraction.

Work with addition and subtraction equations.

Represent and solve equations and inequalities graphically.

Use polynomial identities to solve problems.

Understand the relationship between zeros and factors of polynomials.

Solve real-life and mathematical problems using numerical and algebraic expressions and equations.

Analyze and solve linear equations and pairs of simultaneous linear equations.

Solve problems involving the four operations, and identify and explain patterns in arithmetic.

Understand properties of multiplication and the relationship between multiplication and division.

Work with radicals and integer exponents.

Work with equal groups of objects to gain foundations for multiplication.

Understand and apply properties of operations and the relationship between addition and subtraction.

Use properties of operations to generate equivalent expressions.

Analyze patterns and relationships.

Gain familiarity with factors and multiples.

Apply and extend previous understandings of arithmetic to algebraic expressions.

Write expressions in equivalent forms to solve problems.

Represent and analyze quantitative relationships between dependent and independent variables.

Understand the connections between proportional relationships, lines, and linear equations.

Solve systems of equations.

Understand solving equations as a process of reasoning and explain the reasoning.

Multiply and divide within 100.

Represent and solve problems involving multiplication and division.

Create equations that describe numbers or relationships

Solve equations and inequalities in one variable.

Rewrite rational expressions.

Use the four operations with whole numbers to solve problems.
problem_activity: Activity
This activity gives students another opportunity to write and rearrange equations in two variables and to do so in context.
Unlike in previous activities, students no longer start by computing one quantity given numerical values of the other quantity and then generalizing the process. Instead, they are prompted to write one equation, solve for each variable, and interpret the solution. Students also articulate why one model might be more helpful than the other under a certain circumstance. Along the way, students reason quantitatively and abstractly (MP2) and engage in aspects of modeling (MP4).
Making spreadsheet technology available gives students an opportunity to choose appropriate tools strategically (MP5).
Launch
Familiarize students with the idea of organizational budgets, if needed. Explain that every governmental agency has a budget (not unlike a budget for a pizza party) and needs to decide how to allocate a limited amount of funds. Ask students to think of examples of expenses that a city's Department of Streets might have (road repairs, new street construction, additional workers, new equipment, and so on). Given a set budget, spending more on one thing means having less to spend on something else, so the agency would need to weigh their options and prioritize their needs.
Reading, Listening, Conversing: MLR6 Three Reads.
Ask students to keep their books or devices closed and display only the image and the task statement without revealing the questions that follow. Use the first read to orient students to the situation. Ask students to describe what the situation is about without using numbers (a city department has money to resurface roads this year). Use the second read to identify quantities and relationships. Ask students “What can be counted or measured?” Listen for, and amplify, the important quantities that vary in relation to each other in this situation: the total amount of money available, the cost to resurface 1 mile of a 2-lane road, the average worker’s salary, and the number of miles of road to resurface. After the third read, reveal the first question and ask students to brainstorm possible strategies to write an equation. This routine helps students interpret the language within a given situation needed to create an equation.
Design Principle(s): Support sense-making
Representation: Internalize Comprehension.
Provide appropriate reading accommodations and supports to ensure student access to written directions, word problems, and other text-based content.
Supports accessibility for: Language; Conceptual processing
Student Facing
The Department of Streets of a city has a budget of \$1,962,800 for resurfacing roads and hiring additional workers this year.
The cost of resurfacing a mile of 2-lane road is estimated at \$84,000. The average starting salary of a worker in the department is \$36,000 a year.
[IMG]
Write an equation that represents the relationship between the miles of 2-lane roads the department could resurface,
\(m\)
, and the number of new workers it could hire,
\(p\)
, if it spends the entire budget.
Take the equation you wrote in the first question and:
Solve for
\(p\)
. Explain what the solution represents in this situation.
Solve for
\(m\)
. Explain what the solution represents in this situation.
The city is planning to hire 6 new workers and to use its entire budget.
Which equation should be used to find out how many miles of 2-lane roads it could resurface? Explain your reasoning.
Find the number of miles of 2-lane roads the city could resurface if it hires 6 new workers.
Student Response
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Anticipated Misconceptions
Some students may have trouble sorting out the given information because of unfamiliarity with the context or with certain terms. Ask them to explain the setup of the problem as best they understand it, and then point out any information that might be missing.
Activity Synthesis
Verify that the equations students wrote correctly isolate each variable. If students wrote expressions in different forms for a variable (for instance,
\(\dfrac {4,\!907 - 210m}{90}\)
and
\(\dfrac {1,\!962,\!800 -84,\!000m}{36,\!000}\)
for
\(p\)
), discuss how the expressions are equivalent.
If time permits, consider asking students to use their equations to answer these questions:
"How many people could be hired if the department is resurfacing 16 miles of roads?" (17 people) "3 miles?" (47 people)
"How many miles of roads could be resurfaced if the department is hiring 2 new workers?" (22.5 miles) "If no new workers are hired?" (About 23.37 miles)
Make sure students understand that solving for
\(p\)
makes it possible to quickly find the number of people that the department could hire given some miles of road to be resurfaced (and while sticking to the budget). Similarly, solving for
\(m\)
makes it possible to quickly find the miles that could be resurfaced given any number of new hires.


correct_option_index: 30

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