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: Understand ratio concepts and use ratio reasoning to solve problems.

Analyze proportional relationships and use them to solve real-world and mathematical problems.
problem_activity: Optional activity
In this activity, the dots are distributed uniformly in the first square but not in the second square:
[IMG]
However, these dots are drawn so it is not too hard to see that if they were redistributed, each square inch would have 8 dots:
[IMG]
The fact that we have 8 dots per square inch means that if we distributed the dots uniformly throughout the square in the right way, we would see 8 dots in each square inch. This prepares students to be able to interpret the constant of proportionality in the next two activities when working with actual houses per square mile or people per square kilometer. When we speak of “500 houses per square kilometer,” we can think of this as, “If we took all of the houses in the region and spread them out uniformly, then we would see 500 houses in every square kilometer.”
Digital
Launch
Display the image of the two squares with dots. Ask students to describe what is the same and what is different about these squares.
Give students 2–3 minutes of quiet work time, followed by 4–5 minutes of partner work time and whole-class discussion.
Action and Expression: Internalize Executive Functions.
Chunk this task into more manageable parts to support students who benefit from support with organizational skills in problem solving. For example, pause to check for understanding after the first 2–3 minutes of work time.
Supports accessibility for: Organization; Attention
Writing, Conversing: MLR5 Co-craft Questions.
Display only the image of the four dot arrays without revealing any of the questions that follow. Give students 1–2 minutes to write a list of possible mathematical questions they could ask about the arrays. Invite students to share their questions with a partner, and then select 2–3 students to share their questions with the whole class. Highlight any questions that refer to how dots are “distributed,” even if students do not use that particular phrase. Finally, reveal the whole problem with text so that students may begin addressing the questions. This helps amplify language related to the distribution of dots.
Design Principle(s): Cultivate conversation; Maximize meta-awareness
Student Facing
The figure shows two arrays, each enclosed by a square that is 2 inches wide.
[IMG]
Let
\(a\)
be the area of the square and
\(d\)
be the number of dots enclosed by the square. For each square, plot a point that represents its values of
\(a\)
and
\(d\)
. Start by choosing an appropriate scale for each axis, using the sliders to adjust the intervals.
Draw lines from
\((0, 0)\)
to each point. For each line, write an equation that represents the proportional relationship.
What is the constant of proportionality for each relationship? What do the constants of proportionality tell us about the dots and squares?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Print
Launch
Display the image of the two squares with dots. Ask students to describe what is the same and what is different about these squares.
Give students 2–3 minutes of quiet work time, followed by 4–5 minutes of partner work time and whole-class discussion.
Action and Expression: Internalize Executive Functions.
Chunk this task into more manageable parts to support students who benefit from support with organizational skills in problem solving. For example, pause to check for understanding after the first 2–3 minutes of work time.
Supports accessibility for: Organization; Attention
Writing, Conversing: MLR5 Co-craft Questions.
Display only the image of the four dot arrays without revealing any of the questions that follow. Give students 1–2 minutes to write a list of possible mathematical questions they could ask about the arrays. Invite students to share their questions with a partner, and then select 2–3 students to share their questions with the whole class. Highlight any questions that refer to how dots are “distributed,” even if students do not use that particular phrase. Finally, reveal the whole problem with text so that students may begin addressing the questions. This helps amplify language related to the distribution of dots.
Design Principle(s): Cultivate conversation; Maximize meta-awareness
Student Facing
The figure shows two arrays, each enclosed by a square that is 2 inches wide.
[IMG]
Let
\(a\)
be the area of the square and
\(d\)
be the number of dots enclosed by the square. For each square, plot a point that represents its values of
\(a\)
and
\(d\)
.
[IMG]
Draw lines from
\((0, 0)\)
to each point. For each line, write an equation that represents the proportional relationship.
What is the constant of proportionality for each relationship? What do the constants of proportionality tell us about the dots and squares?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Activity Synthesis
The goal of this discussion is for students to make sense of the constant of proportionality in the case where the dots are not uniformly distributed. Invite students to share their interpretations of the constants of proportionality.
Consider asking questions like:
“Why is one of the constants of proportionality larger than the other? How can we see this in the picture of the squares?” (There are more dots in the same area.)
“What are the units of the constants of proportionality?” (The number of dots per square inch).
Students will have more opportunities to think about this when working on the activities that follow.


correct_option_index: 1