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: Represent and solve problems involving multiplication and division.

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

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

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

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

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

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

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

Reason about and solve one-variable equations and inequalities.

Use polynomial identities to solve problems.

Write expressions in equivalent forms to solve problems.

Create equations that describe numbers or relationships

Use properties of operations to generate equivalent expressions.

Multiply and divide within 100.

Represent and analyze quantitative relationships between dependent and independent variables.

Generate and analyze patterns.

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

Work with addition and subtraction equations.

Add and subtract within 20.

Represent and solve problems involving addition and subtraction.

Represent and solve equations and inequalities graphically.

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

Rewrite rational expressions.

Analyze patterns and relationships.

Interpret the structure of expressions.

Solve systems of equations.

Perform arithmetic operations on polynomials.

Write and interpret numerical expressions.

Gain familiarity with factors and multiples.

Apply and extend previous understandings of arithmetic to algebraic expressions.

Work with radicals and integer exponents.

Solve equations and inequalities in one variable.

Use the four operations with whole numbers to solve problems.

Understand the relationship between zeros and factors of polynomials.
problem_activity: Narrative
The purpose of this activity is for students to describe the structure of an array by identifying the number of columns, the number of objects in each column, and the total number of objects. They learn that the columns of an array go up and down. Students begin to reason about how they can use the structure of the array to find the total number of objects without counting by 1.
This activity uses
MLR8 Discussion Supports. Advances: speaking, conversing
Representation: Develop Language and Symbols. S
upport understanding of the rows and columns by inviting students to act it out. For example, use students to create the first array (3 rows of 2). Discuss the students who create the rows. Now switch to talking about columns (2 columns of 3). Discuss the difference in which students create the columns.
Supports accessibility for: Visual-Spatial Processing, Organization
Required Materials
Materials to Gather
Counters
Launch
Groups of 2
Give students counters.
“Use your counters to make an array with 3 rows with 4 in each row. Make sure the counters in each row line up with the counters in the other rows.”
Display counters in an array as shown:
[IMG]
“Arrays are made up of rows, but they also have columns that go up and down. How many columns are in this array?” (4)
30 seconds: quiet think time
Share responses.
“How many counters are in each column of the array you made?” (3)
30 seconds: quiet think time
Share responses.
Activity
“Now you will look at a few arrays to decide how many columns they have, how many in each column, and find the total number of counters. Then you will make arrays using counters and answer a few questions.”
MLR8 Discussion Supports
Display sentence frames to support students when they describe the structure of the array:
“There are _____ columns in the array.”
“There are _____ counters in each column.”
“There are _____ counters in all.”
8 minutes: partner work time
Student Facing
[IMG]
How many columns are in this array?
How many counters are in each column?
How many counters are there in all?
[IMG]
How many columns are in this array?
How many counters are in each column?
How many counters are there in all?
Use 10 counters to make 2 columns with the same number in each column.
How many counters are in each column?
How many rows are in the array?
How could you count these counters without counting by ones?
Use 15 counters to make 3 columns with the same number in each column.
How many counters are in each column?
How many rows are in the array?
How many counters are in each row?
How could you count these counters without counting by ones?
Student Response
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Activity Synthesis
Display student work for an array with 10 counters.
“Describe _____’s array.” (It has 10 counters, it has 2 columns and 5 counters in each column. It has 5 rows with 2 counters in each row.)
“How can we prove there are 10 counters in this array?” (We can skip count by 2. We can add
\(2 + 2 + 2 + 2 + 2\)
. We can add
\(5 + 5\)
.)


correct_option_index: 1