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.


Your response:
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.


Your response:
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.


Your response:
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 the place value system.

Use place value understanding and properties of operations to perform multi-digit arithmetic. (A range of algorithms may be used.)

Generalize place value understanding for multi-digit whole numbers.
Grade 4 expectations in this domain are limited to whole numbers less than or equal to 1,000,000.

Extend the counting sequence.

Perform operations with multi-digit whole numbers and with decimals to hundredths.

Work with numbers 11–19 to gain foundations for place value.

Understand place value.

Use place value understanding and properties of operations to perform multi-digit arithmetic.
Grade 4 expectations in this domain are limited to whole numbers less than or equal to 1,000,000.

Use place value understanding and properties of operations to add and subtract.
problem_activity: Narrative
The purpose of this activity is for students to use their understanding of place value to determine the number described in a riddle. Students then write the number of hundreds, tens, and ones, and represent the value as a three-digit number.
MLR8 Discussion Supports.
Display sentence frames to support partner discussion about each number written on the table: “I agree because . . .” or “I disagree because . . . .”
Advances: Conversing
Engagement: Provide Access by Recruiting Interest.
Provide choice. Invite students to decide the order in which they complete the work. They can start with any of the riddles, as long as they complete all of them.
Supports accessibility for: Attention, Social-Emotional Functioning
Required Materials
Materials to Gather
Base-ten blocks
Launch
Groups of 2
Give students access to base-ten blocks.
“I have 4 hundreds, 3 ones, and 2 tens.”
“Which of these shows the total value written as a three-digit number? Explain how you know.”
Display 432, 234, 423.
30 seconds: quiet think time
1 minute: partner discussion
Share responses.
Activity
“You are going to solve number riddles using base-ten blocks.”
As needed, demonstrate the task with a student.
“Take turns reading the clues, while your partner uses blocks to make the number.”
“Make sure you agree before adding each number to the table.”
10 minutes: partner work time
If students finish early, ask them to write their own riddles and trade them with other groups to solve.
Monitor for students who recognize they need a zero when writing the three-digit number in places where there were no tens or no ones.
Student Facing
Solve each riddle and write the three-digit number. Use the table to help you organize the digits.
| riddle | hundreds | tens | ones | three-digit number |
|--------|----------|------|------|--------------------|
|   1    |          |      |      |                    |
|   2    |          |      |      |                    |
|   3    |          |      |      |                    |
|   4    |          |      |      |                    |
|   5    |          |      |      |                    |
|   6    |          |      |      |                    |
I have 2 ones, 7 tens, and 6 hundreds.
I have 3 ones, 5 tens, and 2 hundreds.
I have 7 hundreds, 5 ones, and 3 tens.
I have 5 hundreds, no tens, and 9 ones.
I have 4 ones, 6 tens, and 3 hundreds.
I have 8 tens, 1 hundred, and no ones.
Student Response
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Activity Synthesis
Invite previously selected students to share what they noticed when writing 509 and 180.
As needed, write 81, 801, and 810 and ask, “What is the difference between 81, 801, and 810?”


correct_option_index: 6