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

Use place value understanding and properties of operations to add and subtract.

Understand the place value system.

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.

Understand place value.

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.

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

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

Extend the counting sequence.
problem_activity: Narrative
In this activity, students compare pairs of numbers with the same number of digits and the same set of digits (for example, 278 and 872, or 1,356 and 3,156). The goal is to elicit the idea that both the placement and the size of the digits matter in determining the value of a number, and that the digits in certain places matter more than others for making comparisons (MP7).
Required Materials
Materials to Gather
Number cards 0–10
Launch
Groups of 3–4
Give each group a set of cards 0–10. Ask students to remove the cards that show 10.
“What three-digit numbers can we make with 5, 7, and 3?” (357, 375, 537, 573, 735, 753)
Activity
“Work with your group to make pairs of numbers using the digits on the cards. First use 3 cards, then 4, then 5, and lastly 6. Compare each pair of numbers.”
“Think about how you go about comparing each pair of numbers.”
5 minutes: group work time
Monitor for students who:
notice that not all digits need to be compared in order to tell which number in a pair is greater
use place-value language in describing the numbers
make two numbers with the same first digit or the same first and second digits
Student Facing
Your teacher will give you a set of cards, each with a single digit, 0–9.
[IMG]
Use the cards for 2, 7, and 8 to make two different three-digit numbers. Use < or > to compare them.
\(\boxed{\phantom{\frac{00}{00}}} \ \boxed{\phantom{\frac{00}{00}}} \ \boxed{\phantom{\frac{00}{00}}} \ \underline{\hspace{0.5in}} \ \boxed{\phantom{\frac{00}{00}}} \ \boxed{\phantom{\frac{00}{00}}} \ \boxed{\phantom{\frac{00}{00}}}\)
Now include the digit 1 to make two different four-digit numbers. Compare the numbers.
\(\boxed{\phantom{\frac{00}{00}}} _{\ \huge{,}} \ \boxed{\phantom{\frac{00}{00}}} \ \boxed{\phantom{\frac{00}{00}}} \ \boxed{\phantom{\frac{00}{00}}} \ \underline{\hspace{0.5in}} \ \boxed{\phantom{\frac{00}{00}}} _{\ \huge{,}} \ \boxed{\phantom{\frac{00}{00}}} \ \boxed{\phantom{\frac{00}{00}}} \ \boxed{\phantom{\frac{00}{00}}}\)
Shuffle the cards. Repeat what you did earlier with new cards.
Four-digit numbers
\(\boxed{\phantom{\frac{00}{00}}} _{\ \huge{,}} \ \boxed{\phantom{\frac{00}{00}}} \ \boxed{\phantom{\frac{00}{00}}} \ \boxed{\phantom{\frac{00}{00}}} \ \underline{\hspace{0.5in}} \ \boxed{\phantom{\frac{00}{00}}} _{\ \huge{,}} \ \boxed{\phantom{\frac{00}{00}}} \ \boxed{\phantom{\frac{00}{00}}} \ \boxed{\phantom{\frac{00}{00}}}\)
Five-digit numbers
\(\boxed{\phantom{\frac{00}{00}}} \ \boxed{\phantom{\frac{00}{00}}} _{\ \huge{,}} \ \boxed{\phantom{\frac{00}{00}}} \ \boxed{\phantom{\frac{00}{00}}} \ \boxed{\phantom{\frac{00}{00}}} \ \underline{\hspace{0.5in}} \ \boxed{\phantom{\frac{00}{00}}} \ \boxed{\phantom{\frac{00}{00}}} _{\ \huge{,}}\ \boxed{\phantom{\frac{00}{00}}} \ \boxed{\phantom{\frac{00}{00}}} \ \boxed{\phantom{\frac{00}{00}}} \ \)
Six-digit numbers
\(\boxed{\phantom{\frac{00}{00}}} \ \boxed{\phantom{\frac{00}{00}}} \ \boxed{\phantom{\frac{00}{00}}} _{\ \huge{,}} \ \boxed{\phantom{\frac{00}{00}}} \ \boxed{\phantom{\frac{00}{00}}} \ \boxed{\phantom{\frac{00}{00}}} \ \underline{\hspace{0.5in}} \ \boxed{\phantom{\frac{00}{00}}} \ \boxed{\phantom{\frac{00}{00}}} \ \boxed{\phantom{\frac{00}{00}}} _{\ \huge{,}} \ \boxed{\phantom{\frac{00}{00}}} \ \boxed{\phantom{\frac{00}{00}}} \ \boxed{\phantom{\frac{00}{00}}} \)
For each pair you compared, how did you decide which number is greater?
Student Response
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Advancing Student Thinking
When answering the question about how they compared the numbers, students may say that they “just know” which one was greater. Encourage them to think about the features of the numbers that gave them an immediate clue about the size of the numbers, or the features that they didn’t find as useful.
Activity Synthesis
Select students to display their number statements and read them. Ask if the class agrees with their comparison.
“How did you decide which number is greater? Did you compare every digit?”
Select students who wrote numbers with the same first digit (or the same first two digits) to share their number statements. Ask them to explain how they compared the numbers.
If no students mentioned that the digits in some places matter more than those in others, ask them about it.
“Did you pay attention only to some digits but not others?”
“Which ones did you prioritize? Were there any you tended to ignore?”


correct_option_index: 5