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 add and subtract.

Understand the place value system.

Understand place value.

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

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

Extend the counting sequence.

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.

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.
problem_activity: Narrative
The purpose of this activity is for students to find combinations of coins with a value of 100 cents and understand that a dollar is the same as 100 cents. They add within 100 by grouping like coins, using a ten, or using counting on strategies. They find coin combinations that add up to 100 cents or 1 dollar and write the leftover amount as cents. This work builds on students’ understanding of a hundred.
Engagement: Develop Effort and Persistence.
Chunk the tasks by question and check in with students to provide feedback and encouragement after each question. Look for the students to be using a strategy and ask them to verbalize their solution to ensure they are on the right track.
Supports accessibility for: Language; Organization
Required Preparation
Add a dollar bill to the money poster showing the front and back to display in the launch.
Launch
Groups of 2
Display the poster with the dollar added.
[IMG]
“We just saw that there are many different ways to use coins to make 100 cents. One dollar is also the same as 100 cents.”
“If I have a dollar and I found a quarter, how much money would I have?” (125 cents or \$1 and 25¢)
30 seconds: quiet think time
Display a dollar and a quarter and write $1 and 25¢.
[IMG]
“When you write dollar amounts you use the \$ symbol. Use the ¢ symbol for the amount that cannot be grouped to make a dollar.”
Activity
“Now, you are going to work with a partner to make dollars and cents.”
10 minutes: partner work time
Monitor for students who show Priya's money in different ways to share in the Lesson Synthesis.
Student Facing
[IMG]
Andre emptied his pockets and found these coins.
[IMG]
How much money does he have? Show your thinking.
Han emptied his pockets and found these coins.
[IMG]
How much money does he have? Show your thinking.
Priya has $1 and 18¢ in her pocket.
If Priya only had coins in her pocket, what coins could she have?
Represent Priya’s coins:
If Priya had 1 dollar bill and some coins, what coins could she have?
Represent Priya's money:
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Advancing Student Thinking
If students show the value of Andre and Han's coins only in cents, consider asking:
“How did you find the total value of the coins?”
“How many dollars does _______ have? How many leftover cents would he have?”
“How could you say how much money _______has using dollars and cents?”
Activity Synthesis
“I noticed that when you were creating a coin collection for Priya, there were different solutions.”
“How is it possible that everyone represented the same amount of money?” (Different coins can show the same amount, but you need a different number of coins.)
Invite students to share different combinations of coins they used to make $1.
Share and record responses.


correct_option_index: 0