few_shots: Example 1:
Problem/activity:
Task
Materials
One 10-frame per pair of students (see example below)
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Counters, 10 per pair of students
Decks of number cards 0-5 (with 4 of each number) or two 0-5 dice
Setup
This activity is done in pairs. The students sit side by side with 10-frame and deck of cards or dice. If they have cards, the cards are shuffled and placed face down in front of them.
Action
Player 1 flips a card or rolls the die. They place that many counters on the 10-frame. Player 2 flips a card or rolls the die. They place that many more counters on the 10-frame next to the counters already on the ten frame. The students count the total together.
A recording sheet can be added when the students have been introduced to equations and number sentences. For those students who can handle higher numbers, a deck of 0-10 cards and two 10-frames can be used.
This game can also be played alone.


Mathematical concept/skill options: 
A. Write and interpret numerical expressions.
B. Generate and analyze patterns.
C. Analyze patterns and relationships.
D. Use properties of operations to generate equivalent expressions.
E. Use the four operations with whole numbers to solve problems.
F. Work with radicals and integer exponents.
G. Solve systems of equations.
H. Represent and solve problems involving multiplication and division.
I. Perform arithmetic operations on polynomials.
J. Understand the relationship between zeros and factors of polynomials.
K. Understand properties of multiplication and the relationship between multiplication and division.
L. Solve real-life and mathematical problems using numerical and algebraic expressions and equations.
M. Solve problems involving the four operations, and identify and explain patterns in arithmetic.
N. Use polynomial identities to solve problems.
O. Apply and extend previous understandings of arithmetic to algebraic expressions.
P. Rewrite rational expressions.
Q. Interpret the structure of expressions.
R. Add and subtract within 20.
S. Analyze and solve linear equations and pairs of simultaneous linear equations.
T. Represent and analyze quantitative relationships between dependent and independent variables.
U. Solve equations and inequalities in one variable.
V. Create equations that describe numbers or relationships
W. Reason about and solve one-variable equations and inequalities.
X. Represent and solve equations and inequalities graphically.
Y. Understand solving equations as a process of reasoning and explain the reasoning.
Z. Understand and apply properties of operations and the relationship between addition and subtraction.
AA. Gain familiarity with factors and multiples.
AB. Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.
AC. Work with addition and subtraction equations.
AD. Multiply and divide within 100.
AE. Work with equal groups of objects to gain foundations for multiplication.
AF. Represent and solve problems involving addition and subtraction.
AG. Write expressions in equivalent forms to solve problems.
AH. Understand the connections between proportional relationships, lines, and linear equations.


Thought: This task teaches students to understand addition as putting together counters, or adding to already placed counters.
Answer: AB

Example 2:
Problem/activity:
Task
 Arrange the fractions in order from least to greatest. Explain your answer with a picture.
 $\frac{1}{5}, \frac{1}{7}, \frac{1}{3} $
 $\frac{2}{5}, \frac{2}{7}, \frac{2}{3} $
 $\frac{5}{6}, \frac{3}{6}, \frac{1}{6} $
 $\frac{5}{12}, \frac{8}{12}, \frac{4}{12} $

Mathematical concept/skill options: 
A. Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.
Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.
B. Extend understanding of fraction equivalence and ordering.
Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.
C. Use equivalent fractions as a strategy to add and subtract fractions.
D. Apply and extend previous understandings of multiplication and division to multiply and divide fractions.
E. Understand decimal notation for fractions, and compare decimal fractions.
Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.
F. Develop Understanding of fractions as numbers.
Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8.


Thought: This problem involves understanding fractions as numbers by comparing their quantities.
Answer: F

Example 3:
Problem/activity:
Problem 1
Solve using the standard algorithm. If you get stuck, use an area model and/or the partial products algorithm to help. Then assess the reasonableness of your answer.
a.
$$5,803\times46$$
b.
$$63\times2,116$$
Problem 2
Dunkin' introduced a croissant doughnut in 2014. In the first three months of its debut, they sold 7,870 dozen of them per day. How many croissant doughnuts were sold in the first three months?


Mathematical concept/skill options: 
A. Use place value understanding and properties of operations to perform multi-digit arithmetic. (A range of algorithms may be used.)
B. Understand place value.
C. Understand the place value system.
D. Extend the counting sequence.
E. Work with numbers 11–19 to gain foundations for place value.
F. 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.
G. Perform operations with multi-digit whole numbers and with decimals to hundredths.
H. 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.
I. Use place value understanding and properties of operations to add and subtract.


Thought: This set of problems asks students to perform multiplication, which is an operation, on multi-digit whole numbers.
Answer: G

Now, assign the following problem or activity to one or more relevant mathematical concepts/skills it teaches.
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options: Represent and solve problems involving addition and subtraction.

Analyze patterns and relationships.

Solve systems of equations.

Represent and analyze quantitative relationships between dependent and independent variables.

Represent and solve equations and inequalities graphically.

Use the four operations with whole numbers to solve problems.

Write and interpret numerical expressions.

Solve equations and inequalities in one variable.

Write expressions in equivalent forms to solve problems.

Rewrite rational expressions.

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

Apply and extend previous understandings of arithmetic to algebraic expressions.

Perform arithmetic operations on polynomials.

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

Gain familiarity with factors and multiples.

Generate and analyze patterns.

Use polynomial identities to solve problems.

Represent and solve problems involving multiplication and division.

Work with radicals and integer exponents.

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

Create equations that describe numbers or relationships

Reason about and solve one-variable equations and inequalities.

Use properties of operations to generate equivalent expressions.

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

Understand the relationship between zeros and factors of polynomials.

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

Work with addition and subtraction equations.

Interpret the structure of expressions.

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

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

Add and subtract within 20.

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

Multiply and divide within 100.

Understand and apply properties of operations and the relationship between addition and subtraction.
problem_activity: Activity
This activity continues the work of using a balanced hanger to develop strategies for solving equations. Students are presented with balanced hangers and are asked to write equations that represent them. They are then asked to explain how to use the diagrams, and then the equations, to reason about a solution. Students notice the structure of equations and diagrams and find correspondences between them and between solution strategies.
Launch
Draw students’ attention to the diagrams in the task statement. Ensure they notice that the hangers are balanced and that each object is labeled with its weight. Some weights are labeled with numbers but some are unknown, so they are labeled with a variable.
Keep students in the same groups. Give 5–10 minutes of quiet work time and time to share their responses with a partner, followed by a whole-class discussion.
Student Facing
Here are some balanced hangers where each piece is labeled with its weight. For each diagram:
Write an equation.
Explain how to figure out the weight of a piece labeled with a letter by reasoning about the diagram.
Explain how to figure out the weight of a piece labeled with a letter by reasoning about the equation.
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Student Response
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Activity Synthesis
Invite students to demonstrate, side by side, how they reasoned with both the diagram and the equation. For example, diagram A can be shown next to the equation
\(7=3x+1\)
. Cross out a piece representing 1 from each side, and write
\(7-1=3x+1-1\)
, followed by
\(6=3x\)
. Encircle 3 equal groups on each side, and write
\(6 \div 3 = 3x \div 3\)
, followed by
\(2=x\)
. Repeat for as many diagrams as time allows. If diagrams A and B did not present much of a challenge for students, spend most of the time on diagrams C and D.
We want students to walk away with two things:
An instant recognition of the structure of equations of the form
\(px+q=r\)
where
\(p\)
,
\(q\)
, and
\(r\)
are specific, given numbers.
A visual representation in their minds that can be used to support understanding of why for equations of this type, you can subtract
\(q\)
from each side and then divide each side by
\(p\)
to find the solution.
Representation: Develop Language and Symbols.
Create a display of important terms and vocabulary. Include the following terms and maintain the display for reference throughout the unit: hanger diagram. For example, display an example of a balanced hanger. With class participation, create step-by-step instructions on how to write and solve an equation based on the hanger.
Supports accessibility for: Memory; Language
Representing, Speaking: MLR8 Discussion Supports.
To invite participation in the whole-class discussion, use sentence frames to support students’ explanations for how they found the weights by reasoning about the hangers and equations. For example, provide the frame “First, I ___ because . . .”, “Then I ___ because . . . ." Be sure to verbalize and amplify mathematical language in the students’ explanations (e.g., “subtracting the constant”, and “dividing by the coefficient”). This will help students explain their reasoning with the diagram and the equation.
Design Principle(s): Optimize output (for explanation); Cultivate conversation


correct_option_index: 13