few_shots: Example 1:
Problem/activity:
Task
Materials
One 10-frame per pair of students (see example below)
[IMG]
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.
shot_info: im_task_001477

K

im_task_000157

3

fl_problem_000674

5
options: Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.

Use properties of operations to generate equivalent expressions.

Perform arithmetic operations on polynomials.

Add and subtract within 20.

Use polynomial identities to solve problems.

Reason about and solve one-variable equations and inequalities.

Write expressions in equivalent forms to solve problems.

Represent and analyze quantitative relationships between dependent and independent variables.

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

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

Represent and solve problems involving addition and subtraction.

Work with radicals and integer exponents.

Write and interpret numerical expressions.

Solve systems of equations.

Apply and extend previous understandings of arithmetic to algebraic expressions.

Interpret the structure of expressions.

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

Rewrite rational expressions.

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

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

Analyze patterns and relationships.

Understand the relationship between zeros and factors of polynomials.

Gain familiarity with factors and multiples.

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

Work with addition and subtraction equations.

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

Represent and solve equations and inequalities graphically.

Represent and solve problems involving multiplication and division.

Solve equations and inequalities in one variable.

Use the four operations with whole numbers to solve problems.

Multiply and divide within 100.

Create equations that describe numbers or relationships

Generate and analyze patterns.

Understand properties of multiplication and the relationship between multiplication and division.
problem_activity: Activity
This activity is a continuation of the previous one. Students match each situation from the previous activity with an equation, solve the equation by any method that makes sense to them, and interpret the meaning of the solution. Students are still using any method that makes sense to them to reason about a solution. In later lessons, a hanger diagram representation will be used to justify more efficient methods for solving. For example, when they are using tape diagrams, they could just say “I subtracted the 9 extra markers and then divided the remaining 21 markers by 7.” Later, when working with hanger diagrams, we can press them to say “I can subtract 9 from each side and then divide each side by 7.”
For each equation, monitor for a student using their diagram to reason about the solution and a student using the structure of the equation to reason about the solution.
Launch
Keep students in the same groups. 5 minutes to work individually or with a partner, followed by a whole-class discussion.
Engagement: Develop Effort and Persistence.
Encourage and support opportunities for peer interactions. Prior to the whole-class discussion, invite students to share their work with a partner. Display sentence frames to support student conversation such as “To find the solution, first I ____ because...”, “I made this match because I noticed...”, “Why did you...?”, or “I agree/disagree because…”
Supports accessibility for: Language; Social-emotional skills
Student Facing
Each situation in the previous activity is represented by one of the equations.
\(7x+9=30\)
\(30=9x+7\)
\(30x+7=9\)
Match each situation to an equation.
Find the solution to each equation. Use your diagrams to help you reason.
What does each solution tell you about its situation?
Student Response
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Student Facing
Are you ready for more?
While in New York City, is it a better deal for a group of friends to take a taxi or the subway to get from the Empire State Building to the Metropolitan Museum of Art? Explain your reasoning.
Student Response
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click here to register or sign in
for free access to Extension Student Response.
Activity Synthesis
For each equation, ask one student who reasoned with the diagram and one who reasoned only about the equation to explain their solutions. Display the diagram and the equation side by side, drawing connections between the two representations. If no students bring up one or both of these approaches, demonstrate maneuvers on a diagram side by side with maneuvers on the corresponding equation. For example, “I subtracted the 9 extra markers and then divided the remaining 21 markers by 7,” can be shown on a tape diagram and on a corresponding equation. It is not necessary to invoke the more abstract language of “doing the same thing to each side” of an equation yet.
Speaking, Representing: MLR3 Clarify, Critique, Correct.
Present an incorrect match of an equation with a situation and/or diagram (e.g.,
\(30 = 9x + 7\)
with Situation 1). Invite students to clarify and then critique the error (e.g., there are 7 packs of an unknown amount of markers in Situation 1; not 9 as illustrated in the equation). Press for details, if needed. For example, students can reference “markers” in Situation 1, not “x”, as the unknown value represent by the variable
\(x\)
. This will help students reflect on the quantity of the unknown values that would be a reasonable solution and improve on their reasoning when matching an equation to a situation.
Design Principle(s): Optimize output (for justification)


correct_option_index: 8