few_shots: Example:
Problem/activity:
Task
Cruz and Erica were both getting ready for soccer.
Cruz ran 1 lap around the school.
Erica ran 3 laps around the playground.
Erica said,
I ran more laps, so I ran farther.
Cruz said,
4 laps around the school is 1 mile, but it takes 12 laps around the playground to go 1 mile. My laps are much longer, so I ran farther.
Who is right? Draw a picture to help you explain your answer.


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. Develop Understanding of fractions as numbers.
Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8.
E. Apply and extend previous understandings of multiplication and division to multiply and divide fractions.
F. 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.


Your response:
Thought: In this problem, students learn to recognize equivalent fractions. That is, three laps around the playground is 3/12 mile, which is equivalent to 1/4, which is one lap around the school.
Answer: B

Now, assign the following problem or activity to one or more relevant mathematical concepts/skills it teaches.
shot_info: im_task_000212

4
options: Analyze patterns and relationships.

Represent and solve equations and inequalities graphically.

Perform arithmetic operations on polynomials.

Work with radicals and integer exponents.

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

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.

Create equations that describe numbers or relationships

Rewrite rational expressions.

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

Use properties of operations to generate equivalent expressions.

Solve systems of equations.

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

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

Gain familiarity with factors and multiples.

Represent and solve problems involving addition and subtraction.

Understand the relationship between zeros and factors of polynomials.

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

Solve equations and inequalities in one variable.

Generate and analyze patterns.

Multiply and divide within 100.

Use the four operations with whole numbers to solve problems.

Represent and solve problems involving multiplication and division.

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

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

Add and subtract within 20.

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

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

Interpret the structure of expressions.

Use polynomial identities to solve problems.

Work with addition and subtraction equations.

Apply and extend previous understandings of arithmetic to algebraic expressions.

Write and interpret numerical expressions.
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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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: 4