few_shots: Example:
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.


Your response, in json format:
{
  "explanation": "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"
}

Now, assign the following problem or activity to one or more relevant mathematical concepts/skills it teaches.
shot_info: fl_problem_000972

6
options: Understand the connections between proportional relationships, lines, and linear equations.

Use properties of operations to generate equivalent expressions.

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

Represent and solve problems involving addition and subtraction.

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

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

Represent and solve problems involving multiplication and division.

Analyze patterns and relationships.

Use polynomial identities to solve problems.

Work with addition and subtraction equations.

Interpret the structure of expressions.

Perform arithmetic operations on polynomials.

Solve equations and inequalities in one variable.

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

Rewrite rational expressions.

Reason about and solve one-variable equations and inequalities.

Work with radicals and integer exponents.

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

Write expressions in equivalent forms to solve problems.

Solve systems of equations.

Add and subtract within 20.

Gain familiarity with factors and multiples.

Represent and analyze quantitative relationships between dependent and independent variables.

Multiply and divide within 100.

Use the four operations with whole numbers to solve problems.

Represent and solve equations and inequalities graphically.

Create equations that describe numbers or relationships

Write and interpret numerical expressions.

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

Apply and extend previous understandings of arithmetic to algebraic expressions.

Understand the relationship between zeros and factors of polynomials.

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

Generate and analyze patterns.

Understand solving equations as a process of reasoning and explain the reasoning.
problem_activity: Warm-up
This warm-up picks up where the previous lesson ended asking students to calculate the sum of the first 50 terms of two different sequences, but with less scaffolded support. The goal of this activity is to build student flexibility in using the formula for the sum and thinking about the terms in a sequence by comparing two related sequences, specifically, two sequences where one has values double that of the first.
Monitor for students who identify the sum of the second sequence in different ways, such as by
thinking of it as a sequence with the same
\(r\)
and
\(n\)
values as the first but a different
\(a\)
value.
noticing it must be twice the sum of the first.
seeing it as 1 plus the sum of the first sequence, only where the first sequence has
\(n=49\)
instead of
\(n=50\)
.
Student Facing
Recall that for any geometric sequence starting at
\(a\)
with a common ratio
\(r\)
, the sum
\(s\)
of the first
\(n\)
terms is given by
\(s=a\frac{1-r^{n}}{1-r}\)
. Find the approximate sum of the first 50 terms of each sequence:
\(\frac12\)
,
\(\frac14\)
,
\(\frac18\)
,
\(\frac{1}{16}\)
, . . .
1,
\(\frac12\)
,
\(\frac14\)
,
\(\frac18\)
,
\(\frac{1}{16}\)
, . . .
Student Response
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Activity Synthesis
Once students are agreed on the
\(a\)
,
\(r\)
, and
\(n\)
values for the first sequence, invite previously selected students to share their work for the second sequence. While students describe their thinking, record any expressions used for all to see. If not brought up by a student for the second strategy, display the relationship
\(1 \frac{1-\left(\frac12\right)^{n}}{1-\frac12} = 2 \boldcdot \frac12 \frac{1-\left(\frac12\right)^{n}}{1-\frac12}\)
as one way to understand why the second sequence has a sum twice as large.


correct_option_index: 18