options: Work with radicals and integer exponents.

Gain familiarity with factors and multiples.

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

Write and interpret numerical expressions.

Generate and analyze patterns.

Create equations that describe numbers or relationships

Add and subtract within 20.

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

Rewrite rational expressions.

Reason about and solve one-variable equations and inequalities.

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

Solve systems of equations.

Use polynomial identities to solve problems.

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

Use properties of operations to generate equivalent expressions.

Apply and extend previous understandings of arithmetic to algebraic expressions.

Represent and solve problems involving addition and subtraction.

Interpret the structure of expressions.

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

Perform arithmetic operations on polynomials.

Represent and solve equations and inequalities graphically.

Represent and analyze quantitative relationships between dependent and independent variables.

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

Understand the relationship between zeros and factors of polynomials.

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

Write expressions in equivalent forms to solve problems.

Solve equations and inequalities in one variable.

Analyze patterns and relationships.

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

Work with addition and subtraction equations.

Represent and solve problems involving multiplication and division.

Multiply and divide within 100.

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

Use the four operations with whole numbers to solve problems.
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: 25