options: Use polynomial identities to solve problems.

Gain familiarity with factors and multiples.

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

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

Represent and solve problems involving addition and subtraction.

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

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

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

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.

Represent and solve problems involving multiplication and division.

Represent and analyze quantitative relationships between dependent and independent variables.

Use the four operations with whole numbers to solve problems.

Interpret the structure of expressions.

Create equations that describe numbers or relationships

Understand the relationship between zeros and factors of polynomials.

Multiply and divide within 100.

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

Write expressions in equivalent forms to solve problems.

Rewrite rational expressions.

Work with addition and subtraction equations.

Represent and solve equations and inequalities graphically.

Solve systems of equations.

Analyze patterns and relationships.

Write and interpret numerical expressions.

Use properties of operations to generate equivalent expressions.

Reason about and solve one-variable equations and inequalities.

Generate and analyze patterns.

Perform arithmetic operations on polynomials.

Solve equations and inequalities in one variable.

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

Apply and extend previous understandings of arithmetic to algebraic expressions.

Work with radicals and integer exponents.
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: 19