Teacher notes
In Activity 1, students use two types
of sequences to model population
growth over time. The assumption
that population grows by a constant
amount each year leads to an arithmetic
sequence. A more realistic
assumption that population grows
by a constant annual percentage
leads to a geometric sequence. The
context in Activity 2 is creditcard
debt. In the first scenario, no interest
is charged and the $200 payments
lead to debt balances that form an
arithmetic sequence. In the second
scenario, interest is charged on the
balance. In this case, the debt balances
form a mixed sequence (a
combination of arithmetic and geometric
sequences). In Activity 3, students
work with sequences that
describe the rows of Pascal’s triangle.
A search for efficient formulas to
calculate various terms in Pascal’s
triangle leads to several of Pascal’s
identities, which appear in his
Treatise on the Arithmetical Triangle.
Finally, the connection is made
between the terms in the nth row of
Pascal’s triangle and combinations
“n choose k” for k = 0, . . . n. Then,
students use the rows of Pascal’s triangle to construct probability models for
the number of heads in n flips of a coin.
The activities in this PullOut address
the following standards from the
Common Core State Standards for High
School Mathematics:
• FIF3: Recognize that sequences are
functions, sometimes defined recursively,
whose domain is a subset of the
integers.
• FBF2: Write arithmetic and geometric
sequences both recursively and with
an explicit formula, use them to model
situations, and translate between the
two forms.
• FLE2: Construct linear and exponential
functions, including arithmetic
and geometric sequences, given a
graph, a description of a relationship,
or two inputoutput pairs (including
reading them from a table).
Mathematics prerequisites and
discussion:
This PullOut can be used either to
introduce sequences or as enrichment
activities on sequences. The PullOut
does not require any prior knowledge of
sequences. Students should have some
background in percentage increase or in
how to apply an interest rate. If not, consult
the Lesson Notes for Activities 1
and 2 for some suggestions. Although
no background in geometric series is
required for this this unit, in Activity 2,
students who have a working knowledge
of geometric series will be able to
find an explicitlydefined formula for
mixed sequences. All other students will
use technology (graphing calculator or
Excel) to generate a list of terms in a
mixed sequence.
Materials needed:
Graphing calculators
or access to Excel. Instructions are provided
in the activities for TI84s.
Teachers will need to adapt those
instructions for other graphing calculators.
Instructions for use of Excel are
provided in the Lesson Notes.
