Curriculum / Math / 9th Grade / Unit 6: Exponents and Exponential Functions / Lesson 12
Math
Unit 6
9th Grade
Lesson 12 of 22
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Write recursive formulas for sequences, including the Fibonacci sequence.
The core standards covered in this lesson
F.BF.A.2 — Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.
F.IF.A.2 — Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
F.IF.A.3 — Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥ 1.
The foundational standards covered in this lesson
8.F.B.4 — Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.
The essential concepts students need to demonstrate or understand to achieve the lesson objective
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Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding
Consider the sequence below.
$${1, \space1,\space 2,\space 3,\space 5,\space 8,\space 13,\space 21, \space34, …}$$
a. Describe the pattern that you notice. How is each next term determined?
b. This pattern is famously called the Fibonacci sequence. Write a recursive formula to represent the Fibonacci sequence. Write a formula in sequence notation using $${a_n}$$ and in formula notation using $${f(n)}$$.
c. If $${a_{15}=610}$$ and $${a_{16}=987}$$, what are the values of $${a_{17}}$$ and $${a_{14}}$$?
For each sequence below, write a recursive formula, in both sequence notation and function notation, to represent the sequence.
a. $${18,\space 13,\space 8,\space 3,\space -2,\space …}$$
b. $${4,\space 12,\space 36,\space 108,\space 324,\space …}$$
A sequence is given by an explicit formula, as shown below, where $$n$$ represents the term number.
$$a_n=4n-1$$ for $$n\geq1$$
a. Write the first five terms in the sequence.
b. Write a recursive formula to represent the sequence.
A set of suggested resources or problem types that teachers can turn into a problem set
Give your students more opportunities to practice the skills in this lesson with a downloadable problem set aligned to the daily objective.
A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved
Consider the sequence following a minus $$8$$ pattern: $${9,\space1,\space-7,\space-15\space,...}$$
Write a recursive formula for the sequence.
Algebra I > Module 3 > Topic A > Lesson 2 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds. © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US license. Accessed Dec. 2, 2016, 5:15 p.m..
Consider the sequence given by the formula $${a(n+1)=5a(n)}$$ and $${a(1)=2}$$ for $${n\geq1}$$
The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.
Lesson 11
Lesson 13
Topic A: Exponent Rules, Expressions, and Radicals
Use exponent rules to analyze and rewrite expressions with non-negative exponents.
8.EE.A.1
Add and subtract polynomial expressions using properties of operations.
A.APR.A.1
Multiply polynomials using properties of exponents and properties of operations.
Solve mathematical applications of exponential expressions.
Use negative exponent rules to analyze and rewrite exponential expressions.
8.EE.A.1 A.SSE.A.2
Define rational exponents and convert between rational exponents and roots.
N.RN.A.1 N.RN.A.2
Write equivalent radical and rational exponent expressions. Identify quantities as rational or irrational.
N.RN.B.3
Simplify radical expressions.
N.RN.A.2
Multiply and divide rational exponent expressions and radical expressions.
N.RN.A.2 N.RN.B.3
Add and subtract rational exponent expressions and radical expressions.
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Topic B: Arithmetic and Geometric Sequences
Describe and analyze sequences given their recursive formulas.
F.BF.A.2 F.IF.A.2 F.IF.A.3
Define arithmetic and geometric sequences, and identify common ratios and common differences in sequences.
F.BF.A.2 F.LE.A.2
Write explicit rules for arithmetic sequences and translate between explicit and recursive formulas.
F.BF.A.2 F.IF.A.3 F.LE.A.2
Write explicit rules for geometric sequences and translate between explicit and recursive formulas.
Topic C: Exponential Growth and Decay
Compare rates of change in linear and exponential functions shown as equations, graphs, and situations.
A.SSE.A.1 F.IF.C.9 F.LE.A.1 F.LE.A.3
Write linear and exponential models for real-world and mathematical problems.
A.SSE.A.1 F.LE.A.1 F.LE.A.2 F.LE.B.5
Graph exponential growth functions and write exponential growth functions from graphs.
F.BF.B.3 F.IF.C.7.E
Write exponential growth functions to model financial applications, including compound interest.
F.IF.C.8.B F.LE.A.2 F.LE.B.5
Write, graph, and evaluate exponential decay functions.
F.BF.B.3 F.IF.C.7.E F.IF.C.8.B F.LE.A.1.C
Identify features of exponential decay in real-world problems.
F.IF.C.8.B F.LE.A.1.C
Solve exponential growth and exponential decay application problems.
F.IF.C.8.B F.LE.A.1 F.LE.A.2
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