Curriculum / Math / 9th Grade / Unit 6: Exponents and Exponential Functions / Lesson 14
Math
Unit 6
9th Grade
Lesson 14 of 22
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Write explicit rules for arithmetic sequences and translate between explicit and recursive formulas.
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.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.
F.LE.A.2 — Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).
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
The formula for an explicit rule for an arithmetic sequence is given by $${a_n=a_1+d(n-1)}$$, where $$d$$ is the common difference and $$n$$ is the term number.
Write an explicit rule for the number of tiles in each pattern shown below.
a.
b.
c.
An arithmetic sequence is shown below.
$${29,\space 22,\space15,\space8,\space1,\space…}$$
An arithmetic sequence is described by either a recursive or explicit rule below. If given a recursive rule, provide the explicit rule. If given the explicit rule, give the recursive rule.
a. $${f(n)=6(n-1)+1}$$, for $${n\geq1}$$
b. $${a_{n+1}=a_n-{3\over2}}$$, $${a_1=10}$$
A set of suggested resources or problem types that teachers can turn into a problem set
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An arithmetic sequence and its recursive formula are shown below.
Sequence: $${2,11,20,29,38,…}$$
Recursive formula: $${a_{n+1}=a_n+9}$$, where $${a_n=2}$$
Your teacher asks you to find the 100th term in the sequence. Max, a student in your class, says that he plans to use the recursive formula to determine the 100th term.Â
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 13
Lesson 15
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
Write recursive formulas for sequences, including the Fibonacci sequence.
Define arithmetic and geometric sequences, and identify common ratios and common differences in sequences.
F.BF.A.2 F.LE.A.2
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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