Curriculum / Math / 9th Grade / Unit 6: Exponents and Exponential Functions / Lesson 15
Math
Unit 6
9th Grade
Lesson 15 of 22
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Write explicit rules for geometric 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
Suggestions for teachers to help them teach this lesson
Use technology to show the graphs of geometric functions along with their explicit formulas. This is a preview of the work that students will do in upcoming lessons on exponential growth and decay.Â
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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 a geometric sequence is given by $${a_n=a_1(r)^{n-1}}$$, where $$r$$ is the common ratio and $$n$$ is the term number.
Write an explicit formula for the sequence shown in each table.
Act 1: Watch the Act 1 video of the “Incredible Shrinking Dollar.” If Dan shrinks the dollar nine times like this, how big will it be? Will you still be able to see it?
a. Give a guess you know is too big.
b. Give a guess you know is too small.
c. What information would help solve this problem?
Act 2: Share the dimensions of the dollar bill provided on the website.
d. Do the changing lengths of the dollar bill follow a geometric sequence pattern or an arithmetic sequence pattern?
e. Write an explicit formula to represent the length of the dollar bill as it shrinks.
Act 3: Watch Act 3 video.
f. Calculate the actual length of the dollar bill after Dan shrank it nine times.
Incredible Shrinking Dollar by Dan Meyer is licensed under the CC BY 3.0 license. Accessed Feb. 27, 2018, 8:56 a.m..
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
Given the following information, determine the explicit equation for each geometric sequence.Â
a.  $${f(1)=8}$$, common ratio $${r=2}$$
b.  $${f(1)=4,f(n)=3f(n-1)}$$
c.  $${f(n)=4f(n-1);f(1)={5\over3}}$$
Which geometric sequence above has the greatest value at $${f(100)}$$?
Module 1: Sequences from Secondary Mathematics One: An Integrated Approach made available by Mathematics Vision Project under the CC BY 4.0 license. © 2016 Mathematics Vision Project. Accessed May 16, 2018, 4:29 p.m..
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 14
Lesson 16
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
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
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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