Students extend their understanding of properties of exponents to include rational exponents and radicals, and investigate rates of change in linear and exponential sequences and functions.
In Unit 6, Exponents and Exponential Functions, students compare linear and exponential functions in novel ways to reveal new information about and applications of each one. They also extend their understanding of properties of exponents from eighth grade to include rational exponents and radicals.
In Topic A, students recall the properties of exponents and operations that enable complex-looking expressions to be written more simply. Students are briefly introduced to polynomials, and they add, subtract, and multiply polynomials using properties. Students learn how to write rational exponential expressions from radicals, and vice versa, and they hone their skills in simplifying and computing with both forms.
In Topic B, students are introduced to sequences, specifically arithmetic and geometric sequences. They write recursive and explicit formulas for both types of sequences, focusing on precision in their notation and language. By identifying the common difference or ratio and looking at tables and graphs of sequences, students make connections between the rate of growth of a sequence and whether it is linear or exponential.
In Topic C, students make more explicit connections between increasing or decreasing rates of change and exponential functions. They graph exponential functions and transformations of exponential functions, and they write equations for those functions using key features from the graphs. Students look at several real-world applications of exponential growth and decay, in particular the concept of compound interest.
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Properties of exponents | Polynomial, binomial, trinomial | Degree of term or polynomial | Leading coefficient |
Exponential expression | Radical | Root | Radicand |
Index | Recursive formula | Explicit formula | Sequence |
Sequence notation, $${a_n}$$ | Fibonacci sequence | Arithmetic sequence | Geometric sequence |
Common difference | Common ratio | Exponential growth/decay function | Compound interest |
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Internalization of Standards via the Unit Assessment
Internalization of Trajectory of Unit
This assessment accompanies Unit 6 and should be given on the suggested assessment day or after completing the unit.
8.EE.A.1
Use exponent rules to analyze and rewrite expressions with non-negative exponents.
A.APR.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.
8.EE.A.1
Solve mathematical applications of exponential expressions.
A.SSE.A.2
8.EE.A.1
Use negative exponent rules to analyze and rewrite exponential expressions.
N.RN.A.1
N.RN.A.2
Define rational exponents and convert between rational exponents and roots.
N.RN.A.2
Write equivalent radical and rational exponent expressions. Identify quantities as rational or irrational.
N.RN.A.2
Simplify radical expressions.
N.RN.A.2
N.RN.B.3
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.
F.IF.A.2
F.IF.A.3
F.BF.A.2
Describe and analyze sequences given their recursive formulas.
F.IF.A.2
F.IF.A.3
F.BF.A.2
Write recursive formulas for sequences, including the Fibonacci sequence.
F.BF.A.2
F.LE.A.2
Define arithmetic and geometric sequences, and identify common ratios and common differences in sequences.
F.IF.A.3
F.BF.A.2
F.LE.A.2
Write explicit rules for arithmetic sequences and translate between explicit and recursive formulas.
F.IF.A.3
F.BF.A.2
F.LE.A.2
Write explicit rules for geometric sequences and translate between explicit and recursive formulas.
A.SSE.A.1
F.IF.C.9
F.LE.A.1
F.LE.A.3
Compare rates of change in linear and exponential functions shown as equations, graphs, and situations.
A.SSE.A.1
F.LE.A.1
F.LE.A.2
F.LE.B.5
Write linear and exponential models for real-world and mathematical problems.
F.IF.C.7.E
F.BF.B.3
Graph exponential growth functions and write exponential growth functions from graphs.
F.IF.C.8.B
F.LE.A.2
F.LE.B.5
Write exponential growth functions to model financial applications, including compound interest.
F.IF.C.8.B
F.LE.A.1.C
Write, graph, and evaluate exponential decay functions.
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
Solve exponential growth and exponential decay application problems.
Key: Major Cluster Supporting Cluster Additional Cluster
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