Curriculum / Math / 9th Grade / Unit 6: Exponents and Exponential Functions / Lesson 7
Math
Unit 6
9th Grade
Lesson 7 of 22
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Write equivalent radical and rational exponent expressions. Identify quantities as rational or irrational.
The core standards covered in this lesson
N.RN.B.3 — Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.
The foundational standards covered in this lesson
8.EE.A.1 — Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 3² × 3<sup>-5</sup> = 3<sup>-3</sup> = 1/3³ = 1/27.
8.EE.A.2 — Use square root and cube root symbols to represent solutions to equations of the form x² = p and x³ = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.
8.NS.A.1 — Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.
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
For the problems below, determine whether each equation is True or False.
a. $${\sqrt{32}=2^{5\over2}}$$
b. $${16^{3\over2}=8^2}$$
c. $${4^{1\over2}=\sqrt[4]{64}}$$
d. $${2^8=(\sqrt[3]{16})^6}$$
e. $${(\sqrt{64})^{1\over3}=8^{1\over6}}$$
MAT.HS.SR.1.00NRN.A.152 from Development and Design: Item and Task Specifications made available by Smarter Balanced Assessment Consortium. © The Regents of the University of California – Smarter Balanced Assessment Consortium. Accessed May 17, 2018, 11:29 a.m..
In each of the following problems, a number is given. If possible, determine whether the given number is rational or irrational. In some cases, it may be impossible to determine whether the given number is rational or irrational. Justify your answers.
a. $${4+\sqrt7}$$
b. $${\sqrt{45}\over\sqrt{5}}$$
c. $${6\over \pi}$$
d. $${\sqrt2 + \sqrt3}$$
e. $${{2+\sqrt{7}}\over{2a+\sqrt{7a^2}}}$$, where $$a$$ is a positive integer
f. $${x+y}$$, where $$x$$ and $$y $$ are irrational numbers
Rational or Irrational?, accessed on May 17, 2018, 11:34 a.m., is licensed by Illustrative Mathematics under either the CC BY 4.0 or CC BY-NC-SA 4.0. For further information, contact Illustrative Mathematics.
A set of suggested resources or problem types that teachers can turn into a problem set
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A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved
Provide a written explanation for each question below.Â
a.  Is it true that $${\left(1000^{1\over3}\right)^3=(1000^3)^{1\over3}}$$? Explain or show how you know.Â
b.  Is it true that $${\left(4^{1\over2}\right)^3=(4^3)^{1\over2}}$$? Explain or show how you know.Â
c.  Suppose that $$m$$ and $$n$$ are positive integers and $$b$$ is a real number so that the principal $$n^{th}$$ root of $$b$$ exists. In general, does $$\left(b^{1\over n}\right)^m=(b^m)^{1\over n}$$? Explain or show how you know.Â
Algebra II > Module 3 > Topic A > Lesson 3 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..
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 6
Lesson 8
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
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
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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