Unit 1: The Number System and Exponents (8.NS/8.EE)

Big Ideas: In this unit, students will know that there are numbers that are not rational, and approximate them by rational numbers. Students should know that numbers that are not rational are called irrational and understand that every number has a decimal expansion. For rational numbers students should show that the decimal expansion repeats eventually. Students should also use rational approximations of irrational numbers to compare the size of irrational numbers and approximately locate them on a number line diagram. (CCSS Grade 8 page 53)

Students will apply properties of the law of exponents to simplify expressions. Students will understand the meaning behind square root and cubed root symbols. Numbers will be expressed in scientific notation so students can compare very large and very small quantities and compute with those numbers.

Overview (Big Ideas), Enduring Understandings, Essential Questions, Common Misconceptions:

Note: Students may need review of fraction operations. Check student understanding and review as needed. Resources are available on the Common Core 6 and Common Core 7 wikis.

Common Core Content Standards:

Know that there are numbers that are not rational, and approximate them by rational numbers.

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.

PARCC Assessment Limit/Clarification:
This standard is part of supportingcontent cluster assessed on PARCC. This content cluster supports the work in the Expressions and Equations domain.

8.NS.A.2. Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., (pi)^2). For example, by truncating the decimal expansion of the square root of 2, show that the square root of two is between 1 and 2, then between 1.4 and1.5, and explain how to continue on to get better approximations.

PARCC Assessment Limit/Clarification:
This standard is part of supportingcontent cluster assessed on PARCC. This content cluster supports the work in the Expressions and Equations and Geometry domains. This cluster is intimately related to the work with radicals (8.EE.A.2) and both of these may be connected to the Pythagorean theorem (8.G.B.6-8) as well as to volume problems (8.G.C.9), e.g., in which a cube has known volume but unknown edge lengths.

Teacher Professional Development Resource:

8.NS.A.2 Approximating Square Roots

Web Resources: http://illuminations.nctm.org/ActivityDetail.aspx?ID=161
NCTM Illuminations Activity: Computing Pi. Students approximate pi by inscribing and circumscribing polygons about a circle and calculating their perimeters.

NCTM Journal Article Lesson Ideas/Resources:
Lewis, Leslie. (April 2007). "Irrational Numbers can "In-Spiral" You." Mathematics Teaching in the Middle School. Vol. 12. No. 8. p. 442-446.
This article shows how to make an artistic project allowing students to visualize, discuss, and create a product that displays irrational and rational numbers.

Brown, R.E. & Owens, A. (August 2009). "Tilted Squares, Irrational Numbers, and the Pythagorean Theorem." Mathematics Teaching in the Middle School. Vol. 15. No. 1. p. 57-62.
This mathematical exploration shows two activities that have students discovering with squares and irrational segment lengths.

Work with radicals and integer exponents. 8.EE.A.1. Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, (3^2)(3^-5)=3^-3=1/27. (Note: Be sure to teach laws of exponents with variable as the base.)

PARCC Assessment Limit/Clarification:
This standard is part of the major content cluster assessed on PARCC.

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 the square root of 2 is irrational.

Disciplinary Literacy Resources:
Green, P. (2011, June 22). On city rooftops, scrappy green spaces in bloom. New York Times. Retrieved from http://www.nytimes.com/2011/06/23/garden/on-city-rooftops-scrappy-green-spaces-in-bloom.html?_r=1&pagewanted=all
There are several stories about various rooftop gardens. You could assign each group a garden to read about and then share with the class to save time. Then, you could assign each group an area for their garden and stipulate that it must be square. Students could "shop" for astro turf for their space and then buy square planters to fill their garden. Students should sketch their design and include specifications for the area of each plant.

PARCC Assessment Limit/Clarification:
This standard is part of the major content cluster assessed on PARCC. By working with equations such as x2=2, students enlarge their concept of number beyond the system of rationals to include irrational numbers. They represent these numbers with radical expressions and approximate these numbers with rationals.

8.EE.A.3. Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. For example, estimate the population of the United States as 3X10^8 and the population of the world as 7X10^9, and determine the world population is more than 20 times larger.

PARCC Assessment Limit/Clarification:
This standard is part of the major content cluster assessed on PARCC. Web Resources: http://illuminations.nctm.org/Reflections_9-12.html
This lesson connects exponents to Alice in Wonderland (how she doubles in size each time she eats an ounce of cake.

8.EE.A.4. Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology.

PARCC Assessment Limit/Clarification:
This standard is part of the major content cluster assessed on PARCC. Examples of Connections to Standards for Mathematical Practices: Scientific notation presents opportunities for strategically using appropriate tools (MP5). For example, the computation (1.73x10-4)(1.73x10-5) can be done quickly with a calculator by squaring 1.73 and then using properties of exponents to determine the exponent of the product by inspection.

Unit 1: The Number System and Exponents (8.NS/8.EE)Big Ideas:In this unit, students will know that there are numbers that are not rational, and approximate them by rational numbers. Students should know that numbers that are not rational are called irrational and understand that every number has a decimal expansion. For rational numbers students should show that the decimal expansion repeats eventually. Students should also use rational approximations of irrational numbers to compare the size of irrational numbers and approximately locate them on a number line diagram. (CCSS Grade 8 page 53)

Students will apply properties of the law of exponents to simplify expressions. Students will understand the meaning behind square root and cubed root symbols. Numbers will be expressed in scientific notation so students can compare very large and very small quantities and compute with those numbers.

Overview (Big Ideas), Enduring Understandings, Essential Questions, Common Misconceptions:Unit 1 Starting Points:*New*Common Core 8 Unit 1 Items to Support Formative Assessment (Alfresco) - HCPSS employees log in with your active directory

McCallum Web Resource:*New*The Number System Progressions Document

Maryland State Department of Education Resources:*New*Grade 8 The Number System

Grade 8 Expressions and Equations

EngageNY Resources:Grade 8 Exponents and Scientific Notation Unit ModuleGrade 8 Exponents and Scientific Notation Unit Module

Note: Students may need review of fraction operations. Check student understanding and review as needed. Resources are available on the Common Core 6 and Common Core 7 wikis.Common Core Content Standards:Know that there are numbers that are not rational, and approximate them by rational numbers.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.

HCPSS Math Task:Discovering Pi

HCPSS UDL Lesson:Rational and Irrational Numbers

PARCC Assessment Limit/Clarification:This standard is part of

supportingcontent clusterassessed on PARCC. This content cluster supports the work in theExpressions and Equationsdomain.Web Resources:http://www.learner.org/courses/learningmath/number/session1/part_c/index.html

"Building a Number Line": This applet shows how the number line is constructed with categories of real numbers.

http://www.studyzone.org/mtestprep/math8/a/numbersl.cfm

Contains possible lesson seeds and practice.

http://rhymenlearn.com/math-rap/rational-irrational-number/

Rap about rational and irrational numbers.

8.NS.A.2. Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., (pi)^2).

For example, by truncating the decimal expansion ofthe square root of 2, show that the square root of two is between 1 and 2, then between 1.4 and1.5, and explain how to continue on to get better approximations.HCPSS Math Task:The Name Game

Patio Predicament

PARCC Assessment Limit/Clarification:This standard is part of

supportingcontent clusterassessed on PARCC. This content cluster supports the work in theExpressions and EquationsandGeometrydomains. This cluster is intimately related to the work with radicals (8.EE.A.2) and both of these may be connected to the Pythagorean theorem (8.G.B.6-8) as well as to volume problems (8.G.C.9), e.g., in which a cube has known volume but unknown edge lengths.Teacher Professional Development Resource:Web Resources:http://illuminations.nctm.org/ActivityDetail.aspx?ID=161

NCTM Illuminations Activity: Computing Pi. Students approximate pi by inscribing and circumscribing polygons about a circle and calculating their perimeters.

NCTM Journal Article Lesson Ideas/Resources:Lewis, Leslie. (April 2007). "Irrational Numbers can "In-Spiral" You."

Mathematics Teaching in the Middle School. Vol. 12. No. 8. p. 442-446.This article shows how to make an artistic project allowing students to visualize, discuss, and create a product that displays irrational and rational numbers.

Brown, R.E. & Owens, A. (August 2009). "Tilted Squares, Irrational Numbers, and the Pythagorean Theorem."

Mathematics Teaching in the Middle School.Vol. 15. No. 1. p. 57-62.This mathematical exploration shows two activities that have students discovering with squares and irrational segment lengths.

Work with radicals and integer exponents.8.EE.A.1. Know and apply the properties of integer exponents to generate equivalent numerical expressions.

For example,(3^2)(3^-5)=3^-3=1/27.(Note: Be sure to teach laws of exponents with variable as the base.)HCPSS UDL Lesson:Laws of Exponents Pt. 1

Laws of Exponents Pt. 2: Zero and Negative Exponents

Discovering Power of a Power and Power of a Quotient

PARCC Assessment Limit/Clarification:This standard is part of the

major content clusterassessed on PARCC.8.EE.A.2. Use square root and cube root symbols to represent solutions to equations of the form

x2=pandx3=p, wherepis a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that the square root of 2 is irrational.Disciplinary Literacy Resources:Green, P. (2011, June 22). On city rooftops, scrappy green spaces in bloom.

New York Times.Retrieved from http://www.nytimes.com/2011/06/23/garden/on-city-rooftops-scrappy-green-spaces-in-bloom.html?_r=1&pagewanted=allThere are several stories about various rooftop gardens. You could assign each group a garden to read about and then share with the class to save time. Then, you could assign each group an area for their garden and stipulate that it must be square. Students could "shop" for astro turf for their space and then buy square planters to fill their garden. Students should sketch their design and include specifications for the area of each plant.

HCPSS UDL Lesson:Cube Hotel

PARCC Assessment Limit/Clarification:This standard is part of the

major content clusterassessed on PARCC. By working with equations such asx2=2, students enlarge their concept of number beyond the system of rationals to include irrational numbers. They represent these numbers with radical expressions and approximate these numbers with rationals.8.EE.A.3. Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other.

For example, estimate the population of the United States as3X10^8and the population of the world as7X10^9, and determine the world population is more than 20 times larger.HCPSS UDL Lesson:Scientific Notation

PARCC Assessment Limit/Clarification:This standard is part of the

major content clusterassessed on PARCC.Web Resources:http://illuminations.nctm.org/Reflections_9-12.html

This lesson connects exponents to Alice in Wonderland (how she doubles in size each time she eats an ounce of cake.

8.EE.A.4. Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology.

HCPSS Math Task:Land Purchases in U.S. History

PARCC Assessment Limit/Clarification:This standard is part of the

major content clusterassessed on PARCC. Examples of Connections to Standards for Mathematical Practices: Scientific notation presents opportunities for strategically using appropriate tools (MP5). For example, the computation (1.73x10-4)(1.73x10-5) can be done quickly with a calculator by squaring 1.73 and then using properties of exponents to determine the exponent of the product by inspection.Web Resources:http://www.realworldmath.org/uploads/8/4/6/7/8467908/writing_in_scientific_notation.kmz

This resource provides a Google Earth lesson in which they use scientific notation to find distances between locations on Earth.

http://spacemath.gsfc.nasa.gov/Modules/8Module3.html

This resource includes information about a dust ring around Saturn.