7/8 Compacted Mathematics Curriculum Guide

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1 Unit 3: Exponents No Calculators Time Frame: Quarter 1 about 20 days Connections to Previous Learning: Students in Grades 6 and 7 have learned to use expressions, equations and inequalities to represent problem solving situations. Students in Grade 8 will expand upon those skills to include work with very large and very small numbers involving the use of integer exponents. Focus of this Unit: Beginning with familiar number sense topics helps students transition into the Grade 8 content. Turning decimal expansions into fractions and deepening understanding of the meaning of decimal expansions sets a firm foundation for understanding irrational numbers. Students will learn that the square roots of perfect squares are rational numbers, and that the square roots of non-perfect squares, such as 2 or 7, are examples of irrational numbers. Students will understand the value of square roots and cube roots and use this understanding to solve equations involving perfect squares and cubes. Further work with exponents, including scientific notation, naturally flow from the understanding of squares and cubes. Students will learn that the square roots of perfect squares are rational numbers, and that the square roots of non-perfect squares, such as 2 or 7, are examples of irrational numbers. Students will understand the value of square roots and cube roots and use this understanding to solve equations involving perfect squares and cubes. Further work with exponents, including scientific notation, naturally flow from the understanding of squares and cubes. Connections to Subsequent Learning: Solving equations of the form x² = p reminds students about inverse operations which they will need to explore the topics of solving linear and proportional equations later in the year. Exponents and roots also connect closely to work with the Pythagorean Theorem and volume of rounded objects later in Grade 8. Mathematical Practices 1. Make Sense of Problems and Persevere in Solving Them. 2. Reason Abstractly and Quantitatively. 3. Construct Viable Arguments and Critique the Reasoning of Others. 4. Model with Mathematics. 5. Use Appropriate Tools Strategically. 6. Attend to Precision. 7. Look for and Make Use of Structure. 8. Look for and Express Regularity in Repeated Reasoning. Unit 3 Clover Park School District 5/11/16 Page 1

2 Students will be able to independently use their learning to Stage 1 Desired Results Transfer Goals apply concepts and procedures involving irrational numbers, radicals, integer exponents, and scientific notation to represent, interpret and solve problems for realworld and mathematical situations. Ex: A pet fish needs 1000 in 3 of open water. If you want your fish tank to be a cube, what should the side length be? UNDERSTANDINGS Students will understand that Large and small numbers can be expressed in scientific notation to allow for quicker comparison and computation. Roots and powers are inverses of each other. Properties of operations with whole and rational numbers also apply to all real numbers. Students will know Decimals that terminate actually repeat the digit zero. (2.5= ) (8.NS.1) Numbers that repeat in their decimal form are called rational. (8.NS.1) Numbers that do not repeat in their decimal form are called irrational. (8.NS.1) Exponent operation properties. (8.EE.1) The difference between rational and irrational numbers. (8.EE.2) That square and square root are inverses. (8.EE.2) That cube and cube root are inverses. (8.EE.2) The number 2 is irrational. (8.EE.2) The square root of the area of a square represents the side length of the square. (8.EE.2) Meaning Goals ESSENTIAL QUESTIONS Acquisition Goals How are rational numbers used and applied in real-life mathematical situations? When is it appropriate to use exponents? How are roots and powers related? Why do we use different representations for very large and/or very small numbers? Why are quantities represented in multiple ways? Students will be skilled at Distinguish between rational and irrational numbers. (8.NS.1) Convert a decimal expansion which repeats eventually into a rational number. (8.NS.1) Find rational approximations of irrational numbers. (8.NS.2) Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line, and estimate the value of expressions.(8.ns.2) Apply the properties of integer exponents to generate equivalent numerical expressions. (8.EE.1) Evaluate square roots of small perfect squares and cube roots of small perfect cubes. (8.EE.2) Use square root and cube root symbols to solve and represent solutions of equations. (8.EE.2) Unit 3 Clover Park School District 5/11/16 Page 2

3 Calculators 8.NS.1 8.NS.2 8.EE.1 8.EE.2 8.EE.3 8.EE.4 no no no no no no Materials Needed for Unit Holt Course 3 Holt Course 3 Common Core Companion Teacher Student Holt Algebra 1 Additional Materials Holt Course 2 Essential Skills Required for this Unit Stage 1 Established Goals: Common Core State Standards for Mathematics Number System 8.NS.A 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. 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., "_). For example, by truncating the decimal expansion of 2, show that 2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations. Expressions & Equations 8.EE.A 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, 8.EE.A.2 Use square root and cube root symbols to represent solutions to equations of the formx 2 = p and x 3 = 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.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 and the population of the world as , and determine that the world population is more than 20 times larger. 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. Major Clusters Supporting Clusters Additional Clusters Unit 3 Clover Park School District 5/11/16 Page 3

4 Suggested Assessments See Sample Assessments for Unit 3. Vocabulary Fluency Activities Select Activities from math-grades-6-8-fluency-support.pdf as is appropriate. Quizlet: Square Numbers Cubed Numbers Base (8)- When a number is raised to a power, the number that is used as a factor is the base. 3 Cube Root (8)- A number, written as x,whose cube is x. Expanded Form (5)- A multi-digit number is expressed in expanded form when it is written as a sum of single-digit multiples of powers of ten. For example, 643 = Exponent (8)- The number that indicates how many times the base is used as a factor. Exponential Form (8)- A nonlinear function in which the variable is in the exponent. Integer (6)- A member of the set of whole numbers and their opposites. Irrational Number(8)- A number that cannot be expressed as a ratio of two integers or as a repeating or terminating decimal. Perfect Cube (8)- A number whose positive cube root is a whole number. Perfect Squares (8)- A number whose positive square root is a whole number. Pi π (7)- The ratio of the circumference of a circle to the length of its diameter; π 3.14 or Power (8)- A number produced by raising a base to an exponent. Radical (8)- A quantity expressed as the root of another quantity. Radicand(8)- The expression under a radical sign. Rational Numbers (6)- Any number that can be expressed as a ratio of two integers. Real Numbers (8)- A rational or irrational number. Repeating Decimals (6)- A decimal in which one or more digits repeats infinitely. Scientific Notation(8)-A method of writing very large or very small numbers by using powers of 10. Square Root (8)- One of the two equal factors of a number. Terminating Decimal (6)-A decimal is called terminating if its repeating digit is 0. Truncate (8)- to approximate by ignoring all terms beyond a chosen one. Whole Numbers(4)- The numbers 0, 1, 2, 3, 8.NS.A.1 Vocab factoring, real numbers, terminating decimal, repeating decimal, irrational numbers Students understand that Real numbers are either rational or irrational. They distinguish between rational and irrational numbers, recognizing that any number that can be expressed as a fraction is a rational number. The diagram below illustrates the relationship between the subgroups of the real number system. Unit 3 Clover Park School District 5/11/16 Page 4

5 Students recognize that the decimal equivalent of a fraction will either terminate or repeat. Fractions that terminate will have denominators containing only prime factors of 2 and/or 5. This understanding builds on work in 7th grade when students used long division to distinguish between repeating and terminating decimals. Students convert repeating decimals into their fraction equivalent using patterns or algebraic reasoning. One method to find the fraction equivalent to a repeating decimal is shown below. Example 1: Change 0.4 to a fraction. Let x = Multiply both sides so that the repeating digits will be in front of the decimal. In this example, one digit repeats so both sides are multiplied by 10, giving 10x = Subtract the original equation from the new equation. Solve the equation to determine the equivalent fraction. Additionally, students can investigate repeating patterns that occur when fractions have denominators of 9, 99, or 11. Example 2: Unit 3 Clover Park School District 5/11/16 Page 5

6 8.NS.A.2 Vocab expressions, whole numbers, radical, radicand, square root, perfect square, truncate Students locate rational and irrational numbers on the number line. Students compare and order rational and irrational numbers. Students also recognize that square roots may be negative and written as Students can approximate square roots by iterative processes. Examples: Approximate the value of to the nearest hundredth. 5 Solution: Students start with a rough estimate based upon perfect squares. falls between 2 and 3 because 5 falls between 2 2 = 4 and 3 2 = 9. The value will be closer to 2 than to 3. Students continue the iterative process with the tenths place value. falls between 2.2 and 2.3 because 5 falls between = 4.84 and = The value is closer to 2.2. Further iteration shows that the value of is between 2.23 and 2.24 since is and is Compare 2 and 3 by estimating their values, plotting them on a number line, and making comparative statements. Solution: Statements for the comparison could include: o 2 is approximately 0.3 less than 3 o 2 is between the whole numbers 1 and 2 3 is between 1.7 and EE.A.1 Vocab - law of exponents, power, exponent, base (exponent), exponential form, expanded form, integer In 6th grade, students wrote and evaluated simple numerical expressions with whole number exponents (ie. 5ᵌ = = 125). Integer (positive and negative) exponents are further developed to generate equivalent numerical expressions when multiplying, dividing or raising a power to a power. Using numerical bases and the laws of exponents, students generate equivalent expressions. Unit 3 Clover Park School District 5/11/16 Page 6

7 Students understand: Bases must be the same before exponents can be added, subtracted or multiplied. (Example 1) Exponents are subtracted when like bases are being divided (Example 2) A number raised to the zero (0) power is equal to one. (Example 3) Negative exponents occur when there are more factors in the denominator. These exponents can be expressed as a positive if left in the denominator. (Example 4) Exponents are added when like bases are being multiplied (Example 5) Exponents are multiplied when an exponents is raised to an exponent (Example 6) Several properties may be used to simplify an expression (Example 7) Unit 3 Clover Park School District 5/11/16 Page 7

8 8.EE.A.2 Vocab - perfect square (of an integer), square, square root, radical, cube root, perfect cube, rational number, irrational number, Continued on next page. Unit 3 Clover Park School District 5/11/16 Page 8

9 Unit 3 Clover Park School District 5/11/16 Page 9

10 Stage 2 - Evidence Evaluative Criteria/Assessment Level Descriptors (ALDs): 8.NS.A (SBAC Target A) Level 4 students should be able to approximate irrational numbers to a specified level of precision and should be able to use the approximations to solve problems or estimate the value of an expression. Level 3 students should be able to use rational approximations of irrational numbers to locate them on a number line and to make numerical comparisons; convert between fractions and repeating decimals; and compare rational numbers. Level 2 students should be able to identify approximate locations of familiar irrational numbers on a number line; identify numbers as rational or irrational; and convert between fractions and terminating decimals. Level 1 students should be able to identify square roots of numbers less than 100; identify pi as not rational; and understand that every rational number has a decimal expansion. 8.EE.A (SBAC Target B) Level 4 students should be able to use scientific notation and choose units of appropriate size for realistic measurements, solve binomial quadratic and cubic equations, and represent the solution as a square or cube root, respectively. Level 3 students should be able to identify that the square root of 2 is irrational, calculate or approximate to an appropriate degree of precision the square or cube of a rational number, solve quadratic and cubic monomial equations, and represent the solution as a square or cube root, respectively. They should be able to work with and perform operations with scientific notation and work with and apply the properties of integer exponents in order to produce or identify equivalent numerical expressions. Level 2 students should be able to identify and calculate the cube root of familiar perfect cubes and calculate the cube of integers. They should be able to use appropriate tools (e.g., calculator, pencil and paper) to translate large or small numbers from scientific to standard notation. They should be able to work with and apply the properties of integer exponents of degree 2 or less in order to produce or identify equivalent numerical expressions. Level 1 students should be able to identify and calculate square roots of familiar perfect squares and calculate the square of integers. They should be able to translate between standard form and scientific notation. Stage 3 Learning Plan Sample Summary of Key Learning Events and Instruction that serves as a guide to a detailed lesson planning NOTES: LEARNING ACTIVITIES: **Days may change depending on any tasks or assessing you choose to do. 8.NS.A.1 & 8.NS.A.2 Day 1: Rational Numbers Holt Course 3 Lesson 2-1 Day 2: Squares and Square Roots Holt Course 3 Lesson 4-5 Day 3: Estimating Square Roots (1) Holt Course 3 Lesson 4-6 Day 4: The Real Numbers (1) 8.NS.A.1 & 8.NS.A.2 Suggested Performance Tasks Georgia Department Of Education: Rational or Irrational Reasoning? 8.NS.1 & 8.NS.2 Howard County - The Code Name Organizer 8.NS.2 Unit 3 Clover Park School District 5/11/16 Page 10

11 Holt Course 3 Lesson 4-7 Stage 3 Learning Plan Sample Day 5: Extension: Identifying and Graphing Irrational Numbers (CC) p Holt Course 3 Common Core Companion Lesson 4-7A 8.EE.A.1 Day 6: Exponents Holt Course 3: 4-1 Day 7: Look for a Pattern in Integer Exponents Holt Course 3: 4-2 Day 8: Properties of Exponents Holt Course 3: 4-3 Day 9: Evaluate Powers and Roots (optional) Holt Course 3: 4-6 Technology Lab 8.EE.A.1 Suggested Performance Task Georgia Department Of Education: 8.EE.A.1 A Few Folds (pattern) MP8 Alien Attack (discovery of rules) MP7 Nesting Dolls (advance: variables and create formula) MP1 Day 10: Explore Properties of Exponents Holt Algebra 1: 7-3 Algebra Lab Day 11: Multiplication Properties of Exponents Holt Algebra 1: 7-3 Day 12: Division Properties of Exponents Holt Algebra 1: EE.A.2 Day 13: Square and Square Roots (8.EE.2) Holt Course 3: 4-5 Day 14: Cubes & Cube Roots Holt Course 3 p. 830 Common Assessment 8.EE.A.2 Intervention Holt Course 2: 9-7 Square and Square Roots (8.EE.2) Holt Course 2 P. 776 Cube Roots TI Activities Irrational Numbers Scientific Notation Square Roots Unit 3 Clover Park School District 5/11/16 Page 11

12 Stage 3 Learning Plan Sample Daily Lesson Plan Learning Target: Warm-up: Activities: Whole Group: Small Group/Guided/Collaborative/Independent: Whole Group: Checking for Understanding (before, during and after): Assessments: Common Assessments (available on P: drive) Spaced Learning Over Time (SLOT - entrance & exit slips) Checking for understanding (CFU s) Lesson Quizzes Classroom Assessments. SBA Examples Claim 1 Item Specs Target A & B Unit 3 Clover Park School District 5/11/16 Page 12