The Number System Unit Guide Math 8 Big Idea (Cluster): Know that there are numbers that are not rational, and approximate them by rational numbers (8.NS.1-2, 8.EE.2, and 8.G.6-8) Edited 6/10/14 Renton School District The Number System (8.NS.1-8.NS.2, 8.EE.2) Know that there are numbers that are not rational, and approximate them by rational numbers Standard 8.NS.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. Standard 8.NS.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., π2). For example, by truncating the decimal expansion of √2 (square root 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. Standard 8.EE.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. (The cube root content within this standard will be taught alongside the three-dimensional geometry standard 8.G.9 where students must know the formulas for volume of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.) 2 This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment. (8.NS.1-2, 8.EE.2) Relevant Math Practices and Student Actions MP1 Make sense of problems and persevere in solving them. Perseveres when solving problems. Understands what the problem asks and the relationship among the problem’s parts. Makes a plan for solving a problem. Checks and explains why a solution is reasonable. MP 2 Reason abstractly and quantitatively. Interprets problems in context. Uses representations to make meaning of problems. Translates a problem from situation to equation. Explains connections between equation and situation. Understands the meaning of quantities and units. MP 3 Construct viable arguments and critique the reasoning of others. Uses definitions and draws on prior mathematical knowledge when constructing arguments. Makes conjectures and evaluates their accuracy. Justifies conclusions with mathematical evidence and responds to arguments of others. Asks clarifying and probing questions to improve argument. MP 5 Use appropriate tools strategically. Selects tools strategically for visualizing, exploring, comparing, predicting, and solving problems. MP 6 Attend to precision. Communicates mathematical thinking accurately both orally and in writing. Understands the meaning of mathematical symbols and vocabulary and uses them appropriately. Labels consistently and accurately graphs and diagrams. Specifies the units of measure when labeling. Calculates accurately and efficiently. Pursues a level of precision appropriate to the context of the problem. MP 7 Look for and make use of structure. Looks for, identifies, develops and generalizes patterns and relationships. Makes connections to prior mathematical knowledge to solve new problems. MP 8 Look for and express regularity in repeated reasoning. Notices repeated calculations and looks for general methods and shortcuts to solve a problem. MP 4 Model with mathematics. Identifies patterns to develop algorithm, formula, or calculation. Applies prior mathematical knowledge to describe, analyze, and solve Evaluates reasonableness of results. problems arising in everyday life, society and workplace. Makes assumptions and approximations to make problems easier. Checks to see if an answer makes sense within the context of a situation and improves model when necessary. 3 This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment. (8.NS.1-2, 8.EE.2) SBAC Required Evidence (Claim 1) 8.NS.1-8.NS.2 The student classifies real numbers as rational or irrational (required evidence #1). The student converts a repeating decimal into a fraction (required evidence #2). The student writes approximations of irrational numbers as rational numbers (required evidence #3). The student compares the sizes of irrational numbers by using rational approximations of irrational numbers (required evidence #4). The student approximates the locations of irrational numbers on the number line by using rational approximations of irrational numbers (required evidence #5). A calculator is not an allowable tool for this cluster of standards on an assessment. For more information on the assessment of this set of standards, read the Grade 8 Claim 1 Target A Item Specifications. 8.EE.2 (Calculator allowed on some tasks for this target on Smarter Balanced assessment. See Target B Item Specifications for detail.) The student represents solutions to equations of the form x2 = p and x3 = p, where p is a positive rational number that is not a perfect square or a perfect cube, using square root and cube root symbols (required evidence #2). The student represents solutions to equations of the form x2 = p and x3 = p, where p is a positive rational number that is a perfect square or a perfect cube, and evaluates square roots of small perfect squares and cube roots of small perfect cubes (required evidence #3). (The cube root content within this standard will be taught alongside the three-dimensional geometry standard 8.G.9 where students must know the formulas for volume of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.) A calculator is not an allowable tool for this cluster of standards on an assessment. For more information on the assessment of this set of standards, read the Grade 8 Claim 1 Target B Item Specifications. 4 This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment. (8.NS.1-2, 8.EE.2) Vocabulary Mathematically proficient students communicate precisely by engaging in discussions about their reasoning using appropriate mathematical language. Students should learn the following terms with increasing precision within the cluster. The bolded terms will be used on Smarter Balanced assessment items. Approximate Decimal expansion Estimate Exponent Expressions Integers Irrational number Non-perfect squares Non-square rectangle Number line Perfect square Pi Power Radical Radical equations Rational number Real numbers Repeating decimal Square Square root Terminating decimal (The cube root content within this cluster of standards will be taught alongside the three-dimensional geometry standard 8.G.9 where students must know the formulas for volume of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.) 5 This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment. (8.NS.1-2, 8.EE.2) The Number System (8.NS.1-8.NS.2, 8.EE.2) Know that there are numbers that are not rational, and approximate them by rational numbers Standard 8.NS.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. See Grade 8 Common Core Flip Book pages 4-6 for explanations and examples of this standard. Learning Objectives Convert between fractions and repeating decimals (SBAC Grade 8 Claim 1 Target A Achievement Level Descriptor Level 3). Classify a number as rational or irrational based on its decimal expansion. Justify a square root of a non-perfect square will be irrational. Know that real numbers that are not rational are irrational. SBAC Required Evidence (Claim 1) The student classifies real numbers as rational or irrational (required evidence #1). The student converts a repeating decimal into a fraction (required evidence #2). A calculator is not an allowable tool for this cluster of standards on an assessment. For more information on the assessment of this set of standards, read the Grade 8 Claim 1 Target A Item Specifications. Connections to Curricular Materials CMP2 does not have a unit that supports the 8.NS.1-2 cluster of standards. The Grade 8 curriculum team has suggested supplemental resources in the pacing guide below and within the additional resources section. These resources have been marked with an asterisk (*). Connections to Prior Learning 7.NS.2 a-d Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. During Grade 7, students will convert fraction and mixed numbers to repeating or terminating decimals using long division. In Grade 7, students also develop an understanding of the rational number set to include all values that can be written as a/b where a and b are integers and b ≠ 0. Additional Resources/Technology Resources Repeating Decimals additional task (MAP)* Terminating and Repeating Decimals lesson* Classifying Numbers PPT (SlideShare.doc)* Classifying Numbers II PPT (NMS)* Real Number Race task (NCDPI)* Roots and Classifying Real Numbers student task (NMS) Exponents and Equations module (georgiastandards.org) 6 This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment. (8.NS.1-2, 8.EE.2) Explanations and Examples Students should be able to describe, distinguish and classify rational and irrational numbers. Students can use a graphic organizer to show the relationship between the subsets of the real number system. Any number that can be expressed as a fraction is a rational number. 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 should also have an understanding of irrational values such as, pi, e, and square roots of nonperfect squares, and how to describe them. Students should understand the decimal expansion of an irrational number neither terminates nor repeats. Students will convert repeating decimals into their fraction equivalent using patterns or algebraic reasoning. Students may investigate repeating patterns for denominators of 9, 99, and 11. One method of converting a repeating decimal to a fraction algebraically is shown below. Change ̅ to a fraction. Set x = ̅ Then, multiply both sides of the equation by 10 so the repeating value becomes the whole number. The new equation will be 10x = ̅ . Subtract both equations and solve for x. If the repeating pattern has more than one repeating decimal value, multiply the original equation by the power of 10 for the number of repeating decimal places. 7 This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment. (8.NS.1-2, 8.EE.2) The Number System (8.NS.1-8.NS.2, 8.EE.2) Know that there are numbers that are not rational, and approximate them by rational numbers Standard 8.NS.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., π2). For example, by truncating the decimal expansion of √2 (square root 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. See Grade 8 Common Core Flip Book page 7 for explanations and examples of this standard. Learning Objectives Use rational approximations of irrational numbers to locate them on a number line (SBAC Grade 8 Claim 1 Target A Achievement Level Descriptor Level 3). Use rational approximations of familiar irrational numbers to make numerical comparisons (SBAC Grade 8 Claim 1 Target A Achievement Level Descriptor Level 3). Compare rational and irrational numbers (SBAC Grade 8 Claim 1 Target A Achievement Level Descriptor Level 3). Approximate irrational numbers between two integers to a specified level of precision (SBAC Grade 8 Claim 1 Target A Achievement Level Descriptor Level 4). Use approximations to solve problems or estimate the value of an expression (SBAC Grade 8 Claim 1 Target A Achievement Level Descriptor Level 4). SBAC Required Evidence (Claim 1) The student writes approximations of irrational numbers as rational numbers (required evidence #3). The student compares the sizes of irrational numbers by using rational approximations of irrational numbers (required evidence #4). The student approximates the locations of irrational numbers on the number line by using rational approximations of irrational numbers (required evidence #5). Connections to Prior Learning In Grades 6 and 7, students will have graphed positive and negative rational values on a number line. 7.NS.2 a-d Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. A calculator is not an allowable tool for this cluster of standards on an assessment. For more information on the assessment of this set of standards, read the Grade 8 Claim 1 Target A Item Specifications. 8 This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment. (8.NS.1-2, 8.EE.2) Connections to Curricular Materials Additional Resources/Technology Resources CMP2 does not have a unit that supports the 8.NS.1-2 cluster of standards. The Grade 8 curriculum team has suggested supplemental resources in the pacing guide below and within the additional resources section. These resources have been marked with an asterisk (*). Approximating Square Roots with tiles video Approximating Square Roots (Khan Academy) video Patio Predicament task (HCPSS)* The Code Name Organizer task (HCPSS)* Roots and Classifying Real Numbers student task (NMS)* The Laundry Problem task (NCDPI) Exponents and Equations module: Rational or Irrational Reasoning (georgiastandards.org) Number Systems Progression of Learning Document Illustrative Mathematics tasks Khan Academy Common Core Number Systems student practice site Instructional Strategies Students will 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 with irrational solutions. Students should have experiences identifying the perfect squares the irrational value is between. Then using either a decimal or fraction estimation for the irrational value, students should be able to compare, order, or locate the irrational value on a number line. Students will need to know how to find the irrational value to the nearest tenth and hundredth and explain their process. The two videos in the Additional Resources for 8.NS.2 demonstrate how to find an estimated fraction and decimal value for an irrational number. 9 This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment. (8.NS.1-2, 8.EE.2) The Number System (8.NS.1-8.NS.2, 8.EE.2) Know that there are numbers that are not rational, and approximate them by rational numbers Standard 8.EE.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. (The cube root content within this standard will be taught alongside the three-dimensional geometry standard 8.G.9 where students must know the formulas for volume of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.) See Grade 8 Common Core Flip Book page 11 for explanations and examples of this standard. Learning Objectives Identify that the is irrational (SBAC Grade 8 Claim 1 Target B Achievement Level Descriptor Level 3). Calculate or approximate to an appropriate degree of precision the square or cube root of a rational number (SBAC Grade 8 Claim 1 Target B Achievement Level Descriptor Level 3). Solve simple quadratic monomial equations (ex. 4x2) and represent solution as a square root (SBAC Grade 8 Claim 1 Target B Achievement Level Descriptor Level 3). Solve simple cubic monomial equations (ex. 27x3) and represent the solution as a cube root (SBAC Grade 8 Claim 1 Target B Achievement Level Descriptor Level 3). Represent the solution to equations using square root and cube root symbols. Understand that all non-perfect square roots and cube roots are irrational. SBAC Required Evidence (Claim 1) The student represents solutions to equations of the form x2 = p and x3 = p, where p is a positive rational number that is not a perfect square or a perfect cube, using square root and cube root symbols. The student represents solutions to equations of the form x2 = p and x3 = p, where p is a positive rational number that is a perfect square or a perfect cube, and evaluates square roots of small perfect squares and cube roots of small perfect cubes. Connections to Prior Learning In Grade 6, students will have learned to write and evaluate numerical expressions involving whole-number exponents. In Grades 5 and 6, students will have calculated area of two-dimensional figures and the volume of a rectangular prism with positive rational edge lengths. 10 This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment. (8.NS.1-2, 8.EE.2) For more information on the assessment of this standard, read the Claim 1 Target B SBAC Item Specifications. Connections to Curricular Materials CMP2 does not have a unit that supports the 8.NS.1-2 cluster of standards. The Grade 8 curriculum team has suggested supplemental resources in the pacing guide below and within the additional resources section. These resources have been marked with an asterisk (*). Additional Resources/Technology Resources Roots and Classifying Real Numbers student task (NMS)* Square Roots using area and side length (NMS)* Square Roots PPT (NJCTL)* Pythagorean Theorem module (Newark PS) Geometric Applications of Exponents module (georgiastandards.org) Exponents and Equations module (georgiastandards.org) Expressions and Equations Progression of Learning Document Illustrative Mathematics tasks Khan Academy Common Core Expressions and Equations practice site 11 This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment. (8.NS.1-2, 8.EE.2) Instructional Strategies Students may find prime factorization of the radicand to be helpful when solving for a square root or cube root. Students should have experiences with square roots geometrically and represent the square roots of a number as the side length of a square whose area is equal to the radicand. For example, students represent using the following diagram: Use prime factorization to find the = 5. 125 5 25 5 5 12 This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment. (8.NS.1-2, 8.EE.2) Grade 8 Number System Unit Pacing Guide for 2014-2015 Looking for Pythagoras and Alignment Notes CCSS-M Aligned Lessons Converting fraction to decimal This lesson would be a review for Common Core standard 7.NS.2d. The Grade 7 standard (Supplemental lesson) focuses on converting a fraction to terminating or repeating decimal. The Terminating and Repeating Decimals lesson will help students develop a sense of what fraction values will terminate as decimals and which will repeat as decimals. The Repeating Decimals task from MAP will cover both this lesson on changing from fraction to decimal and the next lesson of converting a repeating decimal back to a fraction. Converting decimal to fraction The Repeating Decimals task from MAP will cover converting a repeating decimal back to a (Supplemental lesson) fraction. Students will need instruction on the algorithm for converting a repeating decimal to a fraction. Converting a repeating decimal to a fraction (rational number) will be a launch into to teaching the Real Number System with rational and irrational numbers. Classifying numbers in the Students will need instruction on the meaning of the rational and irrational numbers within the Real Number System as Real Number System and how they are classified. The Classifying Numbers PowerPoint and rational, irrational, integer, and Classifying Numbers II PowerPoint may help to supplement this content. The Real Number whole number Race task from NCDPI will provide an opportunity for students to make sense of classifying and (Supplemental lesson) identifying numbers as rational, irrational, and a real number. Squares and square roots Students will begin to develop their understanding of squares and square roots before beginning (Supplemental lesson) Looking for Pythagoras. A Square Roots PowerPoint from NJCTL may support your student understanding of squares and square roots. A student task has been provided that connects squares and square roots using area of square and side lengths. These lessons will support some of the missing content in Looking for Pythagoras. Rational approximations for The Patio Predicament task (HCPSS) will develop a connection between 8.EE.2 and 8.NS.2. irrational values and ordering Following the Patio Predicament, The Code Name Organizer (HCPSS) will have students values on a number line practice finding rational approximations for irrational numbers and graphing them on a number (Supplemental Lessons) line. From there, the Square Roots and Classifying Real Numbers student task will tie 8.NS.1, 8.NS.2 and 8.EE.2 together. Additional tasks that will support the development of rational approximations for irrational numbers and ordering on a number line is the The Laundry Problem task (NCDPI). The concept of rational approximations for irrational numbers will be further developed in Looking for Pythagoras when students explore tilted squares. CCSS-M Standards 7.NS.2d 8.NS.1 8.NS.1 8.NS.1 8.EE.2 8.NS.2 13 This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment. (8.NS.1-2, 8.EE.2) The Number System Common Assessment (8.NS.1-8.NS.2 and 8.EE.2) For understanding the content required to be assessed, see the rubrics at the end of the unit guide 8.NS.1, 8.NS.2, which is based on the SBAC Grade 8 Claim 1 item specifications for Target A and B. Based on and 8.EE.2 the SBAC item specifications for Target A and B, a calculator is an not allowable tool during the assessment. *SBAC ALD Level 3 means Smarter Balanced Assessment Consortium Achievement Level Descriptors Level 3 criterion for the SBAC assessment. Hard copies of these lessons can be found at http://staff.rentonschools.us/renton/secondary-math/math-8-ccss-m-resources or within the underlined hyperlinks. The following resources were used to create this curriculum guide: SBAC Claim 1 Target A and BItem Specifications, Grade 8 Common Core State Standards Flip Book compiled by Melisa Hancock, and NC 8th grade Mathematics Unpacked Contents. 14 This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment. (8.NS.1-2, 8.EE.2) Standard 8.NS.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. 4 In addition to being proficient on the standard, student can demonstrate one or more of the following: Applies understanding of standard to unfamiliar situations and/or to solve complex problems. Use precise and relevant communication to justify mathematical thinking. Connects knowledge to other learning targets and/or advance problem sets. 8.NS.1 3 Classify and define real numbers as rational or irrational. Convert rational number to terminating and repeating decimals. Convert repeating decimal into a fraction. 2 Convert rational number to terminating or repeating decimals. Classify real numbers as rational or irrational. Convert simple repeating decimal into a fraction. 1 With help, student has minimal success with classify real numbers as rational or irrational. With help, student has partial success with conversion process. Convert a simple repeating decimal into a fraction with omissions and/or misconceptions. For example, provide a clear explanation of the process of converting a repeating decimal to a fraction. Rubric constructed from the SBAC Claim 1 item specifications and SBAC Achievement Level Descriptors. A PLC may decide to further develop these rubrics. For more information on the assessment of this standard, read the Claim 1 SBAC Item Specifications-Target A. 15 This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment. (8.NS.1-2, 8.EE.2) Standard 8.NS.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., π2). For example, by truncating the decimal expansion of √2 (square root 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. 8.NS.2 4 In addition to being proficient on the standard, student can demonstrate one or more of the following: Applies understanding of standard to unfamiliar situations and/or to solve complex problems. Use precise and relevant communication to justify mathematical thinking. Connects knowledge to other learning targets and/or advance problem sets. 3 Estimate a rational (using decimals) approximation for an irrational number Explain the reasonableness of the rational approximation. Use rational approximations of irrational numbers to locate them on a number line. Make numerical comparisons between rational and irrational numbers. Compare rational and irrational numbers. 2 Identify approximate locations of familiar irrational numbers on a number line. Identify the perfect squares the irrational value lies between. 1 With help, minimal success estimating and ordering irrational numbers on a number line. Identify rational numbers on a number line with omissions and/or misconceptions. Identify square roots of perfect squares less than 100. For example, estimate the value of an expression using a rational approximation or approximate irrational numbers between two integers to a specified level of precision. Rubric constructed from the SBAC Claim 1 item specifications and SBAC Achievement Level Descriptors. A PLC may decide to further develop these rubrics. For more information on the assessment of this standard, read the Claim 1 SBAC Item Specifications-Target A. 16 This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment. (8.NS.1-2, 8.EE.2) Standard 8.EE.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. 4 In addition to being proficient on the standard, student can demonstrate one or more of the following: Applies understanding of standard to unfamiliar situations and/or to solve complex problems. Use precise and relevant communication to justify mathematical thinking. Connects knowledge to other learning targets and/or advance problem sets. 8.EE.2 3 Evaluates solutions to equations in form x2 = p (where p is positive rational number) that is not a perfect square, using square root symbol AND evaluates square roots of perfect squares up to 400. 2 Evaluate solutions to equations in form x2 = p (where p is positive rational number) that is a perfect square using the square root symbol. 1 With help, minimal success evaluating familiar, small perfect squares (1-100). For example, solve binomial quadratic (9x2) and cubic equations (8x3), and represent the solution as a square or cube root. Rubric constructed from the SBAC Claim 1 item specifications and SBAC Achievement Level Descriptors. A PLC may decide to further develop these rubrics. For more information on the assessment of this standard, read the Claim 1 SBAC Item Specifications-Target B. 17 This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment. (8.NS.1-2, 8.EE.2)
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