Grade 8 Mathematics, Quarter 2, Unit 2.1 Rational and Irrational Numbers Overview Number of instructional days: 14 (1 day = 45 60 minutes) Content to be learned Develop a conceptual understanding of rational and irrational numbers. Develop a conceptual understanding of square roots and cube roots. Develop a conceptual understanding of whole numbers or fractional bases with whole number exponents. Use a variety of mental computation strategies to solve problems. Compare and order all rational numbers (integers, scientific notation, absolute value, whole number and fractional bases with whole number exponents, benchmark percents) and all irrational numbers ( 2,! ) using number lines or equality and inequality symbols. Mathematical practices to be integrated Attend to precision. Communicate using proper vocabulary and clear definitions pertaining to types of numbers (real, rational, etc.). State the meaning of the equality/inequality symbols. Calculate accurately and efficiently. Express numerical answers with precision when comparing numbers. Construct viable arguments and critique the reasoning of others. Categorize numbers appropriately and justify reasoning. Order numbers appropriately and justify reasoning. Analyze situations by breaking them into cases such as rational and irrational (i.e., justify why all natural numbers are whole numbers). Essential questions What are the characteristics of rational and irrational numbers? How would you mentally compute irrational numbers? How would you order and compare rational and irrational numbers? What is the relationship between an exponent and its base in a problem? What does it mean to find the square or cube root of a number? Cumberland, Lincoln, and Woonsocket Public Schools C-17
Grade 8 Mathematics, Quarter 2, Unit 2.1 Rational and Irrational Numbers (14 days) Written Curriculum Grade-Level Expectations M(N&O) 8 1 Demonstrates conceptual understanding of rational numbers with respect to absolute values, perfect square and cube roots, and percents as a way of describing change (percent increase and decrease) using explanations, models, or other representations. (Local) M(N&O) 8 2 Demonstrates understanding of the relative magnitude of numbers by ordering or comparing rational numbers, common irrational numbers (e.g., 2,! ), numbers with whole number or fractional bases and whole number exponents, square roots, absolute values, integers, or numbers represented in scientific notation using number lines or equality and inequality symbols. (Local) M(N&O) 8 4 Accurately solves problems involving proportional reasoning (percent increase or decrease, interest rates, markups, or rates); multiplication or division of integers; and squares, cubes, and taking square or cube roots. (Local) (IMPORTANT: Applies the conventions of order of operations.) M(N&O) 8 6 Uses a variety of mental computation strategies to solve problems (e.g., using compatible numbers, applying properties of operations, using mental imagery, using patterns) and to determine the reasonableness of answers; and mentally calculates benchmark perfect squares and related square roots (e.g., 1 2, 2 2,, 12 2, 15 2, 20 2, 25 2, 100 2, 1000 2 ); determines the part of a number using benchmark percents and related fractions (1%, 10%, 25%, (e.g., 25% of 16; 1 33 % 3 of 330). (Local) 1 33 % 3, 50%, 2 66 % 3, 75%, and 100%) (IMPORTANT: The intent of this GSE is to embed mental arithmetic throughout the instructional program, not to teach it as a separate unit.) M(N&O) 8 7 Makes estimates in a given situation (including tips, discounts, tax, and the value of a non-perfect square root as between two whole numbers) by identifying when estimation is appropriate, selecting the appropriate method of estimation; determining the level of accuracy needed given the situation; analyzing the effect of the estimation method on the accuracy of results; and evaluating the reasonableness of solutions appropriate to grade level GLEs across content strands. (Local) (IMPORTANT: The intent of this GLE is to embed estimation throughout the instructional program, not to teach it as a separate unit.) Clarifying the Standards Prior Learning In earlier grades, students developed an understanding of various rational numbers. From kindergarten to grade 2, students focused on understanding the relative magnitude of whole numbers up to 200. They also worked with positive fractional numbers, as well as mentally adding and subtracting with whole numbers to a sum of 20. C-18 Cumberland, Lincoln, and Woonsocket Public Schools
Rational and Irrational Numbers (14 days) Grade 8 Mathematics, Quarter 2, Unit 2.1 In grade 3, students continued to work with and understand the magnitude of equivalent positive fractions as well as larger rational numbers. They also began to work with decimals in a limited context and mentally added and subtracted larger numbers that are multiples of 10. In grade 5, students continued working with decimals, fractions, integers, multiplication, and division of a variety of whole numbers. In grade 6, students began working with rational numbers with respect to ratios and rates, whole-number base exponents, rational numbers within and across formats, and using equality and inequality symbols. Further emphasis was placed on using mental math from previous grades to solve problems involving various rational number representations. From grade 7 on, students worked with square roots, perfect squares, rates, absolute value, scientific notation, and proportional reasoning. From kindergarten through grade 8, students used explanations, models, and other representations when dealing with rational numbers. Current Learning Students extend the learning of relative magnitude of numbers by ordering common irrational numbers, numbers with fractional bases, and square roots on a number line. Students use equality and inequality symbols to compare common irrational numbers, numbers with fractional bases, and square roots. These topics are introduced and reinforced at this level. Students continue practicing mental computation in problem situations as a process embedded throughout the year. Future Learning Students will continue to work with the relative magnitude of various rational and irrational numbers in future grades and will continue to refine and use their mental computation strategies in problem situations. Additional Research Findings According to Principles and Standards for School Mathematics, students should acquire the ability to make reasonable estimates and use benchmarks to order and compare numbers in sixth through eighth grades. Students should be able to think flexibly about the sizes of rational and irrational numbers and should be given the opportunity to share different estimation methods for solving various problems (pp. 214 221). Additionally, A Research Companion to Principles and Standards for School Mathematics states that students should be given the opportunity to represent all numbers in general and abstract ways based upon their prior learning. The goal of problem solving is for students to generalize their learning applicable to various situations (pp. 263 273). Notes About Resources and Materials Cumberland, Lincoln, and Woonsocket Public Schools C-19
Grade 8 Mathematics, Quarter 2, Unit 2.1 Rational and Irrational Numbers (14 days) C-20 Cumberland, Lincoln, and Woonsocket Public Schools
Grade 8 Mathematics, Quarter 2, Unit 2.2 Problem Solving Using Integers Overview Number of instructional days: 10 (1 day = 45 60 minutes) Content to be learned Solve problems involving multiplication and division of integers. Apply field properties (commutative, associative, identity, multiplicative property of one, distributive, inverses). Apply properties of numbers throughout the unit. This is not intended to be taught separately. Essential questions How can you apply the rules for multiplying and dividing integers when solving problems? Mathematical practices to be integrated Look for and make use of structure. Look closely to discern a pattern that leads to a field property (i.e., using the commutative property to reorder numbers to simplify calculations). Look closely to discern a pattern that leads to a number property. Reason abstractly and quantitatively. Extract the symbolic representation (numbers and operations) in order to solve the problem. Put the solution into the context of the problem. How can field properties be used to simplify computations and solve equations? Cumberland, Lincoln, and Woonsocket Public Schools C-21
Grade 8 Mathematics, Quarter 2, Unit 2.2 Problem Solving Using Integers (10 days) Written Curriculum Grade-Level Expectations M(N&O) 8 4 Accurately solves problems involving proportional reasoning (percent increase or decrease, interest rates, markups, or rates); multiplication or division of integers; and squares, cubes, and taking square or cube roots. (Local) (IMPORTANT: Applies the conventions of order of operations.) M(N&O) 8 8 Applies properties of numbers (odd, even, remainders, divisibility, and prime factorization) and field properties (commutative, associative, identity [including the multiplicative property of one, e.g., 2 0 x 2 3 = 2 0+3 = 2 3, so 2 0 = 1], distributive, inverses) to solve problems and to simplify computations, and demonstrates conceptual understanding of field properties as they apply to subsets of real numbers when addition and multiplication are not defined in the traditional ways (e.g., If a " b = a + b! 1, is! a commutative operation?) (Local) Clarifying the Standards Prior Learning Beginning in first grade, students applied properties of numbers (odd, even) and field properties (commutative, additive identity) to solve problems and simplify computations involving whole numbers. In second grade, the associative property of addition was used. In grade 3, the multiplicative property of zero, multiplicative identity, and the commutative property for multiplication were introduced. Grade 4 introduced remainders and expanded student experience with the commutative and associative properties. In grade 5 students learned about divisibility and the distributive properties. Prime factorization was introduced in grade 6 as well as the multiplicative property of one and additive inverses. Inverses were expanded in grade 7 and students demonstrated conceptual understanding of field properties as they apply to subsets of real numbers. The concept of multiplication and the relationship between repeated multiplication and division was introduced in grade 3. In grade 4, students learned about the inverse relationship between multiplication and division as well as about division without remainders. Multiplication was limited to two digits by two digits and division was limited to one-digit divisors. Integers were introduced in grade 5, followed by the addition and subtraction of integers in grade 6. In grade 7, students used their knowledge of adding and subtracting integers to solve problems. Current Learning Students extend their knowledge of repeated addition of integers into understanding and applying the rules for multiplying integers when solving problems. Students use and apply their knowledge of division to dividing integers when solving problems. Properties of numbers and field properties are applied to simplifying computations and solving equations. Students reinforce and apply their knowledge of multiplication and division of integers through practice. C-22 Cumberland, Lincoln, and Woonsocket Public Schools
Problem Solving Using Integers (10 days) Grade 8 Mathematics, Quarter 2, Unit 2.2 Future Learning Ninth-grade students will apply properties of numbers to solve problems and simplify computations. In grade 11 students will apply properties to determine whether a given subset of numbers is closed under a given arithmetic operation and will apply arithmetic properties to matrices. Additional Research Findings: According to Principles and Standards for School Mathematics, students must have an understanding of the inverse relationships of addition, subtraction, multiplication, and division. They must be able to connect integers to everyday experiences while using the associative, commutative, and distributive properties to simplify problems involving integers (p. 214). Notes About Resources and Materials Cumberland, Lincoln, and Woonsocket Public Schools C-23
Grade 8 Mathematics, Quarter 2, Unit 2.2 Problem Solving Using Integers (10 days) C-24 Cumberland, Lincoln, and Woonsocket Public Schools
Grade 8 Mathematics, Quarter 2, Unit 2.3 Problem Solving Using Proportional Reasoning Overview Number of instructional days: 16 (1 day = 45 60 minutes) Content to be learned Use proportional reasoning to solve problems involving percent increase/decrease, interest rates, and markups. Develop an appropriate method of estimation as used in real-life situations including tips, discount, and tax. Develop an understanding of rates in various contexts. Mathematical practices to be integrated Model with mathematics. Apply math to solve problems involving proportions in everyday life, society, and the workplace (i.e., tax, tips, discount, markup). Interpret mathematical results in the context of the situation and reflect on whether the results make sense. Make sense of problems and persevere in solving them. Explain the meaning of a problem, looking for entry points to its solution. Plan a solution pathway rather than jumping into the work. Understand that there are multiple methods for solving problems (proportions, percents, etc.), and identify relationships among these methods (i.e., using a proportion vs. equation method). Look for and express regularity in repeated reasoning. Identify shortcuts when solving problems involving benchmark percents. Essential questions How you can determine a percent of increase or decrease? In the context of this unit, when is it appropriate to use an estimate as opposed to a precise solution? How can you relate interest rates and markups to real life situations? How do you apply the concept of proportional reasoning to solve problems involving percents? How can you describe the concept of rates in various situations? Cumberland, Lincoln, and Woonsocket Public Schools C-25
Grade 8 Mathematics, Quarter 2, Unit 2.3 Problem Solving Using Proportional Reasoning (16 days) Written Curriculum Grade-Level Expectations M(N&O) 8 4 Accurately solves problems involving proportional reasoning (percent increase or decrease, interest rates, markups, or rates); multiplication or division of integers; and squares, cubes, and taking square or cube roots. (Local) (IMPORTANT: Applies the conventions of order of operations.) M(N&O) 8 7 Makes estimates in a given situation (including tips, discounts, tax, and the value of a non-perfect square root as between two whole numbers) by identifying when estimation is appropriate, selecting the appropriate method of estimation; determining the level of accuracy needed given the situation; analyzing the effect of the estimation method on the accuracy of results; and evaluating the reasonableness of solutions appropriate to grade level GLEs across content strands. (Local) (IMPORTANT: The intent of this GLE is to embed estimation throughout the instructional program, not to teach it as a separate unit.) Clarifying the Standards Prior Learning In earlier grades, students developed a basic understanding of benchmark fractions, decimals, and percents. In sixth grade, students became familiar with ratios; they were introduced to multiplying and dividing decimals, and they determined parts of a whole number using benchmark percents. Seventh grade further developed the understanding of percent of a whole using benchmark percents and related fractions. Estimating and computing percents involving discounts, tax, tips, and rates was also introduced in grade 7, as well as proportional reasoning (part to part and part to whole). Current Learning Students apply their understanding of proportional reasoning by estimating and solving problems involving percent increase/decrease, interest rates, markups, and rates. Using proportional reasoning to solve these types of problems is reinforced. Students extend the concept of rates to include constant rates of change (linear) and varying rates of change (nonlinear). Students reinforce the concepts of proportional reasoning through practice with the goal of mastery by the end of the unit. Future Learning In high school, students will solve real-world problems involving proportional relationships, percents, ratios, and rates. Students will need to use a variety of mental computation strategies to solve problems. Additional Research Findings: According to Principles and Standards for School Mathematics, students should acquire computational fluency in sixth through eighth grades. Students should be allowed to solve proportional reasoning problems with whichever method they feel most comfortable. As different methods for solving problems are learned, teachers should help students understand when and how these methods should be used (p. 220). C-26 Cumberland, Lincoln, and Woonsocket Public Schools
Problem Solving Using Proportional Reasoning (16 days) Grade 8 Mathematics, Quarter 2, Unit 2.3 Notes About Resources and Materials Cumberland, Lincoln, and Woonsocket Public Schools C-27
Grade 8 Mathematics, Quarter 2, Unit 2.3 Problem Solving Using Proportional Reasoning (16 days) C-28 Cumberland, Lincoln, and Woonsocket Public Schools