Ordering and Comparing Rational Numbers

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1 Grade 6 Mathematics, Quarter 3, Unit 3.1 Ordering and Comparing Rational Numbers Overview Number of instructional days: 10 (1 day = 45 minutes) Content to be learned Understand statements of inequality as statements about the relative position of two numbers on a number line. Write and explain inequalities in real-world context. Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value for a positive or negative quantity in a real-world situation. Order absolute values on a number line and relate them to real-world situations. Essential questions How do you plot an inequality on a number line? What does that inequality represent? How do you write and explain an inequality for a real-world problem? Mathematical practices to be integrated Model with mathematics. Identify important quantities in a practical situation. Map the relationships of the important quantities. Reason abstractly and quantitatively. Make sense of quantities and their relationships in problem situations. Represent a given situation. What does the absolute value of a negative and positive number mean when compared to its distance from 0? (e.g., 3 and 3 ) How can you model absolute values on a number line? How can you relate them to realworld situations? 29

2 Grade 6 Mathematics, Quarter 3, Unit 3.1 Ordering and Comparing Rational Numbers (10 days) Written Curriculum Common Core State Standards for Mathematical Content The Number System 6.NS Apply and extend previous understandings of numbers to the system of rational numbers. 6.NS.7 Understand ordering and absolute value of rational numbers. a. Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram. For example, interpret 3 > 7 as a statement that 3 is located to the right of 7 on a number line oriented from left to right. b. Write, interpret, and explain statements of order for rational numbers in real-world contexts. For example, write 3 C > 7 C to express the fact that 3 C is warmer than 7 C. c. Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a realworld situation. For example, for an account balance of 30 dollars, write 30 = 30 to describe the size of the debt in dollars. d. Distinguish comparisons of absolute value from statements about order. For example, recognize that an account balance less than 30 dollars represents a debt greater than 30 dollars. Common Core Standards for Mathematical Practice 2 Reason abstractly and quantitatively. Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects. 4 Model with mathematics. Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such 30

3 Grade 6 Mathematics, Quarter 3, Unit 3.1 Ordering and Comparing Rational Numbers (10 days) tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose. Clarifying the Standards Prior Learning Students began work with number lines to use two perpendicular number lines to define a coordinate system. They graphed points on the coordinate plane to solve real-world and mathematical problems. Current Learning Students apply and extend previous understandings of numbers to the system of rational numbers. They extend their work with the system of rational numbers to include using positive and negative numbers to describe quantities, extending the number line and coordinate plane to represent rational numbers and ordered pairs, and understanding ordering and absolute value of rational numbers. Future Learning Going forward, students will add, subtract, multiply, and divide within the system of rational numbers. They will apply and extend this understanding. Additional Findings According to Principles and Standards for School Mathematics, Middle school students should [also] work with integers. In lower grades, students may have connected negative integers in appropriate ways to informal knowledge derived from everyday experiences. Positive and negative integers should be seen as useful for noting relative changes or values (pp. 217 and 218) Also, a mistaken expectation about the magnitude of a computational result is likely to interfere with students [answers] making sense. (Graeber and Tanenhaus, 1993) Through teacher-orchestrated discussions of problems in context, students can develop useful methods to compute with fractions, decimals, percents, and integers in ways that make sense. (p. 220) 31

4 Grade 6 Mathematics, Quarter 3, Unit 3.1 Ordering and Comparing Rational Numbers (10 days) 32

5 Grade 6 Mathematics, Quarter 3, Unit 3.2 Expressions and Equations Overview Number of instructional days: 15 (1 day = 45 minutes) Content to be learned Write and solve numerical expressions involving whole number exponents. Write expressions using variables. Use math terms (e.g., sum, term, product, factor, quotient, coefficient) to identify parts of an expression. For example, describe the expression 2(8 + 7) as a product of two factors. View each part of an expression as a single entity. For example, view (8 + 7) as both a single entity and a sum of two terms. Evaluate expressions using specific values for a variable. Use formulas to describe and solve real-world problems. Use order of operations, including the Distributive Property, with exponents to solve expressions. Apply properties of operations to generate equivalent expressions. For example, apply properties of operations to y + y + y to produce the equivalent expression 3y. Essential questions How do you write and solve expressions with exponents? (e.g., five more than the product of a number and 3) How do you write and solve expressions with variables? (e.g., two less than a number to its fourth power) Given an expression, how do you identify each part using correct math terminology and view each part as a separate entity? Mathematical practices to be integrated Look for and make use of structure. View some algebraic expressions as single objects or as being composed of several objects. Look for and express regularity in repeated reasoning. Look for general methods and shortcuts to solve problems. How do you solve expressions with variables given a specific value for the variable? How do you use formulas to solve real-world problems? How do you solve problems with exponents using order of operations? How do you use each property to make equivalent expressions? (See CCSS Glossary, Table 3, p. 90) 33

6 Grade 6 Mathematics, Quarter 3, Unit 3.2 Expressions and Equations (15 days) Written Curriculum Common Core State Standards for Mathematical Content Expressions and Equations 6.EE Apply and extend previous understandings of arithmetic to algebraic expressions. 6.EE.1 6.EE.2 6.EE.3 Write and evaluate numerical expressions involving whole-number exponents. Write, read, and evaluate expressions in which letters stand for numbers. a. Write expressions that record operations with numbers and with letters standing for numbers. For example, express the calculation Subtract y from 5 as 5 y. b. Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity. For example, describe the expression 2 (8 + 7) as a product of two factors; view (8 + 7) as both a single entity and a sum of two terms. c. Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations). For example, use the formulas V = s 3 and A = 6 s 2 to find the volume and surface area of a cube with sides of length s = 1/2. Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression 3 (2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6 (4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y. Common Core Standards for Mathematical Practice 7 Look for and make use of structure. Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 8 equals the well remembered , in preparation for learning about the distributive property. In the expression x 2 + 9x + 14, older students can see the 14 as 2 7 and the 9 as They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 3(x y) 2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y. 34

7 Grade 6 Mathematics, Quarter 3, Unit 3.2 Expressions and Equations (15 days) 8 Look for and express regularity in repeated reasoning. Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y 2)/(x 1) = 3. Noticing the regularity in the way terms cancel when expanding (x 1)(x + 1), (x 1)(x 2 + x + 1), and (x 1)(x 3 + x 2 + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results. Clarifying the Standards Prior Learning Students have written and interpreted numerical expression. They multiplied and divided to solve word problems involving symbols for an unknown number and distinguishing multiplicative comparison from additive comparison. Students used drawings and equations with a symbol for the unknown number to represent the problem. In grade 5, students used parentheses, brackets, or braces in numerical expressions and evaluated the expressions with these symbols. Current Learning Students apply and extend previous understanding of arithmetic to algebraic expressions. They begin using properties of operations systematically to work with variables, variable expressions, and equations. Students begin to show an ability to write, read, and evaluate expressions in which letters stand for numbers. They apply the previously mentioned knowledge to the volume formulas V = l w h and V = b h. Students use properties of operations that they are familiar with from previous grades work with numbers generalizing arithmetic in the process. Future Learning In future grades, students will use properties of operations to generate equivalent expressions. They will solve multistep problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. This work is the culmination of many progressions of learning in arithmetic, problem solving, and mathematical practices. This is a major capstone standard for arithmetic and its application. Additional Findings According to Principles and Standards for School Mathematics, Students understanding of variable should go far beyond simply recognizing that letters can be used to stand for unknown numbers in equations. (Schoenfeld and Arcavi, 1988). (p. 225) Most students will need extensive experience in interpreting relationship among quantities in a variety of problem contexts before they can they can work meaningfully with variables and symbolic expressions. (p. 225) 35

8 Grade 6 Mathematics, Quarter 3, Unit 3.2 Expressions and Equations (15 days) According to Progressions for the Common Core State Standards in Mathematics, The distributive law is of fundamental importance. Collecting like terms, e.g., 5b + 3b = (5 + 3)b = 8b, should be seen as an application of the distributive law, not as a separate method. (p. 6) 36

9 Grade 6 Mathematics, Quarter 3, Unit 3.3 Equations and Inequalities Overview Number of instructional days: 15 (1 day = 45 minutes) Content to be learned Identify equivalent expressions. For example, the expressions y + y + y and 3y are equivalent because they name the same number regardless of which number y stands for. Use substitution to determine whether a given number in a set makes an equation or inequality true. Use variables to represent numbers. Write expressions when solving a real-world problem. Understand that a variable can represent an unknown number or any number in a set. Essential questions What are two different ways to represent equivalent expressions? Which values of a set make an equality or inequality true? Mathematical practices to be integrated Reason abstractly and quantitatively. Know and flexibly use different properties of operations and objects. Make sense of quantities and their relationships in problem situations. Construct viable arguments and critique the reasoning of others. Understand and use prior learning in constructing arguments. Justify conclusions, communicate them to others, and respond to the arguments of others. How do you represent a number with a variable? How do you use variables to write an expression to solve math or real world problems? 37

10 Grade 6 Mathematics, Quarter 3, Unit 3.3 Equations and Inequalities (15 days) Written Curriculum Common Core State Standards for Mathematical Content Expressions and Equations 6.EE Apply and extend previous understandings of arithmetic to algebraic expressions. 6.EE.4 Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them). For example, the expressions y + y + y and 3y are equivalent because they name the same number regardless of which number y stands for. Reason about and solve one-variable equations and inequalities. 6.EE.5 6.EE.6 Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true. Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set. Common Core Standards for Mathematical Practice 2 Reason abstractly and quantitatively. Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects. 3 Construct viable arguments and critique the reasoning of others. Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and if there is a flaw in an argument explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments. 38

11 Grade 6 Mathematics, Quarter 3, Unit 3.3 Equations and Inequalities (15 days) Clarifying the Standards Prior Learning In grade 5, students wrote and interpreted numerical expressions. When students wrote equations to solve real-world mathematical problems, they drew on the meaning of operations that they were familiar with from previous grades work. They also began to learn algebraic approaches to solving problems. Current Learning Students begin using properties of operations systematically to work with variables, variable expressions, and equations. They resolve problems by writing and solving equations; this involves not only an appreciation of how variables [text missing?], but also some ability to write, read, and evaluate expressions in which letters stand for numbers. Future Learning Students will use properties of operations to generate equivalent expressions. They will solve word problems leading to one-variable equations of the form px + q = r and p(x + q) = r. Students will solve real-world and mathematical problems using numerical and algebraic expressions and equations. Additional Findings According to Principles and Standards for School Mathematics, students should begin to work more frequently with algebraic symbols. They must become comfortable with symbolic expressions and variables. Students should also learn to recognize and generate equivalent expressions, solve equations, and use simple formulas. The teaching and learning of algebra should be integrated with other areas of the curriculum. 39

12 Grade 6 Mathematics, Quarter 3, Unit 3.3 Equations and Inequalities (15 days) 40

Extending Place Value with Whole Numbers to 1,000,000

Grade 4 Mathematics, Quarter 1, Unit 1.1 Extending Place Value with Whole Numbers to 1,000,000 Overview Number of Instructional Days: 10 (1 day = 45 minutes) Content to Be Learned Recognize that a digit