# Integer Operations. Overview. Grade 7 Mathematics, Quarter 1, Unit 1.1. Number of Instructional Days: 15 (1 day = 45 minutes) Essential Questions

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1 Grade 7 Mathematics, Quarter 1, Unit 1.1 Integer Operations Overview Number of Instructional Days: 15 (1 day = 45 minutes) Content to Be Learned Describe situations in which opposites combine to make zero. Represent integer addition on a number line. Evaluate integer sums. Understand subtraction of a number as adding the additive inverse. Show that the distance between two integers is the absolute value of their difference in realworld contexts. Evaluate integer products. Understand quotients of integers with non-zero divisors. Apply properties of operations to integers. Solve real-world problems involving the four operations with integers. Essential Questions How do addition and subtraction of integers compare? When will the sum of two integers be positive? Negative? Or zero? When multiplying and dividing integers, how do you determine the sign? Mathematical Practices to Be Integrated Make sense of problems and persevere in solving them. Read and analyze givens. Plan a solution. Evaluate progress. Change course if necessary. Model with mathematics. Identify important quantities. Use tools such as diagrams (number lines, integer chips). Analyze to draw conclusions. Interpret results, determining if they make sense. Attend to precision. Calculate accurately and efficiently. Specify units/labels. Communicate precisely. How can you use properties to simplify your calculations with integers? In what real-life situations would you need to compute with integers? 1

2 Grade 7 Mathematics, Quarter 1, Unit 1.1 Integer Operations (15 days) Written Curriculum Common Core State Standards for Mathematical Content The Number System 7.NS Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers. 7.NS.1 7.NS.2 7.NS.1 7.NS.2 7.NS.3 Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram. a. Describe situations in which opposite quantities combine to make 0. For example, a hydrogen atom has 0 charge because its two constituents are oppositely charged. b. Understand p + q as the number located a distance q from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts. c. Understand subtraction of rational numbers as adding the additive inverse, p q = p + ( q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts. Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. a. Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as ( 1)( 1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing realworld contexts. b. Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero divisor) is a rational number. If p and q are integers, then (p/q) = ( p)/q = p/( q). Interpret quotients of rational numbers by describing realworld contexts. Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram. d. Apply properties of operations as strategies to add and subtract rational numbers. Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. c. Apply properties of operations as strategies to multiply and divide rational numbers. Solve real-world and mathematical problems involving the four operations with rational numbers. 1 1 Computations with rational numbers extend the rules for manipulating fractions to complex fractions. 2

3 Grade 7 Mathematics, Quarter 1, Unit 1.1 Integer Operations (15 days) Common Core Standards for Mathematical Practice 1 Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, Does this make sense? They can understand the approaches of others to solving complex problems and identify correspondences between different approaches. 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 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. 6 Attend to precision. Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions. 3

4 Grade 7 Mathematics, Quarter 1, Unit 1.1 Integer Operations (15 days) Clarifying the Standards Prior Learning Previously, students learned to compute fluently with multidigit numbers in four operations using standard algorithms. They found and positioned integers on a number line. Students recognized opposites and absolute value as a number s distance from zero. Students assigned and ordered integers to solve realworld problems. They also applied properties of operations. Current Learning Operations with rational numbers are a critical area and a major cluster in grade 7. This unit reinforces the concepts of opposites and absolute value. Students learn integer operations at the developmental level and continue to reinforce these concepts throughout the year to achieve fluency. The unit extends the application of properties of operations to include integers. Units 1.2 and 1.3 will extend the application of the four operations and properties to all rational numbers. Application of rational numbers will continue through all units to achieve fluency by the end of this year. Future Learning In eighth grade, students will apply their knowledge of integers when reasoning about expressions and equations, solving linear equations and systems of equations, grasping the concept of functions, and evaluating integer exponents. Additional Findings According to Principles and Standards of School Mathematics, In grades 6 8, students should deepen their understanding of fractions, decimals, percents, and integers, and they should become proficient in using them to solve problems (p. 215). According to Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics, Students extend understandings of addition, subtraction, multiplication, and division, together with their properties, to all rational numbers, including negative integers. By applying properties of arithmetic and considering negative numbers in everyday contexts, students explain why the rules for adding, subtracting, multiplying, and dividing with negative numbers makes sense (p. 19). According to the PARCC Model Content Frameworks, Because there are no specific standards for rational number arithmetic in later grades and because so much other work in grade 7 depends on rational number arithmetic, fluency with rational number arithmetic should be the goal in grade 7 (p. 31). 4

5 Grade 7 Mathematics, Quarter 1, Unit 1.2 Addition and Subtraction of Rational Numbers Overview Number of Instructional Days: 15 (1 day = 45 minutes) Content to Be Learned Convert between different forms of rational numbers (decimals and fractions). Understand that rational numbers in decimal form either terminate or repeat. Show that a rational number and its opposite have a sum of zero (additive inverses). Evaluate rational sums. Understand subtraction of rational numbers as adding the additive inverse. Show that the distance between two rational numbers is the absolute value of their difference in real-world contexts. Interpret sums and differences of rational numbers in real-world contexts. Solve real-world problems involving addition and subtraction of rational numbers. Essential Questions What is a rational number? How can rational numbers be represented? How do you convert between fractions and decimals? When will the sum of two rational numbers be positive? Negative? Or zero? Mathematical Practices to Be Integrated Make sense of problems and persevere in solving them. Read and analyze the givens. Plan a solution. Evaluate progress. Change course if necessary. Model with mathematics. Identify important quantities. Use tools such as diagrams (number lines, 10 x 10 grids). Analyze to draw conclusions. Interpret results, determining if they make sense. Attend to precision. Calculate accurately and efficiently. Specify units/labels. Communicate precisely. How do you add and subtract rational numbers in the same format? How do you add and subtract rational numbers in different formats (i.e., fraction and decimal)? Where would you use operations with rational numbers in real-life situations? 5

6 Grade 7 Mathematics, Quarter 1, Unit 1.2 Addition and Subtraction of Rational Numbers (15 days) Written Curriculum Common Core State Standards for Mathematical Content The Number System 7.NS Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers. 7.NS.2 7.NS.1 7.NS.3 Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. d. Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats. Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram. b. Understand p + q as the number located a distance q from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts. c. Understand subtraction of rational numbers as adding the additive inverse, p q = p + ( q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts. Solve real-world and mathematical problems involving the four operations with rational numbers. 1 1 Computations with rational numbers extend the rules for manipulating fractions to complex fractions. Common Core Standards for Mathematical Practice 1 Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, Does this make sense? They can understand the approaches of others to solving complex problems and identify correspondences between different approaches. 6

7 Grade 7 Mathematics, Quarter 1, Unit 1.2 Addition and Subtraction of Rational Numbers (15 days) 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 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. 6 Attend to precision. Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions. Clarifying the Standards Prior Learning In grades 5 and 6, students learned to compute fluently with positive rational numbers. They found and positioned rational numbers on a number line, identified opposites, and defined absolute value as a number s distance from zero. Current Learning Operations with rational numbers are a critical area and a major cluster in grade 7. This unit reinforces the concepts of opposites, absolute value, and integer operations. Students understand a rational number as a number expressible in the form a/b or a/b for some fraction a/b. Students identify fractions and repeating or terminating decimals as different representations of rational numbers. Students learn rational number addition and subtraction at the developmental level and continue to reinforce these concepts throughout the year to achieve fluency. The next unit will introduce multiplication and division of rational numbers, and reinforce the four operations and properties of rational numbers. Application of rational numbers will continue through all units to achieve fluency by the end of this year. 7

8 Grade 7 Mathematics, Quarter 1, Unit 1.2 Addition and Subtraction of Rational Numbers (15 days) Future Learning In eighth grade, students will apply their knowledge of rational numbers when reasoning about expressions and equations, solving linear equations and systems of equations, grasping the concept of functions, and evaluating integer exponents. Students will expand their concept of number beyond the system of rationals to include irrational numbers. They will represent these numbers with radical expressions and approximate these numbers with rationals. Additional Findings According to Principles and Standards of School Mathematics, In grades 6 8, students should deepen their understanding of fractions, decimals, percents, and integers, and they should become proficient in using them to solve problems (p. 215). According to Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics, Students extend understandings of addition, subtraction, multiplication, and division, together with their properties, to all rational numbers, including negative integers. By applying properties of arithmetic and considering negative numbers in everyday contexts, students explain why the rules for adding, subtracting, multiplying, and dividing with negative numbers makes sense (p. 19). According to the PARCC Model Content Frameworks, Because there are no specific standards for rational number arithmetic in later grades and because so much other work in grade 7 depends on rational number arithmetic, fluency with rational number arithmetic should be the goal in grade 7 (p. 31). According to the CCSS, students must develop a unified understanding of number, recognizing fractions, decimals (that have a finite or a repeating decimal representation), and percents as different representations of rational numbers (p. 46). 8

9 Grade 7 Mathematics, Quarter 1, Unit 1.3 Multiplication and Division of Rational Numbers Overview Number of Instructional Days: 10 (1 day = 45 minutes) Content to Be Learned Evaluate rational products. Evaluate rational quotients. Understand why the denominator cannot equal zero. Apply properties (including distributive property) when performing operations with rational numbers. Interpret products and quotients of rational numbers in real-world contexts. Solve real-world problems using the four operations with rational numbers. Essential Questions How can you use properties to simplify your calculations with rational numbers? In what situations do you get a smaller product when multiplying two rational numbers? Mathematical Practices to Be Integrated Make sense of problems and persevere in solving them. Read and analyze the givens. Plan a solution. Evaluate progress. Change course if necessary. Construct viable arguments and critique the reasoning of others. Use properties to justify the simplification of calculations. Make conjectures and justify conclusions regarding the magnitude of a solution. Communicate findings and respond to others arguments. Distinguish correct logic or reasoning from that which is flawed. Ask useful questions to clarify, refine, and improve arguments. Look for and make use of structure. Look for a solution, pattern, or structure. Reflect and shift perspectives. Understand structures. See complex tasks as a collection of several simpler objects. In what situations do you get a larger quotient when dividing two rational numbers? Where would you use multiplication and division of rational numbers in real-life situations? 9

10 Grade 7 Mathematics, Quarter 1, Unit 1.3 Multiplication and Division of Rational Numbers (10 days) Written Curriculum Common Core State Standards for Mathematical Content The Number System 7.NS Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers. 7.NS.2 7.NS.1 7.NS.2 7.NS.3 Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. a. Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as ( 1)( 1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing realworld contexts. b. Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero divisor) is a rational number. If p and q are integers, then (p/q) = ( p)/q = p/( q). Interpret quotients of rational numbers by describing realworld contexts. Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram. d. Apply properties of operations as strategies to add and subtract rational numbers. Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. c. Apply properties of operations as strategies to multiply and divide rational numbers. Solve real-world and mathematical problems involving the four operations with rational numbers. 1 1 Computations with rational numbers extend the rules for manipulating fractions to complex fractions. Common Core Standards for Mathematical Practice 1 Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using 10

11 Grade 7 Mathematics, Quarter 1, Unit 1.3 Multiplication and Division of Rational Numbers (10 days) concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, Does this make sense? They can understand the approaches of others to solving complex problems and identify correspondences between different approaches. 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. 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. Clarifying the Standards Prior Learning In grades 5 and 6, students learned to compute fluently with positive rational numbers. Students were also introduced to properties of operations, including the distributive property in sixth grade. Current Learning Operations with rational numbers are a critical area and a major cluster in grade 7. This unit reinforces the concepts of opposites, absolute value, and integer operations. Students add, subtract, multiply, and divide within the system of rational numbers. Application of rational numbers will continue through all units to achieve fluency by the end of this year. Students will apply the four operations and properties to solve real-world problems involving rational numbers. 11

12 Grade 7 Mathematics, Quarter 1, Unit 1.3 Multiplication and Division of Rational Numbers (10 days) Future Learning In eighth grade, students will apply their knowledge of rational numbers when reasoning about expressions and equations, solving linear equations and systems of equations, grasping the concept of functions, and evaluating rational number exponents. Students will 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. Additional Findings According to Principles and Standards of School Mathematics, In grades 6 8, students should deepen their understanding of fractions, decimals, percents, and integers, and they should become proficient in using them to solve problems (p. 215). According to Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics 2006 edition, Students extend understandings of addition, subtraction, multiplication, and division, together with their properties, to all rational numbers, including negative integers. By applying properties of arithmetic and considering negative numbers in everyday contexts, students explain why the rules for adding, subtracting, multiplying, and dividing with negative numbers makes sense (p. 19). According to the PARCC Model Content Frameworks, Because there are no specific standards for rational number arithmetic in later grades and because so much other work in grade 7 depends on rational number arithmetic, fluency with rational number arithmetic should be the goal in grade 7 (p. 31). 12

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