Adding and Subtracting with Rational Numbers Overview Number of instruction days: 12 14 Content to Be Learned Model addition and subtraction of rational numbers using distance on a number line. Describe real-world situations where opposite quantities combine to make zero. Understand how to show addition of rational numbers on a number line, including problems with negative addends. Understand that addition of a number and its opposite results in zero. Interpret sums of rational numbers using realworld scenarios. Understand the relationship between subtracting rational numbers is adding the additive inverse. Apply the properties of operations to addition and subtraction of rational numbers Solve real-world problems involving the addition and subtraction with rational numbers. Essential Questions How can you decide if the sum of two numbers is positive, negative, or zero without actually calculating the sum? How can you decide if the difference of two numbers is positive, negative, or zero without actually calculating the difference? How can the difference of any two numbers be restated using the additive inverse? Mathematical Practices to Be Integrated 1 Make sense of problems and persevere in solving them. Explain addition and subtraction of rational numbers using a number line. Solve real-world problems involving the addition and subtraction with rational numbers. 5 Use appropriate tools strategically. Draw number line models and use distance and direction to make a connection among relationships between addition and subtraction of rational numbers. 6 Attend to precision. Exhibit care when specifying units of measure and label a number line to clarify the relationship with the quantities that pertain to the absolute value drawn on a number line. How do you model addition and subtraction using a number line? What strategies would you use to solve computational problems involving addition and subtraction of rational numbers? What strategies would you use to solve realworld problems involving addition and subtraction of rational numbers? Providence Public Schools D-1
Adding and Subtracting with Rational Numbers (12 14 days) Standards 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.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. d. Apply properties of operations as strategies to add and subtract 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 State 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. D-2 Providence Public Schools
Adding and Subtracting with Rational Numbers (12 14 days) Grade 7 Mathematics, Quarter 1, Unit 1.1 5 Use appropriate tools strategically. Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts. 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 Grade 6, students learned about negative numbers and the kind of quantities they can represent. Students also learned about absolute value and ordering of rational numbers including real-world problems. Students interpreted and computed quotients of fractions and solved word problems involving division of fractions by fractions, and they completed the extension of operations to fractions. Extending multiplication and division to fractions was a sixth grade culminating standard. Current Learning In Grade 7, applying and extending previous understandings of addition and subtraction to add and subtract rational numbers is a major focus. Operations of rational numbers are a culminating standard. Students become fluent with addition and subtraction of rational numbers using horizontal or vertical number lines. They extend addition and subtraction to rational numbers maintaining the properties of operations (i.e., Additive Identity Property of Zero, additive inverses) and the relationships between addition and subtraction. Students show an understanding that p q = p + ( q) is the distance between any two rational numbers and the absolute value of the difference is that distance. Students integrate their skills and understanding of addition and subtraction of rational numbers to solve real-world and mathematical problems. Seventh-grade students extend their understanding of operations with fractions to multiply and divide rational numbers. Providence Public Schools D-3
Adding and Subtracting with Rational Numbers (12 14 days) Future Learning The number system will continue to develop in grade 8, expanding to include real numbers by introducing irrational numbers, and will develop further in high school, expanding to include complex numbers with the introduction of imaginary numbers. In grade 8, students will learn that there are numbers that are not rational; however, they will use rational numbers to approximate their value. Additional Findings From earlier work with whole numbers, students should be familiar with the inverse relationship between the operation pairs of addition subtraction. In the middle grades, they can continue to apply this relationship as they work with fractions, decimals, and integers. (Principles and Standards for School Mathematics, pp. 219 220) Instruction focused solely on symbolic manipulation without understanding is ineffective for most students. It is necessary to correct that imbalance by paying more attention to conceptual understanding as well as the other strands of proficiency and by helping students to connect them. (Adding It Up, p. 241) Assessment When constructing an end-of-unit assessment, be aware that the assessment should measure your students understanding of the big ideas indicated within the standards. The CCSS for Mathematical Content and the CCSS for Mathematical Practice should be considered when designing assessments. Standards-based mathematics assessment items should vary in difficulty, content, and type. The assessment should comprise a mix of items, which could include multiple choice items, short and extended response items, and performance-based tasks. When creating your assessment, you should be mindful when an item could be differentiated to address the needs of students in your class. The mathematical concepts below are not a prioritized list of assessment items, and your assessment is not limited to these concepts. However, care should be given to assess the skills the students have developed within this unit. The assessment should provide you with credible evidence as to your students attainment of the mathematics within the unit. Model addition and subtraction of rational numbers on a number line. Add and subtract rational numbers. Demonstrate that addition of a number and its opposite would result in zero. Describe real-world situations where opposite quantities combine to make zero. Solve real-world problems involving the addition and subtraction with rational numbers. Decide if the sum of two numbers is positive, negative, or zero without actually calculating the sum or difference. Show that the difference of any two numbers can be restated using the additive inverse. D-4 Providence Public Schools
Adding and Subtracting with Rational Numbers (12 14 days) Grade 7 Mathematics, Quarter 1, Unit 1.1 Learning Objectives Students will be able to: Instruction Use the set model and the number line model to generalize strategies for adding rational numbers. Use the set model and the number line model to generalize strategies for subtracting rational numbers. Explore and apply algorithms for inverse relationships with rational numbers. (Learn that they are actually adding the opposite number. ) Understand and use the relationship between addition and subtraction found in fact families and solve simple equations with missing facts by using related fact families. Demonstrate and apply understandings of adding and subtracting rational numbers Resources Connected Mathematics 2, Pearson/Prentice Hall, 2008: Accentuate the Negative Investigation 2: Adding and Subtracting Integers; Student Book (pages 22-41) Teacher s Guide Implementing and Teaching Guide Teaching Transparencies Assessment Resource Book Additional Practice and Skills Workbook Strategies for English Language Learners Special Needs Handbook Parent Guide Prentice Hall Teacher Station Software Exam View Software www.pearsonsuccessnet.com/snpapp/login/login.jsp and www.phschool.com. (Students can enter webcodes.) Teaching with Foldables (Dinah Zike; Glencoe McGraw Hill 2010) Available with the Algebra resources Algebra 1, (Glencoe McGraw Hill) 2010 Section 0-3 pp. P11 to P12 Section 0-4 pp. P13 to P16 Providence Public Schools D-5
Adding and Subtracting with Rational Numbers (12 14 days) Integer Addition and Subtraction Supplemental Unit see the Supplemental Unit Materials section of this booklet Launch-Explore-Summary notes Black line masters for related transparencies and lab sheets Black line masters for integer cards Reflections and assessment Note: The district resources may contain content that goes beyond the standards addressed in this unit. See the Planning for Effective Instructional Design and Delivery and Assessment sections for specific recommendations. Materials Transparencies, lab sheets, number lines, student notebooks, chips or tiles in two colors, rulers or straight edges, poster paper, overhead pens or dry-erase markers for laminated game board (one or two per pair), game markers (optional, one per student), game ball (e.g., penny, small chip), integer cards (one set per pair) or deck of playing cards, large chart paper (two sheets), scissors (one per pair), integer chips (optional), two color map pencils (optional), number lines (optional), centimeter grid paper (one per student minimum), transparent grid paper Instructional Considerations Key Vocabulary additive inverse algorithm Planning for Effective Instructional Design and Delivery Reinforced vocabulary from previous grades or units: absolute value and commutative property. Living word walls will assist all students in developing content language. Word walls should be visible to all students, focus on the current unit s vocabulary, both new and reinforced, and have pictures, examples, and/or diagrams to accompany the definitions. Teachers should review the Mathematics of the Unit found on page 3 of all CMP2 teacher editions. Adding and subtracting rational numbers is a culminating standard so students should work toward becoming fluent in this area. This is the foundation for much of the work with expressions and equations. In investigations 2.1 and 2.2 models provide ways to check students thinking. Have students talk through solutions to problems using a number line. Encourage students to use what they know about number lines and chips to explain their thinking. When appropriate, have students create real world problems to assess their understanding of adding and subtracting integers. In Investigation 2.2, use cues, questions, and advance organizers to activate prior knowledge. Provide students with the number sentence 12 + = 20, then ask the question What is a story that we could write to match this number sentence? This will support students in a review of generalizing strategies for adding integers before beginning to generalize strategies for subtracting integers. In Investigation 2.3 encourage students to refer to the number line and chip board to understand the relationships between positive and negative integers. D-6 Providence Public Schools
Adding and Subtracting with Rational Numbers (12 14 days) Grade 7 Mathematics, Quarter 1, Unit 1.1 In Investigation 2.4 students need to understand that fact families are based on the relationship between addition and subtraction. Create your own fact family to help your students understand this concept if necessary. Students should understand that this is an extension of what they already know from whole numbers. By making this connection to prior learning, students will use this information to make meaning of the more complex concept of finding missing facts by using related fact families. If students have difficulty with the variable, have them replace the variable with a question mark, and express in words what they are trying to find. The supplemental unit provides an opportunity for students to apply their knowledge in a real world problem. In Problems 1.1 and 1.2, use nonlinguistic representations to represent knowledge by having students represent integer addition using physical models (number lines and chip boards). Students use Lab Sheet 1.2B to show their representations. The number line models also relate to the mental picture promoted by the integer ball game that students played in Problem 1.1. The integer ball game is also a kinesthetic activity as students draw cards and/or use chips to add and subtract integers. It is important that students make connections between those physical models and the number sentences that they represent. Throughout this unit of study, reinforce the use of appropriate notation using symbols (=,,,, <, >). There are numerous ACE questions in the Accentuate the Negative book (e.g., page 29, items 12-23) that provide additional practice while also reinforcing order and comparison with rational numbers. The strategy of identifying similarities and differences is appropriate in Problem 1.3 as an application/review experience by having students compare addition and subtraction. Sentence stems such as Subtracting a positive is the same as would help students make connections between addition and subtraction and deepen their understanding of subtraction as adding the opposite. The Algebra 1 resource provides students with the opportunity to work with all rational numbers. Students should be able to demonstrate and apply their knowledge of signed numbers with rationals. Incorporate the Essential Questions as part of the daily lesson. Options include using them as a do now to activate prior knowledge of the previous day s lesson, using them as an exit ticket by having students respond to it and post it, or hand it in as they exit the classroom, or using them as other formative assessments. Essential questions should be included in the unit assessment. For planning considerations read through the teacher edition for suggestions about scaffolding techniques, using additional examples, and differentiated instructional guidelines as suggested by the CMP resource. CMP2 has online resources that may be helpful in planning for all units of study. Visit www.phschools.com and sign on to SuccessNet. You will find the Common Core Additional Investigations and Common Core Investigations Teacher s Guide under the worksheet tab. Providence Public Schools D-7
Adding and Subtracting with Rational Numbers (12 14 days) Notes D-8 Providence Public Schools