Overview. Essential Questions. Grade 7 Mathematics, Quarter 1, Unit 1.2 Multiplication and Division With Rational Numbers

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1 With Rational Numbers Overview Number of instruction days: Content to Be Learned Understand how to multiply and divide positive and negative rational numbers to solve problems. Apply the Distributive Property to multiply signed numbers. Understand that every quotient of integers (nonzero divisor) is a rational number. Interpret the products and quotients of rational numbers by describing real-world problems. Convert rational numbers to repeating or terminating decimals. Apply the properties of operations (i.e., Distributive Property, Associative Property, Commutative Property, Multiplicative Identity Property, Multiplicative Inverse Property) to multiply and divide rational numbers. Solve real-world problems using the four operations with rational numbers. Essential Questions How can you decide if the product of two or more numbers is positive, negative, or zero without actually calculating the product using real-world and mathematical problems? How can you decide if the quotient of two numbers is positive, negative, or zero without actually calculating the quotient using realworld and mathematical problems? Mathematical Practices to Be Integrated 1 Make sense of problems and persevere in solving them. Make conjectures about the form and meaning of the solution pertaining to decimal form of a rational number that terminates in zero or eventually repeats. 6 Attend to precision. Calculate accurately and efficiently, and express numerical answers with a degree of precision appropriate for converting a rational number to a decimal that terminates in zero or eventually repeats. Understand and explain the meaning of signed numbers used in multiplication and division of rational numbers with consistency. 8 Look for and express regularity in repeated reasoning. Notice if calculations are repeated, and look for general methods for solving problems involving positive and negative rational numbers. When dividing 25 by 11, notice that the same calculations are repeated over and over again. How do you convert a rational number to a decimal? How do you determine if a rational number is a terminating or repeating decimal? How would you use the Distributive Property to solve multiplication of integers? Providence Public Schools D-9

2 Version 4 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.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. c. Apply properties of operations as strategies 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. 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-10 Providence Public Schools

3 Grade 7 Mathematics, Quarter 1, Unit 1.2 With Rational Numbers (12 14 days) Version 4 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. 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 In grade 6, students learned about negative numbers and the kind of quantities they can be used to represent. Students also learned about absolute value and ordering of rational numbers, including realworld 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, students apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers. Students extend their knowledge of multiplication and division with rational numbers, maintaining the properties of operations. Students integrate their skills and understanding of the four operations to solve problems with rational numbers. Additionally, they learn to convert rational numbers to terminating or repeating decimal numbers. 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. Providence Public Schools D-11

4 Version 4 With Rational Numbers (12 14 days) Additional Findings Middle grade 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, such as below-zero winter temperatures or lost yards on football plays. In the middle grades, students should extend these initial understandings of integers. Positive and negative integers should be seen as useful for noting relative changes or values. (Principles and Standards for School Mathematics, pp ) Students need to be fully proficient with rational numbers and integers. This proficiency forms the basis for much of advanced mathematical thinking, as well as the understanding and interpretation of daily events. (Adding It Up, p. 247) 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. Multiply and divide of positive and negative rational numbers to solve problems. Apply the distributive property to multiply signed numbers. Interpret the products and quotients of rational numbers by describing real-world problems. Convert rational numbers to repeating or terminating decimals. Apply the properties of operations (i.e. distributive property, associative property, commutative property, multiplicative identity property, multiplicative inverse property) to multiply and divide of rational numbers. Solve real world problems using the four operations with rational numbers. D-12 Providence Public Schools

5 Grade 7 Mathematics, Quarter 1, Unit 1.2 With Rational Numbers (12 14 days) Version 4 Learning Objectives Students will be able to: Instruction Represent real-world problems with rational numbers in order to multiply positive and negative numbers. Solve real world problems with rational numbers in order to multiply positive and negative numbers. Use patterns to develop an algorithm for multiplying and dividing rational numbers (positive and negative numbers). Apply order of operations to simplify and solve expressions and equations with rational numbers Use the distributive properties to simplify real world problems with rational numbers. Use division to convert rational numbers to terminating or repeating decimals Apply understandings of integer operations and properties of numbers. Resources Connected Mathematics 2, Pearson/Prentice Hall, 2008: Accentuate the Negative Investigation 3: Multiplying and Dividing Integers; Student Book (pp ) Connected Mathematics 2, Pearson/Prentice Hall, 2008: Bits and Pieces 3 Investigation 3.4: Representing Fractions as Decimals: Students Book (pp.41-42) ACE Problems 25-27; 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 and (Students can enter webcodes.) Teaching with Foldables (Dinah Zike; Glencoe McGraw Hill 2010) Available with the Algebra Resource Materials. Providence Public Schools D-13

6 Version 4 With Rational Numbers (12 14 days) Algebra 1, (Glencoe McGraw Hill) 2010 Section 0-5 pp. P17 to P19 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, transparency markers, colored pens, paper clips (2 per pair), calculators Instructional Considerations Key Vocabulary There is no new vocabulary in this unit. Planning for Effective Instructional Design and Delivery Reinforced vocabulary taught in previous grades or units: distributive property, order of operations, number sentence, product, and quotient. 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, 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. Read note to teachers on page 63 of the Teacher s Guide regarding consistency in talking about integer multiplication. Multiplying and dividing 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 Accentuate the Negative, you may want to review Some Notes on Notation on page 42. Students make common errors by confusing the signs and the operation symbols. This section may help lessen those mistakes. Lesson 3.1 can be skipped or compacted if you use the pattern provided in problem 3.2 to understand the rules of multiplying positive and negative numbers. This modification should be based on your students current level of understanding. The Special Needs Handbook (pp ) provides support for Investigation 3, ACE question 1. ACE question 31 (p. 57) reinforces and deepens understanding of the associative property. Problem 3.3 is a good time to use identifying similarities and differences to apply content. Students compare number sentences and look for patterns among fact families as they develop their understandings of integer multiplication and division. Use a comparison matrix for this activity. Students learned to play the product game in the 6th-grade unit Prime Time. The game included only whole number factors and products, the game in Problem 3.4, includes integer factors and products. The ACE questions for this investigation provide lots of excellent practice with all of the number and operations concepts that students have learned over the years (rational numbers, integers, exponents). D-14 Providence Public Schools

7 Division Multiplication Subtraction Addition Grade 7 Mathematics, Quarter 1, Unit 1.2 With Rational Numbers (12 14 days) Version 4 During the Looking Back and Looking Ahead unit review, you can use cues, questions, and advance organizers to activate prior knowledge before starting the assignment. Introduce a graphic organizer, such as a foldable, so that students can organize their work with integer operations. Students can divide a sheet of notebook paper into quadrants. Students could write the rules and some examples for each operation. The following organizer could also be used: Rule(s) Examples The Algebra 1 resource provides students with the opportunity to work with all rational numbers. Bits and Pieces 3, section 3.4 is a good resource for instruction on repeating and terminating decimals. Check the ACE questions and Additional Practice and Skills Workbook for more optional items for students to practice. 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 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-15

8 Version 4 With Rational Numbers (12 14 days) Notes D-16 Providence Public Schools