# Number System Properties Grade Nine

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1 Ohio Standards Connection: Number, Number Sense and Operations Benchmark C Apply properties of operations and the real number system, and justify when they hold for a set of numbers. Indicator 1 Identify and justify whether properties (closure, identity, inverse, commutative and associative) hold for a given set and operations; e.g., even integers and multiplication. Mathematical Processes Benchmarks D. Apply reasoning processes and skills to construct logical verification or counterexamples to test conjectures and to justify and defend algorithms and solutions. E. Use a variety of mathematical representations flexibly and appropriately to organize, record and communicate mathematical ideas. G. Write clearly and coherently about mathematical thinking and ideas. Lesson Summary: Students work independently and in cooperative pairs to review and apply the commutative, associative, identity and inverse properties to a variety of sets of numbers. The closure property is introduced. Students work independently and in cooperative pairs to practice the application of the closure property. As a post-assessment, students demonstrate their comprehension of the properties by selecting an ancient number system or by designing an original number system, and justifying which of the properties hold for the system and why other properties do not. Students present results to the class. Estimated Duration: Two hours and 30 minutes Commentary: Students experience the properties of the real number system early in mathematics when they discover different ways to compose a number, such as = 5 and = 5. In the intermediate and middle grades, more sets of numbers from the real number system are introduced at appropriate times. Students may make conjectures before they are introduced to formal mathematical terms. As they progress in their understanding and use of the subsets of real numbers, conjectures are tested to determine if they hold true for any real number. Pre-Assessment: Instructional Tip: Use Attachment A, Mini Lesson for Understanding Justify and Counter-Example, if students need practice on the concepts of justify and counter-example prior to this lesson. Write the set of natural numbers in set notation, {1, 2, 3, 4 }, on the overhead or board. Ask students to answer a group of questions about the natural numbers, justifying their answers in writing. Distribute copies of the handout, Pre-Assessment Discussions, Attachment B, or write questions on the board or overhead. Also, read them orally. 1

5 Differentiated Instructional Support: Instruction is differentiated according to learner needs, to help all learners either meet the intent of the specified indicator(s) or, if the indicator is already met, to advance beyond the specified indicator(s). For students who have not mastered the basic skills, based on the pre-assessment and/or the class activities, the teacher may want to: Work more examples with different sets of numbers in teacher-led small groups. Supply students with scientific calculators. Allow students to work in pairs for the post-assessment. For Part One, Lesson Closure, students explain the properties using two of the three or one of the three methods in writing, symbols or with examples. For Part Two, Lesson Closure, students choose an operation and three properties to investigate, rather than all the properties (commutative, associative, inverse, identity and closure). Investigate and prepare reports/presentations on why division by zero is undefined. Investigate and prepare reports/presentations on field properties (group theory). Extensions: Students, independently or in pairs, create a column chart, illustrating which properties hold for the operations for these number systems: natural numbers, whole numbers, integers, rational numbers, irrational numbers and the real numbers. For the post-assessment, independently or in pairs students investigate and compare/contrast several ancient number systems to determine which of the properties hold and which do not. In place of a poster in the post-assessment, students use presentation software. Interdisciplinary Connections: The investigation of ancient number systems connects with history or social studies. Writing informative papers has an English language arts connection. Creating symbols for original number systems could involve calligraphy or other art. Studying the Mayan, Aztec or other ancient Hispanic systems connects with foreign language. Materials and Resources: The inclusion of a specific resource in any lesson formulated by the Ohio Department of Education should not be interpreted as an endorsement of that particular resource, or any of its contents, by the Ohio Department of Education. The Ohio Department of Education does not endorse any particular resource. The Web addresses listed are for a given site s main page, therefore, it may be necessary to search within that site to find the specific information required for a given lesson. Please note that information published on the Internet changes over time, therefore the links provided may no longer contain the specific information related to a given lesson. Teachers are advised to preview all sites before using them with students. 5

6 For the teacher: Overhead transparencies, overhead markers, manipulatives such as round circular cereal, candy, or larger square cereal, aquarium gravel in two sizes, or overhead manipulatives in two different sizes For the student: Paper, pencil, overhead transparencies, overhead markers, manipulatives such as round circular cereal, candy, or larger square cereal, aquarium gravel in two sizes, or small and larger circles cut from construction paper, scientific calculator (optional), poster board, markers Vocabulary: associative property closure property communicative property composite counter example even identity property integers inverse property irrational numbers justify natural numbers odd prime rational numbers real numbers whole numbers Technology Connections: Use the overhead projector to model manipulation of the pebbles and rocks. Use computers for research and presentation software. Verify calculations with scientific calculators when determining which property holds for the operation on a set of numbers Research Connections: "BSCS Science: An Inquiry Approach." BSCS Biological Sciences Curriculum Study. 23 Dec Marzano, Robert J., Jane E. Pollock and Debra Pickering. Classroom Instruction that Works: Research-Based Strategies for Increasing Student Achievement, Alexandria, Va: Association for Supervision and Curriculum Development,

7 Sousa, David A. How the Brain Learns: A Classroom Teacher s Guide. Reston, Va: NASSP, General Tips: Even if class periods are long enough to get through more than Part One activities, students may benefit from having the material spread over several days. Though all properties, except closure, are addressed in earlier grades, keep in mind that all students may not have had sufficient experience with the properties. Match the number of practice problems to the needs of the students. Students may need more intense review of the number systems used in the activities. Attachments: Attachment A, Mini Lesson for Understanding Justify and Counter-Example Attachment B, Pre-Assessment Discussion Questions Attachment C, Post-Assessment Scoring Guidelines 7

10 Attachment C Post-Assessment Scoring Guidelines Rubric 5 Explains number system and correctly identifies and justifies which of the properties (closure, identity, inverse, commutative or associative) hold for the number system and which do not and correctly offers further analysis, such as: If a particular property does not hold, the change(s) in the number system would cause that property to hold. OR The consequences of a particular property not holding. 4 Explains number system and correctly identifies and justifies which of the properties hold for the number system and which do not hold. 3 Explains numbers system and correctly identifies and justifies which of the properties hold for the number system and which do not hold, with two or fewer errors. 2 Explains number system and correctly identifies and justifies which of the properties hold for the number system and which do not hold, with more than two errors. 1 Explains number systems, but makes no attempt to identify and justify the properties. OR Does not explain number system, but attempts to identify and justify the properties. 0 Does not explain number system and makes no attempt to identify and justify the properties. 10

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