Number System Properties Grade Nine


 Cameron Powers
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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 postassessment, 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. PreAssessment: Instructional Tip: Use Attachment A, Mini Lesson for Understanding Justify and CounterExample, if students need practice on the concepts of justify and counterexample 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, PreAssessment Discussions, Attachment B, or write questions on the board or overhead. Also, read them orally. 1
2 Students work with partners to compare and discuss their answers. Monitor them as they work. As a whole class, discuss students results, correcting errors and misconceptions. Scoring Guidelines: Informal teacher observation PostAssessment: Students independently research an ancient number system, such as the Mayan or the Egyptian systems, and investigate and justify which of the properties (commutative, associative, inverse, identity or closure) hold for the system. OR Students invent an original number system and explain/justify which properties hold for the system. Each student prepares a poster explaining the system and its properties. The poster serves as a visual aid during the student s presentation to the class. Scoring Guidelines: Use the rubric, PostAssessment Scoring Guidelines, Attachment C, to determine the student s level of understanding. Students who score zero, one or two need to have moderate to intense intervention. Instructional Procedures: Part One 1. Complete the preassessment activities as appropriate to the level of students understanding. 2. Review four of the five properties by writing the property names, key phrases words or symbols on the board or overhead: commutative property order a + b = b + a associative property grouping (a + b) + c = a + (b + c) inverse property opposite a + (a) = 0 and (a) + a = 0 identity property  value does not change a + 0 = a and 0 + a = a 3. Ask students to refer to the preassessment questions. Discuss the connections between the properties and the questions. For example, the question referring to the order of adding the numbers deals with the commutative property. 4. Explain the closure property, which is new to the students. If two numbers from a set are added (or subtracted, multiplied or divided) and the answer is in the same set, that set of numbers is said to be closed with respect to addition (or subtraction, multiplication, or division). In this case, if any two natural numbers are added, the sum is a natural number; so the set of natural numbers is closed with respect to addition. 5. Add to the property list: Closure: a. The set of whole numbers is closed with respect to addition. For example: if a, b {whole numbers}, then a + b {whole numbers}. b. If the sum of any two elements of A is also in A, then A is closed under addition. 2
3 Instructional Tip: Checking for closure property in division may lead to division by zero. Instruct students to consider only numbers for which division is defined, ignoring the case of dividing by zero. 6. As a class, determine and justify which properties hold for the remaining operations (subtraction, multiplication, and division) on the natural numbers, and which do not. Elicit student answers correcting misconceptions. Record results on the board or overhead, naming the property and using symbolic form. [Example: commutative property of multiplication [(a) (b) = (b) (a)] 7. Given a list of properties (commutative, associative, identity, inverse and closure), students independently write explanations for each property, write the property in algebraic symbols and give numerical examples of each. Part Two 8. Distribute manipulatives to students. Each student needs approximately 15 pebbles (small pieces) and one rock (large piece). Instructional Tip: Use this number system to provide the students experience with nontraditional number systems, as well as to get them thinking about designing their own number systems. 9. Tell the following story to the students while demonstrating with manipulatives: An ancient civilization has just been discovered still living in a remote area. Having been kept from the modern world, their ways are quite primitive. As their numbers are used primarily for counting only, their number system is very simple. They use small pebbles and larger rocks to indicate numbers. One pebble means the number one. (Hold up one pebble or place one pebble on the overhead.) Two pebbles mean the number two. (Hold up two pebbles or place two pebbles on the overhead.) Continue through to the number 10. The symbol for any amount greater than 10 is the rock and means many. (Hold up the larger rock.) 10. Discuss which operation(s) these primitive people would most likely use. Using the pebbles, model addition problems, such as and 5 + 7, for the students. Group three pebbles on the overhead; group four pebbles on the overhead; move the two groups together. Repeat this process for The answer to 5+7 is the rock. 11. Allow students time to practice their own addition problems, using their pebbles and rock. 12. Discuss which properties hold for addition on this set of numbers, and why other properties do not hold. Elicit student response for justification and counterexamples. Record results on the board or overhead, correcting students misconceptions. 3
4 13. Continue instruction by repeating the procedures and modeling examples for the other operations: subtraction, multiplication and division. To model a subtraction problem, such as 73, begin with seven pebbles on the overhead and remove three. To model a multiplication problem, such as 4x2, place four groups of two pebbles on the overhead. To model a division problem, such as 8 4, begin with eight pebbles on the overhead and separate them into four equal groups. 14. Write the set of even integers: { 4, 2, 0, 2, 4 } on the overhead projector or board. Direct students to find a partner Instruct students to consider the commutative, associative, inverse, identity and closure properties in relation to the set of even integers. Assign one operation (addition, subtraction, multiplication or division) to each pair, making certain each operation is assigned to at least one pair of students. Each pair determines and justifies which of the properties (commutative, associative, inverse, identity and closure) hold for its assigned operation, and why other properties do not hold. 16. Observe pairs as they work, monitoring their progress and helping as needed. 17. When pairs have completed the task, explain to students that one corner of the room is the addition corner, another, the subtraction corner, another, the multiplication corner, and the fourth corner, a division corner. Students report to the corner of the operation they were assigned and discuss results with that group. 18. Bring students back together to discuss results as a class. Record results on the board or overhead, correcting misconceptions. On the board or overhead, write the set of rational numbers. Ask what types of numbers would be included in this set. Elicit student responses, recording correct answers on the overhead or board and correcting misconceptions. 19. Assign each student an operation (addition, subtraction, multiplication or division). Each student independently determines and justifies which of the properties (commutative, associative, inverse, identity and closure) hold for their operation on the set of rational numbers and why other properties do not hold. Monitor student progress, helping as needed. 20. Have the students report to the corner designated for their assigned operations to discuss results. (See Step 17.) 21. Bring students back together as a class to discuss results, eliciting student responses. Record class results on the board or overhead, correcting misconceptions. 22. Have students summarize concepts and reflect on learning a. List on board or overhead in set notation the set of even whole numbers {0, 2, 4, 6 }. b. All students consider the set of even whole numbers. Each student chooses an operation (addition, subtraction, multiplication or division). c. Independently, students investigate the set of even whole numbers, determining and justifying, in writing, which of the properties (commutative, associative, inverse, identity or closure) hold for the set of even whole numbers and why other properties do not hold. 4
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 preassessment and/or the class activities, the teacher may want to: Work more examples with different sets of numbers in teacherled small groups. Supply students with scientific calculators. Allow students to work in pairs for the postassessment. 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 postassessment, 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 postassessment, 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: ResearchBased 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 CounterExample Attachment B, PreAssessment Discussion Questions Attachment C, PostAssessment Scoring Guidelines 7
8 Attachment A Mini Lesson for Understanding Justify and CounterExample 1. Write the terms justify and counterexample on the board or overhead and ask students to think about the meanings of these terms, when they might have heard the terms, or how they might have used them. 2. Write this statement on the board or overhead: All prime numbers are odd. Ask students to independently decide if the statement is true or false and to write on their paper why they think so. 3. Ask student volunteers to share their ideas. If no one mentions that the number two is even and prime, prompt the students by asking, Is there an even prime number? 4. Explain that since the number two is prime and even, it is a counter example to the statement: All prime numbers are odd. That is, it is one example that shows the statement is false. Only one counter example is needed to show that a statement is false. 5. Write this statement on the board or overhead: If a figure is a square, then it is a rectangle. Ask students to independently decide if the statement is true or false and to write why they think so. 6. Ask student volunteers to share their ideas. Record on the board or overhead correct student explanations. 7. Explain to students that to justify means to explain why something is true or why something works. Diagrams, calculations or definitions are some of the tools used to justify. One example alone does not prove a statement is true. 8
9 Attachment B PreAssessment Discussion Questions Directions: Answer the following questions. Record your responses in a notebook or journal. If two numbers from the set are added, does it matter which number comes first? Is this true for any two numbers in the set? If three of these numbers are added, does it matter which two are added together first? Is this true for any three numbers in the set? Pick a number from the set. Add it to any number in the set. Is the answer the original number? Will this ever happen using the numbers in this set? Does each number in this set have an opposite? Pick two numbers from the set and add them. Is the answer in this set? Will the answer to every such addition problem result in a number in this set? 9
10 Attachment C PostAssessment 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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