Inductive and Deductive Reasoning - If-you-give-a-moose-hall

Mathematics Enhanced Scope and Sequence – Geometry

Inductive and Deductive Reasoning Organizing topic Overview Related Standard of Learning

Reasoning and Proof Students practice inductive and deductive reasoning strategies. G.1

Objectives • • • •

The student will use inductive reasoning to make conjectures. The student will use logical arguments to prove or disprove conjectures. The student will justify steps while solving linear equations, using properties of real numbers and properties of equality. The student will solve linear equations as a form of deductive proof.

Prerequisite Understandings/Knowledge/Skills • Students must be able to differentiate between inductive and deductive reasoning. • Students must be familiar with the definitions of addition and multiplication properties. • Students must know how to read and analyze word problems. • Students must be able to recognize and identify the use of variables in equations. Instructional activity 1. 2. 3.

Review the basic vocabulary included on the activity sheets. Have students work in pairs or small groups to complete the activity sheets. Use the algebraic properties of equality (shown on Activity Sheet 3) for matching, concentration, or filling in the steps of a proof in addition to writing.

Follow-up/extension • •

Have students investigate practical problems involving inductive or deductive reasoning. Have students create their own conjectures to prove or disprove.

Sample assessment • • •

Have students work in pairs to evaluate strategies. Use activity sheets to help assess student understanding. Have students complete a journal entry comparing and contrasting inductive and deductive reasoning strategies.

Specific options for differentiating this lesson Technology • Allow students to use a calculator to make simple calculations and compare their findings. Multisensory • Use an overhead projector to illustrate the steps in deductive and inductive reasoning. • Place two extra large triangles on the board. One triangle is red; place it on the board, point side up. The other triangle is blue; place it on the board point side down. Place 4 green boxes Virginia Department of Education 2004

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(called information blocks) inside the triangles, at the top of each triangle, keeping them in a straight line: 1 in the red and 3 in the blue. Then place 4 yellow boxes (called clue blocks) inside the triangles and in the middle of each triangle, keeping them in a straight line: 2 in the red and 2 in the blue. Then place 4 orange boxes (called conclusion blocks) inside the triangles and in the middle of each, keeping them in a straight line: in the red and 1 in the blue. After this process, explain to the class that the inductive process begins with few facts, but ends with many possible conclusions. Then explain that the deductive process begins with many facts, but ends with few possible conclusions. Community Connections • Invite a politician or political analyst to visit the class. Ask the guest speaker to explain the relationship among facts, trends, and educated guesses. Small Group Learning • Write the names of properties on one set of index cards and their definitions on another set of index cards. Have students pick one or more cards and match themselves with the student who has the corresponding property or definition. Check for correctness and ask students to post their answers on the board or orally share their answers with the class. Vocabulary • Students need to know the following vocabulary: deductive reasoning, inductive reasoning, pattern, fact, definition, property, logical argument, conjecture, verify, modify, proof, prove, disprove, observation, prior experience, conclusion, addition property, subtraction property, multiplication property, division property, reflexive property, symmetric property, transitive property, substitution property, distributive property, acute angle, obtuse angle, addends, sum. • Have students use PowerPoint software to create presentations of the vocabulary terms. This will allow them to manipulate software so that sound, pictures, graphs, colors, and motion are available during the learning process. • Post the properties in the room with the statements. • Have students write the topics and statements on index cards and practice their understanding with one another before and after the lesson. • Have students draw illustrations to represent the mathematical properties. Illustrations may be groupings of people at work or play, animals in habitat scenes, and (but not limited to) daily or current events. Student Organization of Content • Have students use a FISH diagram to illustrate the information (each student could have his/her own FISH diagram). • Have students use the same graphic organizer as on Activity Sheet 1 to organize material learned in this lesson.

Virginia Department of Education 2004

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Activity Sheet 1: Inductive and Deductive Reasoning Inductive reasoning works from the more specific observations to broader generalizations.

Deductive reasoning works from the more general to the more specific. Deductive Reasoning

Inductive Reasoning

Pattern Facts

Definitions

Accepted Properties Conjecture

Logical Argument

Verify/Modify

•

• •

Example of Deductive Reasoning Tom knows that if he misses the practice the day before a game, then he will not be a starting player in the game. Tom misses practice on Tuesday. Conclusion: He will not be able to start in the game on Wednesday.

• • • •

Example of Inductive Reasoning Observation: Mia came to class late this morning. Observation: Mia’s hair was uncombed. Prior Experience: Mia is very fussy about her hair. Conclusion: Mia overslept.

Complete the following conjectures based on the pattern you observe in specific cases: Conjecture: The sum of any two odd numbers is ________.

1+1=2 1+3=4 3+5=8

7 + 11 = 18 13 + 19 = 32 201 + 305 = 506

Conjecture: The product of any two odd numbers is ________. Conjecture: The product of a number (n – 1) and the number (n + 1) is always equal to ________. Prove or disprove the following conjecture: 2 Conjecture: For all real numbers x, the expression x is greater than or equal to x.


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Activity Sheet 2: Inductive and Deductive Reasoning 1.

John always listens to his favorite radio station, an oldies station, when he drives his car. Every morning he listens to his radio on the way to work. On Monday when he turns on his car radio, it is playing country music. Make a list of valid conjectures to explain why his radio is playing different music.

2.

∠M is obtuse. Make a list of conjectures based on that information.

3.

Based on the table to the right, Marina concluded that when one of the two addends is negative, the sum is always negative. Write a counterexample for her conjecture.

Statement 5x – 18 = 3x + 2 2x – 18 = 2 2x = 20 x = 10

Reason Given Subtraction Property of Equality Addition Property of Equality Division Property of Equality

Addends –8 –10 –17 –5 15 –23 –26 22

Sum –18 –22 –8 –4

The Algebraic Properties of Equality, as shown on Activity Sheet 3, can be used to solve 5x – 18 = 3x + 2 and to write a reason for each step, as shown in the table on the left.

Using a table like this one, solve each of the following equations, and state a reason for each step. 4. –2(–w + 3) = 15 5. p – 1 = 6 6. 2r – 7 = 9 7. 3(2t + 9) = 30 8. Given 3(4v – 1) –8v = 17, prove v = 5. Match each of the following conditional statements with a property: A. B. C. D. E.

9. 10. 11. 12. 13.

Multiplication Property Substitution Property Transitive Property Addition Property Symmetric Property

F. G. H. I.

Reflexive Property Distributive Property Subtraction Property Division Property

If JK = PQ and PQ = ST, then JK = ST. _____ If m ∠S = 30°, then 5° + m∠S = 35°. _____ If ST = 2 and SU = ST + 3, then SU = 5. _____ If m ∠K = 45°, then 3(m∠K) = 135°. _____ If m ∠P = m ∠Q, then m ∠Q = m ∠P. _____


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Activity Sheet 3: Algebraic Properties of Equality a, b, and c are real numbers

Addition Property

If a = b, then a + c = b + c

Subtraction Property

If a = b, then a – c = b – c

Multiplication Property

If a = b, then ac = bc

Division Property

If a = b and c ≠ 0, then a÷c=b÷c

Reflexive Property

a=a

Symmetric Property

If a = b, then b = a

Transitive Property

If a = b and b = c, then a = c

Substitution Property

If a = b, then a can be substituted for b in any equation or expression.

Distributive Property

a(b + c) = ab + ac


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