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Make conjectures based on inductive reasoning. Find counterexamples. conjecture inductive reasoning counterexample Lesson 1 MI/Vocab

An educated guess based on known information. CONJECTURE - An educated guess based on known information. INDUCTIVE REASONING - Reasoning that begins with knowledge of SPECIFIC Facts to develop a GENERAL CONCLUSION (from SPECIFIC to GENERAL).

GEOMETRY 2.1 INDUCTIVE Reasoning & Conjecturing Some Examples of Inductive Reasoning Specific Observations: 3 Consecutive Times Alex Chose Chocolate rather than Vanilla Ice Cream General Conjectures: Alex doesn’t like Vanilla Alex likes Chocolate Ice Cream BEST Alex likes Chocolate more than Vanilla

GEOMETRY 2.1 INDUCTIVE Reasoning & Conjecturing Some Examples of Inductive Reasoning Specific Observations: When it Snows, my car won’t start. General Conjectures: My Car is allergic to Snow I need to push the Snow Button on my Car

Patterns and Conjecture Make a conjecture about the next number based on the pattern. 2, 4, 12, 48, 240 Find a pattern: 2 4 12 48 240 ×2 ×3 ×4 ×5 The numbers are multiplied by 2, 3, 4, and 5. Conjecture: The next number will be multiplied by 6. So, it will be 6 ● 240 or 1440. Lesson 1 Ex1

A. B. C. D. A B C D Lesson 1 CYP1

Geometric Conjecture For points L, M, and N, LM = 20, MN = 6, and LN = 14. Make a conjecture and draw a figure to illustrate your conjecture. Given: points L, M, and N; LM = 20, MN = 6, and LN = 14. Examine the measures of the segments. Since LN + MN = LM, the points can be collinear with point N between points L and M. Conjecture: L, M, and N are collinear Lesson 1 Ex2

Given: ACE is a right triangle with AC = CE Given: ACE is a right triangle with AC = CE. Which figure would illustrate the following conjecture? ΔACE is isosceles, C is a right angle, and is the hypotenuse. A. B. C. D. A B C D Lesson 1 CYP2

ONE counterexample defeats an Inductive Conjecture! An example used to show that a given statement is not always true. ONE counterexample defeats an Inductive Conjecture!

GEOMETRY 2.1 INDUCTIVE Reasoning & Conjecturing Can you Find a COUNTEREXAMPLE?

GEOMETRY 2.1 INDUCTIVE Reasoning & Conjecturing Can you Find a COUNTEREXAMPLE?

GEOMETRY 2.1 INDUCTIVE Reasoning & Conjecturing Can you Find a COUNTEREXAMPLE?

? GEOMETRY 2.1 INDUCTIVE Reasoning & Conjecturing Can you Find a COUNTEREXAMPLE? ?

GEOMETRY 2.1 INDUCTIVE Reasoning & Conjecturing

GEOMETRY 2.1 INDUCTIVE Reasoning & Conjecturing

Find two cities such that the population of the first is UNEMPLOYMENT Based on the table showing unemployment rates for various counties in Texas, find a counterexample for the following statement. The unemployment rate is highest in the cities with the most people. Find two cities such that the population of the first is greater than the population of the second while the unemployment rate of the first is less than the unemployment rate of the second. Lesson 1 Ex3

Answer: Maverick has a population of 50,436 people in its population, and it has a higher rate of unemployment than El Paso, which has 713,126 people in its population. Lesson 1 Ex3

The greater the population of a state, the lower the number of drivers CONJECTURE: The greater the population of a state, the lower the number of drivers per 1000 residents. COUNTEREXAMPLE? A. Texas & California B. Vermont & Texas C. Wisconsin & West Virginia D. Alabama & West Virginia Lesson 1 CYP3

GEOMETRY 2.1 INDUCTIVE Reasoning & Conjecturing SUMMARY Inductive Reasoning tries to move from one or more Specific Observations to a General Conjecture or Conclusion If a General Conjecture is proposed, test it’s TRUTH by looking for a COUNTEREXAMPLE QUESTION: If no COUNTEREXAMPLE is FOUND is the Conjecture TRUE?

Determine truth values of conjunctions and disjunctions. Construct truth tables. statement truth table truth value negation compound statement conjunction disjunction Lesson 2 MI/Vocab

Lesson 2 KC1

COMPOUND Statement – Notation – A statement formed by joining two or more statements. Notation – p ^ q is sometimes written, CONJUNCTION is sometimes called INTERSECTION Lesson 2 KC2

TRUTH VALUE – The truth or falsity of a statement. TRUTH Table – A table used as a convenient method for organizing the truth values of statements.

p: q: A TRUTH Question – I need a LADY Volunteer! T R U T H T A B L E

p q p ^ q Truth Values of Conjunctions A. Use the following statements to write a compound statement for the conjunction p and q. Then find its truth value. p: One foot is 14 inches. q: September has 30 days. r: A plane is defined by three noncollinear points. Statement: p q p ^ q Lesson 2 Ex1

p r ~p ~p ^ r Truth Values of Conjunctions B. Use the following statements to write a compound statement for the conjunction ~p  r. Then find its truth value. p: One foot is 14 inches. q: September has 30 days. r: A plane is defined by three noncollinear points. Statement: p r ~p ~p ^ r Lesson 2 Ex1

A. A square has five sides and a turtle is a bird; false. A. Use the following statements to write a compound statement for each conjunction. Then find its truth value. p: June is the sixth month of the year. q: A square has five sides. r: A turtle is a bird. p and r A. A square has five sides and a turtle is a bird; false. B. June is the sixth month of the year and a turtle is a bird; true. C. June is the sixth month of the year and a square has five sides; false. D. June is the sixth month of the year and a turtle is a bird; false. A B C D Lesson 2 CYP1

A. A square has five sides and a turtle is not a bird; true. B. Use the following statements to write a compound statement for each conjunction. Then find its truth value. p: June is the sixth month of the year. q: A square has five sides. r: A turtle is a bird. ~q  ~r A. A square has five sides and a turtle is not a bird; true. B. A square does not have five sides and a turtle is not a bird; true. C. A square does not have five sides and a turtle is a bird; false. D. A turtle is not a bird and June is the sixth month of the year; true. A B C D Lesson 2 CYP1

Notation – p V q is sometimes written, DISJUNCTION is sometimes called UNION Lesson 2 KC3

p: q: A TRUTH Question – I need a LADY Volunteer! T R U T H T A B L E p v q

p q p V q Truth Values of Disjunctions A. Use the following statements to write a compound statement for the disjunction p or q. Then find its truth value. p: is proper notation for “line AB.” q: Centimeters are metric units. r: 9 is a prime number. Statement: p q p V q Lesson 2 Ex2

q r q V r Truth Values of Disjunctions B. Use the following statements to write a compound statement for the disjunction q  r. Then find its truth value. p: is proper notation for “line AB.” q: Centimeters are metric units. r: 9 is a prime number. Statement: q r q V r Lesson 2 Ex2

A. 6 is an even number or a cow has 12 legs; true. A. Use the following statements to write a compound statement for each disjunction. Then find its truth value. p: 6 is an even number. q: A cow has 12 legs r: A triangle has 3 sides. p or r A. 6 is an even number or a cow has 12 legs; true. B. 6 is an even number or a triangle has 3 sides; true. C. A cow does not have 12 legs or 6 is an even number; true. D. 6 is an even number or a triangle does not have 3 side; true. A B C D Lesson 2 CYP2

B. A cow has 12 legs or a triangle has 3 sides; true. ~q  ~r B. Use the following statements to write a compound statement for each disjunction. Then find its truth value. p: 6 is an even number. q: A cow has 12 legs r: A triangle has 3 sides. A. A cow does not have 12 legs or a triangle does not have 3 sides; true. B. A cow has 12 legs or a triangle has 3 sides; true. C. 6 is an even number or a triangle has 3 sides; true. D. A cow does not have 12 legs and a triangle does not have 3 sides; false. A B C D Lesson 2 CYP2

Use Venn Diagrams DANCING The Venn diagram shows the number of students enrolled in Monique’s Dance School for tap, jazz, and ballet classes. Lesson 2 Ex3

9 is the INTERSECTION Use Venn Diagrams A. How many students are enrolled in all three classes? The students that are enrolled in all three classes are represented by the intersection of all three sets. 9 is the INTERSECTION Lesson 2 Ex3

121 is the UNION Use Venn Diagrams B. How many students are enrolled in tap or ballet? The students that are enrolled in tap or ballet are represented by the union of these two sets. 121 is the UNION Lesson 2 Ex3

C. How many students are enrolled in jazz and ballet but not tap? Use Venn Diagrams C. How many students are enrolled in jazz and ballet but not tap? The students that are enrolled in jazz and ballet and not tap are represented by the intersection of jazz and ballet minus any students enrolled in tap. Answer: 25 students enrolled in jazz and ballet and not tap. Lesson 2 Ex3

PETS The Venn diagram shows the number of students at Manhattan School that have dogs, cats, and birds as household pets. Pets Lesson 2 CYP3

Pets A. How many students in Manhattan School have one of three types of pets? A. 226 B. 311 C. 301 D. 110 A B C D Lesson 2 CYP3

B. How many students have dogs or cats? Pets B. How many students have dogs or cats? A. 57 B. 242 C. 252 D. 280 A B C D Lesson 2 CYP3

C. How many students have dogs, cats, and birds as pets? Lesson 2 CYP3

Construct Truth Tables A. Construct a truth table for ~p  q. Lesson 2 Ex4

Construct Truth Tables B. Construct a truth table for p  (~q  r). Lesson 2 Ex4

A. Which sequence of Ts and Fs would correctly complete the last column of the following truth table for the given compound statement? (p  q)  (q  r) A. T B. T C. T D. T F F F F F T F T F F F F T T F T F F F F T T F F F F F F A B C D Lesson 2 CYP4

B. Which sequence of Ts and Fs would correctly complete the last column of the following truth table for the given compound statement? (p  q)  (q  r) A. T B. T C. T D. T T T F T T T T T F T F F T T T T F T F T T T F T F F F F A B C D Lesson 2 CYP4

Analyze statements in if-then form. Write the converse, inverse, and contrapositive of if-then statements. conditional statement inverse if-then statement hypothesis conclusion converse contrapositive deductive logic Lesson 3 MI/Vocab

GEOMETRY 2.3 If – Then Statements & Postulates For Inductive Reasoning: QUESTION: If no COUNTEREXAMPLE is FOUND is the Conjecture TRUE? ANSWER: INDUCTIVE Conjectures are NEVER certain BUT – in some circumstances, THAT’S OKAY!

DEDUCTIVE LOGIC – GEOMETRY 2.3 If – Then Statements & Postulates To Find CERTAIN Conjectures Use DEDUCTIVE Reasoning KEY: IF – THEN Statements DEDUCTIVE LOGIC – Reasoning from GENERAL facts known to be true to SPECIFIC Conclusions (from GENERAL To SPECIFIC)

HYPOTHESIS – CONCLUSION – In a conditional statement, the statement that immediately follows “if”. The part of the conditional assumed to be TRUE. CONCLUSION – In a conditional statement, the statement that immediately follows “then”. The part of the conditional who truth depends on the hypothesis.

If p, then q GEOMETRY 2.3 If – Then Statements & Postulates Parts of IF-THEN Statements: IF this is September THEN this is FOOTBALL Season Hypothesis Conclusion CONDITIONAL STATEMENT In Symbols: If p, then q

GEOMETRY 2.3 If – Then Statements & Postulates In IF – THEN Statements We ASSUME the Hypothesis statement is TRUE.

GEOMETRY 2.3 If – Then Statements & Postulates In IF – THEN Statements We ASSUME the Hypothesis statement is TRUE. We SEEK a CONCLUSION statement that is TRUE.

GEOMETRY 2.3 If – Then Statements & Postulates In IF – THEN Statements We ASSUME the Hypothesis statement is TRUE. We SEEK a CONCLUSION statement that is TRUE. IF this is SUNDAY, THEN there is NO SCHOOL!

GEOMETRY 2.3 If – Then Statements & Postulates In IF – THEN Statements We ASSUME the Hypothesis statement is TRUE. We SEEK a CONCLUSION statement that is TRUE. IF you are GEORGE W. Bush, THEN you are President

GEOMETRY 2.3 If – Then Statements & Postulates In IF – THEN Statements We ASSUME the Hypothesis statement is TRUE. We SEEK a CONCLUSION statement that is TRUE. Name of Student Driver:

GEOMETRY 2.3 If – Then Statements & Postulates In IF – THEN Statements We ASSUME the Hypothesis statement is TRUE. We SEEK a CONCLUSION statement that is TRUE. IF this is : car THEN:

GEOMETRY 2.3 If – Then Statements & Postulates In IF – THEN Statements We ASSUME the Hypothesis statement is TRUE. We SEEK a CONCLUSION statement that is TRUE. IF you get an A in Geometry, THEN

Now for some GEOMETRIC IF – THEN’s GEOMETRY 2.3 If – Then Statements & Postulates Now for some GEOMETRIC IF – THEN’s If M is the Midpoint between A and B THEN

Now for some GEOMETRIC IF – THEN’s GEOMETRY 2.3 If – Then Statements & Postulates Now for some GEOMETRIC IF – THEN’s If m ABC < 90 THEN

Now for some GEOMETRIC IF – THEN’s GEOMETRY 2.3 If – Then Statements & Postulates Now for some GEOMETRIC IF – THEN’s If line m and n are PERPENDICULAR THEN

A. Hypothesis: you will cry Conclusion: you are a baby A. Which of the choices correctly identifies the hypothesis and conclusion of the given conditional? If you are a baby, then you will cry. A. Hypothesis: you will cry Conclusion: you are a baby B. Hypothesis: you are a baby Conclusion: you will cry C. Hypothesis: babies cry Conclusion: you are a baby D. none of the above A B C D Lesson 3 CYP1

B. Which of the choices correctly identifies the hypothesis and conclusion of the given conditional? To find the distance between two points, you can use the Distance Formula. A. Hypothesis: you want to find the distance between 2 points Conclusion: you can use the distance formula B. Hypothesis: you are taking geometry Conclusion: you learned the distance formula C. Hypothesis: you used the distance formula Conclusion: you found the distance between 2 points D. none of the above A B C D Lesson 3 CYP1

Write a Conditional in If-Then Form A. Identify the hypothesis and conclusion of the following statement. Then write the statement in the if-then form. Distance is positive. Sometimes you must add information to a statement. Here you know that distance is measured or determined. Answer: Hypothesis: a distance is measured Conclusion: it is positive If a distance is measured, then it is positive. Lesson 3 Ex2

Write a Conditional in If-Then Form B. Identify the hypothesis and conclusion of the following statement. Then write the statement in the if-then form. A five-sided polygon is a pentagon. Answer: Hypothesis: a polygon has five sides Conclusion: it is a pentagon If a polygon has five sides, then it is a pentagon. Lesson 3 Ex2

A. If an octagon has 8 sides, then it is a polygon. A. Which of the following is the correct if-then form of the given statement? A polygon with 8 sides is an octagon. A. If an octagon has 8 sides, then it is a polygon. B. If a polygon has 8 sides, then it is an octagon. C. If a polygon is an octagon, then it has 8 sides. D. none of the above A B C D Lesson 3 CYP2

A. If an angle is acute, then it measures less than 90°. B. Which of the following is the correct if-then form of the given statement? An angle that measures 45° is an acute angle. A. If an angle is acute, then it measures less than 90°. B. If an angle is not obtuse, then it is acute. C. If an angle measures 45°, then it is an acute angle. D. If an angle is acute, then it measures 45°. A B C D Lesson 3 CYP2

Truth Values of Conditionals A. Determine the truth value of the following statement for each set of conditions. If Yukon rests for 10 days, his ankle will heal. Yukon rests for 10 days, and he still has a hurt ankle. The hypothesis is true, but the conclusion is false. Answer: Since the result is not what was expected, the conditional statement is false. Lesson 3 Ex3

Truth Values of Conditionals B. Determine the truth value of the following statement for each set of conditions. If Yukon rests for 10 days, his ankle will heal. Yukon rests for 3 days, and he still has a hurt ankle. The hypothesis is false, and the conclusion is false. The statement does not say what happens if Yukon only rests for 3 days. His ankle could possibly still heal. Answer: In this case, we cannot say that the statement is false. Thus, the statement is true. Lesson 3 Ex3

Truth Values of Conditionals C. Determine the truth value of the following statement for each set of conditions. If Yukon rests for 10 days, his ankle will heal. Yukon rests for 10 days, and he does not have a hurt ankle anymore. The hypothesis is true since Yukon rested for 10 days, and the conclusion is true because he does not have a hurt ankle. Answer: Since what was stated is true, the conditional statement is true. Lesson 3 Ex3

Truth Values of Conditionals D. Determine the truth value of the following statement for each set of conditions. If Yukon rests for 10 days, his ankle will heal. Yukon rests for 7 days, and he does not have a hurt ankle anymore. The hypothesis is false, and the conclusion is true. The statement does not say what happens if Yukon only rests for 7 days. Answer: In this case, we cannot say that the statement is false. Thus, the statement is true. Lesson 3 Ex3

Conditional Truth Table p q

A. Determine the truth value of the following statement for each set of conditions. If it rains today, then Michael will not go skiing. It does not rain today; Michael does not go skiing. A. true B. false A. B. Lesson 3 CYP3

B. Determine the truth value of the following statement for each set of conditions. If it rains today, then Michael will not go skiing. It rains today; Michael does not go skiing. A. true B. false A. B. Lesson 3 CYP3

C. Determine the truth value of the following statement for each set of conditions. If it rains today, then Michael will not go skiing. It snows today; Michael does not go skiing. A. true B. false A. B. Lesson 3 CYP3

D. Determine the truth value of the following statement for each set of conditions. If it rains today, then Michael will not go skiing. It rains today; Michael goes skiing. A. true B. false A. B. Lesson 3 CYP3

Lesson 3 KC2

Lesson 3 KC2

Lesson 3 KC2

Lesson 3 KC2

First, write the conditional in if-then form. Related Conditionals Write the converse, inverse, and contrapositive of the statement All squares are rectangles. Determine whether each statement is true or false. If a statement is false, give a counterexample. First, write the conditional in if-then form. Conditional: If a shape is a square, then it is a rectangle. The conditional statement is true. Write the converse by switching the hypothesis and conclusion of the conditional. Converse: If a shape is a rectangle, then it is a square. The converse is false. A rectangle with and w = 4 is not a square. Lesson 3 Ex4

Related Conditionals Inverse: If a shape is not a square, then it is not a rectangle. The inverse is false. A 4-sided polygon with side lengths 2, 2, 4, and 4 is not a square. The contrapositive is formed by negating the hypothesis and conclusion of the converse. Contrapositive: If a shape is not a rectangle, then it is not a square. The contrapositive is true. Lesson 3 Ex4

A. All 4 statements are true. Write the converse, inverse, and contrapositive of the statement The sum of the measures of two complementary angles is 90. Which of the following correctly describes the truth values of the four statements? A. All 4 statements are true. B. Only the conditional and contrapositive are true. C. Only the converse and inverse are true. D. All 4 statements are false. A B C D Lesson 3 CYP4

IF the conditional is true, THEN the contrapositive is true!

End of Lesson 3