section 1.4 proving conjectures: deductive reasoning pages/teacher pages/margot crewe...41 september...

1.4ProvingConjectures(DeductiveReasoning).notebook

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Section 1.4Proving Conjectures: Deductive Reasoning


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Definition:

Proof: A mathematical argument showing that a statement is valid in all cases, or that no counterexample exists.

Generalization: A principle, statement or idea that has general application.


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Example 1: (Text page 27)Jon discovered a pattern when adding consecutive integers:a) 1 + 2 + 3 + 4 + 5 =15b) (-15) + (-14) + (-13) + (-12) + (-11) = -65c) (-3) + (-2) + (-1) + 0 + 1 = -5

He claims that whenever you add five consecutive integers, the sum is always 5 times the median of the numbers.Prove that Jon‛s Conjecture is true for all integers.


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Definition:

Median: the "middle number" (in a sorted list of numbers).

To find the Median, place the numbers you are given in value order and find the middle number.

Consecutive Numbers: Numbers which follow each other in order, without gaps, from smallest to largest.

12, 13, 14 and 15 are consecutive numbers.


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Check Jon‛s examples:

a) 1 + 2 + 3 + 4 + 5 =15Median # = 3 x 5 = 15

b) (-15) + (-14) + (-13) + (-12) + (-11) = -65Median # = -13 x 5 = -65

c) (-3) + (-2) + (-1) + 0 + 1 = -5Median # = -1 x 5 = -5


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Try a sample with larger numbers:

1233 + 1234 + 1235 + 1236 + 1237 = 6185

1235 x 5 = 6175


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How can you prove that Jon's conjecture is true for all integers?

To prove a conjecture is true for all cases, we use deductive reasoning.

Definition:Deductive Reasoning: Drawing a specific conclusion through logical reasoning by starting with general assumptions that are know to be valid.


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Prove using generalizations:Let x = the first number

x + (x+1) + (x+2) + (x+3) + (x+4)

= 5x + 10

= 5 (x+2)Since the sum equals five times the median number, Jon‛s conjecture is true.


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If you can prove Jon‛s conjecture using generalizations, then you can prove it is true for all integers.

Reflect: We used deductive reasoning to prove Jon‛s conjecture. Jon used inductive reasoning to develop his conjecture.


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We will use three different strategies to prove conjectures.

1. Venn Diagrams (a visual representation)

2. Number Theory Proofs (choosing a variable to algebraically represent a situation).

3. Two-Column Proofs (using statements and reasons in an organized list).


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Venn Diagrams

- An illustration that uses overlapping or non-overlapping circles to show the relationship between groups of things.


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Example:All zips are zaps. All zaps are zops. Shaggy is a zip.

What can be deduced about Shaggy?

Since Shaggy is a Zip, we also know he is a zap and a zop.

Draw a Ven diagram.


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You can think of this example in different terms and it might help you to understand:• All dogs are mammals. All mammals are vertebrates. Shaggy is a dog. What can be deduced about Shaggy?

Shaggy is a dog, a mammal and a vertebrate.


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Hypothesis:Casey voted in the last election.Only people over 18 years old vote.Conclusion:Casey is over 18 years old.

Example:


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Example:Mammals have fur (or hair). Lions are classified as mammals.

What can be deduced about lions?

Illustrate your conjecture with a Venn diagram.


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Number Theory

This strategy involves choosing a variable or variables to algebraically represent a situation.

Example:Two consecutive integers: x and x + 1Even integer: 2mOdd integer: 2n + 1


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Even Integers can be written in the form: 2m, where m is an integer.(Even numbers have a factor of two.)

Odd Integers can be written in the form: 2n + 1, where n is an integer. (One more than an even number will always be an odd.)

Consecutive Integers can be written in the form: n, n + 1, n + 2, etc, where n is an integer. (By adding one more to the previous number you will get the next consecutive integer.)

Consecutive Odd Integers can be written in the form: 2n + 1, 2n + 3, 2n + 5, etc, where n is an integer. (By adding two more to the previous number you will get the next consecutive odd integer.)

Consecutive Even Integers can be written in the form: 2n, 2n + 2, 2n + 4, etc, where n is an integer. (By adding two more to the previous number you will get the next consecutive even integer.)


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Multiply:


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Write an expression for the following:

1) an even number


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2) an odd number


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3) the sum of two consecutive even integers


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4) the sum of two positive consecutive odd integers



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5) the sum of 5 consecutive natural numbers



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6) the sum of three consecutive even integersWrite an expression for the following:


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7) the sum of the squares of two consecutive integers



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8) The sum of the squares of two consecutive even integers (2 versions)


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9) The sum of 7 consecutive integers


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10) The product of two consecutive integers.


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11)The difference of squares of two consecutive integers.


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12) a two digit


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13) a three digit number


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14) The sum of a two digit and a three digit number



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15) The sum of a two digit number and the number formed by reversing its digits.


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Write an expression for the following:16) The sum of a three digit number and

the number formed by reversing its digits.


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17) The product of two consecutive integers .


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18) The product of an even and an odd integer.


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19) The square of an odd integer.


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Inductive Reasoning and

Deductive Proofs


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What happens when you multiply two even integers?

Conjecture: If you multiply two even integers then the product will be even.


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Now use deductive reasoning to prove your conjecture:Three Steps:1. Define the variables.2. Use algebra to prove the conjecture.3. State what you have proven.


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Conjecture: If you multiply two even integers then the product will be even.


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Let 2m = one even integerLet 2n = a second even integer

The product = 2m X 2n= 4mn= 2(2mn)

By showing the product has a factor of 2 you are proving that it is even.

Therefore, if you multiply two even integers then the product will be even.


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What happens when you multiply two odd integers?

Conjecture: If you multiply two odd integers then the product will be odd.


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Conjecture: If you multiply two odd integers then the product will be odd.


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Let 2m + 1 = one odd integerLet 2n + 1 = a second odd integer

The product = (2m + 1) X (2n + 1)= 4mn + 2n + 2m + 1= 2(2mn + n + m) + 1

By showing the product is 2 times an integer plus 1 you are proving that it is odd.Therefore, if you multiply two odd integers then the product will be odd.


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Conjecture: The sum of four consecutive integers is equivalent to the first and last integers added, then multiplied by two.

a) defend the conjecture inductively by showing 2 examples.

b) prove the conjecture deductively.


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Conjecture: The sum of four consecutive integers is equivalent to the first and last integers added, then multiplied by two.


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Support the following inductively(showing three examples) and then prove it deductively:

The square of the sum of two positive integers is greater than the sum of the squares of the same two integers.


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The sum of a 2-digit number and the number formed by reversing its digits will always be divisible by 11.



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The difference between an odd integer and an even integer is odd.



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An odd number times an even number is even.

Prove the following inductively(showing at least three examples) and then prove it deductively:

Let 2n represent an even numberLet 2m + 1 be an odd number2n(2m + 1) = 4nm + 2n = 2(2nm + n)Since 2(2nm + n) is divisible by 2 , it is even.

Inductive

Deductive


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Consider the following problem: Think of any number. Multiply that number by 2, then add 6, and divide the result by 2. Next subtract the original number. What is the result?(a) Use inductive reasoning to make a

conjecture for the answer.(b) Use deductive reasoning to prove your

conjecture.


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Inductive

So we might form a conjecture that the result will always be the number 3. But this doesn‛t prove the conjecture, as we‛ve tried only two of infinitely many possibilities.


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Deductive

Let x = a number


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Select a number. Add 50 to the number. Multiply the sum by 2. Subtract the original number from the product.What is the result?

(a) Use inductive reasoning to arrive at a general conclusion.(b) Use deductive reasoning to prove your

conclusion is true.


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Inductive


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Deductive

Let x = a number


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Two-Column Proof

Two column proofs will be studied in depth in chapter 2. The following will be an introductory example only.


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Example:Prove that when two straight lines intersect, the vertically opposite angles are equal.


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1.4 Assignment: Nelson Foundations of Mathematics 11, Sec 1.4, pg. 31‐33Questions: 1‐11, 13, 14, 16,17, 19, 20

Mid‐Chapter Review: page 35#'s 2, 5, 8‐11

section 1.4 proving conjectures: deductive reasoning pages/teacher pages/margot crewe...41 september...

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