6th Grade Math Textbook, Fundamentals

Lesson 13-3 for exercise sets. &KDSWHU 

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Test the conjecture to decide whether it is true or false.

If true, explain why. If false, provide a counterexample.

1.The sum of two even numbers is always an even number.

2.Discuss and Write Explain a systematic approach you can use to test

the conjecture that every real number is an integer.

Conjecture:The sum of two odd numbers is always an odd number.

To test the conjecture, add the first two odd numbers: 1  1  2

Immediately, a counterexample was found, so your work is done.

The conjecture is false.

The sum of two odd numbers is notalways an odd number.

1

Conjecture:All prime numbers are odd.

To test the conjecture, list prime numbers: 2, 3, 5, 7, 11, 13, 15,...

The first prime number, 2, is notodd.

The conjecture is false.

Not all prime numbers are odd.

2

To prove a conjecture is true, you cannot simply try a large number of cases and

conclude that since no counterexamples could be found, the conjecture must be

true. Sometimes you need to make generalizations to exhaust all possibilities.

Conjecture:A multiple of 6 is always also a multiple of 3.

Start with a few random tests and extreme cases.

Think more generally: Any multiple of 6 can be written as 6x,

where xis any nonzero whole number.

Rewrite 6xas the equivalent expression (2 • 3)x.

So 6x(2 • 3)x

3(2x) Apply the Commutative Property of Multiplication.

3(2x) is a multiple of 3.

So any multiple of 6 is also a multiple of 3.

This proves the conjecture is true.

Choose a

multiple of 6.

Express the number

as a multiple of 6.

Can you express the

number as a multiple of 3?

Can you express the number

as a multiple of 6 and 3?

Try 36. 6 • 6 3 • 12 Ye s

Try 60. 6 • 10 3 • 20 Ye s

Try 336. 6(56) 3(112) Ye s