3. Consecutive numbers are adjacent integers on a number line, such as 5 and 6. Nathan, Trina, Kasib,
and Ivana were trying to prove the following statement.
The sum of any two consecutive number is always an odd number
Study each of the arguments below.
Nathan: If the first number is even, then the second number must be odd. This combination will always
add up to an odd number.
Kasib: Two consecutive numbers are of the form n and n+1. n + (n+1) = 2n +1 which is, by definition,
the form of an odd number.
Trina: No matter what two consecutive numbers you take, their sum is always odd as shown below.
5+6=11
22+23=45
140+141=281
Ivana: For any two consecutive numbers, one will be even and the other will be odd. In Part c of the
Check Your Understanding task on page 9 I gave an argument justifying that the sum of an even number
and an odd number is always an odd number. So, the sum of any two consecutive numbers is always an
odd number.
a.
Which proof is the closest to the argument you would give to prove that the sum of any two
consecutive numbers is always an odd number?
b. Which arguments are not correct proofs of the statement? Explain your reasoning.
c. Give a “visual proof” of the statement using arrays of counters.
d. How would you prove or disprove the assertion, “The sum of three consecutive numbers is
always an odd number”?
9. Tonja made the following conjecture about consecutive whole numbers.
For any four consecutive whole numbers, the product of the middle two numbers is always two
more than the product of the first and last numbers.
a. Test Tonja’s conjecture for a set of four consecutive whole numbers.
b. Find a counterexample or give a deductive proof of Tonja’s conjecture.