# Problem 1

1. For every integer prove that x(x+1) is an even integer (Problem related to mathematical proofs in Chapter 1)

Approaches to solutions

## The concepts that a pupil must know are

1. What is an integer?
2. What is an even integer?
3. What is an odd integer?
4. X and ( x+1) are consecutive integers and x(x+1) is the representation of the product
5. Pupil should have the concept of distributive property of integers
6. The pupil must have an opportunity for an arguement that the proof is true even of negative integers
7. Pupil must have a sound understanding of Euclid's lemma
8. The difference between mathematical proofs and Verification/Justification-the scope of mathematical proofs is beyond verification-Higher order skill in problem solving

## Method -1: Solution by cases

Proposition: x is an integer. i.e., x=m, where m is an integer
Conclusion : x(x+1) is an even integer. i.e., x(x+1) =2K

Pupil can solve this in several ways viz., proofs by cases.

### Case -1: x is an even integer

Then x=2k+0 = 2K, $K in Z$ by Euclid's lemma

Discussion on Euclid's lemma: What is division? Discuss on the process of division rather than on the procedure of division. Consider the example . Is it (12-5)=7 then (7-5)=2
Discussion on Euclid's lemma: What is division? Discuss on the process of division rather than on the procedure of division. Consider the example . Is it (12-5)=7 then (7-5)=2 Abstract thinking:
b) a (q
-bq
.........
(a - bq)
.........