Activities-Real numbers

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

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) .........