By James Dean Brown
This article presents a realistic, finished assessment of different levels and actions concerned with constructing and enforcing a valid, rational, and powerful language application. It systematically describes and exemplifies all of the parts of language curriculum layout. actions and workouts, picture organizers, and pattern language courses illustrate and advertise pedagogically sound perform and potent integration of fabric.
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Extra resources for Elements of Language Curriculum: A Systematic Approach to Program Development
An integer a is prime if a 2 and its only divisors are ˙1 and ˙a; if a 2 has other divisors, then it is called composite. The first few primes are 2, 3, 5, 7, 11, 13; : : : : We will soon see that there are infinitely many primes. The reason we do not consider 1 to be a prime is that theorems about primes would then require special cases treating the behavior of 1. For example, we will prove later that every positive integer a 2 has exactly one factorization of the form a D p1 p2 p t , where p1 Ä p2 Ä Ä p t are primes.
We first prove another lemma from Greek times. 27. Let a and b be integers. r; a/. b a; a/. Proof. 20, it suffices to show that an integer c is a common divisor of a and b if and only if it is a common divisor of a and r . 46 on page 29. b a/. 27 is not necessary; it is there only to put you in the mood to accept the next example showing a method the Greeks probably used to compute gcd’s. 29. 28. b; a/. b; a/ is simple when a and b are small. b qa; a/ as long as b qa > 0. Since the natural numbers b a; b 2a; : : : ; b qa are strictly decreasing, the Least Integer Axiom (or Infinite Descent) says that they must reach a smallest such integer: r D b qa; that is, 0 < r < a.
Answer: True. Answer: True. (ii) 2 j 6. (iv) 0 j 6. Answer: True. Answer: False. 38 True or false, with reasons. n; n C 1/ D 1 for every natural number n. n; n C 2/ D 2 for every natural number n. (iii) 113 is a sum of distinct powers of 2. Answer: True. Answer: False. Answer: True. a; b/ D sa C t b. Answer: False. (v) If an integer m is a divisor of a product of integers ab, then m is a divisor of either a or b (or both). Answer: False. 39 Prove, or disprove and salvage if possible. a; b/ D gcd.
Elements of Language Curriculum: A Systematic Approach to Program Development by James Dean Brown