a Follow the steps of the algorithm to find the corresponding answer to the chosen integer.
B
b Repeat the steps from Part A for integer x.
C
c Review the definition of inductive and deductive reasoning.
A
a The result is two more than the chosen integer.
B
b x+2
C
c The expression in Part B is equivalent to the conjecture in Part A. In Part A inductive reasoning was used to make a conjecture based on pattern. In Part B deductive reasoning was used in order to write and simplify an expression.
Practice makes perfect
a Let's choose four different integers and complete the algorithm for them first. Then we will look for a pattern, and make a conjecture that relates the chosen integers to the answers.
Completing the algorithm
To complete the algorithms, we have to choose four different integers. Let's take 1, 2, 3, and 4.
Integer
Multiply by 3
Add 6
Divide by 3
Answer
1
1 * 3=3
3 +6 = 9
9 Ă· 3 =3
3
2
2 * 3 =6
6 +6 = 12
12 Ă· 3=4
4
3
3 * 3 =9
9 +6 = 15
15 Ă· 3=5
5
4
4 * 3 =12
12 +6 = 18
18 Ă· 3=6
6
Conjecture
Analyzing the chosen integers and the corresponding answers, we can tell that to get each answer, we have to add2 to the chosen integer.
1, 2, 3, 4
↓ +2 ↓ +2 ↓ +2 ↓ +2
3, 4, 5, 6
Therefore, our conjecture is that after choosing an integer, we can add 2 to it to get the same answer as if we applied the algorithm for the integer.
b Let's apply the algorithm to x. We will do it step by step.
c In Part A our conjecture was that we add 2 to the chosen integer. In Part B, however, we took x as the integer and resulted in x+2, which means that our conjecture was correct. Now we can describe how inductive and deductive reasoning are exhibited in Part A and Part B.
Inductive reasoning
Inductive reasoning is reasoning based on patterns we observe. In Part A, we observed how the chosen integers change when applying the algorithm. We concluded that since in each case the answer is the integer plus 2, then this is the pattern. Thus, in Part A we used inductive reasoning.
Deductive reasoning
Deductive reasoning is the process of reasoning logically from given statements, or facts to a conclusion. In Part B we have a direct proof of our conjecture. We show that if x is the chosen integer, then x+2 is the corresponding answer. Therefore, in Part B we used deductive reasoning.
