1.4 Proving Conjectures: Deductive Reasoning
Applied Mathematics
Practice:
Page 31-33 Questions 4, 5,6, 7, 12, 13, 17
4. Prove that the sum of two even integers is always even.
2n (2 times any variable) is used to represent an even number.
Let 2n and 2m represent any two even numbers.
2n + 2m = 2(n + m)
Since 2 is a factor of the sum, the sum is even.
5. Prove that the product of an even integer and an odd integer is always even.
Let 2n +1 represent any odd number, and 2m represent any even number.
2m(2n + 1) = 4mn + 2m
or
2m(2n + 1) = 2(2mn + m)
In either case, 2 is a factor of the product, so the product is even.
6.
Prove that triangles a, b, and c are equal.
Use the Pythagorean theorem to show that the first and third triangles have a right angle opposite the
hypotenuse.
a. 42 + 32 = 16 + 9
c.
62 + 82 = 36 + 64
42 + 32 = 25
62 + 82 = 100
42 + 32 = 52
62 + 82 = 102
The theorem shows that angles a and c are 90o.
Triangle b shows that it has a right angle or 90o, and angle b is supplementary, meaning that
angle b is 180o – 90o = 90o.
Since angles a, b, and c are all 90o, they are equal to each other.
1.4 Proving Conjectures: Deductive Reasoning
Applied Mathematics
7. a. Show inductively, using three examples, that the result is always 3 more than the chosen number.
Choose any number
4
0
-5
X4
16
0
-20
+10
26
10
-10
/2
13
5
-5
-5
8
0
-10
/2
4
0
-5
+3
7
3
-2
b. Prove deductively that the result is always 3 more than the chosen number.
Let n represent any whole number
X4
+10
/2
-5
/2
+3
The proof shows that the answer will be n+3 for any value of n.
n
4n
4n +10
2n + 5
2n
n
n+3
12. Choose a combination of addition, subtraction, multiplication and division steps to create a
5 step number trick (similar to questions 7 and 9 on page 32) that results in a final answer of 6.
Use a two column proof similar to the answer shown in 7b. The proof will show that the trick works for
any number.
13. Prove that any four-digit number is divisible by 2 when the last digit in the
number is divisible by 2.
Let abcd represent any four digit number.
abcd = 1000a + 100b + 10c + d
abcd = 2(500a + 50b + 5c) + d
The number abcd is divisible by two only when d is divisible by 2.
1.4 Proving Conjectures: Deductive Reasoning
Applied Mathematics
17. Which student has the strongest proof?
Joan and Garnet used inductive reasoning to provide more evidence for the conjecture, but their
solutions aren’t proofs since they do not use algebra. They simply add more evidence to support the
conjecture.
Jamie uses deductive reasoning by using algebra to prove the answer for all sets of three numbers.