Math 3: Unit 1 – Reasoning and Proof
Inductive, Deductive
Quick Check on Lesson 1 Name:_______________________
Objective: To use inductive reasoning to make conjectures. To use deductive reasoning to prove conjectures.
1. If n is a positive integer, then (𝑛 3
Case 1: n is even
− 𝑛) is even.
Case 2: n is odd
2. If n is odd, then n 2 is odd.
3. If n is an element of the integers and n is even, then 3𝑛 + 5 is odd.

4. The difference of the squares of any two consecutive integers is odd. Write in the If-Then form and then either prove or disprove.
5. The sum of any three consecutive integers is divisible by three. Write in the If-Then form and then either prove or disprove.
6. Proved by a deductive proof. If n is even, then 7n + 4 is even.

7. Shane made the following assertion: All numbers that are divisible by 4 are even numbers. a. Write Shane’s assertion in if-then form. b. Check Shane’s assertion by using inductive proof. c. Give a deductive proof of Shane’s assertion.
8. Give a deductive proof. If a and b are consecutive integers, then (𝑎 + 𝑏) 2
is an odd number.
9. True or false. If true, provide a deductive proof. If false, provide a counter example.
If m and n are consecutive integers, then 4 divides (𝑚 2 + 𝑛 2 ) .

 x
 y

2
11. x
 y if and only if xy

4 a. Write the statement as 2 if-then statements. b. Give a deductive proof for one of the if-then statements.
12. Suppose it is true that “all members of the senior class are at least 5 feet 2 inches tall.” What, if anything, can you conclude with certainty about each of the following students? a. Darlene, who is a member of the senior class. b. Trevor, who is 5 feet 10 inches tall c. Anessa, who is 5 feet tall d. Ashley, who is not a member of the senior class
13. In 1742, number theorist Christian Goldbach (1690 – 1764) wrote a letter to mathematician
Leonard Euler in which he proposed a conjecture that people are still trying to prove or disporve. Goldbach’s Conjecture states:
Evey even number greater than or equal to 4 can be expressed as the sum of 2 prime numbers. a. Verify Goldbach’s conjecture is true for 12 and 28. b. Write Goldbach’s conjecture in if-then form.

c. Write the converse of Goldbach’s Conjecture. Is the converse true? Give a counterexample or a deductive proof.
14. If right triangle XYZ with side lengths x and y, and hypotenuse z, has area z
2
4 isosceles. Is the converse true? If so, give a deductive proof. (8 – 9 steps) then

XYZ is
15. Determine whether the following are valid or invalid. Justify your reasoning. a. If someone buys a new Lamborghini, she or he will pay over $200,000. Marie does not buy a new Lamborghini. Therefore, Marie does not pay over $200,000 for her new car. b. In china, job applicants do not ask how much they will be paid when they are hired. When
Jin Tai was hired, he asked his employer how much he would be paid. Jin Tai must have been hired outside of China.