1. Question 38

2. Question 39
A student made this conjecture and found two examples to support the conjecture: If a rational number is not an integer, then the square root of the rational number is irrational. For example, is irrational and is irrational.
Provide two examples of non-integer rational numbers that show that the conjecture is false.
For this problem, the students are being tested on if they understand rational and irrational numbers, as well as integers and non-integers. To define some words:
Integer: All positive and negative whole numbers
Non-Integer: All positive and negative numbers (that are whole numbers)
Rational Number: All positive and negative numbers, including fractions, numbers, etc; they have decimals that stop or repeat
Irrational Number: Real number that aren’t rational; they have decimals that go on forever that never repeat or stop.
It just so happened that the two examples given were irrational but there are numbers that will be rational when you square root them. The answers will vary so you can ask the students to help out. But, anything that is a square number that happens to be in decimal for will work. For example, 2.25 works because when it is square rooted, you get 1.5 as your answer. Another example would be ¼, because the square root of that is ½.
The main thing the students need to look for is using numbers that you know are square to begin with. I included other answers as well.
2.25; ¼; 5.29; 84.64; 1/9