8.NS.2 - Approximating Irrational Numbers - Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., π 2 ). For example, by truncating the decimal expansion of √2, show that √2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations.
What is a square?A square is you take a number and multiply it by itself. A visual representation would be that you are making a square that is comprised of many smaller squares.
As you can see on the left side of the top image, when you square a value, you have a base with an exponent of 2. This means that you take the base and make a square with sides that long.
Looking again at the top image, you can see on the bottom we have 5 with an exponent of 2. This means we make a square that is 5 x 5. When you multiply 5x5 you get 25, which means that you need to have 25 "little squares." This means that 5 squared is equal to 25.
A square root is the "inverse operation" of squaring. In other words, it "undoes" a square. As you can see on the right side of the top image, you put a value "under" the √ symbol in order to take the square root.
Let's look at the bottom square root. You have √25. This means that I need to take 25 "little blocks" and try to make them into a perfect square. We can do this by making a square that has size that are 5 each. This is because we can divide 25 into 5 equal rows that are 5 long. Therefore, this tells us the square root of 25 is 5.
Because 25 broke up into two sides of the exact same length, 5, this makes is a perfect square, and it is a rational number. But if you cannot divide the number up into two numbers exactly the same, then you have an irrational number. Read more below
It's not too hard to come up with a "ballpark estimate" of the value on a non-perfect square. Since it will be an irrational number, you can never get the exact answer, but we can come up with a reasonable guess. Below are two example on how to estimate the value of a non-perfect square.