8th Grade Common Core Math Booklet 1 The Number System
Main Idea of the Number System: Know that there are numbers that are not rational, and approximate them by rational numbers. What this means: There are two types of numbers, rational and irrational. You can use rational numbers to find the general value of the irrational numbers. Rational Numbers vs. Irrational Numbers Rational numbers are numbers that can be made by dividing two integers (a whole number): (b cannot be 0). Examples of rational numbers: 5 = 8 = or " 1.8 = or " " A decimal number that ends (terminates) is rational. A decimal that repeats forever is rational as long as it repeats in a pattern. A repeating pattern in math is shown by a bar over the numbers that repeat. Examples: 1.375 = 0.0625 = " 0. 6 = Irrational numbers are numbers that cannot be made by dividing two integers. They are numbers that aren t rational. Examples: π (can t be written as a fraction, goes on forever without repeating) 5 2 (can t be written as a fraction, goes on forever without repeating)
8 th Grade Common Core Math Standards: Standard 8.NS.A.1: Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number. What the student learns: Students learn that numbers that aren t rational are irrational. Every number has a decimal form. If it is rational the decimal stops or repeats in a pattern. Irrational numbers have nonterminating decimals with no pattern. Standard examples: Find the decimal expansion of the following rational numbers: 17-8 Answers: 17 = 17.0 = 0.33-8 = -8.0 = 0.428571 Find the decimal expansions of the following irrational numbers (to 5 decimal digits): π 8 24 Answers: π = 3.14159 (irrational number, continues forever with no pattern) 8 = 2.82842 (continues to repeat with no pattern) 24 = 4.89897 (continues to repeat with no pattern) Now we will convert 0.555555 into a fraction (Fractions are rational and a repeating decimal is as well.) Answer: A rational number requires the numerator to be an integer. There cannot be a repeating decimal in the numerator, so we cannot convert 0.555555 into a fraction this way:.
We need the numerator to not have a repeating decimal, so we need to remove it from the fraction we are trying to make. In order to do that, we need to assign a variable to 0.555555. Let s call it x. x = 0.555555 We need to work with a numerator larger than 1, so we are going to multiply x by 10. 10 * x = 10x and 10 * 0.555555 = 5.555555 Now, we can remove the.555555 from the expression by subtracting x from 10x. 10x - x = 5.555555 - x Remember that x = 0.555555 10x - x = 5. 555555-0. 555555 9x = 5 We want x to be by itself so we divide both sides by 9. When we do that we see that x (which is 0. 555555) is equivalent to So 0. 555555 in fraction form is
Standard 8.NS.A.2: 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., π2). 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 the student learns: Students learn how to approximate the value of an irrational number using rational numbers and can locate the value of an irrational number on a number line between two rational numbers. Standard example: The 10 is approximately what number? Round the answer to the nearest hundredth. To find the answer we are going to use a number that isn t exact but is very close to the exact value. Answer: We know that the 10 will not be a whole number because the 9 = 3 and the 16 = 4. Since 10 is a value between the 9 and the 16, we know the approximate value of the 10 is between 3 and 4. The number 10 is closer to 9 than 16 so the 10 will be closer to 3 than it is to 4. 2 If we start with a guess that 10 is 3.25, we see that 3.25 (3.25 * 3.25) is 10.56, so the 10 is lower than 3.25. 2 If we guess 3.1, we do 3.1 (3.1 * 3.1) and get 9.61, so the 10 is larger than 3.1. If we guess 3.15 we do 3.15 2 (3.15 * 3.15) which = 9.92 so, the 10 it is slighter larger than 3.15. 3.16 is our final guess because 3.16 2 (3.16 * 3.16) is 9.9856 which is approximately 9.99 which is about 10. So we know that the 10 is approximately 3.16. On a number line, 10 is approximately between 3.15 and 3.17 3.15 3.16 3.17 3 10 4
WHY THIS IS IMPORTANT Irrational numbers are important because they are used in everyday life. For example, if you are making a coffee mug and you want the circumference of the mug to be a certain size, you will have to deal with Pi, which is an irrational number. If you want to see how far your car will travel after 12 wheel rotations, you would need to find the circumference of the wheel and multiply it by 12 to figure out the distance traveled. Any time you find the circumference of an object, you would use Pi. Knowing how to approximate numbers is important because not all numbers are exact in every day life. If you have ever wanted to install a light fixture you may be working with a circular mounting base and a square cutout in your ceiling. You need to approximate the size of the square cutout so that the fixture will fit into the opening and the circular mounting base also covers the hole. If the diameter of the light fixture is 6", a square cutout in your ceiling that is 18" (or about 4.24") on each side will just touch the edges of the circle, so a square cutout that is slightly smaller than 4.24 on each side will be completely covered by the circular base of the light fixture.