8th Grade Common Core Math

8 GradeCommon CoreMaththBooklet 1The Number System

Main Idea of the Number System:Know that there are numbers that are not rational, andapproximate them by rational numbers.What this means: There are two types of numbers, rational andirrational. You can use rational numbers to find the general value of theirrational numbers.Rational Numbers vs. Irrational NumbersRational 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 thatrepeats forever is rational as long as it repeats in a pattern. Arepeating pattern in math is shown by a bar over the numbers thatrepeat.Examples: 1.375 !!!0.0625 !!"0. 6 !!Irrational numbers are numbers that cannot be made by dividing twointegers. They are numbers that aren’t rational.Examples: 𝜋  (can’t be written as a fraction, goes on forever without repeating)52 (can’t be written as a fraction, goes on forever without repeating)

8th Grade Common Core Math Standards:Standard 8.NS.A.1: Know that numbers that are not rational are calledirrational. Understand informally that every number has a decimalexpansion; for rational numbers show that the decimal expansionrepeats eventually, and convert a decimal expansion which repeatseventually into a rational number.What the student learns: Students learn that numbers that aren’trational are irrational. Every number has a decimal form. If it is rationalthe 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! 0.428571Find the decimal expansions of the following irrational numbers(to 5 decimal digits):𝜋824Answers: 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 arerational 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 cannotconvert 0.555555 into a fraction this way:!.!!!!!!!

We need the numerator to not have a repeating decimal, so we need toremove it from the fraction we are trying to make. In order to do that, weneed to assign a variable to 0.555555. Let’s call it x.x 0.555555We need to work with a numerator larger than 1, so we are going tomultiply x by 10.10 * x 10x and 10 * 0.555555     5.555555Now, we can remove the .555555 from the expression by subtracting xfrom 10x.10x - x 5.555555 - xRemember that x 0.55555510x - x 5.  555555 - 0.  5555559x 5We 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 numbersto compare the size of irrational numbers, locate them approximately ona number line diagram, and estimate the value of expressions (e.g.,π2). For example, by truncating the decimal expansion of 2, show that2 is between 1 and 2, then between 1.4 and 1.5, and explain how tocontinue on to get better approximations.What the student learns: Students learn how to approximate the valueof an irrational number using rational numbers and can locate the valueof an irrational number on a number line between two rational numbers.Standard example:The 𝟏𝟎 is approximately what number? Round the answer to thenearest hundredth.To find the answer we are going to use a number that isn’t exact but isvery close to the exact value.Answer: We know that the 10 will not be a whole number becausethe 9 3 and the 16   4. Since 10 is a value between the 9 andthe 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 thanit is to 4.If we start with a guess that 10 is 3.25, we see that 3.252 (3.25 * 3.25)is 10.56, so the 10 is lower than 3.25.If we guess 3.1, we do 3.12 (3.1 * 3.1) and get 9.61, so the 10 is largerthan 3.1.If we guess 3.15 we do 3.152 (3.15 * 3.15) which 9.92 so, the 10 it isslighter larger than 3.15.3.16 is our final guess because 3.162 (3.16 * 3.16) is 9.9856 which isapproximately 9.99 which is about 10. So we know that the 10 isapproximately 3.16.On a number line, 10 is approximately between 3.15 and 3.173.1533.16 103.174

WHY THIS IS IMPORTANTIrrational numbers are important because they are used in everydaylife. For example, if you are making a coffee mug and you want thecircumference of the mug to be a certain size, you will have to deal withPi, which is an irrational number. If you want to see how far your carwill travel after 12 wheel rotations, you would need to find thecircumference of the wheel and multiply it by 12 to figure out thedistance traveled. Any time you find the circumference of an object,you would use Pi.Knowing how to approximate numbers is important because not allnumbers are exact in every day life. If you have ever wanted to install alight fixture you may be working with a circular mounting base and asquare cutout in your ceiling. You need to approximate the size of thesquare cutout so that the fixture will fit into the opening and the circularmounting base also covers the hole. If the diameter of the light fixtureis 6", a square cutout in your ceiling that is 18" (or about 4.24") oneach side will just touch the edges of the circle, so a square cutout thatis slightly smaller than 4.24” on each side will be completely covered bythe circular base of the light fixture.

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 .