Standard: MACC.8.NS.1.1 Depth of Knowledge Know that numbers ...

Standard: MACC.8.NS.1.1Depth of KnowledgeKnow that numbers that are not rational are called irrational.Level 1: RecallUnderstand informally that every number has a decimal expansion; forrational numbers show that the decimal expansion repeats eventually,and convert a decimal expansion which repeats eventually into arational number.Explanations and Ideas to Support:Sample Test/Task Item(s):A rational number is of the form a/b, where aand b are both integers, and b is not 0. In theelementary grades, students learned processesthat can be used to locate any rational numberWhat is the rational number in the form a/b forthe decimal expansion 0.83?on the number line: Divide the interval from 0to 1 into b equal parts; then, beginning at 0,count out those parts. The surprising fact, now,is that there are numbers on the number linethat cannot be expressed as a/b, with a and bboth integers, and these are called irrational Suppose you have a fraction with a denominatornumbers. Students construct a right isosceles of 7. What is the longest string of non-repeatingtriangle with legs of 1 unit. Using thedigits that will occur in the decimal expansions ofPythagorean theorem, they determine that the the number?length of the hypotenuse is √2. In the figureright, they can rotate the hypotenuse back to (Hint: Use the long division algorithm to showthe original number line to show that indeed √2 that for a denominator of n, there are only nis a number on the number line.possible remainders, 0 to n-1.)In the elementary grades, students becomefamiliar with decimal fractions, most oftenwith decimal representations that terminate afew digits to the right of the decimal point. Forexample, to find the exact decimalrepresentation of 2/7, students might use theirGroup the following numbers based on what youcalculator to find 2/7 = 0.2857142857…, andknow about the number system:they might guess that the digits 285714 repeat.To show that the digits do repeat, students inGrade 7 actually carry out the long division and5.3 ̅ √10 2 π 4.010010001…recognize that the remainders repeat in apredictable pattern—a pattern that creates therepetition in the decimal representation (see7.NS.1.2.d).Once students understand that (1) everyrational number has a decimal representationthat either terminates or repeats, and (2) everyterminating or repeating decimal is a rationalnumber, they can reason that on the numberline, irrational numbers (i.e., those that arenot rational) must have decimalrepresentations that neither terminate norrepeat. Given two distinct numbers, it ispossible to find both a rational and anirrational number between them.Convert ̅ to a fraction.

Ideas to support MACC.8.NS.1.1: Use set diagrams to show the relationshipsamong real, rational, irrational numbers,integers, and counting numbers. Thediagram should show that the all realnumbers (numbers on the number line) areeither rational or irrational. The important point here is that students cansee that the pattern will repeat, so they canimagine the process continuing withoutactually carrying it out. Conversely, given a repeating decimal,students learn strategies for converting thedecimal to a fraction.Connections:SMPs to be Emphasized MP2- Reason abstractly and quantitatively. MP6- Attend to precision. MP7- Look for and make use of structure.FCAT 2.0 Connections:Related NGSSS Standard(s)MA.8.A.6.4- Perform operations on realnumbers (including integer exponents, radicals,percents, scientific notation, absolute value,rational numbers, and irrational numbers) usingmultistep and real world problems.Critical Area Working with irrational numbers, integerexponents, and scientific notation.FCAT 2.0 Test Item SpecificationBenchmark Clarification: Students will perform operations on real numbersusing multistep and real-world problems. The student understands real number expressionsand the law of exponents.Common Misconceptions: Some students are surprised that the decimal representation of pi does not repeat. Some studentsbelieve that if only we keep looking at digits farther and farther to the right, eventually a pattern willemerge. A few irrational numbers are given special names (pi and e), and much attention is given to sqrt(2).Because we name so few irrational numbers, students sometimes conclude that irrational numbers areunusual and rare. In fact, irrational numbers are much more plentiful than rational numbers, in thesense that they are “denser” in the real line.