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The Number System (NS) Know that there are numbers that are not rational, and approximate them by rational numbers. STANDARD: CC.8.NS.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. The intent of this standard is…Students understand that Real numbers are either rational or irrational. Learning Targets Notes for Teacher Students can use graphic organizers to show the relationship between the subsets of the real number system. I can convert any number type to a fraction of A/B. I can convert any number to decimal form. I can explain that every number has a decimal expansion. I can recognize decimal expansions as rational or irrational numbers. I can identify a decimal expansion that repeats or terminates as a rational number. I can convert a repeating decimal into rational number in A/B form. I can identify a decimal expansion that does not repeat or terminate as an irrational number. Some students are surprised that the decimal representation of π does not repeat. Some students believe that if only we keep looking at digits farther and farther to the right, eventually a pattern will emerge. Some students will think that a negative fraction is an integer. Vocabulary Rational Numbers Irrational numbers Terminating decimals Repeating decimals Madison County Schools Common Misconceptions/ Vocabulary not to use Samples Mathematical Practices 8.MP.2. Reason abstractly and quantitatively. 8.MP.4. Model with mathematics. 8.MP.7. Look for and make use of structure. 8.MP.8. Look for and express regularity in repeated reasoning. Spring 2014 Fractions in A/B form Bar notation Resources Graphing calculators Dynamic geometry software The Number System (NS) Know that there are numbers that are not rational, and approximate them by rational numbers. STANDARD: CC.8.NS.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., 2). For example, by truncating the decimal expansion of √2, show that √2 is between 1and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations. The intent of this standard is…Students locate rational and irrational numbers on the number line. Students compare and order rational and irrational numbers. Students also recognize that square roots may be negative and written as - 28 . Learning Targets I can find a number’s square and its square root. I can explain that squaring and square roots are inverse operations. I can categorize the square root of a number as irrational or rational. I can explain why an irrational number does not have a position on a number line diagram. I can compare the size of irrational numbers using rational numbers on a number line diagram. Notes for Teacher… Students understand that the value of a square root can be approximated between integers and that non-perfect square roots are irrational. Students can approximate square roots by iterative processes. Examples: Approximate the value of hundredth. 5 to the nearest Solution: Students start with a rough estimate based upon perfect squares. 5 falls between 2 and 3 because 5 falls between 22 = 4 and 32 = 9. The value will be closer to 2 than to 3. Students continue the A few irrational numbers are given special names (pi and e), and much attention is given to A sqrt(2). Because we name so few irrational numbers, students sometimes conclude that irrational numbers are unusual and rare. In fact, irrational numbers are much more plentiful than rational numbers, in the sense that they are “denser” in the real line. iterative process with the tenths place value. 5 falls between 2.2 and 2.3 because 5 falls between 2.22 = 4.84 and 2.32 = 5.29. The value is closer to 2.2. Further iteration shows that the value of Madison County Schools Common Misconceptions/ Vocabulary not to use Samples 5 is between 2.23 Spring 2014 and 2.24 since 2.232 is 4.9729 and 2.242 is 5.0176. Compare √2 and √3 by estimating their values, plotting them on a number line, and making comparative statements. Solution: Statements for the comparison could include: √2 is approximately 0.3 less than √3 √2 is between the whole numbers 1 and 2 √3 is between 1.7 and 1.8 Vocabulary Square root Truncating Estimation Approximation Π (Pi) as an irrational number Degree of accuracy Mathematical Practices 8.MP.2. Reason abstractly and quantitatively. 8.MP.4. Model with mathematics. 8.MP.7. Look for and make use of structure. 8.MP.8. Look for and express regularity in repeated reasoning. Resources Graphing calculators Dynamic geometry software Madison County Schools Spring 2014