+ × a c Table 4. The properties of equality. Here and stand for arbitrary numbers in the a, b c rational, real, or complex number systems. = Reflexive property of equality a a If then Symmetric property of equality a = b, b = a If and then Transitive property of equality a = b b = c, a = c If then Addition property of equality a = b, a + c = b + c If then = Subtraction property of equality a = b, a – c b – c If then × = × Multiplication property of equality a = b, a c b c If and 0, then ÷ = ÷ Division property of equality a = b c a c b c ≠ If = then may be substituted for Substitution property of equality a b, b a in any expression containing a. Table 5. The properties of inequality. Here and stand for arbitrary numbers in the a, b c rational or real number systems. Exactly one of the following is true: < = > a b, a b, a b. If > and > then > a b b c a c. If > then < a b, b a. If > then < a b, –a –b. If > then ± > ± a b, a c b c. If > and > 0, then × > × a b c a c b c. If > and < 0, then × < × a b c a c b c. If > and > 0, then ÷ > ÷ a b c a c b c. If > and < 0, then ÷ < ÷ a b c a c b c. Sample of Consulted Existing state standards documents. Research summaries and briefs provided to the Working by researchers. National Assessment Governing Board, Mathematics Framework for the 2009 National Assessment of Educational Progress. Department of Education, 2008. NAEP Validity Studies Panel, Validity Study of the NAEP Mathematics Assessment: Grades 4 and 8. Daro et al., 2007. Mathematics documents from: Alberta, Canada; Belgium; China; Chinese Taipei; Denmark; England; Finland; Hong Kong; India; Ireland; Japan; Korea, New Zealand, Singapore; Victoria (British Columbia). Adding it Up: Helping Children Mathematics. National Research Council, Mathematics Learning Study Committee, 2001. Benchmarking for Success: Ensuring Students Receive a World-Class Education. National Governors Association, Council of Chief State School Officers, and Achieve, Inc., 2008.
+ a × c Table 4. The properties of equality. Here a, b and c stand for arbitrary numbers in the rational, real, or complex number systems. Reflexive property of equality a = a Symmetric property of equality If a = b, then b = a Transitive property of equality If a = b and b = c, then a = c Addition property of equality If a = b, then a + c = b + c Subtraction property of equality If a = b, then a – c = b – c Multiplication property of equality If a = b, then a × c = b × c Division property of equality If a = b and c ≠ 0, then a ÷ c = b ÷ c Substitution property of equality If a = b, then b may be substituted for a in any expression containing a. Table 5. The properties of inequality. Here a, b and c stand for arbitrary numbers in the rational or real number systems. Exactly one of the following is true: a < b, a = b, a > b. If a > b and b > c then a > c. If a > b, then b < a. If a > b, then –a < –b. If a > b, then a ± c > b ± c. If a > b and c > 0, then a × c > b × c. If a > b and c < 0, then a × c < b × c. If a > b and c > 0, then a ÷ c > b ÷ c. If a > b and c < 0, then a ÷ c < b ÷ c. Sample of Consulted Existing state standards documents. Research summaries and briefs provided to the Working by researchers. National Assessment Governing Board, Mathematics Framework for the 2009 National Assessment of Educational Progress. Department of Education, 2008. NAEP Validity Studies Panel, Validity Study of the NAEP Mathematics Assessment: Grades 4 and 8. Daro et al., 2007. Mathematics documents from: Alberta, Canada; Belgium; China; Chinese Taipei; Denmark; England; Finland; Hong Kong; India; Ireland; Japan; Korea, New Zealand, Singapore; Victoria (British Columbia). Adding it Up: Helping Children Mathematics. National Research Council, Mathematics Learning Study Committee, 2001. Benchmarking for Success: Ensuring Students Receive a World-Class Education. National Governors Association, Council of Chief State School Officers, and Achieve, Inc., 2008.