Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Fundamental theorem of algebra wikipedia , lookup
Linear algebra wikipedia , lookup
Complexification (Lie group) wikipedia , lookup
Heyting algebra wikipedia , lookup
Universal enveloping algebra wikipedia , lookup
Geometric algebra wikipedia , lookup
Exterior algebra wikipedia , lookup
Homological algebra wikipedia , lookup
2-4 Reasoning in Algebra Objectives: 1) To develop an awareness of the structure of a mathematical system, connecting postulates, logical reasoning, and theorems. 2) To connect reasoning in algebra and geometry. 2-4 Reasoning in Algebra Objectives: 1) To develop an awareness of the structure of a mathematical system, connecting postulates, logical reasoning, and theorems. 2) To connect reasoning in algebra and geometry. Properties of Equality c c c c c c c c a a c a ab ac Both properties can be named APE, short for Addition Prop. of Equality. Subtraction is actually adding the negative of the number. Both properties can be named MPE, short for Multiplication Prop. of Equality. Division is actually multiplying by the reciprocal of the number. 2-4 Reasoning in Algebra Objectives: 1) To develop an awareness of the structure of a mathematical system, connecting postulates, logical reasoning, and theorems. 2) To connect reasoning in algebra and geometry. Properties of Congruence AB A AB A EF C 2-4 Reasoning in Algebra Objectives: 1) To develop an awareness of the structure of a mathematical system, connecting postulates, logical reasoning, and theorems. 2) To connect reasoning in algebra and geometry. There are three types of properties in mathematics: assumed, POSTULATES defining, and deduced. DEFINITIONS THEOREMS Corollaries PROPERTIES Simplifying expressions and solving equations both represent a series of justified steps. A proof with a given and only one justified conclusion is often called a one-step proof. Common Types of Proofs: two-column proof flow proof paragraph proof Indirect proof 2-4 Reasoning in Algebra Objectives: 1) To develop an awareness of the structure of a mathematical system, connecting postulates, logical reasoning, and theorems. 2) To connect reasoning in algebra and geometry. Justify each step used to solve for 5x 12 = 32 + x . APE (Addition Property of Equality) APE MPE (Multiplication Property of Equality) Suppose that points A, B, and C are collinear with point B between points A and C. Solve for x if AC = 21, AB =15 x , and BC = 4 + 2x . Justify each step. 15 x A 4 + 2x B Segment Addition Postulate C 15 x 4 + 2x 19 21 Substitution Property 21 2 APE 2-4 Reasoning in Algebra Objectives: 1) To develop an awareness of the structure of a mathematical system, connecting postulates, logical reasoning, and theorems. 2) To connect reasoning in algebra and geometry. Given Angle Substitution Property APE MPE Recall that the length of segment AC is 21. 15 x A 4 + 2x B C Find the lengths of AB and BC by substituting x = 2 into the expressions in The diagram. Check that AB + BC = 21. AB = 13 BC = 8 AB + BC = 13 + 8 = 21 2-4 Reasoning in Algebra Objectives: 1) To develop an awareness of the structure of a mathematical system, connecting postulates, logical reasoning, and theorems. 2) To connect reasoning in algebra and geometry. Name the property that justifies each statement. a. If x = y and y + 4 = 3x, then x + 4 = 3x. The conclusion of the conditional statement is the same as the equation y + 4 = 3x (given) after x Substitution Property has been substituted for y (given).The property used is the ____________________. b. If x + 4 = 3x, then 4 = 2x. The conclusion of the conditional statement shows the result after x is subtracted APE from each side of the equation in the hypothesis. The property used is the ____________________. 2-4 Reasoning in Algebra Objectives: 1) To develop an awareness of the structure of a mathematical system, connecting postulates, logical reasoning, and theorems. 2) To connect reasoning in algebra and geometry. c. If P Q, Q R, and R S, then P S. Transitive Property of Congruence P R Transitive Property of Congruence P S. Transitive Property of Congruence 2-4 Reasoning in Algebra Objectives: 1) To develop an awareness of the structure of a mathematical system, connecting postulates, logical reasoning, and theorems. 2) To connect reasoning in algebra and geometry. Name the property the property of equality or congruence illustrated. Reflexive Property of Congruence Transitive Property of Congruence 2-4 Reasoning in Algebra *Connect reasoning in algebra and geometry. Angle Addition Postulate Substitution Property Simplify APE MPE Segment Addition Postulate Substitution Property Distributive Property Simplify APE MPE textbook page 105 2-4 Reasoning in Algebra *Connect reasoning in algebra and geometry. textbook page 106 MPE Distributive Property Distributive Property APE APE MPE Distributive Prop. Reflexive Prop. of = Symmetric Prop. of ≅ MPE MPE Reflexive Prop. of Equality ( or = ) Symmetric Prop. of = APE Substitution Prop. APE Transitive Property of Congruence ( or ≅ ) 2-4 Reasoning in Algebra *Connect reasoning in algebra and geometry. 15 YU=AB PQR 5x K 3 WYT EF + 7 2-4 Reasoning in Algebra *Connect reasoning in algebra and geometry. 2-4 Reasoning in Algebra *Connect reasoning in algebra and geometry. 2-4 Reasoning in Algebra *Connect reasoning in algebra and geometry. 2-4 Reasoning in Algebra *Connect reasoning in algebra and geometry. End of Notes and Exercises for Lesson 2-4. You may proceed to work on Practice 2-4 Worksheet, DNG page 281. Please make an effort to print copies of the additional exercises for each lesson as they will be good reference when you make your homework. You use the library services to print these pages. I saved them all in PDF file and You may choose to print 2-3 pages for each page of printing paper. Mr. Chapa, The librarian or Ms. A. Rivera can help you on this matter. Once again thank you for all your cooperation and work.