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Mastering Algebra and Geometry: Justifying Mathematical Steps

In this engaging math warm-up session, students will solve for 'x' using various algebraic properties, including addition, subtraction, multiplication, and division. They will also learn to link reasoning to both algebra and geometry, justifying their mathematical steps through definitions, theorems, and logical reasoning methods. By the end, students will apply their knowledge to solve problems involving supplementary angles and segment relationships, enhancing their understanding of geometric principles and algebraic operations. Homework will reinforce these concepts.

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Mastering Algebra and Geometry: Justifying Mathematical Steps

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  1. 2.4 & 2.5 Geometry Brett Solberg AHS ‘11-’12

  2. Warm-up • Solve for x • 2(x – 5) = -4 • Solve for x

  3. Today’s Objectives • Link reasoning to algebra and geometry. • Justify steps of math. • No Name Basket/Missing Assignments

  4. Mr. Fix-It • Who is Mr. Fix-It? • What tools does he need to do his job?

  5. Tools For Math • To reach conclusions we’ll use: • Inductive/Deductive Reasoning • Properties • Postulates • Theorems

  6. Addition Property • If a = b, then a + c = b + c. • Examples • If 2 = 2, then 2 + 4 = 2 + 4. • If x – 2 = 4, then x – 2 + 2 = 4 + 2

  7. Subtraction Property • If a = b, then a – c = b – c. • Examples • If 4 = 4, then 4 – 2 = 4 – 2. • If x + 6 = 10, then x + 6 – 6 = 10 – 6.

  8. Multiplication Property • If a = b, then a*c = b*c • Examples • If 4 = 4, then 4*3 = 4*3 • If x = 3, then *2x = 3*2

  9. Division Property • If a = b and c not equal o, then a/c = b/c. • If 7 = 7, then = . • If 4x = 16, then = .

  10. Distributive Property • a(b + c) = ab + ac • 2(x + 1) = 2x + 2 • x(x – 5) = x2 – 5x

  11. Reflexive Property • a = a • 2 = 2 • ∠a = ∠a

  12. Symmetric Property • If a = b, then b = a. • 4 = 4 • If 2x = 6, then 6 = 2x. • Biconditional

  13. Transitive Property • If a = b and b = c, then a = c. • If angle a = 45, and angle b = 45, then a = b. • Law of Syllogism

  14. Substitution Property • If a = b, then b can replace a. • Angle A and B are complimentary • Angle A = 2x + 1 Angle B = 4x • 2x + 1 +4x = 90

  15. Theorems and Definitions • You can use definitions and theorems to help reach conclusions. • M is the midpoint of AB. • AM congruent to MB by definition.

  16. Using Your Tools • Solve for x • 4x – 2 = 10 • +2 +2 Addition Property • 4x = 12 • /4 /4 Division Property • x = 3

  17. Justify Each Step • Solve For x • ∠CDE & ∠EDF are supplementary • x + (3x + 20) = 180 • 4x + 20 = 180 • 4x = 160 • x = 40 • Angle Addition Post. • Substitution Property • Simplify • Subtraction Property • Division Property

  18. Example 2 Prove x = 6 • AB = 2x, BC = 3x – 9, AC = 21 Given • AB + BC = AC Segment Addition Postulate • 2x + 3x – 9 = 21 Substitution • 5x – 9 = 21 Simplify • 5x – 9 + 9 = 21 + 9 Addition Property • 5x = 30 Simplify • 5x/5 = 30/5 Division Property • x = 6

  19. Theorem • A statement that you prove to be true.

  20. Vertical Angles Theorem • Vertical Angles are Congruent. • ∠1 ≅ ∠3 • ∠2 ≅ ∠4

  21. Given: ∠1 and ∠2, ∠3 and ∠2 are supplementary Prove ∠1 ≅∠3 • ∠1 + ∠ 2 = 180 given • ∠3 + ∠2 = 180 given • ∠1 + ∠2 = ∠2 + ∠3 substitution • ∠1 = ∠3 subtraction

  22. Homework • 2-4 Worksheet • 2-5 pg. 112 #1-13 on back of worksheet

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