Transcript 2-4 Notes
2.4 Objective: The student will be able to: recognize and use algebraic properties What are some ways or situations in which we use the word “prove” or “proof”? Have you ever use the phrase “prove it”? Proof That Dogs Are Evil First, we state that dogs require time and money. Dogs Time Money And as we all know “time is money”. Therefore: Time Money Dogs Money Money (Money ) 2 And because “money is the root of all evil.” By substitution, we see that: And we are forced to conclude: Money Evil Dogs ( Evil ) 2 Dogs Evil Intro to Proof We need proof when the connection between two statements is not obvious. Obvious: If Jill is smart, then she will get good grades. Not so Obvious: If Jill is smart, then she will eat a lot of fish. Intro to Proof she will get good grades If Jill is smart, then __________________. she will get into a very fine university If Jill gets good grades, then ___________________________. she will excel at the very fine university If Jill is smart, then _____________________________. she will receive many fine job offers If Jill excels at the university, then __________________________. she will be able to pick one that pays well If Jill receives many job offers, ______________________________. she will be able to retire early If Jill has a high paying job, then _____________________. she will be bored If Jill retires early, then _____________. she takes up fishing as a hobby. If Jill is bored, then _______________________. Intro to Proof she will become very proficient at fishing If Jill is smart, then ______________________________. she catches a whole bunch of fish If Jill is proficient at fishing, then _________________________. the fish will soon fill every part of her house If Jill catches a lot of fish, then ________________________________. her husband will become angry If the fish fill every part of her house, then _______________________. she will want peace in her home If Jill is smart, then ________________________. she will eat a lot of fish If Jill wants peace in her home, then _________________. This line of argument proves: If Jill is smart, then she will eat a lot of fish. Intro to Proof Logical Thinking is critical to construct proofs that make sense. Activity: After dividing into groups of two or three people, each group will receive a cartoon strip that has been cut apart into individual frames. Your task as a group will be to put the frames together in their original order, so the story makes sense. Addition and Subtraction Properties 1) Addition Property For all numbers a, b and c if a = b, then: a+c=b+c 2) Subtraction Property For all numbers a, b and c if a = b, then: a - c= b - c Multiplication Property: For all numbers a, b and c if a = b, then: a • c= b • c Division Property: For all numbers a, b and c if a = b and if c ≠ 0, then: a b c c Algebraic Properties 1)Reflexive Property: For every number a, a = a, 2) Symmetric Property: For all numbers a and b, if a = b, then a = b and b = a If 4 = 2 + 2 then 2 + 2 = 4. More Properties 3) Transitive: If a = b and b = c, then a = c. If 4 = 2 + 2 and 2 + 2 = 3 + 1 then 4 = 3 + 1. 4) Substitution: If a = b, then a can be replaced by b. (5 + 2)x = 7x The Distributive Property The process of distributing the number on the outside of the parentheses to each term on the inside. a(b + c) = ab + ac and (b + c) a = ba + ca a(b - c) = ab - ac and (b - c) a = ba - ca Example #1 5(x + 7) 5•x+5•7 5x + 35 Name the Property 1. If 2x+1= 4, then 2x = 3 Substraction 2. (10 + 2) 3 = 12 3 Substitution 3. 2 + 3 = 5 then 5 = 2 + 3 Symmetric 4. If 5 2 = 10 & 10 = 5 + 5 then 52=5+5 Transitive 5. If 7x = 21, then x = 3 Division Property 6. 2( 5+x) = 2• 5 +2 • x Distributive 7. k + 7 = k + 7 Reflexive 8. 2+k = k+ 2 Symmetric Example 1: Writing Reasons Solve 5x – 18 = 3x +2 1. 5x – 18 = 3x + 2 1. Given 2. Subtraction property 2. 2x – 18 = 2 3. Addition property 3. 2x = 20 4. Division property 4. x = 10 Example 2: Writing Reasons Solve 55z – 3(9z + 12)= -64 1. 55z – 3(9z + 12)= -64 2. 55z – 27z – 36 = -64 3. 28z – 36 = -64 4. 28z = -28 5. z = -1 1. 2. 3. 4. 5. Given Distributive property Simplify Addition property Division property Example 4: Using properties of length Given: AB = CD Prove: AC = BD A B C 1. AB = CD 1. Given 2. 3. 4. 5. 2. 3. AB + BC = BC + CD AC = AB + BC BD = BC + CD AC = BD 4. 5. D Addition property Segment addition postulate Segment addition postulate Substitution property