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. And as we all know “time is money”. Therefore: And because “money is the root of all evil.” By substitution, we see that: And we are forced to conclude:

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 If Jill is smart, then __________________. If Jill gets good grades, then ___________________________. If Jill is smart, then _____________________________. If Jill excels at the university, then __________________________. If Jill receives many job offers, ______________________________. If Jill has a high paying job, then _____________________. If Jill retires early, then _____________. If Jill is bored, then _______________________. she will get good grades she will get into a very fine university she will excel at the very fine university she will receive many fine job offers she will be able to pick one that pays well she will be able to retire early she will be bored she takes up fishing as a hobby.

Intro to Proof If Jill is smart, then ______________________________. If Jill is proficient at fishing, then _________________________. If Jill catches a lot of fish, then ________________________________. If the fish fill every part of her house, then _______________________. If Jill is smart, then ________________________. If Jill wants peace in her home, then _________________. she will become very proficient at fishing she catches a whole bunch of fish the fish will soon fill every part of her house her husband will become angry she will want peace in her home she will eat a lot of fish 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: 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:

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 = then = 4.

More Properties 3)Transitive: If a = b and b = c, then a = c. If 4 = and = then 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 - acand(b - c) a = ba - ca Example #1 5(x + 7) 5 x x + 35

Name the Property 1. If 2x+1= 4, then 2x = 3 Substraction 2. (10 + 2)  3 = 12  3 Substitution = 5 then 5 = Symmetric

4. If 5  2 = 10 & 10 = then 5  2 = Transitive 5. If 7x = 21, then x = 3 Division Property

6. 2( 5+x) = x Distributive 7. k + 7 = k + 7 Reflexive 8. 2+k = k+ 2 Symmetric

Example 1: Writing Reasons Solve 5x – 18 = 3x x – 18 = 3x x – 18 = 2 3.2x = 20 4.x = 10 1.Given 2.Subtraction property 3.Addition property 4.Division property

Example 2: Writing Reasons Solve 55z – 3(9z + 12)= z – 3(9z + 12)= z – 27z – 36 = z – 36 = z = z = -1 1.Given 2.Distributive property 3.Simplify 4.Addition property 5.Division property

Example 4: Using properties of length 1.AB = CD 2.AB + BC = BC + CD 3.AC = AB + BC 4.BD = BC + CD 5.AC = BD 1.Given 2.Addition property 3.Segment addition postulate 4.Segment addition postulate 5.Substitution property ABCDABCD Given: AB = CD Prove: AC = BD