Download 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
52=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