Warm-Up

•How do you balance your life?

Section 2.5 (10/07/2013)Learning Target

• I am learning the properties of algebra and geometry.

a

a = weighted blocks

a

Reflexive Property (Copy this)

• Let “a” be a real number.• The property states that:• a = a • Example:•5 = 5

a

a = weighted blocks b = weighted blocks

b

What would happen to the scale if I were to switch the blocks to different sides?Since the blocks are the same weight to start out with, no weight would change.

Symmetric Properties (copy this down)

• Let “a” and “b” be any real number.• The property say: • If a = b, then b = a.• Example:• 5x + 10 = 20• 20 = 5x + 10

a

a = weighted blocksc = weighted blocks

b

What would happen if I were to add an “c” block to the left hand side?

b = weighted blocks

c

a

b

The scale would tip downward on the left hand side.

c

Now, without removing any blocks. What can you do to make the scale balance?You can add “c” block to the right hand side.

abc c

Addition (copy this down)

• Let a, b, and c be any real number.• If a = b, then a + c = b + c• Example:•2x = 10•2x + 5 = 10 + 5

abbb aa

What would happen to the scale if I replace two of “a” block with two “b” block?

bb

Nothing, since “b” block is equal weight with “a” block.

Substitution Property (copy this)• Let a, b, and c be any real number.• The property states:• If a = b, then b can replace a in any expression

• Example:• Let a = b and 5a = 10• Then we can say 5a = 5b = 10

bbb a bb

What would happen if I take 2 “b” blocks away from the right hand side?

bbb

a

The scale would tip downward on the left hand side since it’s heavier.

Without adding any blocks, how would you balance the scale?

You can take away 2 “b” blocks on the left hand side.

b a

Subtraction Property (copy this)• Let a, b, and c be real numbers.• The property states that:• If a = b, then a – c = b – c.

• Example:• 5x = 20• 5x – 10 = 20 - 10

b ac

Let the weight of block “b” and the weight of block “a” be equal.

Let the weight of block “b” and the weight of block “c” be equal.

What would happen if replace “b” block with an “a” block?

Using the substitution property, we didn’t do anything to change the weight.

a

Transitive Property

• Let a, b, and c be real numbers.• The property states:• If a = b and b = c, then a = c

• Example:• Let 3x = 5 and let 5 = 2y.• Then 3x = 2y

bbb b a

What would happen if I make the left hand side 3 times as heavy?

bbb b

a

It would tilt downward on the left hand side because it’s heavier.

bbb b a aaa

To balance out the scales, you multiply the weight on the right hand side by 3.

Multiplication Property (copy this down)

• Let a, b, and c be real number.• The property states that:• If a = b, then .

• Example:• 5x + 2 = 10•

bbb b a

What would happen I divide the weight on the left hand side in half??

a aa

bb

a

It would tilt downward on the right hand side because it’s heavier.

aa a

Without adding blocks on the left hand side, what can I do to balance out the scale?

bb a

To balance out the scale, divide the weight on the right hand side in half.

a

Division Property

• Let “a” and “b” be real numbers.• The property states:• If a = b and c 0, then .

• Example:

What is the area???

3

4 6

3∗4=12 3∗6=18

𝐴=3∗10=3 (4+6 )=3∗4+3∗6

Distributive Property

• Let a, b, and c be any real numbers.• The property states:

• Example:• 5(2 + 4) = 5(2) + 5(4)

a

b c

𝑎∗𝑏 𝑎∗𝑐

Properties of Congruence

Reflexive Property:

Symmetric Property:

Transitive Property:

1)

3)

2)

Example: At each step, indicate the properties (congruence or algebraic) that was use.

1)

2)

3)

4)

5)

Recap• Properties of Equality

(a.k.a. Algebraic Reasoning)• Additional• Subtraction• Multiplication• Division• Reflexive• Transitive• Symmetric• Substitution

• Properties of Congruence (a.k.a. Geometric reasoning)• Reflexive• Symmetric • Transitive

Reflection

• In your own words, what were the learning targets?• On a scale from 1-5• 1 for not understanding the learning target at

all.• 5 for completely understanding the learning

target.• Explain