Download Document

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
= weighted blocks
a
a
Reflexive Property (Copy this)
• Let “a” be a real number.
• The property states that:
•a = a
• Example:
•5 = 5
Since the
What
would
blocks
happen
are the
to the
same
scale
weight
if I were
to start
to switch
out with,
the no
blocks to different
sides? would change.
weight
a
a
= weighted blocks
b
b
= weighted blocks
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
What would happen if I were to add an “c” block to the
left hand side?
a
c b
c
= weighted blocks
b
= weighted blocks
a
= weighted blocks
Now,
without
removing
any blocks.
What
can
you
do to
The
scale
would
tip
downward
on
the
left
hand
side.
You can add “c” block to the right hand side.
make the scale balance?
a
c b
c b
a 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
What
would
happen
to the
scale weight
if I replace
“a”
Nothing,
since
“b” block
is equal
withtwo
“a”of
block.
block with two “b” block?
b b b
a b
a b
a
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
What would happen if I take 2 “b” blocks away from
the right hand side?
b b b
a b b
You
can
take
away
2downward
“b” blocks
left
Without
The
scale
adding
would
anytipblocks,
how would
onon
thethe
you
left balance
handhand
side
side.
the
since
scale?
it’s heavier.
a
b b b
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
Let the weight of block “b”
Let the weight of block “b”
Using the substitution property,
we didn’t do
and
the weight
block if replace
and“b”
the weight
of block “c”
What
wouldofhappen
block with
anything
to change the weight.
“a”
be
equal.
be equal.
an “a” block?
b a
c 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
What would happen if I make the left hand
side 3 times as heavy?
b b b b
a
It would tilt downward on the left hand side
because it’s heavier.
a
b b b b
To balance out the scales, you multiply the
weight on the right hand side by 3.
b b b b
a a a
a
Multiplication Property (copy this
down)
• Let a, b, and c be real number.
• The property states that:
• If a = b, then a ∙ c = b ∙ c.
• Example:
• 5x + 2 = 10
• 4 5x + 2 = 4 ∙ 10
What would happen I divide the weight on the left
hand side in half??
b b b b
a
a a
a
Without adding
blockstilt
ondownward
the left handon
side,
what
canhand
I do toside
It would
the
right
balance outbecause
the scale?it’s heavier.
b b
a
a a
a
To balance out the scale, divide the weight on the
right hand side in half.
b b
a a
Division Property
• Let “a” and “b” be real numbers.
• The property states:
𝑎
𝑐
• If a = b and c ≠ 0, then =
• Example:
• 3𝑥 − 5 = 10
•
3𝑥−5
2
=
10
2
𝑏
.
𝑐
What is the area???
4
3 3 ∗ 4 = 12
6
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:
• 𝑎 𝑏 + 𝑐 = 𝑎𝑏 + 𝑎𝑐
• Example:
• 5(2 + 4) = 5(2) + 5(4)
a
b
c
𝑎∗𝑏
𝑎∗𝑐
Properties of Congruence
Reflexive Property:
1)
Symmetric Property:
2)
Transitive Property:
3)
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