Transcript Properties of Equality, Identity, and Operations

Properties of Equality, Identity,
and Operations
September 11, 2014
Essential Question: Can I justify
solving an equation using
mathematical properties?
Commutative Property
a+b=b+a
(a)(b) = (b)(a)
• The Commutative Property states that the order
of the numbers may change and the
sum/product will remain the same.
• This property applies to only addition and
multiplication; NOT subtraction and division.
2+3=3+2
(2)(3) = (3)(2)
Associative Property
(a + b) + c = a + (b + c)
(a · b) · c = a · (b · c)
• The Associative Property states that the
grouping of numbers can change and the
sum/product will remain the same.
• This property also applies to both addition and
multiplication.
(2 + 4) + 5 = 2 + (4 + 5)
(2 · 4) · 5 = 2 · (4 · 5)
Distributive Property of Multiplication
a (b + c) = a(b) + a(c)
a (b – c) = a(b) – a(c)
• The Distributive Property takes a number and
multiplies it by everything inside the
parentheses.
• This property works over addition and
subtraction.
2(3 + 4) = 2(3) + 2(4)
2 (5 – 2) = 2(5) – 2(2)
Identity Properties
n·1=n
n+0=n
• This property shows how a given number is itself
when multiplied by 1 or added to 0.
• The one and zero act like mirrors.
4·1=4
5+0=5
Zero Property of Multiplication
n·0=0
Simply stated, any number times zero
equals zero.
Multiplicative Inverse Property
½ (2) = 1
• This property is helpful when solving equations
where there is a fraction “attached” to a variable
by multiplication. The normal inverse operation
for multiplication is division, but in this case, you
will multiply both sides of the equation by the
reciprocal of the fraction.
½n–3=4
½ n -3 + 3 = 4 + 3
½n=7
½ n (2) = 7(2)
n = 14
Addition Property of Equality
 If a = b, then a + c = b + c or a + (-c) = b + (-c)
 The addition property of equality says that if you
may add equal quantities to each side of the
equation & still have equal quantities
 Example
 In if-then form:
 If 6 = 6 ; then 6 + 3 = 6 + 3 or 6 + (-3) = 6 + (3).
Subtraction Property of Equality
• If a = b, then a – c = b – c.
 The subtraction property of equality says that if
you may subtract equal quantities to each side
of the equation & still have equal quantities
 Example
 In if-then form:
 If 6 = 6 ; then 6 - 3 = 6 - 3
Multiplication Property of Equality
• If a = b, then ac = bc
• The multiplication property of equality says that if
you may multiply equal quantities to each side of
the equation & still have equal quantities.
• In if-then form:
 If 6 = 6 ; then 6 * 3 = 6 * 3.
Division Property of Equality
• If a = b and c ≠ 0, then a ÷ c = b ÷ c.
• Dividing both sides of the equation by the same
number, other than 0, does not change the
equality of the equation.
• In if-then form:
 If 6 = 6 ; then 6 ÷ 3 = 6 ÷ 3
 Why can’t C be 0?
Properties of Equality
Turn and Talk
Notice, after using any of the properties of
equality, the numbers are still equal.
Why do you think it is important to learn
these properties?