• How should numbers be added, subtracted, multiplied, and divided? What latitude do we have?
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• Does it matter what we do first? second? third? . . .
• 
  
  
• Can we add a series of numbers together in any order? Will the final answer be the same regardless of the order we choose?
• 
  
  
• Can we multiply a series of numbers together in any order? Will the final answer be the same regardless of the order we choose?
• 
  
  

• Associative Property
• 
  
  
• Commutative Property
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• Distributive Property
• 
  
  
• Identity Property
• 
  
  
• Inverse Property

• Reflexive Property
• 
  
  
• Symmetric Property
• 
  
  
• Transitive Property

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Equivalence Relationships -

What is an Equivalence Relationship ?

Equivalence simply means two objects are alike in some way .

Equivalence means that two objects have a common characteristic . They are the same in that one respect.

However, equivalence does not mean two objects are alike in every respect.

For example:

is equivalent to

in three ways

Both of these numbers have the same value, mathematically. Their values are equivalent.

Both of these numbers have the same color. Their colors are equivalent.

Both of these numbers are displayed on the same background color. Their background colors are equivalent.

However, the format of the numbers is not the same. One number is a fraction and the other number is a decimal. Only the numerical values, color, and background color are equivalent. Nothing else about the numbers is equivalent: not the format, not the shape, not the numerals used, etc.

Why is an equivalence relation an important math property?

If an equivalence relation exists, substitution is possible . If two items are equivalent, one item can be replaced with the other , or vice versa.

For example:

(1) The number ¹ / ₂ can be replaced with 0.5 to simplify the following arithmetic problem:

3.0367 + ¹ / ₂ = ?

Substituting .5 for ¹ / ₂

3.0367 + 0.5 = 3.5367

(2) The number 0.5 can also be replaced with the fraction ¹ / ₂ to simplify a problem:

0.5 + ³ / ₂ = ?

Substituting ¹ / ₂ for 0.5

¹ / ₂ + ³ / ₂ = ?

¹ / ₂ + ³ / ₂ = ⁴ / ₂ = 2

This point cannot be overemphasized : equivalence relations make substitution possible .

Being able to substitute equivalent values, expressions, formulas, or quantities, simplifies problem solving .

Math Properties - Other Examples of Equivalence

Measurements of completely disparate objects are equivalent if they have the same length, weight, or volume - even if those measurements are expressed in different units:

Equivalence Relationships Involving Measurements

.	.	.
1 meter of ribbon	length is equivalent to	3.28 feet long fishing rod
1 ton of crushed ice	weight is equivalent to	2000 pounds of beach sand
1 gallon of antifreeze	volume is equivalent to	3.79 liters pink lemonade

The examples shown above have been deliberately selected so that the objects which are being compared (ribbon vs fishing rod, crushed ice vs beach sand, antifreeze vs pink lemonade) have almost nothing in common except their length, weight, or volume. Yet, they have something in common - they are equivalent in at least one respect.

This is true of other characteristics as well:

Miscellaneous Equivalence Relationships

.	.	.
5 green beads	color is equivalent to	a mixture of blue and yellow paint
4 year old horse	height is equivalent to	5 year old oak tree
95% average grade in PE class	percentage grade is equivalent to	95% average grade in a calculus class

Some equivalent relationships in mathematics are shown below. Note that the numbers do not refer to any special objects such as 5 ping pong balls, or 3 planets. That is the point - and the value - of using numbers or variables (such as "x"). Numbers and variables are an abstraction. They are not limited to any one application. The same numbers can be used over and over. They can be applied to any situation.

Some Equivalent Relationships in Mathematics

	.		.		.
	5 + 1		=		3 + 3
	2x		=		x + x
	15÷5		=		3