InterMath Dictionary

Properties

 Addition Property of Equality Distributive Property Operation Associative Properties Dividing Powers Property Order of Operations Canceling Property Identity Properties Reciprocal Property Commutative Properties Inverse Operation Subtraction Property of Equality Cross Product Property Inverse Properties Zero Property of Addition

Field Axioms

 Associative Properties - Denoting an operation is independent of grouping. Commutative Properties - Denoting an operation is independent of the order of combination. Distributive Property - The sum of two addends multiplied by a number is the sum of the product of each addend and the number. For example, a(b+c)= ab + ac. Identity Properties - A number that can be added to (or multiplied by) any second number without changing the second number. Inverse Properties - An number combined with its inverse equals the identity.

What happens when you add two numbers that are opposites, like 5 and -5? In a sense, they "cancel" each other, resulting in zero. In other words, the sum of a number and its additive inverse is zero.

2 + (-2) = 0

x + (-x) = 0

-0.5 + 0.5 = 0

2 + (-1) = 1

5 + (-6) = -1

Check for Understanding

 Determine the property shown in the examples below. Choose each answer from the menus. Graded Response a) 5 x 1 = 5 a) Choose one associative propertiy commutative property distributive property identity property inverse property b) 3 + 2 = 2 + 3 b) Choose one associative propertiy commutative property distributive property identity property inverse property c) (5 x 3) x 4 = 5 x (3 x 4) c) Choose one associative propertiy commutative property distributive property identity property inverse property d) 3 x 1/3 = 1 d) Choose one associative propertiy commutative property distributive property identity property inverse property e) 3(a + b) = 3a + 3b e) Choose one associative propertiy commutative property distributive property identity property inverse property

External Links to Extend Basic Understanding

Associative Property for Addition: Definition with example.
Associative Property for Multiplication: Definition with example.
Commutative Property for Addition: Definition with example.
Commutative Property for Multiplication: Definition with example.
Distributive Property: Definition with example.
Where Will I See These Properties Again?: An answer, and a nice discussion of how the distributive property works.

Related Investigations on InterMath to Challenge Teachers

Formulaic Properties: Find values of x, y, and z, so that a given expression equals 18.

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Operations

 Inverse Operation - Pairs of operations that undo each other. Operation - A mathematical process such as addition, subtraction, multiplication, and division. Order of Operations - The rules to be followed when simplifying expressions.

Examples of Inverse Operations

Addition and subtraction are inverse operations.

Multiplication and division are inverse operations.

Order of Operations

When you are simplifying expressions, operations should be performed in the following order to ensure accuracy. Also any expression in parenthesis should be simplified first.

1. Multiplication
2. Division
4. Subtraction

This order is an arbitrary order that was established so that we would all simplify expressions in the same way.

Check for Understanding

 Enter your answer in the left column, then press tab Graded Response Solve: 5 + 3 x 3 - 6

External Links to Extend Basic Understanding

Order of Operations Definition: Clarify your understanding of order of operations.
Practice with Order of Operations: A lesson on order of operations.

Related Investigations on InterMath to Challenge Teachers

Formulaic Properties: Find values of x, y, and z, so that a given expression equals 18.
Multiple Solutions: Find the set of solutions of a system of two equations with three variables.

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Properties of Equality

 Addition Property of Equality - Adding the same number to each side of an equation produces an equivalent expression. Subtraction Property of Equality - Multiplying both sides of an equation by the same nonzero number produces an equivalent expression.

Examples of Addition Property of Equality
Suppose we had the equation a = b. When we add the same number, say 3, to both sides of the equation, we still have an equivalent equation:

a + 3 = b + 3.

To look at it in a different way, consider the following balance scale.

The scale is balanced with 1 box on each side. If we place 3 balls on both sides of the scale, the scale remains balanced.

We could write this mathematically, using x to represent the weight of one box as:

x = x
x + 3 = x + 3.

Non-Examples
Adding different numbers will not produce an equivalent equation.

Here we added 5 balls to one side of the scale and only 3 balls to the other side. As a result, the scale becomes unbalanced.

External Links to Extend Basic Understanding

Simplifying with Addition and Subtraction: General tips for using these properties to solve equations.

Related Investigations on InterMath to Challenge Teachers

Counterfeit Coin: How can you determine which coin is counterfeit?

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Other Properties

 Canceling Property - When a common factor can be found in both the denominator and the numerator of an expression, the common factor can be "factored out" or cancelled. Cross Product Property - For two ratios, if a/b = c/d, then ad = bc. Dividing Powers Property - To divide two powers of the same base, subtract the exponent of the denominator from the exponent of the numerator. Reciprocal Property - For two ratios, if a/b = c/d, then b/a = d/c. Zero Product Property - If ab = 0, then a = 0 or b = 0 or both.

"Canceling" is usually indicated by using slashes through the numbers, as the following examples show.

Examples of Canceling

Example 1: First, the numerator and denominator are written as a product of their factors. Then common factors are "cancelled."

Example 2: When common factors exists on opposite sides of an equation, you can also cancel as the following shows:

Example 3:
We can also "cancel" using addition and subtraction.

In the above example, since there is a 3 on both sides of the equation, we can "cancel" both 3's.

Non-Examples of Canceling
In order to "cancel" a number, the operations being performed on the numbers must be the same. Consider the following expression.

What do you think can be canceled?

Can you determine why?

Check for Understanding

 Determine the property shown in the examples below. Choose each answer from the menus. Graded Response a) Choose one canceling propertiy cross product property dividing powers property reciprocal property zero product property b) Choose one canceling propertiy cross product property dividing powers property reciprocal property zero product property c) Choose one canceling propertiy cross product property dividing powers property reciprocal property zero product property d) Choose one canceling propertiy cross product property dividing powers property reciprocal property zero product property e) Choose one canceling propertiy cross product property dividing powers property reciprocal property zero product property

External Links to Extend Basic Understanding

Put these Properties to Use: See an online lesson on solving equations that use these properties!

Related Investigations on InterMath to Challenge Teachers

Reciprocal Sums: Given the sum of two positive integers, determine the smallest possible sum of their reciprocals. Reciprocal Functions: Describe any patterns you find when comparing a function to its reciprocal function.

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