Properties of Numbers

Distributive Property

The Distributive Property is easy to remember, if you recall that "multiplication distributes over addition". Formally, they write this property as "a(b + c) = ab + ac". In numbers, this means, that 2(3 + 4) = 2×3 + 2×4. Any time they refer in a problem to using the Distributive Property, they want you to take something through the parentheses (or factor something out); any time a computation depends on multiplying through a parentheses (or factoring something out), they want you to say that the computation used the Distributive Property.

• Why is the following true? 2(x + y) = 2x + 2y

Since they distributed through the parentheses, this is true by the Distributive Property.

• Use the Distributive Property to rearrange: 4x – 8

The Distributive Property either takes something through a parentheses or else factors something out. Since there aren't any parentheses to go into, you must need to factor out of. Then the answer is "By the Distributive Property, 4x – 8 = 4(x – 2)"

"But wait!" you say. "The Distributive Property says multiplication distributes over addition, not subtraction!  What gives?" You make a good point. This is one of those times when it's best to be flexible. You can either view the contents of the parentheses as the subtraction of a positive number ("x – 2") or else as the addition of a negative number ("x + (–2)"). In the latter case, it's easy to see that the Distributive Property applies, because you're still adding; you're just adding a negative.
The other two properties come in two versions each: one for addition and the other for multiplication. (Note that the Distributive Property refers to both addition and multiplication, too, but to both within just one rule.)

Associative Property

The word "associative" comes from "associate" or "group";the Associative Property is the rule that refers to grouping. For addition, the rule is "a + (b + c) = (a + b) + c"; in numbers, this means
2 + (3 + 4) = (2 + 3) + 4. For multiplication, the rule is "a(bc) = (ab)c"; in numbers, this means 2(3×4) = (2×3)4. Any time they refer to the Associative Property, they want you to regroup things; any time a computation depends on things being regrouped, they want you to say that the computation uses the Associative Property.   Copyright © Elizabeth Stapel 2000-2011 All Rights Reserved

• Rearrange, using the Associative Property: 2(3x)

They want you to regroup things, not simplify things. In other words, they do not want you to say "6x". They want to see the following regrouping: (2×3)x

• Simplify 2(3x), and justify your steps.

In this case, they do want you to simplify, but you have to tell why it's okay to do... just exactly what you've always done. Here's how this works:


2(3x)	original (given) statement
(2×3)x	by the Associative Property
6x	simplification (2×3 = 6)

• Why is it true that 2(3x) = (2×3)x?

Since all they did was regroup things, this is true by the Associative Property.

Commutative Property

The word "commutative" comes from "commute" or "move around", so the Commutative Property is the one that refers to moving stuff around. For addition, the rule is "a + b = b + a"; in numbers, this means 2 + 3 = 3 + 2. For multiplication, the rule is "ab = ba"; in numbers, this means 2×3 = 3×2. Any time they refer to the Commutative Property, they want you to move stuff around; any time a computation depends on moving stuff around, they want you to say that the computation uses the Commutative Property.

• Use the Commutative Property to restate "3×4×x" in at least two ways.

They want you to move stuff around, not simplify. In other words, the answer is not "12x"; the answer is any two of the following:

4 × 3 × x, 4 × x × 3, 3 × x × 4, x × 3 × 4, and x × 4 × 3

• Why is it true that 3(4x) = (4x)(3)?

Since all they did was move stuff around (they didn't regroup), this is true by the Commutative Property.


• Simplify 3a – 5b + 7a. Justify your steps.

I'm going to do the exact same algebra I've always done, but now I have to give the name of the property that says its okay for me to take each step. The answer looks like this:


3a – 5b + 7a	original (given) statement
3a + 7a – 5b	Commutative Property
(3a + 7a) – 5b	Associative Property
a(3 + 7) – 5b	Distributive Property
a(10) – 5b	simplification (3 + 7 = 10)
10a – 5b	Commutative Property

The only fiddly part was moving the "– 5b" from the middle of the expression (in the first line of the table above) to the end of the expression (in the second line). If you need help keeping your negatives straight, convert the "– 5b" to "+ (–5b)". Just don't lose that minus sign!

• Simplify 23 + 5x + 7y – x – y – 27.   Justify your steps.
     

23 + 5x + 7y – x – y – 27	original (given) statement
23 – 27 + 5x – x + 7y – y	Commutative Property
(23 – 27) + (5x – x) + (7y – y)	Associative Property
(–4) + (5x – x) + (7y – y)	simplification (23 – 27 = –4)
(–4) + x(5 – 1) + y(7 – 1)	Distributive Property
–4 + x(4) + y(6)	simplification
–4 + 4x + 6y	Commutative Property

• Simplify 3(x + 2) – 4x.   Justify your steps.
     

3(x + 2) – 4x	original (given) statement
3x + 3×2 – 4x	Distributive Property
3x + 6 – 4x	simplification (3×2 = 6)
3x – 4x + 6	Commutative Property
(3x – 4x) + 6	Associative Property
x(3 – 4) + 6	Distributive Property
x(–1) + 6	simplification (3 – 4 = –1)
–x + 6	Commutative Property

http://www.purplemath.com/modules/numbprop.htm

Order of Operations

Order of Operations

Problem:	Evaluate the following arithmetic expression:  3 + 4 x 2
Solution:	Student 1       Student 2 3 + 4 x 2 3 + 4 x 2 = 7 x 2 = 3 + 8 = 14 = 11

It seems that each student interpreted the problem differently, resulting in two different answers. Student 1 performed the operation of addition first, then multiplication; whereas student 2 performed multiplication first, then addition. When performing arithmetic operations there can be only one correct answer. We need a set of rules in order to avoid this kind of confusion. Mathematicians have devised a standard order of operations for calculations involving more than one arithmetic operation.

Rule 1:	First perform any calculations inside parentheses.
Rule 2:	Next perform all multiplications and divisions, working from left to right.
Rule 3:	Lastly, perform all additions and subtractions, working from left to right.

The above problem was solved correctly by Student 2 since she followed Rules 2 and 3. Let's look at some examples of solving arithmetic expressions using these rules.

Example 1:	Evaluate each expression using the rules for order of operations.
Solution:	Order of Operations Expression Evaluation Operation 6 + 7 x 8 = 6 + 7 x 8 Multiplication = 6 + 56 Addition = 62   16 ÷ 8 - 2 = 16 ÷ 8 - 2 Division = 2 - 2 Subtraction = 0   (25 - 11) x 3 = (25 - 11) x 3 Parentheses = 14 x 3 Multiplication = 42

In Example 1, each problem involved only 2 operations. Let's look at some examples that involve more than two operations.

Example 2:	Evaluate 3 + 6 x (5 + 4) ÷ 3 - 7 using the order of operations.
Solution:	Step 1:   3 + 6 x (5 + 4) ÷ 3 - 7  =  3 + 6 x 9 ÷ 3 - 7 Parentheses Step 2:   3 + 6 x 9 ÷ 3 - 7  =  3 + 54 ÷ 3 - 7 Multiplication Step 3:   3 + 54 ÷ 3 - 7  =  3 + 18 - 7 Division Step 4:   3 + 18 - 7  =  21 - 7 Addition

                       

Step 5:

21 - 7

=

14

Subtraction

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Example 3:	Evaluate 9 - 5 ÷ (8 - 3) x 2 + 6 using the order of operations.
Solution:	Step 1:     9 - 5 ÷ (8 - 3) x 2 + 6  =  9 - 5 ÷ 5 x 2 + 6 Parentheses Step 2:   9 - 5 ÷ 5 x 2 + 6  =  9 - 1 x 2 + 6 Division Step 3:   9 - 1 x 2 + 6  =  9 - 2 + 6 Multiplication Step 4:   9 - 2 + 6  =  7 + 6 Subtraction Step 5:   7 + 6  =  13 Addition

In Examples 2 and 3, you will notice that multiplication and division were evaluated from left to right according to Rule 2. Similarly, addition and subtraction were evaluated from left to right, according to Rule 3.

When two or more operations occur inside a set of parentheses, these operations should be evaluated according to Rules 2 and 3. This is done in Example 4 below.

Example 4:	Evaluate 150 ÷ (6 + 3 x 8) - 5 using the order of operations.
Solution:	Step 1:   150 ÷ (6 + 3 x 8) - 5  =  150 ÷ (6 + 24) - 5 Multiplication inside Parentheses Step 2:   150 ÷ (6 + 24) - 5  =  150 ÷ 30 - 5 Addition inside Parentheses Step 3:   150 ÷ 30 - 5  =  5 - 5 Division Step 4:   5 - 5  =  0 Subtraction

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Example 5:   Evaluate the arithmetic expression below:   Solution:   This problem includes a fraction bar (also called a vinculum), which means we must divide the numerator by the denominator. However, we must first perform all calculations above and below the fraction bar BEFORE dividing.   Thus   Evaluating this expression, we get:   Example 6:   Write an arithmetic expression for this problem. Then evaluate the expression using the order of operations.   Mr. Smith charged Jill $32 for parts and $15 per hour for labor to repair her bicycle. If he spent 3 hours repairing her bike, how much does Jill owe him? Solution:   32 + 3 x 15   =   32 + 3 x 15    =   32 + 45 =   77   Jill owes Mr. Smith $77. Summary:   When evaluating arithmetic expressions, the order of operations is: Simplify all operations inside parentheses. Perform all multiplications and divisions, working from left to right. Perform all additions and subtractions, working from left to right. If a problem includes a fraction bar, perform all calculations above and below the fraction bar before dividing the numerator by the denominator.               http://www.mathgoodies.com/lessons/vol7/order_operations.html