It seems that each student interpreted the problem differently, resulting in two different answers.

Who is correct? Write who you think is right? _____________________________

When performing calculations , 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.

You may have predicted correctly, that student 2 solved the problem correctly since she followed Rules 2 and 3. Let's look at some examples of solving arithmetic expressions using these rules.

Evaluate each expression using the rules for order of operations.

 

In the examples above, 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 3 below.

Example 3:

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

 

 

 

 

 

 

 

Example 4:	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.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

As you work with operations on rational numbers, you may notice certain characteristics that hold true. For example, you know that 7 + 3 = 10 and 3 + 7 = 10. There are three basic number properties (or laws) that apply to arithmetic operations: Commutative Property, Associative Property and Distributive Property.

Commutative Property

An operation is commutative if a change in the order of the numbers does not change the results. This means the numbers can be swapped.

Numbers can be added in any order.

For example:	4 + 5 = 5 + 4
	x + y = y + x

Numbers can be multiplied in any order.

For example:	5 × 3 = 3 × 5
	a × b = b × a

Numbers that are subtracted are NOT commutative.

For example:	4 – 5 ≠ 5 – 4
	x – y ≠ y –x

Numbers that are divided are NOT commutative.

For example:	4 ÷ 5 ≠ 5 ÷ 4
	x ÷ y ≠ y ÷ x

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Associative Property

An operation is associative if a change in grouping does not change the results. This means the parenthesis (or brackets) can be moved.

Numbers that are added can be grouped in any order.

For example:	(4 + 5) + 6 = 5 + (4 + 6)
	(x + y) + z = x + (y + z)

Numbers that are multiplied can be grouped in any order.

For example:	(4 × 5) × 6 = 5 × (4 × 6)
	(x × y) × z = x × (y × z)

Numbers that are subtracted are NOT associative.

For example:	(4 – 5) – 6 ≠ 4 – (5– 6)
	(x – y) – z ≠ x – (y – z)

Numbers that are divided are NOT associative.

For example:	(4 ÷ 5) ÷ 6 ≠ 4 ÷ (5÷ 6)
	(x ÷ y ) ÷ z ≠ x ÷ ( y ÷ z)

 

Distributive Property

Distributive property allows you to remove the parenthesis (or brackets) in an expression. Multiply the value outside the brackets with each of the terms in the brackets.

For example:	4(a + b) = 4a + 4b
	7(12 – 4) = (7 x 12) – (7 x 4)

 

Number properties can help you perform mental calculations quickly. For example, to simplify 13 + 19 + 7 mentally, you might use the Commutative Property of Addition to add 13 and 7 before adding 19.

Use the properties of numbers to find the value of the variable in each equation.

A. 3 + 9 + 12 = a + 3 + 12

Use the Commutative Property of Addition to reorder the first two numbers, so 3 + 9 + 12 = 9 + 3 + 12. Therefore, a = 9.

 

B. 25 x (4 x 12) + 3 = (4 x k) x 12 + 3. Use the Commutative Property of Multiplication to reorder the multiplication, so (25 x 4) x 12 + 3 = (4 x 25) x12 + 3. So, k = 25.

 

 

 



Consider the two rectangles at the right. You can find the total area of the rectangles by writing (3 x 4) + (3 x 6) = 12 + 18, or 30.

 

You can also find the total area by combining the smaller rectangles into one large rectangle as shown. The rectangle area is 3(4 + 6) = 3(10), or 30.

 

This model illustrates another property of numbers. The Distributive Property combines multiplication with addiction and subtraction. To multiply a sum or difference, you multiply each number within the parentheses by the number outside the parentheses. For example, you can write 3(4 + 6) = 3(4) + 3(6).

 

 

The Distributive Property can help us to solve problems that seem really hard at first. For example, 16 x 97 can seem super hard. Using the Distributive Property you can write 97 as a difference of two integers, so 16 (97) = 16(100 – 3) = 16(100) – 16(3).

 

 

 

 

Exercises

Name the property illustrated in each equation.

1. 15 + 27 + 25 = 27 + 15 + 25        _______________

 

2. 14 x 20 – 14 x 5 = 14 x 15        _______________

 

3. (5 x 7) + (3 x 2) = (7 x 5) + (3 x 2)      _______________

 

4. 9 + (11 + 6) = (9 + 11) + 6        _______________

 

5. 4(x + 2) = 4x + (8)          _______________

 

6. + ( + y) = ( + ) + y      _______________

 

7. 0.68(5) = 5(0.68)          _______________

 

 

Use mental math and number properties to simplify each expression.

 

8. 25 x 102            9. (51 + 13) + (9 + 7)

 

 

10. 4 x (17 x 25)          11. (20 x 19) x 5

 

 

12. Given that 212 x 4 + 7 = 855, simplify 4 x 212 + 7.

 

 

13. Draw a model to show that 6(7) = 6(5) + 6(2).

 

 

 

 

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