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Chapter 1—Whole Numbers 1-9 Ordering Operations with Whole Numbers.
You have been doing a lot of work here with basic arithmetic problems. You are given two whole-number values and asked to add them, subtract them, multiply them, or divide them. Or you are given a group of numbers and asked to add them or to multiply them. This lesson introduces the procedures for combining more than two numbers and using more than one kind of arithmetic operation.
Combining Addition and Subtraction You have been doing a lot of work here with adding numbers and subtracting numbers. Here you see now to combine addition and subtraction operations.
It is important that you always solve problems of this type from left to right. Why? Sometimes the answer is different, depending upon which way you go. Example of Right and Wrong 12 – 4 + 6 – 1 = ______
Solving from right to left (the incorrect way):
It can make a big difference whether you solve combinations of addition and multiplication from left to right (the correct way), or from right to left (the incorrect way). Check out these examples of operations that include both addition and subtraction.More Examples Exercises
Combining Multiplication and Division You have been doing a lot of work here with multiplying and dividing numbers. Here you see now to combine multiplication and division operations.
You must always solve problems of this type from left to right. Otherwise, you will likely get a different (incorrect) result. Example of Right and Wrong 4 x 12 ÷ 6 x 2 = _____
More Examples Check out these examples of operations that include both multiplication and division.Exercises
Combining Addition, Subtraction, Multiplication, and Division
Example 4 x 3 + 8 ÷ 2 – 1 = _____ — from left to right:
Then do the addition and subtraction — from left to right:
4 x 3 + 8 ÷ 2 – 1 = 15 More Examples Exercises Complete the operations.
Continue these exercises until you can solve them without errors. Introducing Signs of Grouping Signs of grouping are often used for clarifying and simplifying expressions that have a mix of operations. They "group" operations into distinct sets of operations. The first, and most common, sign of grouping is a set of parentheses ( ). Here is an example of a problem that includes signs of grouping: 2 x (8 – 4) + 1 = _____ When a problem includes signs of grouping, you must complete the operations within the group first. Then you handle any multiplication/division followed by addition/subtraction. Rule Operations enclosed in a sign of grouping are always completed first. Example Problem 2 x (8 – 4) + 1 = _____ Complete the operation in parentheses first: 2 x (8 – 4) + 1 = 2 x (4) + 1 When there is just one number inside the signs of grouping, you can just omit the signs. So: 2 x (4) + 1 = 2 x 4 + 1 There are no longer any signs of grouping, so you now use the usual rules for the order of operations: Solution 2 x (8 – 4) + 1 = 9 Note A sign of grouping can be omitted when it contains only a single number. More Examples Exercises Signs of grouping may be nested — signs of grouping placed within other signs of grouping. Here is an example of two sets of groupings within another. 2 x [ 3 + 4 x ( 12 – 10 ) + 15 ] If there were yet another outer group that enclosed the bracket group, the sign of grouping would be a set of braces, or "curly brackets" { } When you are working out a problem that has nested signs of grouping, you should always clear the inside groups first ... no matter what kinds of operations they contain. In our example here, the first thing to do is the (12 - 10) — that is the innermost group. 2 x [ 3 + 4 x ( 12 – 10 ) + 15 ] = 2 x [ 3 + 4 x 2 + 15 ] The inner sign of grouping is now gone. Then work the operations enclosed in brackets: Do multiplication first 2 x [ 3 + 4 x 2 + 15 ] = 2 x [ 3 + 8+ 15 ] Then the addition 2 x [ 3 + 8+ 15 ] = 2 x 26 Now the outer sign of grouping is gone, too. Finish the operation 2 x 26 = 52 So 2 x [ 3 + 4 x ( 12 – 10 ) + 15 ] = 52 Rule When working with nested signs of grouping, always clear the innermost groups first. Examples Exercises |
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| Author:
David L. Heiserman Publisher: SweetHaven Publishing Services |
Copyright © 2007, David L. Heiserman |
