Math Background

Comparing and Ordering Integers and Decimals: Overview

Your students have already had some practice estimating, comparing, and ordering numbers and decimal numbers to tenths. Whenever possible, draw on past experiences to help students learn new concepts.

For example, while students may not know exactly how far Earth is from the Sun or the diameter of a strand of hair, they will most likely be eager to give you estimates of these distances. Your students will also be able to give you situations in which whole numbers and decimal numbers have been estimated. Use their knowledge of estimated whole and decimal numbers to introduce rounding.

Rounding is a method we can use to find an estimate. A number line provides your students with a visual method of rounding numbers (see below).

numberline

Rounded to the nearest hundred thousand, 3,651,114 is 3,700,000.

numberline

Rounded to the nearest tenth, 8.03 is 8.0.

Many students, however, will prefer to follow this list of rules for rounding.

  1. First, locate the place you want to round to.
  2. Then look at the first digit to the right of that place.

3,651,114 — The digit in this place is equal to or greater than 5. So 6 is increased by 1. hundred thousands

The number 3,651,114 rounds to 3,700,000.

8.03 — The digit in this place is less than 5. So 0 is not increased by 1. tenths The number 8.03 rounds to 8.0.

Given some hints, your students can name situations in which they have already compared and ordered whole numbers. You can brainstorm with your students to list some examples, such as whose parent is older, how many miles one store is from another, and so on. Be sure to extend your list to include examples in which larger numbers are compared and ordered. Population totals, voting tallies, and distances should be included in your list. Try to use students' understanding of comparing and ordering lesser numbers to help them compare and order greater numbers.

The most important concept for students to remember is that to compare or order numbers, they must first align them by place value. Then they compare the digits from left to right until they are different. The number with the greater different digit is the greater of the two numbers. We always compare from left to right because the place values on the left are the greatest.

Students' understanding of comparing and ordering whole numbers can be smoothly transferred to comparing and ordering decimal numbers. Giving students examples of decimals, such as batting averages or money amounts, to compare will increase their comfort with the process of comparing and ordering decimals. Many students will have already made such comparisons without having given it a second thought.

Students can compare and order decimals by locating them on a number line (see below) or by following the rules for comparing the digits.

numberline
rules for comparing the digits

The symbols used in math may cause discomfort for some students. Review the meaning of >, <, and =. Be aware that as soon as you write positive and negative signs in front of numbers, some students will feel that math has immediately gotten too difficult for them. Remind students that they have heard temperatures and wind chills reported in negative degrees. Use their understanding of temperatures to explain the concept of negative numbers. Numbers less than zero are known as negative numbers. Numbers to the left of zero on a number line are negative numbers.

numberline

Every number has an opposite. Since the number 4 (negative four) is the number 4 units to the left of zero on a number line, the opposite of 4 can be found by counting 4 units to the right of zero on a number line. Therefore, the opposite of 4 is +4 (positive 4). Likewise, the opposite of +4 is 4. Be sure to explain to students that it is OK to write positive numbers without the positive sign. Thus, +4 can also be written as 4. Zero is its own opposite. Zero is written without a negative sign or a positive sign.

numberline

Integers are defined as “the set of positive whole numbers and their opposites (negative numbers) and 0.” The set of integers is written as “…, 3, 2, 1, 0, +1, +2, +3, …,” since the set of integers is infinite, extending without end to the left and to the right of zero on the number line.

The ordering of integers is best shown on a number line. The value of an integer increases as you move to the right along the number line. To compare two integers, have students first locate each number on the number line. The number on the left is always less than any number to its right.

numberline

Houghton Mifflin Math Grade 5