Mathematical properties are often used to simplify computation. Below are three addition properties stated in words, shown with a numeric example, and shown with an algebraic example. The Zero Property of Addition is also called the Identity Property of Addition.

When numbers or variables are added, for example (2 + 3) + 4 = 2 + (3 + 4) and (a + b) + c = a + (b + c),the addends can be grouped in different ways without changing the result.

When numbers or variables are added, for example 2 + 3 = 3 + 2 and a + b = b + a,the order of the addends can be changed without changing the result.

When 0 is added to a number or variable, for example, 2 + 0 = 2 and a + 0 = a, the result is the same number or variable.

These properties can be used when adding numbers.

Example: 3,378 + 2,983 = (3,000 + 300 + 70 + 8) + (2,000 + 900 + 80 + 3)
= (3,000 + 2,000) + (300 + 900) + (70 + 80) + (8 + 3)
= 5,000 + 1,200 + 150 + 11
= 5,000 + (1,000 + 200) + (100 + 50) + (10 + 1)
= (5,000 + 1,000) + (200 + 100) + (50 + 10) + 1
= 6,000 + 300 + 60 + 1
= 6,361

The method used above is long and cumbersome. The use of a base-ten positional number system for writing numbers allows simpler algorithms for arithmetic operations to be developed. An algorithm is an organized procedure for performing a given type of calculation.

In the addition and subtraction algorithms, digits are aligned according to place value, and the computation is completed from right to left.

 Add the ones Add the tens Add the hundreds Add the thousands 11 ones =1 ten + 1 one 16 tens =1 hundred + 6 tens 13 hundreds =1 thousand + 3 hundreds

Subtraction of Whole Numbers
Subtraction of whole numbers can also be completed using the addition properties and expanded notation, but this method is long and cumbersome. The subtraction algorithm provides a simple, compact method for completing such calculations. As in addition, digits are aligned according to place value, and the computation is completed from right to left.

 Rename. Subtract the tens Rename. Subtract the thousands 5 ten 0 ones =4 tens 10 onesSubtract the ones. 16 tens =1 hundred + 6 tens 3 thousands 1 hundred =2 thousands 11 hundredSubtract the hundreds.

Estimating Sums and Differences
When an exact answer is not necessary, an estimate can be used. The most common method of estimating sums and differences is to round each number to a specific place and then add or subtract the rounded numbers.

Examples:
 Estimate 4,894 + 2,429. 4,894 → 5,000 2,429 → +2,000 7,000

Round each number to the
nearest thousand.

 Estimate 6,209 − 383. 6,209 → 6,200 383 → −  400 5,800

Round each number to the
nearest hundred.
Subtract the rounded numbers.

Mental math can often be used to complete estimates. At this grade level, however, errors can be more easily identified if students write down their work when estimating answers.

As students gain experience and confidence estimating sums and differences, point out that estimates can often be used to check computations. Students should realize that if both addends are rounded up, the estimated sum will be greater than the actual sum, and if both addends are rounded down, the estimated sum will be less than the actual sum. Such generalizations are not possible with subtraction.

Expressions and Equations
An arithmetic expression consists of numbers, grouping symbols, and operation symbols. An algebraic expression is like an arithmetic expression, but contains at least one variable. A variable is a letter that represents a number.

When evaluating expressions at this grade level, students are told to first evaluate within parentheses and then complete the evaluation from left to right. An algebraic expression is evaluated by substituting a given value for the variable and then simplifying the resulting arithmetic expression.

The equality of two expressions gives an equation. If two expressions are not equal, an inequality is created. The inequality symbols used at this grade level are ≠ (is not equal to), > (is greater than), and < (is less than).

When one side of an equation contains the variable and the other side contains a number, the equation can be solved. To solve an equation means to find the value of the variable that will make the equation true. The equations in this chapter are simple enough to be solved by inspection, by using related number sentences, or by using a guess-and-check strategy. Sometimes an equation may have two variables. In this case, a function table is used to find pairs of numbers that make the equation true. Students will learn more about evaluating expressions and solving equations in future chapters.

Teaching Model 3.3: Estimate Sums and Differences