Completing Subtraction Images and Statements

A subtraction statement has three parts: the starting amount (minuend), the amount taken away (subtrahend), and what is left (difference). Sometimes a problem shows a picture or an equation with one of these three parts missing, and the job is to figure out what number belongs there. This skill, completing subtraction images and statements, is really about understanding what subtraction means well enough to work backward or forward to find any missing piece.

A complete subtraction statement looks like \( 9 - 4 = 5 \). In practice problems, one of these three numbers is often replaced with a blank or a question mark. There are three types of missing number problems:

• Missing difference: \( 9 - 4 = \, ? \)
• Missing subtrahend: \( 9 - \, ? \, = 5 \)
• Missing minuend: \( ? \, - 4 = 5 \)

Each type is solved a slightly different way, even though the picture usually looks similar: a group of objects with some crossed out or removed.

A subtraction image usually shows a full group of objects, with some of them crossed out to show what was taken away. The objects that are not crossed out show what is left. Look at the picture below.

Here, the minuend (9) and subtrahend (4) are shown, and the picture reveals the missing difference by counting the objects that are left: 5.

Sometimes the picture shows the start and the end, and the missing part is how many were removed. For example, a picture might show 9 objects total, with 5 left uncrossed. The subtraction statement is \( 9 - \, ? \, = 5 \). To find the missing subtrahend, count how many objects would need to be crossed out to go from 9 down to 5, which is 4. You can also check this using addition, since \( 5 + 4 = 9 \).

A missing minuend problem gives the number of objects crossed out and the number left, but not the starting total. For a statement like \( ? \, - 4 = 5 \), think about how many objects there must have been altogether if 4 were removed and 5 remain. Adding the removed amount and the remaining amount gives the missing minuend: \( 4 + 5 = 9 \).

There are a few reliable ways to find a missing number in a subtraction statement, whether or not a picture is included:

• Use the addition connection: every subtraction statement has a matching addition fact. If \( 9 - 4 = 5 \), then \( 5 + 4 = 9 \). Rewriting a subtraction statement as addition often makes the missing number easier to see.
• Count up or count back: starting from a known number, count forward or backward to reach the other known number, keeping track of how many steps it takes. This works well when using a number line to subtract.
• Break numbers apart with place value: for larger numbers, thinking in terms of tens and ones can make it easier to see what is missing, similar to subtracting with place value models.
• Count objects directly: when a picture is given, simply count the objects in each group, an approach practiced in counting to subtract.

Once a missing number is found, it helps to check the picture and the statement against each other. Count the total objects, count the crossed out objects, and count the remaining objects, then make sure all three numbers fit the pattern minuend minus subtrahend equals difference. If any number does not match the picture, recheck the counting before writing a final answer.

A picture shows 12 objects. 7 of them are crossed out. Complete the statement \( 12 - 7 = \, ? \).

Count the objects that are not crossed out: there are 5 remaining. So the missing difference is 5, and the completed statement is \( 12 - 7 = 5 \). As a check, add the subtrahend and the difference: \( 7 + 5 = 12 \), which matches the minuend shown in the picture.

Related lessons

• Math 2