What Is a Number Bond?

Instructor: Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.

A number bond is an extremely useful tool in teaching and understanding numbers and basic arithmetic. In this lesson, we will explore number bonds and their uses through definition and example.

Numbers as a Whole Made Up of Parts

Let me ask you a question. What would I have to add to 5 in order to get 11? There are a couple of ways to figure this out. We can subtract 5 from 11 to get 6 (11 - 5 = 6), or we can count how much we have to add to 5 in order to get 11.

5 + 1 +1 + 1 + 1 + 1 + 1 = 11

We see we have to add 1 to 5 six times to get to 11, so we have to add 6 to 5 to get 11.

Notice that this problem illustrates the fact that 11 is a whole number with parts that make it up. That is, numbers can be considered to be wholes made up of parts, where the parts are what we need to add together to get the whole. Let's consider the different pairs, or parts, that make up the whole, 11.

  • 1 + 10 = 11
  • 2 + 9 = 11
  • 3 + 8 = 11
  • 4 + 7 = 11
  • 5 + 6 = 11

After this, the pairs repeat themselves in reverse order.

  • 6 + 5 = 11
  • 7 + 4 = 11
  • 8 + 3 = 11
  • 9 + 2 = 11
  • 10 + 1 = 11

All numbers can be considered this way; as a whole made up of parts.

Number Bonds

In the example above, we looked at the different pairs, or parts, that make 11 when we add them together. The parts, or pairs, in this example, are called number bonds. Number bonds are the relationship between a number and the parts that make it. They are the different pairs of numbers that add up to the same number. For example, if we add together 4 and 5, we get 9. Thus, the pair 4 and 5 are number bonds that make up the number 9. This is often illustrated using a circle and line diagram as shown in the image.


Number Bond Diagrams
bonds1


We can also illustrate number bonds horizontally as follows.

  • 4 + 5 = 9
  • 10 + 8 = 18
  • 22 + 7 = 29
  • 73 + 27 = 100

Understanding number bonds allows us to better understand how numbers work and how they are put together.

Number Bonds to Teach Addition and Subtraction

Another thing that number bonds are extremely useful for is familiarizing oneself with addition and subtraction. We can use number bonds to teach basic mathematical facts. Notice that when we have a number bond a and b that add together to give the number c, we have the following information.

a + b = c

b + a = c

c - a = b

c - b = a

That is, when we add the pair a and b, we get c, so it follows that when we subtract a from c, we get b, and when we subtract b from c, we get a. We can use this fact to learn or teach basic addition and subtraction facts.

For instance, consider our initial question that asked what we need to add to 5 in order to get 11. This is essentially asking what number bond makes up 11 if one of the numbers in the bond is 5.


Example
bonds2


As we said, to figure this out, we can find this number by figuring out what to add to 5 to get 11, or what we have to subtract from 11 to get 5.

5 + ___ = 11

11 - ___ = 5

We see that 6 fills in both of these blanks, so 5 and 6 make up a number bond to create 11.

Examples

Let's consider a couple examples.

1.) Fill in the blank.

39 + ___ = 100

Solution: This is asking for the number bond that makes up 100 when one of the numbers in the pair is 39.


Example
bonds3


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