Part-Whole Thinking & Number Bonds - First Grade Mathematics

Every number is not just a single lonely quantity; it is a whole made up of smaller parts! For example, the number 6 can be broken into 5 and 1, or 4 and 2, or 3 and 3. In mathematics, seeing how numbers split into parts and come back together into wholes is called part-whole thinking. To draw and organize this relationship, mathematicians use a wonderful visual tool called a number bond. In this chapter, you will master number bonds, discover how parts connect to wholes, and learn how one number bond can generate multiple math facts.

The Anatomy of a Number Bond

A number bond uses circles and connecting lines (branches) to show the relationship between a whole and its parts.

The Number Bond Diagram:

             [ WHOLE ]
              /                  /                [ PART ]   [ PART ]

Example with 7:
               ( 7 )   <- Whole
              /                 ( 4 )   ( 3 ) <- Parts

The Three Circles

A standard number bond has one circle for the Whole (the total amount) and two circles branching out for the Parts.

Direction Does Not Matter!

A number bond can face any direction: the whole can be on top, on the bottom, on the left, or on the right. You can always spot the whole because both connecting lines lead directly into it! The parts only have one line connecting them to the whole.

Rotated Number Bonds:
  Whole on Bottom:       Whole on Left:
    ( 2 )   ( 3 )            ( 3 )
        \   /               /
        ( 5 )          ( 8 )
                                                         ( 5 )

Decomposing Numbers with Number Bonds

Let us explore all the different ways you can decompose the number 5 using number bonds.

All the Ways to Make 5:
     (5)           (5)           (5)           (5)           (5)           (5)
    /   \         /   \         /   \         /   \         /   \         /     (0)   (5)     (1)   (4)     (2)   (3)     (3)   (2)     (4)   (1)     (5)   (0)

Number Partners

Each pair of parts that adds up to the whole is called a pair of number partners. For the whole 5, the number partners are: 0 and 5, 1 and 4, 2 and 3.

Turning a Number Bond into Addition Sentences

Every single number bond tells a story that you can write down as addition equations.

From Number Bond to Equations:
        ( 8 )
       /          ( 6 )   ( 2 )

Equation 1 (Part + Part = Whole):  6 + 2 = 8
Equation 2 (Switch the Parts!):    2 + 6 = 8
Equation 3 (Whole = Part + Part):  8 = 6 + 2
Equation 4 (Whole = Part + Part):  8 = 2 + 6

Switching the Parts

Notice that 6 + 2 = 8 and 2 + 6 = 8 use the exact same parts to make the exact same whole. Changing the order of the parts does not change the whole!

Practice Exercises

Exercise 1: Draw or describe a number bond where the whole is 9, one part is 6, and the other part is 3.

Exercise 2: Look at a number bond with a whole of 10. If one part is 7, what must the missing part be?

Exercise 3: Write two different addition equations that match a number bond with parts 4 and 1 and whole 5.

Exercise 4: List all the possible pairs of whole-number parts that make the number 4.

Solutions and Explanations

Solution to Exercise 1: In the number bond, the whole circle contains 9. Two branches connect from 9 to the two part circles: one part circle contains 6, and the other part circle contains 3.

Solution to Exercise 2: The missing part is 3. Since 7 + 3 = 10, the part that joins with 7 to make 10 is 3.

Solution to Exercise 3: Two addition equations matching this number bond are 4 + 1 = 5 and 1 + 4 = 5 (you could also write 5 = 4 + 1 or 5 = 1 + 4).

Solution to Exercise 4: The pairs of parts that make 4 are: 0 and 4, 1 and 3, and 2 and 2 (as well as 3 and 1, and 4 and 0).