The Importance of Number Bonds

The Importance of Number Bonds

Number bonds help students see that numbers can be "broken" into pieces to make computation easier (decomposing/composing). With number bonds, students recognize the relationships between numbers through a written model that shows how the numbers are related. A number bond helps student clearly visualize the Part/Whole relationship. 

First Grade

A number bond for the numbers 2, 3, and 5 might look like the model below. Students will also see the whole on top with the parts branching down. The circles or squares are just a visual representation that students should begin with.  In first grade, some students may move away from the shape visual and opt for the upside down V.

part n         s

Number bonds are similar to fact families.  However, there are a couple of differences. First, fact families are often taught through rote memorization.  Students can quickly tell you that 2 + 3 = 5, 3 + 2 = 5, 5 - 2 = 3, and 5 - 3 = 2, but they often don't really understand how the addition and subtraction sentences are related.


Second, we don't usually focus on all the fact families for a given number, for example, 5. Through working with number bonds, children learn that 2 and 3 make 5, but so do 4 and 1.  In this way, students experience multiple ways to decompose the same number. 

Students start using number bonds with smaller numbers and gradually work toward larger ones. In addition, they need LOTS of concrete practice.  The part/part/whole mat pictured below is a great tool.


The example below shows one way a first grader might determine the sum of 8 and 5. The student knows he wants 2 more to put with the 8 to make 10, and he knows he can decompose 5 into 2 and 3.  So now he is thinking of 10 + 3 = 13. 

 ddd ddddd aa

Second Grade

In second grade, students are expected to "add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction". Once students can make a “ten”, they can easily make “the next ten,” or 60, as shown below with the problem 59 + 36 = 60 + 35. This empowers students to complete a unit of a hundred.  A second grader can see that just as 9 or 59 is close to the “ten” or “next ten”, 390 is only 10 away from the next hundred, in this case, 400 (shown below).

Third Grade

In third grade, students use number bonds to help them decompose numbers using the distributive property. The decomposition of 6 apples as 1 yellow apple and 5 green apples in kindergarten and first grade sets the stage for the distributive property in grade 3 when the unit changes from ones to nines: 6 nines = 1 nine + 5 nines. 

Why do we bother teaching students to decompose multiplication facts using the number bond model? It is important for students to use what they know, rather than learning “tricks”, to help them solve more challenging problems. Multiplying by nines can be difficult for students, but multiplying by fives is more manageable. Six nines can be broken into two smaller parts, 1 nine and 5 nines.

Third-grade students also begin to work with unit fractions, e.g., 1/2, 1/3, 1/4, 1/5, etc. Using number bonds, they can see that any fraction can be decomposed into smaller unit fractions. Below, you can see how students decompose 5/5 into 5 units of 1/5. Then, students can break apart 6 fifths, such that 6/5 = 5/5 + 1/5.

Fourth Grade

In fourth grade, students learn that they can manipulate fractions to complete simple arithmetic, adding and subtracting with like units.  Just as the first graders can complete “1 ten,” a fourth-grade student can complete “1 one.” 

Fourth-grade students understand that 8 ninths is close to 1 one (9 ninths).  Six ninths can be decomposed such that 6/9 = 1/9 + 5/9. Then, 8/9 and 1/9 can be composed to make 1 one. The remaining parts are combined to make 3 5/9 as shown in the example below.

Fourth-grade students can also use their understanding of decomposing a unit (i.e., taking from a ten or a hundred) to take from the one.

Fifth Grade                      

In fifth grade, the new complexity arises when students face units that are not equivalent. Now, students must create equivalent units in order to successfully add or subtract. As illustrated in the problem below, students find a like unit, which is a multiple of both denominators. In this problem, the largest like unit is ninths. Students multiply both the numerator and the denominator by 3 to rename 2/3 as a number of ninths.

Fifth-grade students also use the number bond to decompose fractions to take from the one. In the problem below, the student decomposes 5 2/3 to subtract 8/9 from 1 one, or 9/9. Now, the student solves a simpler addition problem, 1/9 + 4 2/3, changing thirds for smaller equal units, e.g., ninths, that can then be added.

In conclusion, the number bond is a powerful model that supports number sense and our students' ability to think flexibly, strategically, and creatively.