Common Core @ Home: Math - What the heck is a number bond?



Part One of a continuing series


Dear Jeff, What the heck is a number bond? 

If you're like me, you learned arithmetic shortly after the number “1” was invented — at least, it feels that way, especially when you look over your child's math homework. 

And while you might not be able to add 375 + 298 in your head, you probably feel that you could do 8 + 7 without reaching for the calculator. 

But then your child brings home a problem like the following, from a 2nd grade worksheet:

Solve 8 + 7 by recording make 10 solutions with number bonds.

You might wonder if they just discovered that 1 + 1 = 3, and you missed the podcast. 

Fear not, for 1 + 1 is still 2, and the new math is still the old math. 

The important change is that in addition to being asked questions like “What is 8 + 7?” new questions are being asked that rely on a conceptual understanding of mathematics, to prepare students for more advanced work. 


Thus, in addition to being asked to find a numerical value, your child (and indirectly, you) might be asked to elaborate on your answer: How did you obtain it? Why did you do it that way?

Conceptually, addition involves joining two collections together to make one collection. For example, if you have 5 apples in one box, and 3 apples in another, then when you put the apples together in the same box you have 8 apples. We can show this relationship with a number bond:

 

One important observation is that if you rotate the screen, so that “8” is on the left, it's still the same number bond: orientation doesn't matter. What this means is that any number bond can be read in at least two ways. First, if we have 5 and 3, we can join (add) them to form 8. But (turning the screen upside down), if we have 8, we can break it apart into 5 and 3. 

One particularly useful set of number bonds are those that add to 10. Thus, since 7 + 3 = 10, we have this number bond:

      

Now we're ready to add 8 + 7 by making 10 with number bonds. Our work will look like this:



Let's go through the steps one at a time. First, since one of the addends is 7, we can complete 7 to make 10. A familiarity with the number bonds of 10 should cause us to remember 10 = 7 + 3.  What this means is that if we could find a 3 somewhere, we could complete the 7 to 10.  But the number bonds of 8 tell us that there is a 3 embedded in the 8. This means we can add this way: First, break 8 into 5 and 3. Next, combine the 7 and the 3 to make 10; then combine 10 with the leftover 5 to make 15, our total.


There's no reason we had to complete 7 to make 10.  We could just as easily completed 8 to make 10. In this case, we'd show our work as: 



One final note: You might wonder why students don’t just memorize 8 + 7 = 15. There are many reasons, derived from thousands of pages of research on this question, but here’s one for now. If students must memorize 8 + 7 = 15 before they can do this addition, this means that teachers face a choice: waiting until all their students to have memorized this fact before proceeding to more advanced topics; or forging ahead, letting the students who haven’t memorized the fact fall behind. Neither option is palatable. Instead, if students can develop their own ways of finding 8 + 7 = 15, then all students can reach higher levels of mathematics. And, just as you eventually remember how to drive to work without a map, students will eventually remember 8 + 7 = 15.

Jeff Suzuki teaches mathematics at Brooklyn College, and is one of the founders of the Mid-Hudson Valley Math Teachers Circle, a group of teachers, professors, and mathematics aficionados working to promote mathematics education in the Hudson Valley.