Number Bonds = Better Understanding

[Rescued from my old blog.]

number bondsA number bond is a mental picture of the relationship between a number and the parts that combine to make it. The concept of number bonds is very basic, an important foundation for understanding how numbers work. A whole thing is made up of parts. If you know the parts, you can put them together (add) to find the whole. If you know the whole and one of the parts, you take away the part you know (subtract) to find the other part.

Number bonds let children see the inverse relationship between addition and subtraction. Subtraction is not a totally different thing from addition; they are mirror images. To subtract means to figure out how much more you would have to add to get the whole thing.

A Picture Is Worth More than Many Words

You can draw number bonds on paper using circles or bar diagrams. Imagine each circle to be a pile of blocks or other manipulatives, and think of the bar as the blocks lined up in a row. Even a young student who does not understand math notation can clearly see the connection between these numbers: the whole (6) has been pulled apart into two piles (4 and 2), and the piles can be pushed back together to make the whole.

Math textbooks often try to communicate the same concept using four-fact families. A four-fact family looks like this:

4 + 2 = 6
2 + 4 = 6
6 – 4 = 2
6 – 2 = 4

The idea of the four-fact family is to help students realize that once they know one of the facts in the family, they know all of them. Many students never see the connection, however, and think of these equations as separate little bits of abstract information, all of which have to be memorized. This can overload their minds and make them give up on math. On the other hand, number bonds connect to the student’s understanding at a deeper level, showing all four of the fact family relationships in a single picture.

How to Teach Number Bonds

When you start teaching number bonds, use M&Ms or popsicle sticks or whatever you have on hand to make physical piles that can be pulled apart and pushed back together, then pulled apart in another way. When I introduced number bonds to my youngest daughter, I set out six blocks on the bedspread. (We often do school work sprawled on the bed.)

“How many blocks do we have here?” I asked.

Kitten counted carefully. “1, 2, 3, 4, 5, 6.”

“I’m going to move these blocks over to make a new pile,” I said. I pushed two of the blocks to one side. “How many blocks did I move?”

She did not have to count those. “Two.”

“And how many are left?”

“1, 2, 3, 4.”

“And if I push them back together…” I did so. “Two and four are how many in all?”

“1, 2, 3, 4, 5, 6.”

“Two and four are six. You are so good at counting! Now, this time I’m going to move three blocks over here.” I moved three blocks to the side. “How many are left?”

Again, she did not need to count. “Three.”

I pushed the blocks back together. “And three and three are how many?”

I could see her lips move as she counted silently. She looked up and smiled. “Six.”

“That’s right. Three and three are six. What will happen if I move just one block over?” I did so. “How many are left?”

Kitten started to speak, but then she stopped with a puzzled look on her face. She bent over to hide the blocks with her hand so I could not see her count. “1, 2, 3, 4, 5.”

“Oh, you’re tricky!” I said. “Five blocks. And if I put them back together, one and five are how many?”

I pushed the blocks together again. She looked at the pile. “Six.”

“One and five are six. Now you try it. What piles will you make?”

Kitten pushed all six blocks to the side. She looked up guiltily, not sure that was allowed.

“Okay,” I said. “You moved six blocks. How many are left?”

She looked at the empty spot. “Zero.”

“That’s right. Zero blocks. And if you put the piles back together, six and zero make how many?”

She answered triumphantly, “Six!”

We played this game with blocks many times, using different numbers in the original pile, throughout Kitten’s kindergarten year. The blocks give her a concrete, hands-on way to check herself, building confidence in her understanding of how numbers work. My goal was not for her to memorize specific math facts, but that she understand and be able to use the concept of taking a number apart and then putting it back together.

When teaching number bonds to an older student, I may still start with blocks, but after one or two piles of demonstration, we move quickly to the number bond pictures and games. I usually start older students with the bonds for ten:

1 + 9
2 + 8
3 + 7
4 + 6
5 + 5

Ten is the most important number in our decimal (base ten) number system, so it is vital that our children learn to recognize it in any disguise. When the student knows the bonds for ten automatically, we move on to 20, then 100, then any number we desire.

Don’t “Drill.” Play Games!

Here are a few number bond games to enjoy with your children:

Throw two dice and tell how many more you would need to make 10.
(On the rare throws of 11 or 12, the answer is a negative number.)

Throw 3 dice and tell how many more it takes to make 20.

One player names any number 0-100, and the other tells how many more it takes to make 100.

You could also play the last game with math cards [take out the jokers and face cards, leaving just ace (1) through 10], turning up one for the tens place and one for the ones, to make a two digit number.

10’s Concentration — Turn all the math cards face down on the table. On her turn, each player turns up two cards. If they add up to 10, she gets to keep them and try again. If one of the cards is a 10, she gets to keep it and turn up another card. Whoever takes the most cards, wins.

CountingGames-300This post is an excerpt from my book Counting & Number Bonds: Math Games for Early Learners, available now at your favorite online book dealer.

25 thoughts on “Number Bonds = Better Understanding

  1. I like games. These are neat. But it’s really ok to do some drill, as well. It doesn’t have to be either/or.

    I give my niece and nephew the answer, and ask for a problem to go along with it. They seem willing to play that game. Adaptable by choice of number and operation and number of factors or addends required.


  2. Yes, of course some drill is okay. Necessary, even, since any math practice page could be considered drill—even if what one needs to practice is factoring trinomials. One of my faults as a writer is that I tend to overstate what I mean.

    I like your game. It reminds me of the Children’s Mathematics Calendar that Theoni Pappas used to publish. My math club had fun with that.

  3. Number bonds is a good way to teach number relation. Another key thing is the learning of basic maths operations, the Addition and Subtraction. Also if the numbers are colored, the kids will like them more.

  4. Hi there,

    Just found your blog, I was trying to find some helpful ways to teach my daughter number bonds as we’ve just reached it in her workbook. Your post has been really helpful, and I’m gonna try with the blocks or something similar tomorrow, I think it will help her alot more to understand the concept, as she has struggled a little bit with it. Thankyou.

  5. Hi! I first encountered number bonds through my son’s textbook. He’s in K2 now. I wasn’t quite sure if this concept really works, but now that I’ve read your blog, I’m convinced! Hope to read more articles about how we can teach our kids Math in a fun and exciting way.

  6. Just found your site. Can’t wait to try some of these ideas out on my young ones I watch in my family child care home. May make it easier when they enter school. Also help after school students with homework, one struggles with math, I feel these suggestions will help me to teach math in a fun way, not just drilling and math sheets. Some kids need to learn math in a fun environment first. I never was good in Math, this is what I need to get the kids excited about numbers/math facts

  7. Well, you posted this a long time ago, but I see others are still replying. I wanted to mention a picture book I love that is great for the number pairs that add to 7. It’s called Quack and Count. I wish there were more books as fun as this one for 10 and5 and 6 and 12…

  8. P.S. I hadn’t noticed I was logged in as mathmomma. This is Sue VanHattum, and I’ve started a blog at

  9. Hi, Tisha!

    If you want to read more about number bonds, you may enjoy the other blog posts I listed at the end of this article. Also, I’ve written a recent post that explains a little more about how we use the concept as the kids get older:
    Mental Math: Addition

    I like to use the Singapore Primary Math program with my homeschooled children. The books focus on teaching number bonds in the early grade levels, and I find that the program gives my kids a great foundation of mathematical understanding. I have used the 3rd Edition and U.S. Edition versions, and both are good; the new CA Standards version is probably fine, too, but I have no experience with it.

  10. As a one-room teacher with students in grades 1-8, I encounter many levels of learning. I have found that those students who start with number bonds advance in math faster than those who depend on finger-counting or rote memorization. Sometimes a student will join my school in the later grades. They are already dependent on counting on their fingers and are resistent to change. I demonstrate the number bonds using blocks, items, pictures, etc., but could use more games that are suitable for older students. Do you have any ideas?

  11. My older students still enjoy the Concentration game. I’ve found that mental math work with big numbers helps build (or reinforce) skill with number bonds, especially when we talk about strategies.

    With my kids, when they are causing trouble by poking each other or fidgeting too much, I will tell them to sit on their hands. You might try that, if your students wouldn’t be offended, as a way to enforce the mental challenge of the math.

    I’ll think a bit and see if I can come up with other ideas…

  12. I had no idea Math could be so fun! I would have had a blast learning Math using these ideas! Thank you so much for providing these resources. You will never know what a blessing this is to our family.

  13. Pamela
    This has been posted so long ago,but I have just found your site and I am truly greatful for the simple method you have used. I can now use same to help my second grade child who attends my family Day Care.Thank you so much.This is fun and the way it should be, so more students would gain confidence in themselves,as math is made simple.

  14. Its the best phrase that there is to say, since its easy to understand stuff from pictures than just simple text. I use creately to create infographics and other diagram for kids to teach math.

  15. I do not know of a special Chinese name. Sometimes number bonds might be called “Partitions.” But normally, we think of number bonds as being a pair of numbers that add up to the sum. Partitions can be any set of whole numbers that add up to the sum. So you can partition 10 into 1+2+3+4, but we wouldn’t call that a number bond.

    1. Good question! I don’t know what family numbers are, so I can’t answer it.

      I’ve heard of fact families, as I mentioned in the blog. Those are the relationships between simple numbers (often memorized as “math facts”) that are connected, like:
      3 + 4 = 7
      4 = 3 = 7
      7 − 4 = 3
      7 − 3 = 4
      A number bond picture showing that three and four together make seven represents all four of these equations at once, in an easy-to-understand visual form.

Leave a Reply to Celeste Cancel reply

Fill in your details below or click an icon to log in: Logo

You are commenting using your account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s

This site uses Akismet to reduce spam. Learn how your comment data is processed.