[Rescued from my old blog.]
A 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.
This post is an excerpt from my book Counting & Number Bonds: Math Games for Early Learners, available now at your favorite online book dealer.