What Are Number Bonds?

A number bond is a picture of a number and two parts that make it: a circle for the whole, joined by lines to a circle for each part, such as 10 split into 7 and 3. Part-part-whole is the idea the picture shows, that the two parts together make the whole. One bond holds an addition fact and a subtraction fact at once: 7 + 3 = 10 and 10 − 3 = 7. That is why schools draw the bond before asking children to memorize facts.

The most useful thing to do at home is to cover one part and ask how many are hiding, because finding a missing part is the start of subtraction. Kindergartners split numbers up to 10 into pairs, and first graders use those splits to add and subtract within 20. A free blank number bond template and number bond worksheets with answer keys are at the end of this guide.

What is a number bond?

A number bond is a diagram with three circles: one for a whole number and one for each of its two parts, joined by lines from the whole to the parts. The whole can sit on top, at the bottom, or at the side.

Part-part-whole means that two parts together make a whole: 7 and 3 make 10, and 10 can be split back into 7 and 3. Every number bond holds two addition facts and two subtraction facts.

Number bond	Whole	Parts	The facts it holds
5 is 2 and 3	5	2 and 3	2 + 3 = 5, 3 + 2 = 5, 5 − 2 = 3, 5 − 3 = 2
10 is 7 and 3	10	7 and 3	7 + 3 = 10, 3 + 7 = 10, 10 − 3 = 7, 10 − 7 = 3
13 is 10 and 3	13	10 and 3	10 + 3 = 13, 3 + 10 = 13, 13 − 3 = 10, 13 − 10 = 3

The words mean something different in British schools. England’s national curriculum uses “number bonds” for the facts themselves, so “number bonds to 20” there means every pair that makes 20. American worksheets mean the picture: Eureka Math lists “number bond (mathematical model)” among the terms of its kindergarten Module 4. Both describe the same parts and wholes. The Common Core standards do not use the term at all. Kindergarten standard K.OA.A.3 asks children to “decompose numbers less than or equal to 10 into pairs in more than one way”, and the Common Core’s progressions document calls the two parts “partners”.

A number bond is one drawing of part-part-whole, not the only one. A part-part-whole mat, a large box for the whole above two boxes for the parts, shows the same idea, and a tape diagram draws it as bars, which suits word problems and bigger numbers.

Number bonds vs. traditional addition: why schools teach them

The difference between number bonds and traditional addition shows in a single problem, 8 + 5.

• The way most parents learned it: remember that 8 + 5 = 13, or count on 5 from 8: 9, 10, 11, 12, 13.
• With a number bond: the 8 needs 2 more to make 10, so split the 5 into 2 and 3. Then 8 and 2 make 10, and 10 and 3 make 13.

The second way is the strategy the Common Core calls making ten. Its own example in first-grade standard 1.OA.C.6 is 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14. Texas standard 1.3(D) names “making 10 and decomposing a number leading to a 10”. Subtraction works the same way. For 13 − 5, split the 5 into 3 and 2: 13 take away 3 is 10, and 10 take away 2 is 8. On a number line, the split for 8 + 5 is a jump of 2 to 10 and then 3 more.

Schools teach the bond for two reasons. A child who knows how 10 splits can work out a forgotten fact instead of counting from 1. And the bond shows addition and subtraction as one relationship: standard 1.OA.B.4 treats subtraction as “an unknown-addend problem”, which on a bond is the hidden part.

Swedish researchers have tested the approach. Angelika Kullberg and colleagues taught four first-grade classes to add and subtract through part-whole relations for eight months, with making ten at the center. The 86 children they interviewed improved more than 35 children in comparison classes. A year later, they relied more on part-whole reasoning with larger numbers, while the comparison children more often counted one by one.

Number bonds are the route to memorized facts, not a replacement for them. The same standards that teach the bond ask that by the end of second grade children “know from memory all sums of two one-digit numbers” (2.OA.B.2). Texas asks second graders to “recall basic facts to add and subtract within 20 with automaticity” (2.4(A), TEKS grade 2). Let the bond work out a fact, then practice that fact until it is quick. Why school math looks different in general is the subject of new math explained for parents.

What grade do kids learn number bonds?

Children begin putting numbers together and taking them apart with objects in pre-K, split numbers up to 10 into pairs in kindergarten, and use those splits to add and subtract within 20 in first grade.

Grade	What children do	Common Core	Texas (TEKS)
Pre-K	Begin to see that numbers to 10 can be put together and taken apart, with objects	None	PK4.V.A.7
Kindergarten	Split numbers to 10 into pairs in more than one way; find the partner that makes 10; add and subtract within 5 fluently	K.OA.A.3, K.OA.A.4, K.OA.A.5	K.2(I)
1st grade	Use the splits to add and subtract within 20, including making ten; solve subtraction as a missing part	1.OA.B.4, 1.OA.C.6	1.3(C), 1.3(D)
2nd grade	Know all sums of two one-digit numbers from memory	2.OA.B.2	2.4(A)

Texas asks the same things in its own words. Its kindergarten standard K.2(I) is to “compose and decompose numbers up to 10 with objects and pictures” (TEKS kindergarten math). First grade adds 1.3(C), to “compose 10 with two or more addends” (TEKS grade 1). The Texas Prekindergarten Guidelines start a year earlier: a 4-year-old “begins to understand” that the numbers from 0 to 10 can be put together and taken apart in more than one way.

A 4-year-old who shares 5 crackers between two plates is doing the pre-K version, and a preschooler who has never seen the drawn bond is on time. Eureka Math brings in the diagram in kindergarten Module 4 and calls the number pairs of 6–10 “foundational to Grade 1’s fluency”. Breaking numbers apart is a number sense skill that comes after counting and comparing.

How to teach number bonds at home

Teach number bonds at home in the order schools do: real objects first, then the drawn bond, then numbers alone. The What Works Clearinghouse practice guide Assisting Students Struggling with Mathematics (2021) is written for intervention teachers, and it rates using concrete objects and drawings before abstract numbers as having strong evidence. These five steps adapt that order for a kitchen table, 10 minutes at a time.

1. 1

   Split real things

   Put 5 crackers on the table and two plates beside them. Move some crackers onto each plate and say the bond: 5 is 3 and 2. Use the words whole and parts, so the 5 is the whole and the 3 and the 2 are the parts.

2. 2

   Find all the ways

   Move one cracker at a time from one plate to the other and say each bond as you go: 4 and 1, 3 and 2, 2 and 3, 1 and 4. Then do the same with 6, 7, and on up to 10.

3. 3

   Draw it

   Put the crackers or counters in the circles of a number bond, all of them in the whole circle, then moved into the two part circles. Then draw the same bond with numerals in place of the counters.

4. 4

   Hide a part

   Show 7 counters, cover some of them with a cup, and leave the rest in view. Ask how many are hiding. Your child is finding a missing part, which is the start of subtraction.

5. 5

   Say the sentences

   For each bond, say the addition and the subtraction it holds: 3 + 2 = 5 and 5 − 2 = 3. Once your child can say them, write them under the bond.

Step 1 assumes your child can count the crackers and say how many there are. Step 2 is what the Common Core progressions document calls finding “all of the partners”. The same document advises leaving out pairs with 0, such as 5 and 0, once a child knows they exist.

Everyday life supplies more bonds:

• Socks. Seven socks from the wash: 4 white and 3 striped make 7.
• The table. Six people at dinner: 2 adults and 4 children.
• Snacks. Eight grapes split between two bowls, then split a different way.

Fingers carry the bonds of 5 and 10 everywhere. A whole hand is 5, so 7 is a hand and 2 more. A rekenrek shows the same bonds on beads: 7 is 5 dark beads and 2 light ones.

Number bonds to 10 (and to 5 and 20)

The number bonds to 10 are the pairs of whole numbers that add up to 10: 1 and 9, 2 and 8, 3 and 7, 4 and 6, 5 and 5, the same pairs the other way around, and 0 and 10. Kindergarten standard K.OA.A.4 asks children to find, for any number from 1 to 9, the number that makes 10 with it. The bonds of 10 get practice of their own because first graders add past 10 by making ten first, as in 8 + 5 = 8 + 2 + 3.

The bonds of 5 come first, because they fit on one hand: 1 and 4, 2 and 3, and the same two pairs turned around. Kindergartners are expected to add and subtract within 5 fluently (K.OA.A.5), and every one of those facts sits inside a bond of 5 or less.

“Number bonds to 20” means two different things. In British sources it is every pair that makes 20, such as 13 and 7. On American worksheets it is a teen number split into 10 and some more, such as 13 as 10 and 3, together with make-ten pairs such as 8 and 5 making 13.

A ten frame, two rows of five boxes, shows the bonds of 10 at a glance: 7 counters leave 3 empty boxes, so 7 and 3 make 10.

Number bond games and activities

Number bond games need only a cup, a deck of cards, a handful of beans, or your fingers, and each one ends with your child saying the bond aloud.

• How many are hiding? Put out 6 counters, cover some with a cup while your child looks away, and ask how many are under it. Then your child hides them for you.
• Make ten Go Fish. Use the cards from ace (1) to 9. Each player asks for the card that makes 10 with one in their hand (“Do you have a 3?” when holding a 7) and lays down the pair, saying “7 and 3 make 10.”
• Fingers behind the back. Say “I have 5 fingers,” show some, and hide the rest behind your back. Your child says how many are hidden, then 10 with two hands.
• Shake and spill. Mark one side of 5 dried beans with a marker, shake them in a cup, and spill them. Your child reads the bond: “5 is 3 marked and 2 plain.”
• Domino bonds. Each domino is a bond, with its two halves as the parts. Your child names the parts and the whole: “4 and 2 make 6.”

Rolling two dice is a number bond every turn.

Common number bond mistakes

Each of these five number bond mistakes is something you can see your child do, and each one has a fix you can make on the spot.

• Thinking the whole is always the top circle. With the whole at the bottom, the child adds the wrong two numbers. Draw some bonds with the whole at the bottom or the side.
• Adding the whole and a part. Given a whole of 10 and a part of 3, the child writes 13 in the empty part. Point to the whole and ask which number in the bond is the biggest: the whole is never smaller than a part.
• Counting from 1 every time. To find what goes with 6 to make 10, the child counts every counter from 1. Start from a bond the child knows, 5 and 5, and move one counter across: 6 and 4.
• Treating 3 and 2 and 2 and 3 as different bonds. They are one bond, because the order of the parts does not change the whole; first-grade standard 1.OA.B.3 calls it a property of addition. Swap the two plates and ask whether the crackers changed.
• Seeing the bond as addition only. A child who only fills in the whole never meets a missing part. Cover a part in about half of the bonds you practice.

When number bonds are hard, go back to objects and to the bonds of 5 before 10, and keep the counters on the mat for longer before moving to numerals. If splitting numbers up to 5 is still hard at the end of kindergarten, talk to your child’s teacher.

Free number bond worksheets and template

Our free number bond worksheets come as three PDFs, in the order a child uses them. The Blank Number Bond and Part-Part-Whole Mat is five pages: a large bond with circles that hold 10 counters, a part-part-whole mat the same size, six blank bonds with the whole on top, six with the whole at the bottom or the side, and a page of instructions for grown-ups. Use it with beans in pre-K and kindergarten, and with written numbers in first grade.

Ways to Make a Number is step 2 on paper: one whole per half page, from 5 to 10, with an empty bond for every pair, and a key listing the pairs. Number Bonds is the practice sheet from late kindergarten to second grade, with a page each of bonds of 5, of 6–9, of 10, and within 20, then an answer key. The missing number is sometimes a part and sometimes the whole, and the whole moves around the bond.

The bonds are not numbered, and the key matches each answer to its bond by position. Every sheet prints in black and white, with no account and no email.

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Free printable number bond worksheets and template

Three free PDFs in black and white, with no account and no email, in teaching order: the blank number bond and part-part-whole mats, then ways to make 5 to 10, then number bond practice from bonds of 5 to bonds within 20. Start with the one your child can already do. Ways to Make a Number and Number Bonds each have an answer key, and every PDF ends with a page of instructions for grown-ups on why, what for, and how to use it.

1. 1Blank Number Bond and Part-Part-Whole Mat (PDF)14 number bonds · 5 pagesDownload
2. 2Ways to Make a Number (PDF)21 number bonds · answer key included · 5 pagesDownload
3. 3Number Bonds (PDF)36 number bonds · answer key included · 6 pagesDownload