Kindergarten · K.OA.A.3 · K.OA.A.4

Parts, wholes and number bonds

Some things go into two groups. What can the two groups be, and if you can see one of them, how do you know the other without counting?

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Addition as putting togetherKindergarten

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Fact familiesGrade 1 Making ten and bridging through tenGrade 1

Where you are 12 in Addition and subtraction

Number9

Arithmetic5

Addition and subtraction12

▴ 1 above Addition as putting together Finding the missing number in an equation

Parts, wholes and number bonds

Subtraction as an unknown addend Subtraction as taking away ▾ Show all 12 (6 more)

When you reach for it

A child who can count can find how many counters are on the table. They can also count just the dark ones, and just the light ones. Those look like three separate jobs with three separate answers. The idea of parts and the whole is that they are one fact: the dark counters and the light counters together are all the counters, so the three numbers are tied to each other, and knowing any two of them decides the third.

That one picture is where adding and subtracting meet. Adding asks for the whole when both parts are known. Subtracting asks for a part when the whole and the other part are known. A child who sees both as questions about the same picture never has to learn subtraction as a new and harder thing, and Fact families and story problems all rest on it.

Number bonds are the pairs of parts that make one particular whole, known so well that a child who sees three counters knows, without counting, that two more make five. This lesson builds them at three sizes, each on the one before. The pairs that make 5 are 5 and 0, 4 and 1, and 3 and 2. The pairs that make 10 are 10 and 0, 9 and 1, 8 and 2, 7 and 3, 6 and 4, and 5 and 5. Each pair can also be turned around. The pairs that make 20 are hardly new at all: each one is a pair that makes ten with a full ten added to one part, so 3 and 7 becomes 13 and 7.

Ten matters more than any other whole in early arithmetic. It is two hands of fingers, one full ten frame, and later the size of one ten in every two-digit number. A child who knows that 7 goes with 3 to make 10 can find 7 + 5 by first making ten (Making ten and bridging through ten), can find 10 - 7 without counting back, and later sees 13 as ten and three. Five is where the habit starts, since one hand is five fingers and a five frame is half of a ten frame. Twenty, fingers and toes together or two ten frames side by side, is where it carries a child into adding and subtracting within twenty.

Read everything here aloud with the child, and let them build it with counters, cubes, buttons or fingers.

The procedure

There are three things to be able to do with a whole and its parts: see the whole and the parts in a picture, find every way to split a whole, and find a missing part when one part is known. The first is the same at every size. The other two are done below for five, then ten, then twenty and the numbers just below it.

Parts and the whole

A whole is an amount of things. Split it into two groups, without losing any and without adding any, and each group is a part. One group may even be empty (Example 5). Put the two parts back together and you have the whole again.

Figure 1The whole is 7. Its two parts are 4 and 3.

Here the whole is 7 counters. The dark counters are one part, 4, and the light counters are the other part, 3. A child can say it in words: “7 is 4 and 3.” With the signs from Addition as putting together it is

4 + 3 = 7

The same fact drawn as a number bond puts the whole in a circle at the top and the two parts in circles joined to it. The bond has no counters in it, only the three numbers and how they belong together, so it works for any amount of anything. Some books draw it sideways; the meaning is the same.

Figure 2The same fact as a number bond. The whole sits on top, the parts below.

To a child, the whole is “all of them” and a part is “some of them”. Those words, said while pointing at real things, do more than any sign.

Put seven counters in a line and lay a pencil between two of them. The counters on one side of the pencil are one part and the counters on the other side are the other part. Slide the pencil along and the parts change, 4 and 3, then 5 and 2, but the whole stays 7, because no counter came or went. Finding every place the pencil can go is the work of the bonds below.

Now hide one part under a cup. The child can still see the other part, and knows how many there were in all, so the hidden part is not a mystery: it is whatever the part they can see needs to make the whole. This hiding game is the heart of the idea. It is how a five-year-old meets subtraction before meeting the word.

Many children soon see small parts at a glance, “that is 2 and that is 4”, without counting one by one. Seeing a group as smaller groups is part of Subitizing and ten frames, and it is worth encouraging: it is the part and whole idea happening in the eye.

Four facts follow, and every bond in this lesson uses them.

Two parts make the whole.

4 + 3 = 7

Take one part away from the whole and the other part is left.

7 - 4 = 3 \qquad\qquad 7 - 3 = 4

The order of the parts does not change the whole. 4 and 3 make 7, and so do 3 and 4: they are the same two groups, looked at from the other end.

Any two of the three numbers decide the third. So every question about one whole and two parts is one of two kinds:

What we know	What we find	How it can be written
both parts	the whole	4 + 3 = \square
the whole and one part	the other part	4 + \square = 7 or 7 - 4 = \square

The box stands for a number we have not found yet, read aloud as “what number”. The second row is why a subtraction can be worked out by thinking of an addition: 7 - 4 asks “4 and what make 7?” That link has its own lesson, subtraction as a missing part.

Why the parts always make the whole

Every counter in the whole lands in exactly one of the two parts. It is dark or light, on this side of the pencil or that side, never both and never neither.

So counting all of them is the same as counting the first part and then carrying on counting through the second. Counting the dark counters reaches 4. Carrying on through the light ones, “five, six, seven”, counts 3 more. That is exactly what 4 + 3 means, so the whole is 4 + 3.

Take the first part away, and the counters left are exactly those of the second part, again because each counter was in one part or the other. So the whole take away one part is the other part. Nothing in this argument uses the number 7: it holds for any whole split into two parts, which is why every bond below can be read as an adding fact and as a taking-away fact.

Bonds that make five

Figure 3A number bond with dots and with numerals. Read it as “3 and 2 make 5”.

A child starts with the dots version of the bond, where they can count every part and see that nothing was lost or added. The numeral version is the same picture once a numeral can stand in for its dots.

Every split of five. Put five counters in a row, all the same color. Move one across a gap, then a second, and keep going until all five have crossed. Each move gives a new split, and none is skipped or repeated, because the group on the left shrinks by exactly one each time: 5, 4, 3, 2, 1, 0. The group on the right is always whatever is left over, so it grows 0, 1, 2, 3, 4, 5. Laid out in rows, it makes a staircase. In the picture the counters that have crossed are drawn light, so the two groups are easy to tell apart.

Figure 4Move one counter across the gap at a time and every split of 5 appears, once each.

Six rows, so six ways. The top and bottom rows have all five counters in one group and none in the other. That still counts as a split: a group is allowed to be empty, and 5 + 0 = 5 is as true as 3 + 2 = 5.

The rows also come in matching pairs. 4 + 1 and 1 + 4 use the same two numbers, only in the other order, and sliding the two groups past each other does not change how many counters there are. So a child who knows one of them knows the other, and there are really only three pairs to know: 5 with 0, 4 with 1, 3 with 2.

The smaller wholes 3 and 4 split in exactly the same way with fewer counters, one row fewer each time: 4 has 5 splits and 3 has 4. A child meets them in the same games.

A missing part of five. If the whole is 5 and one part is 3, the other part is whatever fills the gap from 3 up to 5. A child can find it in three ways, and all three rest on the same fact, that the two parts together make the whole. They can count on from 3: “four, five”, that is two more. They can put 3 counters in a five frame, a row of five boxes, and count the empty boxes. Or they can simply remember the pair: 3 goes with 2. The first two work before the facts are known; the third is where practice with the first two leads. Written down, the question is 3 + \square = 5.

Bonds that make ten

A ten frame is two rows of five boxes, filled along the top row first.

Figure 5Seven counters and three empty boxes fill the frame, so 7 and 3 make 10.

The ten frame is what makes ten easy to see. Its two rows of five mean a child can read seven counters as “a full row and two more” without counting them, and the empty boxes show the other part as space. In the picture, 7 counters and 3 empty boxes make the bond 10, 7, 3.

Every split of ten. The staircase works just as it did for five. Put ten counters in a row and move them across the gap one at a time. The left group shrinks by exactly one each time, from 10 down to 0, and the right group is always whatever is left over. Nothing is added and nothing is taken away, so every row still has ten counters.

Figure 6The same staircase with ten counters. Every split of 10 appears, once each.

That makes 11 rows, so 11 ways. The top and bottom rows, 10 + 0 and 0 + 10, still count. The rows come in matching pairs, such as 8 + 2 and 2 + 8. The only row with no partner is the middle one, 5 + 5, whose two parts are the same. So there are really six pairs to know.

The wholes between five and ten work the same way, with one row fewer each time: 9 has 10 splits, 8 has 9, 7 has 8 and 6 has 7. Here are the pairs for every whole from 3 to 10, each of which can also be turned around:

Whole	The pairs that make it
3	3 + 0, 2 + 1
4	4 + 0, 3 + 1, 2 + 2
5	5 + 0, 4 + 1, 3 + 2
6	6 + 0, 5 + 1, 4 + 2, 3 + 3
7	7 + 0, 6 + 1, 5 + 2, 4 + 3
8	8 + 0, 7 + 1, 6 + 2, 5 + 3, 4 + 4
9	9 + 0, 8 + 1, 7 + 2, 6 + 3, 5 + 4
10	10 + 0, 9 + 1, 8 + 2, 7 + 3, 6 + 4, 5 + 5

The wholes 4, 6, 8 and 10 each have a pair with two equal parts, a double; 3, 5, 7 and 9 have none. That difference is the start of even and odd numbers.

A missing part of ten. If the whole is 10 and one part is 7, the other part is whatever fills the gap from 7 up to 10. The same three ways work, and one more. A child can count on from 7: “eight, nine, ten”, three more. They can put 7 counters in a ten frame and count the empty boxes, as in the picture above. They can use five: 7 is a full row of five and 2 more, and 2 needs 3 to finish its row, so the missing part is 3. Or they can simply remember that 7 goes with 3. Written down, the question is 7 + \square = 10. The same moves work for any whole up to ten: for 9 and 4, counting on from 4 or picturing a hand of five and four more both give 5.

Bonds that make twenty

Twenty turns up naturally: fingers and toes together, two ten frames, twenty stickers on a sheet. The key picture is two ten frames side by side, which hold exactly twenty.

Figure 7Twenty is two full ten frames. The bond says that 14 and 6 make 20.

The picture shows the bond 20, 14, 6 both ways: as counters in two frames and as a number bond. Look at the right-hand frame on its own. It holds 4 dark counters and 6 light ones, which is a bond of ten. That is the whole secret of twenty: every bond of twenty is a bond of ten with a full ten added to one of its parts. (This holds because the parts are whole numbers of things, as they always are here.)

Every split of twenty. Start with a fact worth checking with the child. Could both groups be smaller than ten? The biggest they could then be is 9 and 9, and 9 + 9 = 18, which is two short of twenty. So in every split of 20 at least one group is 10 or more. In the frames, that bigger group fills the whole left frame, and whatever is left of it spills into the right frame. The smaller group fills the rest of the right frame. The right frame is full, so what it holds is a bond of ten.

Figure 8Put a full ten in front of a bond of ten and it becomes a bond of twenty.

This means the bonds of twenty do not have to be learned as new facts. Take any bond of ten, put a full ten in front of its first part, and it becomes a bond of twenty. Going down the bonds of ten in order gives every bond of twenty, each exactly once:

A bond of ten	Add a ten to the first part	A bond of twenty
0 + 10	10 + 10	10 and 10
1 + 9	11 + 9	11 and 9
2 + 8	12 + 8	12 and 8
3 + 7	13 + 7	13 and 7
4 + 6	14 + 6	14 and 6
5 + 5	15 + 5	15 and 5
6 + 4	16 + 4	16 and 4
7 + 3	17 + 3	17 and 3
8 + 2	18 + 2	18 and 2
9 + 1	19 + 1	19 and 1
10 + 0	20 + 0	20 and 0

That is 11 pairs. Each pair can also be turned around, 13 + 7 or 7 + 13, since sliding two groups past each other does not change how many there are. Only 10 and 10 turns into itself. So if a left group and a right group count as different, there are 21 ways to split 20: the first group can hold 0, 1, 2 and so on up to 20, and the second group is always whatever is left.

An adult will notice a pattern in the table: apart from 10 + 10 and 20 + 0, the last digits of the two parts make ten, as 3 and 7 do in 13 and 7. That is not a separate trick to learn; it is the right-hand frame, written in digits.

A missing part of twenty. Written down, the question looks like 16 + \square = 20. There are two cases, and both come from the frames.

If the part you know is 10 or more, it fills the left frame and some of the right. Only the right frame needs thinking about: 16 puts 6 there, and 6 needs 4 to make ten. So the missing part is 4.

If the part you know is less than 10, it cannot fill even one frame. It needs its bond of ten to fill the first frame, and then a whole second frame on top. For 3 + \square = 20: 3 needs 7 to make ten, and another full ten makes twenty, so the missing part is 17. Counting on from 3 would also get there, but it takes seventeen counts; the bond takes one thought.

Wholes from eleven to nineteen

The same two frames show every whole from 11 to 19, only not filled to the end. Every one of these numbers is a ten and some ones (Tens and ones), and that split is itself a bond: 13 is 10 and 3 (see Ten more and ten less). A whole such as 13 can be split in 14 ways, from 0 and 13 to 13 and 0, one more way than the whole, for the same reason 20 has 21. Nobody needs to list all fourteen. What matters is finding a missing part, and again there are two cases.

If the part you know is 10 or more, the tens match and only the ones need a bond. For 12 + \square = 15: 12 is 10 and 2, 15 is 10 and 5, and 2 and 3 make 5. So the missing part is 3. This works the same way whenever the known part holds a full ten.

If the part you know is less than 10, there are two possibilities. When it is no bigger than the ones of the whole, the full ten stays in the missing part: in 3 + \square = 15, 3 and 2 make 5, so the missing part is 12. When it is bigger than the ones, as in 7 + \square = 12, there is no ten to match. Put 7 counters in the frames and add counters until there are 12: the added counters are the missing part, and there are 5 of them. Those added counters finish the first frame and spill into the second; using that on purpose is what making ten and bridging through ten teaches.

Worked examples

Read each problem aloud. If the child has counters, cubes or buttons, let them build it; two ten frames drawn on paper hold everything up to twenty.

Example 1

Seeing the parts in a picture

Look at the picture. How many counters are there in all? How many are dark? How many are light?

Figure 9How many in all, and what are the two parts?

Solution

Count them all: there are 6 counters, so the whole is 6. The dark counters are one part, 2. The light counters are the other part, 4. So “6 is 2 and 4”. As a bond it has 6 on top and 2 and 4 below, and as an addition it is

2 + 4 = 6

A child who starts with the light counters says “4 and 2 make 6”. That is just as right: it is the same two groups, read from the other end.

Example 2

Finding the whole

Sam has 3 red blocks and 5 blue blocks. How many blocks does Sam have?

Solution

The red blocks are one part and the blue blocks are the other. The question asks for all the blocks, so it asks for the whole: put the parts together and count them all.

3 + 5 = 8

Sam has 8 blocks. A child who says “five” and carries on “six, seven, eight” is counting on, which has its own lesson, Counting on to add and counting back to subtract.

Example 3

Five fingers on two hands

Hold up 5 fingers. How many are up on this hand, and how many on that hand? Now show 5 fingers a different way. How many different ways can you find?

Solution

The two hands are the two parts and the five fingers are the whole. A child might start with 4 fingers on one hand and 1 on the other, then try 3 and 2. Every split of 5 works, because one hand has room for all five:

5 + 0 \quad 4 + 1 \quad 3 + 2 \quad 2 + 3 \quad 1 + 4 \quad 0 + 5

That is 6 ways, the six rows of the staircase for five. The splits 5 + 0 and 0 + 5 are a whole hand of five and a closed fist, which children enjoy finding once they see it counts. Swapping hands turns 4 + 1 into 1 + 4: the same pair, held the other way round.

Example 4

Filling a five frame

There are 3 counters in the five frame. How many more counters will fill it?

Figure 10The empty boxes are the missing part, drawn as space.

Solution

The frame has 5 boxes and 3 of them are full, so the missing part is the empty boxes. There are 2 of them, so 2 more counters fill the frame. The counters and the empty boxes together make the whole frame, which is the bond

3 + 2 = 5

This is why the five frame is so useful here: the missing part is not hidden, it is drawn as space, and the child can see it before they know the fact.

Example 5

When one part is zero

Ana has 5 grapes and two plates, one red and one blue. She puts all 5 grapes on the red plate. How many grapes are on the blue plate?

Solution

The whole is 5 and one part is 5. Ask: “5 and what make 5?” Nothing more is needed, so the other part is 0. The blue plate is empty.

5 + 0 = 5

As a number bond this is 5 on top with parts 5 and 0. It sounds odd to a child to call an empty plate a part, but allowing it keeps the rule true every time: two parts make the whole, even when one of them is nothing. Writing “none” as the numeral 0 lets it sit in the bond like any other number. The number for nothing has its own place in number names, numerals and zero.

Example 6

Ten fingers

Hold up 6 fingers. How many of your fingers are folded down?

Solution

The ten fingers are the whole, the fingers up are one part and the fingers down are the other. Six fingers up is one whole hand, 5, and one finger on the other hand. That other hand has 1 finger up, so the rest of it, 4 fingers, is folded down. The first hand has none folded. So 4 fingers are down, and

6 + 4 = 10

The child did not count the folded fingers one by one. They used two bonds they may already know: 6 is 5 and 1, and 1 and 4 make 5. Two hands of five fingers are laid out just like the two rows of a ten frame, which is why fingers work so well here.

Example 7

Hiding under a cup

There are 9 counters. Some of them are under the cup. You can see 4 counters. How many counters are under the cup?

Figure 11You can see 4 counters. The bond says how many are hiding.

Solution

The whole is 9 and one part, the counters you can see, is 4. The hidden part is the number that goes with 4 to make 9:

4 + \square = 9

Counting on from 4 gives “five, six, seven, eight, nine”, which is five more, so 5 counters are under the cup. A child who knows that 5 and 4 make 9 (a hand and four more) can say it at once.

The same picture can be read as taking away. Take the 4 you can see away from all 9, and what is left is the hidden part:

9 - 4 = 5

Lift the cup and count to check. One picture gave an adding sentence and a taking-away sentence, and they agree because both describe the same whole and the same two parts.

This works well as a game, with whatever whole the child is learning: five counters at first, then ten. The adult hides some of the counters and the child says how many are hiding. The hidden counters cannot be counted, so the only way to answer is through the bond, and that is exactly the habit the game builds.

Example 8

A take-away story is a part and whole story

There were 9 birds on a fence. 6 flew away. How many birds are still on the fence?

Solution

At the start all 9 birds are on the fence, so 9 is the whole. The birds end up in two groups, the ones that flew away and the ones that stayed. So 6 is one part, and the birds still on the fence are the other part.

9 - 6 = 3

3 birds are still on the fence.

Check by putting the parts back together: 6 that flew and 3 that stayed make 6 + 3 = 9, the whole we started with. Seeing the birds that left as a part, even though they are gone, is what lets every take-away story be checked by adding. Taking away itself has its own lesson, Subtraction as taking away.

Example 9

The whole and its parts as bars

Mia read 10 pages. She read 6 pages before lunch and the rest after lunch. How many pages did she read after lunch?

Solution

Pages are awkward to lay out as counters, so draw bars instead. The long bar on top is the whole, 10 pages. Under it the same length is cut in two: 6 pages before lunch, and a piece with a question mark for the pages after lunch. The bottom row is exactly as long as the top row, because together the two parts are the whole.

Figure 12The whole on top, its two parts underneath, drawn to size.

The whole and one part are known, so the question asks for the other part: “6 and what make 10?” Count on from 6: “seven, eight, nine, ten”. That is four counts.

10 - 6 = 4

Mia read 4 pages after lunch. Check: 6 + 4 = 10.

The bars are the number bond with the sizes drawn in. Children keep using this picture for story problems long after the counters are put away.

Example 10

Every way to make eight

Make a train of 8 cubes. Put red cubes at the front and blue cubes at the back. A train can be all red or all blue. How many different trains can you make?

Solution

Start with all 8 cubes red, then change one cube at the back to blue, then another, just as counters crossed the gap in the staircase. Writing red first and blue second, the trains are

8 + 0 \quad 7 + 1 \quad 6 + 2 \quad 5 + 3 \quad 4 + 4

3 + 5 \quad 2 + 6 \quad 1 + 7 \quad 0 + 8

That is 9 trains, one for each number of red cubes from 8 down to 0. Five red and three blue is a different train from three red and five blue, but it uses the same pair of numbers, so there are 5 pairs to remember. The middle one, 4 + 4, is a double, and it has no partner because turning it around gives the same thing.

Example 11

Is this a bond of ten?

This bond says 6 and 5 make 10. Is that true? Check with a ten frame.

Figure 13Six and five do not fit in ten. One counter is left over.

Solution

No. Put 6 counters in a ten frame: a full top row and one more. There are 4 empty boxes left. Now put in 5 more counters. The first 4 fill the frame and 1 is left over. So 6 and 5 make one more than ten:

6 + 5 = 11

There are two ways to make it a bond of ten. Keep the 6 and change the other part to 4, because 6 + 4 = 10. Or keep the 5 and change the other part to 5, because 5 + 5 = 10. Counting in the frame settles the question every time; once the pairs are known, a child can also answer at once, because the partner of 6 in ten is 4 and nothing else.

Example 12

Sharing fairly

Two friends share 10 stickers. Can they each get the same number? What if there are only 9 stickers?

Figure 14Ten makes two equal rows. Nine always leaves one counter over.

Solution

With 10 stickers, yes: 5 each, since

5 + 5 = 10

Line the stickers up in two rows, one for each friend, and every sticker in one row has a partner in the other.

With 9 stickers, no. Look down the list of pairs for 9: 9 and 0, 8 and 1, 7 and 2, 6 and 3, 5 and 4. The turned-around pairs use the same numbers, and the list has every split, so checking it settles the question: in none of the pairs are the two numbers the same. The equal shares miss 9 on both sides, since 4 each makes only 8 and 5 each makes 10. Put the stickers in two rows and one sticker is always left without a partner.

Five stickers fail in the same way. The pairs for 5 are 5 and 0, 4 and 1, 3 and 2, and none has two equal numbers: 2 and 2 make only 4, and 3 and 3 make 6. However five stickers are shared, one friend gets at least one more than the other.

So among the wholes from 3 to 10, the ones that can be shared fairly are exactly those with a double in the table of pairs, 4, 6, 8 and 10; 3, 5, 7 and 9 cannot. This is a first look at even and odd numbers.

Example 13

From ten to nine

Six and four make ten. Use that to find two numbers that make 9.

Solution

Picture the bond as counters: 6 in one group and 4 in the other, 10 in all. Take one counter away from either group. The whole is now one less, 9, and the other group has not changed. If the counter comes from the 4, the groups are 6 and 3. If it comes from the 6, they are 5 and 4. Either way

6 + 3 = 9 \qquad 5 + 4 = 9

This connects the whole family of bonds. Every pair for 9 can be made from a pair for 10 by taking one away from a part, and every pair for 10 from a pair for 9 by adding one to a part. The same move works at every size: take one dark counter away from any of the top five rows of the staircase for five, and you have a row of the staircase for four, so 3 + 2 = 5 becomes 2 + 2 = 4. A child who knows the bonds of ten well can reach the bonds of 9, 8 and the rest from them.

Example 14

Filling two ten frames

There are 12 counters in the ten frames. How many more counters will fill both frames?

Figure 15How many more counters fill both frames?

Solution

The left frame is full, so it needs nothing. The right frame has 2 counters, and a frame holds 10, so the empty boxes are the missing part. 2 and 8 make 10, so there are 8 empty boxes, and 8 more counters fill both frames:

12 + 8 = 20

The bond of ten, 2 and 8, did all the work. The full frame just came along. A child can count the empty boxes to check, and before long will see the 8 without counting.

Example 15

A part bigger than ten

16 + \square = 20

Solution

16 is a full ten and 6 more. The full ten fills one frame of twenty. The 6 sits in the second frame, and 6 needs 4 to make 10. So \square = 4:

16 + 4 = 20

Counting on from 16 also works here, “seventeen, eighteen, nineteen, twenty”, which is four counts. When the known part is close to twenty, counting on is short, and it is a good check on the bond.

Example 16

A part smaller than ten

3 + \square = 20

Solution

3 does not fill even one frame. It needs 7 to make its ten, since 3 and 7 make 10. Then a whole second ten is still empty. So the missing part is 7 and another 10, which is 17:

3 + 17 = 20

A child might first say 7, thinking only of the first ten. Put out the two frames and fill them: 3 counters, then 7 to finish the first frame, then 10 more for the second. The answer has to be more than ten, because 3 and any number up to 10 make at most 13.

Example 17

Crayons in a box

A box holds 20 crayons. 9 crayons are out on the table. The rest are in the box. How many crayons are in the box?

Solution

The whole is 20 crayons and one part, on the table, is 9. The part in the box goes with 9 to make 20. 9 needs 1 to make ten, and another ten makes twenty, so the box has 1 and 10, which is 11 crayons:

9 + 11 = 20

As a number bond it is 20 on top with parts 9 and 11. The same bond answers “20 take away 9”, since the crayons in the box are what is left when the 9 come out: 20 - 9 = 11.

Example 18

Marbles in two bags

Ivy has 15 marbles in two bags. One bag has 12 marbles. How many marbles are in the other bag?

Solution

The whole is 15 and one part is 12. Both numbers are a ten and some ones: 15 is 10 and 5, and 12 is 10 and 2. The tens match, so only the ones need a bond. 2 and 3 make 5, so the other bag has 3 marbles:

12 + 3 = 15

This is the bond 5, 2, 3 from the bonds that make five, with a ten added to the whole and to one part. Counting on from 12, “thirteen, fourteen, fifteen”, gives 3 as well.

Example 19

Ducks on the grass

There are 14 ducks. 6 ducks are in the pond. The rest are on the grass. How many ducks are on the grass?

Solution

The whole is 14 and one part is 6. This time the known part is less than ten but bigger than the 4 ones in 14, so the tens cannot simply be matched. Build it instead: 6 dark counters for the ducks in the pond, then light counters, one duck at a time, until there are 14 counters in all.

Figure 16The 8 light counters are the ducks on the grass.

The light counters are the ducks on the grass. There are 8 of them, so 8 ducks are on the grass:

6 + 8 = 14

Counting on from 6, “seven, eight, nine, ten, eleven, twelve, thirteen, fourteen”, gives 8 as well, though it is a long count. Look where the light counters sit: 4 finish the first frame and 4 start the second. Seeing the 4 and the 4 without counting is the start of making ten and bridging through ten.

Boundaries

The parts are whole numbers. Counters, fingers, cubes and stickers are never cut up here, so every part is one of 0, 1, 2, 3, \dots, the whole numbers. That is the only reason 5, 9 and 19 have no fair share: no whole number added to itself makes them, since 2 + 2 = 4 and 3 + 3 = 6 miss 5, 4 + 4 = 8 and 5 + 5 = 10 miss 9, and 9 + 9 = 18 and 10 + 10 = 20 miss 19. If a sticker could be cut in half, two friends could share 9 as four and a half each, but halves are a much later idea.
A part is never bigger than the whole. Splitting a group can only make smaller groups, or leave all of it in one group and none in the other. So 6 + \square = 5, 12 + \square = 10 and 23 + \square = 20 have no answer among the whole numbers, and every part of 10 is one of 0, 1, 2, \ldots, 10. This is also why 3 - 5 has no answer among the numbers a child uses now: there is no way to take 5 counters out of 3.
Each thing belongs to exactly one part. If 4 children in a group have a dog and 3 have a cat, the group need not have 7 children: a child with both pets has been counted twice, so “dog” and “cat” are not two parts. The parts must not share anything, and they must use everything; a counter pushed aside and forgotten breaks the whole just as badly.
“How many ways” depends on the question. With a left group and a right group, 5 has 6 splits, 10 has 11 and 20 has 21, one more than the whole each time. If 7 and 3 is treated as the same pair as 3 and 7, then 5 has 3 pairs, 10 has 6 and 20 has 11. If both groups must have something in them, the splits with 0 drop out and 10 has 9. Each answer is right for its question; say which question is being asked.
Matching the tens needs a full ten in the known part. For 15 and 12 the tens match and 2 + 3 = 5 finishes the job. For 14 and 6 there is no ten to match, and the ones cannot be matched either, since 6 is more than the 4 ones in 14; building it in the frames, or counting on, finds the 8.
Two parts is a choice, not a law. A whole can be split into three or more parts, as in 2 + 3 + 1 = 6, or ten as 2, 3 and 5. This lesson keeps to two, because two parts is what every single adding or subtracting fact describes. Putting three parts together is taken up in adding three numbers.

Where this goes next

Every bond is an addition fact and a subtraction fact at once. Read with the parts first, the bond 10, 7, 3 says 7 + 3 = 10, which is adding within five and ten. Read from the whole, it answers “10 take away 7”, which is Subtracting within ten, and the missing-part question 7 + \square = 10 is the heart of subtraction as a missing part. One bond, read every way it can be read, gives the four sentences of a fact family: 4 + 3 = 7, 3 + 4 = 7, 7 - 4 = 3 and 7 - 3 = 4, or only two when both parts are the same. Together the bonds of ten and twenty cover a large share of adding and subtracting within twenty.

The bonds then do real work. In Making ten and bridging through ten, 8 + 5 becomes 8 + 2 + 3, because 8 needs 2 to make ten: the 5 is split into the parts 2 and 3, a part and whole move inside a bigger problem. Bridging back through ten, in the same lesson, turns 13 - 5 into 13 - 3 - 2. Ten as a unit leads to teen numbers as ten and some ones and to ten more and ten less. Later still the same idea works with tens in place of ones: 30 and 70 make 100 because 3 tens and 7 tens make 10 tens, which is where Tens and ones and place value take over.

Story problems differ in which number is missing. Stories that give both parts ask for the whole. Stories like Mia’s pages give the whole and one part and ask for the other part. Stories where some are added or taken away can hide any of the numbers, the result, the change or the starting amount (Problems with the start, change or result missing), and drawing such a story as a whole and two parts shows which number is missing, and so whether to add or to subtract. The bars of Example 9 grow into the drawings used for story problems in later grades and, years on, the tape diagram.

Exercises

Find the parts

Use the rows for Exercises 1 to 3. For each row, say how many counters there are in all, and what the two parts are.

Figure 17Rows for Exercises 1 to 3.

Row A
Row B
Row C

Find the whole

There are 2 cats and 3 dogs. How many pets are there?
A plate has 6 grapes and 3 strawberries. How many pieces of fruit are on the plate?
A number bond has the parts 7 and 2. What number goes in the top circle?

Wholes up to five

4 + \square = 5
\square + 2 = 5
5 = 0 + \square
A number bond has 5 in the top circle and 3 in one part. What number goes in the other part?
A five frame has 2 counters in it. How many boxes are empty?
Hold up 5 fingers on one hand. Fold down 1 finger. How many fingers are still up?
There are 5 counters. Some are under a cup. You can see 2. How many are under the cup?
There are 5 children at the playground. 3 are on the swings and the rest are on the slide. How many are on the slide?
A train is made of 5 cubes. Each cube is blue or red. 0 cubes are blue. How many cubes are red?
Find every way to split 3 counters into two groups.
\square + 1 = 4
2 + \square = 4
There are 3 counters. Some are under a cup. You can see 1. How many are under the cup?

Ten frames

Use the ten frames for Exercises 20 to 22. For each one, ask: “How many counters are in the frame? How many more will fill it?” Then say the bond.

Figure 18Ten frames for Exercises 20 to 22.

Frame A.
Frame B.
Frame C.

Fill in the bond

Use the number bonds for Exercises 23 to 25. What number goes where the question mark is?

Figure 19Number bonds for Exercises 23 to 25.

Bond A.
Bond B.
Bond C.

Wholes from six to ten

There are 10 birds on a wire. 3 are red and the rest are blue. How many birds are blue?
Maya has 8 crayons. 2 are in the box and the rest are on the table. How many crayons are on the table?
A tower needs 10 blocks. Leo has put on 7. How many more blocks does he need?
There are 9 grapes on a plate. 4 are red and the rest are green. How many grapes are green?
There are 6 ducks. Each duck is in the pond or on the grass. How many could be in the pond, and how many on the grass? Find every way.
The whole is 6. One part is 4. What is the other part?
There are 10 counters. Some are under a cup. You can see 6. How many are under the cup?
4 + \square = 10
\square + 1 = 9
7 = 3 + \square
10 = \square + 10
5 + \square = 8
\square + 2 = 7

Taking away

There were 8 ducks on the pond. 2 swam away. How many ducks are still on the pond?
9 - 5 = \square

Make twenty

17 + \square = 20
\square + 5 = 20
20 = 10 + \square
2 + \square = 20
A number bond has 20 in the top circle and 14 in one part. What number goes in the other part?

Use the pairs of ten frames for Exercises 46 to 48.

Figure 20Pairs of ten frames for Exercises 46 to 48.

Look at frames A. How many more counters will fill both frames?
Look at frames B. How many more counters will fill both frames?
Look at frames C. How many more counters will fill both frames?
There are 20 children on a bus. 13 children are sitting down. The rest are standing. How many children are standing?

Wholes from 11 to 19

10 + \square = 16
14 + \square = 18
9 + \square = 12
A number bond has 19 in the top circle and 13 in one part. What number goes in the other part?

Tell it and show it

Draw a number bond with 9 in the top circle and 5 and 4 in the part circles. Tell a story that matches it.
Ava says 3 and 3 make 5. Show with counters why that cannot be right.
Ben says 4 and 5 make 10. Is he right? Show it on a ten frame.
Jo holds 2 counters in one hand and 3 in the other. Then he swaps hands, so the 3 are in the first hand and the 2 in the other. How many counters does he have now? How do you know without counting again?
Look at the bond with 6 on top and the parts 2 and 4. How does it show that 6 - 2 = 4?

Explain

Can a part be bigger than the whole? How do you know?
You found 6 ways to split 5. How do you know there is not a seventh way?
Find every way to split 7 counters into two groups. How do you know you found them all?
Why are there more ways to split 10 counters than to split 9?
Can two numbers that are each smaller than 10 make 20? How do you know?
Omar says: “12 and 8 make 20, so 2 and 8 make 10.” Is he right? Show it with two ten frames.
Can 20 stickers be shared fairly between two friends? Can 19 stickers? How do you know?
How many ways can you split 20 counters into a left group and a right group? How do you know you have them all?

Answers

5 in all. The parts are 3 dark and 2 light: 3 + 2 = 5.
8 in all. The parts are 5 dark and 3 light: 5 + 3 = 8.
7 in all. The parts are 1 dark and 6 light: 1 + 6 = 7.
5 pets. The cats and the dogs are the two parts, and all the pets are the whole: 2 + 3 = 5.
9 pieces of fruit. The grapes and the strawberries are the parts: 6 + 3 = 9.
9. The top circle holds the whole, and the parts together make it: 7 + 2 = 9.
1. 4 and 1 make 5, so 4 + 1 = 5. Counting on from 4: “five” is one more.
3. 3 and 2 make 5, so 3 + 2 = 5.
5. All five are in one part and none in the other, so 5 = 0 + 5.
2. The two parts must make 5, and 3 + 2 = 5.
3 boxes. The frame has 5 boxes and 2 are full, and 2 + 3 = 5.
4 fingers. The hand is the whole, 5; the folded finger is one part, 1; the fingers still up are the other part, and 1 + 4 = 5.
3 counters. You see 2, and 2 + 3 = 5.
2 children. The swings have 3 and the whole is 5, and 3 + 2 = 5.
5 red cubes. The blue part is 0, so the red part is all of them: 0 + 5 = 5.
4 ways: 3 + 0, 2 + 1, 1 + 2, 0 + 3. Start with all 3 in the first group and move one across at a time.
3. 3 and 1 make 4, so 3 + 1 = 4.
2. 2 and 2 make 4, so 2 + 2 = 4. Both parts are the same: this is a double.
2 counters. The whole is 3 and you see 1, and 1 + 2 = 3. Counting on from 1: “two, three” is two more.
8 counters, 2 more. 8 + 2 = 10: the top row is full and the bottom row has 3 counters and 2 empty boxes.
5 counters, 5 more. 5 + 5 = 10: the top row is full and the whole bottom row is empty.
1 counter, 9 more. 1 + 9 = 10: nine boxes are empty.
5. The whole is 7 and one part is 2, and 2 + 5 = 7.
4. The whole is 10 and one part is 6, and 4 + 6 = 10.
9. The parts are 5 and 4, and 5 + 4 = 9: a hand and four more.
7 blue birds. The red part is 3 and the whole is 10, and 3 + 7 = 10.
6 crayons. The box has 2 and the whole is 8, and 2 + 6 = 8.
3 more blocks. 7 + 3 = 10: count on from 7, “eight, nine, ten”.
5 green grapes. 4 + 5 = 9.
7 ways: 6 + 0, 5 + 1, 4 + 2, 3 + 3, 2 + 4, 1 + 5, 0 + 6, with the pond first. Start with all 6 in the pond and move one duck at a time to the grass. If both places must have a duck, leave out the first and last and there are 5 ways.
2. Ask “4 and what make 6?” and count on: “five, six”. So 6 - 4 = 2.
4. The whole is 10 and the part we can see is 6: 6 + 4 = 10, so 10 - 6 = 4.
6. 4 + 6 = 10.
8. 8 + 1 = 9: one less than 9 is 8.
4. 7 = 3 + 4: the box is on the right, but the parts 3 and 4 still make the whole 7.
0. 10 = 0 + 10: one part is already all ten, so the other part is empty.
\square = 3. “5 and what make 8?”: “six, seven, eight”. Check: 5 + 3 = 8.
\square = 5. The whole is 7 and one part is 2. Put out 7 counters and take 2 away: 5 are left, so 7 - 2 = 5. Check: 5 + 2 = 7.
6 ducks. The whole is 8 and the ducks that swam away are one part: 8 - 2 = 6. Check: 2 + 6 = 8.
\square = 4. The whole is 9 and one part is 5. “5 and what make 9?”: “six, seven, eight, nine”, four counts. So 9 - 5 = 4.
3. 17 is a ten and 7, and 7 needs 3 to make ten, so 17 + 3 = 20.
15. 5 needs 5 to make ten, and another ten makes twenty, so 15 + 5 = 20.
10. Two tens make twenty: 20 = 10 + 10. Both parts fill a whole frame.
18. 2 needs 8 to make ten, and another ten makes twenty, so 2 + 18 = 20.
6. 14 puts 4 in the second frame, and 4 and 6 make ten, so 14 + 6 = 20.
5 more. The frames hold 15: a full frame and 5. 5 and 5 make ten, so 15 + 5 = 20.
9 more. The frames hold 11: a full frame and 1. 1 and 9 make ten, so 11 + 9 = 20.
2 more. The frames hold 18: a full frame and 8. 8 and 2 make ten, so 18 + 2 = 20.
7 children. The whole is 20 and the sitting part is 13; 3 and 7 make ten, so 13 + 7 = 20 and 20 - 13 = 7.
6. 16 is a ten and 6, so 10 + 6 = 16.
4. The tens match, so only the ones matter: 4 and 4 make 8, so 14 + 4 = 18.
3. 9 is more than the 2 ones in 12, so the tens do not match. Put 9 counters in the frames and add counters until there are 12: that is 3 more. Counting on, “ten, eleven, twelve”, gives 3 too, so 9 + 3 = 12.
6. The tens match, so only the ones matter: 3 and 6 make 9, so 13 + 6 = 19.
Any story with 9 things in all, split into a group of 5 and a group of 4. For example: “9 children are at the park. 5 are on the swings and 4 are on the slide.”
Put out 3 counters and another 3 and count them all: there are 6, one too many. Or start from 5 counters, take 3 for one group, and count what is left: only 2. Either way 3 and 3 make 6, not 5.
No. Put 4 counters in a ten frame, then 5 more: there are 9 counters and one box is still empty, so 4 + 5 = 9. Ten needs 4 and 6, or 5 and 5.
5 counters. They are the same counters, only in different hands, so the whole has not changed: 2 + 3 = 5 and 3 + 2 = 5.
The whole is 6. Take away the part 2 and the other part, 4, is what is left. So 6 - 2 = 4.
No. A part is made by splitting the whole, so it is some of the whole or all of it, never more. The biggest a part can be is the whole itself, and then the other part is 0.
The first group can only hold 0, 1, 2, 3, 4 or 5 counters, since there are only five. That is six choices, and once the first group is chosen the second group is whatever is left, so each choice gives exactly one split. Six choices, six splits, and the staircase already shows all six.
8 ways: 7 + 0, 6 + 1, 5 + 2, 4 + 3, 3 + 4, 2 + 5, 1 + 6, 0 + 7. The first group can only hold 0, 1, 2, 3, 4, 5, 6 or 7 counters, and once it is chosen the second group is whatever is left, so there are exactly eight splits, and the list has all of them.
10 has 11 splits and 9 has 10. The first group can hold any number from 0 up to the whole. For 10 that is 11 choices (0 to 10), for 9 it is 10 choices (0 to 9), so ten has one more split. In the staircase, ten needs one more row.
No. The biggest two numbers smaller than 10 are 9 and 9, and 9 + 9 = 18, which is still two short of 20. Any smaller numbers make less. So one part of 20 is always 10 or more.
Yes. Put 12 counters in two ten frames: one full frame and 2 more. Adding 8 fills the second frame, so 12 + 8 = 20. The second frame on its own shows 2 and 8 filling ten: 2 + 8 = 10. The full frame is the same in both.
20 can: 10 each, since 10 + 10 = 20. 19 cannot: 9 each makes only 18, and 10 each makes 20, so one friend always gets at least one more.
21 ways: 20 + 0, 19 + 1, 18 + 2, and so on down to 0 + 20. The left group can hold any number from 0 to 20, which is 21 choices, and once it is chosen the right group is whatever is left, so each choice gives exactly one split.