Kindergarten · K.OA.A.2 · K.OA.A.5

Adding within five and ten

Two groups make ten or fewer altogether. How can a child know the total by looking and thinking, instead of counting every one from the start?

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Addition as putting togetherKindergarten

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Doubles and near doublesGrade 1 Subtracting within tenKindergarten

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Adding and subtracting within 207

Adding within five and ten

Doubles and near doubles Subtracting within ten Ten more and ten less Counting on to add and counting back to subtract ▾ Show all 7 (2 more)

When you reach for it

A child who knows what adding means can already find 3 + 2 or 6 + 3: put out one group, put out the other, push them together and count everything from one. That always works, and it is where every child starts. But it is slow, and the longer the count the easier it is to skip a counter or count one twice.

This lesson is about the step after that, where the sums with a total of ten or less stop being counted and start being known. It happens in two stretches. First come the sums within five, a total of 5 or less. These are small enough to see: two counters and three counters are a picture of five that the eye takes in at once. Past five that stops working. Few children can see eight loose counters at a glance, so the sums within ten are reached a different way, by thinking from facts already known: five as a landmark (one hand, one row of a ten frame), the number bond of a whole and its two parts, two equal rows for a double, and a full ten frame for ten.

Knowing a fact is not the same as reciting it. A child who knows 3 + 2 = 5 can see it in a picture, show it on a hand, tell a story for it, and say why it is right, and nothing in this lesson is timed. These sums come up constantly, in games with dice and dominoes, on fingers, in sharing a few snacks, which is exactly where they are best learned.

The procedure

There is no single method for these facts, and there should not be. A child who knows several ways to see a total picks whichever one the numbers make easy. None of the ways below is a list of steps; each is a reason the answer has to be what it is, resting on something already true about joining groups.

Adding within five

“Within five” means the total is 5 or less. With whole numbers there are not many such sums. Every one of them sits in this table, where the row is the first number and the column is the second:

+	0	1	2	3	4	5
0	0	1	2	3	4	5
1	1	2	3	4	5
2	2	3	4	5
3	3	4	5
4	4	5
5	5

That is 21 facts, and they sort out quickly. Eleven of them have a 0 in them; 0 + 1 and 1 + 0 count here, because they have a 0 as well as a 1. Of the ten left, seven have a 1 in them. That leaves only three, 2 + 2, 2 + 3 and 3 + 2. So the whole table rests on three ideas, each of which a child can see in a picture.

See it. A very small group can be known just by looking, with no counting (seeing one to five at a glance), and a tidy pattern makes that easier. The dice patterns are the tidiest of all.

Figure 1The dice patterns for 1 to 5. Each one can be seen at a glance, without counting.

When both parts are small patterns, a child can see each part, and then the whole as one pattern: two dots and three dots on a domino are seen as two and three, and then as five. Seeing a group as made of smaller groups is the first way a sum is known rather than counted.

One more, or nothing more. Adding 0 changes nothing, because an empty group puts nothing into the pile: 4 + 0 = 4 and 0 + 3 = 3. Adding 1 gives the next number in the count, because counting itself goes up by one each time. A child who can say what comes after 3 already knows 3 + 1 = 4, with no adding at all.

Figure 2Each tower is the one before it with one more cube on top. Adding 1 gives the next number in the count.

These two ideas cover 18 of the 21 facts. They also cover facts like 1 + 4, where the 1 comes first. The order of the parts never changes the total (this is shown in adding as joining), so 1 + 4 is the same amount as 4 + 1, and 4 + 1 is one more than 4: five.

The pairs that make four and five. What is left is 2 + 2, 2 + 3 and 3 + 2. These are the bonds that make five and one double. A five frame shows them clearly, because a full frame is five without counting, and an empty box shows how far short of five the total is.

Figure 3A full frame is 5, so 2 + 3 = 5. One empty box means one short of 5, so 3 + 1 = 4.

The frame also links the facts to each other. 3 + 1 leaves one box empty, so it is one less than five; 2 + 2 leaves one box empty too, so it is four as well. A child who notices that has found two facts that make 4 without being told.

Adding within ten

From six to ten, seeing alone runs out, and the same few ideas grow into six ways of thinking.

Start from what you already know. In 7 + 2 the seven are already counted, so there is no need to count them again: say “seven” and count on two more, “eight, nine”. Why that is allowed is the lesson on counting on; it is quickest when the second part is small: adding 1 is the next number, exactly as within five, and adding 2 is the one after that.

Start from the bigger part. 2 + 7 and 7 + 2 are the same amount, because joining two groups gives the same pile whichever group goes in first. So a child may always start from the larger part and count on the smaller one: “seven, eight, nine”. This is the same move that turned 1 + 4 into 4 + 1.

Use five. Five is a full hand and a full row of a ten frame, and a child who sees six as five and one can use that inside a sum. 5 + 3 is “five and three”, read straight off the frame. 6 + 3 is five and one and three, which is five and four, which is nine.

Figure 4A full row is five, and the bottom row has four more, so 6 + 3 = 9. One empty box says it is one short of ten.

In the frame above nobody needs to count. The top row is full, so it is five. The bottom row holds four counters, one dark and three light. Five and four is nine, and the one empty box confirms it: nine is one less than ten. The same picture shows the bond: 9 is made of 6 and 3. The full ten frame does here what the full five frame did for the smallest sums.

Use a double. Two equal groups line up in two equal rows, each counter with a partner, and that picture is easy to remember: 2 + 2 = 4 from the first stretch, then 3 + 3 = 6, 4 + 4 = 8, 5 + 5 = 10. A sum with parts one apart is one more than the double of the smaller part, because one row has one extra counter: 4 + 5 is one more than 4 + 4, so it is 9. These are doubles and near doubles.

Use ten. The pairs that fill a ten frame, 10 and 0, 9 and 1, 8 and 2, 7 and 3, 6 and 4, 5 and 5, are the bonds to ten; they grow from the bonds to five the way a ten frame grows from a five frame. A sum with one part one smaller than in such a pair is one short of ten: 3 + 6 is one less than 3 + 7, so it is 9.

Adding zero changes nothing, as within five: an empty group adds no counters, so 0 + 9 = 9 and 7 + 0 = 7.

How few facts there really are

Counting every sum of two whole numbers from 0 up to a total of 10, turned-around sums included, gives 66 facts. That sounds like a lot, but the ways above shrink it a lot.

21 of the 66 have a zero in them, and adding zero changes nothing.
Turning a sum around gives the same total, so 6 + 3 and 3 + 6 are one fact to know, not two. The 66 facts are only 36 pairs.
Sort the 36 pairs by their smaller part. 11 pairs have a zero (0 with 0 up to 0 with 10), 9 more have a one as the smaller part (1 with 1 up to 1 with 9), and 7 more have a two (2 with 2 up to 2 with 8). These 27 are easy for one reason: from the bigger number they take a count of at most two steps, or no count at all.

That leaves 9 pairs, the ones whose parts are both 3 or more. Every one of them has a total from 6 to 10 (the few facts that stay at five or less are all in the first three groups, as the table of sums within five shows), and each of them has a picture:

Fact	A way to see it
3 + 3 = 6	a double
4 + 4 = 8	a double
5 + 5 = 10	a double: two full hands
3 + 4 = 7	one more than 3 + 3
4 + 5 = 9	one more than 4 + 4
3 + 5 = 8	five and three
3 + 6 = 9	six is five and one, so it is five and four
3 + 7 = 10	a pair that makes ten
4 + 6 = 10	a pair that makes ten

A child does not need this table. The adult may find it reassuring: fluency with these facts is not sixty-six separate things to learn by heart. It is a handful of pictures, used again and again.

What knowing a fact means

A child knows 4 + 5 = 9 when they can say it without counting from one, and can say why: “four and four is eight, and one more is nine”, or “five and four is a hand and four more”. Which way a child takes to a fact does not matter, and it is good to see the same fact more than one way. Speed is not the aim and comes by itself with use. A child who answers “nine” and can explain it knows more than one who answers “nine” quickly and cannot. When a child is unsure, counting the counters is always allowed, and it is always the final check.

Worked examples

Example 1

A domino

Read aloud: “How many dots are on this domino? Can you tell without counting them one by one?”

Figure 5Three and two. Both patterns can be seen without counting.

Solution

The dark side shows three dots on a slant, the same as the three on a dice face, so a child who knows that pattern sees three without counting. The light side shows two. Two groups, three and two, make one group of five:

3 + 2 = 5

A child who does not yet know the pair can count on: start at “three” for the dark side, then count the light dots, “four, five”. Either way, touching and counting every dot from one gives the same five, which is the check.

Example 2

One more cube

Read aloud: “Leo has a tower of three cubes. He puts one more cube on top. How many cubes are in his tower now?”

Solution

Build the tower of three and add one more cube. The question is how many are in the tower now, the three that were there and the one that came:

3 + 1 = 4

No counting all is needed. Adding one more is the same as saying the next number in the count, and the number after three is four. The towers pictured in the procedure show this for every number up to five: each tower is the one before it with one cube on top.

Example 3

Sorting cards by the total

Read aloud: “Here are nine cards: 2 + 1, 0 + 3, 1 + 3, 4 + 1, 2 + 2, 3 + 2, 1 + 2, 5 + 0, 4 + 0. Put each card in a pile for 3, a pile for 4 or a pile for 5.”

Solution

Take each card and say its total, using whichever way fits it:

2 + 1 = 3 and 1 + 2 = 3: one more than two.
0 + 3 = 3: nothing more than three.
1 + 3 = 4: one more than three.
4 + 0 = 4: nothing more than four.
2 + 2 = 4: the double.
4 + 1 = 5: one more than four.
5 + 0 = 5: nothing more than five.
3 + 2 = 5: a bond that makes five.

So the piles are:

Total	Cards
3	2 + 1, 0 + 3, 1 + 2
4	1 + 3, 2 + 2, 4 + 0
5	4 + 1, 3 + 2, 5 + 0

Each pile shows that one number can be made in several ways. The equals sign can join any two cards from the same pile, such as 1 + 3 = 2 + 2, because both sides are the same amount (the equal sign means the same amount).

Example 4

Which is more?

Read aloud: “Which is more, 2 + 2 or 2 + 3? Can you tell before you find the totals?”

Solution

Both start with two. The second part is two in one and three in the other, and three is one more than two. So 2 + 3 has one more than 2 + 2, and it is the bigger total, without working out either one.

Now check by finding the totals:

2 + 2 = 4 \qquad 2 + 3 = 5

Five is one more than four, as the reasoning said. The same kind of looking settles 3 + 1 and 1 + 3 at once: the same two numbers in the other order, so they are the same amount, four. Comparing the parts instead of the totals works just as well for bigger sums, and it is often quicker than finding either total.

Example 5

Birds on a wire

Read aloud: “Seven birds sit on a wire. Two more birds land on the wire. How many birds are on the wire now?”

Solution

There is no need to count the seven birds again: we already know there are seven. Point to the seven, say “seven”, and count on one step for each bird that lands: “eight, nine”. On a number line this is a start at 7 and two hops forward.

Figure 6Start at the seven you already have and count on two. The last number you land on is the total.

The last number said is 9, so there are nine birds on the wire.

7 + 2 = 9

A child counting all says every number from one to nine; counting on says only two. That difference grows as the numbers do.

Example 6

Nine in the ten frame

Read aloud: “Put six dark counters and three light counters in the ten frame. How many counters are there? How many more would fill the frame?”

Solution

Fill the top row first, then the bottom row, as in the ten frame pictured under Use five. The top row is full, so it holds five, and no counting is needed for it. The bottom row holds one dark and three light counters, which is four. Five and four is nine:

6 + 3 = 9

One box is empty, so one more counter would fill the frame. Nine is one less than ten. The frame shows three facts at once: 6 + 3 = 9, nine is five and four, and nine and one make ten.

Example 7

Two rows of cars

Read aloud: “Pat lines up three toy cars in one row and three toy cars in another row. How many cars are there? Then Pat puts one more car at the end of the second row. How many cars are there now?”

Solution

Figure 7Two equal rows make a double. One more counter makes one more in all.

The two rows are the same length, and every car in the top row has a partner below it. Three and three is six, a double:

3 + 3 = 6

When one more car joins the second row, only that one car is new. Everything else is the same, so the total is one more than six:

3 + 4 = 7

A child who knows the double does not need to count the seven cars. The picture of one row with an extra counter is exactly the reason that 3 + 4 is one more than 3 + 3, a near double.

Example 8

Frogs on a log

Read aloud: “Two frogs sit on a log. Seven more frogs jump onto the log. How many frogs are on the log now?”

Solution

The story starts with two, and counting on from two means seven more counts: “three, four, five, six, seven, eight, nine”. That works, but it is long and easy to lose track of.

The two frogs and the seven frogs make the same group whichever we think of first. So start from the bigger part instead. Say “seven” and count on two: “eight, nine”. Nine frogs.

2 + 7 = 9 \qquad 7 + 2 = 9

The story is about 2 + 7, and it is fine to write it that way. Starting from the seven is a choice about how to find the total, not a change to the story. The same choice works within five: a story about one sticker and then four more is 1 + 4 = 5, found as “four, and one more is five”.

Example 9

Is it ten?

Read aloud: “Is three and six the same as ten? Show it on a ten frame.”

Solution

Figure 8Three and seven fill the frame. Three and six leave one box empty, so they make nine.

A child who knows the bonds of ten knows that 3 goes with 7 to make 10. Put three dark counters in a frame, then seven light ones, and every box is full:

3 + 7 = 10

Six is one less than seven. So three and six fill the frame all but one box, as the second frame shows. Three and six is not ten; it is one less than ten:

3 + 6 = 9

The fact 3 + 6 was found without counting at all, from a nearby fact the child already knew. A known bond is a good place to stand when the sum is next to it.

Example 10

Two dice

Read aloud: “Roll two dice. There are eight dots in all. How many dots could each one show?”

Solution

A die shows 1, 2, 3, 4, 5 or 6 dots. Go through what the first die might show, and ask what the second must show to make eight.

If the first shows 1, the second would need 7. No die has a side with seven dots, so that cannot happen. If the first shows 2, the second shows 6. If it shows 3, the second shows 5. If it shows 4, the second shows 4. And 5 with 3 and 6 with 2 are the first two turned around.

2 + 6 = 8 \qquad 3 + 5 = 8 \qquad 4 + 4 = 8

So the two dice show 2 and 6, 3 and 5, or 4 and 4. Each check used a fact the child knows (“five and three is eight”), not a count of dots. Dice and domino games give a child these facts again and again, in a setting where nobody is timing anything.

Boundaries

Within five means a total of five or less; within ten, ten or less. 3 + 3 is small numbers, but its total is 6, so it is a fact within ten, not within five. Every fact here has a total that fits in one ten frame. In 7 + 5 the parts do not fit, and a different move takes over: fill the ten first, then add the rest. That is Making ten and bridging through ten, part of adding within twenty.
Whole numbers of things. The parts are counts of counters, birds and cars, each 0, 1, 2 and so on. There are no halves in these facts, and adding never makes the total smaller than either part.
Seeing at a glance needs a small, tidy group. A dice face or a full five frame can be taken in by looking. A loose scatter of five is harder, and a child may need to count it, or split it into smaller groups first. Past five, seeing needs the help of a frame’s full row.
Starting from the bigger part is a choice the total allows. Counting on from 7 to find 2 + 7 works only because 2 + 7 and 7 + 2 are the same amount. The story may still start with the smaller number, and so may its number sentence.
One more than a double needs parts one apart. 4 + 5 is one more than 4 + 4. For 4 + 6 the parts are two apart, so it is two more than 4 + 4, or a pair that makes ten, which is quicker.
A known fact is still about counting. Every fact here is true for any objects at all, and counting real objects carefully is always the final check of it. Knowing a fact saves the count; it never overrules one.

Where this goes next

Each fact here is also a subtraction fact. Read from the total, 6 + 3 = 9 says that nine take away three leaves six, which is taking away and subtracting within ten, and the question “three and what make nine?” is subtraction as a missing part. The two additions and two subtractions from one bond make a fact family, and each fact is a way of reading a whole and its two parts.

The facts then carry bigger sums. Making ten turns 8 + 5 into 8 + 2 + 3 using the bond of 8 and 2, doing for bigger sums what a full frame does here. Adding ten to a number makes 10 + 6 a teen number, and adding three numbers looks for a double or a ten among them first. All of this sits inside adding and subtracting within twenty and feeds word problems. The two rules used again and again here, that order does not change a total and that zero changes nothing, are named much later in the properties of addition.

Exercises

Dominoes and ten frames

For each domino, ask: “How many dark dots? How many light dots? How many dots in all?” Then say the number sentence.

Figure 9Dominoes for Exercises 1 to 3.

Domino A.
Domino B.
Domino C.

For each ten frame, ask: “How many dark? How many light? How many in all?” Then say the number sentence.

Figure 10Ten frames for Exercises 4 to 6.

Frame D.
Frame E.
Frame F.

Number sentences

Find the number that goes in the box. Say how you found it.

3 + 1 = \square
1 + 4 = \square
\square = 1 + 1
2 + 3 = \square
2 + 6 = \square
5 + 4 = \square
0 + 9 = \square
4 + 6 = \square
4 + 5 = \square

Stories

Read each story aloud and show it with counters or fingers.

Two cats are on the bed. Two more cats jump up. How many cats are on the bed now?
There are three apples in a bowl and two apples on the table. How many apples are there in all?
Kim has five red beads and three blue beads. How many beads does she have?
Two children are on the slide. Seven more children come to play on the slide. How many children are on the slide now?
Ben puts four grapes on his plate. Then he puts on four more. How many grapes are on his plate?
A box holds ten crayons. There are six crayons in the box. Omar puts in three more. How many crayons are in the box now? Is it full?

Compare the cards

Which is more, 2 + 1 or 2 + 2?
Which is more, 3 + 1 or 1 + 3?
Which of these cards make 4? 2 + 2, 3 + 2, 4 + 0, 1 + 2, 1 + 3
Which is more, 4 + 4 or 4 + 5? Can you tell before you find the totals?

Explain

How can you know that 4 + 1 = 5 without counting all the counters?
Show 4 + 3 on a ten frame. How do you know how many there are without counting every counter?
Ava knows that 5 + 5 = 10. How can that help her find 5 + 4?
Leo says 8 + 1 is ten. Is he right? Show it on a ten frame.
Can you add two numbers that are each five or less and get more than ten? How do you know?

Answers

1 + 3 = 4. One dark dot and three light dots; start at three and one more is four.
4 + 1 = 5. Four dark dots and one light dot; one more than four is five.
2 + 0 = 2. Two dark dots and a blank half; adding nothing leaves two.
7 counters. 5 + 2 = 7: the top row is full, so five, and two more make seven. Three boxes are empty.
8 counters. 4 + 4 = 8: two rows of four, every counter with a partner, a double.
10 counters. 6 + 4 = 10: five on top and one dark below make six, four light make ten, and every box is full.
4. 3 + 1 = 4: the number after three.
5. 1 + 4 = 5: start from four, and one more is five.
2. 2 = 1 + 1: one and one more is two. The box is on the left, but the equals sign still says both sides are the same amount.
5. 2 + 3 = 5: two and three fill a five frame, a bond that makes five.
8. 2 + 6 = 8: start at the bigger part, six, and count on two: seven, eight.
9. 5 + 4 = 9: a full hand and four more; or one more than 4 + 4 = 8.
9. 0 + 9 = 9: an empty group changes nothing.
10. 4 + 6 = 10: four and six are a pair that fills the ten frame.
9. 4 + 5 = 9: one more than the double 4 + 4 = 8.
Four cats. 2 + 2 = 4: two and two, a double.
Five apples. 3 + 2 = 5: apples in the bowl and apples on the table are one group of apples.
8 beads. 5 + 3 = 8: five and three.
9 children. 2 + 7 = 9: start at seven, the bigger part, and count on two: eight, nine.
8 grapes. 4 + 4 = 8: a double.
9 crayons, and the box is not full. 6 + 3 = 9, and nine is one less than ten, so there is room for one more crayon.
2 + 2. Both start with two, and two is one more than one, so 2 + 2 is one more: 2 + 1 = 3 and 2 + 2 = 4.
Neither; they are the same. The same two numbers in the other order make the same amount: 3 + 1 = 4 and 1 + 3 = 4.
2 + 2, 4 + 0 and 1 + 3. Each makes 4: 2 + 2 = 4, 4 + 0 = 4, 1 + 3 = 4. The other two do not: 3 + 2 = 5 and 1 + 2 = 3.
4 + 5. Both start with four, and five is one more than four, so 4 + 5 has one more, and that can be told before finding either total. Then check: 4 + 4 = 8 and 4 + 5 = 9.
Adding one is saying the next number in the count, and the number after four is five. Or: four dark counters and one light one fill a five frame, and a full frame is five.
4 + 3 = 7. Four counters and one more fill the top row, so that row is five. The other two go in the bottom row. Five and two is seven, and three boxes are still empty.
5 + 4 = 9. Four is one less than five, so 5 + 4 is one less than 5 + 5. One less than ten is nine.
No. 8 + 1 = 9. Put eight counters in a ten frame, then one more: one box is still empty, so there are nine. Eight needs two to make ten.
No. The biggest each number can be is five, and 5 + 5 = 10. Any smaller number in place of a five makes the total smaller, so the total is never more than ten.