Grade 1 · 1.OA.C.6 · 2.OA.B.2

Doubles and near doubles

Two groups have the same number of things in them. How many are there altogether, and why are those sums often the first ones a child knows without counting? And once a child knows that six and six make twelve, does she still have to count to find six and seven?

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Adding within five and tenKindergarten

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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 double is a sum in which both parts are the same number: {4 + 4}, {7 + 7}, {10 + 10}. The world is full of them, because so many things come in two matching halves. Two hands hold {5 + 5} fingers. A spider has 4 legs on each side. A box of twelve eggs has two rows of 6. Two dice that land on the same face, the left and right wings of a butterfly, a child’s own reflection: each one shows the same amount twice.

This is why doubles are usually the first sums a child knows without counting. The picture is simple and it is always the same shape: two equal rows, each thing in one row with a partner in the other.

Figure 1A double is two equal rows. Every counter has a partner, and the bond says 4 and 4 make 8.

Figure 1 shows {4 + 4} both ways: as two rows of counters, and as a number bond with the whole in the top circle and the two parts below, the same picture as in Parts, wholes and number bonds. The only thing special about a double’s bond is that both part circles hold the same number.

A child reaches for a double in three kinds of situation. When the two parts are the same, the double is the answer. When two amounts are the same size and need putting together, such as two teams of 9, the double finds the total. And when something is to be shared fairly between two, or a double is hiding inside a take-away, the double read backwards gives the answer.

The doubles then do a second job, for the sums that sit right beside them. In {6 + 7} the two parts are next-door numbers: seven comes straight after six when we count. A sum like that is called a near double. It is a double with one more on top, or, looked at from the other side, a double with one taken away. A child who knows {6 + 6 = 12} can find {6 + 7} in one short step instead of counting on six more from seven.

The move is worth teaching because it is cheap. Within twenty there are only nine pairs of next-door numbers from {1 + 2} up to {9 + 10} (and {0 + 1}, which needs no help), and each can be turned around, so the doubles a child already has unlock close to twenty more facts with nothing new to learn. It is also one of the first times a child builds a fact they do not know out of a fact they do, which is the habit behind every later mental strategy.

The signal to reach for a near double is simple: look at the two parts and ask, “Are these two numbers next to each other when we count?” Four and five, yes. Eight and nine, yes. Six and eight, no, though the last worked example shows how the same idea stretches to cover them.

The procedure

There is nothing to learn by rote here. There are eleven doubles from {0 + 0} to {10 + 10}, and each one can be seen, built and checked. The work is to see each double, to use one double to find the next, to read a double backwards, and then to lean on the doubles for the near doubles beside them.

Doubles

Seeing each double. Here are the doubles from {1 + 1} to {10 + 10}, with a place a child might meet each one:

The double	The total	Where you might see it
{1 + 1}	2	a pair of socks
{2 + 2}	4	a car’s wheels, two at the front and two at the back
{3 + 3}	6	an ant’s legs, three on each side
{4 + 4}	8	a spider’s legs, four on each side
{5 + 5}	10	the fingers on two hands
{6 + 6}	12	a box of twelve eggs, two rows of six
{7 + 7}	14	the days in two weeks
{8 + 8}	16	the legs on two spiders
{9 + 9}	18	the players in a baseball game, nine on each team
{10 + 10}	20	all the fingers and all the toes

Fingers are the handiest picture of all for the small doubles. Hold up the same fingers on both hands, one little finger on each, then two fingers on each, and so on: the fingers come in matching pairs, and two whole hands are {5 + 5 = 10}. Dice and dominoes give the same pictures in dots, since a child who knows the dot patterns sees four at a glance and then sees it again.

Using one double to find the next. Put the doubles in order and a pattern appears. The totals go 2, 4, 6, 8, 10, and on through 12, 14, 16, 18, 20. Each total is two more than the one before.

Figure 2Each double is one more dark counter and one more light counter than the one before, so its total is two more.

Figure 2 shows why. To go from {3 + 3} to {4 + 4}, one more counter joins the dark row and one more joins the light row. That is two more counters in all, so the total goes from 6 to 8. This is the reason, and it is worth saying aloud with the child: each part gets one more, so the whole gets two more. The totals are the numbers said when counting by twos. (They are also the even numbers, which have their own lesson, Even and odd numbers.)

The pattern works both ways. A child who knows {5 + 5 = 10} can reach {6 + 6} by adding two, and can reach {4 + 4} by taking two away.

Doubles past five. From {6 + 6} to {10 + 10}, each part is bigger than a hand, and counting starts to get long. The ten frame (see Subitizing and ten frames) helps here. Put one part in one frame and the other part in a second frame. Each frame then holds a full row of five and some more.

Figure 3The two full rows make 10 and the two short rows make 4, so 7 + 7 = 14.

In Figure 3 the two full rows are {5 + 5}, which is 10. The two short rows are {2 + 2}, which is 4. Ten and four make fourteen, so {7 + 7 = 14}. Every double from {6 + 6} to {10 + 10} splits the same way, into {5 + 5} and a small double:

The double	Two fives	Two extras	Altogether
{6 + 6}	10	{1 + 1 = 2}	12
{7 + 7}	10	{2 + 2 = 4}	14
{8 + 8}	10	{3 + 3 = 6}	16
{9 + 9}	10	{4 + 4 = 8}	18
{10 + 10}	10	{5 + 5 = 10}	20

So the big doubles are built from the small ones and ten. Putting ten together with a small number to make a teen number is its own lesson, Ten more and ten less.

Reading a double backwards. Sometimes the total is known and the two equal parts are not. Written down, it looks like {\square + \square = 12}, read aloud as “what number, added to itself, makes 12?” Each box stands for a number we have not found yet, and both boxes stand for the same number. This is the question behind sharing fairly between two: 12 crayons shared so that two children have the same number. A child who knows the doubles can go through them, {5 + 5 = 10} is too few and {6 + 6 = 12} is right, so each part is 6. A child who does not yet know them can deal the crayons out, one to each child in turn, and count one child’s pile.

The same backwards reading helps with taking away. {12 - 6} asks what is left when 6 are taken from 12, and since {6 + 6 = 12}, what is left is the other 6. That is subtraction as a missing part, and doubles make it easy because both parts are already known.

Near doubles

A child may already have met the first of these, {3 + 4} and {4 + 5}, as one more than a double in Adding within five and ten. Here the idea is taken all the way to twenty and seen from both sides.

Find the double hiding inside. Line up the two parts in two rows, dark counters for the first part and light counters for the second part underneath, one counter under another.

Figure 4Six and seven is six and six with one counter left over, so 6 + 7 = 13.

Every dark counter has a light partner under it, and exactly one light counter has no partner. The pairs are a double: six and six, which is twelve. The spare counter is one more. So six and seven is thirteen:

6 + 7 = 6 + 6 + 1 = 13

This is allowed because seven is six and one more; that is simply what “next-door number” means. So the counters for {6 + 7} are exactly the counters for {6 + 6} with one extra, and adding one to a number is a single count forward. In words for the child: double the smaller number, then one more.

Or use the bigger double. The same counters can be seen a second way. Imagine the top row were as long as the bottom row. Then it would be a double of the bigger number, but one of those counters is not really there, so it has to come off again.

Figure 5One set of counters, two doubles. Seven and seven is one short; eight and eight is one too many.

For seven and eight both ways work, and they must agree, because they describe one set of counters:

7 + 8 = 7 + 7 + 1 = 15 \qquad 7 + 8 = 8 + 8 - 1 = 15

The double of the smaller number is always one short, and the double of the bigger number is always one too many. That is the whole of the method. Which double to use is the child’s choice, and the right choice is the double they are surest of. Many children know {5 + 5 = 10} and {10 + 10 = 20} best of all, so {5 + 6} tends to go through {5 + 5} and {9 + 10} through {10 + 10}. Taking one away is one count back, so the “one less” route is no harder than the “one more” route once counting back is familiar.

The order of the parts does not matter. {7 + 6} is the same amount as {6 + 7}, since the two rows of counters can be swapped without adding or losing any (see Addition as putting together). So the double to look for is fixed by the two numbers, not by which is written first: for {7 + 6} it is still {6 + 6} and one more, or {7 + 7} and one less.

Here are all the near doubles within twenty, each built both ways.

Near double	Smaller double, one more	Bigger double, one less	Total
{1 + 2}	{1 + 1 + 1}	{2 + 2 - 1}	3
{2 + 3}	{2 + 2 + 1}	{3 + 3 - 1}	5
{3 + 4}	{3 + 3 + 1}	{4 + 4 - 1}	7
{4 + 5}	{4 + 4 + 1}	{5 + 5 - 1}	9
{5 + 6}	{5 + 5 + 1}	{6 + 6 - 1}	11
{6 + 7}	{6 + 6 + 1}	{7 + 7 - 1}	13
{7 + 8}	{7 + 7 + 1}	{8 + 8 - 1}	15
{8 + 9}	{8 + 8 + 1}	{9 + 9 - 1}	17
{9 + 10}	{9 + 9 + 1}	{10 + 10 - 1}	19

An adult will notice that every total in the last column is odd. That is not a coincidence. A double always shares out into two equal rows with nothing left over, which is what makes it even, and a near double is a double with one counter left without a partner. So its total is always odd, and a child who gets an even total for a near double knows to look again.

A near double also finds a missing part. Suppose the whole is 13 and one part is 6, written {6 + \square = 13}. The double of six is twelve, and thirteen is one more than twelve, so the missing part is one more than six: seven. This is the idea of subtraction as a missing part with a near double doing the work, and it answers {13 - 6} as well.

Worked examples

Read each problem aloud. Counters in two colors, two dice, a small mirror and two ten frames drawn on paper let a child build every one of these. The first six are doubles; the last six use a double for the sums beside it.

Example 1

Two dice

Sam rolls two dice. Both dice show 4. How many dots are there in all?

Figure 6Both dice show 4, and every dot on one has a partner on the other.

Solution

Both dice show the same number, so this is a double: 4 and 4. Look at Figure 6. Each dot on one of the dice has a partner on the other, in the same corner. Four dots and four more dots make eight dots:

4 + 4 = 8

To check, point at a matching pair of dots each time and count by twos: “two, four, six, eight”. Four pairs, eight dots. Before long a child sees the two dice and says “double four is eight” without counting at all.

Example 2

A mirror

Put 5 counters in a row next to a mirror. How many counters can you see, counting the ones in the mirror?

Figure 7A mirror shows exactly as many again, so it always makes a double.

Solution

A mirror shows exactly the same number of counters as there are in front of it. So there are 5 counters in the row and 5 in the mirror. Five and five make ten:

5 + 5 = 10

This is the same double as two hands. A child can hold up one hand, five fingers, to the mirror and see ten fingers. It is also the two rows of a ten frame, one full row and then another.

Example 3

Seven and seven

What is {7 + 7}?

Solution

Put 7 counters in one ten frame and 7 more in a second frame, as in Figure 3. Each frame has a full top row of 5 and then 2 in the bottom row. The two full rows make 10, because {5 + 5 = 10}. The two short rows make 4, because {2 + 2 = 4}. Ten and four make fourteen:

7 + 7 = 14

Counting on from 7 also works, “eight, nine, ten, eleven, twelve, thirteen, fourteen”, but that is seven counts to keep track of. The frames turn it into two small facts the child already has.

Example 4

A tug of war

In a tug of war, 9 children pull on one end of the rope and 9 children pull on the other end. How many children are pulling?

Solution

Both teams have 9, so this is the double {9 + 9}. There are two good ways to find it, and it is worth showing both.

The first way uses ten. A team of 9 is one child short of 10. Two teams of 10 would be 20, since {10 + 10 = 20}. Each team is one short, so together they are two short of 20, and two less than 20 is 18. In two ten frames, each frame has one empty box, two empty boxes in all.

The second way splits each team into 5 and 4. The two fives make 10 and the two fours make 8, so there are 10 and 8, which is 18 children:

9 + 9 = 18

The two ways agree, which is a good check. A child who likes one way better should use it.

Example 5

Sharing crayons

Ava has 12 crayons. She shares them with her brother so that they both have the same number. How many crayons does each child get?

Solution

The whole is 12 and it is split into two equal parts, so the question is a double read backwards:

\square + \square = 12

Try the doubles in order. {5 + 5 = 10} is too few; two crayons would be left over. {6 + 6 = 12} is exactly right. So each child gets 6 crayons. As a number bond, it is 12 on top with 6 and 6 below.

A child can also act it out: give one crayon to Ava, one to her brother, one to Ava, one to her brother, until the crayons run out. Each pile then has 6.

Example 6

Children going out to play

There are 16 children in a class. Then 8 children go out to play. How many children are still in the classroom?

Solution

The whole is 16 and one part, the children outside, is 8. The children still inside are the other part: 8 and how many make 16? The double {8 + 8 = 16} answers it straight away. So 8 children are still in the classroom:

16 - 8 = 8

To be sure of {8 + 8}, split each 8 into 5 and 3: the fives make 10 and the threes make 6, so {8 + 8 = 16}. This is the idea of subtraction as a missing part, with a double doing the work.

Example 7

Cars on the mat

Kai has 4 red cars and 5 blue cars. How many cars does he have?

Solution

Four and five are next-door numbers, so there is a double inside. Line the cars up in two rows, red above blue. Each red car has a blue car under it, and one blue car is left over. The pairs are four and four, which is eight, and the spare car makes nine:

4 + 5 = 4 + 4 + 1 = 9

Kai has 9 cars. Check. Counting on from five, “six, seven, eight, nine”, is four counts and lands on nine too.

Example 8

The bigger number first

What is {6 + 5}?

Solution

The bigger number comes first here, but the double to find does not change: the smaller number is five, and five and five is ten, the two full hands. One more than ten is eleven:

6 + 5 = 5 + 5 + 1 = 11

A child who knows six and six better can go the other way. Six and six is twelve, and five is one less than six, so take one away: eleven. Both routes give 11, as they must.

Example 9

Eight and nine in two ten frames

Put 8 counters in one ten frame and 9 counters in the other. How many counters are there?

Solution

Figure 8The right frame holds everything the left one does, and one more.

The right frame has every counter the left frame has, and one more at the end. So the two frames together hold eight and eight, which is sixteen, and one more, which is seventeen:

8 + 9 = 8 + 8 + 1 = 17

Or: if the left frame also had nine, there would be nine and nine, which is eighteen; it has one fewer, so there are seventeen. The frames give a third check as well. The two full top rows make ten, and the bottom rows hold 3 and 4, which make seven, and ten and seven is seventeen.

Example 10

Apples in a bowl

There are 13 apples in a bowl. 6 apples are red. The rest are green. How many apples are green?

Solution

The whole is 13 apples and one part is the 6 red ones, so the question is “six and what make thirteen?” Six and six make twelve. Thirteen is one more than twelve, so the green part must be one more than six, which is seven:

6 + 7 = 13 \qquad\text{so}\qquad 13 - 6 = 7

There are 7 green apples. Check. {6 + 7 = 6 + 6 + 1 = 13}. The near double turned a take-away question into one step from a known double.

Example 11

Which double?

Find {9 + 10}.

Solution

Nine and ten are next-door numbers, so either double will do. Nine and nine is eighteen, and one more is nineteen. Ten and ten is twenty, and one less is nineteen. A third way needs no double at all: ten and nine is a ten and nine ones, which is nineteen (see Ten more and ten less).

9 + 10 = 9 + 9 + 1 = 10 + 10 - 1 = 19

All three agree, so the child should take the one they are surest of. Many children know {10 + 10 = 20} more firmly than {9 + 9 = 18}, and for them the “one less” route is the natural one.

Example 12

Two apart

Find {6 + 8}.

Solution

Six and eight are not next-door numbers: seven sits between them, so this is not a near double. It is a stretch of the same idea, because a double is still close. Build six dark counters over eight light ones, then move the last light counter up to the end of the top row.

Figure 9Moving one counter from the bigger part to the smaller turns 6 + 8 into the double 7 + 7.

Now there are seven counters in each row. No counter was added and none was taken away, only moved, so the total has not changed. Seven and seven is fourteen:

6 + 8 = 7 + 7 = 14

Check. Six and six is twelve, and eight is two more than six, so two more than twelve is fourteen. Making ten agrees as well: eight and two make ten, and four more make fourteen. Moving one across to make two parts equal is a first taste of compensation, changing the parts while keeping the total.

Boundaries

A double needs both parts the same. {4 + 5} is not a double, because the parts are different. It is one more than the double {4 + 4}, which makes it a near double.
Only some totals come from a double. The totals of doubles are 0, 2, 4, 6 and so on up to 20. There is no double that makes 7: {3 + 3 = 6} is one short and {4 + 4 = 8} is one too many, and the things being counted come whole, so no number between 3 and 4 will do. The totals of doubles are the even numbers, which is where Even and odd numbers picks this up.
Zero counts. {0 + 0 = 0} is a double too: two empty groups make an empty group.
A double has only two facts in its family. For most bonds, swapping the parts gives a new fact, as {3 + 5} and {5 + 3} do. For a double, swapping changes nothing, so {6 + 6 = 12} and {12 - 6 = 6} are the whole fact family.
Doubling adds a number to itself, once. Double three is {3 + 3 = 6}: two groups of three. Three rows of three is a different thing, 9, and belongs to Square numbers much later.
A near double needs next-door numbers. “One more” or “one less” is exactly right only when the two numbers differ by one. When they differ by two, as in {6 + 8}, the sum is not a near double, but the idea stretches: moving one counter across gives the double of the number in the middle, or the smaller double and two more also works. When they differ by three or more, as in {5 + 8}, the double of the smaller number is still true ({5 + 5 + 3}) but the step after it is no longer small, and making ten is usually quicker.
The double comes first. A near double borrows from a double the child already knows. If {7 + 7} is not yet known, {7 + 8} has nothing to lean on, and counting on from eight is the honest way to find it.
One more goes on the smaller double; one less comes off the bigger. The double of the smaller number is one short of the near double and the double of the bigger number is one over, so each route has its own direction.
The total of a near double is always odd. It is a double, which is even, and one more. An even total for a near double means something has slipped.
This lesson stops at ten and ten. Everything here counts whole things: counters, cars and apples. Bigger numbers can be doubled too, and the same ideas carry over: {30 + 30} is three tens and three tens, which is six tens, 60, and {25 + 26 = 25 + 25 + 1} as soon as the bigger doubles are known. That comes with place value.

Where this goes next

With doubles and near doubles a child has a quick way to every sum whose two parts are the same or one apart. They belong with making ten as the main ways of turning an unknown fact into a known one, and together with the bonds within twenty they cover most of adding and subtracting within twenty. Every near double is a full fact family of four facts: {6 + 7 = 13} gives {7 + 6 = 13}, {13 - 6 = 7} and {13 - 7 = 6}.

Looking for a double speeds up adding three numbers, where {4 + 3 + 4} is easier as the double {4 + 4 = 8} and then 3 more. The “move one across” of the last example grows into compensation for bigger numbers.

Read backwards, doubles are sharing fairly between two, which is the start of halving and dividing. Read forwards, a double is two equal groups, and “two groups of seven” is the first step toward multiplication as equal groups. The near-double habit returns there too: seven sixes are found from six sixes and one more six, in Facts for three, four and six. The totals of doubles, 2, 4, 6, \ldots, are the numbers of counting by twos, and with the odd totals of the near doubles they lead to even and odd numbers.

Exercises

Find the double

{1 + 1}
{3 + 3}
{5 + 5}
{8 + 8}
{10 + 10}

Pictures and stories

Figure 10Pictures for Exercises 6 to 8.

Look at picture A in Figure 10. How many dots are there on the two dice?
Look at picture B. How many counters can you see, counting the ones in the mirror?
Look at picture C. How many counters are in the two ten frames?
A box of eggs has two rows. Each row holds 6 eggs. How many eggs does the full box hold?

Doubles backwards

{\square + \square = 14}. Both boxes are the same number. What is it?
{\square + \square = 6}. Both boxes are the same number. What is it?
Jack has 18 stickers. He shares them with his friend so that they both have the same number. How many stickers does each of them get?
{20 - 10}

Near doubles in pictures

Use the pictures in Figure 11 for Exercises 14 to 16. For each one, ask: “Where is the double? How many are there in all?” Then say the number sentence.

Figure 11Pictures for Exercises 14 to 16.

Picture A.
Picture B.
Picture C.

Find the double that helps

Say the double that helps, then find the total.

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

Missing parts and stories

Read each one aloud and show it with counters.

{7 + \square = 15}
{5 + \square = 11}
Ben builds a tower of 7 blocks. Ada builds a tower with one more block than Ben’s. How many blocks are in the two towers together?
There are 17 grapes in two bowls. One bowl has 8 grapes. How many grapes are in the other bowl?
Find {5 + 7}. Can you move one counter to make a double?

Explain

Is {4 + 5} a double? How do you know?
Can you put 7 counters into two groups with the same number in each? Show it with counters.
Say the doubles {1 + 1}, {2 + 2}, {3 + 3}, {4 + 4} and {5 + 5}. How much bigger is the total each time? Why?
Kim knows that {5 + 5 = 10}. How can that help her find {6 + 6}?
Show {4 + 4} and {4 + 5} with counters. How do you know {4 + 5} is one more than {4 + 4}?
To find {8 + 9}, Zoe thinks “{9 + 9} and take one away.” Ravi thinks “{8 + 8} and one more.” Who is right? What is {8 + 9}?
Put out counters for {6 + 7} and pair them up, two by two. Is one left over? Will a near double always leave one over? How do you know?
Can a double help you find {5 + 8}? Is it a near double? Find the total.

Answers

2. One and one more: {1 + 1 = 2}, a pair.
6. Three fingers on each hand, or an ant’s legs: {3 + 3 = 6}.
10. Two whole hands: {5 + 5 = 10}.
16. Each 8 is 5 and 3. The fives make 10 and the threes make 6, so {8 + 8 = 16}.
20. Two full ten frames: {10 + 10 = 20}.
4 dots. Both dice show 2, so {2 + 2 = 4}.
8 counters. There are 4 in the row and the mirror shows 4 more, so {4 + 4 = 8}.
18 counters. Each frame holds 9, one short of ten, so two boxes are empty and the frames hold two less than 20: {9 + 9 = 18}. Or the full rows make 10 and the rows of 4 make 8.
12 eggs. Each 6 is 5 and 1. The fives make 10 and the ones make 2, so {6 + 6 = 12}.
7, because {7 + 7 = 14}. Trying 6 gives {6 + 6 = 12}, too few, and 8 gives 16, too many.
3, because {3 + 3 = 6}: three fingers on each hand.
9 stickers each. The two equal parts make 18, and {9 + 9 = 18}.
10. Taking 10 from 20 leaves the other part of the double {10 + 10 = 20}, so {20 - 10 = 10}.
7. Three dark counters each have a light partner, and one light counter is spare: {3 + 3 = 6} and one more, so {3 + 4 = 7}.
11. Five light counters each have a dark partner, and one dark counter is spare: {5 + 5 = 10} and one more, so {6 + 5 = 11}.
15. The right frame has seven, the left has one more: {7 + 7 = 14} and one more, so {8 + 7 = 15}.
3. {1 + 1 = 2} and one more: {1 + 2 = 3}.
5. {2 + 2 = 4} and one more: {2 + 3 = 5}.
9. {4 + 4 = 8} and one more: {5 + 4 = 9}.
13. {6 + 6 = 12} and one more: {7 + 6 = 13}. Or {7 + 7 = 14} and one less.
17. {8 + 8 = 16} and one more: {9 + 8 = 17}. Or {9 + 9 = 18} and one less.
8. {7 + 7 = 14}, and 15 is one more than 14, so the missing part is one more than 7: {7 + 8 = 15}.
6. {5 + 5 = 10}, and 11 is one more than 10, so the missing part is one more than 5: {5 + 6 = 11}.
15 blocks. Ada’s tower has 8 blocks, so the towers hold {7 + 8}. {7 + 7 = 14} and one more: {7 + 8 = 15}.
9 grapes. {8 + 8 = 16}, and 17 is one more than 16, so the other bowl has one more than 8: {8 + 9 = 17}.
12. Five and seven are two apart. Move one counter from the 7 to the 5 and both rows have six: {5 + 7 = 6 + 6 = 12}.
No. In a double both parts are the same, and 4 and 5 are different. Put 4 dark counters in a row and 5 light counters under them: one light counter has no partner.
No. Put out 7 counters and share them into two rows one at a time. After 3 each, {3 + 3 = 6} counters are used and one is left over. Giving that one to a row makes 4 and 3, which are not the same. 4 each would need {4 + 4 = 8} counters, one more than there are.
2 more each time: 2, 4, 6, 8, 10. Going from one double to the next, each part gets one more, one more in each row, so the whole gets two more.
In {6 + 6} each part is one more than in {5 + 5}, so the total is two more than 10: 12. Or split each 6 into 5 and 1: {5 + 5 = 10} and {1 + 1 = 2}, so {6 + 6 = 12}.
Put four dark counters over four light ones: that is {4 + 4 = 8}. Add one more light counter to the bottom row and the picture shows {4 + 5}. It is the same eight counters and one more, so {4 + 5 = 9}.
Both are right, and both get 17. Ravi: {8 + 8 = 16} and one more is 17. Zoe: {9 + 9 = 18}, but there are only eight in one part, so one comes off: 17. They used different doubles of the same counters, so {8 + 9 = 17}.
Yes, one is left over. {6 + 7 = 13}, and the 13 counters make 6 pairs and 1 spare, so 13 is odd. Every near double leaves one over: its counters are a double, which pairs up exactly, and one more counter with no partner. So a near double always makes an odd total.
13. It is not a near double, because 5 and 8 are three apart. A double still helps a little: {5 + 5 = 10} and three more is 13. Making ten works too: {8 + 2 = 10} and three more is 13. So {5 + 8 = 13}.