Grade 2 · 2.OA.C.3

Even and odd numbers

Some piles of things can be put into pairs with nothing left over, and some always leave one thing on its own. How can we tell which kind a number is, even a big number we would rather not count out, and what happens when two such numbers are added together?

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Counting to 100 and beyondKindergarten

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Even and odd numbers

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When you reach for it

On a field trip, children often walk in a buddy line: two by two, each child holding a partner’s hand. Before the bus leaves, the teacher has one question. Will every child have a partner, or will one child be left on their own?

The answer does not depend on how big the group is. Twelve children pair up exactly; thirteen leave one child alone. Forty pair up exactly; three do not. Every whole number falls into one of these two kinds, and the kinds have names.

> A number is even when that many things can be put into pairs with nothing left over. A > number is odd when putting that many things into pairs leaves one thing over.

Pairing is the test, and the definition says nothing else. Every other way of telling even from odd in this lesson, the doubles, the ones digit and the rules for sums, comes back to this picture and is true because of it.

Two facts follow at once. First, the leftover is never more than one. If two things were still without partners, they could be partners for each other, so the pairing was not finished. When pairing is done there is either nothing left, and the number is even, or exactly one thing left, and the number is odd. Second, zero is even. With no things at all there is nothing to pair and nothing left over, and “nothing left over” is exactly what even means.

A child reaches for this test whenever things go two by two: partners for a game, socks into pairs, two teams of the same size, a bag of marbles shared fairly between two friends. “Will everyone have a partner?” and “can this be shared equally between two?” are the same question, since sharing between two is handing out one pair at a time, one thing to each.

The procedure

The lesson runs over three days. Part 1 builds the meaning by pairing real things and connects it to the doubles. Part 2 finds a shortcut for big numbers: the ones digit. Part 3 puts two numbers together and asks whether the total is even or odd before adding.

Part 1: pair them up

Pairing objects. Give a child some counters, marbles or linking cubes and ask them to push the things together two at a time, setting each pair aside. When nothing more can be paired, look at what is left: nothing, or one.

The pairs are easiest to see lined up the way a buddy line walks, side by side. Put one thing from each pair in a top row and its partner just below it. Figure 1 shows 17 children set out this way.

Figure 1Seventeen children in a buddy line make 8 pairs, and one child has no partner.

There are 8 columns with a child on top and a child below: 8 pairs, which is 16 children. The seventeenth child has an empty place underneath. So 17 is odd.

Children who count by twos will notice that counting the pairs two at a time from 0 lands on exactly the numbers that pair up with nothing left; counting by twos has its own lesson.

Pairs make a double. Look at the two rows of a picture that pairs up exactly. The top row holds one child from each pair and the bottom row holds the other, so the two rows hold the same number. The total is therefore a double, a sum of two equal numbers, the sums met in Doubles and near doubles. For 16 children in 8 pairs, the rows hold 8 and 8:

16 = 8 + 8

It works the other way too. A double such as 9 + 9 is two equal rows, so every thing in the top row has a partner below it, and nothing is left over.

> An even number is a double: it is the sum of two equal numbers. An odd number is not a double. It > is a double and one more.

So 17 is not a double. 8 + 8 = 16 is one too few and 9 + 9 = 18 is one too many. Its two rows hold 9 and 8, and 17 = 8 + 8 + 1.

A child may check “even” either way, by pairing things or by finding the double. They are one picture read two ways, so they always agree. The double is not being taught here as a way to add fast; it is simply what the two rows of pairs say.

Taking turns. Pair up each number of counters from 0 to 20 and a pattern shows:

Even: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20.
Odd: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19.

Even and odd take turns, and the pairing says why. Each number is one more than the number before it, so it is the same pile with one more thing added. If the pile was even, the new thing has no partner, and the number is odd. If the pile was odd, the new thing becomes the partner of the one that was alone, and the number is even.

Part 2: the ones digit tells

Pairing 24 marbles one pair at a time is slow, and pairing 86 is slower. Place value gives a shortcut, the same one that makes it quicker to count a big pile by grouping it into tens.

Think about one ten. Put ten counters in a ten frame, two rows of five. Each column holds one counter on top and one below: a pair. So a full ten frame is 5 pairs with nothing left over, and 10 = 5 + 5. Ten is even.

Now take a number made of tens and ones, such as 24: two tens and four ones (Figure 2).

Figure 2Each column of a ten frame is a pair, so a full frame is 5 pairs. Only the ones can leave a counter over.

Each ten pairs up by itself, with nothing left. That is true however many tens there are: two tens, five tens, nine tens, all of them pair up. The only counters that might be left over are the ones. Here the 4 ones make 2 pairs, so nothing is left, and 24 is even. To tell whether a number is even or odd, then, only its ones digit matters.

> A number is even when its ones digit is 0, 2, 4, 6 or 8. It is odd when its ones digit > is 1, 3, 5, 7 or 9.

This is not a new rule to set beside the pairing. It is the pairing done in a hurry: the tens are already known to pair up, so only the ones need checking. The tens digit has no say at all, so 74 is even and 47 is odd. The same holds past a hundred, since a hundred is ten tens and every ten pairs up. 131 is odd for the same reason 1 is.

A big even number as a double. To split an even number into two equal rows, share out its tens and ones between the rows. When the number of tens is even, as in 24, each row gets half the tens and half the ones: one ten and two ones each, so 24 = 12 + 12. When the number of tens is odd, one ten is left with no row to go to. That ten is 5 pairs, so break it into 5 and 5 and give each row five. Example 5 does this for 34.

Part 3: adding even and odd numbers

Now pour two piles together. Without adding, can we tell whether the total is even or odd? The pairing picture answers it (Figure 3).

Figure 3The spare from one odd number pairs with the spare from the other. With only one odd number, its spare stays alone.

Even plus even. Both piles pair up with nothing left. Pushed together, the pairs stay pairs and there is still nothing left over, so the total is even. In the first picture, 4 + 6 = 10.

Odd plus odd. Each pile pairs up with one thing left over. Pushed together, the two left-over things become partners for each other. Nothing is left, so the total is even. In the second picture the spare dark counter from the 5 and the spare light counter from the 3 make the pair inside the dashed loop, and 5 + 3 = 8.

Even plus odd. One pile has no spare and the other has one. That one spare has nobody to pair with, so the total is odd. In the third picture 4 + 3 = 7, and the last light counter stands alone. Odd plus even is the same two piles added in the other order, which by the commutative property gives the same total, so it is odd too.

First number	Second number	The total
even	even	even
odd	odd	even
even	odd	odd
odd	even	odd

The table does two jobs. It tells a child whether a sum is even or odd before working it out, and it gives a quick check afterward: 9 + 8 is odd plus even, so its total must be odd, and the total 17 is. A total that came out even would send the child back to add again.

With three or more numbers, look at the spares. Each odd number brings one spare, and every two spares make a pair. If the spares pair off with none left, the total is even; if one spare is left, the total is odd.

Worked examples

Examples 1 to 3 go with Part 1, Examples 4 and 5 with Part 2, and Examples 6 and 7 with Part 3.

Example 1

The buddy line

Casimiro’s class has 17 children going on the field trip. They walk in a buddy line, two by two. Will every child have a partner?

Solution

Pair the children up. Figure 1 shows them in two rows, each child beside a partner. There are 8 full pairs, which is 16 children, because 8 + 8 = 16. One child is left over, since 16 + 1 = 17. So 17 is odd, and one child has no partner. (A grown-up on the trip can walk with that child, or three children can walk together.)

Check. The two rows hold 9 and 8. They are not equal, so 17 is not a double, which agrees: an odd number is never a double.

Example 2

Sharing marbles

Tuyet has a bag of 15 marbles. She wants to share all of them with Ragna so that the two girls have the same number. Can she?

Solution

Sharing fairly between two is the same as pairing: each time Tuyet takes out two marbles, one goes to her and one to Ragna. So the question is whether 15 is even. Pair the marbles. 7 pairs use 14 marbles, because 7 + 7 = 14, and one marble is left over. So 15 is odd, and the marbles cannot all be shared equally. Each girl gets 7 and one marble stays in the bag.

Check. Look for a double instead. 7 + 7 = 14 is one short and 8 + 8 = 16 is one too many, so no double makes 15.

Example 3

Which are even?

Sort 6, 11, 0 and 20 into even and odd. Write each even one as a double.

Solution

Pair up counters for each number.

6 counters make 3 pairs with none left. Even, and 6 = 3 + 3.
11 counters make 5 pairs and 1 left over. Odd: 5 + 5 = 10 is one short and 6 + 6 = 12 one too many.
0 counters make no pairs and leave nothing over. Nothing left over is what even means, so 0 is even, and 0 = 0 + 0.
20 counters fill two ten frames, 10 pairs with none left. Even, and 20 = 10 + 10.

So 6, 0 and 20 are even and 11 is odd. Zero is the one where there is nothing to look at, so the definition itself has to settle it, and it does.

Example 4

Ragna’s marbles

Ragna has 24 marbles. Without pairing every marble, decide whether 24 is even or odd. If it is even, split the marbles into two equal groups.

Solution

24 is 2 tens and 4 ones, as in Figure 2. Each ten is 5 pairs, so the tens make 10 pairs with none left. The 4 ones make 2 more pairs with none left. Altogether that is 12 pairs and nothing over, so 24 is even. The ones digit, 4, says the same.

For two equal groups, let each pair give one marble to each group. There are 12 pairs, so each group gets 12 marbles. Read as tens and ones, each group got one ten and two ones.

Check. 10 + 10 = 20 and 2 + 2 = 4, so 12 + 12 = 24.

Example 5

Two classes in one line

Casimiro’s class joins another class for the trip, and 34 children walk in one buddy line. Will every child have a partner? How many children walk on each side of the line?

Solution

The ones digit of 34 is 4, so 34 is even: everyone has a partner. The two sides of the line are the two rows of the pairing picture, so they hold the same number, and we need the double that makes 34.

Share the tens and ones between the two sides. 34 is 3 tens and 4 ones. Each side gets one ten and two ones, 12 children, which uses 2 tens and all 4 ones. One ten is left. A ten is 5 pairs, so it splits into 5 and 5, and each side gets 5 more. Each side has 12 + 5 = 17 children.

Check. 10 + 10 = 20 and 7 + 7 = 14, and 20 + 14 = 34. So 34 = 17 + 17: 17 children on each side, in 17 pairs.

Notice that 17 itself is odd, yet 17 + 17 is even. Every double is even, whatever number is doubled, because its two rows match.

Example 6

Marbles in one bag

Tuyet has 9 red marbles and Ragna has 8 blue marbles. They pour them into one bag. Before adding, decide whether the marbles in the bag can all be put into pairs. Then add to check.

Solution

9 is odd: 4 pairs and 1 left over. 8 is even: 4 pairs and none left. In the bag the pairs stay pairs, and the one red marble that was left over finds no spare blue marble to pair with. Odd plus even is odd, so the marbles cannot all be paired; one will be left.

Now add. Take 1 from the 8 to make ten with the 9: 9 + 1 = 10, and 7 more is 17. So 9 + 8 = 17. The ones digit of 17 is 7, which is odd, as predicted: 8 pairs and 1 left over.

Example 7

Three groups at the bus

On the way back, three groups meet at the bus: Casimiro’s group of 7, Tuyet’s group of 5 and Ragna’s group of 6. All of them make one buddy line. Will anyone be left without a partner?

Solution

Look at the spares. 7 is odd, so pairing its children leaves one alone. 5 is odd and leaves one alone. 6 is even and leaves none. The two spare children, one from Casimiro’s group and one from Tuyet’s, become partners. Nobody is left, so the total is even and everyone has a partner.

Check. Add: 7 + 5 = 12 and 12 + 6 = 18. The ones digit of 18 is 8, even, and 18 = 9 + 9: 9 pairs.

Boundaries

Even and odd are about whole numbers, 0, 1, 2, 3 and so on: amounts that can be counted out as whole things and paired. A half of an apple is not a whole thing to be paired, and the words are not used for parts of things.
Every whole number is exactly one of the two. Pairing always ends with nothing or exactly one left over, never two, because two would make one more pair.
Zero is even. It has no pairs and nothing left over.
The ones digit test works because ten is even, so every ten pairs up. It reads the ones digit of a number written in tens and ones; the tens digit, the hundreds digit and the size of the number have no say.
Odd only means “one left over when sharing into pairs”. An odd number may still share equally in other ways: 15 is odd, yet 15 = 5 + 5 + 5 is three equal groups. Even and odd are only about twos.
The table in Part 3 is about adding. It is not used here for taking away.

Where this goes next

The pairs in this lesson are the first step toward equal groups. Two rows of the same length grow into rows of three, four and five, which is where arrays and multiplication begin, and the even numbers become the answers in the facts for two. The ones digit test comes back as the first of the divisibility rules: a number can be split into twos exactly when it is even.

Much later, algebra writes an even number as 2n and an odd number as 2n + 1, and the rules for sums in Part 3 are proved in symbols in Proving statements about even and odd numbers; the proof is this lesson’s pairing picture written with letters. Pairing also settles questions far from marbles. Every line in a network has two ends, a pair, which is why in counting the ends of lines at each point the total always comes out even, and why the route that uses every line once depends on even and odd counts.

Exercises

Part 1: pairs and doubles

Pictures A, B and C below are for Exercises 1, 2 and 9.

Figure 4Pictures for Exercises 1, 2 and 9.

Look at picture A. How many pairs are there, and is anything left over? Is the number of counters even or odd?
Look at picture B. Is the number of counters even or odd? Write it as a double if you can.
Is 13 even or odd? Say how many pairs and how many left over.
Is 4 even or odd? If it is even, write it as a double.
Is 1 even or odd?
There are 19 children in a buddy line. How many pairs are there? Does anyone walk without a partner?
Which of 10, 7 and 12 can be written as a double? Write each one that can.
Casimiro and Ragna share 8 marbles. Can each get the same number with none left over? How many each?

Part 2: tens and the ones digit

Look at picture C. How many counters are there, and is the number even or odd? Explain with the ten frames.
Is 46 even or odd?
Is 75 even or odd?
Is 90 even or odd?
Is 131 even or odd?
Write 28 as a double.
Write 36 as a double.
Tuyet and Ragna share a bag of 52 marbles equally, with none left over. How many marbles does each girl get?

Part 3: adding even and odd numbers

For Exercises 17 to 20, say whether the total is even or odd before adding. Then add to check.

6 + 8
7 + 9
5 + 8
12 + 7
Casimiro has 13 marbles and Tuyet has 9. They pour them into one bag. Can all the marbles in the bag be put into pairs? How many pairs?
Without adding, is 3 + 5 + 7 even or odd? Then add to check.

Reasoning

Can pairing up a pile of counters ever finish with two counters left over? Explain.
Is every number whose ones digit is 0 even? Explain using tens.
When an even number is added to another number, can it change the total from even to odd or from odd to even? Explain with the pairing picture.

Answers

5 pairs and 1 left over, so 11 counters, odd. Five columns have a counter on top and one below, and the sixth top counter has no partner: 5 + 5 + 1 = 11.
18 is even. Every top counter has a partner below it, 9 pairs, and the rows are equal: 18 = 9 + 9.
Odd. 6 pairs use 12 counters, since 6 + 6 = 12, and 1 is left over.
Even. 2 pairs with none left, so 4 = 2 + 2.
Odd. One counter has nothing to pair with: 0 pairs and 1 left over.
9 pairs, and one child walks without a partner. 9 + 9 = 18 and 18 + 1 = 19, so 19 is odd.
10 = 5 + 5 and 12 = 6 + 6. 7 is not a double: 3 + 3 = 6 is one short and 4 + 4 = 8 is one too many. So 10 and 12 are even and 7 is odd.
Yes, 4 each. The 8 marbles make 4 pairs with none left, and each pair gives one marble to each child: 4 + 4 = 8.
23 counters, odd. The two full frames are 20 counters, 10 pairs. The third frame holds 3: one full column, 1 pair, and one counter alone. That is 11 pairs and 1 left over. The ones digit, 3, is odd.
Even. The ones digit is 6. The 4 tens pair up and the 6 ones make 3 pairs.
Odd. The ones digit is 5: the tens pair up, and 5 ones make 2 pairs and 1 left over.
Even. The ones digit is 0: 90 is 9 tens, every ten pairs up, and there are no ones.
Odd. The ones digit is 1. The hundred and the 3 tens all pair up, and the single one is left over.
28 = 14 + 14. The 2 tens give one ten to each side, and the 8 ones give four to each, so each side has 10 + 4 = 14. Check: 10 + 10 = 20 and 4 + 4 = 8, so 14 + 14 = 28.
36 = 18 + 18. The 3 tens give one ten to each side, and the third ten splits into 5 and 5. The 6 ones give three to each side. Each side has 10 + 5 + 3 = 18. Check: 10 + 10 = 20 and 8 + 8 = 16, and 20 + 16 = 36.
26 marbles each. 52 ends in 2, so it is even. The 5 tens give two tens to each girl, and the fifth ten splits into 5 and 5. The 2 ones give one to each. Each girl gets 20 + 5 + 1 = 26. Check: 20 + 20 = 40 and 6 + 6 = 12, and 40 + 12 = 52.
Even, because even plus even is even. 6 + 8 = 14, which ends in 4.
Even, because odd plus odd is even: the two spares pair up. 7 + 9 = 16, which ends in 6.
Odd, because odd plus even is odd. 5 + 8 = 13, which ends in 3.
Odd, because even plus odd is odd. 12 + 7 = 19, which ends in 9.
Yes, all of them pair up, in 11 pairs. 13 and 9 are both odd, so each leaves one spare, and the two spares pair with each other. 13 + 9 = 22, and 22 = 11 + 11.
Odd. All three numbers are odd, so there are three spares. Two of them pair up and one is left. Check: 3 + 5 = 8 and 8 + 7 = 15, which ends in 5.
No. Two counters left over could be put together as one more pair, so the pairing was not finished. When it is finished, at most one counter is left.
Yes. A number whose ones digit is 0 is made only of tens. Each ten is 5 pairs with nothing left, so all the tens together pair up with nothing left, and there are no ones to leave a spare.
No. An even number brings only pairs and no spare. If the other number had a spare, the total still has exactly that one spare; if it had none, the total has none. So adding an even number keeps an even total even and an odd total odd.