Tens and ones
A heap of thirty-four sticks is slow to count one at a time. What changes if we tie them in tens, and why is the number written with a 3 and a 4?
Where you are 15 in Place value
When you reach for it
A child who can Counting to 100 and beyond can count a heap of 34 sticks one at a time. It is slow, and one lost place means starting again from one. Now tie the sticks into bundles of ten as they are counted, the way Counting a large pile by grouping does. The heap becomes 3 bundles and 4 sticks left over, and nobody has to keep track of a long count: “ten, twenty, thirty” for the bundles, then “thirty-one, thirty-two, thirty-three, thirty-four” for the loose sticks.
That is also exactly what the written number says. In 34 the 3 counts the bundles and the 4 counts the loose sticks. This lesson is about reading a two-digit number that way, as some tens and some ones, and about going back the other way, from a number to the bundles that make it. It is the first real piece of Place value, and most of what comes later about adding, subtracting and comparing bigger numbers is built on it.
Reach for it whenever a number bigger than ten has to be counted, built, read or written down.
The lesson has three parts, and they come in the order a child meets them. The first is bundling ten ones into one ten: ten single things can be tied or snapped together and treated as one thing, a ten. The second is teen numbers as ten and some more: 14 is one ten and 4 more, which the word “fourteen” hides. The third carries the idea past twenty, to every number up to 99.
Straws or lolly sticks with small rubber bands work well, or cubes that snap into trains of ten, and two ten frames (as in Subitizing and ten frames) for the teen numbers. Whatever is used, a bundle should hold exactly ten, and it helps if the child ties some of the bundles themselves.
The procedure
Bundling ten ones into one ten
One thing of a larger kind. Ten single sticks can be held together with a band and then treated as one thing: one bundle. Nothing has been added and nothing taken away; there are still ten sticks. What has changed is how they are counted. Counted one at a time they are ten ones. Counted as a bundle they are one ten. Both are true at once, and the skill is seeing the same sticks either way.
The words “one ten” carry both views. The “one” counts bundles; the “ten” says how many sticks are in each bundle. Example 1 is about exactly this.
Every bundle holds exactly ten. Ten is the size everyone agrees on. A band round nine sticks or round eleven makes a group, but not a ten. Because every bundle is the same size, bundles can be counted by tens later, “ten, twenty, thirty”, without untying them.
Making tens out of a heap. To bundle a heap of loose sticks, count out ten, tie them, set the bundle to one side, and start again from one with the sticks that are left. Stop when fewer than ten are left, since those cannot make a ten. Each count only goes up to ten, so a lost place costs at most one bundle’s worth of counting, not the whole heap.
Untying a bundle. A bundle can always be untied, and then it gives back ten loose ones:
This is how a child checks that a bundle really holds ten. Much later it is also how subtraction gets more loose ones when it needs them, in Regrouping in subtraction.
A ten need not look like ten. Once ten ones are being treated as one thing, that thing need not show its ten parts. A train of ten snapped cubes, a full ten frame, and all the fingers on two hands are tens that still show their ten parts. A 10-cent coin is a ten that does not: it is a single coin, and nothing about it shows ten, yet it can be swapped for ten 1-cent coins. The digit 3 in 34 works the same way. It is a single mark on the page, and it stands for three whole tens.
Teen numbers as ten and some more
What a teen number is. Each of the numbers from 11 to 19 is one ten and some more ones. Thirteen is one ten and three ones:
In two ten frames, thirteen counters fill the left frame and put three in the right one. With sticks, thirteen is one bundle and three loose sticks.
The written number says this plainly. In 13 the 1 is in the tens place and stands for the one ten, the full frame or the bundle. The 3 is in the ones place and stands for the three left over.
What the words hide. The English names do not say it plainly. “Thirteen” says the three first (“thir” is three) and the ten last (“teen” is an old way of saying ten), the reverse of the written order, which puts the 1 first. “Eleven” and “twelve” say neither part: they come from Old English words meaning “one left” and “two left”, that is, left over after ten. So a child has to learn the teen words as names, and learn separately what each one is made of. Not every language hides it: in Chinese, Japanese and Korean, thirteen is said “ten three”.
Saying it the ten way. It helps to say a teen number both ways, “thirteen” and “ten and three”, since the second puts the ten first, the way the digits and the frames do. From twenty-one upwards the ordinary names already work like this: “twenty-three” says the tens first.
| Number | The ten way | Name |
|---|---|---|
| 11 | ten and one | eleven |
| 12 | ten and two | twelve |
| 13 | ten and three | thirteen |
| 14 | ten and four | fourteen |
| 15 | ten and five | fifteen |
| 16 | ten and six | sixteen |
| 17 | ten and seven | seventeen |
| 18 | ten and eight | eighteen |
| 19 | ten and nine | nineteen |
Down the table the tens digit never changes, because each of these numbers has exactly one full ten. Only the ones digit moves, from 1 to 9.
Putting together and taking apart. The idea runs both ways. A ten and 6 ones put together make sixteen, 10 + 6 = 16. And sixteen taken apart is a ten and 6 ones. Finding a missing part of a teen number, such as 12 + \square = 15, is in Parts, wholes and number bonds, in its part on wholes from eleven to nineteen. Adding ten to a one-digit number, 10 + 4 or 4 + 10, and then ten more than a bigger number, is Ten more and ten less.
Twenty. Add one more counter to nineteen and the right frame is full too. Twenty is two full frames: 2 tens and 0 ones. The ten loose ones in the right frame have become a second ten, which is the next part.
Tens and ones up to ninety-nine
What the words mean. A one is a single thing, one stick. A ten is a group of ten ones kept together and counted as one thing, like a bundle of ten sticks. A two-digit number is a number from 10 to 99. It is written with two digits. The left digit is the tens digit: it says how many tens there are, from 1 to 9. The right digit is the ones digit: it says how many ones are left over besides the tens, from 0 to 9. So
Bundle, then count the tens and the ones. Take a heap of 34 sticks and tie them in tens until fewer than ten are left. There are 3 bundles and 4 loose sticks.
Each bundle holds ten, so the bundles can be counted by tens, “ten, twenty, thirty”, as in Counting by tens to one hundred. The loose sticks are then counted on from there: “thirty-one, thirty-two, thirty-three, thirty-four”. The count ends on the same word as counting every stick from one, because tying sticks together does not add or remove any. It only makes them quicker to count.
Write the tens on the left and the ones on the right. The number of bundles is written first and the number of loose sticks second: 3 and 4 make 34. The order is not a choice made fresh each time. Everyone writes the tens on the left, and that shared rule is what lets a reader know, from where a digit stands, what it counts. A small table with a column for each place shows it:
| Tens | Ones |
|---|---|
| 3 | 4 |
Where a digit stands gives its value. The 3 in 34 is in the tens place, so it stands for 3 tens, which is thirty. The 4 is in the ones place, so it stands for 4 ones, which is four. The same digit can be worth very different amounts.
Three loose sticks are written 3. Three bundles are written 30. The digit is the same; the place is different, and so is the number of sticks. In the same way, 34 and 43 use the same two digits but are different heaps: 34 is 3 bundles and 4 loose sticks, while 43 is 4 bundles and 3 loose sticks.
Tens with no ones left over. A heap that ties up exactly, with no loose sticks, is written with a 0 in the ones place. The 0 says “no loose ones”, and it keeps the tens digit in the tens place. These are the numbers a child meets when counting by tens:
| Bundles of ten | Loose sticks | Number | Name |
|---|---|---|---|
| 1 | 0 | 10 | ten |
| 2 | 0 | 20 | twenty |
| 3 | 0 | 30 | thirty |
| 4 | 0 | 40 | forty |
| 5 | 0 | 50 | fifty |
| 6 | 0 | 60 | sixty |
| 7 | 0 | 70 | seventy |
| 8 | 0 | 80 | eighty |
| 9 | 0 | 90 | ninety |
Saying the number. From twenty-one upwards the number names follow the digits: “thirty-four” says the tens first, “thirty”, and then the ones, “four”. The teen numbers are the odd ones out, as the part above showed: “fourteen” says the four first, and “eleven” and “twelve” do not say either part. Saying it both ways helps a child hear the tens: “thirty-four is three tens and four”, and “three tens and four is thirty-four”.
From a number back to the bundles. Reading the digits works the other way round too. To build 52, make 5 bundles of ten, because the tens digit is 5, and put out 2 loose sticks, because the ones digit is 2. Then count them as before to check: “ten, twenty, thirty, forty, fifty, fifty-one, fifty-two”.
Ten loose ones make another ten. The ones digit only ever runs from 0 to 9. That is not luck: as soon as ten loose sticks are lying about, they are tied into one more bundle, because ten ones make one ten. So before a heap is written down, any ten loose sticks become a bundle. Example 11 shows this.
Worked examples
Read each problem aloud with the child and build it with sticks, straws or counters. When a child can say the answer without building it, and then builds it to check, they have the idea.
One bundle, ten sticks
Sam counts out ten sticks and puts a band round them. His friend says, “That is one.” Sam says, “That is ten.” Who is right?
SolutionBoth are, because they are counting different things. Count the bundles: there is one. Count the sticks: slide a finger along the bundle, or untie it, “one, two, three, ..., ten”, and there are ten. The band did not add a stick or take one away. It only changed what is being counted as one thing.
So the friend is counting bundles and Sam is counting sticks, and “one ten” says both at once: one bundle, of ten. Lay a second heap of ten loose sticks beside the bundle. Both heaps hold ten, but the bundle can be counted with a single look, because its counting has already been done.
Tying up a heap
There are 26 loose straws on a table. Tie them into bundles of ten. How many bundles are there, and how many straws are left loose?
SolutionCount out ten straws, “one, two, ..., ten”, and tie them. That is 1 bundle, and 16 straws are still loose. Count out ten more and tie them: 2 bundles, and 6 straws loose. Six is fewer than ten, so they cannot make another bundle, and they stay loose.
There are 2 bundles and 6 loose straws. To check, count the bundles by tens and then the loose straws on from there, “ten, twenty, twenty-one, twenty-two, ..., twenty-six”:
The count ends on 26, the number of straws at the start, since tying them changed nothing about how many there are. Each count went only as far as ten, so a lost place would have meant recounting one bundle, not all twenty-six straws. Two bundles and six loose straws is 2 tens and 6 ones, which is exactly what the digits of 26 say; the last part of the procedure explains why.
Seventeen in two frames
Mo puts 17 counters into two ten frames, filling the left frame first. How many counters go in the right frame? Say 17 as a ten and some ones.
SolutionThe left frame holds 10 counters, and then it is full. The rest go in the right frame, and they are counted on from ten: “eleven, twelve, thirteen, fourteen, fifteen, sixteen, seventeen”. That is seven counts, so 7 counters go in the right frame.
So 17 is 1 ten and 7 ones. The 1 in 17 is the full frame and the 7 is the right frame. Said the ten way it is “ten and seven”; its name, “seventeen”, says the seven first.
Eleven, twelve and fifteen
Write eleven, twelve and fifteen in digits, and say each one as ten and some more.
SolutionEleven is one more than ten: a full frame and one counter. That is 1 ten and 1 one, “ten and one”:
Twelve is two more than ten: a full frame and two counters, “ten and two”:
Fifteen says its parts, if backwards: “fif” is five and “teen” is ten. A full frame and five counters, “ten and five”:
In all three the tens digit is 1, because each has exactly one full ten. Only the ones digit changes. Eleven and twelve are the two that give the child nothing to hear: neither word sounds like “ten”, “one” or “two”, so their meaning has to be built and learned.
Coins
Ten 1-cent coins can be swapped for one 10-cent coin. Nia has one 10-cent coin and four 1-cent coins. How many cents does she have? Could she swap her coins for two 10-cent coins?
SolutionThe 10-cent coin is a ten, even though it is one coin, because it is worth the same as ten 1-cent coins. The four 1-cent coins are four ones. So Nia has 1 ten and 4 ones:
She has 14 cents. Counting it out: “ten” for the big coin, then “eleven, twelve, thirteen, fourteen” for the small ones.
She cannot swap them for two 10-cent coins. A second 10-cent coin needs ten 1-cent coins and she has only four; she would need six more, since 4 + 6 = 10. Fourteen cents is one ten and some more, not two tens.
Thirteen or thirty-one?
Ella hears “thirteen” and writes 31, because she hears the three first. Build 13 and 31 with sticks. Which one is thirteen?
SolutionThirteen is a teen number, one ten and three more. With sticks that is 1 bundle and 3 loose sticks, and written down it has tens digit 1 and ones digit 3:
The number 31 has tens digit 3 and ones digit 1, so it is 3 bundles and 1 loose stick. Counting it gives “ten, twenty, thirty, thirty-one”:
The two heaps look nothing alike: 31 has three bundles and 13 has only one. Thirteen is 13. The word “thirteen” says the three first, but the three counts loose ones, and the written number always puts the tens on the left. To check, count the first heap: “ten” for the bundle, then “eleven, twelve, thirteen”.
Straws in a cup
Leo has 5 bundles of ten straws and 2 loose straws. How many straws does he have?
SolutionEach bundle is a ten, so 5 bundles are 5 tens. The 2 loose straws are 2 ones. Five tens and two ones is written with a tens digit of 5 and a ones digit of 2:
Leo has 52 straws. To check, count them: touch each bundle and say “ten, twenty, thirty, forty, fifty”, then touch the loose straws, “fifty-one, fifty-two”. If the child is not sure the bundles really hold ten, untie one and count it.
Build a number
Build 67 with sticks.
SolutionThe tens digit is 6, so make 6 bundles of ten. The ones digit is 7, so put out 7 loose sticks next to them. Check by counting: “ten, twenty, thirty, forty, fifty, sixty” for the bundles, then “sixty-one” up to “sixty-seven” for the loose sticks. The name agrees with the digits too: sixty-seven is sixty, which is six tens, and seven.
Three and thirty
Look at Figure 4. What number does each row show? Why do both numbers use the digit 3?
SolutionThe top row has no bundles and 3 loose sticks: 0 tens and 3 ones. A number with no tens is written with its ones digit alone, so this is 3.
The bottom row has 3 bundles and no loose sticks: 3 tens and 0 ones. Counting the bundles by tens gives “ten, twenty, thirty”, so this is 30.
Both rows have a 3 in them because both have three of something: three sticks in the top row, three tens in the bottom row. The place tells which. In 30 the 0 fills the ones place to say there are no loose sticks, and that puts the 3 in the tens place. Without the 0 both rows would be written 3, and nobody could tell them apart.
What is each digit worth?
What is the 7 worth in 72, and what is the 2 worth? Then answer the same questions for 27.
SolutionIn 72 the 7 is the tens digit, so it stands for 7 tens, which is 70. The 2 is the ones digit, so it stands for 2 ones, which is 2. Together:
In 27 the places swap. The 2 is now the tens digit and is worth 20; the 7 is the ones digit and is worth 7:
With sticks, 72 is 7 bundles and 2 loose sticks, and 27 is 2 bundles and 7 loose sticks. Build both side by side: the same two digits make two very different heaps.
Too many loose sticks
Ava has 2 bundles of ten sticks and 13 loose sticks. How many sticks does she have, and how is that number written?
SolutionThe ones digit can only be 0 to 9, and 13 loose sticks is more than that. So first tie ten of the loose sticks into a new bundle.
Now there are 3 bundles and 3 loose sticks: 3 tens and 3 ones, which is 33.
Ava has 33 sticks. Check by counting the heap as it was at the start: “ten, twenty” for the two bundles, then on through the 13 loose sticks, “twenty-one, twenty-two”, all the way to “thirty-three”.
Every way to show thirty-four
Using bundles of ten and loose sticks, find every way to show 34 sticks. Which way matches the digits of 34?
SolutionStart with as many bundles as possible. Three bundles are 30 sticks and four bundles would need 40, so the most is 3 bundles, with 4 loose sticks. Now untie one bundle: its ten sticks join the 4 loose ones, leaving 2 bundles and 14 loose sticks. Untie another: 1 bundle and 24 loose sticks. Untie the last: no bundles and 34 loose sticks.
| Bundles | Loose sticks | Sticks in all |
|---|---|---|
| 3 | 4 | 30 + 4 |
| 2 | 14 | 20 + 14 |
| 1 | 24 | 10 + 24 |
| 0 | 34 | 0 + 34 |
There are four ways, and every one of them is 34 sticks: each sum in the last column is 34. There are no others, because every bundle holds exactly ten, so the only choice is how many whole bundles to keep: 3, 2, 1 or none. Only the first has fewer than ten loose sticks, and only the first can be read straight off the digits: 3 tens and 4 ones. The other three still matter. Untying a bundle to get more loose ones is the move behind Regrouping in subtraction, and tying ten loose ones into a bundle, as in Example 11, is the move behind Regrouping in addition.
Boundaries
- A ten holds exactly ten. The digits of 34 mean 3 tens and 4 ones only because every bundle is a ten. Three bundles of nine sticks hold 27, and with 4 loose sticks that is 31 sticks, not 34. The same goes for any other ten: a frame is a ten only when all ten boxes are full, and a coin is a ten only if it is worth ten 1-cent coins.
- The ones digit is always 0 to 9. A heap with ten or more loose sticks has to have ten of them tied into a bundle before it can be written, as in Example 11.
- The tens digit of a two-digit number is 1 to 9. A heap with no bundles has a one-digit number: three loose sticks are 3, never written 03. Ten bundles are ten tens, which is one hundred, and that needs a third digit. Bundling ten tens into one hundred is the same move as bundling ten ones into one ten, one place further left, in Reading and writing numbers to 1,000.
- Only the teen names hide their ten. From twenty-one to ninety-nine the names say the tens first, as the digits do. Ten itself is 1 ten and 0 ones, and twenty is 2 tens and 0 ones; neither is a teen number.
- The order of the digits is fixed; the order of the sticks is not. A heap can be laid out with the loose sticks first, and “four ones and three tens” is still 34. The written number always puts the tens digit on the left.
Where this goes next
Seeing a teen number as ten and some more is what makes making ten work: 8 + 5 becomes a full ten and 3 more, which is 13. Together with the bonds of Parts, wholes and number bonds it is the heart of adding and subtracting within twenty.
Almost everything a child does next with numbers up to a hundred uses tens and ones. To decide which of two numbers is bigger, look at the tens first and then the ones: that is Comparing two-digit numbers. One more bundle or one fewer is Ten more and ten less, and counting by tens from a number such as 7 or 34 is Counting by tens from any number.
Adding two-digit numbers puts tens with tens and ones with ones, which is Adding two-digit numbers without regrouping, and taking away works the same way in Subtracting two-digit numbers without regrouping. When the ones add up to ten or more, they are tied into a new ten, exactly as in Example 11: that is Regrouping in addition.
Base ten blocks swap the bundles for rods of ten and single cubes, in Base ten blocks and Building a number out of blocks. The same idea, one place further left, gives Reading and writing numbers to 1,000. Writing 34 as 30 + 4 is Expanded form, and the difference between a digit and what it is worth is The value of a digit.
Exercises
Bundling ten
- Ten crayons are held together with a band. How many bundles is that? How many crayons?
- A heap of 30 loose sticks is tied into bundles of ten. How many bundles are there?
- A heap of 24 loose sticks is tied into bundles of ten. How many bundles are there, and how many sticks are left loose?
- Kim unties 1 bundle of ten sticks. How many loose sticks does she get?
- Jo has 20 coins worth 1 cent each. Ten of them can be swapped for one 10-cent coin. How many 10-cent coins can she get for all 20?
Teen numbers
- How many counters are in frames A? Say it as tens and ones.
- How many counters are in frames B? Say it as tens and ones.
- How many counters are in frames C? Say it as tens and ones.
- 17 is 1 ten and \square ones.
- 1 ten and 5 ones is \square.
- 10 + 8 = \square
- Say 14 the ten way. Then say its name. Which comes first in the name, the ten or the four?
- A tray holds 10 cupcakes. There are 16 cupcakes: one full tray, and the rest on a plate. How many cupcakes are on the plate?
- Kai has one 10-cent coin and seven 1-cent coins. How many cents does he have?
- Write in digits: nineteen, eleven, twelve.
- Which is more, 13 or 31? Build both with sticks to show it.
Bundles to numbers
- Ben has 4 bundles of ten sticks and 6 loose sticks. How many sticks does he have?
- There are 7 bundles of ten sticks and 1 loose stick. How many sticks are there?
- There are 9 bundles of ten sticks and no loose sticks. How many sticks are there?
- There is 1 bundle of ten sticks and 9 loose sticks. How many sticks are there?
Pictures
- How many sticks are in heap D? Say it as tens and ones.
- How many sticks are in heap E? Say it as tens and ones.
- How many sticks are in heap F? Say it as tens and ones.
Numbers to tens and ones
- 58 is \square tens and \square ones.
- 83 is \square tens and \square ones.
- 40 is \square tens and \square ones.
- 2 tens and 9 ones is \square.
- 9 ones and 2 tens is \square.
- 6 tens is \square.
- What is the 5 worth in 52? What is the 5 worth in 25?
- Crayons come in boxes of 10. Mia has 3 full boxes and 8 loose crayons. How many crayons does she have?
- 3 tens and 12 ones is \square.
Explain
- Lee puts a band round 9 sticks and calls it a ten. Is he right? How do you know?
- One child points at 3 bundles of ten sticks and says “three”. Another says “thirty”. Who is right?
- The word “thirteen” says the three first. Why is 13 written with the 1 first?
- Why does the number 50 need its 0?
- Use the digits 2 and 6 to write two different two-digit numbers. Build each one with sticks. Why are they different?
- Show 34 sticks using only 2 bundles of ten. How many loose sticks do you need?
- Why is the ones digit of a number never more than 9?
- A child has 1 bundle of ten and 9 loose sticks, which is 19. They add one more loose stick. What should they do with the loose sticks, and what number is it now?
Answers
- 1 bundle and 10 crayons. The band makes the ten crayons one bundle, and it does not add or take away any crayons: 1 ten is 10 ones.
- 3 bundles. Each count of ten makes a bundle, and 10 + 10 + 10 = 30, with none left over.
- 2 bundles and 4 loose sticks. Two counts of ten use 20 sticks, and the 4 left are fewer than ten, so they stay loose: 10 + 10 + 4 = 24.
- 10 loose sticks. Untying a bundle gives back its ten ones: 1 ten is 10 ones.
- 2 coins. Each 10-cent coin takes ten 1-cent coins, and 20 is two tens: 10 + 10 = 20.
- 16. The left frame is full, 1 ten, and the right frame has 6, 6 ones: 10 + 6 = 16.
- 11. A full frame and 1 more: 1 ten and 1 one, 10 + 1 = 11.
- 20. Both frames are full: 2 tens and 0 ones, 10 + 10 = 20.
- 7. 17 is a full ten and 7 more: 17 = 10 + 7.
- 15. A ten and 5 ones: 10 + 5 = 15.
- 18. A ten and 8 ones is eighteen: 10 + 8 = 18.
- “Ten and four”, then “fourteen”. The name says the four first and the ten (“teen”) last, but 14 is still 1 ten and 4 ones: 10 + 4 = 14.
- 6 cupcakes. Sixteen is one ten and 6 more, and the full tray is the ten: 10 + 6 = 16.
- 17 cents. The 10-cent coin is a ten and the seven coins are 7 ones: 10 + 7 = 17.
- 19, 11, 12. Each is one ten and some ones: 10 + 9 = 19, 10 + 1 = 11, 10 + 2 = 12.
- 31 is more. 31 is 3 bundles and 1 loose stick, 30 + 1 = 31; 13 is 1 bundle and 3 loose sticks, 10 + 3 = 13. Three tens is more than one ten.
- 46 sticks. 4 bundles are 4 tens and the 6 loose sticks are 6 ones: 40 + 6 = 46.
- 71 sticks. 7 tens and 1 one: 70 + 1 = 71.
- 90 sticks. 9 tens and 0 ones: “ten, twenty, ..., ninety”, so 90 + 0 = 90.
- 19 sticks. 1 ten and 9 ones: 10 + 9 = 19.
- 47. Heap D is 4 bundles and 7 loose sticks, so 4 tens and 7 ones: 40 + 7 = 47.
- 60. Heap E is 6 bundles and no loose sticks, so 6 tens and 0 ones: 60 + 0 = 60.
- 22. Heap F is 1 bundle and 12 loose sticks. Twelve loose sticks is more than nine, so tie ten of them into a bundle: that makes 2 bundles and 2 loose sticks, so 2 tens and 2 ones: 10 + 12 = 22.
- 5 tens and 8 ones: 58 = 50 + 8.
- 8 tens and 3 ones: 83 = 80 + 3.
- 4 tens and 0 ones: 40 = 40 + 0. The 0 says there are no loose ones.
- 29. 2 tens and 9 ones: 20 + 9 = 29.
- 29. The tens are said second, but the tens digit is still written on the left: 2 tens and 9 ones is 20 + 9 = 29.
- 60. 6 tens and no ones: 60.
- 50 and 5. In 52 the 5 is the tens digit, so it is worth 5 tens, 50. In 25 it is the ones digit, so it is worth 5.
- 38 crayons. Each full box is a ten, so 3 boxes are 3 tens, and the 8 loose crayons are 8 ones: 30 + 8 = 38.
- 42. Ten of the 12 ones make one more ten, so it is 4 tens and 2 ones: 30 + 12 = 42.
- No. A ten holds exactly ten, and his bundle holds 9; it needs one more stick to be a ten, since 9 + 1 = 10. If bundles of nine were counted as tens, “ten, twenty, thirty” would be the wrong count of the sticks.
- Both are right. The first child is counting bundles, and there are three: 3 tens. The second is counting sticks by tens, “ten, twenty, thirty”, and there are thirty: 10 + 10 + 10 = 30. “Three tens” and “thirty” name the same heap, just as “one ten” and “ten” do.
- The written number always puts the tens on the left and the ones on the right. Thirteen is 1 ten and 3 ones, so its tens digit, 1, comes first. The English word happens to say the ones first, but the word does not change where the digits go.
- 50 is 5 tens and 0 ones. The 0 fills the ones place and says there are no loose ones. Without it the 5 would stand alone and be read as 5 ones, which is five, not fifty.
- 26 and 62. 26 is 2 bundles and 6 loose sticks, 20 + 6 = 26; 62 is 6 bundles and 2 loose sticks, 60 + 2 = 62. The digits are the same, but each digit counts something different in each number: bundles when it is on the left, loose sticks when it is on the right.
- 14 loose sticks. Two bundles are 20 sticks, and 20 + 14 = 34.
- The ones digit counts the loose ones. As soon as there are ten loose ones they are tied into one more ten, and the tens digit goes up instead. So there are never ten or more loose ones left to count, and the ones digit is at most 9.
- Now there are 10 loose sticks, which is a ten, so they are tied into a bundle. That makes 2 bundles and 0 loose sticks, which is 20: 19 + 1 = 20.