Numbers to 1,000: Hundreds, Tens and Ones

Imagine counting a big pile of straws one by one. You might lose your place! Put ten straws in each bundle instead. Each bundle is one ten.

Make a bigger group

Collect ten bundles of ten straws. Together they make one hundred.

There are still 100 straws when you untie all the bundles. So one hundred is ten tens or one hundred ones. Grouping changes how you count the straws, but keeps every straw.

Base-ten blocks show the same groups. A small cube stands for 1. A rod has ten cubes in a row and stands for 10. A flat has ten rods side by side and stands for 100.

Count groups with different values

Suppose you have 2 flats, 5 rods, and 3 cubes. Start with the biggest groups.

1. Count the flats: 100, 200.
2. Count on for the rods: 210, 220, 230, 240, 250.
3. Count on for the cubes: 251, 252, 253.

The blocks show 253. There are only ten block pieces, but they stand for 253 ones. A flat counts as 100, so counting every piece as one gives the wrong total.

Count whole hundreds

With only flats, each new piece adds a hundred. Count them: 100, 200, 300, 400, 500, 600, 700, 800, 900. These are the whole hundreds. Six flats show 600: six hundreds, with no extra tens or ones.

A tenth flat makes 1,000, read as “one thousand.” Ten hundreds make one thousand. It comes right after 999 and has four digits.

Keep counting tens

You can count rods past ten rods: 90, 100, 110, 120. Twelve rods show 12 tens, which is 120. Trade ten rods for one flat. You now have one flat and two rods, still worth 120.

Twenty rods make 200 because two groups of ten rods make two flats. Thirty-five rods make 350: three flats and five rods.

Look at 626. You see the digit 6 twice. The first 6 stands for 600. The last 6 stands for just 6. Its place tells you what a digit is worth.

Read the three places

A digit is one of these ten symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. We put digits together to write numbers. Three-digit numbers run from 100 to 999.

A place value chart has a column for each place:

Read from left to right. In 472, the 4 shows 4 hundreds, worth 400. The 7 shows 7 tens, worth 70. The 2 shows 2 ones, worth 2.

The digit in the tens place is 7. Its value is 70. Those answer two different questions: “Which digit?” asks for the symbol. “What is it worth?” asks for its value.

Why place matters

In 333, each place has the same digit. Their values are 300, 30, and 3. One hundred is ten tens, and one ten is ten ones. Moving a digit one place left makes its value ten times as much.

The largest digit does not always have the greatest value. In 192, the 1 is worth 100, while the 9 is worth 90. One hundred is more than nine tens.

Keep the zero

Build 706 with seven flats and six cubes. There are no rods. In the chart, write 7 under Hundreds, 0 under Tens, and 6 under Ones.

Zero holds the empty tens place. Leave it out and you get 76. That means seven tens and six ones, a much smaller amount.

Now build 540. There are five flats and four rods. The ones place needs a zero because there are no loose ones. In 800, both the tens and ones places need zeros.

To write a number from its groups, fill every place. For 9 hundreds and 2 ones, write the hundreds digit first. Keep a zero for the empty tens place, then write the ones digit: 902.

You can build a number with blocks or sketch it quickly. You can also use disks and a chart. Each model helps you see the groups inside the number.

Build it, then draw it

To show 213, choose two flats, one rod, and three cubes. You can draw these groups without drawing every tiny cube:

• Draw a square for each hundred.
• Draw a line for each ten.
• Draw a dot for each one.

Your drawing for 213 needs two squares, one line, and three dots. The square stands for a whole hundred, even though you draw it with one shape.

Read disks and charts

Place value disks are round counters with labels such as 100, 10, and 1. The label tells you the value. A disk marked 100 can be the same size as one marked 1.

Three 100 disks, two 10 disks, and four 1 disks show 324. To put that in a chart, write 3 under Hundreds, 2 under Tens, and 4 under Ones.

Sort a mixed pile

A pile has four rods, two flats, and nine cubes. The order of the pieces does not change the number. Sort by value before you count.

1. Two flats give 200.
2. Four rods give 40 more, making 240.
3. Nine cubes make the total 249.

Write 249 with the hundreds first. Writing 429 would put the number of rods in the hundreds place.

Trade before you write

What if your model has 13 rods and 2 cubes? There are more than nine rods, so make a trade. Ten rods become one flat. Three rods remain. You now have one flat, three rods, and two cubes: 132.

You can also show a number with more small pieces. Build 324 using 32 rods and 4 cubes. Thirty rods make three flats, leaving two rods and four cubes. That is the same 324 you built before, with a different number of pieces.

Count pieces when you want to know how many blocks to pick up. Count their values when you want to know the number they show.

The number 472 has four hundreds and seventy-two more. Say “four hundred seventy-two.” Say the hundreds first, then read the last two digits together.

Build the number name

For 864, read the 8 as “eight hundred.” The last two digits make 64, read as “sixty-four.” Put the parts together: “eight hundred sixty-four.”

Use a hyphen between a tens word and a ones word. You need one in “twenty-three” and “ninety-eight.” You do not need one in “forty” or “nineteen.”

In American math writing, leave out “and.” Write “nine hundred fifteen” for 915.

Hear the empty places

You do not say “zero” inside these number names. Read 307 as “three hundred seven.” Read 560 as “five hundred sixty.” Read 800 as “eight hundred.” The zeros still belong in the digits.

To write “six hundred four,” start with 6 hundreds. You hear no tens word, so put 0 in the tens place. Four gives 4 ones. The number is 604.

For “four hundred twenty,” the groups are 4 hundreds, 2 tens, and 0 ones. Write 420. Writing 40020 joins two numbers instead of putting the groups in their places.

Listen to the word ending

“Fourteen” and “forty” sound a little alike. But fourteen is one ten and four ones. Forty is four tens.

• “Three hundred fourteen” is 314.
• “Three hundred forty” is 340.

The teen names are eleven, twelve, thirteen, fourteen, fifteen, sixteen, seventeen, eighteen, and nineteen. Learn their spellings so you can tell them apart from whole tens.

The whole-tens words are ten, twenty, thirty, forty, fifty, sixty, seventy, eighty, and ninety. Watch “forty”: it has no u. “Fifteen” and “fifty” both start with “fif.” “Eighteen” and “eighty” each have only one t.

The name for 100 is “one hundred.” You may hear “a hundred” too. Write 1,000 as “one thousand.”

The number 583 is a short way to write five hundreds, eight tens, and three ones. Expanded form spreads out those values so you can see them.

Name each value

Start with 946. The 9 is worth 900. The 4 is worth 40. The 6 is worth 6. Join the values with plus signs:

The equal sign says both sides have the same value. You can turn the sentence around too:

Writing 9 + 4 + 6 would add the digits themselves. Expanded form uses what the digits are worth in the number.

Four ways to write one number

These forms all name the same amount:

Standard form is the usual way to write a number with digits. Unit form names the groups. Word form uses the number name.

Show an empty place

In 702, the tens digit is zero. You can include its value or leave that part out of the addition:

The zero must stay in 702, even when it is left out of expanded form. In 350, there are no ones, so its expanded form is 300 + 50.

Build the number back

Try 4 + 60 + 200. The parts are not written from largest to smallest, but you can still sort them. Two hundreds go in the hundreds place. Six tens go in the tens place. Four ones go in the ones place. The number is 264.

Now try 800 + 9. There are eight hundreds and nine ones. No part gives any tens, so the tens digit is zero. Write 809. The number 89 has only eight tens, and 8009 has an extra place.

You can check a form with blocks. For 237, use two flats, three rods, and seven cubes. Their values make 200 + 30 + 7, matching the digits.

Suppose you need more rods, but you have only flats. You can trade one flat for ten rods. Both are worth 100. Changing the groups without changing the number is called regrouping.

Trade a hundred for tens

Build 345 with three flats, four rods, and five cubes. Trade one flat for ten rods.

1. Take away one flat. Two flats remain.
2. Put ten rods in its place.
3. Add those rods to the four you already had.

Now you have 2 hundreds, 14 tens, and 5 ones.

You must remove the flat you traded. Keeping it as well as the ten new rods would add another hundred.

Trade a ten for ones

Start again with 3 hundreds, 4 tens, and 5 ones. Trade one ten for ten ones. Three tens remain. Ten new ones join the five loose ones.

So 345 is also 3 hundreds, 3 tens, and 15 ones. You may hear regrouping called renaming or exchanging.

Count all the tens

The tens digit in 480 is 8, but there are more than eight tens in all. Its four hundreds contain 40 tens. Add the eight extra tens to get 48 tens.

For 345, the three hundreds become 30 tens. Four more tens make 34 tens, with 5 ones left over. You can also name it as 345 ones.

Trade back to write digits

A chart for standard form has just one digit in each place. Suppose your groups are 4 hundreds, 12 tens, and 3 ones. You cannot fit “12” into one digit place.

Trade ten of the twelve tens for one hundred. You now have 5 hundreds, 2 tens, and 3 ones. Write 523. Writing 4123 would make a much larger number.

A zero does not stop you trading. Build 305 with three flats and five cubes. Trade a flat for ten rods. Now you have 2 hundreds, 10 tens, and 5 ones. The total is still 305.

You can split 156 into 100, 50, and 6. You can also split it into 150 and 6. Both splits make the same whole. Splitting a number into parts is called partitioning.

Show the parts

A part-whole model has one circle for the whole. Lines join it to circles for the parts. Put 156 in the whole circle and 100, 50, and 6 in the part circles.

A bar model shows the whole as one long bar. Smaller pieces underneath show the parts. All the pieces together must match the whole bar.

Putting parts together is called composing. Breaking a whole into parts is called decomposing.

Change the parts

Start with two parts: 150 and 6. Move ten from the first part to the second. The first gets smaller and the second gets bigger.

Move another ten the same way:

Each split keeps the same total. If you take ten from one part, that ten must go into another part. Writing 140 + 6 loses a ten and makes only 146.

You can choose different numbers of parts too:

In the last split, two parts are hundreds and the last part is 28. The whole stays 428 even though you use three parts.

Find a missing part

Suppose the whole is 726. Two parts are 700 and 6. You still need the two tens, so the missing part is 20.

For 385 split into 300 and a missing part, think about what remains. The 8 tens and 5 ones make 85. Check by putting 300 and 85 back together.

Whole hundreds split just like small groups. Five hundreds can be two hundreds and three hundreds. So 500 splits into 200 and 300. Ten hundreds can be six hundreds and four hundreds: 1,000 splits into 600 and 400.

Count on from 397: 398, 399, 400, 401. The new hundred comes right after 399. You keep taking one step each time, even though several digits change.

Forward and back in ones

The ones digit runs from 0 to 9. After nine ones, one more one makes a new ten. When both the tens and ones digits are 9, one more makes a new hundred.

Count back from 503: 502, 501, 500, 499, 498. To go one below 500, think of the number just before five hundreds. It is four hundreds, nine tens, and nine ones: 499.

At the top of this count, you say 997, 998, 999, 1,000. The last step makes ten hundreds.

Jump in tens

From 670, count 680, 690, 700, 710. At 690, one more ten makes ten tens. Those tens become a new hundred.

You can start at a number that has loose ones too. From 457, say 467, 477, 487, 497, 507, 517. Every jump adds a ten, so the ones digit stays 7.

Counting back works the same way. From 523, say 513, 503, 493. At 503, trade a hundred for ten tens. Taking one ten away leaves nine tens, giving 493.

Jump in hundreds

From 235, count 335, 435, 535, 635, 735, 835, 935. You add one hundred each time. The last two digits stay 35.

Count back from 362 in hundreds: 262, 162, 62. When the last hundred goes, you have a two-digit number.

Follow more than one kind of step

Start at 286. Count on three tens: 296, 306, 316. Then count four ones: 317, 318, 319, 320. Count each jump once; 286 is your starting point.

A number grid can help you see the steps. In a grid with ten numbers per row, moving right within a row adds 1. Moving down to the square directly below adds 10.

“Ten more” means add one ten. “Ten less” means take away one ten. You can find both by moving along a count in tens.

Choose the right place

For 254, one more gives 255. Ten more gives 264. A hundred more gives 354. The size of the step tells you which group to change.

To find ten less than 254, remove one rod from 2 hundreds, 5 tens, and 4 ones. You have 2 hundreds, 4 tens, and 4 ones: 244.

For a hundred less, remove one flat. You have 1 hundred, 5 tens, and 4 ones: 154. In a hundred-jump, the tens and ones keep their values.

Cross a hundred in a ten-jump

Build 293: two flats, nine rods, and three cubes. Add a rod for ten more. You now have ten rods.

Trade those ten rods for one flat. The result has three flats, no rods, and three cubes.

The ones stay 3, but both the tens and hundreds digits change. Writing a two-digit “10” in the tens place would not work. Make the trade and write one digit in each place.

For ten less than 403, start with four flats and three cubes. Trade one flat for ten rods. Take away one rod. Three flats, nine rods, and three cubes remain.

Near zero and one thousand

A hundred more than 900 is 1,000. A hundred less than 1,000 is 900. A hundred less than 120 is 20, because taking its only hundred leaves two tens. Ten less than 100 is 90.

Use more and less in a story

Ben has 238 stickers. He gets 10 more, making 248. Then he gets 100 more, making 348. Notice the size of each change. Adding ten and adding a hundred give different totals.

Skip counting means taking equal jumps. Each jump counts one more group. Choose the jump size that matches the group you are counting.

Count by fives

From 0, say 5, 10, 15, 20, 25, 30. These numbers end in 0 or 5. Keep going across a hundred: 485, 490, 495, 500, 505.

Count back by fives too: 315, 310, 305, 300, 295. Every backward jump takes away five.

Count by fifties

Two fifties make a hundred. From 0, count 50, 100, 150, 200, 250, 300. The endings take turns: 50, then 00. Twenty jumps of 50 take you from 0 to 1,000.

Suppose seven bags each hold 50 marbles. Count once for each bag:

1. One bag: 50.
2. Two bags: 100.
3. Three bags: 150.
4. Four bags: 200.
5. Five bags: 250.
6. Six bags: 300.
7. Seven bags: 350.

There are 350 marbles. The first bag gives the first jump from zero.

You can start elsewhere. From 120, count 170, 220, 270, 320. The jump is still 50, even though these numbers do not end in 00 or 50.

Count by twenty-fives

Four jumps of 25 make a hundred. From 0, say 25, 50, 75, 100, 125, 150, 175, 200. The last two digits repeat 25, 50, 75, 00.

After 50 comes 75. Saying 70 would make that jump only 20. Check the size of the jump when a pattern seems wrong.

Count by two hundreds

From 0, count 200, 400, 600, 800, 1,000. You are counting two hundreds at a time. Five equal jumps reach 1,000.

Going back, say 1,000, 800, 600, 400, 200, 0. For backward jumps of 50 instead, start 1,000, 950, 900, 850. Read or listen for the jump size before you begin.

A number line puts numbers in order. Numbers get greater as you move right. Equal spaces between marks stand for equal jumps in value.

Find the jump size

Look at the labels before you count. A number line need not start at zero. A line from 300 to 400 shows just one part of a longer line.

Suppose that line has ten equal spaces. The distance from 300 to 400 is 100. Ten equal jumps must share that distance. Each jump is 10.

Count the marks after 300: 310, 320, 330, 340, 350, 360, 370, 380, 390, 400. An arrow seven jumps after 300 points to 370.

Count spaces, not marks

Draw a short line with a mark at each end and one in the middle. You have three marks but only two spaces. Each space is one jump.

On the 300-to-400 line, there are eleven marks when you include both ends. There are only ten spaces. Put your finger at 300, then count each move to the next mark.

The same distance can have different steps

Between 500 and 600, two equal jumps are worth 50 each. The middle mark is 550. With ten equal jumps instead, each would be worth 10.

Between 200 and 300, five equal jumps are worth 20 each. The marks read 200, 220, 240, 260, 280, 300.

Between 240 and 250, ten equal jumps are worth 1 each. Count 241, 242, and so on. Do not assume every number line counts in tens.

Read a longer line

A line from 0 to 1,000 with ten equal spaces counts in hundreds. The fifth jump reaches 500. The tenth reaches 1,000.

You can check a step by counting all the way to a labeled mark. If your count misses its label, look again at the spaces and your starting number.

To place a number on a line, first find numbers you already know nearby. Whole tens, whole hundreds, and halfway numbers make useful starting spots.

Place a number between marks

Put 463 on a line from 400 to 500 with marks every 10. Find 460 and 470. Your number belongs between them.

Start at 460 and count on three ones to reach 463. Put the point a little less than halfway to 470. It should be closer to 460.

The same number is between 400 and 500. The hundreds tell you the larger part of the line to look at; the tens help you find a smaller part.

Find halfway

A halfway number is the same distance from both ends. Between 300 and 400, the middle is 350. It is 50 from each end.

Between 200 and 400, halfway is 300. Here the ends are two hundreds apart, so the middle is a whole hundred. Only neighboring whole hundreds have a halfway number ending in 50.

Between 240 and 250, the middle is 245. On a line from 0 to 1,000, the middle is 500.

Which hundred is closer?

Place 437 between 400 and 500. It is left of the middle, 450. It is closer to 400: the distance is 37, while the distance to 500 is 63.

At 450, the two distances are both 50. From 451 to 499, numbers are closer to 500.

Make a sensible estimate

An estimate is a sensible guess near the exact value. When only the ends are labeled, use the middle to help. On a line from 0 to 1,000, a point a little left of the middle might be about 450.

To place 820 on that line, think about the hundreds. It is between 800 and 900, close to 800. Put it a little right of where 800 belongs.

Keep the whole line in mind. On a 0-to-1,000 line, 100 is close to the left end. It takes ten jumps of 100 to reach the right end; one jump does not take you near the middle.

Which is greater: 352 or 325? Both start with 3, so both have three hundreds. You need to look at the next place to decide.

Start with the largest place

Compare the hundreds first. More hundreds means a greater three-digit number. For example, 301 is greater than 298. Three hundreds are more than two hundreds, even though 298 has larger tens and ones digits.

When the hundreds match, compare the tens. In 352 and 325, the hundreds match. Five tens are more than two tens, so 352 is greater. You can stop there.

When the hundreds and tens both match, compare the ones. In 671 and 674, only the ones differ. Four ones are more than one, so 674 is greater.

If all three digits match, the numbers are equal.

Write the comparison

Use these symbols to record what you found:

• > means “is greater than.”
• < means “is less than.”
• = means “is equal to.”

The open side faces the greater number. The point faces the smaller number.

Read left to right: “352 is greater than 325.” You can write the same comparison the other way:

That says “325 is less than 352.”

Watch zeros and different forms

Compare 609 and 690. Their hundreds match. Now compare zero tens with nine tens. So 609 is less than 690. The 9 ones in 609 do not outweigh nine tens.

For numbers in different forms, first find each value. The addition 400 + 7 makes 407. Compare that with 470. Both have four hundreds, but 407 has fewer tens.

By contrast, 300 + 20 + 9 and 329 show equal values. Use the equal sign.

Any three-digit number is greater than any two-digit number. A three-digit number has at least one hundred. The greatest two-digit number, 99, has no hundreds. So 120 is greater than 99.

A list can go from least to greatest or from greatest to least. Check which direction you need before you start comparing.

Sort by place value

Ascending order means least to greatest. To sort 534, 345, 543, 354, and 435, group them by hundreds first.

1. The numbers with three hundreds are 345 and 354. Compare their tens: 345 comes first.
2. The only number with four hundreds is 435.
3. The numbers with five hundreds are 534 and 543. Compare their tens: 534 comes first.

The ascending list is 345, 354, 435, 534, 543. On a number line, you would meet them in this order moving right.

Descending order means greatest to least. For the same numbers, reverse the list: 543, 534, 435, 354, 345.

Include numbers with different lengths

Sort 98, 605, 560, 1,000, and 506 from greatest to least. One thousand is greatest. Next compare the three-digit numbers. They go 605, 560, 506. The two-digit number 98 is least.

So the list is 1,000, 605, 560, 506, 98. A first digit of 9 does not make 98 greater than 506. In 98, that 9 stands for tens.

Make numbers with digit cards

Use the cards 4, 7, and 2 once each. To make the greatest number, put the largest digit in the hundreds place. Then put the next largest in the tens place. You get 742.

To make the least, start with the smallest digit: 247.

What if one card is zero? With 0, 5, and 3, the greatest number is 530. For the least three-digit number, start with the smallest digit that is not zero. Put 3 first, then 0, then 5: 305. Writing 035 gives only 35.

Find a number between two others

Between 395 and 402 are 396, 397, 398, 399, 400, and 401. The end numbers are not included.

A number greater than 680 and less than 700 could be 685, 690, or 699. Check both ends: your number must be above the smaller one and below the greater one.