Reading and writing numbers to 1,000

When you reach for it

Radu is reading a thick book. The number at the bottom of his page is 247. At a fun run, Pendo pins on a race bib that says 315, and as she crosses the finish line the announcer reads her number out loud. Each time, three digits stand for one number. Someone has to turn the digits into words, or turn spoken words back into digits, and get the same number both ways.

This lesson reads and writes whole numbers from 100 up to 1{,}000. A two-digit number is some tens and some ones, as in Tens and ones. A third digit, on the left, counts a bigger group: hundreds. The lesson runs over three days. Day 1 builds a hundred out of ten tens and reads a three-digit number from base-ten blocks. Day 2 writes the same numbers in words and turns words back into digits. Day 3 handles a zero in the tens or ones place, as in 805, 508 and 580, and ends at one thousand, which is ten hundreds.

You reach for it whenever a count goes past 99: page numbers, bib numbers, seats in a hall, the steps on a long stair.

The procedure

Day 1: hundreds, tens and ones

A hundred is ten tens. Base-ten blocks come in three sizes. A small square is one. A rod is a column of ten small squares: one ten. A flat is a big square made of ten rows of ten small squares: one hundred. Lay ten rods side by side and they cover exactly one flat (Figure 1). So

This is the step from Tens and ones, one size up: there, ten ones made one ten; here, ten tens make one hundred. Since each rod is 10 ones, a flat is also 100 ones. Swapping ten rods for a flat, or a flat back for ten rods, is the lesson Base ten blocks and trading ten for one. Here we only need the fact that a flat is worth ten tens.

Counting by hundreds. Put out flats one at a time and count them: one hundred, two hundred, three hundred, and on up to nine hundred. Each new flat adds 100. Counting by hundreds also comes up in Counting by tens to one hundred.

The first digit counts the flats. The two zeros after it say that no tens and no ones are left over. The words say “two hundred”, never “two hundreds”.

Reading a number from blocks. Radu shows his page number with 2 flats, 4 rods and 7 small squares (Figure 2). The digits are counts, written in a fixed order: the first digit counts hundreds, the second counts tens, the third counts ones.

The order is what lets one row of digits carry three different counts. Nothing in “247” says “hundreds” or “tens”; the place each digit sits in says it.

Why each digit is 0 to 9. Ten of any block make one of the next size up: ten ones make a ten, and ten tens make a hundred. So when a number is shown with the fewest blocks, there are never ten or more of one kind, and every count fits in one digit, from 0 to 9.

What a digit is worth. A digit is worth a different amount in each place. In 247, the 2 counts flats, so it is worth 200. The 4 counts rods, so it is worth 40. The 7 counts small squares, so it is worth 7. Writing a number as the sum of these amounts, 247 = 200 + 40 + 7, has its own lesson, Expanded form.

Day 2: number words

The words follow the digits. To say a three-digit number, say the first digit with the word “hundred”. Then say the last two digits as the two-digit number they make. So 247 is “two hundred” followed by “forty-seven”: two hundred forty-seven.

The last two digits are said exactly as in Counting to 100 and beyond. A tens digit from 2 to 9 has a decade word:

The ones word follows the decade word, joined by a hyphen: forty-seven, sixty-two. Watch the spelling of “forty”, which has no “u”. When the tens digit is 1, the last two digits make ten or a teen number, and those have their own words: ten, eleven, twelve, thirteen, and on to nineteen. These words hide the tens digit, as Teen numbers as ten and some more explains. So 213 is two hundred thirteen: the “thirteen” carries both the 1 ten and the 3 ones.

No “and”. Written math leaves out the word “and”: 213 is “two hundred thirteen”, not “two hundred and thirteen”. Plenty of people say the “and” out loud, and listeners still understand. In math writing the word “and” is saved for the decimal point, which comes much later in Decimal place value.

Where the hyphen goes. Only between a decade word and a ones word, as in “eighty-one”. There is no hyphen after “hundred”: “four hundred twelve” has none, and neither has “four hundred twenty”.

Words to digits. Run the same steps backward. The word in front of “hundred” gives the first digit. The words after “hundred” give the last two digits. For “six hundred thirty-eight”, “six hundred” gives 6 and “thirty-eight” gives 38, so the number is 638. To check, read the digits back: 638 says six hundred thirty-eight, the words we started with.

Day 3: zeros, and one thousand

A zero holds a place. Itzayana’s book has a page eight hundred five. The words name 8 hundreds and 5 ones. They say nothing about tens, because there are none. But the digits still have to show that the tens place is empty. If she wrote 85, the 8 would sit in the tens place and count tens: 85 is eighty-five, 8 tens and 5 ones. The zero in 805 says “no tens here”, and that keeps the 8 in the hundreds place.

So to turn words into digits, fill all three places, and give a 0 to any place the words skip:

• “eight hundred five” names hundreds and ones, so the tens place gets 0: 805.
• “eight hundred fifty” names hundreds and tens, so the ones place gets 0: 850.
• “eight hundred” names only hundreds, so both other places get 0: 800.

Going the other way, a 0 in the tens or ones place is never read out. The number 805 is read “eight hundred five”, and 850 is read “eight hundred fifty”.

One thousand. Count flats past nine hundred. Nine flats are 900. One more flat makes ten flats, and ten hundreds is the next size of group: one thousand (Figure 3).

A digit can only count up to nine, so ten hundreds cannot sit in the hundreds place. One thousand needs a fourth place, to the left of the hundreds: 1{,}000 is 1 thousand, 0 hundreds, 0 tens and 0 ones. The comma marks off the last three digits. Numbers past one thousand, and what the comma does there, are in Numbers to one million.

Worked examples

The first two examples go with Day 1, the next two with Day 2, and the last three with Day 3.

Boundaries

• Three digits reach from 100 to 999. One thousand needs a fourth place, and every number past it does too.
• Each place holds one digit, 0 to 9, and that is only true once the number is shown with the fewest blocks. A heap of 3 flats, 12 rods and 4 small squares is a real amount, but the counts 3, 12, 4 cannot be written side by side as digits. Ten of the rods must first become a flat, giving 4 hundreds, 2 tens and 4 ones, which is 424. That trade is Base ten blocks and trading ten for one.
• A zero in the tens or ones place changes the number: 805, 850 and 85 are three different numbers. A zero in front changes nothing. A bib printed “085” carries the number 85, because it has no hundreds.
• The number words here leave out “and”, and they put a hyphen only between a decade word and a ones word.

Where this goes next

Writing 247 as 200 + 40 + 7, the sum of what its digits are worth, is Expanded form. The The place value chart sets the digits in labeled columns, one column for each place. Reading the hundreds first is also how Comparing and ordering whole numbers decides that 930 is more than 903: both have 9 hundreds, and then 3 tens beat 0 tens. Adding and subtracting numbers up to 1{,}000 with blocks rests on the same three groups and the same trades. Past one thousand, Numbers to one million adds thousands, ten thousands and hundred thousands, each a group ten times the one before, and the comma that keeps them readable.

Exercises

Day 1: hundreds, tens and ones

1. How many tens make one hundred? How many ones make one hundred?
2. What number do the blocks in picture A show?

3. What number do the blocks in picture B show?
4. What number do the blocks in picture C show?
5. Radu shows his page number with 7 flats, 3 rods and 9 small squares. Which page is he on?
6. How many hundreds, tens and ones are in 682?
7. What is each digit of 574 worth?
8. How many flats show 800? Write 800 in words.

Day 2: number words

9. Write 263 in words.
10. Write 418 in words.
11. Write 555 in words.
12. Write 871 in words.
13. Write six hundred forty-one in digits.
14. Write nine hundred seventy-seven in digits.
15. Write eight hundred seventeen in digits.
16. One runner’s bib is one hundred nineteen, and the next runner’s bib is one hundred ninety. Write both in digits.

Day 3: zeros and one thousand

17. Write four hundred nine in digits.
18. Write six hundred in digits.
19. Write 702 and 720 in words.
20. Itzayana builds page 309 and page 390 with the fewest blocks. How many rods does each one need?
21. Which blocks show 470 with the fewest blocks? Write 470 in words.
22. Pendo counts by hundreds from 600 until she reaches one thousand. Write every number she says, in digits.

Explain

23. The words “eight hundred five” have no word for tens. Explain why 805 still needs its 0.
24. Use the digits 0, 4 and 6, each once, to write every three-digit number you can. Read each one in words. Why can the 0 not go first?
25. What is the smallest three-digit number? What is the largest? Explain each.