Expanded form

Why this exists

A museum has a counter at its front door. Every time a visitor walks in, the number on the counter goes up by one. At noon it shows 438. Alemayehu, Oyuun and Kiele are on a class trip, and they want to know what that number is made of.

The The place value chart lesson answered one part of that: each digit is worth an amount that depends on its column. The 4 is worth 400, the 3 is worth 30 and the 8 is worth 8. This lesson puts those amounts back together. The number is the three amounts added up:

Writing a number this way is called expanded form. It shows in one line what is hidden in the short way of writing. The short form 438 is quick to write and read. The expanded form 400 + 30 + 8 shows how much each digit carries. Both name the same number.

Going the other way matters just as much. A sum of hundreds, tens and ones can be put back into a single number, even when the parts come in a jumble or a part is too big for one column. That is how blocks, bags and crates get counted, and it is the first step of adding and subtracting three-digit numbers by place value.

The lesson runs over three days.

• Day 1, from a number to its parts. Writing a three-digit number as hundreds plus tens plus ones, from the digits, from blocks and from words.
• Day 2, zero digits. A digit 0 is worth nothing, so its part is left out: 706 = 700 + 6.
• Day 3, from the parts back to the number. Parts given out of order, like 40 + 5 + 600, and parts that are not one per place, like 400 + 17.

Base ten blocks, a sheet of paper for the sums and a set of digit cards are all the lesson needs.

Definition

The expanded form of a number is the number written as a sum of the values of its digits, one part for each digit that is not 0, largest part first.

For a three-digit number the parts are:

• the hundreds part, the hundreds digit followed by two zeros, such as 400;
• the tens part, the tens digit followed by one zero, such as 30;
• the ones part, the ones digit itself, such as 8.

So 438 = 400 + 30 + 8. A digit 0 has the value 0, and adding 0 changes nothing, so its part is left out: 706 = 700 + 6, and 900 is its own expanded form.

The same number can also be written in unit form, with the place names in words: 4 hundreds + 3 tens + 8 ones. Unit form and expanded form say the same thing. Unit form names the block; expanded form says what that many blocks are worth.

Intuition

Day 1: one part for each kind of block

Build 438 with base ten blocks from Base ten blocks and trading ten for one: 4 flats, 3 rods and 8 small squares. Now look at the blocks one kind at a time. The flats are worth 400. The rods are worth 30. The small squares are worth 8. Those three amounts are the three parts of the expanded form, and pushing the three groups together is adding them (Figure 1).

So a sum in expanded form is a written record of a pile of blocks sorted by kind. The plus signs stand where the gaps between the groups are. Reading the sum aloud gives the number name almost word for word: “four hundred” is 400, “thirty” is 30, “eight” is 8. Number names were the work of Reading and writing numbers to 1,000.

Day 2: no blocks, no part

Some numbers have a kind of block missing. The museum counter shows 706 at closing time. That is 7 flats, no rods and 6 small squares. There is nothing in the tens group to write down, so the expanded form has only two parts (Figure 2).

Writing 700 + 0 + 6 is not wrong: adding zero changes nothing, so it still equals 706. The shorter 700 + 6 is the usual way, the same way nobody lists an empty group of blocks.

Here is something worth stopping on. In the number 706 the 0 must be written, because it holds the tens column open and keeps the 7 in the hundreds column (The place value chart). In the sum 700 + 6 no zero digit is needed in the middle. Each part carries its own zeros at its right end, so 700 already says “hundreds” without any column around it.

Day 3: every part knows its own place

That last point is what makes going backward easy. A part such as 600 is worth 6 hundreds wherever it sits in the sum. So the parts can come in any order. The sum 40 + 5 + 600 still has 6 hundreds, 4 tens and 5 ones, and the number is 645. Changing the order of the parts leaves the sum the same; this is the commutative property of addition.

A part may also be bigger than one digit’s worth. Four crates of 100 ears of corn and 17 loose ears make 400 + 17. The 17 is 1 ten and 7 ones, so the sum is 400 + 10 + 7, which is 417. The trade of ten ones for one ten is the one from Base ten blocks and trading ten for one; here it is done in the numbers instead of with the blocks (Figure 3).

Results

Worked examples

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

Boundaries

• Expanded form here is for whole numbers up to 999. Bigger numbers get more parts, one for each further place, as in Numbers to one million.
• The expanded form has exactly one part for each digit that is not 0. A sum such as 400 + 17 or 200 + 160 + 3 is equal to the number, but it is not its expanded form until every part is one digit followed by zeros, with at most one part for each place.
• A digit 0 gives no part, but the 0 is still needed when the number is written the short way: 706 and 76 are different numbers.
• The number 0 has no digits other than 0, so it has no parts at all; it is simply written 0.
• The parts are joined with + only. Every part is a whole number of hundreds, tens or ones.

Where this goes next

Adding two three-digit numbers can start from their expanded forms, hundreds with hundreds, tens with tens and ones with ones; that is The partial sums method. Subtracting works the same way, in Subtracting three-digit numbers by place value, including across zero, and comparing two numbers part by part, largest part first, is Comparing and ordering whole numbers. Later, each part is written as a digit times 100, 10 or 1, which is how Partial products breaks up a multiplication, and the same idea reaches to the right of the decimal point in Thousandths: reading, writing and expanded form.

Exercises

Day 1: from a number to its parts

1. Write 563 in expanded form.
2. Write 218 in expanded form.
3. Write 581 in expanded form.
4. Alemayehu builds a number with 3 flats, 8 rods and 1 small square. Write it in expanded form and as a number.
5. Write “6 hundreds, 4 tens, 9 ones” in expanded form and as a number.
6. A museum counter reads 925. Write it in expanded form. Which part is the biggest, and which digit gives it?
7. Which number has the expanded form 100 + 80 + 9?

Day 2: zero digits

8. Write 803 in expanded form.
9. Write 360 in expanded form.
10. Write 900 in expanded form.
11. Write 102 in expanded form.
12. The museum counter shows 780 in the morning. Later it shows 708. Write both numbers in expanded form.
13. Is 300 + 0 + 8 equal to 308? What is the expanded form of 308?

Day 3: from the parts back to the number

14. What number is 50 + 4 + 300?
15. What number is 6 + 500?
16. What number is 90 + 700?
17. What number is 800 + 19?
18. What number is 200 + 40 + 30 + 5?
19. What number is 600 + 150 + 3?
20. Fill in the box: 782 = 700 + \square + 2.
21. At the corn maze farm, Kiele counts 5 crates of 100 ears, 13 bags of 10 ears and 4 loose ears. Write the total as a sum of three parts, then as a number.

Explain

22. Explain why a digit 0 gives no part in expanded form.
23. Are 300 + 40 + 2 and 2 + 40 + 300 the same number? Explain, and name the property.
24. A three-digit number has no digit 0. How many parts does its expanded form have? Explain.
25. Write 540 and 504 in expanded form. Use the parts to explain which number is bigger.