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Building Numbers in Base Ten

Composing & Decomposing Numbers

Place value tells us what each digit represents. But understanding place value goes beyond identifying the hundreds, tens, and ones places. A strong understanding of number means recognizing that quantities can be built, broken apart, and regrouped in many different ways. For example, 347can be thought of as:

• 3 hundreds, 4 tens, and 7 ones

• 34 tens and 7 ones

• 3 hundreds, 3 tens, and 17 ones

• 300 + 40 + 7

• 200 + 140 + 7

All of these representations describe the same quantity.

In this lesson, we will explore how numbers are composed and decomposed using the structure of the base-ten system—and why flexible representations are so important for later arithmetic.

What You'll Learn

By the end of this lesson, you should be able to:

• Build whole numbers using base-ten units.

• Explain what it means to compose and decompose a number.

• Represent the same number in multiple equivalent ways.

• Explain why 10 units of one place can be regrouped as 1 unit of the next place.

• Move flexibly among hundreds, tens, and ones.

• Connect base-ten models to symbolic representations.

• Recognize the difference between standard and nonstandard decompositions.

• Explain how flexible place-value thinking supports addition, subtraction, multiplication, and division.

• Analyze student representations for evidence of place-value understanding.

Start Here: There Is More Than One Way to Build a Number

Consider the number 243. A familiar representation is 2 hundreds + 4 tens + 3 ones. That is correct. But it is not the only way to represent 243. We could also write 24 tens + 3 ones, or 2 hundreds + 3 tens + 13 ones, or 1 hundred + 14 tens + 3 ones. Every representation still has a total value of 243. This flexibility is one of the most important ideas in place value.

Building Numbers with Base-Ten Units

One way to make place value visible is to build numbers using physical or visual models. With base-ten blocks, we might represent:

• one small unit as 1

• one rod as 10

• one flat as 100

So the number 326 could be represented with:

• 3 hundreds

• 2 tens

• 6 ones

A model helps make the structure of the written number visible. The 3 does not simply mean “three.” It means three hundreds. The 2 means two tens. The 6 means six ones.

The Fundamental Trade in Base Ten

One of the most important relationships in our number system is that 10 ones = 1 ten, 10 tens = 1 hundred, and 10 hundreds = 1 thousand. These are not different quantities. They are different ways of grouping the same quantity.

Imagine ten individual units. If we group all ten together, we can treat them as one ten. Nothing has been added. Nothing has been removed. Only the grouping has changed.

Composing and Decomposing Numbers

To compose a number means to put smaller units or parts together to form a larger quantity. To decompose a number means to break a quantity into smaller parts. Consider the number 352. We can compose it from 300 + 50 + 2 or 3 hundreds + 5 tens + 2 ones. That is one decomposition of 352.

But we can decompose the same number differently. For example 300 + 40 + 12 because 40 + 12 = 52. So 300 + 40 + 12 = 352. The number has not changed. Only the way we have broken it apart has changed.

Standard and Nonstandard Decompositions

We often teach students the standard decomposition first. For the number 468, the standard decomposition is 4 hundreds + 6 tens + 8 ones or 400 + 60 + 8. But there are many other valid decompositions. As an example, 468 could also be 3 hundreds + 16 tens + 8 ones because 300 + 160 + 8 = 468. Or it could be 4 hundreds + 5 tens + 18 ones because 400 + 50 + 18 = 468. It could even be 46 tens + 8 ones because 460 + 8 = 468.

These are sometimes called nonstandard decompositions. They are extremely useful because they show whether a student understands the relationships among places rather than simply memorizing a fixed format.

Try It

Represent the number 274 in at least four different ways.

Standard Form

2 hundreds + 7 tens + 4 ones

Another Way

_____ hundreds + _____ tens + _____ ones

Using Only Tens and Ones

_____ tens + _____ ones

Your Own Decomposition

How can you check that each representation still equals 274?

Trading Without Changing the Amount

Suppose we have 1 hundred + 4 tens + 3 ones. That represents 143. Now trade one hundred for ten tens. We have: 14 tens + 3 ones. The representation looks different. The quantity remains 143. We could also trade one ten for ten ones to get 13 tens + 13 ones. Again, the number is still 143. This idea of trading equivalent place-value units becomes essential when students later learn regrouping in addition and subtraction.

Flexible Thinking in Place Value

A student with flexible place-value understanding does not see a number in only one fixed way. Consider the number 1,250. It can be thought of as:

• 1 thousand + 2 hundreds + 5 tens

• 12 hundreds + 5 tens

• 125 tens

• 1,200 + 50

• 1,000 + 250

• 1,100 + 150

Different representations are useful for different purposes.

For example, if we want to subtract 50 from 1,250, thinking of the number as 1,200 + 50 may be especially useful. If we want to divide 1,250 into groups of 10, thinking of it as: 125 tens may be more useful. Flexible representations help students choose methods strategically.

Why Flexible Thinking Matters

Place value is not only about naming positions. It supports efficient computation. Consider 398 + 25. A student might think 398 + 2 = 400, and then 400 + 23 = 423. That strategy depends on understanding how numbers can be decomposed and recomposed.

Or consider 503 - 198. A student might think 198 is close to 200. So 503 - 200 = 303, then add back 2 to get 305. Again, flexible number structure supports mental reasoning.

Representation Matters

A number can be represented in several ways:

• Symbolically - 426

• Expanded Form - 400 + 20 + 6

• Place-Value Units - 4 hundreds + 2 tens + 6 ones

• Base-Ten Model - 4 flats, 2 rods, and 6 units

• Nonstandard Decomposition - 3 hundreds + 12 tens + 6 ones

These are not different numbers. They are different ways of seeing the same quantity.

A Teacher's View: Go Beyond “How Many Hundreds?”

Suppose a student looks at the number 352 and correctly says 3 hundreds, 5 tens, and 2 ones. That is useful evidence. But try asking “Can you build 352 without using any hundreds?”

A student with flexible place-value understanding might say 35 tens and 2 ones. Then ask “Can you show it another way?” These questions reveal much more than asking only for the standard decomposition.

Student Thinking: What Do You Notice?

Situation A

A student says “324 is 3 hundreds, 2 tens, and 4 ones.”

Think About:

• Is the representation correct?

• What does it tell you?

• What could you ask next to determine whether the student's thinking is flexible?

Situation B

You ask a student to represent 324 another way. The student says “It can't be represented another way because the 3 has to stay in the hundreds place.”

Think About:

• What misconception might this reveal?

• What physical model could help?

Situation C

A student says “324 is 32 tens and 4 ones.”

Think About:

• Is the student correct?

• How could you verify the representation?

• What place-value relationship does this demonstrate?

Situation D

A student models 152 using:

• 1 hundred

• 4 tens

• 12 ones

Think About:

• Is the model correct?

• What evidence of understanding does it provide?

• Why might this representation be useful when learning subtraction?

Common Misconceptions

“A number has only one correct decomposition.”

The standard decomposition is useful, but many equivalent decompositions are possible.

“You can't have more than 9 ones or 9 tens.”

In standard written notation, each place contains one digit from 0 through 9. But when representing or decomposing a quantity, we can absolutely have:

• 14 ones,

• 17 tens,

• 12 hundreds,

as long as the total value remains correct.

“Trading changes the number.”

Trading ten ones for one ten changes the representation—not the quantity.

“Expanded form always has to look exactly one way.”

The standard expanded form of 352 is 300 + 50 + 2 but other equivalent expressions can represent the same number.

“Base-ten blocks are only for young children.”

Concrete models can reveal important mathematical structure at many levels. The goal is not merely to manipulate blocks, but to connect physical grouping to symbolic reasoning.

Check Your Understanding

Try answering these questions without looking back.

1. What does it mean to compose a number?

2. What does it mean to decompose a number?

3. Why is 300 + 40 + 12 a valid decomposition of 352?

4. How many tens are in 470?

5. How could you represent 263 without using any hundreds?

6. Why does trading 10 ones for 1 ten not change the total?

7. What is the difference between a standard and nonstandard decomposition?

8. Why is flexible decomposition useful when performing arithmetic?

Teaching Challenge

Choose a three-digit number. Represent it in four ways.

• Standard Place-Value Form

• Expanded Form

• Without Using Hundreds

• A Nonstandard Decomposition

Now imagine a student says “Only the first representation is correct.” How would you respond? What model or question could you use to help the student see that all four representations describe the same quantity?

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