Extending Place Value
College Mathematics | Math for Teachers | Part I | Math for Teachers I
Decimals & Powers of Ten
Place value does not stop at the ones place. The same base-ten pattern that gives us tens, hundreds, and thousands also continues in the other direction to give us tenths, hundredths, thousandths, and beyond. Decimals are not a completely new kind of number. They are an extension of the place-value system you already know. In this lesson, we will explore how decimals fit into the number system, how place values continue to the right of the decimal point, how powers of ten organize the pattern, and how place value helps us compare decimal quantities accurately.
What You'll Learn
By the end of this lesson, you should be able to:
Explain how decimals extend the base-ten place-value system.
Identify common types of numbers and describe how decimals fit among them.
Name decimal places and determine the value represented by a digit.
Explain the relationship between neighboring places on both sides of the decimal point.
Connect place values to powers of ten.
Read and write decimal numbers accurately.
Compare and order decimals using place-value reasoning.
Explain why adding zeros to the end of a decimal does not change its value.
Recognize common misconceptions students have about decimal size.
Use representations and questions that support conceptual decimal understanding.
Start Here: Place Value Keeps Going
You already know this pattern:
1,000 → 100 → 10 → 1
Each place is one-tenth of the place to its left. So what happens if we continue past 1? We get:
1 → 0.1 → 0.01 → 0.001
These places are called:
ones,
tenths,
hundredths,
thousandths.
The decimal point does not create a completely new system. It marks the location of the ones place and helps us distinguish whole-number places from fractional places.
Where Do Decimals Fit?
Decimals appear in many parts of mathematics. For example, 0.5 can also be written as 1/2 and 2.75 can be written as 2 3/4. Some decimals terminate. Some repeat. Some never terminate or repeat. The important idea for this lesson is that decimal notation is one way of representing numbers using the base-ten system.
A Quick Look at Number Types
Different number systems build on one another. For example:
Natural numbers: 1, 2, 3, ...
Whole numbers: 0, 1, 2, 3, ...
Integers: ..., -2, -1, 0, 1, 2, ...
Rational numbers: numbers that can be written as a fraction of integers
Irrational numbers: numbers that cannot be written as such a fraction
Real numbers: rational and irrational numbers together
Many familiar decimals are rational numbers. For example, 0.25 = 25/100 = 1/4.
Decimal Place Value
Consider 4.372. The digits represent:
4 ones
3 tenths
7 hundredths
2 thousandths
Notice how the pattern continues smoothly across the decimal point.
| Place | Thousands | Hundreds | Tens | Ones | Tenths | Hundredths | Thousandths |
|---|---|---|---|---|---|---|---|
| Value | 1,000 | 100 | 10 | 1 | 0.1 | 0.01 | 0.001 |
Each place is 10 times the value of the place to its right. Equivalently, each place is one-tenth the value of the place to its left.
Digit, Place, and Value Still Matter
Consider 5.684. Focus on the digit 8.
Digit - 8
Place - Hundredths
Value - 0.08
The same distinction you used with whole numbers still applies. Now focus on the 6.
Digit - 6
Place - Tenths
Value - 0.6
Nothing about the idea of place value changed just because we crossed the decimal point.
Reading Decimals Carefully
Consider 7.304. One useful way to read this is seven and three hundred four thousandths. The word and marks the decimal point in this style of reading. Another way to understand it is 7 + 3/10 + 4/1000. Notice the zero in the hundredths place. That zero is important because it keeps the 4 in the thousandths place.
The Pattern of Powers of Ten
The entire base-ten system can be described using powers of ten. For example:
1,000 = 10³
100 = 10²
10 = 10¹
1 = 10⁰
Then the pattern continues:
0.1 = 10⁻¹
0.01 = 10⁻²
0.001 = 10⁻³
The exponents may look different, but the place-value pattern is the same. Each step to the right divides the value by 10. Each step to the left multiplies the value by 10.
Why 10⁰ = 1 Matters
Look at the pattern:
10³ = 1,000
10² = 100
10¹ = 10
Each time the exponent decreases by 1, the value is divided by 10. So 10⁰ = 1. Then 10⁻¹ = 0.1 and 10⁻² = 0.01. This connects whole-number place value, decimal place value, exponents, and eventually scientific notation.
Try the Pattern
Complete the sequence:
1,000 → 100 → 10 → 1 → _____ → _____ → _____
Now label each with a power of ten.
10³ → 10² → 10¹ → 10⁰ → _____ → _____ → _____
What happens to the value each time you move one place to the right?
Why the Decimal Point Matters
The decimal point helps us identify the position of the ones place. Consider 37.5. The 7 is in the ones place. The 5 is in the tenths place. Now compare 3.75. The same digits appear, but they represent different values. This is another reminder that the value of a digit depends on its position.
Equivalent Decimal Representations
Consider 0.5. We can also write 0.50 or 0.500. These all represent the same number. Why? Because 5 tenths = 50 hundredths = 500 thousandths. Adding zeros to the right end of a decimal does not change the value. This is similar to writing 5 = 5.0 = 5.00.
But Zero Can Matter
Compare 0.5 and 0.05. These are not equal. The 5 moved from the tenths place to the hundredths place. So 0.5 = 5 tenths while 0.05 = 5 hundredths. The zero is important because it changes the place occupied by the 5.
Comparing Decimals
Suppose we compare 0.7 and 0.65. A common mistake is to think 65 is greater than 7, so 0.65 must be greater. But decimals are not compared by treating the digits after the decimal point as whole numbers. Instead, use place value. Write 0.70 and 0.65. Now compare:
tenths: 7 tenths vs. 6 tenths
Since 7 tenths is greater than 6 tenths 0.70 > 0.65. Therefore 0.7 > 0.65.
Compare One Place at a Time
Consider 3.482 and 3.491.
Ones: 3 = 3
Tenths: 4 = 4
Hundredths: 8 < 9
So 3.482 < 3.491. The first place where the digits differ determines the comparison, just as it did with whole numbers.
Another Example
Compare 5.08 and 5.8. It can help to write 5.08 and 5.80. Now compare tenths. The first number has 0 tenths. The second has 8 tenths. Therefore 5.08 < 5.8. The number with more digits after the decimal point is not automatically larger.
A Teacher's View: Watch for Whole-Number Thinking
Many decimal misconceptions come from applying whole-number rules where they do not belong. For example, a student may say 0.125 > 0.8 because 125 > 8. That response tells us something useful. The student may be interpreting the digits after the decimal point as a separate whole number rather than as place-value units.
A helpful question might be “How many tenths are in each number?” or “Could you write 0.8 using thousandths?” Since 0.8 = 0.800 the comparison becomes 0.125 < 0.800.
Using Benchmarks
Sometimes we can compare decimals using familiar benchmarks. For example 0.48 is slightly less than 0.5 and 0.72 is greater than 0.5. So we immediately know 0.48 < 0.72. Benchmarks such as:
0
0.5
1
can support flexible decimal reasoning.
Connecting Decimals and Fractions
Decimal place values are fractional units.
0.1 = 1/10
0.01 = 1/100
0.001 = 1/1000
So 0.37 means 3 tenths + 7 hundredths or 37 hundredths or 37/100. This connection becomes especially important when students later work with fractions and decimals together.
Student Thinking: What Do You Notice?
Situation A
A student says “0.62 is greater than 0.7 because 62 is greater than 7.”
Think About:
What misconception is revealed?
What place should be compared first?
What representation might help?
Situation B
A student says “0.4 and 0.40 can't be equal because 40 is bigger than 4.”
Think About:
What whole-number idea is being misapplied?
How could tenths and hundredths help explain the equivalence?
Situation C
A student reads 2.305 as “two point three hundred five.”
Think About:
What does the student understand?
What place-value language could strengthen the interpretation?
Situation D
A student correctly says 0.09 < 0.1 and explains “Nine hundredths is less than one tenth.”
Think About:
What place-value understanding is demonstrated?
How could you connect the explanation to equivalent forms?
Common Misconceptions
“More decimal digits means a larger number.”
Not necessarily. 0.125 < 0.8 even though 0.125 has more digits.
“Treat everything after the decimal point as a whole number.”
Decimal digits occupy different place values. 0.7 means seven tenths, not “point seven” as an isolated string of digits.
“Adding a zero always changes the number.”
Adding zeros to the far right of a decimal does not change its value. 0.5 = 0.50 = 0.500.
“Zeros after the decimal never matter.”
They can matter when they determine the position of another digit. 0.5 ≠ 0.05.
“The decimal point moves when multiplying by 10.”
A more meaningful interpretation is that each digit's value changes relative to the place-value positions. Saying that “the decimal moves” can hide the underlying mathematics.
Check Your Understanding
Try these without looking back.
What place comes immediately to the right of the ones place?
How are neighboring place values related?
What is the value of the 6 in 4.062?
Write 0.37 as a fraction with denominator 100.
Why are 0.5 and 0.50 equal?
Which is greater: 0.74 or 0.703? Explain.
What power of ten represents one hundredth?
Why does 0.05 represent a smaller quantity than 0.5?
What misconception might lead someone to say 0.125 > 0.8?
Teaching Challenge
Consider 0.46 and 0.406
Part 1: Compare
Write the correct comparison.
Part 2: Explain with Place Value
Which place determines the comparison? Explain.
Part 3: Show an Equivalent Form
Write each number to the thousandths place.
0.46 = __________
0.406 = __________
Part 4: Choose a Representation
Which could help a student understand the comparison? Why?
Place-value chart
Hundred grid
Number line
Base-ten model
Fraction representation
Other: __________________
Part 5: Anticipate Student Thinking
Write one incorrect student explanation. What misconception does it reveal? What follow-up question would you ask?
Reflect
Decimals can sometimes feel like a separate topic from whole-number place value. But after this lesson, consider: What actually changes when we move across the decimal point - and what stays the same? How might emphasizing the continuous base-ten pattern help students see decimals as an extension of ideas they already understand? And from a teaching perspective, why is “the decimal point moves” a less powerful explanation than talking about how the value of each digit changes by a factor of ten?