How to Explain Place Value and Powers of 10 to Fifth Graders

"Why does multiplying by 10 just add a zero?"

When a fifth grader asks this, they're ready to understand the beautiful secret behind our entire number system: powers of 10.

Why Place Value and Powers of 10 Matter

Fifth grade is when students finally see the complete picture. They've learned place value for whole numbers—now they'll understand why it works and extend that understanding in both directions: to massive numbers AND tiny decimals.

This matters because:

• It's the foundation for all decimal operations
• It explains why "moving the decimal point" works
• It prepares students for scientific notation
• It builds true number sense, not just rule-following

The Big Revelation: Everything Is Powers of 10

Here's what makes our number system elegant:

Each place value is a power of 10.

Thousands    = 10 × 10 × 10      = 10³ = 1,000
Hundreds     = 10 × 10           = 10² = 100
Tens         = 10                = 10¹ = 10
Ones         = 1                 = 10⁰ = 1
Tenths       = 1 ÷ 10            = 1/10 = 0.1
Hundredths   = 1 ÷ 10 ÷ 10       = 1/100 = 0.01
Thousandths  = 1 ÷ 10 ÷ 10 ÷ 10  = 1/1000 = 0.001

The pattern goes both directions! Moving left multiplies by 10. Moving right divides by 10.

Understanding Powers of 10

What the Exponent Tells Us

The small number (exponent) tells us how many 10s to multiply:

10¹ = 10                    (one 10)
10² = 10 × 10 = 100         (two 10s)
10³ = 10 × 10 × 10 = 1,000  (three 10s)
10⁴ = 10,000                (four 10s)
10⁵ = 100,000               (five 10s)
10⁶ = 1,000,000             (six 10s)

The shortcut: The exponent equals the number of zeros!

• 10³ has 3 zeros → 1,000
• 10⁶ has 6 zeros → 1,000,000

The Special Case: 10⁰ = 1

This surprises students! Why does anything to the zero power equal 1?

Follow the pattern down:

10³ = 1,000
10² = 100   (÷10)
10¹ = 10    (÷10)
10⁰ = 1     (÷10)

Each step divides by 10. Following that pattern, 10⁰ must equal 1.

Multiplying and Dividing by Powers of 10

The Multiplication Pattern

Multiplying by 10, 100, 1000 shifts digits LEFT:

34 × 10   = 340       (digits shift 1 place left)
34 × 100  = 3,400     (digits shift 2 places left)
34 × 1000 = 34,000    (digits shift 3 places left)

With decimals:

2.5 × 10   = 25       (decimal shifts 1 place right)
2.5 × 100  = 250      (decimal shifts 2 places right)
2.5 × 1000 = 2,500    (decimal shifts 3 places right)

The rule: Multiply by 10ⁿ → shift decimal n places RIGHT (or add n zeros to whole numbers).

The Division Pattern

Dividing by 10, 100, 1000 shifts digits RIGHT:

3,400 ÷ 10   = 340     (digits shift 1 place right)
3,400 ÷ 100  = 34      (digits shift 2 places right)
3,400 ÷ 1000 = 3.4     (digits shift 3 places right)

The rule: Divide by 10ⁿ → shift decimal n places LEFT.

Decimal Place Value: Going Right of the Decimal

The Mirror Pattern

Decimal places mirror whole number places—but they get SMALLER:

      Ones    .    Tenths    Hundredths    Thousandths
       1      .     1/10       1/100         1/1000
       1      .     0.1        0.01          0.001

Just as hundreds are 10× tens, hundredths are 1/10 of tenths.

Reading Decimals by Place Value

In 3.456:

• 3 is in the ones place = 3
• 4 is in the tenths place = 4/10 = 0.4
• 5 is in the hundredths place = 5/100 = 0.05
• 6 is in the thousandths place = 6/1000 = 0.006

Expanded form: 3 + 0.4 + 0.05 + 0.006 = 3.456

The Place Value Chart

Thousands | Hundreds | Tens | Ones | . | Tenths | Hundredths | Thousandths
    1000  |   100    |  10  |  1   | . |  0.1   |    0.01    |    0.001

Notice the symmetry around the ones place!

Comparing Decimals: The Tricky Part

Why 0.9 > 0.12

Students often think 0.12 > 0.9 because "12 is bigger than 9."

The fix: Compare place by place, starting from the left.

0.9  = 0.90  (9 tenths)
0.12 = 0.12  (1 tenth, 2 hundredths)

Compare tenths: 9 > 1
Therefore: 0.9 > 0.12

Key insight: 9 tenths is way more than 1 tenth, regardless of what comes after.

The Money Connection

• 0.9 = 90 cents (9 dimes)
• 0.12 = 12 cents (1 dime, 2 pennies)

Would you rather have 90 cents or 12 cents?

Visual Models for Powers of 10

The Zoom Model

Imagine zooming in and out:

Zooming OUT (×10 each step):
[1 cube] → [10 cubes in a row] → [100 cubes in a flat] → [1000 cubes in a big cube]

Zooming IN (÷10 each step):
[1 whole] → [1/10 slice] → [1/100 tiny piece] → [1/1000 speck]

Base-10 Blocks Extended

Thousands cube = 1,000 (big cube)
Hundreds flat  = 100   (10×10 flat)
Tens rod       = 10    (row of 10)
Ones unit      = 1     (single cube)

Now go smaller:
Tenths        = 0.1    (if the flat is now "1 whole")
Hundredths    = 0.01   (one small cube of the flat)
Thousandths   = 0.001  (imagine splitting that cube into 10)

Hands-On Activities

Powers of 10 Flip Book

Create a flip book showing:

• Page 1: 1 dot
• Page 2: 10 dots
• Page 3: 100 dots (10×10 grid)
• Page 4: 1,000 dots (imagine!)

Students see how quickly numbers grow.

Decimal Number Line

Draw a number line from 0 to 1:

0----0.1----0.2----0.3----0.4----0.5----0.6----0.7----0.8----0.9----1

Then zoom into 0 to 0.1:

0--0.01--0.02--0.03--0.04--0.05--0.06--0.07--0.08--0.09--0.1

Then zoom into 0 to 0.01... and so on!

Calculator Patterns

Have students use calculators to discover patterns:

Start with 5. Multiply by 10 repeatedly:
5 → 50 → 500 → 5,000 → 50,000...

Start with 5000. Divide by 10 repeatedly:
5,000 → 500 → 50 → 5 → 0.5 → 0.05 → 0.005...

Money Math

Use actual or play money:

• $1.00 = 1 whole
• $0.10 = 1 tenth (dime)
• $0.01 = 1 hundredth (penny)

"If you have $3.47, how many tenths is that? How many hundredths?"

Common Mistakes and How to Fix Them

Mistake 1: Adding Zeros Instead of Shifting

Wrong: 2.5 × 10 = 2.50

Fix: Multiplying by 10 makes the number 10 times BIGGER. 2.50 = 2.5 (no change). The answer is 25.

Mistake 2: Confusing Decimal Place Names

Wrong: Calling 0.01 "one tenth"

Fix: Use the place value chart. The first place after the decimal is tenths, the second is hundredths. "Hundredths" has more letters, and it's farther from the decimal—just like "hundreds" is farther from the decimal than "tens."

Mistake 3: Thinking More Digits = Bigger Number (for decimals)

Wrong: 0.125 > 0.3 because 125 > 3

Fix: Compare starting from the decimal point. 0.125 has 1 tenth. 0.3 has 3 tenths. More tenths = bigger number.

Mistake 4: Moving the Decimal the Wrong Way

Wrong: 45 ÷ 100 = 4,500

Fix: Dividing makes numbers SMALLER. Ask: "Is 4,500 smaller than 45?" No! The answer should be 0.45.

Practice Ideas for Home

Daily Decimal Hunt

Find decimals in daily life:

• Gas prices ($3.459 per gallon)
• Sports statistics (batting average 0.325)
• Measurements (3.5 inches)

Ask: "What place is each digit in?"

Powers of 10 Challenge

"What is 3.7 × 100?" Give 5 seconds to answer.

Mix multiplication and division:

• 450 ÷ 10 = ?
• 0.08 × 1000 = ?
• 23,000 ÷ 100 = ?

Build the Number

"Build 4.725 using place values."

Answer: 4 ones + 7 tenths + 2 hundredths + 5 thousandths

Comparison Battles

Flash two decimals. First person to correctly say "greater than" or "less than" wins:

• 0.8 vs 0.75
• 0.099 vs 0.1
• 2.50 vs 2.5 (trick—they're equal!)

Connecting to Future Concepts

Scientific Notation (Middle School)

Understanding 10⁶ = 1,000,000 leads directly to writing:

• 3,400,000 = 3.4 × 10⁶

Negative Exponents (Middle School)

The pattern continues:

• 10⁻¹ = 0.1
• 10⁻² = 0.01
• 10⁻³ = 0.001

Metric System

The metric system IS powers of 10:

• Kilo- = 10³ = 1,000
• Centi- = 10⁻² = 0.01
• Milli- = 10⁻³ = 0.001

Decimal Operations

Understanding that 0.3 × 0.4 involves tenths × tenths = hundredths requires deep place value knowledge.

The Bottom Line

Place value and powers of 10 aren't just topics to memorize—they're the DNA of our number system. When fifth graders truly understand that every place is simply 10 times the place to its right (or 1/10 the place to its left), they've unlocked the pattern that makes all of arithmetic make sense.

The student who sees 3.456 as "3 ones, 4 tenths, 5 hundredths, 6 thousandths" will handle decimal operations with confidence. The student who understands why multiplying by 100 shifts digits two places will never again wonder whether to add or remove zeros.

Powers of 10 is the key that unlocks everything else.