Understanding Place Value: Ten Times Greater

Grade 4 Number and Operations in Base Ten: 4.NBT.A.1

In this lesson:

• Compare the same digit in different places
• Explain why dividing by the next place gives 10

Learning Objectives for This Lesson

You will learn to:

1. Compare places ten at a time, left to right
2. Compare them one tenth at a time, right to left
3. Compare a repeated digit across two places
4. Model the pattern on a place value chart
5. Explain why 700 ÷ 70 = 10

10 Dimes, 10 Dollars — One Rule?

• You already know 10 dimes trade for 1 dollar.
• Does a "trade ten for one" rule work everywhere in a number?
• Even in places you've never used blocks for — like ten-thousands?

Quick Check: What Does 3 × 10 Mean?

• 3 × 10 means 3 groups of 10, not "3 plus 10."
• Hold onto that "groups of" idea — it's the key to today's lesson.

Bundling Ones, Tens, and Hundreds

Same idea as the dimes and the dollars — bundle 10 of one unit to get 1 of the next.

One Pattern Repeats at Every Step

• 10 ones = 1 ten. 10 tens = 1 hundred. 10 hundreds = 1 thousand.
• Does this pattern ever stop? No — it keeps going forever.
• Rule: each place is worth ten times the place to its right.

The Chart Shows the Pattern: ×10 Left

Every arrow says the same thing: move one place left, multiply by 10.

How Many Times Greater on the Chart

• If 1 hundred is 10 tens, how many times greater is 1 thousand than 1 hundred?
• If 1 thousand is 10 hundreds, how many times greater is 1 ten-thousand than 1 thousand?

Use the chart's arrows to answer before advancing.

Predict First: Which One Is "10 Times"?

Which bar shows "10 times as much as 100" — 110, or 1,000? Pick before advancing.

"10 More" Is Not "10 Times as Much"

• 100 + 10 = 110. That's ten more.
• 100 × 10 = 1,000. That's ten times as much.
• The two bars from the last slide are almost 10 times apart in size.

Fill in the Blanks: Unit Relationships

• 10 hundreds = ___ thousand(s)
• 1 ten-thousand = ___ thousands
• A hundred is ___ times as great as a ten.
• 10 thousands = ___ ten-thousand(s)

From Comparing Places to Comparing Digits

So far you've compared different digits in neighboring places on the chart.

Next: what if the same digit shows up twice in one number?

Predict First: Are Both 4s Worth the Same?

The number 4,400 has a 4 in the hundreds place and a 4 in the thousands place. Are they worth the same?

Same Digit, Different Places: 400 vs. 4,000

• The 4 in the hundreds place is worth 400.
• The 4 in the thousands place is worth 4,000.
• Thousands is one place left of hundreds, so 4,000 = 10 × 400.

Three Places, One Digit: 55,500

• The digit 5 appears in the hundreds, thousands, and ten-thousands places.
• 500 (hundreds), 5,000 (thousands), 50,000 (ten-thousands).
• Each move left multiplies by 10: 500 × 10 = 5,000, and 5,000 × 10 = 50,000.

Adjacent or Not? 2,200 vs. 2,020

• In 2,200: the 2s are in thousands and hundreds — adjacent places, 10 times apart.
• In 2,020: the 2s are in thousands and tens — two places apart.
• Two places apart means two "times 10" jumps: 10 × 10 = 100 times.

Predict First: Does Counting Zeros Work?

The number 3,300 has a 3 in the hundreds place and a 3 in the thousands place.

If you just count the zeros in "3,300," does that tell you how the two 3s compare? Predict before advancing.

Why Zero-Counting Isn't the Real Reason

• The written numeral "3,300" has exactly two zeros total.
• But the 3s compare as 10 times, not 100 times.
• The zeros that matter belong to 300 and 3,000 separately, not to "3,300" as one numeral.

State the Value, Then the Relationship

For each number, name the value of the repeated digit in every place, then state how the values compare:

• 7,770
• 88,800

Every Move So Far Went Left

• Every comparison today has moved from a smaller place to a bigger one.
• What happens if you move the other direction — right, instead of left?

Reading the Place Value Chart Backward

700 is 7 hundreds. 70 is 7 tens. Hundreds is one place left of tens, so 700 = 10 × 70.

More Quotients Using the Same Reasoning

• 5,000 ÷ 500 = 10 — thousands is one place left of hundreds.
• 60,000 ÷ 6,000 = 10 — ten-thousands is one place left of thousands.
• 300 ÷ 30 = 10 — same logic, one place apart.

Saying the Relationship Both Ways

• Moving left: multiply by 10.
• Moving right: divide by 10 — or say "one tenth of."
• 1 thousand is 10 times 1 hundred. 1 hundred is one tenth of 1 thousand.

Explain Without Computing: 40,000 ÷ 4,000

Without dividing, explain why 40,000 ÷ 4,000 must equal 10.

Hint: where do the two 4s sit on the chart?

Predict First: Does This Always Equal 10?

8,000 ÷ 80 — is the answer 10, following the pattern from the last few slides? Predict before advancing.

Counting How Many Places Apart

• 8,000 is in the thousands place. 80 is in the tens place.
• Thousands to tens is two places apart, not one.
• Two places apart means 10 × 10 = 100, so 8,000 ÷ 80 = 100.

Building a Staircase of Powers of Ten

Every step up multiplies by 10. Every step down divides by 10.

The Names Mirror the Math

• "Ten-thousand" literally means ten thousands.
• "Hundred-thousand" literally means one hundred thousands.
• The place-value names are not arbitrary — they describe the math directly.

Extending the Pattern to Hundred-Thousands

• The digit 4 appears in hundred-thousands, ten-thousands, and thousands places.
• Values: 400,000, then 40,000, then 4,000.
• Each is 10 times the one to its right — same rule, bigger numbers.

Solo Commit: One Value, No Help

In the number 660,000, what is the value of the 6 in the hundred-thousands place?

Write your answer alone before comparing with a partner.

Integrative Challenge: 6,660, No Scaffolding

The digit 6 appears three times in 6,660. List the value of each 6, and explain how each value relates to the one in the place to its right.

No fill-in-the-blanks this time — work it out completely on your own.

Error Analysis: Checking a Claim About 330,000

A student claims: "In 330,000, the 3 in the ten-thousands place is worth twice the 3 in the hundred-thousands place."

Is this student correct? Explain why or why not.

Four Common Mistakes to Double-Check

Same digit, different amounts — check the place, not the symbol.

100 + 10 = 110, but 100 × 10 = 1,000 — very different.

The pattern keeps working past thousands, in both directions.

Zeros in the written number aren't the reason — decompose first.

Key Takeaways From Today's Lesson

✓ Each place is ten times the place to its right.
✓ Each place is one tenth of the place to its left.
✓ The same digit is worth different amounts in different places.
✓ Counting how many places apart two values sit tells you their exact ratio.

Coming Up Next: Comparing Numbers

Next lesson uses this exact rule to compare and order multi-digit numbers by their place values.

When two numbers disagree in several places at once, which place decides who's bigger — and why does starting from the left always work?