Understanding Place Value Relationships

Times 10 and Divided by 10

In this lesson:

• Every place is exactly 10 times the place to its right
• Every place is 1/10 of the place to its left
• This pattern works the same way everywhere - including decimals

Learning Objectives

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

1. Explain that each place in a multi-digit number is 10 times the value of the place to its right
2. Explain that each place in a multi-digit number is 1/10 the value of the place to its left
3. Identify the value of a digit in any place from thousands to thousandths and relate it to the same digit in neighboring places
4. Predict how a digit's value changes when it shifts one or more places to the left or right
5. Apply the times-10 and divide-by-10 relationships to connect whole number place value to decimal place value

Quick Review: What You Already Know

From Grade 4, you learned:

• A digit in the tens place represents 10 times what it represents in the ones place
• A digit in the hundreds place represents 10 times what it represents in the tens place

Example: In the number 333:

• The first 3 is worth 300 (hundreds)
• The second 3 is worth 30 (tens) - that's 10 times 3
• The third 3 is worth 3 (ones)

Today: We'll extend this pattern in both directions!

The Pattern Going Left: Times 10

Every time we move one place left, the value is 10 times as much:

The "Times 10" Relationship Is Exact

• Not "bigger" or "the next place" - it's exactly 10 times
• Not addition (10 more) - it's multiplication (10 times as much)

Key phrase: "10 times as much"

Example: Comparing the Same Digit in Different Places

In the number 6,364, the digit 6 appears twice:

• The 6 in the thousands place is worth 6,000
• The 6 in the tens place is worth 60

Question: How many times as much is 6,000 compared to 60?

Answer: - so 6,000 is 100 times 60

Why? We moved 2 places to the left, and

Quick Check

Imagine we extended the place value chart one more place to the left of hundred thousands.

What would that place be called?

What would the digit 4 be worth in that place?

Think for a moment before the next slide...

Answer: The place is millions. The digit 4 would be worth 4,000,000 (four million) - because it's 10 times 400,000.

The Universal Rule: Every Place Is 10 Times the Place to Its Right

This pattern is always true:

• No matter how large the number
• No matter which places you're comparing
• The factor is always exactly 10 for adjacent places

Moving left = multiplying by 10

This is not just a fact about specific places - it's the structure of base ten.

Your Turn

Given the number 88,808:

1. What is the value of the 8 in the ten thousands place?
2. What is the value of the 8 in the hundreds place?
3. How many times as much is the first 8 compared to the second 8?

Pause and try before the next slide

Answers:

1. (eighty thousand)
2. (eight hundred)
3. - the first 8 is 100 times the second 8

Moving Right: What About the Other Direction?

We just saw that moving left means times 10.

Question: What happens when we move right?

If moving left multiplies by 10, then moving right must...

...divide by 10!

Or we can say: moving right gives us 1/10 of the value.

The Pattern Going Right: Divide by 10

Every time we move one place right, the value is 1/10 of what it was:

"1/10 of" Means Divide by 10

Fraction language: "One tenth of 400 means one out of ten equal parts of 400"

Division language:

These are two ways of saying the same thing:

• Finding 1/10 of a number
• Dividing by 10

What Happens If We Keep Going Right?

We've reached the ones place: the digit 4 is worth 4.

The pivotal question: What if we move one more place to the right?

What is 1/10 of 4?

This is where the pattern extends into decimals...

The Pattern Continues: Into Decimals

The same pattern - divide by 10 - works across the decimal point!

Visual: Understanding 1/10 of 4

• Take 1 whole unit and partition it into 10 equal slices
• Each slice is 1/10 (one tenth), written as 0.1
• 4 slices = 4 tenths = 0.4

So:

Quick Check: Divide by 10

1. What is 1/10 of 300?
2. What is 1/10 of 50?
3. What is 1/10 of 7?

Pause and try each one

Answers:

3. (seven tenths)

Putting It All Together: Both Directions

• Left arrow (→): times 10
• Right arrow (←): divide by 10

The Pattern Doesn't Break at the Decimal Point

Moving left = ×10 (everywhere - whole numbers AND decimals)
Moving right = ÷10 (everywhere - whole numbers AND decimals)

Examples across the decimal point:

• - ones is 10 times tenths
• - tenths is 10 times hundredths
• - hundredths is 1/10 of tenths

The decimal point is a marker, not a wall.

Comparing Non-Adjacent Places

In the number 2,222.222, compare these two 2s:

• The 2 in the hundreds place is worth 200
• The 2 in the tenths place is worth 0.2

Question: How many times as much is 200 compared to 0.2?

Answer: We moved 3 places to the left.

So is 1,000 times

Real-World Connection: Money

The same pattern:

• 1 dollar = 10 dimes (dime is 1/10 of a dollar)
• 1 dime = 10 pennies (penny is 1/10 of a dime)

Times 10 and divide by 10 show up everywhere!

Key Takeaways

✓ Every place is 10 times the place to its right (moving left = ×10)
✓ Every place is 1/10 of the place to its left (moving right = ÷10)
✓ This pattern works everywhere - whole numbers AND decimals
✓ The decimal point is a marker, not a wall - the pattern continues seamlessly

Watch out:

• The pattern doesn't stop at ones - it continues into decimals
• Place value is about "times," not "plus" (10 times, not 10 more)
• The pattern doesn't change at the decimal point - same rule applies
• Left = times 10 (bigger), Right = divide by 10 (smaller)

Coming Up Next

Now that you understand place value relationships, you're ready for:

5.NBT.A.3: Reading and writing decimals to thousandths using base-ten numerals, number names, and expanded form

You'll use today's times-10 and divide-by-10 understanding to:

• Read decimal numbers correctly
• Write decimals in expanded form
• Compare and order decimals