Dividing Decimals and Mixed Operations

Lesson 2 of 2

In this lesson:

• Divide decimals using equivalent whole numbers
• Understand why dividing by less than 1 gives more
• Combine all four operations and explain reasoning

Learning Objectives for This Lesson

1. Divide decimals by reasoning about groups or converting to whole-number equivalents
2. Use estimation to judge reasonableness before and after calculating
3. Represent operations with models and connect to written methods
4. Explain reasoning behind each step using place value language

How Many Pieces of Rope Can You Cut?

You have 4.5 m of rope. Each piece is 0.9 m.

How many pieces can you cut?

• Count by 0.9: 0.9, 1.8, 2.7, 3.6, 4.5
• That is 5 groups of 0.9

Check: 5 × 0.9 = 4.5

Convert Decimal Division to Whole Numbers

Multiply both dividend and divisor by the same power of 10.

• 4.5 ÷ 0.9 → multiply both by 10 → 45 ÷ 9 = 5
• Same answer, easier computation

Converting and Checking with Estimation

What is 7.2 ÷ 0.6?

Estimate: 7 ÷ 0.5 = 14

Convert: multiply both by 10

Check: 12 × 0.6 = 7.2

12 is close to 14 — reasonable (0.6 > 0.5, so fewer groups)

When the Quotient Is a Decimal

What is 3.5 ÷ 0.4?

Estimate: 3.5 ÷ 0.5 = 7

Convert: multiply both by 10

Check: 8.75 × 0.4 = 3.5

8.75 is reasonable — 0.4 < 0.5, so more groups fit

Worked Example: Division by a Whole Number

Share $8.40 equally among 3 friends.

Estimate: $9 ÷ 3 = $3

• 3 into 8 ones → 2 ones, remainder 2
• 24 tenths ÷ 3 = 8 tenths
• 0 hundredths ÷ 3 = 0

Check: 3 × $2.80 = $8.40

Check-In: Use the Conversion Strategy

Solve:

• Estimate first
• Convert to whole numbers
• Verify with multiplication

Pause and try before the next slide.

Answer: The Quotient Exceeds the Dividend

Estimate: 6 ÷ 1 = 6

Convert: 63 ÷ 7 = 9

Check: 9 × 0.7 = 6.3

The quotient (9) is larger than the dividend (6.3) — correct when dividing by less than 1!

Why Dividing by Less Than One Gives More

Smaller pieces means more groups fit.

• 6 ÷ 1 = 6 groups (each piece is 1 whole)
• 6 ÷ 0.5 = 12 groups (half-size → twice as many)

Rule: Dividing by < 1 → quotient > dividend

Practice: Convert and Divide Decimals

Solve each. Convert, compute, and check.

1. 5.6 ÷ 0.8 = ?
2. 9.36 ÷ 4 = ?

Estimate first. Verify by multiplying. Advance for solutions.

Answers: Division with Estimation Checks

Problem 1: 5.6 ÷ 0.8

• Estimate: 6 ÷ 1 = 6
• Convert: 56 ÷ 8 = 7
• Check: 7 × 0.8 = 5.6

Problem 2: 9.36 ÷ 4

• Estimate: 10 ÷ 4 = 2.5
• Compute: 2.34
• Check: 4 × 2.34 = 9.36

From Single Operations to All Four Combined

You now know how to add, subtract, multiply, and divide decimals.

The same tools work for all four:

• Place value reasoning
• Models (charts, area models, number lines)
• Estimation before and after

Notebook Shopping: Multiply and Subtract

Notebooks cost $2.75 each. You buy 3.

Total: Estimate: 3 × $3 = $9

Change from $10: Estimate: $10 − $8 = $2

Total is $8.25; change is $1.75

Notebook Shopping: Divide and Compare

Split $8.25 between 2 friends:

Each pays $4.13 (rounded to nearest cent)

Another store: $2.49 per notebook. How much cheaper?

Each notebook is $0.26 cheaper.

How to Explain Your Decimal Reasoning

Use place value language and estimation in explanations.

Example: "I aligned decimal points so tenths were above tenths. My estimate was 8; I got 8.25."

Sentence starters:

• "I aligned decimal points because..."
• "The product is in hundredths because..."
• "I checked by multiplying..."

Mixed Practice: Three Word Problems

Solve each. Estimate, compute, and explain.

1. 0.75 cups of sugar × 2.5 batches = ? (Multiply)
2. $15.00 − $8.63 = ? (Subtract)
3. 4.8 liters ÷ 0.6-liter bottles = ? (Divide)

Write one sentence explaining your reasoning for each.

Answers: Mixed Practice with Explanations

1. 0.75 × 2.5 = 1.875 cups

• Estimate: 0.75 × 3 = 2.25 — reasonable

2. 15.00 − 8.63 = $6.37

• Estimate: 15 − 9 = 6 — reasonable

3. 48 ÷ 6 = 8 bottles; check: 8 × 0.6 = 4.8

Key Takeaways from Lesson Two

• Divide: Convert to whole numbers — multiply both by 10
• Magnitude: Dividing by < 1 → quotient larger than dividend
• All operations: Place value, models, estimation

Watch out:

• Getting 0.8 instead of 8 for 4.8 ÷ 0.6
• Dividing by < 1 gives bigger results

What Comes Next in Your Math Journey

• Grade 6 brings fluency with standard algorithms (6.NS.B.3)
• The place value reasoning from today is the foundation
• Every algorithm step has a model counterpart

Keep this habit: Estimate → Compute → Check → Explain