Decimal Operations: All Four Algorithms

Standard Algorithms for Add, Subtract, Multiply, and Divide

In this lesson:

• Align decimal points to add and subtract
• Count decimal places to multiply
• Place the decimal (or convert) to divide

Learning Objectives for This Lesson

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

1. Add and subtract decimals by aligning decimal points
2. Multiply decimals by counting total decimal places in the factors
3. Divide a decimal by a whole number using the long division algorithm
4. Divide a decimal by a decimal by converting to a whole-number divisor
5. Use estimation to verify that decimal point placement is correct

Everyday Decimals — Can You Add These?

A grocery receipt shows three items:

• Bread: $4.75
• Juice: $12.09
• Gum: $0.89

What is the total cost?

Think about how you'd set this up before the next slide.

Decimal Alignment: Why It Matters

Align decimal points — not the rightmost digits.

Adding Decimals: Padding with Zeros

Problem:

Step 1: Align decimal points, pad with a trailing zero:

Step 2: Add column by column from right to left → 5.64

Estimate check: — answer 5.64 is close ✓

Subtracting Decimals: Regrouping Across the Point

Problem:

Step 1: Align and pad:

Step 2: Regroup as needed — identical to whole-number subtraction:

Estimate check: — answer 2.42 is close ✓

More Addition and Subtraction Examples

Three addends:

Pad to thousandths:

Estimate: ✓

Subtracting from a whole number:

Write — estimate: ✓

Quick Check: Estimate and Solve

Problem:

1. First, write your estimate (round to the nearest whole number)
2. Then solve with the standard algorithm
3. Check: does your answer match your estimate?

Try it before the next slide.

Multiplying Decimals: The Three-Step Rule

1. Ignore decimal points — multiply as whole numbers
2. Count total decimal places in both factors
3. Place decimal that many positions from the right

Example: : ; 2 places → 3.22

Why the Rule Works: Fraction Connection

Why does counting decimal places work?

• and
• Product denominator: → 2 decimal places

Three Decimal Multiplication Examples Worked

Ex 1: : → 2 places → 3.22 ✓

Ex 2: : → 3 places → 0.024 (pad zero!)

Ex 3: : → 3 places → 8.478 ✓

Watch Out: The Leading-Zero Pitfall

• , but 3 places are needed
• Wrong: ✗ — Right: ✓

Your Turn: Multiply with Estimation

Solve each problem. First estimate, then compute.

For each: (a) estimate the product, (b) multiply the whole-number versions, (c) count decimal places, (d) check your answer matches your estimate.

Dividing Decimal by Whole Number: One Rule

Place the decimal point in the quotient directly above its position in the dividend — before any division steps.

Then divide as with whole numbers.

Worked Example: 9.45 ÷ 3 = 3.15

• Decimal placed first in quotient above 9.45's decimal
• Divide: , then → 3.15

Check: ✓

Worked Example: 13.72 ÷ 4 = 3.43

• Place decimal point in quotient above the decimal in 13.72
• R; bring down : R
• Bring down : → quotient 3.43

Check: ✓

Dividing by a Decimal: Conversion Rule

• 1 decimal place in divisor → multiply both by 10
• 2 decimal places in divisor → multiply both by 100
• Always multiply both — never just the divisor

Worked Example: 1.32 ÷ 0.4

Divisor 0.4 has 1 decimal place → multiply both by 10:

Now apply Case A:

Check: ✓

Estimate: — answer 3.3 is close ✓

Worked Example: 5.76 ÷ 0.08

Divisor 0.08 has 2 decimal places → multiply both by 100:

Now apply whole-number division:

Check: ✓

Estimate: — answer 72 is in the right range ✓

Quick Check: Find the Right Multiplier

For each, count divisor's decimal places → multiply both by that power of 10:

1. → ×___
2. → ×___
3. → ×___

Key Takeaways: All Four Operations

• Add/Sub: Align decimal points; pad with trailing zeros
• Multiply: Multiply whole numbers; count places in factors
• Divide by whole: Place decimal in quotient first
• Divide by decimal: Multiply both by power of 10 first
• Always estimate to verify placement ✓

Watch Out: Four Common Decimal Errors

Add/Sub: Align decimal points — , not

Multiply: Count places in the factors — 1+1=2 → 3.22, not 32.2

Multiply: Pad zeros — , not

Divide: Multiply both numbers — , not

What Comes Next: Factors and Multiples

Next lesson: 6.NS.B.4

• Greatest Common Factor (GCF) and Least Common Multiple (LCM)
• Applying GCF and LCM to real-world problems
• Decimal fluency from today supports ratio and percent work ahead