Expanded Form or Expanded Notation? - Leaf and STEM Learning - Free Printable

You're looking at a math worksheet titled “Writing decimals with place value”, using the decimal 0.689 as an example.

The image shows multiple equivalent ways to express 0.689 by breaking it down using place value — that is, understanding what each digit represents based on its position after the decimal point.

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🔢 Step-by-Step Breakdown:

#### 1. Original Decimal:
> 0.689

This means:
- 6 in the tenths place → 0.6
- 8 in the hundredths place → 0.08
- 9 in the thousandths place → 0.009

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#### 2. Expanded Form (Addition of Place Values):
> 0.6 + 0.08 + 0.009

This is simply adding the value of each digit according to its place.

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#### 3. Multiplication Form (Place Value Multiplied by Digit):
> (6 × 0.1) + (8 × 0.01) + (9 × 0.001)

Here, we’re showing that:
- The digit 6 is multiplied by 0.1 (which is 1/10) → 6 tenths
- The digit 8 is multiplied by 0.01 (which is 1/100) → 8 hundredths
- The digit 9 is multiplied by 0.001 (which is 1/1000) → 9 thousandths

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#### 4. Fraction Multiplication Form:
> (6 × 1/10) + (8 × 1/100) + (9 × 1/1000)

Same idea as above, but using fractions instead of decimals for the place values.

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#### 5. Fraction Addition Form:
> 6/10 + 8/100 + 9/1000

Now we’ve written each part as a fraction.

To add these, you’d need a common denominator — which is 1000.

So:
- 6/10 = 600/1000
- 8/100 = 80/1000
- 9/1000 = 9/1000

Add them:
600 + 80 + 9 = 689 → 689/1000

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#### 6. Final Fraction Form:
> 689/1000

This is the fraction equivalent of 0.689.

✔ So, 0.689 = 689/1000

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🎯 Summary:

The image teaches how to decompose a decimal into its place value components and shows how those components can be expressed as:

- Decimals added together
- Multiplications of digits by their place values
- Fractions added together
- A single simplified fraction

This helps students understand what decimals really mean — they’re just another way to write fractions with denominators that are powers of 10 (10, 100, 1000, etc.).

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✔ Final Answer:

> 0.689 = 6/10 + 8/100 + 9/1000 = 689/1000

All forms shown in the image are mathematically equivalent — they’re just different ways of expressing the same number using place value concepts.

Let me know if you want to try this with another decimal! 🖍️📚