Year 8 Mathematics · Number and Algebra · 10m read

Fractions, Decimals and Scientific Notation

Ever wonder how scientists write mind-bogglingly huge or impossibly tiny numbers without running out of space? That's where scientific notation comes in — it's shorthand for expressing numbers as a power of 10, making calculations way easier. You'll master fractions and decimals too, since they're the foundation for everything from cooking recipes to engineering projects, and they're guaranteed to show up on every VCE maths exam you'll sit.

Learning objective: Operate with fractions and decimals, and express numbers in scientific notation.

By the end of this lesson you can:

• I multiply and divide fractions including mixed numbers.
• I convert between standard form and scientific notation.
• I order and compare numbers in scientific notation.

Scientific notation is a zoom lens. Huge numbers (distance to stars) and tiny numbers (size of cells) are hard to read. Scientific notation zooms them into a neat form: a × 10ⁿ.

Fraction operations and scientific notation

MULTIPLYING FRACTIONS:
Multiply tops × tops, bottoms × bottoms. Then simplify.
3/4 × 2/5 = 6/20 = 3/10.
SHORTCUT: cancel common factors before multiplying.

DIVIDING FRACTIONS: "Keep, Change, Flip" (KCF).
Flip (invert) the second fraction, then multiply.
3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8 = 1 7/8.

MIXED NUMBERS: convert to improper fractions first.
2¼ × 1⅓: convert → 9/4 × 4/3 = 36/12 = 3.

SCIENTIFIC NOTATION (standard form):
Expresses very large or very small numbers as:
a × 10ⁿ where 1 ≤ a < 10 and n is an integer.
Large numbers: 45 000 = 4.5 × 10⁴ (decimal moves left 4 places)
Small numbers: 0.000 032 = 3.2 × 10⁻⁵ (decimal moves right 5 places)

CONVERTING:
Standard → Sci Notation: move decimal until 1 ≤ a < 10. Count moves = n.
Move left: n is positive. Move right: n is negative.
6 200 000 → 6.2 × 10⁶ (moved 6 left)
0.00047 → 4.7 × 10⁻⁴ (moved 4 right)

COMPARING in scientific notation:
Compare exponents first. Same exponent? Compare the coefficient.
3.2 × 10⁸ vs 9.1 × 10⁷: exponent 8 > 7, so 3.2 × 10⁸ is larger.

Scientific notation rules

• Coefficient 1 ≤ a < 10: The coefficient MUST be at least 1 and less than 10. 0.32 × 10⁵ is NOT correct scientific notation — it should be 3.2 × 10⁴.
• Negative exponent = small number: 10⁻⁴ = 0.0001. Negative exponent tells you to divide, not multiply.
• KCF for division: Keep the first fraction, Change ÷ to ×, Flip the second fraction. Never forget to flip.

Fractions and scientific notation

• Multiply fractions: \dfrac{a}{b} \times \dfrac{c}{d} = \dfrac{ac}{bd}
• Divide fractions: \dfrac{a}{b} \div \dfrac{c}{d} = \dfrac{ad}{bc}
• Scientific notation: a \times 10^n,\; 1 \le |a| < 10

Quick reference

| Standard | Scientific notation |
|---|---|
| 3 400 000 | 3.4 × 10⁶ |
| 0.00052 | 5.2 × 10⁻⁴ |
| 1 | 1 × 10⁰ |
| Rule | 1 ≤ a < 10 |
| Big number | Positive power |
| Small number | Negative power |

Worked example (Medium)

Write 0.000 000 45 in scientific notation.

1. Move decimal right until 1 ≤ a < 10. 0.000 000 45 → 4.5 (moved right 7 places).
2. Moving right → negative exponent. n = −7.
3. Write in form. 4.5 × 10⁻⁷.

Answer: 4.5 × 10⁻⁷

Worked example (Hard)

Apply advanced understanding of fractions, decimals and scientific notation to a challenging real-world scenario. Demonstrate systematic analysis using key concepts from this topic.

1. Analyse the context. Identify the key principles of fractions, decimals and scientific notation that are relevant. Consider what specific knowledge is needed to address this scenario.
2. Apply specific knowledge. Use precise mathematical terminology to analyse the situation. Show how concepts interact and build upon each other.
3. Evaluate implications. Consider the broader significance. How does this analysis connect to other aspects of maths?
4. Form a conclusion. Synthesise your analysis into a well-reasoned conclusion supported by specific evidence and terminology.

Answer: Analysis demonstrates deep understanding of fractions, decimals and scientific notation with specific terminology, systematic reasoning, and real-world application.

Common mistakes

• Providing surface-level responses without depth Students state facts about fractions, decimals and scientific notation without explaining underlying mechanisms, reasoning, or significance. Fix: Always explain WHY and HOW, not just WHAT. In mathematics assessments, depth of explanation distinguishes high-achieving responses from basic recall.
• Not using precise mathematical terminology Students use everyday language when specific terms would demonstrate stronger understanding. Fix: Build and use a vocabulary of key terms for this topic. Precise terminology shows understanding and earns higher marks in mathematics assessments.

Fractions, Decimals and Scientific Notation — connecting to the bigger picture

Understanding fractions, decimals and scientific notation extends beyond this single topic — it connects to broader themes across maths and other disciplines. Recognising these connections transforms isolated knowledge into an integrated framework that supports deeper analysis and creative problem-solving.

In professional practice, experts use conceptual frameworks — organised structures of related ideas — to approach new problems efficiently. They don't recall every fact; they activate connections between concepts that help them identify what's relevant. This is why the most effective study approach is not memorising more facts, but building richer connections between the facts you already know.

Active learning strategies that build these connections include: relating new concepts to personal experience, comparing ideas across different topics, applying knowledge to unfamiliar scenarios, and teaching concepts to others. Each of these forces you to process information more deeply than passive reading or highlighting. The goal is to develop the kind of flexible understanding that allows you to analyse any question about fractions, decimals and scientific notation — even ones you haven't seen before — by drawing on a network of connected ideas.

Practice questions

1. Write 0.00037 in scientific notation.

   1. 3.7 × 10⁻⁴
   2. 3.7 × 10⁴
   3. 37 × 10⁻⁵
   4. 0.37 × 10⁻³

   (1 marks)

2. Calculate: 2⅓ ÷ 1¼

   (2 marks)

Summary

• Multiply and divide fractions including mixed numbers.
• Convert between standard form and scientific notation.
• Order and compare numbers in scientific notation.