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§ Fractions

Fraction Representations

CCSS.3.NFCCSS.4.NF3 min read

When Sophie draws 3/4 as three-quarters of a circle but places it at 0.34 on a number line, she's struggling with fraction representations. Teaching students to visualise the same fraction across multiple formats—shapes, number lines, decimals, and real-world contexts—builds deeper mathematical understanding that goes far beyond rote calculation.

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§ 01

Why it matters

Fraction representations form the foundation for proportional reasoning used throughout GCSE mathematics and beyond. Year 5 pupils who can confidently place 710 at 0.7 on a number line develop stronger decimal sense, essential for percentage calculations in Year 6 SATs. Restaurant managers use fraction representations when calculating that 23 of 48 customers equals 32 people for table planning. Architects rely on equivalent fractions when scaling blueprints, knowing that 14 inch represents the same proportion as 312 inches. Shop assistants apply fraction-to-decimal conversion when customers pay £7.50 for items marked as 34 off the £30 original price. These visual and numerical connections prepare students for advanced topics like ratio, probability, and algebraic fractions in KS3 and GCSE specifications.

§ 02

How to solve fraction representations

Fraction Representations

  • Show fractions as shaded parts of shapes (circles, bars).
  • Place fractions on a number line between 0 and 1.
  • Equivalent fractions: multiply/divide numerator and denominator by the same number.
  • 12 = 24 = 36 = 48 (all the same amount).

Example: 23 on a number line: divide 0–1 into 3 parts, mark the 2nd.

§ 03

Worked examples

Beginner§ 01

You got 1 out of 5 questions right. Write your score as a decimal.

Answer: 0.2

  1. Understand what we need to do → 1/5 → decimal — A fraction is just a division problem in disguise. 1/5 means '1 divided by 5'.
  2. Divide the top number by the bottom number → 1 ÷ 5 = 0.2 — Divide 1 by 5. Think: 1 out of 5 equal parts is 0.2 of the whole.
  3. Check: does the decimal make sense? → 0.2 < 0.5 → less than half — 1/5 is less than half of the whole. Our decimal 0.2 is less than 0.5. Makes sense!
  4. Write the answer → 1/5 = 0.2 — The fraction 1/5 equals the decimal 0.2.
Easy§ 02

A download is 18 complete. Where is the progress bar?

Answer: 0.12 (close to 0)

  1. Turn the fraction into a decimal → 1 ÷ 8 = 0.12 — To find where 1/8 sits on a number line, convert to a decimal. 1 ÷ 8 = 0.12.
  2. Think about where this falls between 0 and 1 → 0 ← 0.12 → 1 — The number line goes from 0 (nothing) to 1 (the whole thing). 0.5 is exactly in the middle (that is 1/2). Our number 0.12 is close to 0.
  3. Mark the position → 1/8 = 0.12 → close to 0 — Place a dot at 0.12 on the number line. It is close to 0. It is less than half.
  4. Verify with a benchmark → 1/2 = 0.5, 1/8 = 0.12 — Compare to 1/2 (0.5): 0.12 is less than 0.5. This matches our position: close to 0. ✓
Medium§ 03

A shelf holds 16 flowers. 8 are red. What fraction is red?

Answer: 816 = 12

  1. Find the part and the whole → Part = 8, Whole = 16 — We are looking at 8 flowers out of 16 total. The part goes on top (numerator), the whole goes on the bottom (denominator).
  2. Write as a fraction → 8/16 — 8 on top, 16 on bottom gives us 8/16.
  3. Look for a common factor to simplify → GCF of 8 and 16 = 8 — Can both numbers be divided by the same thing? Yes! Both 8 and 16 are divisible by 8. Think of cutting a pizza into fewer, bigger slices — same amount of pizza.
  4. Divide both by the common factor → 8 ÷ 8 = 1, 16 ÷ 8 = 2 → 1/2 — Simplify: 8/16 = 1/2. Simpler fraction, same value!
  5. Check: does this make sense? → 8/16 = 0.5 — As a decimal, 8/16 = 0.5. That means about 50% of the flowers. Does that feel right? ✓
§ 04

Common mistakes

  • Students write equivalent fractions incorrectly, such as claiming 1/2 = 2/3 because they add 1 to both numerator and denominator instead of multiplying both by the same number.
  • Pupils place fractions wrongly on number lines, positioning 3/8 closer to 1 than to 0, not recognising that 3/8 = 0.375 falls in the first half.
  • Children convert 1/4 to 0.14 instead of 0.25, treating the fraction bar as separation rather than division.
  • Students shade fraction diagrams inconsistently, dividing a circle into 6 parts but shading 2 parts to show 2/8 instead of 2/6.
Practice on your own
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§ 05

Frequently asked questions

How do I help Year 3 pupils understand that 1/2 and 2/4 represent the same amount?
Use concrete materials like pizza slices or chocolate bars. Cut one bar in half (1/2) and another identical bar into quarters, taking 2 pieces (2/4). Let pupils see and touch that both amounts are identical. This visual proof builds understanding before introducing the multiply-both-by-same-number rule.
What's the easiest way to teach fraction placement on number lines?
Start with benchmark fractions: 1/4, 1/2, 3/4. Use a metre stick where pupils can see 0cm, 25cm, 50cm, 75cm, 100cm positions. Convert fractions to decimals first: 3/8 = 0.375, so it sits between 1/4 (0.25) and 1/2 (0.5), closer to 1/2.
How can I connect fractions to real-world contexts for Year 4?
Use familiar situations: 'You ate 3 slices of an 8-slice pizza (3/8)', or 'Your phone battery shows 5 bars out of 10 total (5/10 = 1/2)'. Sports work well too: 'United scored in 2 out of 5 matches (2/5)'. Make the denominator the total and numerator the part.
Why do students struggle with improper fractions and mixed numbers?
They don't visualise that 5/3 means 'five thirds' which is more than one whole. Use pie charts: draw one complete circle plus 2/3 of another circle. Show that 5/3 = 1⅔ because you have 1 complete pie plus ⅔ remaining. Physical models make this concept concrete.
What order should I teach different fraction representations?
Begin with visual models (shapes, diagrams) in Year 3, add number lines in Year 4, then decimals in Year 5. Always connect new representations back to previous ones. When introducing 3/5 = 0.6, show the shaded shape, number line position, and decimal simultaneously to build multiple pathways.
§ 06

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