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Topic ladders · Mathematics

Fractions, Decimals & Percentages

One topic, four rungs: from Classes 1–3 to Classes 11–12. Start at your own stage, or climb from the bottom. Every rung follows the same six steps: Notice, Guess, Try, Observe, Explain, Ask another question.

Print the family table: all four rungs side by side on one sheet, so everyone at home can find their own rung.

Choose your class stage

Rung 1 · Classes 1–3

Sharing Equally (Early Fractions)

Before you start: safety and help

For the grown-up who helps.

  • Safety: Use real food only if hands and surfaces are clean; otherwise use paper models. Any cutting of food is done by a grown-up. Small objects must be too big to fit in a mouth, and kept away from babies and toddlers.
  • Good questions: "Are the parts equal?"

What you need

  • Paper circles and paper strips, or 12 small objects (or a real chapati or fruit if hygiene is managed)
  • Paper and pencil

Child steps

  1. Notice. Sometimes we need to share things equally.
  2. Guess. If 2 children share 1 chapati, how much does each get? What about 4 children?
  3. Try. Fold paper circles in half and in half again, or share 12 objects equally among 2 and among 4.
  4. Observe. Look at the equal parts. How many parts make the whole? Are they really equal?
  5. Explain. Say "one half" or "one quarter" and show it.
  6. Ask another question. What if 3 children share? Try it with 12 objects, or fold a paper strip into three equal parts.
For teachers
  • Learning aims: Understand half and quarter as equal parts of a whole; meet one third.
  • Key vocabulary: whole, half, quarter, third, equal parts, share.
  • Note: Two unequal pieces are not "halves". Children often call any piece a half; checking equality is the key idea.
  • Review: Children show one half and one quarter of a shape.

Rung 2 · Classes 4–7

Fractions, Decimals and Simple Percentages

Before you start: safety and help

For the grown-up or teacher who helps. Students: read it too, so you know the rules.

  • Help with accurate colouring of squares.

What you need

  • Squared paper with 10 × 10 grids (100 squares each), coloured pencils
  • Ruler

Student steps

  1. Notice. The same amount can be written as a fraction, a decimal, or a percentage.
  2. Guess. What is ½ as a decimal and as a percentage?
  3. Try. On a 100-square grid, colour squares to show ½, ¼, ¾, 0.1 and 25%, and write each in all three forms (½ = 50 squares = 0.5 = 50%).
  4. Observe. Compare the different ways of writing the same amount.
  5. Explain. Explain how to convert between fraction, decimal and percentage for simple cases ("per cent" means "out of 100").
  6. Ask another question. How would you show 20% on the grid? Can you shade exactly ⅓ using whole squares only? What goes wrong, and how could you do it if you split one square into three?
For teachers
  • Learning aims: Convert between simple fractions, decimals and percentages; represent them visually.
  • Key vocabulary: fraction, decimal, percentage, equivalent, convert.
  • Step 6: 20% = 20 squares. ⅓ would be 33⅓ squares, so it cannot be shaded exactly using whole squares only: ⅓ = 0.333… (recurring) ≈ 33.3%. Splitting one square into three equal strips and shading one of them makes it exact.
  • Review: Students show one quantity in all three forms.

Rung 3 · Classes 8–10

Using Percentages in Real Situations

Before you start: safety and help

For the grown-up or teacher who helps. Students: read it too, so you know the rules.

  • Encourage estimation before exact calculation.

What you need

  • Paper, pencil, calculator (optional)
  • Real examples (discounts, exam marks, interest, recipes)

Student steps

  1. Notice. Percentages appear in shops, results, bank interest and news.
  2. Guess. How do you calculate a percentage of a quantity or find the original amount?
  3. Try. Solve real problems: discount prices, percentage increase/decrease, simple interest, exam percentages.
  4. Observe. Check answers using different methods (mental, written, calculator).
  5. Explain. Write clear steps and explain the method used.
  6. Ask another question. What is the difference between successive percentage changes and a single overall change?
For teachers
  • Learning aims: Apply percentages to practical contexts; calculate increase, decrease and reverse percentages.
  • Key vocabulary: percentage increase/decrease, discount, profit, loss, simple interest, reverse percentage.
  • Useful facts: Simple interest = P × R × T ÷ 100. Reverse percentage: if a shirt costs ₹400 after a 20% discount, the original price is ₹400 ÷ 0.8 = ₹500 (not ₹400 + 20% = ₹480). Successive changes: +10% then −10% gives 1.1 × 0.9 = 0.99, an overall fall of 1%.
  • Review: Students solve and explain one real-life percentage problem.

Rung 4 · Classes 11–12

Advanced Applications of Percentages and Compound Change

What you need

  • Calculator, paper (or a spreadsheet)

Student steps

  1. Notice. Many real quantities change by percentages repeatedly (compound interest, population, depreciation).
  2. Guess / Hypothesise. How does compound change differ from simple change over several periods?
  3. Try. Calculate compound interest, population growth, or depreciation over multiple periods. Compare with simple interest.
  4. Observe. Tabulate results and plot growth curves if possible.
  5. Explain. Derive or use the compound growth formula and discuss assumptions (is the rate really the same every year?).
  6. Ask another question. How sensitive are long-term results to small changes in the percentage rate?
For teachers
  • Learning aims: Apply compound percentage change; interpret results in financial or scientific contexts.
  • Formulae: growth A = P(1 + r/100)ⁿ; depreciation A = P(1 − r/100)ⁿ. Quick check: at 8% a year an amount roughly doubles in 9 years (the "rule of 72": 72 ÷ 8 = 9).
  • Use made-up numbers. This is maths practice, not advice on real loans or investments.
  • Review: Students present a multi-period calculation with clear interpretation.

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