Math // Fractions

Understanding Percentages

Understand the per cent symbol (%); know that per cent means ‘number of parts per hundred’; write percentages as a fraction with denominator 100 and as a decimal

Ages
9 to 10
Type
Concept
Builds on
2 topics
Leads to
3 topics
Standards
1 code

Quick check

If a shop advertises 50% off, can the learner explain that this means half price — and write 50% as both a fraction and a decimal?

Ask this to see whether the topic is secure.

Topic // Mastery

What mastery looks like

What a learner can do once this topic is secure.
  • Write 35% as 35/100 and as 0.35
  • Shade 40% of a 10×10 grid and write the fraction 40/100
  • Explain that 100% means the whole, 50% means half

Topic // Map

Where this fits

What to learn first and what this topic opens up, with the reason for each link. Required links come first.
  • RequiredMust come first
  • HelpfulHelps, but not essential

Before this

2 topics
  • Decimal & Percent NotationRequired

    Fractions // Ages 8 to 11

    Understanding the % symbol and 'per cent means parts per hundred' is the LANGUAGE node content

  • Decimals for Tenths & HundredthsRequired

    Fractions // Ages 9 to 10

    Decimal notation for fractions is prerequisite to understanding % as parts per 100

You are here

Understanding Percentages

Ages 9 to 10 // Concept

Leads to

3 topics

Topic // Research-informed

Teaching notes

From the Marble Skill Taxonomy. These notes have not been validated in classrooms, so treat them as a starting point for your own judgement.

The approach

A percentage is a proportion expressed as parts per hundred: 40% means 40 out of every 100, so it equals 40/100 and 0.40, with 100% as the whole and 50% as half of it. The decisive idea — and the one children most often miss — is that when we find a percentage OF an amount, the percentage acts on a base rather than standing for a fixed amount: 50% is not 'fifty' (it equals 50/100 = 0.5), it is half of whatever the whole is, so 50% of 20 is 10 while 50% of 200 is 100. Teach percentage on the hundred square (40 of 100 shaded) tied to the fraction n/100, the decimal, and a 0%–100% number line, establish landmarks (100%, 50%, 25%, 10%, 1%) and that percentages can exceed 100%, keep asking '% of what?', and confront base-neglect directly through deliberate comparisons.

Common misconceptions

  • Base-neglect: treating a percentage as a fixed number independent of the whole ('50% is always 50 / a lot').
  • Reading a percentage as a fraction with the number as denominator ('24% means 1/24', '40% means 1/40').
  • Percentage–decimal placeholder error: writing 9% as 0.9 instead of 0.09.
  • Believing 100% is the maximum — that you cannot have more than 100%.
  • Thinking a bigger percentage always means a bigger amount, regardless of the base.

Useful representations

  • 10×10 hundred square shaded to the percentage
  • Four-way link: percentage + fraction n/100 + decimal + 0%–100% number line
  • Bar model of a whole partitioned to show a percentage of it
  • Landmark strip (0%, 10%, 25%, 50%, 100%) against fractions/decimals
  • Two different-sized wholes each showing '50%' to expose the role of the base

Teaching practices

  • Anchor 'per cent = parts per hundred' on a 10×10 hundred square: shade 40 squares to show 40%, and read off 40/100 and 0.40 from the same picture.
  • Foreground the base from the outset: a percentage is a proportion OF a whole, so always ask '% of what?'; show that 50% of a small set and 50% of a large set are the same proportion but different amounts.
  • Link the three forms explicitly — 40% = 40/100 = 0.40 — and simplify where natural (40/100 = 2/5) so percentage sits inside the fraction–decimal web.
  • Establish landmarks: 100% = the whole, 50% = a half, 25% = a quarter, 10% = a tenth, 1% = a hundredth; and seed that percentages can exceed 100%.
  • Confront base-neglect with deliberate comparisons (10% of 500 = 50 versus 90% of 60 = 48) so children see a smaller percentage of a bigger base can be the larger amount.

Activities to try

  1. 1

    Same percentage, different whole: resize the base

    Why it works: A percentage fixes the PROPORTION of a whole, not a fixed number; the actual amount depends entirely on the base, so you must always ask '% of what?'.

    1. Set a percentage (e.g. 50%) and shade that proportion of a 10×10 hundred square.
    2. With the whole set to 20, read off the actual amount (50% of 20 = 10); then change the whole to 200 and read the new amount (50% of 200 = 100).
    3. Describe what stays the same (the proportion / the shaded half of the square) and what changes (the actual amount) as the base changes.
    4. Alongside the shading, write the percentage as a fraction n/100 and as a decimal, and locate it on a 0%–100% number line.
  2. 2

    Bigger percentage, smaller amount: compare across different bases

    Why it works: A bigger percentage does not guarantee a bigger amount, because the amount also depends on the size of the base being taken.

    1. Work out 10% of 500 and 80% of 60, showing each amount.
    2. State which is the larger AMOUNT and explain how you know without guessing.
    3. Reflect on why the 'smaller' percentage (10%) gave the bigger amount (50 beats 48).
  3. 3

    Writing and checking the three forms

    Why it works: Percentage, fraction (over 100) and decimal are three names for the same proportion; 'per cent' means parts per hundred, key landmarks (100%, 50%, 25%, 10%, 1%) link to familiar fractions and decimals, 100% is the whole, and percentages can exceed 100%.

    1. Complete links such as 40% = __/100 = 0.__ and 35% = 35/100 = 0.35.
    2. Shade 40 of 100 squares to match 40%, and simplify 40/100 to 2/5 where asked.
    3. Place landmark percentages on a 0%–100% strip and match them to their fraction and decimal: 100% = the whole = 1, 50% = a half = 0.5, 25% = a quarter = 0.25, 10% = a tenth = 0.1, 1% = a hundredth = 0.01.
    4. Convert single-digit percentages carefully: write 9% as a decimal and as a fraction, and 5% as a decimal.
    5. Decide whether a percentage can exceed 100% and say what 150% would mean.

Topic // Standards

Curriculum standards

Codes for the curriculum statements this topic maps to. Each publisher owns its standard text.
The national curriculum in England: Key stages 1 and 2 framework document
Ma5/2.4k

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