Math // Fractions

Adding fractions (different denominators)

Add and subtract fractions with denominators that are multiples of the same number by finding a common denominator

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

Quick check

If the learner needs to add 1/3 and 1/6, can they find a common denominator and work out the total — showing their working?

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.
  • Calculate 1/3 + 1/6 by converting to sixths: 2/6 + 1/6 = 3/6 = 1/2
  • Calculate 3/4 − 1/8 by converting to eighths
  • Add 2/5 + 3/10 and simplify the answer

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

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Adding fractions (different denominators)

Ages 9 to 10 // Method

Leads to

2 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

Adding and subtracting fractions whose denominators are RELATED — one denominator is a multiple of the other (1/3 + 1/6, 3/4 − 1/8, 2/5 + 3/10). The central new idea: when you combine fractions by counting parts, the parts must be the same size, so one fraction is renamed as an equivalent fraction sharing a common unit (common denominator), after which like parts are counted as before. Because the denominators are related, only ONE fraction needs converting (e.g. 1/3 = 2/6), making this a gentle bridge from same-denominator work to fully unlike denominators. Renaming must scale numerator AND denominator together — the heart of getting it right.

Common misconceptions

  • Adds across the unlike denominators: 1/3 + 1/6 = 2/9, combining numerators and denominators (NFER's 3/9-type error).
  • Renames the denominator without scaling the numerator: converts 1/3 to 1/6 and gets 1/6 + 1/6 = 2/6 — 'just inserts the larger denominator' (IES).
  • Forgets to simplify the result (leaves 3/6 rather than 1/2), or doesn't recognise the answer can be reduced.

Useful representations

  • Two fraction strips with related partitions (thirds and sixths) overlaid so one must be re-partitioned to match
  • Number line where thirds are subdivided into sixths to align the two fractions
  • Fraction circles/area model showing 1/3 recut into 2/6 to give a common denominator (IES Fig 6)
  • Equivalence chain 1/3 = 2/6 shown as cutting each third in two (numerator doubles with denominator)

Teaching practices

  • Establish the NEED for a common unit first (you can't add thirds and sixths as they are) using a representation that makes different-sized parts obvious, before any rule.
  • Exploit the related structure: spot that the larger denominator is already the common one, so only one fraction is renamed.
  • Insist that renaming scales BOTH numerator and denominator (1/3 → 2/6, not 1/6) — the commonest slip is changing only the denominator.
  • Use estimation to judge reasonableness (1/3 + 1/6 must be more than 1/3, so 2/9 can't be right).
  • Simplify the result and check it against a model; connect back to counting like units once the denominators match.
  • Choose representations that hold the unit constant (fraction strips, number lines); avoid ones (e.g. dot paper) shown to reinforce adding without a common denominator.

Activities to try

  1. 1

    Align the strips: cut thirds into sixths to make a common unit

    Why it works: When you add by counting parts, thirds and sixths must be made the same size first; cutting each third in two makes it two sixths (so the numerator doubles exactly as the denominator doubles), and only then do both fractions share one common unit and combine by counting like parts.

    1. Try to combine 1/3 and 1/6 using a thirds strip and a sixths strip laid together; notice the pieces are different sizes and do not line up as they are.
    2. Cut each third-piece in half so each single third becomes TWO sixth-pieces; watch the label change 1/3 → 2/6 as the one piece splits into two and the numerator ticks 1→2.
    3. Now that every piece is a sixth, combine: 2/6 + 1/6 = 3/6.
    4. Simplify 3/6 to 1/2 and check on the strip that 3/6 and 1/2 are the same length.
    5. Repeat the same cut-to-match process for 3/4 − 1/8 (recut quarters into eighths, giving 6/8 − 1/8 = 5/8) and 2/5 + 3/10 (recut fifths into tenths, giving 4/10 + 3/10 = 7/10), simplifying where possible.
  2. 2

    Equivalence-and-combine practice that surfaces common errors

    Why it works: Checking equivalence and combining in a scorable format can reveal whether the learner renames both numerator and denominator together and whether they look for a common unit at all; it does not by itself show they can carry out the re-partition unaided.

    1. Complete the chain: 1/3 = __/6, so 1/3 + 1/6 = __/6 = __ (simplest form).
    2. Choose the correct value of 1/3 + 1/6 from options that include the correct answer, 2/9 (the add-across error) and 2/6 (denominator renamed but numerator left at 1).
    3. Match each fraction to its equivalent written over the common denominator (e.g. 1/3 ↔ 2/6, 3/4 ↔ 6/8).
    4. Explain in words why only one of the two fractions needed changing.
  3. 3

    Read a worked conversion, then perform your own

    Why it works: A clear worked example of renaming can anchor the 'scale both parts together' idea, but the learner must then carry out the re-partition themselves to secure it; reading a conversion does not by itself elicit producing one.

    1. Study a single worked conversion where 1/3 is shown recut as 2/6 over the common denominator, with the numerator and denominator both doubling.
    2. Explain what happened to the top number and why.
    3. Then perform the conversion yourself on a fresh related pair (e.g. rename 2/5 as tenths, giving 4/10) and complete the calculation.

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
Ma5/2.4d

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