Math // Fractions

Adding Fractions (Unlike Denominators)

Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions to produce a common denominator

Ages
10 to 11
Type
Method
Builds on
3 topics
Leads to
4 topics
Standards
2 codes

Quick check

If a recipe uses 2/3 of a cup of one ingredient and 3/4 of another, can the learner add those fractions together — even though the denominators are different?

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.
  • Compute 2/3 + 5/4 by converting to twelfths: 8/12 + 15/12 = 23/12
  • Subtract 1 3/5 from 3 1/4 using a common denominator
  • Explain the strategy of finding a common denominator and verify using a visual model

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

3 topics

You are here

Adding Fractions (Unlike Denominators)

Ages 10 to 11 // Method

Leads to

4 topics
  • Understanding fractionsRequired

    Data & Statistics // Ages 10 to 11

    Solving problems with fraction data requires unlike-denominator addition

  • Fraction Word ProblemsRequired

    Fractions // Ages 10 to 11

    Word problems require the computational skill of adding/subtracting unlike-denominator fractions

  • One Quantity as a FractionRequired

    Ratio & Proportion // Ages 11 to 12

    Adding/subtracting fractions with unlike denominators is needed when comparing fractional parts

  • Mixed & Improper FractionsRequired

    Fractions // Ages 11 to 13

    Four operations with all number types builds on adding/subtracting fractions with unlike denominators

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

General case of adding and subtracting fractions with unlike denominators where neither denominator is a multiple of the other (e.g. 2/3 + 5/4), extended to mixed numbers (e.g. 3 1/4 − 1 3/5). Both fractions must be renamed to a shared unit — a common multiple of the denominators (twelfths for thirds and quarters) — which scales BOTH numerators, then like parts are combined. Mixed-number subtraction adds a further demand: the fraction part of the minuend can end up smaller than that of the subtrahend, forcing a regroup across the whole (borrow a whole) or a conversion to improper fractions. The unifying idea remains 'make the units the same, then count', now with no shortcuts.

Common misconceptions

  • Adds across the unlike denominators: 2/3 + 5/4 = 7/7, combining numerators and denominators.
  • Finds a common denominator but scales only one (or neither) numerator, or uses a wrong common denominator (2/3 = 2/12).
  • In mixed-number subtraction, subtracts the smaller fraction from the larger in each column regardless of position (does 12/20 − 5/20), never regrouping across the whole when the minuend's fraction part is smaller.

Useful representations

  • Two strips/number lines re-gridded to a common denominator (thirds and quarters → twelfths)
  • Area/fraction-circle model showing both fractions recut to the shared unit
  • Mixed-number bars showing a whole tile broken into the common unit to enable borrowing (1 whole → 20/20)
  • Improper-fraction conversion as an alternative route (3 1/4 = 13/4) laid beside the mixed form

Teaching practices

  • Reinforce that BOTH fractions must be renamed to a common unit; a common multiple of the denominators (often the product, sometimes the LCM) gives it.
  • Insist both numerators scale with their denominators (2/3 → 8/12, 5/4 → 15/12) — the multi-conversion is where partial-scaling slips appear.
  • For mixed numbers, teach two secure routes and let pupils choose: regroup/borrow a whole, OR convert to improper fractions first (often cleaner when a borrow is needed).
  • Use estimation and benchmarks to gate the answer (2/3 + 5/4 is more than 2, so 7/7 = 1 is impossible).
  • Verify with a visual model or number line, per the topic's own evidence; simplify and write mixed-number answers.
  • Keep units constant in representations; avoid manipulatives that encourage adding without a common denominator.

Activities to try

  1. 1

    Shared-grid combiner: rename both fractions, then combine

    Why it works: Unlike fractions can only be combined once BOTH are rewritten over the same unit; each numerator scales by the same factor as its own denominator, and estimation gates whether a result is even possible.

    1. Set up 2/3 + 5/4 on two strips or two number lines, one gridded in thirds and one in quarters, and notice they will not combine while the units differ.
    2. Choose a shared unit (a common multiple of 3 and 4) and re-grid both strips into twelfths, re-tiling each fraction: 2/3 becomes 8/12 and 5/4 becomes 15/12.
    3. State explicitly that cutting each third into four pieces turns 2/3 into 8/12 — the top is multiplied by the same factor as the bottom — and check the same for 5/4 (each quarter cut into three, so 5/4 = 15/12) before combining.
    4. Combine the re-tiled pieces to get 23/12 and regroup into the mixed number 1 11/12.
    5. Before computing, estimate: 5/4 is already more than 1, so adding a positive 2/3 makes the sum more than 1 (indeed more than 1.5). That already rules out answers like 7/7 = 1; the true sum lands just below 2 (23/12, just under 24/12).
  2. 2

    Mixed-number subtraction with a regroup across the whole

    Why it works: When the minuend's fraction part becomes smaller than the subtrahend's after finding a common denominator, you must regroup a whole into the common unit (or work with improper fractions) rather than subtracting the smaller fraction from the larger in each column.

    1. Set up 3 1/4 − 1 3/5 and rename both fractions to a common denominator of twentieths: 3 5/20 − 1 12/20.
    2. Notice that 5/20 is smaller than 12/20, so the fraction part cannot be taken directly — a regroup across the whole is needed.
    3. Regroup one whole from the 3 into twentieths: break a whole tile into 20/20 so 3 5/20 becomes 2 25/20, then subtract to get 1 13/20.
    4. As an alternative route, convert both mixed numbers to improper fractions first (3 1/4 = 13/4 = 65/20 and 1 3/5 = 8/5 = 32/20), subtract to 33/20, and convert back to 1 13/20.
    5. Compare the two routes and decide which you find cleaner when a borrow is needed.
  3. 3

    Explain and check with a model

    Why it works: A learner secure in this strand can rename both fractions to a genuine common denominator, handle a mixed-number regroup by either route, simplify, and verify a result against a benchmark estimate — not just execute steps.

    1. Work out 2/3 + 5/4 and 3 1/4 − 1 3/5, showing every step and giving each answer in simplest form as a mixed number where appropriate.
    2. Explain how you chose the common denominator and why BOTH numerators changed when you did.
    3. Answer the challenge: a friend says 2/3 + 5/4 = 7/7 = 1 — but 5/4 on its own is more than 1, so can the total be exactly 1? Explain what went wrong.
    4. For the subtraction, after reaching 3 5/20 − 1 12/20, explain what you do because you can't take 12/20 from 5/20 as it stands.
    5. Verify 23/12 against the benchmark: confirm it is more than 1 (since 5/4 alone exceeds 1) and a little less than 2 — does that match adding 2/3 and 5/4?

Topic // Standards

Curriculum standards

Codes for the curriculum statements this topic maps to. Each publisher owns its standard text.
Common Core State Standards for Mathematics
5.NF.1
The national curriculum in England: Key stages 1 and 2 framework document
Ma6/2.3c

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