Math // Fractions

Fraction Addition Concepts

Understand addition and subtraction of fractions as joining and separating parts; decompose a fraction into a sum of fractions with the same denominator in more than one way

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

Quick check

If the learner has 7/8 of a chocolate bar and gives away 3/8, can they work out what fraction is left — and then show two different ways to split 7/8 into two parts?

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.
  • Show that 3/8 = 1/8 + 1/8 + 1/8 and also 3/8 = 1/8 + 2/8
  • Write two different decompositions of 5/6
  • Use a visual model to justify a decomposition of 2 1/8 into whole-number and fraction parts

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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Fraction Addition Concepts

Ages 9 to 10 // Method

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

Children aged 9–10 learn to see fraction addition and subtraction as JOINING and SEPARATING the same unit parts, with the denominator (the unit) held fixed throughout. The central skill is decomposing a fraction into a sum of same-denominator parts in MORE THAN ONE way (5/6 = 1/6+4/6 = 2/6+3/6 = 1/6+1/6+3/6), and expressing a mixed number as a single count of like parts (2 1/8 = 8/8+8/8+1/8 = 17/8). The aim is to shift from 'apply the numerator rule' to 'a fraction is a flexible collection of unit parts I can split and recombine', which later makes mixed-number arithmetic, common denominators, and regrouping in subtraction coherent rather than separate tricks.

Common misconceptions

  • Alters denominators when decomposing (a reverse add-across): writes 5/6 = 2/3 + 3/3 because 2+3=5 and 3+3=6, so the parts don't actually total 5/6.
  • Believes a fraction has only one 'correct' break-up (3/8 must be 1/8+1/8+1/8) and cannot produce a different valid decomposition.
  • Treats a mixed number's whole and fraction parts as disconnected numbers, unable to express 2 1/8 as a single count of eighths (17/8).

Useful representations

  • Fraction bar that can be sliced into groups in multiple ways, total length fixed
  • Number line showing a fraction reached by different combinations of unit-fraction jumps
  • Part-part-whole / bar model linking a fraction to its decompositions
  • Mixed-number bar showing whole sets of the unit plus the extra part (2 1/8 = 8/8+8/8+1/8)

Teaching practices

  • Frame + and − explicitly as joining and separating the SAME unit parts, keeping the denominator (the unit) fixed throughout.
  • Demand MORE THAN ONE decomposition of the same fraction so flexibility, not a single memorised split, is the goal (5/6 = 1/6+4/6 = 2/6+3/6 = ...).
  • Connect decomposition to the whole: 2 1/8 is two whole sets of eighths plus one eighth (16/8 + 1/8 = 17/8), so mixed numbers are seen as one quantity of like parts.
  • Justify each decomposition on a visual model (bar/number line) where the total length is invariant — different sums, same measure.
  • Use decomposition to explain, not just compute: e.g. why 5/6 − 2/6 separates two sixths from five.

Activities to try

  1. 1

    Slice one bar many ways

    Why it works: A fraction is not one fixed break-up: any grouping of its unit parts that uses the same denominator and the same total count measures the same quantity. Addition joins those parts and subtraction separates them, while the unit stays fixed.

    1. Start with a bar whose total length represents 5/6 of a whole, marked off into five equal sixths (each one-sixth of the whole) against a fixed ruler so its length cannot change.
    2. Group the sixths one way (e.g. 1 and 4) and write the matching sum of sixths underneath: 5/6 = 1/6 + 4/6.
    3. Regroup the SAME bar differently (2 and 3; then 1, 1 and 3) and write each new sum: 2/6 + 3/6, 1/6 + 1/6 + 3/6.
    4. Line up all the sums you have written against the single unchanged bar and say aloud: 'all of these are 5/6'.
    5. Repeat the whole process with a fresh fraction you have not seen (e.g. 7/8) to produce at least two distinct decompositions.
  2. 2

    Mixed number as one count of like parts

    Why it works: A mixed number's whole part and fraction part are the same kind of unit part; the wholes can be recounted as eighths so the whole quantity is a single count of like parts (17/8).

    1. Lay out 2 1/8 as two whole bars, each cut into eight eighths, plus one extra eighth.
    2. Count the eighths in each whole bar (8/8 and 8/8), then join the extra eighth.
    3. Write the quantity as a single count of eighths: 8/8 + 8/8 + 1/8 = 17/8, confirming the 2 wholes contribute 16/8, not 2/8.
    4. State in words why the whole and the fraction are one quantity: 2 1/8 is seventeen eighths altogether.
  3. 3

    Check and sort candidate decompositions

    Why it works: A valid same-denominator decomposition must keep the denominator fixed AND have parts that actually total the original fraction; the reverse add-across (2/3+3/3 because 2+3=5 and 3+3=6) produces parts that do not sum to 5/6.

    1. Given several candidate sums for 5/6, decide which equal 5/6 and which do not — e.g. 1/6+4/6, 2/6+3/6, and the trap 2/3+3/3.
    2. For each candidate, add the parts and compare the total to 5/6 to justify the decision, not just guess.
    3. Explain why 2/3 + 3/3 fails: adding them gives 5/3 (more than a whole), not 5/6, and the parts are not measured in sixths.
    4. Fill in a second valid decomposition of a given fraction, and match each mixed number to its improper form (e.g. 2 1/8 ↔ 17/8).

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
4.NF.3.a4.NF.3.b

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