Interpret n x a/b as n copies of a/b, connecting to repeated addition and to scaling.
RAT.MULTIPLY_FRACTION_WHOLE“I can multiply a fraction by a whole number and show it as repeated copies.”
Compute 6 x 2/3 and draw the repeated copies that justify the answer.
| Course | Unit | Depth | Role | Checkpoint |
|---|---|---|---|---|
| Grade 4 | Fraction arithmetic | intro | introduce | ✓ |
Equal-length bars split into halves, thirds, fourths… stacked for direct comparison.
Equivalence is seen before it is computed.
engine: stacked segmented bars with snap comparison
A measured line for locating, comparing, ordering, and jumping between numbers.
One representation carries whole numbers, fractions, integers, and irrationals.
engine: axis scene + tick generator + hop actor
Rectangular bars segmented to show parts inside a whole, with the unknown drawn as a labelled gap.
Word problems become a picture you can read the equation off.
engine: segmented bar actor with labelled brace
Scene archetype: fraction_bar
RAT.MULT.BOTH_PARTSMultiplies both numerator and denominator by the whole number.
Repair: Count the copies on a number line and compare.
One row per instructional system. Rows marked cluster carry authored treatment for this family of topics; rows marked template are derived from the system’s general pattern and are not topic-specific research. Confidence is recorded on every row.
Fraction addition is framed as expressing both in a common unit rather than as a denominator rule, multiplication is developed as area and as scaling, and division by a fraction is introduced through measurement questions before any reciprocal rule.
Each operation gets its own pictorial stage before symbols: strips for addition, a partitioned rectangle for multiplication, and a "how many fit" strip for division. The bar model carries the word problems throughout so the operation choice stays visible.
Fraction arithmetic is drilled to fluency and then immediately deployed in multi-step problems about parts of parts, which is where the real work lies. Mixed-number regrouping is treated as a technique with its own set of exercises.
Levels E and F are fraction operations at length: common denominators, mixed numbers, and complex fraction computation, all as bare arithmetic on the page. Fluency is high and the reasoning is absent by design.
Division by a fraction is a canonical lesson-study problem: students are given a context, produce several justifications, and the class compares them before the invert-and-multiply rule is named. The tape diagram carries the argument.
The rules are stated (find a common denominator; multiply across; invert and multiply), demonstrated on worked examples, and practised in long sets that then recur in mixed review. Justification of invert-and-multiply is usually brief or omitted.
Operations are performed on fraction insets, with the child physically combining or splitting pieces and discovering the need for a common denominator when the pieces do not match. Symbolic recording follows the physical operation.
Operations are practised inside problems where the fraction arithmetic is incidental to a harder structural question, so fluency is required but never the point. Complex fractions and nested expressions appear early.
Small steps build from same-denominator addition through equivalence to unlike denominators, with bar models kept alongside the symbols. Multiplying and dividing fractions arrive in Year 6 with area and scaling interpretations.
Division of fractions is developed over a sequence of lessons from "how many groups" and "how much in each group" contexts, with diagrams preceding the algorithm and the algorithm justified against them in the synthesis.