Multiply and divide signed fractions and decimals, and justify the sign rules rather than memorising them.
RAT.RATIONAL_OPERATIONS“I can compute with any positive or negative rational number and justify the sign.”
Compute (−3/4) ÷ (−2/5) and explain why the quotient is positive.
Fluency standard: 20 mixed rational computations in 8 minutes, >=95% accurate
| Course | Unit | Depth | Role | Checkpoint |
|---|---|---|---|---|
| Grade 7 | Rational number operations | intro | introduce | ✓ |
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
A rectangle partitioned by place value or by terms, with each sub-rectangle a partial product.
The same picture explains 23x47, 3/4 x 2/5, and (x+3)(x+5).
engine: partitioned rectangle with labelled regions
A dense page of graded exercises with a completion-time target and no scaffolding.
Automaticity, which frees working memory for reasoning.
engine: worksheet builder + print layout
Scene archetype: number_line
RAT.SIGN.ADD_RULES_TO_MULTApplies addition sign reasoning to multiplication.
Repair: Separate the two rule sets and practise them interleaved.
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.
Signed numbers arrive in Grade 6 as positions and opposites on a number line, with operations deferred to Grade 7 where subtraction is defined as adding the opposite and the rules are expected to be justified rather than recited.
Negative numbers are introduced through contexts with a natural zero (temperature, elevation) and stay tied to the number line, with operations developed after the ordering is secure.
Negative numbers come earlier than in the US sequence and are immediately used in equations and coordinate work rather than practised in isolation. The sign rules are derived from the requirement that the distributive law keep working.
Level G opens with positive and negative numbers as computation rules to be automated, immediately followed by algebraic expressions. The transition from arithmetic to algebra is a worksheet increment rather than a conceptual event.
Negative numbers open lower secondary (中1) as the first topic of the year, framed as extending the number system so that subtraction is always possible. The extension itself is the lesson, not just the new symbols.
The sign rules are stated as rules, drilled heavily, and reinforced by mixed review; the number-line justification is presented briefly and then set aside in favour of fluency.
Signed numbers are handled with coloured counters where a positive and a negative cancel to nothing, making the zero pair physical before it is symbolic.
The extension to negatives is motivated by asking what must be true for the familiar laws to survive, so the sign rules are derived as consequences rather than announced.
Directed numbers appear in Year 6 and are developed across KS3, with number lines and contextual models, and a lot of attention paid to distinguishing the operation sign from the sign of the number.
Grade 6 units use elevation and temperature contexts to establish order and opposites; Grade 7 develops the operations through number-line movement and chip models, with rules named after the models have done the work.