Model motion with d = rt, including problems with two moving objects, opposite directions, and changing rates.
MODEL.RATES_MOTION“I can solve distance-rate-time problems, including two objects moving at once.”
Two trains leave stations 300 km apart toward each other at different speeds; find when they meet.
Depth is a column, not a second topic. The same topic is taught at these depths in different years; the path table below places each rung.
Single-object problems with one of d, r, t unknown.
Two objects, same or opposite directions, meeting or catching up.
Changing rates, round trips with different speeds, and average speed over unequal legs.
| Course | Unit | Depth | Role | Checkpoint |
|---|---|---|---|---|
| Grade 6 | Modelling and decisions | intro | introduce | |
| Algebra I (Grade 8-9) | Data and modelling | intermediate | develop | |
| Algebra II (Grade 10-11) | Modelling and decisions | advanced | master | ✓ |
Animated objects moving along a track with a synchronised distance–time graph.
Slope of a distance–time graph felt as speed.
engine: existing scene runtime + graph binding
A spare labelled line-segment schematic of the quantities and relations in a problem, drawn before any arithmetic.
Forces analysis of structure before computation.
engine: labelled segment graph with relation braces
A ruled table of quantity, rate, and total (or was/change/now) filled in from the problem statement.
Makes motion, work, and mixture problems mechanical rather than mysterious.
engine: editable table widget with derived cells
The Cartesian grid for plotting points, lines, curves, and regions.
Algebra and geometry become the same subject.
engine: axis scene + plotted path renderer
Scene archetype: motion_track
MODEL.MOTION.AVG_SPEEDAverages the two speeds on a round trip.
Repair: Compute total distance over total time and compare.
MODEL.MOTION.NO_TABLEAttempts a two-object problem without organising the quantities.
Repair: Fill a distance/rate/time table row per object before writing equations.
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.
Modelling is one of the practice standards and a high-school conceptual category, so motion, mixture, and rate problems are expected to be set up from context rather than pattern-matched to a template.
Complex multi-step word problems are the signature assessment task, and the bar model is the standard tool for making their structure visible before any equation is written.
Motion, work, and mixture problems are a central genre with their own taught techniques: the condition is tabulated, a segment schematic is drawn, and only then is an equation formed. Students meet these years earlier than in the US sequence and in far greater volume.
Word problems are largely absent from the sequence. This is the clearest structural gap in the method: a student may be several levels ahead in computation while unable to set up a two-object motion problem.
Context problems anchor whole lessons, with the tape diagram used to expose structure and student solutions compared for generality. The problem is chosen so that several routes exist.
Word problems appear as classified types with a demonstrated set-up for each (distance-rate-time, coin, mixture), practised in sets and then recurring in mixed review.
Applied problems come from the practical-life and "going out" strands — measuring, budgeting, planning a trip — rather than from a set of classified word-problem types.
Contexts are chosen so that the standard template fails, forcing genuine modelling, and solutions are compared for insight rather than for adherence to a set-up.
Problem solving is one of the three stated aims, with multi-step contextual questions standard at KS2 SATs and GCSE, commonly scaffolded by bar models.
Contexts are the entry point for new mathematics rather than an application at the end of a unit, and the modelling cycle is made explicit in high-school units.