Solve ax = b and x/a = b, including fractional and negative coefficients.
EQ.ONE_STEP_MULT_DIV“I can solve a one-step multiplicative equation, including with fractions.”
Solve (2/3)x = 10 and −4x = 22, checking each answer.
Fluency standard: 20 one-step equations in 5 minutes, >=95% accurate
| Course | Unit | Depth | Role | Checkpoint |
|---|---|---|---|---|
| Grade 6 | One-step equations and inequalities | intro | introduce | ✓ |
A pan balance where both sides must stay level, so any operation must be done to both.
Kills "= means the answer comes next".
engine: rigged balance actor with tilt physics
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
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: balance_scale
EQ.ONE.FRACTION_COEFFDivides by the numerator only when the coefficient is a fraction.
Repair: Multiply both sides by the reciprocal instead.
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.
Equation solving is grounded in the idea that operations preserve equality, and Grade 8 explicitly requires students to give examples of equations with one, no, or infinitely many solutions rather than only to solve.
Many problems that other systems solve algebraically are solved with bar models first, so when formal solving arrives students already know what the answer should look like. The bar and the equation are often shown side by side.
Equation solving is fluent and early, but the emphasis is on constructing the equation from a complicated condition. Multi-step motion, work, and mixture problems are standard rather than optional applications.
Equation solving is drilled as an isolation procedure across Levels G and H, moving quickly to simultaneous equations. Word problems are largely absent, so procedural speed far outruns modelling ability.
Solving is introduced through the balance idea and then applied to problems where the class compares an arithmetic solution with an algebraic one, so students see what the algebra buys them.
A stated procedure (simplify each side, collect variables, isolate) is demonstrated and drilled, with checking by substitution required, and the skill spiralled through cumulative review for the rest of the course.
Equation work is approached through balance materials and the algebraic manipulation of concrete quantities, keeping the "same to both sides" idea physical for as long as possible.
Solving is quickly subordinated to problems where setting up the equation is the difficulty, and where a clever substitution or a symmetry observation shortcuts the algebra entirely.
Solving is built in small steps from function machines through inverse operations to formal balancing, with a deliberate stage where the same problem is solved both ways so the equivalence is visible.
Equations are developed from balanced hanger diagrams, which give a physical justification for doing the same thing to both sides, and the hangers are retained until the symbolic moves are secure.