LESSON PLAN

Solving Multi-Step Linear Equations

Apothem Team

Grade 9 · Algebra & Patterning

LESSON AT A GLANCE

Warm-up

5 min

Explore

15 min

Formalize

10 min

Practice

12 min

Exit ticket

3 min

Warm-up

The equation autopsy wall: five solved equations posted, each with ONE buried error (a sign teleport, a half-distributed bracket, a fraction cleared into only two of three terms, an illegal division by $x$ , and one perfectly correct). Pairs get three minutes to find and NAME each crime — or certify innocence.

Grade 9 equation-solving is fluency plus FORENSICS: multi-step equations with brackets, fractions, and decimals, solved cleanly and audited sharply.

Explore

The gauntlet, difficulty stacked: (1) $5(2x - 3) - 2(x + 4) = 9$ — double distribution with a negative; (2) $\frac{2x - 1}{3} = \frac{x + 4}{2}$ — cross-multiply-or-clear (both legal; clearing by 6 shows WHY cross-multiplication works); (3) $0.4(x - 5) = 0.3x + 1.2$ — decimals cleared by ×10 up front; (4) the rearrangement finale: solve $P = 2\ell + 2w$ for $w$ — same moves, letters as constants, formulas as equations.

Rung 4's reveal: rearranging formulas IS equation-solving where the "numbers" are letters — one skill, not two.

Formalize

Formalize the full pipeline with the pre-clean options:

$\text{pre-clean (×LCD or ×10}^k\text{)} \to \text{distribute} \to \text{collect} \to \text{peel} \to \text{check}$

The cross-multiplication demystified: $\frac{a}{b} = \frac{c}{d} \to ad = bc$ is just "multiply both sides by $bd$ " with the cancelling pre-done. Students who know the engine can extend it ( $\frac{a}{b} = \frac{c}{d} + 2$ does NOT cross-multiply — the shortcut's licence requires lone fractions on each side).

Practice

Practice: the four-rung gauntlet fresh; two formula rearrangements ( $C = 2\pi r$ for $r$ ; $v = u + at$ for $t$ ); one "licence check" item (a cross-multiplication attempted where illegal); one full autopsy write-up.

Exit ticket: solve $\frac{3(x - 2)}{4} - \frac{x}{2} = 1$ , moves labelled. (×4: $3(x-2) - 2x = 4 \to 3x - 6 - 2x = 4 \to x = 10$ .)

Exit ticket

Exit ticket: solve $\frac{3(x - 2)}{4} - \frac{x}{2} = 1$ , moves labelled. (×4: $3(x-2) - 2x = 4 \to 3x - 6 - 2x = 4 \to x = 10$ .)

TIP Grade the CHECK line as heavily as the solve: substitution back into the ORIGINAL equation is the only step that certifies everything above it, and students skip it exactly as often as it isn't graded.

WORKED EXAMPLES

Example 1 — The full gauntlet in one:

\frac{x + 3}{2} - \frac{2x - 1}{5} = 3

Step 1: Pre-clean — LCD is 10, EVERY term rides: $5(x + 3) - 2(2x - 1) = 30$ .

Step 2: Distribute with sign care: $5x + 15 - 4x + 2 = 30$ (note $-2 \times -1 = +2$ — the classic casualty).

Step 3: Collect: $x + 17 = 30$ .

Step 4: Peel: $x = 13$ .

Step 5: Check in the ORIGINAL: $\frac{16}{2} - \frac{25}{5} = 8 - 5 = 3$ ✓.

Step 6: Count the danger points survived: the LCD covering all three terms, the negative distribution, the constant on the right getting its ×10. Three traps, three clean escapes — that's what "multi-step" actually tests.

Example 2 — Rearrangement as solving: the speed formula for time

The formula: $d = vt + \frac{1}{2}at^2$ … too rich for now — take $v = u + at$ and free the $t$ .

Step 1: Treat $v, u, a$ as "numbers whose values we don't happen to know": isolate the $t$ -term: $v - u = at$ .

Step 2: Peel the coefficient: $t = \frac{v - u}{a}$ .

Step 3: Dimension-check the result (the physicist's substitution-check): velocity difference over acceleration = $\frac{m/s}{m/s^2} = s$ ✓ — it's a time.

Step 4: Use it: $u = 5$ m/s, $v = 25$ , $a = 4$ : $t = \frac{20}{4} = 5$ s.

Step 5: The unification said aloud: every move was a Grade 8 equation move; the only novelty was nerve. Formulas are equations wearing lab coats — rearrange first, substitute last, and every physics class ahead gets easier.

Example 3 — Forensics finale: the three-error solve

The evidence: $2(3x - 4) + 6 = 4x + 10$ ; line 2: $6x - 4 + 6 = 4x + 10$ ; line 3: $6x + 2 = 4x + 10$ ; line 4: $2x = 12$ ; line 5: $x = 6$ .

Step 1: Audit line 2: $2 \times -4 = -8$ , not $-4$ — half-distribution, error one. (Honest line: $6x - 8 + 6$ .)

Step 2: Given their line 2, line 3 collects correctly ( $-4 + 6 = 2$ ) — no NEW error, but it inherits the first.

Step 3: Line 4 from line 3: $6x - 4x = 10 - 2 \to 2x = 8$ , but they wrote 12 — arithmetic slip, error two. Their line 5 divides correctly.

Step 4: Reconstruct the honest solve: $6x - 2 = 4x + 10 \to 2x = 12 \to x = 6$ — the "answer" was RIGHT after all: two errors cancelled!

Step 5: The uncomfortable, crucial lesson: a correct answer does not certify a correct method — and only the line-by-line audit (or a substitution check paired with process review) catches compensating errors. Marks for method exist precisely because luck exists.

MATERIALS

•Autopsy wall posters

•Gauntlet card sets

•Formula rearrangement cards

•Practice set (PDF)

WATCH FOR

!Fraction-clearing multiplied into SOME terms — every term rides, including lonely constants.

!Cross-multiplication over-licensed (used with extra terms present). The engine (×both denominators) reveals when the shortcut's conditions fail.

!Dividing both sides by a variable expression, silently discarding the

x = 0

world. At this grade: flag it as "dangerous move, ask why" — full treatment comes with quadratics.