Lesson 6 — Writing and Solving Two-Step Equations

Summary

Learners write a two-step equation for a real situation (2 mystery boxes plus 3 equals 11 → 2x + 3 = 11) and solve it by undoing the operations in reverse order — first the add, then the multiply — keeping the equation balanced. They check by substitution.

Objectives

• Write and solve a two-step equation for a real situation and check the solution. (D05.S2.07.01)

Connection

A vendor sells two identical bundles of fruit plus 3 extra pieces — 11 pieces in all. How many in each bundle? The unknown becomes x: 2x + 3 = 11. To find x you undo in reverse: subtract the 3 first, then halve. 2x = 8, so x = 4. A taxi fare (a flat start fee plus a per-kilometre charge), a phone plan (a base cost plus a per-minute rate), a wage (a base plus a rate per hour) — all are two-step equations. The word “algebra” comes from al-jabr in the 9th-century book of al-Khwarizmi, and the method of balancing both sides is a human achievement shared across cultures, not one person’s or one region’s invention.

Materials

• Balance-scale cards
• Equation recording page
• Small balance or pan scale (where available)

Preparation

• Copy balance-scale cards and equation recording pages.
• Have worked examples ready: 2x + 3 = 11 → 2x = 8 → x = 4; 3x − 4 = 14 → 3x = 18 → x = 6.
• Recall from Grade 6: one-step equations and inverse operations (from U.06.005, Lesson 6).

Facilitator note

This lesson is written to the learner (“you”). The ideas to land: (1) a two-step equation has two operations on the unknown; (2) to solve, undo them in reverse order — the opposite of “add/subtract first, then multiply/divide” is “undo multiply/divide last”; concretely, undo the addition first, then the multiplication; and (3) always check by substitution. Because solving equations is a foundational skill, use explicit instruction, worked examples, and guided practice before independent work (Kirschner, Sweller & Clark, 2006). Watch for the slip of undoing in the wrong order (dividing before subtracting) and of doing a step to only one side. Retrieval: ask learners to recall one-step equations from Grade 6 — a two-step equation is two one-steps chained. The global lens: “algebra” derives from al-jabr in al-Khwarizmi’s 9th-century work; balancing both sides is a shared human method. The critical-thinking lens (docs/philosophy.md §7): checking by substitution is honest reasoning, never trusting a guess. The egalitarian lens: equations turn a fair price or wage into solvable form — a tool of clear dealing for everyone.

Procedure

1. Gather (5 min). In Grade 6 you solved one-step equations by undoing one operation. Today there are two operations — and you undo them in reverse.
2. Meet the two-step equation (10 min). 2x + 3 = 11 means “2 times an unknown, plus 3, equals 11.” The unknown went through multiply by 2 then add 3. To get back to x, undo in reverse: undo the add first, then undo the multiply. Whatever you do, do it to both sides.
3. Worked example (10 min). Solve 2x + 3 = 11. Step 1 — undo “+ 3”: subtract 3 from both sides: 2x + 3 − 3 = 11 − 3, so 2x = 8. Step 2 — undo “× 2”: divide both sides by 2: 2x ÷ 2 = 8 ÷ 2, so x = 4. Check: 2 × 4 + 3 = 8 + 3 = 11 ✓.
4. Worked example — with subtraction (10 min). Solve 3x − 4 = 14. Step 1 — undo “− 4”: add 4 to both sides: 3x = 18. Step 2 — undo “× 3”: divide both sides by 3: x = 6. Check: 3 × 6 − 4 = 18 − 4 = 14 ✓.
5. Practice with a partner (10 min). Take your balance-scale cards: write each balance as a two-step equation, solve by undoing in reverse order, and check by substituting. Trade and check each other’s two steps.
6. Close (5 min). A two-step equation is two operations on an unknown. Undo them in reverse order — the add/subtract first, then the multiply/divide — on both sides, and always check.

Differentiation

• Support: Use only equations of the form 2x + 3 = 11 with a real balance to act them out; describe each step aloud so learners with low vision can follow by listening.
• Extension: Solve equations with a negative or fractional coefficient (e.g., −2x + 5 = 1, x/3 + 2 = 7) and write a two-step equation for a situation of the learner’s own.

Assessment

• Formative (observation/self): Can the learner write a two-step equation from a situation and solve it by undoing in reverse order on both sides, then check by substitution?
• Self-check: The learner asks, “Did I undo in the right order (add/subtract first, then multiply/divide)? Did I do each step to both sides, and did my answer check when I substituted it back?”

Home connection

Find a real two-part situation at home (a fare with a flat fee, a bill with a base charge, a wage with a base plus a rate) and write and solve the two-step equation for it.

Resources

• Solving two-step equations is standard in middle-grades mathematics; see John A. Van de Walle, Elementary and Middle School Mathematics (10th ed., 2019). The note that “algebra” derives from al-jabr in al-Khwarizmi’s 9th-century work is documented history (MacTutor, “Al-Khwarizmi,” S-293); the habit of checking answers is a critical-thinking value (docs/philosophy.md §7). The claim that equations make fair dealing solvable is a value commitment.