Objective
Students will be able to solve multi-step linear equations with variables on both sides, including equations with distribution and combining like terms. Students will correctly solve at least 8 out of 10 equations on an exit ticket and verify solutions by substitution.
Standards
- CCSS.MATH.CONTENT.8.EE.C.7 — Solve linear equations in one variable.
- CCSS.MATH.CONTENT.8.EE.C.7.B — Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.
Materials
- Algebra tiles (1 set per student or virtual algebra tiles on Chromebooks)
- Whiteboard and marker per student
- Guided notes handout
- Practice worksheet (15 problems, tiered difficulty)
- Exit ticket (10 problems)
- Error analysis cards (6 incorrectly solved equations for students to find and fix)
Warm-Up (5 minutes)
Solve two review problems on whiteboards: 3x + 7 = 22 and 2(x - 4) = 10. Review answers (x = 5 and x = 9). Ask a volunteer to explain each step. Then write: 3x + 5 = x + 13. Ask: "What is different about this equation?" (There is a variable on both sides.) "Can we solve it the same way?" Today's lesson: solving equations where the variable appears on both sides, and equations that require multiple steps.
Direct Instruction (15 minutes)
Problem 1: Variables on both sides. Solve 3x + 5 = x + 13.
- Goal: get all x terms on one side, all constants on the other.
- Subtract x from both sides: 2x + 5 = 13.
- Subtract 5 from both sides: 2x = 8.
- Divide by 2: x = 4.
- Check: 3(4) + 5 = 17, and 4 + 13 = 17. Both sides equal, so the solution is correct.
Problem 2: Distribution required. Solve 2(3x - 1) = 4x + 6.
- Distribute: 6x - 2 = 4x + 6.
- Subtract 4x: 2x - 2 = 6.
- Add 2: 2x = 8.
- Divide by 2: x = 4.
- Check: 2(3(4) - 1) = 2(11) = 22, and 4(4) + 6 = 22. Correct.
Problem 3: Combining like terms first. Solve 5x + 3 - 2x = 2x + 12.
- Combine like terms on the left: 3x + 3 = 2x + 12.
- Subtract 2x: x + 3 = 12.
- Subtract 3: x = 9.
- Check: 5(9) + 3 - 2(9) = 45 + 3 - 18 = 30, and 2(9) + 12 = 30. Correct.
Special cases (brief introduction):
- No solution: 2x + 4 = 2x + 7 simplifies to 4 = 7, which is never true. No value of x works.
- Infinite solutions: 3(x + 2) = 3x + 6 simplifies to 3x + 6 = 3x + 6, which is always true. Every value of x works.
Throughout instruction, emphasize: whatever you do to one side, you must do to the other. And always check your answer by substituting back into the original equation.
Guided Practice (10 minutes)
Students solve 5 problems on their whiteboards, holding up answers after each:
- 5x - 3 = 2x + 9 (x = 4)
- 4(x + 2) = 3x + 11 (x = 3)
- 7x - 1 = 3x + 15 (x = 4)
- 2(x - 5) + 3 = x + 4 (x = 11)
- 6x + 2 = 6x + 2 (infinite solutions)
After each problem, select a student to present their solution process. Address common errors: sign errors when subtracting negative terms, forgetting to distribute to all terms inside parentheses, and not checking the answer.
Independent Practice (10 minutes)
Students complete the tiered practice worksheet:
- Level 1 (problems 1–5): Variables on both sides, no distribution (e.g., 4x + 1 = 2x + 7)
- Level 2 (problems 6–10): Distribution required (e.g., 3(2x - 4) = 2(x + 3))
- Level 3 (problems 11–15): Multi-step with combining like terms and distribution, plus one special case
Students work at their own pace and must show all steps. When they finish a level, they self-check with the answer key posted in the room before moving to the next level. Error analysis cards are available as an extension — students find the mistake in an incorrectly solved equation and correct it.
Assessment
- Formative: Monitor whiteboard responses for correct procedures and common errors. Check for consistent answer-checking behavior.
- Summative: Exit ticket with 10 equations (3 Level 1, 4 Level 2, 2 Level 3, 1 special case). Target: 8 out of 10 correct.
Differentiation
- Struggling learners: Use algebra tiles or a virtual manipulative to model moving terms across the equals sign. Provide a step-by-step solving checklist (1. Distribute, 2. Combine like terms, 3. Move variable terms to one side, 4. Move constants to the other side, 5. Divide, 6. Check). Limit to Level 1 and 2 problems.
- ELL students: Provide a math vocabulary reference sheet with examples (variable, coefficient, constant, distribute, like terms, solution, substitute). Use color-coding: variable terms in blue, constants in red. Allow students to explain their process in their home language to a partner.
- Advanced learners: Include equations with fractions and decimals (e.g., 0.5x + 1.2 = 0.3x + 2.6 or x/3 + 2 = x/4 + 5). Assign error analysis cards as required work. Challenge them to write equations with specific solutions (create an equation where x = -3).
- Students with IEPs: Provide guided notes with blanks to fill in during instruction. Allow a calculator for arithmetic. Reduce exit ticket to 6 problems. Offer extended time and a reference card with the solving steps.
Closure (5 minutes)
Display an equation on the board: 4(2x - 1) + 3 = 5x + 8. Give students 2 minutes to solve it independently as a "challenge check" (x = 3). Review the solution together. Then ask: "When you see an equation with variables on both sides, what is your first move?" (Get all variables on one side.) "What should you always do after solving?" (Check by substituting.) Give a thumbs-up/down confidence check. Preview tomorrow's lesson on solving and graphing inequalities.