Equation Case Detectives

Overview

Students solve multi-step linear equations and classify equations as having one solution, no solution, or infinitely many solutions. Through modeling, discussion, physical representation, and game-based practice, students explain why each algebraic process preserves equality.

Learning intentions

• Students will solve multi-step linear equations accurately.
• Students will identify whether an equation has one solution, no solution, or infinitely many solutions.
• Students will justify each step using the properties of equality.
• Students will communicate mathematical reasoning using precise vocabulary.

Success criteria

• I can distribute, combine like terms, and isolate the variable in the correct order.
• I can check a solution by substituting it into the original equation.
• I can recognize a false statement as no solution and a true statement as infinitely many solutions.
• I can explain my reasoning to a partner or group.

Curriculum links

• Algebraic reasoning: Solve linear equations and interpret the meaning of solutions.
• Mathematical practice: Construct arguments, critique reasoning, and attend to precision.
• MYP key concepts: Form, relationships, communication, and perspective.
• IB approaches to learning: Thinking, communication, collaboration, and self-management skills.
• IB inquiry: Students investigate how equivalent operations preserve equality and use evidence to classify solution sets.

Lesson structure (90 minutes)

1. 0–10 minutes — Bell ringer and retrieval

Open with the hook and bell-ringer slides. Students independently solve:

• (x+7=15)
• (3x=21)
• (2x+5=17)
• (4(x+2)=24)

Students compare answers with a partner, then the teacher reviews solutions quickly. Ask: “What operation keeps an equation balanced?”

2. 10–25 minutes — Mini-lesson: model the process

Use the worked-example slides and model (3(x+4)-5=16). Think aloud: distribute, combine like terms, move constants, isolate the variable, and check by substitution.

Introduce the three cases:

• One solution: the variable equals a number.
• No solution: the variable disappears and a false statement remains.
• Infinitely many solutions: the variable disappears and a true statement remains.

Model one example of each and emphasize that students must show the point at which the classification becomes clear.

3. 25–40 minutes — Guided practice

Distribute the multi-step equations practice sheet. Solve the first examples together, prompting students to name the operation used at each step. Students then complete selected problems independently before comparing with a partner.

Pause after each special-case example and ask students to defend their classification using a complete sentence, such as, “There is no solution because the result is (4=9), which is false.”

4. 40–55 minutes — Snack equation challenge

Organize students into five groups of four. Provide clearly labeled crackers, cereal pieces, cups, napkins, and a recording area. Open the snack challenge instruction slide.

Students use crackers to represent variable terms, cereal pieces to represent constants, and cups to represent groups or parentheses. Each group physically models a teacher-provided equation, describes the inverse operations, and records the algebraic solution on the challenge recording section.

State the rule: students complete and check the mathematics before eating. Follow school allergy, hygiene, and food-sharing policies; offer a non-food alternative if needed.

5. 55–75 minutes — Algebra Jeopardy

Use the Algebra Jeopardy game slides. Teams choose from:

• Distribute It!
• Combine Like Terms
• Solve It!
• One, None, or Infinite?
• Vocabulary

Use 100–500 point questions, increasing in complexity. Require teams to show work and explain one reasoning step before receiving points. Invite other teams to agree, challenge, or correct the explanation respectfully.

6. 75–85 minutes — Strategy debrief and error analysis

Return to the discussion and error-analysis slides. Display two common errors, such as distributing incorrectly or changing only one side of an equation. Students identify the error, correct it, and explain why the correction preserves equality.

Ask students to summarize: “How can you tell whether an equation has one, no, or infinitely many solutions?” Record a concise class explanation.

7. 85–90 minutes — Exit ticket

Students complete the exit ticket on the final exit-ticket section:

• Solve (4(x-2)+3=19).
• Classify (3x+6=3x+6) as one, none, or infinitely many solutions.

Collect responses as students leave. Do not provide the answer before collection.

Resources

• the complete lesson slide deck
• the equations practice and exit-ticket worksheet
• the Linear Equations Step Cards for a visual process reminder during guided practice
• Board or projector
• Markers or colored pens
• Crackers, cereal pieces, small cups, napkins, and trays
• Non-food modeling materials for students who cannot eat snacks
• Five team score areas for Jeopardy
• Pencils and calculators only if permitted by classroom policy

Assessment

• Monitor bell-ringer and guided-practice work for procedural accuracy and misconceptions.
• During the snack challenge and Jeopardy, assess whether students can model, justify, and classify equations rather than only state answers.
• Use the exit ticket to determine next steps: reteach multi-step operations, special cases, or checking solutions.

Differentiation

• Provide the Linear Equations Step Cards and color-coded operation marks for students who need sequencing support.
• Read directions aloud, chunk problems, and allow students to explain reasoning verbally before writing; provide vocabulary frames such as “The equation has ___ because ___.”
• Pair students strategically and assign group roles: modeler, recorder, checker, and spokesperson.
• Extend ready students by asking them to create an equation with no solution and another with infinitely many solutions, then prove each classification.