Unknowns and Equations

Overview

Lesson 4 of 5 in Solve, Share, Scale. Students use balance and mystery-number contexts to represent unknown quantities, write equations, solve one-step problems and verify solutions by substitution.

Learning intentions

• WALT represent an unknown quantity using a box, symbol or letter.
• WALT write equations that match diagrams and word problems.
• WALT solve missing-factor, missing-group and missing-total problems.
• WALT check that a solution makes the equation true.

Success criteria

• I can identify the unknown in a problem.
• I can represent the unknown with a box, symbol or letter.
• I can write and solve an equation that matches the relationship.
• I can substitute my answer back and explain whether it is correct.

Curriculum links

• Use mathematical relationships and equations to represent and solve problems.
• Apply multiplication, division, addition and subtraction strategies with whole numbers.
• Communicate mathematical thinking using diagrams, symbols, equations and explanations.
• Develop the competencies of thinking, using language and symbols, and participating and contributing.

Lesson structure (60 minutes)

1. 0–8 minutes – Hook: What is hidden? Open with the mystery-balance introduction. Show a balanced model such as three identical bags and 6 loose counters on one side, with 21 counters on the other. Ask: “What could the bag represent? How do we know?” Students think, pair and share. Emphasise that an unknown can be shown by a box, question mark or letter.

2. 8–18 minutes – Model translating relationships Use the equation modelling slides and the Algebra Balance Mat to model: 3 × □ + 6 = 21. Build the relationship, write the equation, solve it in steps and check by substitution: 3 × 5 + 6 = 21. Also model a missing-total equation, such as 18 + 7 = □. Highlight the difference between an equation and an answer.

3. 18–25 minutes – Guided examples Present three contexts from the guided examples: missing factor (4 × □ = 28), missing group (□ × 5 = 35), and missing total (□ + 16 = 40). Students show answers on mini-whiteboards or by drawing. Prompt them to name the unknown, choose a representation, write the equation and explain how they checked it.

4. 25–45 minutes – Partner practice Distribute the unknowns and equations practice sheet. Pairs solve a graduated set of diagram and word problems, including missing-factor, missing-group and missing-total questions. Require four records for each problem: identify the unknown, write the equation, solve it and verify it. Circulate and run a short teacher-led workshop for students who confuse the operation or write an equation that does not match the story.

5. 45–53 minutes – Compare and explain Return to the discussion and comparison slides. Select two solutions, including one with a different recording method. Ask: “Are both equations equivalent?” and “How can substitution prove the answer?” Students explain why, for example, 4 × □ = 28 and □ = 28 ÷ 4 describe the same relationship.

6. 53–60 minutes – Independent check and plenary Show the final problems on the exit and plenary slides. Students independently solve: “Five bags contain the same number of shells. There are 35 shells altogether. How many shells are in each bag?” They write 5 × □ = 35, solve and verify. Invite two students to share different representations and collect the worksheet for assessment.

Resources

• the complete introduction, modelling, practice and plenary slide deck
• the unknowns and equations practice sheet
• the Algebra Balance Mat
• Counters, linking cubes or coins for balance models
• Balance scales, if available
• Mini-whiteboards, pens and erasers
• Pencils and highlighters
• Chart paper or board space for shared equations

Assessment

• Observe whether students identify the unknown and select an operation that matches the context during guided examples and partner practice.
• Check worksheets for a matching equation, correct solution and verification by substitution.
• Use the final problem to identify students ready for extension or needing further support in Lesson 5.

Differentiation

• Support: begin with counters and the balance mat; use one-step equations with small whole numbers and provide frames such as “The unknown is ___”, “The equation is ___”, and “Check: ___ = ___”.
• Support: provide key vocabulary and visuals for unknown, equation, equal, factor, group, total; allow drawing, oral explanation or concrete modelling before symbolic recording.
• Targeted teaching: work with a small group on distinguishing missing factor, missing group and missing total, using the same four-step routine.
• Extension: create an equation with two possible solution pathways, solve a problem with more than one unknown, or explain why two equations are equivalent.