Design and Defend Problems

Overview

This is lesson 5 of 5 in Solve, Share, Scale. Students design, represent, solve and justify a real-life mathematical problem involving sharing, grouping, scaling or an unknown quantity, then test and improve their work through peer feedback.

Learning intentions

• WALT create a clear real-life problem using sharing, grouping, scaling or an unknown quantity.
• WALT represent and solve a problem using materials, a diagram, table, digital tool or equation.
• WALT explain and justify a strategy, including how we know the answer is reasonable.
• WALT give and use constructive feedback to improve mathematical communication.

Success criteria

• I can write a clear problem with appropriate numbers and context.
• I can represent my problem and write and solve an equation.
• I can explain and justify my strategy using mathematical language.
• I can check that my solution fits the context and revise my work when needed.

Curriculum links

• Number and Algebra: use the four operations and multiplicative thinking to solve problems involving equal groups, sharing, scaling and unknown quantities.
• Number and Algebra: represent relationships with diagrams, tables, words and simple equations.
• Mathematical processes: communicate mathematical ideas, explain strategies and evaluate whether answers are reasonable.
• Key competencies: thinking, managing self, participating and contributing, and using language, symbols and texts.

Lesson structure (60 minutes)

1. 0–8 minutes – Analyse strong examples

Open with the hook and example slides. Present two or three short problems, such as: “Six teams share 48 drink bottles equally” or “A recipe for four people is scaled to serve ten.” Include one strong example and one unclear example.

Ask: What makes a problem clear? Are the numbers realistic? What information is known or unknown? How could we represent and check it? Briefly model identifying the operation and writing an equation.

2. 8–15 minutes – Plan the challenge

Explain that students may work individually or with a partner. Distribute the problem design and solving worksheet. Students choose a context such as sport, kai, shopping, gardening, travel or a class event, then choose sharing, grouping, scaling or an unknown quantity.

Model a simple planning sequence: context, question, known information, unknown quantity, equation, representation and check. Remind students that the problem must be solvable and must have enough information.

3. 15–30 minutes – Design and solve

Students draft their own problem on the worksheet, select appropriate numbers, and solve it. They must include a representation, such as counters, a diagram, table, number line or digital model, as well as an equation and written explanation.

Circulate for teacher check-ins. Ask: What does the unknown represent? Why is this operation appropriate? How will you check the answer? Encourage students to test their numbers before moving on.

4. 30–40 minutes – Peer-test and revise

Pair students with another individual or pair. Each student gives their problem to a peer without explaining it first. The peer attempts to solve it and marks whether the instructions, numbers and question were clear.

Use the peer-testing and feedback prompts to structure feedback: “I understood…”, “I was unsure about…”, and “You could make this clearer by…”. Designers revise their wording, numbers, representation or solution as required.

5. 40–53 minutes – Solve classmates’ problems

Students exchange revised problems with a different classmate. They solve the problem and record their strategy on the worksheet. The designer compares the classmate’s method with their own, checking whether the answer and reasoning fit the context.

Invite selected students to display their work or present from the front. Ask the class to identify the equation, representation, efficient strategy and evidence that the answer is reasonable.

6. 53–60 minutes – Present and reflect

Use the presentation and reflection prompts. Invite two or three students to briefly defend their problem and solution: explain the context, equation, representation, strategy and check.

Students complete the reflection on the worksheet: Which strategy was most efficient? What did peer testing improve? How did you know the answer was reasonable? Collect worksheets for assessment.

Resources

• the complete introduction, modelling, activity instruction, feedback and reflection slide deck
• the problem design and solving worksheet
• Pencils, rulers and coloured pencils
• Counters or linking cubes for modelling equal groups and sharing
• Mini-whiteboards or scrap paper
• Optional tablets or computers for digital representations
• Display board or document camera for sharing student work

Assessment

• Observe planning and discussion: can students identify the unknown, choose an appropriate operation and explain their reasoning?
• Check worksheets for a clear context, suitable numbers, representation, equation, solution, justification and contextual check.
• Listen to peer feedback and presentations, noting whether students communicate clearly and evaluate the efficiency and reasonableness of strategies.

Differentiation

• Provide choice of context, a planning template, number banks and partially completed examples. Offer sentence starters such as “I used ___ because…”, “The unknown is…”, and “I checked my answer by…”.
• Allow students to use counters, diagrams, tables, digital tools, speech-to-text, peer scribing or an oral presentation instead of completing every explanation in writing.
• Pair students strategically and provide frequent teacher check-ins. Read problems aloud and clarify vocabulary for EAL learners; use visual examples and chunked instructions for students needing additional support.
• Extension students design a multi-step problem, include a remainder, or create a problem with more than one valid strategy. They evaluate classmates’ solutions and compare which strategy is most efficient and why.

Extension

Students create a second solution using a different representation or operation pathway, then write a short teacher-style feedback comment explaining the strengths and possible improvements in a classmate’s reasoning.