Reasoning with Operations

Overview

Lesson 7 of 8 in the unit Multiplication and Division Strategies. Students apply multiplication and division strategies to practical Australian contexts, represent situations, explain their reasoning and check that answers match the question.

Learning intentions

Students will:

• identify whether a practical problem requires multiplication or division
• represent problems using drawings, arrays, equal groups, equations or number lines
• solve multiplication and division problems, including unknown-group and unknown-size division
• communicate and justify a strategy using mathematical language
• give and respond to peer feedback

Success criteria

• I can identify the operation by understanding the situation, not just by looking for key words.
• I can represent the problem and write an equation.
• I can solve, label my answer and check that it answers the question.
• I can explain my strategy and respond to feedback.

Curriculum links

• Solve practical multiplication and division problems involving equal groups, arrays and sharing.
• Use a range of mental, written and visual strategies to calculate and check answers.
• Communicate mathematical reasoning using diagrams, equations, labels and mathematical vocabulary.
• Develop problem-solving, reasoning and mathematical communication proficiency.

Lesson structure (60 minutes)

1. 0–5 minutes – Hook and activate prior learning Open with the problem-solving hook and learning intention slides. Display: “A shop has 4 packs of 23 stickers. How many stickers are there altogether?” Students make a quick estimate, then discuss what information matters and what the question is asking.

2. 5–15 minutes – Model unpacking a multiplication problem Use the worked multiplication example to model the routine: read, identify known information, identify the question, choose an operation, represent, solve, label and check. For “There are 4 teams with 23 students. How many students altogether?”, demonstrate an array or partitioning strategy, such as 4 × 20 + 4 × 3 = 92. Emphasise that the situation—not a key word—determines the operation.

3. 15–23 minutes – Model division contexts Display the two division contexts and representation prompts. Model “36 pencils are packed into groups of 4. How many groups?” as 36 ÷ 4 = 9, using equal groups. Contrast this with an unknown-size problem: “36 pencils are shared equally among 9 students. How many pencils does each student get?” Discuss how both use division but ask different questions. Check by multiplying.

4. 23–42 minutes – Partner problem solving Distribute the practical multiplication and division reasoning worksheet. Students work in pairs, then complete selected responses independently. Contexts include an Australian school canteen, football teams, a community garden and classroom materials. Problems include multiplication, unknown-group division and unknown-size division. Students must draw or write a representation, choose an operation, solve, label the answer and explain their strategy. Encourage materials such as counters, grid paper or number lines where helpful.

5. 42–52 minutes – Two strategies and peer feedback Use the strategy comparison and feedback prompts. Students choose one multiplication or division problem and solve it a second way, such as an array and partitioning, or equal groups and a related multiplication fact. Partners compare responses using: “My representation shows…” and “I agree/disagree because…”. Partners check whether the operation fits the situation and whether the answer matches the question.

6. 52–60 minutes – Share, assess and reflect Invite two or three students to share different representations. Revisit the misconception: a word such as “altogether” or “groups” does not automatically determine the operation. Collect one multiplication and one division response from each student, selected from the worksheet, using the reasoning rubric. Students complete the final reflection on the worksheet: “The most useful check was…”

Resources

• the multiplication and division reasoning lesson deck
• the practical multiplication and division reasoning worksheet
• Counters, linking cubes or small classroom objects
• Mini-whiteboards and markers
• Grid paper or blank paper
• Pencils, highlighters and rulers
• Reasoning rubric for collected responses

Assessment

• Observe partner discussion and representations, noting whether students identify operations from the situation rather than key words.
• Collect one multiplication and one division response. Assess whether each student represents the situation, selects a suitable operation, calculates accurately, labels the answer and justifies the strategy.
• During feedback, listen for accurate use of equal groups, sharing, multiplication, division and checking through a related fact.

Differentiation

• Read each problem aloud, highlight known information and circle what the question asks. Provide counters, diagrams, arrays and a partially completed worked example.
• Offer sentence starters: “I chose ___ because…”, “My representation shows…”, “The answer must be ___ because the question asks…”, and “I checked using…”.
• Support EAL learners and students requiring additional support with visual vocabulary, partner rehearsal and reduced-number versions of the same contexts.
• Extend confident students with a multi-step problem or an open-ended task: “Find different equal arrangements for 24 objects. Write a story problem for two of your arrangements and explain how the answers are related.”