Scaling Number Structure

Overview

This is lesson 3 of 5 in the unit Solve, Share, Scale. Students explore how multiplication and division are connected when quantities are scaled up or down, using recipes, maps and classroom quantities. They explain their strategies, identify patterns and check whether answers are reasonable.

Learning intentions

• WALT use number structure and efficient strategies to solve scaling problems.
• WALT recognise and use scale factors when quantities increase or decrease.
• WALT explain how multiplication and division are connected.
• WALT communicate and compare mathematical strategies.

Success criteria

• I can identify the scale factor in a problem.
• I can use known facts, place value or a proportional diagram to solve.
• I can explain how multiplication and division are connected.
• I can check whether my answer is reasonable.

Curriculum links

• Number and Algebra: use multiplication and division strategies with whole numbers and decimals.
• Number and Algebra: recognise and describe patterns and relationships in number.
• Mathematical communication: explain, justify and compare strategies.
• Key competencies: thinking; using language, symbols and texts; relating to others; managing self.

Lesson structure (60 minutes)

1. 0–7 minutes – Hook and connect

Open with the scaling hook and learning intentions. Display a simple recipe for four people and ask: “How much of each ingredient would we need for eight people? What changes, and what stays the same?” Students make a quick estimate, then share with a partner. Connect to earlier learning about multiplication, division and equal groups.

2. 7–15 minutes – Model number structure

Use the slides to model a scaling table, such as:

People	Cups of flour
2	3
4	6
8	12

Think aloud: “The number of people doubled, so the flour doubled.” Show that multiplying by 2 and dividing by 2 are inverse operations. Also model a scale factor of 10 using place value, and ask students to identify the factor and describe the pattern.

3. 15–25 minutes – Explore in pairs

Distribute the scaling tables and strategy worksheet. In pairs, students choose or are assigned contexts such as recipes, map distances or classroom equipment. They complete scaling tables, showing at least one strategy: known facts, place value, repeated multiplication, division or a proportional diagram.

Circulate and ask: “What is the scale factor?” “How do you know?” “Could you solve this by working backwards?” Record useful strategies for the class.

4. 25–37 minutes – Digital pattern investigation

Open the digital investigation instructions and use a projected spreadsheet, virtual counters or another available digital manipulative. Change one quantity by factors such as 2, 5, 10 and 100. Students predict the related quantity before it is revealed, then describe the pattern.

Pairs investigate one scaling factor and record examples on the worksheet. Include scaling down, such as halving 18 items or dividing 240 cm by 10. Pause to discuss why dividing by a factor reverses scaling up.

5. 37–49 minutes – Strategy share

Display the strategy-sharing prompts. Select three pairs to explain different methods, for example a diagram, known multiplication facts and place-value reasoning. The class compares methods using the prompts: “Which method is efficient?” “Would it work for a different number?” “How can we check it?”

Emphasise that the same relationship can be represented in a table, diagram, equation or words. Invite students to challenge or build on ideas respectfully.

6. 49–56 minutes – Independent application

Students complete the final problems on the independent scaling and checking section. Include one scaling-up problem, one scaling-down problem and one reasonableness question. Students must state the scale factor, show their strategy and use an inverse operation or estimate to check.

7. 56–60 minutes – Plenary and exit response

Return to the plenary questions. Students answer orally or on the back of their worksheet: “What does the scale factor tell us?” and “How are multiplication and division connected?” Invite two students to share a clear explanation. Collect worksheets to identify next steps for lesson 4.

Resources

• the scaling lesson slide deck
• the scaling tables and strategy worksheet
• Board and markers
• Pencils and rulers
• Recipe, map-distance or classroom-quantity examples
• Projector or interactive whiteboard
• Optional spreadsheet, virtual counters or digital manipulatives
• Calculators for selected learners

Assessment

• Observe whether students identify the scale factor and select an efficient strategy during pair work.
• Check completed tables for accurate scaling up and down, clear representations and reasonable answers.
• Use the plenary responses to assess understanding of inverse operations and mathematical explanation.

Differentiation

• Support learners with partially completed tables, smaller scale factors, double/halve prompts and concrete counters or grouping materials.
• Provide a visual proportional diagram, calculator or digital manipulative for students who need support with working memory, place value or recording.
• Conference with targeted students using one question at a time: “What is changing?” “What is the factor?” “What operation undoes it?”
• Extend confident learners with multi-step scaling problems, comparisons of two methods and a general rule written in words or symbols, such as “to scale by 5, multiply every quantity by 5”.
• Support EAL learners with illustrated contexts, explicit vocabulary, sentence frames such as “The scale factor is ___ because ___”, and opportunities to rehearse explanations with a partner.