Understanding Scales

Lesson 2 of 29 in Unit: Measuring Chance and Scale

Duration: 60 minutes
Class Size: 14 Year 12 students

---

Learning Objectives

By the end of the lesson, students will be able to:

• Explain the concept of scale in mathematical and real-world contexts.
• Distinguish between different types of scales, specifically linear and logarithmic scales.
• Interpret graphs and data representations using linear and logarithmic scales.
• Explain the appropriate contexts and reasons for using each scale type.
• Apply knowledge of scales to solve practical problems involving measurement and data interpretation.

Alignment with Australian Curriculum (v9) — Mathematics

• AC9M10M02: Interpret and use logarithmic scales in applied contexts involving small and large quantities and change; identify and interpret data representations using logarithmic scales; discuss appropriateness of scale choice
• AC9M10M05: Use mathematical modelling to solve practical problems involving proportion and scaling of objects; investigate and apply scales and ratios in practical contexts

---

Lesson Outline

Time	Activity	Details
5 min	Introduction & Hook	Brief class discussion: Ask students for examples where scale matters (maps, music volume, earthquake magnitude). Use visual aids showing contrasting scales (e.g., map of Australia at different scales, Sydney Harbour Bridge diagram). Connect to everyday experiences.
10 min	Input: Concept of Scale & Linear Scale	Explain "scale" — ratio of measurement on a diagram/model to actual measurement. Introduce linear scale via examples: rulers, maps, architectural drawings. Demonstrate how linear scales work through a simple construction problem or map interpretation. Visual representations and mini whiteboard sketches encouraged.
10 min	Input: Logarithmic Scale	Introduce logarithmic scales: concept of orders of magnitude, exponential compression of data. Use examples such as Richter scale (earthquake magnitudes), decibel scale (sound intensity), and pH scale (acidity). Interactive demonstration with a graphing tool to compare linear vs logarithmic plotting using the same data set. Relate to AC9M10M02 elaborations.
10 min	Guided Practice (Pair Work)	Distribute small tasks where students analyse graphs or data sets — some linear, some logarithmic. Pairs identify the scale type, explain why each scale is appropriate or not, and interpret specific points. Use data from real-world contexts (e.g., growth of bacteria, population change, sound levels).
10 min	Technology Integration	Use graphing software or a graphing calculator app to plot data sets both linearly and logarithmically. Students manipulate parameters and observe effects on graph shape and data interpretation, reinforcing concepts of scale and usefulness of logarithmic view. Emphasize digital tool use aligns with curriculum capabilities.
8 min	Class Discussion & Reflection	Discuss key takeaways. Ask students when they would choose each scale type. Highlight importance in fields like geology, acoustics, finance, and AI algorithms. Relate to real-life applications, including First Nations Australia contexts if possible (e.g., discussing dating techniques involving logarithmic scales).
7 min	Formative Assessment & Exit Ticket	Students individually write one paragraph explaining a real-world scenario where a logarithmic scale is preferred over a linear scale, justifying why. Collect for teacher feedback to inform future lessons.

---

Resources Required

• Whiteboard and markers
• Graphing calculators or laptops/tablets with graphing software
• Sample graphs/data sets (print or digital)
• Visual aids illustrating scales (maps, photos, charts)
• Exit ticket papers

---

Differentiation

• Support: Provide step-by-step instructions and examples for pair work tasks. Use simplified graphs or partially completed tables.
• Extension: Challenge advanced students to create their own logarithmic graphs of datasets, or explore the mathematics of exponent laws related to scales (linking to AC9M9A01 content).
• Use peer teaching strategies to support varied learning speeds.

---

Cross-Curricular Connections

• Science: Use of logarithmic scales in measuring earthquakes or sound.
• Technology: Use of graphing tools, introducing concepts in AI and machine learning where scaling data matters (refer to AC9M10M05 elaborations on AI).
• Indigenous perspectives: Discuss the importance of scales in dating archaeological sites related to First Peoples in Australia, reflecting on the cultural significance of such applications (linked to AC9M10M02_E5).

---

Assessment

• Formative: Exit ticket with written explanation of scale application, evaluated for understanding of scale types and justification.
• Observation during activities for participation and conceptual grasp.
• Use responses to identify misconceptions or areas requiring reteach.

---

Teacher Notes

• Emphasise the difference between the purpose of linear and logarithmic scales rather than just the mechanics.
• Encourage students to verbalise their understanding and reasoning during discussions.
• Use authentic and culturally relevant examples, such as scales used in indigenous contexts or Australian data.
• Encourage students to connect scales to probability and chance concepts from the unit context where appropriate.

---

This lesson builds foundational understanding of scales relevant to Year 12 mathematics, using hands-on, technology-enhanced strategies aligned with the Australian Curriculum (v9), covering critical real-world applications and cognitive skills required for higher order thinking in mathematical contexts.