Lesson Plan for Year 13 Students: Understanding Graphs for AS91400

Subject: Mathematics (Calculus)

Duration: 60 minutes

Class Size: 48 students

Location: New Zealand

Curriculum Area and Level:

This lesson plan is designed for Year 13 students in New Zealand, following the NCEA Level 3 calculus standards, specifically targeting the achievement standard AS91400 (Apply differentiation methods in solving problems).

Learning Objectives:

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

1. Identify and sketch key types of graphs covered in AS91400.
2. Understand the properties of these graphs, including asymptotes, points of inflection, maxima, and minima.
3. Apply their knowledge to solve typical problems that involve these graphs.

Materials Needed:

• Graph paper
• Calculators
• Whiteboard and markers
• Laptops or tablets (optional for graphing software)
• Handouts with graph examples and problem sets

Lesson Overview:

1. Introduction and Review (10 minutes)

   • Brief review of relevant differentiation concepts.
   • Discuss the importance of understanding graph shapes in calculus.

2. Graph Theory Introduction (15 minutes)

   • Introduce common functions and their graphs as related to differentiation: polynomial, exponential, logarithmic, and trigonometric functions.
   • Highlight characteristics such as increasing and decreasing intervals, extrema, and asymptotic behaviour.

3. Graph Sketching Activity (15 minutes)

   • Students will be given specific functions from the AS91400 standard to sketch. These functions include polynomials of higher degrees, trigonometric functions, and their transformations.
   • Focus on critical points, end behaviour, and intervals of increase and decrease.

4. Problem Solving with Graphs (15 minutes)

   • Present problem scenarios where students need to use their sketches to deduce answers.
   • Problems will involve real-life applications, like finding maximum profit or minimum cost, reflecting typical AS91400 exam questions.

5. Conclusion and Recap (5 minutes)

   • Quick recap of the day’s lesson and lessons learned.
   • Discuss next steps and provide additional resources for further study.

Assessment:

• Formatively assess students through their participation in the graph sketching and problem-solving activities.
• Collect the sketches and solved problems to evaluate understanding and provide individual feedback.

Additional Resources:

• New Zealand Qualifications Authority: NCEA Calculus Resources
• Desmos Graphing Calculator - useful for visualising different function graphs.

Reflection:

After the class, reflect on which concepts students grasped quickly and which they struggled with. Use this insight to adjust future lessons or to provide targeted review sessions.

This lesson plan addresses key competencies in the New Zealand Curriculum and helps prepare students for assessments in calculus, ensuring they are well-equipped to tackle NCEA Level 3 problems efficiently.