These DP Mathematics common core subtopic 5.1 materials contain a few explorations designed to help students make connections between limits and rates of change and mathematical concepts they will have studied earlier in the course. All of these resources and activities revolve around the importance and usefulness of graphing—seeing really is believing!

Overview

Teacher Notes

Lesson 1: Limits Exploration

Lesson 2: Rate of Change and Gradient Exploration

Lesson 3: Rate of Change Applications

In this DP Mathematics common core 5.2 set, students explore the graphical meaning of increasing and decreasing functions. They participate in group challenges (e.g., treasure hunt and quiz) and investigative tasks (e.g., coding and modelling the tides and stock market). Teaching unit 5.3 before unit 5.2 might fit better, but the mixture of resources provided work so that they could be taught in either order.

Overview

Lesson 1: Treasure Hunt for Increasing and Decreasing Functions / Solution

Lesson 2: Quiz on Increasing and Decreasing Functions / Solution

Lesson 3: Coding Investigation / Solution

Lesson 4: Investigations of Increasing and Decreasing Functions / Solution

In this DP Mathematics common core 5.03 subtopic, students explore a variety of tasks to build and then consolidate their understanding of differentiating polynomials. They begin to make links between this skill and its uses in understanding gradients and increasing and decreasing functions. This set also includes investigative elements to allow students to build up their exploration toolkit to help with their coursework.

Overview

Lesson 1: Murder in the Maths Department / Solution

Lesson 2: Speed Calculus / Solution

Lesson 3: Differentiation Quiz / Solution

Lesson 4: Differentiation Investigation / Solution

In this DP Mathematics common core subtopic 5.4 set, students determine the equations of the tangent and the normal to a curve at a point and apply these concepts to real-life situations. Learning to expand the application of tangent lines will help them approximate values and find the roots of functions using Newton’s method. Coupling analytical and geometric/visual-spatial understanding allows students to solidify their understanding of the concepts and become stronger students of calculus.

Overview

Lesson 1: Applications of Tangent Lines and Normals / Solution

Lesson 2: Approximations Using Tangent Lines / Solution

Lesson 3: Price Elasticity / Solution

Through this DP Mathematics AI common core 5.5 set, students showcase their knowledge of indefinite and definite integration of polynomial functions. Students solidify their understanding of integration, both analytically and geometrically. HL students have an opportunity to solve a problem in kinematics as an application of integration/area under the curve.  

Overview

Lesson 1: Integrals—Indefinite and Definite / Solution

Lesson 2: Area Under a Curve / Solution

Throughout this DP Mathematics AI 5.7 subtopic, students learn about optimisation and its application in real-life contexts. We engage with a mathematics inquiry cycle, particularly through the Desmos activity, to ensure that conceptual understanding is taking place. Inflexion and describing stationary points will be confusing initially, but through graphical tools and differentiated tasks students will be able to distinguish between them.  

Overview

Unit Plan

Lesson 1: Optimization Problems in Real-Life Contexts / Solution

Lesson 2: Desmos Reflection Form

Lesson 3: Final Reflection Form

This DP Mathematics AI subtopic 5.8 activity set is designed in four sections, starting from the base of the content to complicated, sophisticated and open-ended applications of the trapezoid rule. Irregular areas that are not described by mathematical functions, such as lakes, are calculated using dynamic graphing software. Calculating the approximate area under a curve is included in Activity 4. 

Subject guide content covered includes: 

• Approximating areas using the trapezoidal rule.

Overview

Lesson 1: Formula, Integration or Trapezoid Rule / Solution

Lesson 2: Reaching Pi / Solution

Lesson 3: Approximating Area Under the Curve Without Knowing f(x) / Solution

Lesson 4: Trapezoid Rule for Lake Salda Campers / Solution

In this set of DP Mathematics AI HL subtopic 5.9 materials, students will explore additional derivative rules using Desmos activities and differentiated Practice of the Day (POD) tasks, following the Inquiry, Drill, Apply, Reflect cycle. Reflection is supported with Desmos and end-of-session documents.

Content covered: 

• Derivatives of sin x, cos x, tan x, e^x, ln x^n, 

• Chain rule, product rule, quotient rule

• Related rates of change

Overview

Lesson 1: The Chain Rule of Differentiation / Solution

Lesson 2: Product and Quotient Rules / Solution

Lesson 3: The Derivative of Trigonometric Functions / Solution

Lesson 4: The Derivative of Trigonometric Functions / Solution

Lesson 5: Rate of Change / Solution

Lesson 6: Desmos Reflection Form

Lesson 7: Final Reflection Form

This set of DP Mathematics AI subtopic 5.10 materials focuses on 1st and 2nd derivatives and uses the second derivative test to determine the nature of stationary points. Syllabus requires an awareness of concavity and inflection points, and that is covered in the final activity.

Overview

Subtopic Notes

Lesson 1: Finding First and Second Derivatives / Solution

Lesson 2: Second Derivative Testing / Solution

Lesson 3: Concavity Activity / Solution

In this DP Mathematics AI HL subtopic 5.10, students learn about the second derivative. The lessons follow a mathematics inquiry cycle to ensure conceptual understanding. Through the Desmos activity, students explore connections between f, f′ and f′′. 

Subject guide content includes: 

• The second derivative

• Use of the second derivative test to distinguish between a maximum and a minimum point

Overview

Lesson 1: Second Derivatives / Solution

After the basic power rule for integration was covered in an earlier subtopic, this DP Mathematics AI HL 5.11 subtopic deals with the entire remainder of the integration techniques that students need – going all the way from the rest of the basic formulae to an option of a reverse chain rule, to full u-substitution. Students have plenty of opportunities here to practice these techniques and discover how the chain rule plays a role in integration through an investigation. 

Overview

Subtopic Notes

Lesson 1: Using Basic Integration Formulae / Solution

Lesson 2: Integration By Substitution Activity / Solution

In this DP Mathematics AI HL subtopic 5.11, students learn about the second derivative. The mathematics inquiry cycle ensures conceptual understanding. Inquiry into definite and indefinite integration takes place through Desmos classroom activities so students can identify the connection. Students are given differentiated tasks, and reflection is encouraged throughout the learning journey.

Overview

Teacher Notes / PPT

Lesson 1: Definite and Indefinite Integration

This DP Mathematics AI HL subtopic 5.12 set includes two activities:

Paper 3 Skills: Develops problem-solving skills through complex, scaffolded questions using words, symbols, graphs, and tables. 

Five Different Glasses: Enhances 3D thinking by calculating volumes of glasses shaped by rotating curves around the x and y axes. 

Subject guide content includes: 

• Area of the region enclosed by a curve and the x or y-axes in a given interval

• Volumes of revolution about the x−axis or y− axis

Overview

Lesson 1: Hand Carved Wooden Trinket Box / Solution

Lesson 2: Five Different Glasses / Solution

This DP Mathematics AI HL 5.13 subtopic set This set includes two activities:

1. 2D Kinematics of a Dust Particle examines random movement in x and y dimensions with separate equations, exploring position changes, and using calculus to graph velocity, speed, acceleration, and total distance, comparing with given graphs.

2. Kinematics of Usain Bolt's 100 m World Record analyzes data, modeling velocity with v(t) and v(s) functions, and calculating acceleration using both formulas.

Overview

Lesson 1: Movement of Dust Particles and Brownian Motion / Solution

Lesson 2: Kinematics Behind Usain Bolt's 100 m World Record / Solution

This DP Mathematics AI HL subtopic 5.14 set has three examples of how to model exponential growth using different approaches, each more rigorous than the previous, so that the teacher can scaffold the task. This set can be used by teachers when starting this topic or as exemplars when they are preparing for the "exploration" or when doing a "tool kit" task. 

Subject guide content covered includes: 

• Setting up a model/differential equation from a context. 

• Solving by separation of variables.  

Overview

Unit Plan

Lesson 1: Exponentials in Context

Slope fields allow us to analyse differential equations graphically. This task will scaffold how to draw a slope field and use it to find specific solutions. Students will be shown how to draw a slope field for a differential equation in x and y. The first step is to draw a segment with dy/dx as slope at any point (x, y). That segment is the slope field of the equation at any point. Students will be able to match an equation to its slope field by considering the various slopes in the diagram.

Overview

Unit Plan

Lesson 1: Slope Field

Lesson 2: Slope Field Scaffolding

This DP Mathematics AI HL subtopics 5.16 and 5.18 set uses technology to better understand differential equations. First, students review Euler’s method using exam-style questions. This analytical approach gives us a way to find the unknown function given an initial condition (x0, y0). Then, students use formulas on a spreadsheet to calculate a numerical solution, applying knowledge to Excel and TI-Nspire GDC.

Overview

Unit Plan

Lesson 1: Introductory Activity for Euler’s Method / Questions / Data

Lesson 2: Create a GeoGebra File to Represent a Numerical Solution

This DP Mathematics AI HL subtopics 5.16 and 5.18 set walks students through the process of using technology to better understand coupled systems of differential equations. The activities ask students to use a variety of technologies to enhance their understanding of the topic—Excel spreadsheets, TI-Nspire and Geogebra. They use the data supplied to model data sets using Euler’s method, SIR modelling and Lotka-Volterra equations.

Overview

Unit Plan

Lesson 1: Introductory Activity for Coupled Systems

Lesson 2: Exponential Decay Curve with a Slope Field

Lesson 3: The Euler Method for 2nd Order Differential Equations

Lesson 4: 2nd Order Differential Equation Spreadsheet Calculations

Lesson 5: 2nd Order Differential Equation Spreadsheet Calculations 

Lesson 6: Data for Predator-Prey Modelling Task

Lesson 7: SIR Modelling Task

Lesson 8: Data for SIR Modelling Task

This DP Mathematics AI set for HL subtopics 5.17 and 5.18 covers phase portraits and coupled differential equations. To start, students review slope fields, Jacobian matrices, eigenvalues and eigenvectors. Then they use Geogebra tools to review and sketch phase portraits, using the spread of infectious diseases as a context. Students can use a phase portraits approach to explore problems for HL 5.16 and 5.18, linking to exponential and logistic growth models and the Lotka-Volterra model.

Overview

Teacher Notes

Lesson 1: Introductory Activity for Phase Plane and Portraits

Lesson 2: Phase Portraits