# SL Scheme - Calculus

The new Mathematics: Applications and Interpretation syllabus is out and published on the IB website. Log in to your MyIB account and head to this link . It is a new course with a new name and renewed focus on understanding the relationship between mathematics and its application and interpretation. It is still intended very much that the SL course will cater for the same group of students that might currently opt for Mathematical Studies. It is hoped that some of the current SL students will opt for the applications HL. Clearly all sorts of permutations are possible. Read more about the development of this course through the links in the New Syllabus section of the website. This page will focus on the Calculus unit and evolve over time as the new applications website develops.

### Calculus

So on this page we will just have a quick comparison between the two syllabi

Current Syllabus

- Concept of derivative as rate of change
- Deriving polynomials
- Finding gradients of curves
- Increasing and decreasing functions
- Solving gradient = 0
- Optimisation problems

Syllabus 2019

- Concept of a limit & derivative as gradient
- Increasing and decreasing functions
- Tangents and Normals
- Integration as anti differentiation
- Definite Integrals using technology
- Solving gradient = 0
- Optimisation problems
- Trapezium/trapezoid rule

### Points of Observation

There is an obvious and significant addition to this unit in the form of Integration. It has been in and out of the Maths Studies course over the iterations and now its back! For maths teachers this is a mixed blessing I suspect. Its great to teach, but will be an extra challenge, particularly for our weaker students. Its worth noting though that the implication is that students will need to do some basic integration with given boundary conditions and then there will be an emphasis on technology. The numerical method of the trapezium/trapezoid rule is another addition, but one that is quite approachable. It will be an extra demand on time though.

## An SL Scheme of work

Based on these guidelines, here is an idea about how we might spend the time. Remember that the Toolkit hours can come anywhere we want them. Clearly there are a number of ways in which this can be done, but this is just a suggestion to get us started.

I have included 2 hours of the toolkit time

A note about assessment - Each week shows a suggested assignment and there are a number of ASSESSMENT points through the unit. Many teachers like to use regular quizzes and clearly, all of this can be weaved in and out as the teacher chooses.

**Week 1 - Introduction and Concept**

Clearly a significantly difficult concept to move in to and I think it is important to spend some time focusing on rate of change and gradient.

- A first lesson might involve an activity like Distance Time Graphs and Rates of Change to properly explore the key ideas behind calculus.
- A second lesson might focus more specifically on Measuring Gradients as a key step in recognising the idea of rate of change and how that changes.
- In a third lesson, I favour the exposition of the differentiation of y=x
^{2}from first principles. It is manageable and is an imprtant bridge in understanding how the the results of measuring gradients can be justified by other means.

Other resources from Focus on Calculus Concept will be useful here.

* Assignment* - I recommend a review exercise based on something from the functions unit given the need for functional fluency in this calculus unit.

#### Week 2 - Deriving Polynomials

Students have understood how gradient measures rate of change and then way in which gradient changes can be summarised first by spotting patterns and then by algebraic methods from first principles. Then it makes more sense when formal demonstration of the the formula for deriving polynomials achieves the same thing.

- Formal demonstration of the formula for deriving polynomials followed by some practice
- Working with negative indices and more practice of deriving functions
- Working on problems in and out of context that can be solved by deriving polynomials (Beginning to use the GDC)

Other resources from Focus on Calculus Concept will be useful here.

* Assignment* - A summary exercise based on the work of the last 2 weeks

**Week 3 - Tangents, Normals and Stationary Points**

Building on the previous week's work, this week the focus is on tangents and normals. resources from Focus on Tangents and Normals will be useful here.

- A formal exposition of how the equation of a tangent can be found both with algebraic methods and the GDC followed by practice.
- Similar for equations of normals
- Problems involving tangents and normals

* Assignment* - A summary exercise on Tangents and Normals

**Week 4 - Optimisation**

This week might start with a good exercise like the 'max box' activity to see first hand how calculus can be used to optimise.

Then a number of other examples can be used to show the different ways the process is used. (See slides on Focus on stationary points and optimisation)

Further practice of optimisation problems. The Optimal Jumble puzzle could be used here.

* Assignment* - A range of Optimisation problems

#### Week 5 - Integration

- A key conceptual activity to introduce this idea. Students will likely have had experience of the area under a speed time graph being distance, and this might be a nice link with the activity used to introduce the start of the unit.
- Exemplification of problem solving with Integration
- Practice problem solving

* Assignment* - A practice exercise based on the idea of integration.

#### Week 6 - Trapezium Rule

These week focuses on this numerical method for estimating the area under a graph.

- Demonstration
- Exemplification
- Practice

* Assignment* - A practice exercise based on the idea of the trapezium rule.

#### Week 7 - Review and Assess

As with the other units, this is an opportunity to review and work with the ideas in this unit and prepare for an end of unit assessment at the end of the week.

**ASSESSMENT POINT** - A 60 minute assessment on the Calculus unit.