# SL Scheme - Functions

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 Functions unit and evolve over time as the new applications website develops.

### Mathematical Modelling

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

Current Syllabus

• Concept of a function
• Linear models
• Exponential models
• Other models
• Accurate graph drawing

Syllabus 2019

• Linear functions and their manipulation
• Concept of a function and function as a model.
• Graphing and sketching functions with and without technology.
• Technology for finding key features.
• Different models - Linear, quadratics, exponential, cubic, Sinusoidal and direct/inverse variation.
• Modelling skills

### Points of Observation

• There is a clear move here to make the focus of the unit to be the purpose and practise of modelling, with the different types of models coming out of that rather than the other way around.
• Sinusoidal models are back! Also, the specific mention of cubic models and direct/inverse variation.

## 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 suggest that we add 5 hours of the toolkit time to this unit as this is likely to be a really productive area for Internal Assessment. That total of 36 hours is very long for a single unit, so I also propose having it in 2 sections. I have set this plan out in terms of 'weeks' where we might consider that a week consists of 3, 1 hour lessons.

In addition, the syllabus item 2.6 listed as modelling skills is something I would expect to be a running theme through out the units. In each sub section, there is Cleary the potential for exmplification of modelling in this context.

#### Week 1 - Introduction

I think this is a terrific opportunity for students to experience some mathematical activity that brings many of the aspects of modelling to the fore. Ideally the activity would be something that we could come back to through the module.

Example activity -  This Modelling World Population Growth activity, students are given a hands on experience of trying to fill some gaps in a model. In doing so they are inevitably having the kind of conversations that lead us to modelling concepts. When students are invited to use a graph plotter to try and fill in some gaps and talk about shapes and likely types of function, much of the intended scene is set for what follows. Some of these models can be returned to at a later date which provides a nice link.

Assignment - Students could be asked to write a summary of their experiences with this task as a good introduction to writing about mathematics that could be useful ground work for the Internal assessment

#### Week 2 - Concept, notation, gradients and straight lines

This week would be more formal than the last with a focus on the following specific skills...

• Defining  what is meant by a function, its inverse and the associated notation
• Defining and understanding how to identify and work with Domain and Range
• An exploration of straight lines in different forms looking at gradients, intercepts and perpendicular lines

The activity  Meeting Functions provides an excellent context for this work and a great opportunity to establish the power of the GDC. There are a number of practice tasks on the  Functions Practice page that could be used here as well as the coordinate geometry tasks on the  G & T Practice page where there is a string crossover.

Assignment - This is likely to be a busy week and so I would suggest the completion of practice tasks as an independent assignment.

#### Week 3 - Key Features and Graphing

The previous week's work might spill over in to this wek and the  Meeting Functions activity will also be useful here as well. The part that asks students to 'present the functions' gives us a good opportunity to practice the identification of key points and the idea of sketching.

Assignment - A practice activity that encompasses all the features of the last 2 weeks work and a few extra challenges.

#### Week 4 - Linear Models

This week will draw the first 3 together in many ways. The leap from abstract linear functions to linear models is an important one and this will require some good contextual examples to bring this out. The focus will be on helping students to understand what the gradient and y - intercept of a given linear function mean in context. Activities might include

• Review and practice of linear functions in the abstract
• Examples of Linear models in context
• Problems involving intersecting linear models.
• References to the meaning of domain and range in context here.

See the  Functions Practice page as well as the  Half My Age activity. Returning to the  Modelling World Population Growth activity, students might be asked to develop and justify a linear model for population growth in Europe over the last 200 years.

Assignment - A set of exam style questions testing the understanding of linear models (As part of preparation for the part 1 unit test coming up.)

#### Week 5 - Quadratic models

Over the next two weeks, students will explore quadratic models in context in much the same way as they have just done with linear models. The key objectives are to...

• Explore the key features of quadratic models in the abstract
• Examples of Linear models in context
• Problems involving intersecting with other models
• References to the meaning of domain and range in context here.

Assignment - A set of exam style questions testing the understanding of quadratic models (As part of preparation for the part 1 unit test coming up.)

#### Week 6 - More quadratic models and a part 1 Assessment

The start of this week will be a continuation of last weeks work and review of the last 2 weeks assignments.

ASSESSMENT POINT - This week will culminate in a Unit test based on the modelling work done so far

#### Week 7 - Exponential growth and Decay Models

Probably this week will start with a brief review of the previous week's assessment

The next 2 weeks will focus on exponential growth and decay models and the key features of intersect and asymptote and what they mean in context. We will pay particular attention to what happens with exponential functions when x =0 which sets them apart from polynomials. The following activities could fit in to this week.

Assignment - A set of exam style questions testing the understanding of exponential models (As part of preparation for the part 1 unit test coming up.)

#### Week 8 - Exponential growth and decay models

This week will be a continuation of last weeks work. Students can revisit the  Modelling World Population Growth activity to fit exponential models to population growth patters over the last 200 years. this provides an excellent opportunity to discuss the idea about extrapolation beyond the given data range.

Assignment - Students could be asked to revisit their initial write up of the population growth activity and elaborate to include the work they have done to find models

#### Week 9 - Direct and Inverse variation

This week students will explore some rational function models and their implications in context. Here we will introduce the idea of vertical asymptotes and their significance.

Assignment - A set of exam style questions testing the understanding of rational models (As part of preparation for the part 1 unit test coming up.)

#### Week 10 - Cubic Models

This week, the focus will be on cubic models and their key features. Students could work with some volume related models and focus on the meaning and relevance of local minima and maxima along with exploring the notion of 'valid domain' in that context.

Assignment - A set of exam style questions testing the understanding of cubic models (As part of preparation for the part 1 unit test coming up.)

#### Week 11 - Sinusoidal models

This week, the focus will be on sinusoidal models and their key features. We will explore these models in some of their variety of real contexts.

Assignment - A set of exam style questions testing the understanding of sinusoidal models (As part of preparation for the part 1 unit test coming up.)

#### Week 12 - Review and test

Following 11 weeks of work on modelling where students have experienced a wide range of modelling problems and contexts, this week will be devoted to the review and practice of these key ideas ahead of another big assessment point.

ASSESSMENT POINT - The end of this week will be an end of unit assessment for the Modelling unit of the course.

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