# Mathematical Models

'A guide for for teaching! Ideas and resources and lessons to look forward to'

*This page is good for the 'Mathematical models' unit in the new syllabus and the 'Functions' unit from the previous one!*

This is a big, key topic in the syllabus that covers a variety of function families, both in the abstract and in context. Its a fabulous opportunity to show where the abstract and contextual meet, using functions as a tool to model the world around us. Its key because of the presumed knowledge involved, the presence of which, or not as the case may be, has a big impact on the way its delivered! This topic is the one that involves the most use of the GDC and as such provides the opportunity to help students become fluent with that tool.

This page is a guide to the topic, ideas for classroom activity that encourage critical thinking, practise exercises and some of the best resources available on the Internet to help! The overview written below aims to help teachers think about the main objectives of the module and potential opportunities and issues there are with teaching it! Please follow links to the pages on Ideas and resources, exercises and the internet guide.

#### Ideas and resources

Classroom experiences are the most important thing that we as teachers are responsible for. This links to a page of activities and ideas for teaching this unit that are aimed to be both engaging and effective in encourage students to enjoy, discover and understand mathematics. The list is a brief outline of an activity that links to a page that gives more detail and the associated resources. This is an area that should develop regularly. Think about subscribing to the RSS feed to get notifications of updates.

#### Exercises

This page has a variety of tasks designed for practise and revision. Practice questions are an important part consolidating students' understanding of a concept. They can be found from numerous sources and the following is not intended as a 'fix all' solution. Most teachers appreciate that questions gathered from numerous different sources make the best diet!

#### The Internet Guide

This page is intended to list and outline some of the best resources available online to support the teaching and learning of this module. These will mostly be videos, virtual manipulatives and self help sites. Where appropriate, there will be a short commentary of what they are and how they might best be employed! It is also intended that this selection grow over time and that users contribute.

## Overview

As with many of the topics in the syllabus, this topic blends a raft of previously seen material with new ways of looking at that knowledge and some completely new ideas. Going from linear, to quadratics, to exponentials and sine waves, the topic exposes students to a variety of different functions and invites them to understand how they are formed and how they are used to model real situations. As such its a really appealing topic to teach, whilst the depth of understanding required provides a significant challenge. The calculators should be worked hard during this topic and the use of other graphing software can help this topic be really engaging.

### Key elements

- conceptual understanding of functions, associated language and notation
- linear functions - properties and applications
- quadratic functions - properties and applications
- exponential functions - properties and applications
- sine and cosine waves - properties and applications
- accurate graph drawing
- GDC and unfamiliar functions - sketching and solving

Please refer to the subject guide for full details

### Teaching time

The recommended teaching time for this topic is 24 hours. From my experience this is feasible, but depends entirely on how much of the 'presumed knowledge' is present. The 3 hours that are allocated for getting to know the GDC can be enveloped in this topic as well since so much can be done on the calculator. Ideally, there is time to really play with all of these functions. In practice it is more feasible to take a couple of the function families and go into great depth in ways that prompt students to look at the other families in a similar way. It is important to spend quality time on the first section either as a whole or as a part of the study of each function family, because the language and notation can present significant barriers if not properly understood.

### Sticking points

#### Language and notation

Needless to say, this is crucial, not only because it is the means of communication for this topic, but because without it, whole exam questions will be left not translatable. I mention this often because I am aware of how easy it is to use the variety of associated language there is and to be content that students actually understand the concept without being tight on on the notation and language they use to express their working, thinking and solutions. The syllabus is clear about the language used and required and it is a great help to students if we can pass that clarity on. An example is the use of set notation to express the domain and range of functions. This will be treated differently depending on whether or not students have done the work on the sets topic already. Different brackets, punctuation, arrows and set notation can seem daunting if they are not properly defined.

#### Axes and scales

Interestingly, alarmingly or perhaps both, the practise of constructing axes and choosing scales should not be presumed. Neither should the ability to read accurately from other scales be assumed. This is a deep seated issue and I suggest taking great care with these ideas at the beginning of this topic and to make sure that appropriate paper etc. is available to help with this.

#### Calculator fluency

At times it feels that teaching the use of GDC should be done elsewhere and that teachers should just be able to use the calculators to explore and solve problems. The teaching of 'how it works' does need to be considered but can easily be incorporated into teaching the mathematics as well. If it is not then students will be frustrated that they know exactly what they want the calculator to do for them, just not how to ask it. This would of course be a terrible shame and so again I invite teachers not to underestimate the importance of this fluency. I have found that if careful patient instruction is given from the beginning, then students begin to understand how the GDC thinks and are better able to help themselves and sometimes even teach us things we didn't know they could do!

#### Awkward numbers

When studying linear equations it is easy and probably common to stick to integer values between -10 and 10, moving on to some common fractions. Applications, however, often use bigger less friendly numbers and so students are suddenly faced with y = 2140x -134.5 instead of y=3x + 2. It is easy for teachers to assume that the understanding will translate when often it does not. This is true across all of the families of functions and I suggest it is worth exposing students to 'awkward' numbers from the beginning.

#### Common features

In studying this topic, much use is generally made of the 'general expression' for functions. For liner, y=mx+c, for quadratics, y=ax^{2}+bx+c and so on. With linear and quadratic functions, there is a constant (often 'c') at the end that can be equated to the y-intercept. This, of course, does not translate to translate to exponentials and trig functions in the same way. As a result it is necessary to be very careful when arriving at and using such generalities. Ironically it can be hard for students to generalise, but often, once they have managed, they may over generalise by applying their conclusions to different function families. This is a good opportunity to explore the very nature of generalisation and limits.

#### Indices

Its always worth being careful not to assume that this concept is understood. Order of operations may also need reviewing. Where unfamiliar functions are concerned, beware of negative indices in particular. Studying exponentials is an excellent opportunity to show the dramatic effect of indices.

#### Applications

Maths teachers will often debate the use of context to teach new concepts and use a variety of different approaches. Many will argue that context must be used from the outset so that the mathematics will be given meaning. Others will note that it is often easier for students to work in the abstract before making the leap to applications. There is unlikely to be a definitive view, but it is very important to consider the interplay between the two. A class that learn quadratics in the cartesian plane purely without context may struggle to make the link between that work and a real situation that can be modelled by a quadratic. Conversely a diet of application only can block the understanding of the general nature of the functions. Considering the interplay is a good approach.

#### Algebraic manipulation skills

A typical example of an applied problem that can create barriers is one where a quadratic is formed from a geometric situation. This relies both on a thorough understanding of the geometry and sound algebraic manipulation skills. Another example is where to functions need to be combined to form a quadratic in examples about cost and profit. In both these cases it is important to make students aware of the different skills. A student can often believe that they are weak with functions because their manipulation or geometry skills are not good. They should be made aware of what is blocking them and helped to distinguish the skills so that the specific weakness is identified.

#### Substitution

Traditionally, substitution skills are wrapped up in the teaching of functions, and whilst the two are inextricably linked, the substitution can often get in the way of the study of functions. For teachers and students of Maths Studies, the key point here is that the GDC should take care of all substitution issues. Specifically, students should enter functions and read from lists. This applies to filling blanks from tables and finding corresponding x and y coordinates. Students really must learn to use their calculators to their advantage in this area.

### Teaching Opportunities

#### Investigation

With some of the above hurdles in mind, investigation is an extremely powerful tool for learning about functions. Students should be given the opportunity to experiment and play with the general forms of the function families and begin to discover and shape their own understanding of their properties. Certainly, it is possible to simply teach the generalities and their meanings, but the understanding will be limited by that experience. Playing with abstract mathematics in this way can provide very satisfying challenges and its very important for students to experience this freedom and satisfaction.

#### Modelling

Showing how functions can be used to model real situations is important for empowering students to understand the world around them. Intersecting linear functions can help compare different pricing structures and conversions. Quadratic models show the beginnings of optimisation. Exponential functions are arguably the least understood by the general public and those with the biggest consequences. and decay problems are very relevant. The modelling part of this topic can and should bring a very engaging aspect.

#### ICT

The GDCs are powerful tools and much use of them should be made in this topic, with a particular emphasis on helping students get to know them and become fluent. This is a requirement of the course and a massive advantage in the exams.

Computers and other graphing packages can offer a more versatility and an improved classroom experience though and their use is highly recommended. Please see the resources page to learn more about the options here. The ideas and resources page contains a number of ideas for using ICT. The key thing here is that ICT can provide more efficient and engaging ways to do tasks that could be done without, but also offers completely new tasks and ways of learning.

## Selected Pages

### Functions Internet Guide 20 February 2018

This page is intended to list and outline some of the best resources available online to support the teaching and learning...

more

### Functions Teaching ideas 12 February 2018

Here you should find a wealth of resources that can be used by you and your students for teaching this topic. The resources...

more

### Dancing Quadratics 8 November 2017

'Call the dancing! Get these quadratics in order and make them dance the way they should!'Watch the videos below and work...

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### Quadratic Links 3 October 2016

'Use logical deduction to piece this quadratic puzzle together and make lots of quadratic links!'How much information do...

more

### Modelling World Population Growth 29 September 2016

How has the World's population been changing?Page under constructionMathematical models are thrown about all over the place...

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### Focus on - Exponential models 1 July 2018

The exponential function is set apart from other families of functions because of the fundamental impact of having the variable...

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