# Calculus

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

*This page is good for the units from both the new and previous syllabi 'Introduction to differential calculs'.*

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.

#### Practice

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 grows over time and that users contribute.

## Overview

Calculus unlocks a whole new world of fabulous mathematics and is often studied as a course in its own right. This topic is a brave attempt to give Maths Studies students an introduction to the concept and its applications. Its very much in the spirit of the course philosophy and as such a key component of the overall course. Working with calculus at this level requires a thorough combination of algebraic skills and logical deduction and presents a rewarding challenge both for the student and the teacher. Most appealing is the bringing together of pure and applied mathematics. The concepts can be explored and generalised in the abstract and then applied to practical optimisation problems that have a huge role to play in society. Which ever order these ideas are presented in it's powerful for students to experience them both and the link between them.

### Key elements

- gradients of approximations to curves
- derivatives of polynomials
- equations of tangents
- increasing and decreasing functions
- maxima and minima

Please refer to the subject guide for full details

### Teaching time

This topic is given 15 hours as a recommended time. For the calculus involved this is just about reasonable. The catch is that many other mathematical skills are inherently embroiled in this topic. See the sticking points below for details, but preparatory work on these may well be required.

### Sticking points

In time these sticking points will be illustrated with a video or screen cast....

#### Concept

Not a good start! When applied to real situations about rates of change like distance, speed and acceleration, the concept can be clear, but translating that to the more abstract idea of the derivative of any given function is a whole new level. A good amount of time exploring the idea of rates of change and relating it to functions in and out of context is important.

#### Notation

f(x), f'(x), dy/dx, d^{2}y/dx^{2 }and so on... Combine a difficult new concept with a set of unfamiliar notation and you are bound to create some confusion. In applications of course, x and y are interchangeable with the variables in the situation. All of this can be handled with patience and a sensitivity to that confusion. A simple example is of a student not answering a question like what is f '(1) given f(x) when they are able to both differentiate the function and substitute an x value into a function. Not understanding the notation, in this case, prevents students from demonstrating their understanding.

#### Distinguishing

Tied in with the above points about notation, it becomes very important for students to clearly distinguish between a function and its derivative. For example, working out the coordinates of a point with a given gradient involves working with the derivative to find the x coordinate, but with the original function to then find the corresponding y coordinate. To rely on routines here is risky, when emphasising the meaning of a function and its derivative can give clarity. This presents a challenge!

#### Negative indices

This is a notorious barrier to progress in the teaching of this topic and often requires some very specific review. Essentially the required skill is to know that ax^{-b} and a/x^{b} are interchangeable. A review of this relation and why it is so is highly recommended before asking students to differentiate reciprocals. The danger here is that students will confuse their difficulty with this idea with a difficulty with calculus!

#### Applications

Questions on applications of calculus often involve a large amount of logical deduction and algebraic manipulation before the calculus even begins. This often presents a barrier that, whilst still of considerable importance, should not get in the way of a student understanding the principles of calculus. An typical example is of a maximisation problem involving a rectangle having 4 equal square corners cut out and then being folded into an open top cuboid. The question is, what size square generates the box with the largest volume and what is that volume. To answer this question students must;

- define the variable as the length of the square (lets use x)
- deduce the length and width of the resulting cuboid in terms of x (forming algebraic expressions)
- multiply the terms for length width and height of the box together to generate an expression for the volume of the box (algebraic manipulation)
- recognise that the resulting cubic function will have a 'valid' domain containing a local maximum and that the gradient of that function will equal zero at that point (calculus)
- differentiate the function (calculus)
- solve the gradient function equal to zero for the value of x (solving equations)
- substitute that value of x into the volume function to work out the maximum volume (applied algebraic substitution)

The point here is not that it is not an important exercise, but that a 'calculus' question is likely to involve a global level of mathematical understanding and dexterity that far supercedes the level of calculus used. It is important to help students distinguish between the different skills that they are using at different times. For example solving equations relevant across most of the syllabus and is not a discrete idea when looked at through a calculus question.

### Teaching Opportunities

#### First principles

The syllabus requires that students are able to measure the gradient of a function over a small variation in x and and if you have gone this far it is really worth students exploring the derivative of y = x^{2} from first principles. Just doing this once can add much more meaning to the topic, apart from being a deeply satisfying piece of mathematics. With the aid of ICT you can then show a more difficult derivation from first principles so that students understand the power and beauty of the general derivative!

#### ICT

Graphing software really brings with some great possibilities for investigating this topic. I have included an activity called 'measuring gradients' that involves students discovering that the order of a derivative is always one lower that the function (where the power is not equal to 1). This too brings a sense of meaning to the process of differentiating.

#### Optimisation

A very real and significant application of Calculus! Let students see and experience the relevance of this mathematical tool.

### Ideas and resources

This links to a page of activities and ideas for teaching this unit. The list is a brief outline of an idea; some are just ideas and others link to a page that gives more detail and some resources. This is an area that should develop regularly. Think about subscribing to the RSS feed to get notifications of updates.

## Selected Pages

### Measuring Gradients 4 October 2016

'Measure gradients and see for yourself the relationship between a function and its gradient'This activity is all about...

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### Calculus Teaching Ideas 4 October 2016

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

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### Calculus Practice 4 October 2016

Practice questionsThis page has a variety of tasks designed for practise and revision. Practice questions are an important...

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### Optimal Jumble 4 October 2016

'Piece this puzzle together by splitting up the three problems into a logical order'This problem is all about the 9 steps...

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### Rates of Change 4 October 2016

'Distance, speed and acceleration - how does one affect the other?'The concepts behind calculus begin with understanding...

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### Classifying Sequences 4 October 2016

'What makes one sequence different from another?'This activity takes the study of sequences on to a new level. Look at all...

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