A graph showing an inverse relations
Graphs are fun!

Welcome and orientation

Hello everyone from around the world!

We are going to be looking at some core senior-level high school mathematics, but from a somewhat novel, applied orientation. We will focus on inverse relations and power laws that occur in many areas of the modern world, in science, economics, mathematics and daily life. This will let us strengthen both our skills at algebraic manipulations and our geometrical understanding, while connecting with lots of interesting and instructive real-life examples.

Course structure

The course covers four weeks. All of the activities of a given week are connected by a common theme. For example, in Week 1 we will be having a first look at hyperbolas and their geometry. The first two weeks concentrate on inverse relations and interesting applications of them, while in the last two weeks of the course we move up to power laws and other more advanced relations and in particular relations to biology.

Week 1: Inverse relations and hyperbolas

Week 2: Zipf’s law and Mobius transformations

Week 3: Cubics, polynomials and power laws

Week 4: Allometry, big animals live longer, and beyond

Each week consists of several topics, or activities. An activity begins with a short video that orients you to the topic, covers some historical aspects, and gives an overview of the key mathematical ideas. The activity is developed in more detail through a number of steps, each focusing on a particular aspect of that topic. These steps involve you reading carefully the prepared text we have put together, studying the diagrams, of which there are quite a few, and attempting to answer a series of questions that will be interspersed with the text.

At the end of each activity there will be a short quiz to let you check that you have understood the main ideas. You should be able to do at least half the questions in the quizzes easily, and if you can do all of them, you can be confident that you have a good grasp of the topic. At the end of the week, there will be a more official test.

The importance of active participation

Mathematics is, ideally, an active subject. We want you to learn actively. An online environment runs the danger of encouraging passivity — but don’t let that apply to you! Please have a book on hand to write in, or a laptop that you can record in. Jot down some summary notes of the main facts you learn from the videos and the individual steps. Make sure you tackle questions actively — by trying to write out correct solutions. We will be giving you some guidance as to what that looks like as we go.

Download GeoGebra and play with it

GeoGebra is a free dynamic geometry package that can help you to visualise lines, parabolas and higher degree curves. We have used it to create a lot of the nice diagrams you will find in this course. You can download it for free at geogebra.org, and we encourage you to do this and play around with it.

Using discussion and comments

We also encourage you to employ the opportunities that FutureLearn offers in terms of discussion, asking and answering questions, making comments and reading other participants’ inputs. On most steps you can find a place to add comments and engage with the comments of others. We would suggest that you pay particular attention to answers offered by the instructors of this course: Norman Wildberger, Anna Tomskova, Daniel Mansfield and Joshua Capel. You should follow us so that you can easily see our replies to people’s questions.

Why not get started right now? At the bottom of this step you can see a comments section, and we invite you to say a few words about yourself by way of hello. You might like to comment also on someone else’s comment.

Remember that the essence of mathematics is clear thinking and careful use of language, and so it is a good habit to try to cultivate these skills by wrestling with ideas yourself to try to understand them. It’s also a good idea to have a look at prior questions: if something is unclear to you, it is likely unclear to others as well, and someone may have already asked a similar question.

Answering questions

The answers to each of the questions in a step can be found at the end of that step — but please don’t just skip ahead to the answers! We expect you to put in effort to answer all the questions that are at your level. And what is your level? That will depend on you, your background, the educational system in which you have learnt, and your mathematical maturity. We recognise that in a large online course like this, there will be a wide variety of levels of participants.

To simplify this, questions will be marked as E (Easy), M (Medium) and C (Challenging). It is important that you are able to solve all the Easy questions correctly, and hopefully on your own. Some of you will find some of these perhaps too easy — which is no problem. Just read them anyway, and skip to the Medium or perhaps Challenging questions. For those of you whose mathematics needs some careful nurturing at this stage, don’t be discouraged if you find the Challenging questions and even the Medium questions too hard. It is not necessary to be able to answer those on a first go. But read them in any case, and give yourself the option of thinking about something for a few hours, or days (yes it can be fun!), and coming back to it later.

Getting started in FutureLearn

If this is your first time taking a FutureLearn course, you can visit How it Works for extra support. If you have further questions about how the site works, take a look at the FutureLearn help pages, or click the grey question mark icon at the bottom right corner of the page to ask for help.

Would you like a certificate?

If you want a record of your course, you can buy a Certificate of Achievement from FutureLearn.

The Certificate of Achievement is a great way to prove what you have learned on the course and as evidence of your Continuing Professional Development. This is a personalised certificate and transcript, detailing the syllabus and learning outcomes from the course. It comes as a printed certificate as well as a digital version which you can add to your LinkedIn profile. To be eligible, you must mark at least 90% of the steps in this course as complete.

There is also the option to purchase a personalised Statement of Participation, to celebrate taking part. To be eligible for the Statement of Participation, you must mark at least 50% of the steps on the course as complete. This also comes in a printed and digital format and you can add it to your LinkedIn profile.

Course content as a future resource

Remember that once you are enrolled in this FutureLearn course, the contents will be accessible to you long after the course is officially over. So the course can be a place where you can locate information and understanding for future reference as well. Let’s have fun! Good luck everyone, and we hope you enjoy the course, learn some interesting mathematics, and perhaps make some friends.

Norman and Daniel

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This article is from the free online course:

Maths for Humans: Inverse Relations and Power Laws

UNSW Sydney