Unit 2 - Growth and Decay
In unit 2 we will get a good look at the ideas of linear and exponential growth through the study of arithmetic and geometric sequences. These are key ideas that link well with experiences that students have had as well as laying some good ground work for further study in the context of mathematical models. The unit links geometric sequences with compound interest and depreciation and the effects of repeated percentage change before then moving in to more complex financial arrangements with loans and annuities where payments are made a long the way. The unit depends heavily on knowing and understanding how the finance app works on the GDC.
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What is this unit about?
The syllabus items in this unit are addressed as follows
Week 5 - Arithmetic Sequences - Introducing this fundamental idea of linear growth and relating it to students previous experiences. Start with this Visual Sequences activity as a way of engaging with sequences through concrete experiences. Then formalise the the ideas that were involved with that activity and move in to some of the practise resources on the SL Arithmetic Sequences topic page.
Week 6 - Geometric Sequences - We will return to arithmetic sequences, but first we will introduce geometric ones through this In the money activity. Then, as with arithmetic, we will formalise our thinking and practise. The week will draw on the resources on this SL Geometric Sequences page.
Week 7 - Problems with Sequences - Having learned about the basics of these sequences, we will spend this week looking specifically at applied problems involving these sequences using various resources from the two topic pages above and culminating in exam style questions.
Week 8 - Compound Interest - This week we will build on previous understanding of compound interest and depreciation as repeated percentage change. This links well with items from unit 1. We will go back to 'e' which was introduced in unit 1 when looking at the Compound Interest Calculator with which we can show the limits of mans greed! Resources are on the SL Compound Interest & depreciation topic page. The week will involve showing how the finance app can be used to solve compound interest problems which sets us up well for next week.
Week 9 - Loans and Annuities - This week we will develop our understanding of the various financial contracts that people enter in to using resources from the SL Amortization and Annuities topic page. Much work needs to be done here on properly understanding the finance app on the GDC!
Week 10 - Practise and test - We will spend the week reviewing the unit and working with lots of exam style questions from the topic pages. See below.
Why does it matter?
Sequences provide a lovely model for showing how mathematical thinking can lead ius to helpful, labour saving generalisations. In that sense it is a good opportunity to think about mathematics as an Area of Knowledge. Then we can get stuck in to the applications and dwell on the professor from Colorado (see the SL Geometric Sequences topic page) who describes the human race's biggest single failing as our inability to understand the exponential function! It is quite easy to get in to important consequences, especially in the current climate. Then the shift in to understanding complex finances gets pretty relevant with most of us likely to do some borrowing in our lives as well as a world wrapped up in tough economic decisions and their consequences. We can even revisit the beginning of banking as an idea! It is also great to give the GDC such a high profile early in the course!
Opportunities for broader goals of education
The IB philosophy as detailed under 'Approaches to teaching and learning' (ATTL and planning) , invites is to always be thinking about the broader goals of teaching these units. It can be really hard to detail all the things that we do, big and small, planned and spontaneous, to do this, but it is important to try and reference some of them so that we get a strong sense of how we are doing against these objectives. Each of the pages and activities referenced above will also include references specific to those activities.
Each of the links above have extensive details about the nature of the tasks and the opportunities within them to address the broader goals of teaching and learning. For example..
- Both this Visual Sequences and this In the money task involve a good amount of enquiry based and collaborative activity where students will be expected to reason with what they already know in order to make some useful generalisations.
- As teachers we can make sure we teach the formal language of this (the new part) carefully to helps students bridge that gap, along with modelling examples carefully.
- The two activities are also rich feeding ground for thinking about intuition and the ways in which knowledge is produced.
- The lesson on the Compound Interest Calculator is another such moment where programming the spreadsheet asks us to think carefully about what is happening and then produce an algorithmic tool that solves problems for us. This builds a nice bridge to the heavy use of the GDC in this unit.
- At the same time it allows us a chance to explore the enigmatic Euler's number and one context where it turns up in a good ethical discussion about the origins of banking and the limit of mans greed.
- Although many of our students are not too involved with finances, they are about to be and so Financial mathematics does get pretty relevant.
There is more written about all of this on the individual pages, but the unit is richly charged with opportunity to be broad in our attention to ATTL..
Assessments
Students will use the practise exercises listed above as assignments during the unit and build towards an 'Unit test' that aims to test their understanding in exam conditions of the kind of questions they are building towards. This will be marked on an IB scale from 1 to 7 and used in the feedback for induction.
Here is an End of unit test assessment that can act as a summary assessment for the unit. Solutions are in the making..... now also embedded below.
solutions
Help and support
Many of the tasks in the unit have different access points that should help students to engage at all levels. In addition, students have access to their text book and studyIB.net/mathapplications, where they can watch video lessons and get help and support. This is of course in addition to help from the teacher during lessons.