Unit 3 - Relationships
In unit 3, we build on the work with sequences in unit 2. From arithmetic sequences to linear models we look at variables varying with a constant rate of change and the significance of that rate. We move on from this to bivariate data that might approximate to a linear model, working with scatter diagrams, Pearson's correlation coefficient, linear regression and then moving to Spearman's when the data is not linear, but still monotonic. Then we go on to look at Quadratic and then Exponential models and their key features and significance in context.
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What is this unit about?
The syllabus items in this unit are addressed as follows
Week 11 - Linear models - Start this unit with Modelling World Population Growth (or similar) as this raises questions about the different ways in which variables can change with each other. This task is unlikely to be finished this week but can be returned to at a later date. Use resources from SL Linear models to make sure students are up to speed with the key features of a linear model.
Week 12 - Bivariate data and scatter diagrams - We can build on the linear models work by looking at the data for European population growth from last week as we go through the process of approximating linear data using resources from the SL Linear correlation page. We might look specifically at Educating Linear regression to give some good context.
Week 13 - Spearman's - This week might be a good opportunity to look at 15 Countries and 200 years given that the scales in these graphs are logarithmic. This makes it a good opportunity to consider the role of Spearman's Rank. Other resources are here on the SL Spearman's Rank page.
Week 14 - Quadratic models - This week the focus is on resources in the SL Quadratic models page and looking at contexts that can be modelled by quadratics.
Week 15 - Exponential models - There are some good experiments on the SL Exponential models page to introduce these. We will also return to Modelling World Population Growth to consider the exponential functions that model some parts of these population curves.
Week 16 - Tool kit Exploration - This week is reserved for trying out 2 or 3 different scenarios that will model how these skills might be used in an Internal Assessment. See Human Development Index Exploration and Marathon Records for examples.
Week 17 - Practise and test - This week is reserved for practise exam style questions and then a test (see below)
Why does it matter?
This unit gets to the heart of one of the most obvious applications of mathematics to the world around us in the mathematical modelling. Analysing and recognising how one quantity changes with respect to another so that we might better understand the relationship between them and use it to diagnose, to predict and to make decisions. The abstract algebra of functions is a deeply absorbing topic and on this course the goal is to take the behaviour that we learn from this algebra and see what it means in practise. When two quantities vary in a linear fashion, what is the meaning of the gradient of that model? What does the vertex of a quadratic model or the asymptote of an exponential one mean in context? How is any of this information useful to us? It is a huge area of mathematics that will be seen across a large range of future employment or study and here, we get an important introduction. This unit also builds a bridge between the functions and statistics by introducing the idea of linear regression and with it the important statistical questions about correlation and causation. This part of a bigger bridge between abstract and applied mathematics where a critical eye is needed to understand the limits of the validity of such techniques in a given context. It could be a course on its own!
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..
- Modelling World Population Growth is a terrific exercise in thinking conceptually about what it means to model and invites students to think and reason collaboratively as they do so. It is an activity that ultimately raises questions and generates a need for the processes and understanding we go on to teach.
- Similarly, Educating Linear regression and Scattertastic give us excellent contexts for generating discussion about correlation and the different techniques there are for analysing those relationships.
- There are further proposed experiments for exponential and quadratic modelling that allow students to get concrete experiences of how these kinds of functions might manifest in the world around us.
- The above open a significant number of doors for students to think about exploration topics and we need to allow for this as we teach, hence the specific week recommended for a focus on the toolkit.
- The unit also relies heavily on understanding the functions of the GDC. Teachers have to balance the use of modelling software like Desmos and the use of the GDC. The former will offer more flexibility for teaching and learning, but the latter is the tool students hve to know best.
- Please note that more attention to other models is given during the Geometry and Trigonometry unit - Cubic and a quadratic models related to area and volume and the Calculus unit where these and other models come in to play with optimisation.
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
This is a test written for the third unit in this SL Scheme 2 that I have planned for this year. The unit is the first on concept of mathematical modelling and covers linear models that lead on to scatter graphs, correlation and linear regression as well as Spearman's rank. Then there is some broader modelling with Quadratic and Exponential models. The questions can be used as an assessment as they are or taken and used for different purposes. Full solutions follow in the document.
SL - Unit 3 - relationships unit test - 50 marks over 6 short and 2 longer questions. Solutions included at the end of the document.
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.