Week 1 - Financial Geometric Sequences: compound interest and depreciation

This lesson highlights for students how poor our "human intuition/gut instincts" are when it comes to the concept of compounding growth:  SL Investment Intuition - Compounding and the importance of precise measurement of relative returns. Students are given 10 000$/£/€ to invest in a range of companies, awarding differeing dividends on shares (to help provide a lump sum on their first house later). Depreciation examples can include nuclear decay, crypto currency prices (at certain times!), car magasines/websites showing depreciation of resale price over time for a given make & model (most countries have a magasine specifically devoted to this, in US (&UK): Autotrader, UK: Parker's Car Price Guide or France: L'Argus), dampening of vibrating materials (physics experiments they can do/use data from), a swing (once the person is at the highest point that they dare reach! model effect of gravity and air resistance slowly bring them to a standstill), bouncing balls etc.

Use the finance package to calculate Future value of investments from Present value with different interest rates. 

Problem solving with compound interest and annual depreciation - The above lessons will include the different ways in which these kind of questions can be asked, setting up equations and solving for different variables. It is easy to overlook that understanding of the key ideas is not the same as the experience in solving related problems.

Use this context to introduce the more formal notation of geometric sequences and series: U₁, U₂, U_n, r, S_n etc. with further contexts such as growth of bacteria, populations, crypto currency prices (at certain times) and nuclear decay or dampening of vibrating materials, bouncing balls etc.

Start to sum these geometric sequences (geometric series).

The page  Focus on - Geometric Sequences has some excellent, IB ATL and 'mathematics toolkit' investigation style activities, with applications, on this topic.

Problem solving with geometric sequences including use of GDC, table function and equation solver - Based on the formulae that have been derived for Geometric Sequences, problems should include a range of unknown variables

Assignment - Set a practice assignment, or exam style questions, for compound interest and geometric sequences, all SL tasks, here:  SL Geometric Sequences or arithmetic or mixed exam style arithmetic and geometric sequences and series(sums) questions from the options here: SL Arithmetic Sequences    

Week 2 - Sum of Geometric Sequence & Graphical representations of sequences - the Exponential & Logistic function

• Finishing off the work done with sums of Geometric sequences and introducing sigma notation i.e. get students to produce the first 5 terms from notation such as: \(\sum\limits_{1}^{n=k}{(5){{2.1}^{(n-1)}}}\)to ensure students understand how to translate the mathematics into a sequence of numbers.
• Plot some of the Geometric sequences from Week 1 then students use the GDC's in-built "dynamic graphs" (Casio), "sliders" (TiNspire), "explorer app" (HPprime) to fit an appropriate model ('power' function versus 'exponential' function) to each one. Explore the graphing analysis tools to predict future results.
• The concept of 'e' arises naturally from the previous lessons' contexts, as we demonstrate that continuous growth, defined by ever larger values of 'n' in the compound interest formula: \({{\left( 1+\frac{1}{n} \right)}^{n}}\) leads to \(e\approx 2.7182...\)
• Do students think the sequence will keep progressing like this forever (can disease, populations, returns on investment continue expanding exponentially forever?). Leads into the Logistic function. 

Week 3 - Sum of infinite sequence, Arithmetic Sequences and Linear Models

• Finish off Geometric series, from week 2, with a look at the sum of an infinite sequence (HL only).
• Introduction of some real contexts/applications with constant differences and plotting the data to model it using linear functions e.g. taxis and mobile phone costs, or anything that has a fixed cost, followed by known, variable costs, can be modelled using arithmetic sequences and linear functions (linear programming (not explicitly in syllabus). Trig functions complete one wave every period (constant difference), in fact anything that is periodic e.g. Old faithful geyser at Yellowstone national park explodes 21 to 23 times a day, can be approximated using an arithmetic sequence (in modelling, it's important students appreciate the role of approximation), sales and production (x people to produce y amount of goods. With approximations to arithmetic sequences there is a strong possible link with linear regression. The idea is that students would be looking at the data in terms of finding an approximation for the likely 'common difference' e.g; Old Faithful geyser explosion times at the US Yellowstone national park.
• Exploring patterns in arithmetic sequences leading to generalisations about the nth term and sums of an arithmetic sequence - This topic provides an excellent opportunity to focus on the mathematics that leads to generalisation.  Visual Sequences and Arithmetic Sum are great ways to do this.
• Formalising and applying the generalisations form the previous lesson - The teacher will likely demonstrate and number of examples of the formulae in action and provide practice exercises.

Links: to linearising data using logarithms (HL 2.10) which simplifies the analysis of exponentials through linear techniques.

Assignment - Set an assignment involving practice of Arithmetic sequences

Week 4 - Measuring rate of change - concept of derivative

• A good starting point can be looking at the differences between terms in the previously covered Arithmetic and Geometric sequences. This links to first and second difference methods (perhaps only HL and Level 5+ students will have seen the second differences method pre-IB) for sequences, that many will have encountered in their pre-IB courses.
• I always like to introduce concept of 'rates of change' in the context of displacement-velocity-acceleration, because all humans have lots of experience of what this feels like. Videos of rollercoaster rides make a great "in" & "hook", with students sketching how their vertical height is changing against time (play from 2m25 and 5m43 to experience the Kingda Ka, Six Flags amusement park in New Jersey and Superman Ride of Steel). This activity: Distance-Time graph, gets students building a physical memory (and concrete application) of a distance-time grpahs, whilst collaborating, communicating, thinking and developing social and teamwork skills. Once built, discussion commences, and modifications made to the track with graphs pre-drawn on how students predict this will effect velocity, acceleration etc.
• This introductory lesson leads nicely into the  Classifying Sequences then  Rates of change , just the matching parts of the task. The final part can be returned to once students have completed:  Measuring Gradients. The TiNspire, Casio CG20/50 or HPPrime can equally well be used in place of Geogebra,Desmos,Autograph for these activities.

The above introduces the concept of 'rate of change' and 'how to measure it' for curves, where a function, rather than a single value is required. This lays the foundations for modules 3 and 4 where the power rule (introduced in the   Measuring Gradients activity above), chain rule, tangents and normals etc. are investigated in more detailed.

Assignment - Set an assignment involving distance-velocity-accelearation and finishing with a few questions on predicting the gradient function's highest power of x (given the original function).

Week 5 - Scientific notation, Indices and Logs

Refer back to the work covered in week 2 above on exponential functions. The numbers in such sequences can get very big (growth) or very small (decay). Focus on understanding how scientific notation (standard index form) helps us to work, and understand, more effectively these very large and small numbers. Demonstrate the utility of this form of notation - Big & Small Numbers might be a useful catalyst for discussion.

This makes a nice introduction to logarithms with base '10' understanding equivalences, since we can now fit, plotting powers of 10 on the y-axes, instead of the number itself, exponential graphs onto standard axes, see activity:  Logarithms: an introduction (that also includes a section on good internal assessment ideas around logarithms).  

Refer back to the lesson material from the natural number 'e' from week2. As we saw in week 2, the number 'e' is often found when describing natural phenomenon since it describes, and is the limit of, 'continuous' growth. Given its prevalence in nature, we often have to solve equations involving 'e' and hence need to use the natural logarithm (Ln).

The  The 6 million question activity is an excellent opening activity for estimation (and the related: percentage error). Percentage Error is an excellent way to zoom in on the importance of the size of one thing relative to another e.g. 1€/$/£ increase on the price of a Mars bar is more a source of concern than a €1 increase on the price of a new car, and provides the necessary "proportions" conceptual foundation for understanding Pearson's Coefficient (r) and [HL only] the Coefficient of Determination (R²) (in week 6).

Assignment - Set a practice assignment for scientific notation (standard index form) from the  N & A Practice page.

Week 6 - Pearson's correlation coefficient and Least squares linear regression using GDC and SSE

During this module we have looked at exponential and linear models. How can we make good decisions on the relative merits of competing models? This week's lesson provide an introduction into measures of "goodness of fit" with Pearson's correlation coefficient and the concept of least squares regression (linear only in the case of Pearson's (r), but the activities go on to look at other regression models (GDC and applets): quadratic, cubic, exponential, trig etc.).

Least Squares Regression and Pearson's correlation coefficient are challenging to understand in detail, but the concepts are more easily accessible.  Scattertastic is a very interactive (and fun!) way for students to really get under the skin of the the concepts behind the calculations.

The presentation at the top of this page:   Focus on Correlation walks students through very clearly a summary of correlation, and then how to calculate least squares linear regression and Pearson's product moment correlation coefficient on their calculators (GDCs).

HL (SL Level 5+ optional) Specific resources
The topic is an example of where I would envisage teaching the same material, slightly differently, with an HL only or HL/SL level 5+ class than with a purely SL or SL Low level 5 and below classes. I think HL students (and some SL students) will need answers to the sorts of questions mentioned below to prepare a sufficiently deep understanding ahead of looking at the sums of the squares of the residual and coefficient of determination (HL4.13), unbiased estimators of population parameters (HL4.14) etc.

This activity provides resources for teaching from scratch, or reviewing (via practice questions and summary slides) Standard Deviation, a good understanding of which underpins the conceptual understanding of Covariance and Pearson's correlation coefficient (r).

These resources:  Covariance, Pearson's Correlation Coefficient and   Pearson's (r) Limitations are designed to provide this deep understanding of covariance, Pearson's (r) and its limitations, leading to the need for other measures such as Sum of the Squares of residuals (HL4.13) and the Coefficient of Determination (R²) (HL4.13). Making explicit reference to the work done in the previous week 5, on percentage error, can help students get a firmer grasp on the concepts used here.

For SL students the explanation “Pearson’s (r) is a measure of how closely the data points fit the linear regression”, is likely sufficient, and more wouldn’t necessarily be required to get very good marks at SL on this topic (beyond an awareness of its limitations, without necessarily needing more than a superficial understanding). However, if HL don’t understand, in detail, the concept of co-variance, and hence how Pearson’s is calculated, then for the limitations: it is only valid for linear data, outliers can have a disporportionate effect on Pearson’s as a useful measure of correlation, the coefficient of determination, R², is the square of pearson’s ‘r’ (for linear data), but the coefficient of determination can be used to measure goodness of fit for non-linear data whereas Pearson’s can’t . . . a lot of confusion can arise.

[These resources are also excellent, focused more on SL, but providing good explanation, practice and support for HL also:   Focus on Correlation e.g.  World Statistics gives examples of the sorts of questions scatter graphs can help students to answer (this resource can be used at anytime, but it also takes an in-depth look at what we can infer about World War II from the statistics: good for cross-curricula or IB induction events): ] 

Both of these techniques, and their associated concepts, are revisited and built on in the next module. 

Assignment - lots of exam style practice questions available by scrolling down to "Practice Exercise" on the  Focus on Correlation page.

Week 7 - Amortization

• The activity: SL Amortization and Annuities provides materials and activities for introducing amortisation. The presentation 'Teaching slides - Understanding' is another example of where I would envisage teaching the same material, slightly differently with an HL only or HL/SL level 5+ class than with a purely SL or SL Low level 5 and below classes. It includes 'mathematical toolkit' extension type activities for HL who will probably inquire after a deeper understanding of the 'black box' working out going on inside the calculator - making a nice link with sums of geometric sequences. These type of extensions, to take a deeper look into the mathematics, facilitate the teaching of SL and HL students together (thought level 3 SL to level 7 HL maybe a step too far?). The slides "Teaching slides - Calculations" would likely be sufficient for SL students to score good marks. Aims:
• Introduce the idea of amortization in the context of loan repayments
• Problem Solving with Amortization
• Review of some of the ideas in the unit in the context of solving equations - For example - when solving questions like 'what term of the geometric sequence will be equal to.... or 'After how many years will the investment have doubled' etc might efficiently be done using the equation solving tools, graphing function (x-solve, y-solve and calculate) or the table function.
• In the modelling process we start off with simple, but effective models, and then add additional layers of complexity if required. Adapting our geometric sequence/exponential financial models for changing interest and inflation rates is one, good, example of this process. 

Assignment - Set a review exercise for the whole unit working towards a unit test.

Week 7 - Review and Assess

1 hour devoted to helping students review the syllabus items - You might use some fill in the gap notes, or decide to go over some examples from the review exercise set as the last assignment. Organisation of files and clarity about exactly what students might expect to see in an exam

ASSESSMENT POINT - A 1 hour unit test on the syllabus items above