Key Ideas - Chapter 8: Functional relations between variables
A list of resources to support the book and website "Key Ideas in the Teaching of Mathematics"
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Algebra arrows
This Java Applet on the Freudenthal Institute for Science and Mathematics Education (FIsme) website allows learners to set up their own arrow diagrams to construct functions and inverse functions.
The boxes need to be dragged onto the screen and linked by clicking and dragging the arrow heads to join the boxes. Input can be numbers or letters, and output can be given as values or as expressions involving numbers or letters as input. All the operations are unary (this means they act in one variable only) so can be turned around to solve linear equations.
The Applet also offers squares and square roots. Some critics have said that the function machine/arrow diagram approach can embed a ‘left-to-right’ misconception of reading algebraic expressions. However, because this Applet can be set to give the correct conventional expression for the selected sequence of operations it can be used to contradict a ‘left-to-right’ tendency rather than embed it.
Number and Algebra 3
The relevant section in this Nuffield Foundation textbook is the function game on pages 91-92.
In silence learners try to identify a hidden function by putting in various values and observing, and conjecturing about, the output. The silence is very helpful because it encourages everyone to think as more and more data is developed on the board. The propensity for learners to form their own understandings of possible functions is a major ingredient of this approach.
There are potential pitfalls in introducing functions as mappings involving particular values – this starting point can embed the idea that the graph is a ‘join the dots’ exercise. However, in the example given the function is expressed in general words as well as number pairs, so can help move towards an understanding of the function in its general form.
The balance strategy
This Java Applet on the Freudenthal Institute for Science and Mathematics Education (FIsme) website allows exploration of solving linear equations with the unknown on both sides using the balance strategy.
Learners can devise their own questions as well as solve those given by the software. There is little difference mathematically between doing this with pencil and paper and using the Applet, but the Applet appeals to some learners as it removes the need for writing and also has a games-like feel to it. Learners can also deal with errors immediately.
The Applet does not provide any justification for the balance method, and in particular it assumes that learners will understand how to deal with negative amounts within the balance metaphor.
Coordinate Geometry
This activity is part of a collection of 40 open-ended investigative activities. The relevant activity is
Risp 17: Six Parabolas.
By varying the coefficients of a quadratic function you can explore the effects of different coefficients. This is a critical step in understanding functions and families of functions. Learners first meet functions as sets of points generated by an independent variable and a calculable function. At this stage the numbers that change are the variables.
As part of the process of shifting towards seeing the function as an object in itself, learners have to focus on the parameters that shape the function. This task triggers exploration of this process, and begins to relate the coefficients to physical characteristics of the parabolae.
Maths shop window
This task on the NRICH website, pushes learners further towards seeing functions as objects with their own characteristics by asking them to find examples of functions with certain characteristics unusual for school mathematics.
In order to use functions in the fullest way possible, learners need to develop a repertoire and this kind of task can help them to do that. The task encourages them to think ‘outside the box’ of what they may already know, and goes beyond the polynomial and ‘named’ functions of the middle years of school.
It is not necessary for learners to know in advance about the functions that will fulfill the conditions. It would be possible to invent some by conjecturing about particular values, or overall shape by sketching, to get some way towards completing the task.
Population dynamics collection
This task, on the NRICH website, is the beginning of a fairly advanced collection of explorations into population dynamics. It is probably best undertaken in groups, so that each stage can be thoroughly discussed.
The beginning tasks are not inherently difficult mathematically, involving algebraic expressions of understandable situations. Exponential growth appears soon, and here the algebraic expression gets more unfamiliar, but the underlying sense of exponential growth should be familiar to learners from contexts outside school or in science.
This set of tasks is an introduction to the modelling process in a real context, so that students can appreciate the power of mathematics to provide a window on phenomena and predictive tools.
Keeping the Pizza Hot
This Bowland Maths task asks students to solve a practical problem, probably over several lessons, using a purposeful modelling cycle. It require them to set up relations between variables and use them to establish the effects of different design decisions. Students can experience the value of mathematical approaches in the realistic context of a cooling curve.