Data-loggers

Perhaps it is surprising to see these included when their more obvious curriculum `homes’ would seem to be science, D&T and ICT. Of course the forging of cross-curricular links using

ICT as a catalyst cannot be a bad thing. But there are many aspects of the mathematics curriculum whose origins lie in modelling scientific phenomena and which can be brought to life through practical contexts. The most obvious of these is the interpretation of distance-time graphs which can be greatly facilitated with the use of a motion detector (i.e. a distance

measurer). Distance data may be captured from a variety of activities which include someone walking to and fro, or a pendulum swinging, or a model car running down a ramp, or a ball bouncing, or a spring-mass system in oscillation. Once the data have been captured using a data-logger then they can be manipulated, displayed, analysed etc. using any of the normal ICT tools for data-handling. Thus an experiment can be performed in front of the whole class, and the data then distributed for group, paired or individual work.

There is a wide range of data-capture systems and accompanying probes in use in other subjects, but we will concentrate on two particular devices designed for use both with graphical

calculators and PCs. These are the Texas Instruments Calculator Based Ranger (CBR) and Calculator Based Laboratory (CBL). The first of these is a device about the size of a pack of three golf-balls which emits and collects an ultra-sound signal (like the way a bat navigates in flight). The second is a more versatile control unit which can be connected to a number of probes simultaneously – such as for temperature, light intensity and voltage. The Key Stage 3 Framework for mathematics includes examples of the use of both devices, e.g. for the

successive bounces of a ball, and for the cooling of a liquid. They can be used throughout the 11-19 curriculum providing data for modelling using linear, quadratic, inverse, sine and exponential functions. One of the DfES video case studies for BETT 2005 shows Y8 pupils working in groups with laptops, CBRs and TI InterActive! for activities with distance-time graphs. The corresponding lesson materials will also be freely available to schools as part of the DfES’ KS3 Offer to schools from BETT 2005.

An account of their use by mathematics and science teachers at KS3 can be found in the Becta publication `Data-capture and modelling in mathematics and science’. This can be downloaded from: http://vtc.ngfl.gov.uk/uploads/application/datacapture-16796.pdf

The TI-83 Plus graphical calculator, as used in RM Maths Alive, has a built-in application `CBL/CBR’ to facilitate the use of data- loggers. In this example we capture distances walked in a straight line.

You can use the left and right cursor keys to trace and read off from the graph. You can also make `qualitative’ observations, such as `walking away’, `standing still’, `walking toward’, `constant speed’, `faster’, `slower’, `longer’, `shorter’ etc. You can take measurements from the graph and thus can make `quantitative’ remarks such as “in the first 4.4 seconds you moved from 0.46m to 2.09m away” or “your average speed was 0.37 m/s”. A popular activity is to draw a target distance-time graph on the whiteboard, or OHT overlay, and to ask students to try to match it. When you choose to "QUIT" the application, the time data (in seconds) are stored

in list L1 and displacement data (in metres) are in list L2. Pressing [GRAPH] redisplays the last graph shown, and by pressing [TRACE] you can take readings from it. So you can now try to `fit’ a model to any part of the motion. Enter a function in Y1 which you think might match the first part of the distance-time graph. The screens above show a first attempt - can you improve on it?

Now think hard about the various terms in the definition of Y1 above. Y1(X) is a measurement in metres of the distance from the CBR at any time X in seconds. So when you use a function like Y1(X) = 0.37X+0.46 it is clear that 0.46 must also be measured in metres, and that Y1(0)=0.46, so that it is the distance away from the CBR when the time is 0, i.e. the starting, or `initial’, displacement. Similarly 0.37X must be in metres, but X is in seconds, so 0.37 must be in metres/sec (or ms-1) and so is a speed (or, since it can be negative, strictly a

`velocity’). Hence the gradient of the line represents the average, or constant, velocity. An alternative to hand-held technology is given by Tim Morland:

“A lot of emphasis has been given to the place of ‘handheld technology’ in school Mathematics, and for good reasons. We have chosen another route, largely because we have reasonable access for our students to desktop PCs and also because it seems unnecessary to duplicate resources when the school’s science department are equipped with data-logging equipment. To show the sort of thing that we have done, I would refer enquirers to my article “Modelling a Simple Mechanical System” in the IMA journal “Teaching Mathematics and its Applications” (OUP), Volume 18, No.2.

The necessary mathematical background required for students is • Hooke’s Law

• Newton’s Laws

• Solution of linear differential equations

This would be covered by most students preparing for a Further Mathematics qualification.

One experiment uses the physical apparatus shown opposite, and the displacement of the lower body is recorded after the system is set in motion according to a variety of initial conditions.

The actual motion of the body is compared to the predictions of a mathematical model constructed by students. After some

Model _ _ _ _ _ Experiment _ -0.10 -0.05 0.00 0.05 0.10 0 2 4 6 8 10 12 14 time (s 61

2.9 Sources of data and mathematical stimulus: the Internet and CD-ROMS